Wikiversity talk:News

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      <text bytes="1059" sha1="8hu8l8v2j50yxkkfix15442oaz9qjuk" xml:space="preserve">If this is a page for meta-news about Wikiversity maybe it should go un the Wikiversity namespace. --[[User:JWSchmidt|JWSchmidt]] 18:25, 15 August 2006 (UTC)

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==Where are announcements about Content?==

Good News!  The first '''[[WikiFilmSchool_Thumbnail_Storyboard_by_Robert_Elliott | Motion Picture Storyboard]]''' has been created for the '''WikiU Film School's''' short movie called '''''Seduced by the Dark Side!'''''

Where do I tell people about it?  All the news and announcement pages that I can find are only about the site itself, not about the new and great lessons inside.  [[User:Robert Elliott|Robert Elliott]] 16:06, 9 October 2006 (UTC)
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&quot;The &quot;news&quot; list here isn't new. Is someone supposed to keep this thing current?&quot; [[User:IDangerMouse|IDangerMouse]] 14:28, 25 January 2011 (UTC)
:see also [[Main Page/News]] + [[Colloquium]], ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] &lt;small&gt;uses the [[Wikiversity:Chat]] + [[Identi.ca]]&lt;/small&gt; 18:38, 25 January 2011 (UTC)

== mechanical engineering projects ==
project ideas on hydrogen energy</text>
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As we progress through this tour, various pages from Wikiversity will appear on the screen as if in a slideshow. Keep clicking on the &quot;back&quot; or &quot;next&quot; buttons in the green box above, and watch as the pages fly past.
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'''R''' is a
* powerful, free and open source, cross-platform, statistical and graphing software package;
* programming language;
* software environment for statistical computing. 

== Downloading R ==
Visit [http://www.r-project.org the R Project home page].

== Integrated development environments for R == 

R can be accessed from a {{w|command-line interface}}.  There are a variety of {{w|graphical user interface}}s that work well with R, including one that ships with it.  

{{w|RStudio}} is a very popular {{w|integrated development environment}} that works well with R and other languages, as does [[w:Jupyter|Jupyter]].&lt;ref name=&quot;R_gui&quot;&gt;{{cite web|title=7 of the Best Free Graphical User Interfaces for R|url=http://www.linuxlinks.com/article/20110306113701179/GUIsforR.html|website=linuxlinks.com|accessdate=9 February 2016}}&lt;/ref&gt;&lt;ref&gt;{{cite web|title=List of R Editors|url=https://r-dir.com/blog/2013/01/list-of-r-editors.html|website=r-dir|accessdate=2018-08-07}}&lt;/ref&gt;

RStudio supports RMarkdown, which makes it relatively easy for researchers to make their work [[w:Reproducibility|reproducible]] by allowing them to mix text with code of different languages, most commonly R, in the same document.  The platform supports mixing R with [[Python (programming language)]], {{w|shell script}}s, [[w:SQL|SQL]], {{w|Stan (software)}}, {{w|JavaScript}}, [[w:CSS|CSS]], {{w|Julia (programming language)}}, {{w|C (programming language)}}, {{w|Fortran}}, and other languages in the same RMarkdown document.&lt;ref&gt;{{cite Q|Q76441281}}&lt;!-- R Markdown: The Definitive Guide --&gt;.&lt;/ref&gt;  [[w:Jupyter|Jupyter]] notebooks provide a similar capability but may be more popular among Python users while RStudio may be more popular among people primarily using R.

Both RStudio and Jupyter are {{w|free and open-source software}}, but RStudio is developed and maintained by a commercial enterprise that offers paid support options.

== RStudio in the cloud == 

There are web-based portals for RStudio, including the following:  
* RStudio Cloud&lt;ref&gt;{{cite Q|Q100799903}}&lt;!-- RStudio Cloud --&gt;.&lt;/ref&gt; As of 2020-10-24 users can &quot;get started for free&quot;.  
* RStudio Server Pro for AWS&lt;ref&gt;{{cite Q|Q100800280}}&lt;!-- RStudio Server Pro for AWS --&gt;.&lt;/ref&gt;

==Tutorials==
{{Note|There are many well developed and maintained R tutorials on the web, e.g., http://www.cyclismo.org/tutorial/R}}
* [[How to use R/Installation|Installation process]]
* [[/Tutorials/]]
** [[/Tutorials/Algebra/]]
** [[/Tutorials/Connecting Fortran and R/]]
** [[/Tutorials/Introduction/]]
** [[/Tutorials/Linear algebra/]]
** [[/Tutorials/Loading data/]]
** [[/Tutorials/Plotting/]]

== Books that are Helpful When Learning R ==
* An R Companion for Applied Regression (2nd ed.) - {{ISBN|141297514X}}
* [http://cran.r-project.org/doc/manuals/R-intro.pdf An Introduction to R]{{font|size=x-small|, by W. N. Venables, D. M. Smith, and the R Development Core Team (PDF)}}
* R in Action - {{ISBN|1935182390}}
* Efficient programming in R &lt;ref name=&quot;OuelletteVisser2015&quot;&gt;{{cite journal|last2=Visser|first2=Marco D.|last3=McMahon|first3=Sean M.|last4=Merow|first4=Cory|last5=Dixon|first5=Philip M.|last6=Record|first6=Sydne|last7=Jongejans|first7=Eelke|title=Speeding Up Ecological and Evolutionary Computations in R; Essentials of High Performance Computing for Biologists|journal=PLOS Computational Biology|volume=11|issue=3|year=2015|pages=e1004140|issn=1553-7358|doi=10.1371/journal.pcbi.1004140}}&lt;/ref&gt;.
* [http://adv-r.had.co.nz/Introduction.html Advanced R by Hadley Wickham]

==See also==
* [[Wikipedia: R (programming language)]]
* [[Wikibooks:R Programming]]
* [[Searching R Packages]]:  A discussion of current search capabilities, inviting readers to help draft a proposal for improvements that might be funded, e.g., by [https://www.r-project.org/foundation/ the R Foundation].

==External links==
* [http://www.r-project.org/ R] (Official site)
* [http://cran.r-project.org/ R archive] (Manuals, downloads, etc.)
* [http://wiki.r-project.org/rwiki/ R wiki]
* [http://www.r-project.org/mail.html R mailing lists] - [https://stat.ethz.ch/mailman/listinfo/r-help R-help] - [https://stat.ethz.ch/mailman/listinfo/r-sig-teaching R-teaching] - [http://www.r-project.org/posting-guide.html Posting guide]
* [http://www.personality-project.org/r/ Using R for psychological research: A simple guide to an elegant package]
* [http://rstatx.com/ Guide to R for Analysts]
* [http://www.programmingr.com Programming R]
* [http://www.r-bloggers.com/ R Bloggers]
* [http://www.datacamp.com/ R Tutorial]

===Books===
* [http://www.r-project.org/doc/bib/R-books.html Books on R and S] - list
* [http://www.ling.uni-potsdam.de/~vasishth/SFLS.html Shravan Vasishth and Michael Broe - The foundations of statistics: A simulation-based approach] - draft
* [http://www.zoology.ufl.edu/bolker/emdbook/index.html Benjamin M. Bolker - Ecological Models and Data in R]
* [http://cran.r-project.org/doc/manuals/R-lang.pdf R Development Core Team - R Language Definition] - (aka `R Language Manual'), the basics of R, draft, 60 pages
* [http://cran.r-project.org/other-docs.html R manuals, tutorials and samples] - fairly large number of documentation available/downloadable for free, also non-English learning materials 

[[Category:Statistical software]]
[[Category:R (programming language)]]
[[Category:Howtos]]
[[Category:Free software]]</text>
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      <comment>Undo revision [[Special:Diff/2718558|2718558]] by [[Special:Contributions/103.181.254.156|103.181.254.156]] ([[User talk:103.181.254.156|talk]]) wrong place</comment>
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{{Comparative law and justice project|region=Asia}}

== Basic Information ==
[[File:Indonesian Flag.JPG|thumb|right|Flag of Indonesia]]


Indonesia is part of Southern Asia. It is an island located between the Indian and the Pacific Ocean.The area of Indonesia is about 1,904,569 sq km. 1,811,569 sq km is land and the rest is water. If you want to compare that to a state in the United States it would be a little less than three times the size of Texas.  It is made up of around 17,00 Islands. Indonesia is a neighboring country to Timor-Leste, Malaysia, Papua New Guinea. The climate is tropical with very humid, hot weather. One major city is Jakarta, which is also Indonesia's capital. The population here is 9.121 million people. The population of Indonesia is 242,968,342 people; 28.1% are between the ages of of 0-14, 66% between the ages of 15-64, and only 6% over 65 years old. 
Indonesia is a very diverse area. They have eight main ethnic groups. Javanese makes up 40.6% of Indonesia's population. The Sundanese makes up 15%, the Madurese makes up 3.3% of the population, the Minangkabau makes up 2.7% of the population, the Betawi makes up 2.4%, the Bugis make up 2.4%,the Banten only make up 2%,the Banjar make up 1.7%,while the left over 29.9% of the population's ethic background is categorized as other or unspecified. The main language spoken in Indonesia is Bahasa Indonesia. This is the country's official language. other languages that are spoken here are English, Dutch and Javanese. 
Indonesia has four main religions. This is made up of 86.1% Muslim, 5.7% Protestant, 3% Roman Catholic, and 1.8% Hindu The last 3.4% of the population is considered other or unspecified.  Some key industries that Indonesia has are petroleum and natural gas, textiles, apparel, footwear, mining, cement, chemical fertilizers, plywood, rubber, food. Their key agriculture is rice, cassava (tapioca), peanuts, rubber, cocoa, coffee, palm oil, copra; poultry, beef, pork, eggs. Some things that Indonesia exports are oil and gas, electrical appliances, plywood, textiles, rubber   imports- machinery and equipment, chemicals, fuels, food products. &lt;ref&gt;https://www.cia.gov/library/publications/the-world-factbook/geos/id.html&lt;/ref&gt;
[[File:LocationIndonesia.svg|250px|right|Location of Indonesia]]

==Brief History==

Between 1942 and 1945 Indonesia was ruled by Japan, but was first colonized by the Dutch. Indonesia declared it's independence on August 17th of 1945 &lt;ref&gt;https://www.cia.gov/library/publications/the-world-factbook/geos/id.html&lt;/ref&gt; The first people that lived in Indonesia were the ancestors of the Malays, Javanese and other Malayo-Polynesian groups and still today these people make up most of Indonesia. &lt;ref&gt;http://www.indo.com/indonesia/history.html&lt;/ref&gt;

==Economic Development, Health, and Education==
Life expectancy in Indonesia is pretty similar between males and females. Males average life expectancy is 73.69 years and for females it is 68.53 years. The infant mortality here is 28.94 deaths per 1,000 live births. Most people in Indonesia have an average education level of thirteen years in school. This is the same for males and females. The literacy rate for males is higher than females. 94% of males can read while only 86.8% of females. 
Indonesia's economic development is 1.033 trillion GDP. The GDP per capita is $4,300. The highest 10% of the populations income is 32.3% of the income, while the lowest 10% of population's income is only 3%. &lt;ref&gt;https://www.cia.gov/library/publications/the-world-factbook/geos/id.html&lt;/ref&gt;

==Governance==
[http://www.embassyofindonesia.org/about/pdf/IndonesianConstitution.pdf Indonesia Constitution ]
The president is elected by a majority vote. When they are elected into office, this is for a five year period. After those five years are up the president is able to be reelected if he chooses he wants to be. In order to become a president you must be an native born in Indonesia.

===Judicial Review===
The supreme court in Indonesia is the final court to look things over. this court is called the Mahkamah Agung and has no power of the judicial review. Once they make a decision it is final. The constitutional court does have power for judicial review. &lt;ref&gt;https://www.cia.gov/library/publications/the-world-factbook/geos/id.html&lt;/ref&gt;

==Courts and Criminal Law==
Indonesia has 8 main types of courts. These types of courts are general, industrial relation dispute courts, fishery, religion, state administrative, military, tax and constitutional. Most disputes appear before general court. Industrial relation dispute courts take care of settling disputes that have to deal with employment. the fishery court is similar to the industrial relation dispute court but it deals with subjects that are more serious. The state administrative court is designed to challenge the public administrative. The military court is used when crimes are done by a soldier or someone in the military. when their is a problem between a tax payer and a taxing authority the tax court jumps in to fix it. Lastly the constitutional court deals with constitutional problems and their decisions can not be appealed.&lt;ref&gt;http://www.nyulawglobal.org/globalex/Indonesia.htm#typesofcourts&lt;/ref&gt; The law in Indonesia is that there is an assumption of innocence until you are proven guilty. When it comes to the courts thought they are not one hundred percent honest. Sometimes they take bribes in order for the grant of bail. &lt;ref&gt;http://www.state.gov/g/drl/rls/hrrpt/2005/61609.htm&lt;/ref&gt;
In Indonesia advocates are the same as our lawyers in the United States. Usually they are private practiced. They do have to go through legal training though. &lt;ref&gt;http://www.llrx.com/features/indonesia.htm#legal%20professionals&lt;/ref&gt; Foreign lawyers are not allowed to come to Indonesia and practice their law. even though this is said, many foreign lawyers do practice law in Indonesia &lt;ref&gt;http://www.nyulawglobal.org/globalex/Indonesia.htm#central&lt;/ref&gt;
Indonesia has a republican type of government. Like the United States they have three main branches; the judicial branch, the legislative branch and the executive branch. Indonesia does have a constitution that was established in august of 1945 after they declared independence. This constitution had had many amendments up until 2002.&lt;ref&gt;https://www.cia.gov/library/publications/the-world-factbook/geos/id.html&lt;/ref&gt;

===Punishment===
Only one part of Indonesia uses corporal punishment. This is in the Aceh Province. This punishment does not apply to anyone of Christian religion. It usually happens in front of a Mosque. Corporal punishment is allowed to men and women but is not allowed for juveniles. The kind of corporal punishment is canning, which leaves bruises and welts on the persons body. Corporal punishment is usually used to punish people for gambling or alcohol use. This is not seen often but it is still around.&lt;ref&gt;http://www.corpun.com/rules2.htm#indonesia&lt;/ref&gt;
A juvenile in Indonesia is considered anyone under the age of 18. From January until May of 2003 approximately 4,325 juveniles were held in jail throughout Indonesia and about 84% of them were in adult prisons. When it comes to juveniles the judge has three options. The first is that the judge can choose to send the juvenile back to his parents without any punishment. The judge can also choose to send the child to a government institution but not punish them. Last the judge can consider the juvenile guilty and punish them. These juveniles may be punished by some sort of fine, being put in jail or a detention facility, or some kind of supervision.&lt;ref&gt;http://www.unafei.or.jp/english/pdf/RS_No68/No68_16PA_Mardite.pdf&lt;/ref&gt;
The prison conditions in Indonesia are dirty and very overpopulated. There is poor health care while in prison and maybe people suffer from HIV. &lt;ref&gt;http://healthdev.net/site/post.php?s=972&lt;/ref&gt;


[[Image:Banda Aceh's Grand Mosque, Indonesia.jpg|250px|center|A centre-aligned thumbnail image.]]

===Law Enforcement===
Indonesia fits into the taxonomy of types of police structure as a decentralized multiple coordinated structure. This means that there are many local forces not just one main force. The military and the police used to work together, but since 1999 the police and the armed forces are now separate. 3% of the GDP in Indonesia join their military. The requirements to join are that you have to be 18 years old. You also have to be active for two years and in the reserve until you are 45 years old. &lt;ref&gt;http://www.polri.go.id/organisasi/op/tp/&lt;/ref&gt;The police officials in Indonesia are very close to the community. their mission is to keep the people safe and have a bond with them. The people of Indonesia respect the police officers and help them out whenever possible.  &lt;ref&gt;http://www.polri.go.id/organisasi/op/vm/&lt;/ref&gt; There is some corruption in Indonesia. when asked in 2010, 45% of police officials have admitted to accepting some kind of bribe.&lt;ref&gt;http://www.transparency.org/regional_pages/asia_pacific/about/south_east_asia/indonesia&lt;/ref&gt;

===Crime Rates and Public Opinion===

Many crimes in Indonesia go unreported, but Indonesia's crime rate is rising due to increasing urbanization. So while looking at crime rates they tend to be lower than the actual crime that is going on there. Some of this could be blames by the poor law enforcement they have.&lt;ref&gt;http://www.expat.or.id/info/crimeratesinindonesia.html&lt;/ref&gt;
Indonesia crime is considerably high in area of tourism. These crimes are not huge crimes more along the lines of credit card fraud or pickpocketing. Robbery is also a crime that tends to occur in Indonesia.&lt;ref&gt;http://www.travelersdigest.com/indonesia_fact_sheet.htm&lt;/ref&gt; The intentional homicide rate per 100,000 for Indonesia is 8.9. In comparison to the United States we are at about 5.6&lt;ref&gt;http://www.unodc.org/documents/data-and-analysis/IHS-rates-05012009.pdf&lt;/ref&gt;
In 2002, Indonesia found that around 49,500 prostitutes were under the age of 18. &lt;ref&gt;http://en.wikiversity.org/w/index.php?title=Indonesia&amp;action=edit&amp;section=11&lt;/ref&gt;


===Family Law===


Under the Marriage Laws of Indonesia a marriage is not official unless the female is over the age of 16 and the male is over the age of 19. Even if they are over this age but under 21 they must have parents consent in order for a marriage to be recognized. If you get divorced in Indonesia and you are a female you must wait 100 days before you are able to remarry. Also in Indonesia all marriages must be performed in a religious ceremony before they are recognized by the Civil Register Office.  &lt;ref&gt;http://ukinindonesia.fco.gov.uk/en/help-for-british-nationals/living-in-indonesia/marriage-in-indonesia&lt;/ref&gt; Polygamy is allowed in Indonesia under religions that support it as long as you have permission by your wife and judicial permission. You also have to be financially able to support more than one family, and you must treat both wives and children equally. &lt;ref&gt;http://www.law.emory.edu/ifl/legal/indonesia.htm&lt;/ref&gt;
A divorce may happen by either partner in Indonesia, but only if the other partner has committed adultery, Alcoholism, addiction to drugs, a gambling problem, or cannot support the family.&lt;ref&gt;http://www.law.emory.edu/ifl/legal/indonesia.htm&lt;/ref&gt;
Inheritance in Indonesia is usually male dominant. Most inheritance goes to children or the spouse of the deceased. &lt;ref&gt;http://family.jrank.org/pages/867/Indonesia-Inheritance.html&lt;/ref&gt;
Regarding adoption in Indonesia the adoptive child must be of the same religion of the adoptive parent. If the religion of the child is not known then they are considered to be Muslim.&lt;ref&gt;http://adoption.state.gov/country_information/country_specific_info.php?country-select=indonesia&lt;/ref&gt; For people in the United States it is pretty difficult to adopt a child from Indonesia. 
There is a pretty strict adoption law that is followed. Couple from the US trying to adopt must actually live in Indonesia for two years and also be between the ages of 30 and 45. There are also requirements that say the couple must be married for five years. They can have an adopted Indonesian baby or if they already have children the mother must not be able to have anymore. The adoptive parents must also believe in God and appear at a court hearing. The child that is being adopted must be under five years of age and the child must be in company of a welfare program.&lt;ref&gt;http://adoption.state.gov/country_information/country_specific_info.php?country-select=indonesia&lt;/ref&gt;

===Human Rights===
In Indonesia there is a right to freedom of religion.  Recently in Indonesia the freedom of minority religious groups has not been protected by the government or the police, and recently around 114 people have been arrested because of their views. &lt;ref&gt;http://www.amnesty.org/en/library/asset/ASA21/009/2011/en/28e8a786-1123-44e3-b918-4a1435518905/asa210092011en.html&lt;/ref&gt; Also in Indonesia people have human rights.  This states that all people have a right to life and freedom of torture or bad treatment. These human rights sometimes are violated by the police officers. Examples of this are use of unnecessary firearms, torture or inhuman treatment or punishment, and lack of protection against minority groups. &lt;ref&gt;http://www.amnesty.org/en/library/asset/ASA21/005/2011/en/577dff37-4858-4c93-9c0f-c0da0371e86b/asa210052011en.pdf&lt;/ref&gt;

===Works Cited===

&lt;references /&gt;

[[Category:Indonesia]]</text>
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  <page>
    <title>Talk:Wisdom</title>
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      <id>2721190</id>
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      <contributor>
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      <text bytes="5916" sha1="bgpg8m1cacuo3awqw645zd3irswoi5j" xml:space="preserve">== Humanity, Humility, and Compassion ==

There seems to be some confusion over the best path to Wisdom.

Is it Humanity?  Is it Humility?  Is it Compassion?

It occurs to me that Compassion and Humanity are deeper and more difficult goals than Humility.

How do others feel about this?

[[User:Moulton|Moulton]] 14:07, 14 February 2011 (UTC)

: What the author intended the sentence to mean is the only source of confusion on my part. The words &quot;Humanity&quot; and &quot;Humility&quot; are only different by two letters (&quot;an&quot; vs &quot;il&quot;). An accidental slip of the fingers causing a different word to be typed than the one intended seems conceivable to me. Now that you bring it up, it is also conceivable the author intended the sentence to mean &quot;Wisdom is choosing to do good&quot;. --[[User:Darklama|&lt;span style=&quot;background:DarkSlateBlue;color:white;padding:2px;&quot;&gt;&amp;nbsp;dark&lt;/span&gt;]][[User talk:Darklama|&lt;span style=&quot;background:darkslategray;color:white;padding:2px;&quot;&gt;lama&amp;nbsp;&lt;/span&gt;]] 14:23, 14 February 2011 (UTC)

:*In [http://www.thewisepath.org/ this referenced resource], what is the deepest objective for emotional development?  —[[User:Moulton|Moulton]] 14:35, 14 February 2011 (UTC)
:: I mean humanity, I am actively working on this course and expect to make substantial additions over the next few days and weeks. I appreciate your attention, participation, and patience as I construct and refine this. Regarding emotional development, the wise path shows the deepest development as &quot;compassion&quot; Thanks! --[[User:Lbeaumont|Lbeaumont]] 15:57, 14 February 2011 (UTC)
::*Now, if we can just get Darklama to concur, we'll have a consensus on what the referenced sources suggest.  —[[User:Moulton|Moulton]] 16:17, 14 February 2011 (UTC)
::*: Now that Lbeaumont has confirmed no mistake was made, I choose to assume that students in Lbeaumont's class may know what &quot;choosing humanity&quot; means even though I do not from the current context. --[[User:Darklama|&lt;span style=&quot;background:DarkSlateBlue;color:white;padding:2px;&quot;&gt;&amp;nbsp;dark&lt;/span&gt;]][[User talk:Darklama|&lt;span style=&quot;background:darkslategray;color:white;padding:2px;&quot;&gt;lama&amp;nbsp;&lt;/span&gt;]] 16:42, 14 February 2011 (UTC)

== The First Paragraph ==

The first paragraph is from a stub created in 2006. I left it there as a tribute to the stub-creator, but don't really like it. What do you think? Thanks--[[User:Lbeaumont|Lbeaumont]] 16:07, 14 February 2011 (UTC)

{{Quotation|'''Introduction'''&lt;BR&gt;
Over the ages, our ancestors have had to think and rethink the meaning of life. In this search, some came to have insights into important concepts of life. Though it may be found in some ancient writings, the definition of wisdom has never been mentioned. Rather, only the effects and how to search it will be listed.}}
:Yeah, that's atrocious.  Let's rewrite it from scratch.  —[[User:Moulton|Moulton]] 16:23, 14 February 2011 (UTC)
:: Ok I worked on it, what do you think. As I mentioned I have ambitious plans for this course and will be adding substantial material over the next days and weeks. I hope the material will quickly become clear and useful. If you are interested, perhaps you can take a look at my recently &quot;(more nearly) competed&quot; course on [[Dignity]] and let me know your comments on that. Thanks--[[User:Lbeaumont|Lbeaumont]] 18:08, 14 February 2011 (UTC)

Current version now reads:

{{Quotation|'''Introduction'''&lt;BR&gt;
Although the subject of wisdom has been contemplated and debated by philosophers for millennia, there is little consensus on its definition. More importantly, although wise choices could help us increase human well-being, there is little understanding, discussion, or application of wisdom in our lives. This course explores the topic of wisdom with a particular emphasis on applying wisdom to solve practical problems.}}
That's better.  Should it mention that Wisdom is the opposite of Foolishness? Or is that a foolish thing to say?

[[User:Moulton|Moulton]] 20:57, 14 February 2011 (UTC)

== Illustrating Wisdom ==

I am searching for an image illustrating the concept of wisdom. I appreciate your ideas and suggestions. Ideas include a labyrinth, or spiral instead of the present Ouroboros image. --[[User:Lbeaumont|Lbeaumont]] 13:38, 16 February 2011 (UTC)

== Potential Source ==

Consider integrating material from the John Vervaeke video - &quot;[http://www.youtube.com/watch?v=ti4Kj8TcEgk The Cognitive Science of Wisdom]&quot; into the course. --[[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|talk]]) 11:34, 14 September 2012 (UTC)
: For example: As the child matures by studying and internalizing the adult’s perspective, the adult can progress toward wisdom by internalizing the sage’s perspective.

== The Components of Wisdom: A Survey of Expert Opinions ==

Yes, there is a paper titled &quot;The Components of Wisdom: A Survey of Expert Opinions&quot; that was co-authored by Monika Ardelt, Judith Glück, and Jacqui Smith, which includes input from Robert Meeks and Ursula Staudinger. [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 16:36, 19 April 2023 (UTC)

== New Short Phronesis Measure ==

A [https://wisdomcenter.uchicago.edu/news/wisdom-news/new-short-phronesis-measure recently publised paper] claims to update Aristotle’s concept of ''phronesis -'' practical wisdom. It will be good to study that paper and update this course based on any reliable and relevant new information. [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 15:21, 12 March 2025 (UTC)

== Translation request ==

Hi Lbeaumont, I'd like to translate your Applied Wisdom curriculum into Spanish. Would you be okay with this? I'll give you attribution for the original work. [[User:Mackenz1337|Mackenz1337]] ([[User talk:Mackenz1337|discuss]] • [[Special:Contributions/Mackenz1337|contribs]]) 10:43, 10 July 2025 (UTC)</text>
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    <title>Understanding Arithmetic Circuits</title>
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      <id>2721165</id>
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      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
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== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.1.A.CLA.20250708.pdf|A]]||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

== Barrel Shifter ==
* Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]])
&lt;/br&gt;

'''Mux Based Barrel Shifter'''
* Analysis  ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) 
* Implementation 
&lt;/br&gt;

== Multiplier == 
=== Array Multipliers ===
* Analysis  ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]])
&lt;/br&gt;

=== Tree Mulltipliers ===
* Lattice Multiplication  ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]])
* Wallace Tree  ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]])
* Dadda Tree  ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]])
&lt;/br&gt;

=== Booth Multipliers ===
* [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]]
* Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]])
&lt;/br&gt;

== Divider == 
* Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])&lt;/br&gt;

&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
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      <id>2721167</id>
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== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.1.A.CLA.20250709.pdf|A]]||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

== Barrel Shifter ==
* Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]])
&lt;/br&gt;

'''Mux Based Barrel Shifter'''
* Analysis  ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) 
* Implementation 
&lt;/br&gt;

== Multiplier == 
=== Array Multipliers ===
* Analysis  ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]])
&lt;/br&gt;

=== Tree Mulltipliers ===
* Lattice Multiplication  ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]])
* Wallace Tree  ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]])
* Dadda Tree  ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]])
&lt;/br&gt;

=== Booth Multipliers ===
* [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]]
* Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]])
&lt;/br&gt;

== Divider == 
* Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])&lt;/br&gt;

&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Digital Circuit Design]]
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{{Law}}

'''Suicide''' is the act of intentionally causing one's own death. Following the [[Should voluntary euthanasia be legal?|debate about euthanasia]], should suicide be legal as well? Do we own our own lives so completely that we have a right to end them? Or do we owe certain obligations to our communities and thus we should stay alive even when we may want to die?

== Suicide should be legal ==

=== Pro ===

* {{Argument for}} Some people just don't want to live. No specific mental issues, they just don't want to live, especially with the way society is.
** {{Objection}} But why don't some people want to live? Surely, despite disappointments, life can also be fun? And what if everyone has a life purpose and reincarnation exists? Then they are stuck with life again in the next life. Then they'd better make something of the life they have now, right? Maybe you can also see it as a goal to make something of your life in a tough world. You can think of life as a big stage play where everyone plays their own part. A kind of game!
* {{Argument for}} In a free society, people should have the right to do what they want with their lives as long as they don't hurt others. Suicide can be what a rational adult chooses for him/herself. No State should have title over one's life. Taking away such choices is patronizing and dehumanizing.Also day by day, the cost of living like rent, groceries and other subscriptions are becoming unaffordable for the individual if he is already poor or has no wealth.
** {{Objection}} Suicide does hurt others. Maybe not physically, but the emotional and social costs of losing a loved one to suicide are huge.
*** {{Objection}} This is solely because the conditions we set for life as something of utmost value has created an emotional link that runs deeper than mere care; its a socially constructed concept. Similarly, the fear of death is so greatly exaggerated that the anxiety of and actually confronting it warps those emotions even further. This is to say that a large portion of that emotional pain and despair are not about the person in question or the issues they aim to escape from, but rather the potential they have with the person in grief and with the deceased, which many wouldn't argue against being a purely selfish desire. If we lived in a modern society where both life and death are held in equal and balanced regard, the concept of suicide would more than likely be less seen as taboo, and therefore the emotions regarding the action and aftermath would not be as impactful.
*** {{Objection}} So should someone else continue to suffer to keep other people happy?
*** {{Objection}} Just like others aren't responsible for our happiness, those who commit suicide are not responsible for how others feel about it. Don't keep empty people alive for your own selfish needs.
*** {{Objection}} While that is partially true, on a moral level, we should not be prioritizing more the wellbeing of the relatives of a potential suicide over the potential suicide itself, because no individual provides informed consent to being born, so no individual is under any obligation to remain alive for the emotional benefit of others.
*** {{Objection}} Many other life events cause emotional harm but are not illegal. For example, a son cutting off all contact with their parents for life (this can be as traumatizing as death). Or, negligent parenting leading to children developing psychological issues like extreme attention seeking behavior (which can lead to school shootings) are not illegal either.
*** {{Objection}} Even if a person dies naturally, it will still cause emotional harm to others. The cause of death might be a secondary issue. 
**** {{Objection}} You are working under the assumption that everyone belongs to a &quot;tribe&quot; and has people that care for them.
***** {{Objection}} Some people will care even though they are not related or as close. For example, for some, it would be upsetting to hear news about your neighbour's death and the reason of it might hurt you even more, maybe you would question yourself that you should have noticed that they seem a little off or that you would have done at least try something to prevent that. Even if you do not care enough, the thought would have strike across your head at least once.
****** {{Objection}} Perhaps it would be upsetting, but the legal right to choose trumps others' feelings about whatever choice is made.
** {{Objection}} Helping adults seek help for a severe mental illness is not patronizing. Suicide is not simply a &quot;decision&quot;, it is murder. And while it should not be criminalized like regular murder, it should not be supported or promoted by the law.
*** {{Objection}} You are taking a life of someone with out their promises
*** {{Objection}} Suicide is a decision, and not murder. Murder is the killing of another person from a person
*** {{Objection}} Helping adults who need help is not patronizing. It is the forcing of &quot;help&quot; which dehumanizes; forced psychiatric &quot;help&quot; is not help when an adult is actively declining such services. Might such &quot;help&quot; be what psychiatrist Dr. Thomas Szasz refereed to as cruel compassion in his book &quot;Cruel Compassion&quot;?
*** {{Objection}} Whether or not to engage in suicide is ultimately a personal decision. Suicide should not be promoted, however, it should not be prohibited through state sanctioned psychiatric coercion, force, and confinement. Voluntary/consensual psychiatry should remain legal. Nonconsensual psychiatry should be illegal.
**** {{Objection}} Nonconsensual psychiatry is necessary in some cases, legalizing suicide should not automatically mean all nonconsentual psychiatry is unethical. For example, if someone is not self-injurious, but unable to bathe or clothe themself, should they be committed to a psychiatric care facility by the state or allowed to rot simply because it is &quot;without consent&quot;?
* {{Argument for}} It is technically impossible to enforce a law against suicide, so as suicide is virtually legal, it should be also ''legally'' legal. 
** {{Objection}} But it is possible to punish those who attempt suicide and fail.
*** {{Objection}} Punishing someone who attempted suicide and failed would the most cruel and Sadistic act a government could perform, they already don't want to live because of the horrible conditions of their lives and you want them to be FORCED to continue living in even worse conditions? That would only prove their decision to commit suicide to be justified.
*** {{Objection}} Forcing someone to live an unwished life is equivalent to torture, so punishment to failure suicide is unhuman, beside useless.
**** {{Objection}} However, at the time of the suicide the person might be too overwhelmed in their emotions to think straight, this might let them sit through life to properly think if this is what they actually want.
***** {{Objection}} If the person however has any idea or wish to not exist, why should they be swayed against that if it is a conclusion?
***** {{Objection}} So if we do follow this, allow them to “sit and think” as you mention, and they still decide that they want suicide, would they then be permitted to take their life?
***** {{Objection}} Maybe so, but you should also allow the possibility that they were not overwhelmed and were thinking straight. Just because they want to make a decision you disagree with, does not mean they are automatically invalid in their opinion.
* {{Argument for}} We do not know what happens after death. It could be better after death for one who commits suicide. By prohibiting suicide we are imposing implicit theological (and Judeo-Christian) beliefs on others. Prohibitions on suicide impose the implicit theological belief that death is a worse consequences than life, and this imposition is morally incorrect.
** {{Objection}} The ambiguity of what happens after suicide is not a sound argument for why suicide should be legal.
*** {{Objection}} The fact we don't allow suicide because of fantasy religious beliefs is not an argument for it not being legal. Who has the right to tell another what to do with their life? 
** {{Objection}} Precisely since we don't know what happens next, we can just wait and see until later—there is no rush to die now, since anyone who would commit suicide will die of some other cause anyway.
*** {{Objection}} For people in crippling pain where they have no life why force them to suffer , that is cruel sadistic and vile
*** {{Objection}} Forcing adults to wait is dehumanizing and robbing them of personal responsibility and freedom. We should use persuasion and kindness to entice adults to remain alive, not coercion.
*** {{Objection}} &quot;Just wait&quot; is not feasible for someone suffering from mental anguish or any other sickness making life unbearable.
* {{Argument for}} Suicide can be seen as a form of migration. Prohibiting suicide is like a nation stating that its citizens cannot leave its borders.
** {{Objection}} Suicide and migration are totally different concepts. For example, migration is impermanent and reversible, while suicide is permanent.
*** {{Objection}} Nothing is Permanent!!
*** {{Objection}} It is unacceptable to force any being with self awareness to live in despair and despondence just because it's the majority opinion that it's not okay to take oneself out of existence.
*** {{Objection}} Well technically we don’t know that for sure, we don’t know if reincarnation is a possibility after death, also in many situations a person can be forced to stay in a country against their will 
*** {{Objection}} If I make an active choice of leaving the country I was born in settle abroad without the intentions of coming back. The foreign state is ok to take me in, can my country of origin stop me from doing so or would it be morally correct?
**** {{Objection}} Sure you can have the intention of never coming back; however, you can change your mind in the future. With suicide, you cannot change your mind. Once you make that decision, it is permanent.
* {{Argument for}} Sick people —especially mentally ill ones— have no right to force their illness on someone else by having children. Those opposed to suicide need to support euthanasia at a minimum.
** {{Objection}} This argument assumes that mentally ill people have only two options (both implies dying): commits suicide, or must be  euthanized . Still, this argument should go to another debate, not in this one about suicide. I have created that new debate at [[Should mentally ill people be allowed to have children?]]
*** {{Objection}} As one who is mentally ill with many varying degrees of issues, I feel it is no different than having someone financially incapable of supporting their children a part of this question. Anyone who cannot be a provider, caregiver, and guiding force in a new person's life shouldn't be allowed to reproduce. Although the American aspect rewards those who reproduce constantly regardless of means or mentality. Welfare is for the common good right?
* {{Argument for}} Suicides can help to solve overcrowding problems. One suicide is one less person who will not have children.
** {{Objection}} If people think this way, some people will sacrifice themselves just for the sake of others and the world, this would create another problem.
*** {{Objection}} I agree with the first post. Population and overcrowding are issues that somehow are ignored, but are becoming far more relevant. If one is truly in control of said life, why can't one choose to not exist? It is not a viable solution to force the suffering to live indefinitely just to support the emotionally and financially hungry.
*** {{Objection}} People sacrifice themselves for others all of the time in non-suicidal capacities. Professionally people can join the military, or it can be as simple as pushing somebody out of the way of a train.
** {{Objection}} Overcrowding is not created by an increase in population, thus, would not be solved by suicide. Overcrowding is a result of increasing population density due to perception of efficiency and preference. It can be solved through technological and educational advancement. Earth is huge. 
*** {{Objection}} And resources are finite, I don't know off the top of my head how much of my taxes go towards wasteful causes like keeping innocent people in jail, feeding generational welfare recipients, or keeping folks alive that just don't want to be here. That money can't be spent better? I would rather it go to repainting lines on the road than wasted on forcing someone to be alive in any circumstance.
*** {{Objection}} Just because overcrowding isn't caused by a lack of suicide doesn't mean it couldn't be alleviated by suicides. This only makes logical sense. Another example of this logic: depression in many cases isn't caused by a serotonin deficiency, but medications that increase serotonin do alleviate depression. 
*** {{Objection}} Earth is not huge, land is very limited in fact. True, increase in population is also due to longer lifespans, but especially people in the third world continue to have lots of children of which more and more immigrate into Western first world nations, artificially increasing the otherwise naturally decreasing populations.
**** {{Objection}} The third world growing population is not because they commit suicide to little, but because of the historical socioeconomic backwardness (most of which is due to western countries).
* {{Argument for}} If a person commits suicide because he/she is depressed, he/she will not negatively influence other people any more.
** {{Objection}} This argument is insensitive to people suffering from mental illnesses like depression. A person suffering from depression already deals with a lot of guilt. Such statements can inhibit any recovery they have a chance at making. No one wants to &quot;negatively influence&quot; or hurt anyone else. It is not valid to support the suicide of someone suffering from depression just so that there might be lesser &quot;negative influence&quot; to deal with. In fact, we should learn to support others better, whether it is in their right to continue living, or not. But not to burden or force them with either decision.
*** {{Objection}} That depends on the situation. Hitler was very obviously capable of suicide. I'm not a phychologist but had he ate that bullet a few years sooner...
**** {{Objection}} ….Another political figure head would have been selected in his place to lead what was a steadily growing Nazi party, feeding on the resentment and anger of a post-ww1 Germany. 
** {{Objection}} This argument is insensitive to people suffering from mental illnesses like depression. A person suffering from depression already deals with a lot of guilt. Such statements can inhibit any recovery they have a chance at making. No one wants to &quot;negatively influence&quot; or hurt anyone else. It is not valid to support the suicide of someone suffering from depression just so that there might be lesser &quot;negative influence&quot; to deal with. In fact, we should learn to support others better, whether it is in their right to continue living, or not. But not to burden or force them with either decision.
* {{Argument for}} In addition to the fact that suicide should be legal, places where one can be euthanized should be legal and available too. First, this stops a lot of trauma of those 'victims' that find their loved ones whom have committed suicide. Second, it stops botched attempts which add even more trauma to the true individual.
** {{Objection}} Although suicide and euthanasia are highly related, there is a separated debate for euthanasia. Please find it in [[Should voluntary euthanasia be legal?]]
*** {{Objection}} There should and shouldn't be a separate debate for euthanasia. All aspects of control and true freedom of one's life shouldn't be under any restrictions due to other's opinions and influence. If you feel that this world is not for you, than feel no regrets or pressure although the selfish majority may want you to.
* {{Argument for}} Punishing suicide is cruel, as many people who have been driven to this point will have  problems which have better solutions than jail time, which would likely just make them even more suicidal.

=== Con ===

* {{Argument against}} Suicide creates an emotional toll beyond simply the person who is suffering the cause of suicidal ideation. It can create feelings of survivor's remorse and intense mourning and trauma for those who are close to this individual. Any possible form of discouragement should be seen as a positive step.
** {{Objection}} So basically put... Don't control your own life because it affects others..? Isn't that a primarily selfish argument?
** {{Objection}} Any death can cause mourning and trauma for those close to the person that dies. The person does not owe their presence to anyone. This can be seen with how absent parents are not always penalized by the law (they may only be required to offer financial support, and may even lose the ability to provide physical support to their child. This can cause trauma for the family but the law doesn’t explicitly prohibit it).
** {{Objection}} If suicide were legal, the person would likely be able to say “goodbye” and explain themselves or answer any questions others have beforehand (without getting dragged away to a hospital). This would likely lessen the emotional anguish felt by family/friends/acquaintances etc.
*** {{Objection}} Suicide is already legal in most of the world, but this is not what happens in reality.
** {{Objection}} Criminalisation of suicide creates a greater emotional toll. It makes it harder for the person struggling to reach out and get help. if they go through with the act the loved ones have to deal with them being known as a criminal. If the suicidal person fails, their loved ones have to see them get punished or hide what they know out of fear.
** {{Objection}} Emotional toll is not a logical criterion for the illegality of suicide because the origin of one's own emotion is entirely caused by and within the self, through the perceptions and relationships one has of and with self, logic and reality.
*** {{Objection}} Formerly a transphobic objection
** {{Objection}} Although the emotional toll to others exists, the right of an adult to end their own life is more important than the feelings of other adults who were not able to persuade other adults to not engage in suicide.
* {{Argument against}} Suicide is usually a permanent solution to a temporary problem. Since most suicides are a product of mental illness and mental illnesses can be transitory, treated, or even cured, then we should discourage people from killing themselves as a way of addressing mental illness.
** {{Objection}} Mental illnesses are a social construct and are not real illnesses, they are &quot;metaphorical&quot; illnesses. The idea of mental illness is often used as a method of labeling, infantilizing and invalidating an adult's decision. There is no objective means of medical diagnosis for some of the mental conditions described in the ''{{W|Diagnostic and Statistical Manual of Mental Disorders}}''. They are diagnosed using verbal interviews and subjective judgement.
*** {{Objection}} Many mental illnesses are caused by aberrant pathology that can be detected.
*** {{Objection}} To say that mental illness is a social construct and not a medical reality is to ignore and invalidate the entire psychiatric, psychological, and neurological communities. As for many other illnesses, mental illness is not some arbitrary classification decided on by asking someone how they feel on a whim, but rather by observing patterns in mood, behavior, and thought over an extended period of time.
**** {{Objection}} This is an {{W|argument from authority}}. Authorities have been wrong many times in the past. Other arguments are needed.
***** {{Objection}} This is an {{W|argument from fallacy}}. Just because this argument is able to categorize something as a logical fallacy does not mean it is untrue. It was  not shown any evidence at all that the psychiatric and other mental scientists who believe, based on the evidence they have seen, that mental illness is an objective thing are incorrect. 
****** {{Objection}} There is no consensus on the diagnostic labelling of psychology. Many in the scientific community view the term &quot;mental illness&quot; as an inappropriate metaphor for what they believe to be emotional or societal problems, and not legitimate illnesses.
******* {{Objection}} But there is consensus on the diagnostic labelling of traumatic brain injuries, and physical damage to your brain has a high risk of suicide.  Physical damage to the brain is not an &quot;emotional&quot; or &quot;societal&quot; problem.
*** {{Objection}} Mental illnesses are real, objective, and highly prevalent throughout the world. The most common are depression and anxiety, but also posttraumatic stress disorder, schizophrenia, conduct disorder, bipolar disorder, and drug addiction.
** {{Objection}} Just because something ought to be discouraged does not mean it should be illegal. Many things are illegal and this ends up harming people, for instance laws against drugs end up harming drug addicts by criminalizing and punishing them, rather than treating their disease. If mental illness is truly an illness than it ought to be treated, just like any other illness, and not criminalized through illegality.
** {{Objection}} Suicide is not only a permanent solution to a temporary problem, it's a permanent solution to all problems, forever.
*** {{Objection}} I believe that that is the point.
*** {{Objection}} Sometimes it’s not a temporary problem, people can’t control the situation that they are born in. Abusive households, poverty, sickness, and why should someone be forced to work that much harder just to live a bearable  life that they didn’t consent to live in the first place. This is like the equivalent of forced labor.
*** {{Objection}} This statement minimizes the pain of one who is suicidal, it shows the lack of understanding of the pain a suicidal person is enduring and increases their isolation. It legitimizes suicide as an option to permanently end one's pain. It can cause more harm than good to say this to a suicidal person.
**** {{Objection}} To assume how a person would feel after being told suicide is a viable option is irrational, because all people are different, and perceive the world and it's contents in different ways.
* {{Argument against}} If a person is in debt, the act of committing suicide would lead to unpaid debts, this would ultimately hurt someone else financially.&lt;ref&gt;{{Cite web|url=https://suicidepreventioncommunity.wordpress.com/2010/03/05/please-stop-saying-suicide-is-a-permanent-solution/|title=Please Stop Saying, 'Suicide is a permanent solution...'|date=2010-03-05|website=Suicide Prevention News and Comment|language=en|access-date=2020-09-09}}&lt;/ref&gt;
** {{Objection}} Everyone is born with a life. In that life, you can owe money, but in the end, we all have the privilege of losing the life, as a life cannot be compared to the likes of money. When one dies, the blame can be put on the abstract thought of the debtor, akin to the money that they owe. Therefore, the debt will be paid by their life and cannot be put into monetary values, as they are different things entirely. Money is a social construct, life is made from something else.
** {{Objection}} The person will probably do it anyways even if there not legally allowed to
** {{Objection}} The creditor could also be hurt financially by the debtor dying prematurely or otherwise being rendered incapable of paying their debt. That is a risk that the creditor takes on when lending them funds in the first place. And, mitigating those risks is the specific role of insurance companies.

== See also ==

* [[Suicide]]
* [[Should voluntary euthanasia be legal?]]

== External links ==

* [[Wikipedia:Suicide legislation]]

== Further reading ==

* {{Cite book|title=Suicide Prohibition: The Shame of Medicine|last=Szasz|first=Thomas}}
* {{Cite book|title=Fatal Freedom: The Ethics and Politics of Suicide|last=Szasz|first=Thomas}}
* {{Cite book|title=The Contribution of Natural Law Theory to Moral and Legal Debate on Suicide, Assisted Suicide and Euthanasia|last=Paterson|first=Craig}}

== References ==

{{Reflist}}

[[Category:Death]]
[[Category:Politics]]
[[Category:Wikiversity Law Review]]
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      <text bytes="206606" sha1="de11zyxn0losgoq5uijng7u7dwjzppl" xml:space="preserve">[[File:Einstein en Lorentz.jpg|220px|thumb|right|[[w:Hendrik Antoon Lorentz|''Hendrik Antoon Lorentz'']] (right) after whom the [[w:Lorentz group|''Lorentz group'']] is named and [[w:Albert Einstein|''Albert Einstein'']] whose [[w:special theory of relativity|''special theory of relativity'']] is the main source of application. Photo taken by [[w:Paul Ehrenfest|''Paul Ehrenfest'']] 1921.]]

The [[w:Lorentz group|''Lorentz group'']] is a [[w:Lie group|''Lie group'']] of symmetries of the [[w:spacetime|''spacetime'']] of [[w:special relativity|''special relativity'']].  This group can be realized as a collection of [[w:matrix (mathematics)|''matrices'']], [[w:linear transformation|''linear transformations'']], or [[w:unitary operator|''unitary operators'']] on some [[w:Hilbert space|''Hilbert space'']]; it has a variety of [[w:Representations of Lie groups|''representations'']]. In any relativistically invariant physical theory, these representations must enter in some fashion;&lt;ref group=nb&gt;The way in which it enters may take many shapes depending on the theory at hand. While not being the present topic, some details will be provided in footnotes labeled &quot;nb&quot;, and in the section [[#Applications|applications]].&lt;/ref&gt; physics itself must be made out of them.  Indeed, special relativity together with [[w:quantum mechanics|''quantum mechanics'']] are the two physical theories that are most thoroughly established,&lt;ref group=nb&gt;{{harvnb|Weinberg|2002|p=1}} ''&quot;If it turned out that a system could not be described by a quantum field theory, it would be a sensation; if it turned out it did not obey the rules of quantum mechanics and relativity, it would be a cataclysm.&quot;''&lt;/ref&gt; and the conjunction of these two theories is the study of the infinite-dimensional unitary representations of the Lorentz group.  These have both historical importance in mainstream physics, as well as connections to more speculative present-day theories.

The full theory of the finite-dimensional representations of the [[w:Lie algebra|''Lie algebra'']] of the Lorentz group is deduced using the general framework of the representation theory of [[w:semisimple Lie algebra|''semisimple Lie algebras'']]. The finite-dimensional representations of the connected component {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} of the full Lorentz group {{math|O(3; 1)}} are obtained by employing the [[w:Lie correspondence|''Lie correspondence'']] and the [[w:matrix exponential|''matrix exponential'']]. The full finite-dimensional representation theory of the [[w:universal covering group|''universal covering group'']] (and also the [[w:spin group|''spin group'']], a double cover) {{math|SL(2, ℂ)}} of {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} is obtained, and explicitly given in terms of action on a function space in [[#Representations of SL(2, C) and sl(2, C)|representations of SL(2, C) and sl(2, C)]]. The representatives of [[w:T-symmetry|''time reversal'']] and [[w:Parity (physics)|''space inversion'']] are given in [[#Space inversion and time reversal|space inversion and time reversal]], completing the finite-dimensional theory for the full Lorentz group. The general [[#Properties of the (''m'', ''n'') representations|properties of the (''m'', ''n'') representations]] are outlined. [[#Action on function spaces|Action on function spaces]] is considered, with the action on [[#Spherical harmonics|spherical harmonics]] and [[#The Riemann P-functions|the Riemann P-function]] appearing as examples. The infinite-dimensional case of irreducible unitary representations is [[#Clasification|classified]] and realized for the [[#Principal series|principal series]] and the [[w:complementary series|''complementary series'']]. Finally, the [[#Plancherel formula|Plancherel formula]] for {{math|SL(2, ℂ)}} is given.

The development of the representation theory has historically followed the development of the more general theory of representation theory of [[w:semisimple group|''semisimple groups'']], largely due to [[w:Élie Cartan|''Élie Cartan'']] and [[w:Hermann Weyl|''Hermann Weyl'']], but the Lorentz group has also received special attention due to its importance in physics. Notable contributors are physicist [[w:E. P. Wigner|''E. P. Wigner'']] and mathematician [[w:Valentine Bargmann|''Valentine Bargmann'']] with their [[w:Bargmann–Wigner equations|''Bargmann–Wigner programme'']],&lt;ref&gt;{{harvnb|Bargmann|Wigner|1948}}&lt;/ref&gt; one conclusion of which is, roughly, ''a classification of all unitary representations of the inhomogeneous Lorentz group amounts to a classification of all possible relativistic wave equations''.&lt;ref&gt;{{harvnb|Bekaert|Boulanger|2006}}&lt;/ref&gt; The classification of the irreducible infinite-dimensional representations of the Lorentz group was established by [[w:Paul Dirac|''Paul Dirac'']]´s doctoral student in theoretical physics, [[w:Harish-Chandra|''Harish-Chandra'']], later turned mathematician,&lt;ref group=nb&gt;In 1945 Harish-Chandra came to see Dirac in Cambridge. He became convinced that he was not suitable for theoretical physics. Harish-Chandra had found an error in a proof by Dirac in his work on the Lorentz group. Dirac said &quot;I am not interested in proofs but only interested in what nature does.&quot;

Harish-Chandra later wrote &quot;This remark confirmed my growing conviction that I did not have the mysterious sixth sense which one needs in order to succeed in physics and I soon decided to move over to mathematics.&quot;

Dirac did however suggest the topic of his thesis, the classification of the irreducible infinite-dimensional representations of the Lorentz group.

See {{harvnb|Dalitz|Peierls|1986}}&lt;/ref&gt; in 1947.

The [[#Non-technical introduction to representation theory|non-technical introduction]] contains some prerequisite material for readers not familiar with representation theory. The Lie algebra basis and other adopted conventions are given in [[#Conventions and Lie algebra bases|conventions and Lie algebra bases]].

== Non-technical introduction to representation theory ==
The present purpose is to illustrate the role of representation theory of groups in mathematics and in physics. Rigor and detail take the back seat, as the main objective is to fix the ''notion'' of finite-dimensional and infinite-dimensional representations of the Lorentz group. The reader familiar with these concepts should skip by.
{{Hidden begin| titlestyle = color:green;background:lightgrey;|title=Prerequisites outlined}}

=== Symmetry groups ===
{{Main|Group (mathematics)|Symmetry group}}
[[File:group D8 90.svg|140px|left|thumb|A 90° clockwise rotation of the square around its center. This is one of the symmetries of a square.]]

The mathematical notion of a [[w:group (mathematics)|''group'']] and the notion of [[w:symmetry|''symmetry'']] in both mathematics and physics are intimately related. A group has the simple property that if one element of a group is multiplied by another, the result is another element of the group. The same can, mutatis mutandis, be said of symmetries. Apply one symmetry operation (physically or by changing coordinate system), and then another one. The result is that of applying a ''single'' symmetry operation. Else they don't qualify as a symmetry operations the present context. [[w:Group theory|''Group theory'']] is thus the mathematical language in which symmetries of nature are expressed.&lt;ref&gt;{{harvnb|Tung|1985|loc=Preface.}}&lt;/ref&gt; These may relate to very concrete symmetries of physical objects, like the [[w:Group (mathematics)#Second example: a symmetry group|''symmetries of a square'']]. One then speaks of the [[w:symmetry group|''symmetry group'']] associated with the object.

In the case of a square, the symmetry group, called the [[w:Dihedral group|''dihedral group'' {{math|D&lt;sub&gt;4&lt;/sub&gt;}}]], is finite. For instance, only some [[w:rotation|''rotations'']], and some [[w:Reflection (mathematics)|''reflections'']] in the [[w:Euclidean plane|''plane'']], will make the transformed square look exactly like it did before the symmetry operation. Other objects possess higher symmetry. The [[w:sphere|''sphere'']] is the extreme example. It possesses full [[w:rotational symmetry|''rotational symmetry'']] and [[w:Reflection symmetry|''reflectional symmetry'']]. Rotate or reflect a ball with any kind of rotation or reflection about any plane through the origin, and it will look exactly the same as before the symmetry operation.

A central fact is that the symmetry groups can be represented by matrices.&lt;ref group=nb&gt;This is true of all groups encountered in the present context. There are examples of non-compact Lie groups not allowing a matrix representation.&lt;/ref&gt; In the case of {{math|''D''&lt;sub&gt;4&lt;/sub&gt;}} for the square, the [[w:Dihedral group#Matrix representation|''matrix representation'']] is composed of eight {{math|2 × 2}} matrices. In the case of the symmetries of a sphere, the matrix group is the [[w:orthogonal group|''orthogonal group'']] of three dimensions. These are {{math|3 × 3}} matrices.

=== Symmetry of space and time ===

::Main articles: [[w:Symmetry (physics)#Spacetime symmetries|''Symmetry (physics)#Spacetime symmetries'']], [[w:Spacetime symmetries|''Spacetime symmetries'']], and [[w:Lorentz group|''Lorentz group'']]

[[File:Sphere wireframe 10deg 6r.svg|140px|right|thumb|The sphere, a maximally symmetric object.]]

Less obvious is that ''space'' itself possesses symmetry. It too looks the same, no matter how one rotates it, so it has rotational symmetry. In fact, in this case, it is more practical to use [[w:Active and passive transformation|''passive rotations'']], meaning the observer&lt;ref group=nb&gt;It is tacitly always assumed that each [[w:inertial frame|''inertial frame'']] has a dedicated ''Lorentz observer'', I. e. someone who has, in principle, a complete record (i. e. coordinates!) of every event as observed in that frame.&lt;/ref&gt; rotates himself and does not attempt to physically rotate the universe. Mathematically, the ''active'' operation of a rotation is performed by multiplying [[w:position vector|''position vectors'']] by a [[w:rotation matrix|''rotation matrix'']]. A [[w:Passive transformation|''passive rotation'']] is accomplished by rotating only the basis vectors of the coordinate system. (Envisage the coordinate system being fixed in the rotated observer. Then ''actively'' rotate the observer ''only''.) In this way, every point in space obtains new coordinates, just as if it was somehow physically rigidly rotated. The Lorentz group contains all rotation matrices, extended to four dimensions with zeros in the first row and the first column except for the upper left element which is one, as elements. There are, in addition, matrices that effect [[w:Lorentz boost|''Lorentz boosts'']]. These can be thought of in the passive view as (instantly!) giving the coordinate system (and with it the observer) a velocity in a chosen direction. Two special transformations are used to invert the coordinate system in space, [[w:Parity (physics)|''space inversion'']], and in time, [[w:T-symmetry|''time reversal'']]. In the first case, the space coordinate axes are reversed. The latter is reversal of the time ''direction''. This is best though of as just having the observer set his clock at ''minus'' what it shows and then have the clock's hands move counterclockwise. Physical time progresses forward as always.

=== Lorentz transformations ===
::Main article: [[w:Lorentz transformation|''Lorentz transformation'']]
In the [[w:spacetime|''spacetime'']] of [[w:special relativity|''special relativity'']], called [[w:Minkowski space|''Minkowski space'']], space and time are interwoven. Thus the four coordinates of points in spacetime, called [[w:Event (relativity)|''events'']], change in ways unexpected before the advent of special relativity, with [[w:time dilation|''time dilation'']] and [[w:length contraction|''length contraction'']] as two immediate consequences. The four-dimensional matrices of [[w:Lorentz transformation|''Lorentz transformations'']] compose the [[w:Lorentz group|''Lorentz group'']]. Its elements represent symmetries, and just like physical objects can be rotated using rotation matrices, the same physical objects (whose coordinates now include the time coordinate) can be ''transformed'' using the matrices representing Lorentz transformations. In particular, the [[w:four-vector|''four-vector'']] representing an [[w:event (relativity)|''event'']] in Lorentz frame transforms as

:&lt;math&gt; x' = \begin{bmatrix} x'^0 \\ x'^1 \\ x'^2 \\ x'^3 \end{bmatrix} = \begin{bmatrix} \lambda_{00}&amp;\lambda_{01}&amp;\lambda_{02}&amp;\lambda_{03}\\ \lambda_{10}&amp;\lambda_{11}&amp;\lambda_{12}&amp;\lambda_{13} \\ \lambda_{20}&amp;\lambda_{21}&amp;\lambda_{22}&amp;\lambda_{23} \\ \lambda_{30}&amp;\lambda_{31}&amp;\lambda_{32}&amp;\lambda_{33} \end{bmatrix} \begin{bmatrix} x^0 \\ x^1 \\ x^2 \\ x^3 \end{bmatrix},&lt;/math&gt;

or on short form

:&lt;math&gt;x' = \Lambda x.&lt;/math&gt;

=== Multiplication table and representations ===
::Main articles: [[w:Cayley table|''Cayley table'']] and [[w:Representation theory|''Representation theory'']]
The basic feature of every finite group is its ''multiplication table'', also called [[w:Cayley table|''Cayley table'']], that records the result of multiplying any two elements. A ''representation'' of a group can be thought of new set of elements, finite-dimensional or infinite-dimensional matrices, ''giving the same multiplication table'' after mapping the old elements to the new elements in a one-to-one fashion.&lt;ref group=nb&gt;It isn't actually required that the mapping is one-to-one. It is merely required that the mapping is a [[w:group homomorphism|''group homomorphism'']], i. e. {{math|Π(''gh'') {{=}} Π(''g'')Π(''h'')}} into some {{math|GL(''V'')}} the [[w:general linear group|''general linear group'']] of some vector space {{mvar|V}}. (The vector space {{mvar|V}} is allowed to be infinite-dimensional, e.g a [[w:Hilbert space|''Hilbert space'']] {{math|''H''}}, in which case one speaks of {{math|''B''(''H'')}}, linear operators on {{mvar|H}} instead of {{math|GL(''V'')}}.&lt;/ref&gt; The same holds true in the case of an infinite group like the [[w:rotation group SO(3)|''rotation group SO(3)'']] or the Lorentz group. The multiplication table is just harder to visualize in the case of a group of [[w:uncountable|''uncountable'']] size (same size as the set of reals).

=== Ordinary Lorentz transformations matrices do not suffice ===
[[File:Lorentz boost electric charge.svg|300px|thumb|Lorentz boost of an electric charge, the charge is at rest in one frame or the other.]]

The objects to be transformed may be something else than ordinary physical objects extending in three spatial dimensions (and time, unless the frame is the rest frame). It is at this point that representation theory enters the picture. The [[w:electromagnetic field|''electromagnetic field'']] is usually envisaged by assignment to each point in space a three-dimensional vector representing the [[w:electric field|''electric field'']] and another three-dimensional vector representing the [[w:magnetic field|''magnetic field'']]. When space is rotated, the expected thing happens. The electric field and the magnetic field vectors at a designated point rotate with preserved length and angle between them. Under Lorentz boosts they behave differently, and in a way showing that the two vectors certainly aren't separate physical objects. The electric and magnetic components mix. See the illustration on the right. The [[w:electromagnetic field tensor|''electromagnetic field tensor'']] displays the [[w:Lorentz covariance|''manifestly covariant'']] mathematical structure of the electromagnetic field.

=== Finite-dimensional representations by matrices ===

The problem of representation theory of the Lorentz group is, in the finite-dimensional case, to find new sets of matrices, not necessarily {{math|4 × 4}} in size that satisfies the same multiplication table as the matrices in the original Lorentz group. Returning to the example of the electromagnetic field, what is needed here are {{math|6 × 6}}-matrices that can be applied to a {{math|6}}-dimensional column vector containing the all together six components of the electromagnetic field. Thus one is looking for {{math|6 × 6}}-matrices such that

:&lt;math&gt; F' = \begin{bmatrix} E'^1 \\ E'^2 \\ E'^3 \\ B'^1 \\ B'^2 \\ B'^3  \end{bmatrix} = \begin{bmatrix} \Pi(\Lambda)_{00}&amp;\Pi(\Lambda)_{01}&amp;\Pi(\Lambda)_{02}&amp;\Pi(\Lambda)_{03}&amp;\Pi(\Lambda)_{04}&amp;\Pi(\Lambda)_{05}\\ \Pi(\Lambda)_{10}&amp;\Pi(\Lambda)_{11}&amp;\Pi(\Lambda)_{12}&amp;\Pi(\Lambda)_{13}&amp;\Pi(\Lambda)_{14}&amp;\Pi(\Lambda)_{15} \\ \Pi(\Lambda)_{20}&amp;\Pi(\Lambda)_{21}&amp;\Pi(\Lambda)_{22}&amp;\Pi(\Lambda)_{23}&amp;\Pi(\Lambda)_{24}&amp;\Pi(\Lambda)_{25} \\ \Pi(\Lambda)_{30}&amp;\Pi(\Lambda)_{31}&amp;\Pi(\Lambda)_{32}&amp;\Pi(\Lambda)_{33}&amp;\Pi(\Lambda)_{34}&amp;\Pi(\Lambda)_{35} \\ \Pi(\Lambda)_{40}&amp;\Pi(\Lambda)_{41}&amp;\Pi(\Lambda)_{42}&amp;\Pi(\Lambda)_{43}&amp;\Pi(\Lambda)_{44}&amp;\Pi(\Lambda)_{45} \\ \Pi(\Lambda)_{50}&amp;\Pi(\Lambda)_{51}&amp;\Pi(\Lambda)_{52}&amp;\Pi(\Lambda)_{53}&amp;\Pi(\Lambda)_{54}&amp;\Pi(\Lambda)_{55} \end{bmatrix} \begin{bmatrix} E^1 \\ E^2 \\ E^3 \\ B^1 \\ B^2 \\ B^3 \end{bmatrix},&lt;/math&gt;

in short

:&lt;math&gt;F' = \Pi(\Lambda)F,&lt;/math&gt;

correctly expresses the transformation of the electromagnetic field under the Lorentz transformation {{math|Λ}}.&lt;ref group=nb&gt;This transformation is usually expressed differently, see e.g. [[w:Lorentz transformation#Transformation of the electromagnetic field|''transformation of the electromagnetic field'']]. That method can be traced converted to applying a {{math|6 × 6}}-matrix like here and vice versa since the field tensor has {{math|6}} independent components.&lt;/ref&gt; The same reasoning can be applied to Dirac's [[w:bispinor|''bispinors'']]. While these have {{math|4}} components, the original {{math|4 × 4}}-matrices in the Lorentz group will not do the job properly, not even when restricted to mere rotations. Another {{math|4 × 4}}-representation is needed.

The sections dedicated to finite-dimensional representations are dedicated to exposing ''all'' such representations by finite-dimensional matrices that respect the multiplication table.

=== Infinite-dimensional representations by action on vector spaces of functions ===

Infinite-dimensional representations are usually realized as acting on sets of real or complex-valued functions on a set {{mvar|X}} endowed with a [[w:group action|''group action'']]. A set being endowed with a group action {{mvar|A}} means, in essence, that if {{math|''x'' ∈ ''X''}} and {{math|''g'' ∈ ''G''}} that {{math|''A''(''g'')''x'' {{=}} ''y''}} with {{math|''y'' ∈ ''X''}}. Now if {{math|ℂ&lt;sup&gt;''X''&lt;/sup&gt;}} denotes the set of all complex-valued functions on {{mvar|X}}, which is a [[w:vector space|''vector space'']], a representation {{mvar|Π}} of {{mvar|G}} can be defined by&lt;ref name=&quot;Rossmann 2002 loc=Section 6.1&quot;&gt;{{harvnb|Rossmann|2002|loc=Section 6.1.}}&lt;/ref&gt;

:&lt;math&gt;(\Pi(g))f(x) = f(A(g^{-1})x),\quad f \in \mathbb C^{X}, g \in G, x\in X.&lt;/math&gt;

The point to make is that again one has

:&lt;math&gt;(\Pi(g))f\in \mathbb C^{X}.&lt;/math&gt;

and one has a representation of {{mvar|G}}. This representation of {{mvar|G}} is finite-dimensional if and only if {{mvar|X}} is a finite set. This method is very general, and one typically explores vector spaces of more specialized functions on sets close at hand. To illustrate this procedure, consider a group {{mvar|G}} of {{math|''n''}}-dimensional matrices as a subset of Euclidean space {{math|ℝ&lt;sup&gt;''n''&lt;sup&gt;2&lt;/sup&gt;&lt;/sup&gt;}}, and let the space of functions be [[w:polynomial|''polynomials'']], perhaps of some maximum degree {{mvar|d}}, or even [[w:homogeneous polynomial|''homogeneous polynomials'']] of degree {{mvar|d}}, all defined on {{math|ℝ&lt;sup&gt;''n''&lt;sup&gt;2&lt;/sup&gt;&lt;/sup&gt;}}. Then restrict those functions to {{math|''G'' ⊂ ℝ&lt;sup&gt;''n''&lt;sup&gt;2&lt;/sup&gt;&lt;/sup&gt;}}. Now observe that the set {{math|''X'' {{=}} ''G''}} automatically comes equipped with group actions, namely

:&lt;math&gt;L_gh = gh, R_gh = hg, C_gh = ghg^{-1}, g, h \in G.&lt;/math&gt;

Here {{math|''L''&lt;sub&gt;''g''&lt;/sub&gt;}} denotes ''left action (by {{mvar|g}})'', {{math|''R''&lt;sub&gt;''g''&lt;/sub&gt;}} denotes ''right action (by {{mvar|g}})'', and {{math|''C''&lt;sub&gt;''g''&lt;/sub&gt;}} denotes ''conjugation (by {{mvar|g}})''. With this sort of action, the vectors being acted on are functions. The resulting representations are (when the functions are unrestricted), in the first and second cases respectively, the [[w:left regular representation|''left regular representation'']] and the ''right regular representation'' of {{mvar|G}} on {{math|ℂ&lt;sup&gt;''G''&lt;/sup&gt;}}.&lt;ref name=&quot;Rossmann 2002 loc=Section 6.1&quot;/&gt;

The goal in the infinite-dimensional case of the representation theory is to classify all different possible representations, and to exhibit them in terms of vector spaces of functions and the action of the standard representation on the arguments of the functions.

=== Infinite-dimensional representations viewed as infinite-dimensional matrices ===
In order to relate this to the finite-dimensional case, one may chose a basis for the vector space of functions and simply then examine what happens to the basis functions under a given transformation. Take ''image'' of the first basis function under a transformation, expressed as a linear combination the basis functions. Explicitly, if {{math|''f''&lt;sub&gt;1&lt;/sub&gt;, ''f''&lt;sub&gt;2&lt;/sub&gt;, ...}} is a basis, compute

:&lt;math&gt;\begin{align}
\Pi(\Lambda)f_1 &amp;= \lambda_{11}f_1 + \lambda_{21}f_2 + \cdots,\\
\Pi(\Lambda)f_2 &amp;= \lambda_{12}f_1 + \lambda_{22}f_2 + \cdots,\\
&amp;\vdots
\end{align}&lt;/math&gt;

The coefficients of the basis functions in this expression is then the first column in a representative matrix. Proceed. In general, the resulting matrix is [[w:countably infinite|''countably infinite'']] in dimension:

:&lt;math&gt; \Pi(\Lambda) = \begin{bmatrix}\lambda_{11}&amp;\lambda_{12}&amp;\cdots \\ \lambda_{21}&amp;\lambda_{22}&amp;\cdots \\ \vdots &amp; \vdots &amp;\ddots \end{bmatrix}&lt;/math&gt;

Again, it is required that the set of infinite matrices obtained this way stand in one-to-one correspondence with the original {{math|4 × 4}}-matrices ''and'' that the multiplication table is the right one - the one of the {{math|4 × 4}}-matrices.&lt;ref group=nb&gt;It may happen that the multiplication of two representative infinite-dimensional matrices is ill-defined. The right composition rule can be verified by other means though.&lt;/ref&gt; It should be emphasized that in the infinite-dimensional case, one is rarely concerned with these matrices. They are exposed here only to highlight the common thread. But ''individual'' matrix elements are frequently computed, especially for the Lie algebra (below).

=== Lie algebra ===
::Main article: [[w:Lie algebra|''Lie algebra'']]
The Lorentz group is a [[w:Lie group|''Lie group'']] and has as such a [[w:Lie algebra|''Lie algebra'']], The Lie algebra is a vector space of matrices that can be said to model the group near the identity. It is endowed with a multiplication operation, the [[w:Lie bracket|''Lie bracket'']]. With it, the product in the group can near the identity be expressed in Lie algebraic terms (but not in a particularly simple way). The link between the (matrix) Lie algebra and the (matrix) Lie group is the [[w:matrix exponential|''matrix exponential'']]. It is one-to-one near the identity in the group.

Due to this it often suffices to find [[w:Lie algebra representation|''representations of the Lie algebra'']]. Lie algebras are much simpler objects than Lie groups to work with. Due to the fact that the Lie algebra is a finite-dimensional vector space, in the case of the Lorentz Lie algebra the dimension is {{math|6}}, one need only find a finite number of representative matrices of the Lie algebra, one for each element of a basis of the Lie algebra as a vector space. The rest follow from extension by linearity, and the representation of the group is obtained by exponentiation.

The [[w:metric signature|''metric signature'']] to be used below is {{math|(−1,  1 ,  1,  1)}} and the metric is given by {{math|''η'' {{=}} diag(−1,  1,  1,  1)}}. The physics convention for Lie algebras and the exponential mapping is used. These choices are arbitrary, but once they are made, fixed. One possible choice of [[w:basis (linear algebra)|''basis'']] for the Lie algebra is, in the standard representation, given by

:&lt;math&gt;\begin{align}
J_1 &amp;= J^{23} = -J^{32} = i\left(\begin{smallmatrix}
0&amp;0&amp;0&amp;0\\ 0&amp;0&amp;0&amp;0\\ 0&amp;0&amp;0&amp;-1\\ 0&amp;0&amp;1&amp;0\\
\end{smallmatrix}\right),
J_2 = J^{31} = -J^{13} = i\left(\begin{smallmatrix}
0&amp;0&amp;0&amp;0\\ 0&amp;0&amp;0&amp;1\\ 0&amp;0&amp;0&amp;0\\ 0&amp;-1&amp;0&amp;0\\
\end{smallmatrix}\right),
J_3 = J^{12} = -J^{21} = i\left(\begin{smallmatrix}
0&amp;0&amp;0&amp;0\\ 0&amp;0&amp;-1&amp;0\\ 0&amp;1&amp;0&amp;0\\ 0&amp;0&amp;0&amp;0\\
\end{smallmatrix}\right),\\
K_1 &amp;= J^{01} = J^{10} = i\left(\begin{smallmatrix}
0&amp;1&amp;0&amp;0\\ 1&amp;0&amp;0&amp;0\\ 0&amp;0&amp;0&amp;0\\ 0&amp;0&amp;0&amp;0\\
\end{smallmatrix}\right),
K_2 = J^{02} = J^{20} = i\left(\begin{smallmatrix}
0&amp;0&amp;1&amp;0\\ 0&amp;0&amp;0&amp;0\\ 1&amp;0&amp;0&amp;0\\ 0&amp;0&amp;0&amp;0\\
\end{smallmatrix}\right),
K_3 = J^{03} = J^{30} = i\left(\begin{smallmatrix}
0&amp;0&amp;0&amp;1\\ 0&amp;0&amp;0&amp;0\\ 0&amp;0&amp;0&amp;0\\ 1&amp;0&amp;0&amp;0\\
\end{smallmatrix}\right).
\end{align}&lt;/math&gt;
{{Hidden end}}

== Applications ==
Many of the representations, both finite-dimensional and infinite-dimensional, are important in theoretical physics. Representations appear in the description of fields in [[w:classical field theory|''classical field theory'']], most importantly the [[w:electromagnetic field|''electromagnetic field'']], and of [[w:particle|''particles'']] in [[w:relativistic quantum mechanics|''relativistic quantum mechanics'']], as well as of both particles and quantum fields in [[w:quantum field theory|''quantum field theory'']] and of various objects in [[w:string theory|''string theory'']] and beyond. The representation theory also provides the theoretical ground for the concept of [[w:spin (physics)|''spin'']]. The theory enters into [[w:general relativity|''general relativity'']] in the sense that in small enough regions of spacetime, physics is that of special relativity.&lt;ref&gt;{{harvnb|Misner|Thorne|Wheeler|1973}}&lt;/ref&gt;

The finite-dimensional irreducible non-unitary representations together with the irreducible infinite-dimensional unitary representations of the ''inhomogeneous'' Lorentz group, the Poincare group, are the representations that have direct physical relevance.&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Section 2.5, Chapter 5.}}&lt;/ref&gt;&lt;ref&gt;{{harvnb|Tung|1985|loc=Sections 10.3, 10.5.}}&lt;/ref&gt;

Infinite-dimensional unitary representations of the Lorentz group appear by ''restriction'' of the irreducible infinite-dimensional unitary representations of the Poincaré group acting on the [[w:Hilbert space|''Hilbert spaces'']] of [[w:relativistic quantum mechanics|''relativistic quantum mechanics'']] and [[w:quantum field theory|''quantum field theory'']]. But these are also of mathematical interest and of ''potential'' direct physical relevance in other roles than that of a mere restriction.&lt;ref&gt;{{harvnb|Tung|1985|loc=Section 10.4.}}&lt;/ref&gt; There were speculative theories,&lt;ref&gt;{{harvnb|Dirac|1945}}&lt;/ref&gt;&lt;ref name=&quot;Harish-Chandra 1947&quot;/&gt; (tensors and spinors have infinite counterparts in the ''expansors'' of Dirac and the ''expinors'' of Harish-Chandra) consistent with relativity and quantum mechanics, but they have found no proven physical application. Modern speculative theories potentially have similar ingredients per below.

=== Mathematics ===
From the point of view that the goal of mathematics is to classify and characterize, the representation theory of the Lorentz group is since 1947 a finished chapter. But in association with the Bargmann&amp;ndash;Wigner programme, there are (as of 2006) yet unresolved purely mathematical problems, linked to the infinite-dimensional unitary representations.

The irreducible infinite-dimensional unitary representations may have indirect relevance to physical reality in speculative modern theories since the (generalized) Lorentz group appears as the [[w:little group|''little group'']] of the Poincare group of spacelike vectors in higher spacetime dimension. The corresponding infinite-dimensional unitary representations of the (generalized) Poincaré group are the so-called ''tachyonic representations''. [[w:Tachyon|''Tachyons'']] appear in the spectrum of [[w:bosonic string|''bosonic strings'']] and are associated with instability of the vacuum.&lt;ref&gt;{{harvnb|Zwiebach|2004|loc=Section 12.8.}}&lt;/ref&gt;&lt;ref name=&quot;Bekaert 2006 48&quot;&gt;{{harvnb|Bekaert|Boulanger|2006|p=48.}}&lt;/ref&gt; Even though tachyons may not be realized in nature, these representations must be mathematically ''understood'' in order to understand string theory. This is so since tachyon states turn out to appear in [[w:superstring theory|''superstring theories'']] too in attempts to create realistic models.&lt;ref&gt;{{harvnb|Zwiebach|2004|loc=Section 18.8.}}&lt;/ref&gt;

One open problem (as of 2006) is the completion of the Bargmann&amp;ndash;Wigner programme for the isometry group {{math|SO(''D'' – 2, 1)}} of the [[w:de Sitter space|''de Sitter space'']]time {{math|''dS''&lt;sub&gt;''D'' – 2&lt;/sub&gt;}}. Ideally, one would like to see the physical components of wave functions realized on the hyperboloid {{math|''dS''&lt;sub&gt;''D'' – 2&lt;/sub&gt;}} of radius {{math|''μ'' &gt; 0}} embedded in {{math|ℝ&lt;sup&gt;''D'' − 2, 1&lt;/sup&gt;}} and the corresponding {{math|O(''D'' − 2, 1)}} covariant wave equations of the infinite-dimensional unitary representation to be known.&lt;ref name=&quot;Bekaert 2006 48&quot;/&gt;

It is common in mathematics to regard the Lorentz group to be, foremost, the [[w:Möbius transformation|''Möbius group'']] to which it is isomorphic. The group may be represented in terms of a set of functions defined on the [[w:Riemann sphere|''Riemann sphere'']]. These are the [[w:Riemann P-function|''Riemann P-functions'']], which are expressible as [[w:hypergeometric function|''hypergeometric functions'']].

=== Classical field theory ===
While the [[w:electromagnetic field|''electromagnetic field'']] together with the [[w:gravitational field|''gravitational field'']] are the only classical fields providing accurate descriptions of nature, other types of classical fields are important too. In the approach to [[w:quantum field theory|''quantum field theory'']] (QFT) referred to as [[w:second quantization|''second quantization'']], one begins with one or more classical fields, where e.g. the wave functions solving the [[w:Dirac equation|''Dirac equation'']] are considered as classical fields ''prior'' to (second) quantization.&lt;ref name=&quot;Greiner 1996 loc=Chapter 2&quot;&gt;{{harvnb|Greiner|Reinhardt|1996|loc=Chapter 2.}}&lt;/ref&gt; While second quantization and the [[w:Lagrangian (field theory)|''Lagrangian formalism'']] associated with it is not a fundamental aspect of QFT,&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Foreword and introduction to chapter 7.}}&lt;/ref&gt; it is the case that so far all quantum field theories can be approached this way, including the [[w:standard model|''standard model'']].&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Introduction to chapter 7.}}&lt;/ref&gt; In these cases, there are classical versions of the field equations following from the [[w:Euler&amp;ndash;Lagrange equations|''Euler&amp;ndash;Lagrange equations'']] derived from the Lagrangian using the [[w:principle of least action|''principle of least action'']]. These field equations must be relativistically invariant, and their solutions (which will qualify as relativistic wave functions according to the definition below) must transform under some representation of the Lorentz group.

The action of the Lorentz group on the space of ''field configurations'' (a field configuration is the spacetime history of a particular solution, e.g. the electromagnetic field in all of space over all time is ''one'' field configuration) resembles the action on the Hilbert spaces of quantum mechanics, except that the [[w:commutator bracket|''commutator brackets'']] are replaced by field theoretical [[w:Poisson bracket|''Poisson brackets'']].&lt;ref name=&quot;Greiner 1996 loc=Chapter 2&quot;/&gt;

=== Relativistic quantum mechanics ===
For the present purpose one may make the following definition:&lt;ref&gt;{{harvnb|Tung|1985|loc=Definition 10.11.}}&lt;/ref&gt; A ''relativistic wave function'' is a set of {{mvar|n}} functions {{math|''ψ''&lt;sup&gt;''α''&lt;/sup&gt;}} on spacetime which transforms under an arbitrary proper Lorentz transformation {{math|Λ}} as

:&lt;math&gt;\psi'^\alpha(x) = D{[\Lambda]^\alpha}_\beta\psi^\beta(\Lambda^{-1}x),&lt;/math&gt;

where {{math|''D''[Λ]}} is an {{math|''n''}}-dimensional matrix representative of {{math|Λ}} belonging to some direct sum of the {{math|(''m'', ''n'')}} representations to be introduced below.

The most useful relativistic quantum mechanics ''one-particle'' theories (there are no fully consistent such theories) are the [[w:Klein&amp;ndash;Gordon equation|''Klein&amp;ndash;Gordon equation'']]&lt;ref&gt;{{harvtxt|Greiner|Müller|1994|loc=Chapter 1}}&lt;/ref&gt; and the [[w:Dirac equation|''Dirac equation'']]&lt;ref&gt;{{harvtxt|Greiner|Müller|1994|loc=Chapter 2}}&lt;/ref&gt; in their original setting. They are relativistically invariant and their solutions transform under the Lorentz group as [[w:Lorentz scalar|''Lorentz scalars'']] ({{math|(''m'', ''n'') {{=}} (0, 0)}}) and [[w:bispinor|''bispinors'']] respectively ({{math|(0, {{frac|1|2}}) ⊕ ({{frac|1|2}}, 0)}}). The electromagnetic field is a relativistic wave function according to this definition, transforming under {{math|(1, 0) ⊕ (0, 1)}}.&lt;ref&gt;{{harvnb|Tung|1985|p=203.}}&lt;/ref&gt;

=== Quantum field theory ===
In QFT, the demand for relativistic invariance enters, among other ways in that the [[w:S-matrix|''S-matrix'']] necessarily must be Poincaré invariant.&lt;ref&gt;{{harvtxt|Weinberg|2002|loc=Section 3.3}}&lt;/ref&gt; This has the implication that there is one or more infinite-dimensional representation of the Lorentz group acting on [[w:Fock space|''Fock space'']].&lt;ref group=nb&gt;See formula (1) in [[w:S-matrix#From free particle states|''S-matrix#From free particle states'']] for how free multi-particle states transform.&lt;/ref&gt; One way to guarantee the existence of such representations is the existence of a Lagrangian description (with modest requirements imposed, see the reference) of the system using the canonical formalism, from which one may deduce a realization of the generators of the Lorentz group.&lt;ref&gt;{{harvtxt|Weinberg|2002|loc=Section 7.4.}}&lt;/ref&gt;

The transformations of field operators illustrate the complementary role played by the finite-dimensional representations of the Lorentz group and the infinite-dimensional unitary representations of the Poincare group, witnessing the deep unity between mathematics and physics.&lt;ref&gt;{{harvnb|Tung|1985|loc=Introduction to chapter10.}}&lt;/ref&gt; For illustration, consider the definition of some {{math|''n''}}-component [[w:field operator|''field operator'']]:&lt;ref&gt;{{harvnb|Tung|1985|loc=Definition 10.12.}}&lt;/ref&gt; Given a matrix representation as above, a relativistic field operator is a set of {{mvar|n}} operator valued functions on spacetime which transforms under proper Lorentz transformations {{math|Λ}} according to&lt;ref&gt;{{harvnb|Tung|1985|loc=Equation 10.5-2.}}&lt;/ref&gt;&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Equations 5.1.6-7.}}&lt;/ref&gt;

:&lt;math&gt; \Psi^\alpha(x) \rightarrow \Psi'^\alpha(x') = U[\Lambda]\Psi^\alpha(x) U[\Lambda^{-1}] = \mathcal D{[\Lambda^{-1}]^\alpha}_\beta\Psi^{\beta}(\Lambda x)&lt;/math&gt;

By considerations of differential constraints that the field operator must be subjected to in order to describe a single particle with definite mass {{mvar|m}} and spin {{mvar|s}} (or helicity), one finds&lt;ref name=&quot;Tung 1985 loc=Equation 10.5-18&quot;&gt;{{harvnb|Tung|1985|loc=Equation 10.5-18.}}&lt;/ref&gt;&lt;ref group=nb&gt;{{harvnb|Weinberg|2002|loc=Equations 5.1.4-5.}} Weinberg deduces the necessity of creation and annihilation operators from another consideration, the [[w:Cluster decomposition theorem|''cluster decomposition principle'']], {{harvtxt|Weinberg|2002|loc=Chapter 4.}}&lt;/ref&gt;

:&lt;math&gt;\Psi^\alpha(x) = \sum_\sigma \int dp[a(\mathbf p, \sigma) u^\alpha(\mathbf p, \sigma)e^{ip\cdot x} + a^\dagger(\mathbf p, \sigma) v^\alpha(\mathbf p, \sigma)e^{-ip\cdot x}],&lt;/math&gt;

where {{math|''a''&lt;sup&gt;†&lt;/sup&gt;, ''a''}} are interpreted as [[w:creation and annihilation operators|''creation and annihilation operators'']] respectively. The [[w:creation operator|''creation operator'']] {{math|''a''&lt;sup&gt;†&lt;/sup&gt;}} transforms according to&lt;ref name=&quot;Tung 1985 loc=Equation 10.5-18&quot;/&gt;&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Equations 5.1.11-12.}}&lt;/ref&gt;

:&lt;math&gt;a^\dagger(\mathbf p, \sigma) \rightarrow a'^\dagger(\mathbf p', \sigma) = U[\Lambda]a^\dagger(\mathbf p, \sigma) U[\Lambda^{-1}] = a^\dagger(\Lambda \mathbf p, \rho) D^{(s)}{[R(\Lambda, \mathbf p)^{-1}]^\rho}_\sigma,&lt;/math&gt;

and similarly for the annihilation operator. The point to be made is that the field operator transforms according to a finite-dimensional non-unitary representation of the Lorentz group, while the creation operator transforms under the infinite-dimensional unitary representation of the Poincare group characterized by the mass and spin {{math|(''m'', ''s'')}} of the particle. The connection between the two is the ''wave function'', also called ''cofficient function''

:&lt;math&gt;v^\alpha(\mathbb p,\sigma)e^{-ip\cdot x}&lt;/math&gt;

that carries ''both'' the indices {{math|(''x'', ''α'')}} operated on by Lorentz transformations and the indices {{math|('''''p''''', ''σ'')}} operated on by Poincaré transformations. This may be called the Lorentz&amp;ndash;Poincaré connection.&lt;ref&gt;{{harvnb|Tung|1985|loc=Section 10.5.3.}}&lt;/ref&gt; All of the above formulas, including the definition of the field operator in terms of creation and annihilation operators, as well as the differential equations satisfied by the field operator for a particle with specified mass, spin and the {{math|(''m'', ''n'')}} representation under which it is supposed to transform,&lt;ref group=nb&gt;A prescription for how the particle should behave under CPT symmetry may be required as well.&lt;/ref&gt; and also that of the wave function, can be derived from group theoretical considerations alone once the frameworks of quantum mechanics and special relativity is given.&lt;ref group=nb&gt;For instance, there are versions (free field equations, i.e. without interaction terms) of the [[w:Klein&amp;ndash;Gordon equation|''Klein&amp;ndash;Gordon equation'']], the [[w:Dirac equation|''Dirac equation'']], the [[w:Maxwell equations|''Maxwell equations'']], the [[w:Proca equation|''Proca equation'']], the [[w:Rarita&amp;ndash;Schwinger equation|''Rarita&amp;ndash;Schwinger equation'']], and the [[w:Einstein field equations|''Einstein field equations'']] that can systematically be deduced by starting from a given representation of the Lorentz group. In general, these are collectively the quantum field theory versions of the [[w:Bargmann&amp;ndash;Wigner equations|''Bargmann&amp;ndash;Wigner equations'']].

See {{harvtxt|Weinberg|2002|loc=Chapter 5}}, {{harvtxt|Tung|1985|loc=Section 10.5.2}} and references given in these works.

It should be remarked that high spin theories ({{math|''s'' &gt; 1}}) encounter difficulties. See {{harvtxt|Weinberg|2002|loc=Section 5.8}}, on general {{math|(''m'', ''n'')}} fields, where this is discussed in some depth, and references therein. High spin particles do without a doubt ''exist'', e.g. nuclei, the known ones are just not ''elementary''.&lt;/ref&gt;

=== Speculative theories ===
In theories in which spacetime can have more than {{math|''D'' {{=}} 4}} dimensions, the generalized Lorentz groups {{math|O(''D'' − 1; 1)}} of the appropriate dimension take the place of {{math|O(3; 1)}}.&lt;ref group = nb&gt;For part of their representation theory, see {{harvtxt|Bekaert|Boulanger|2006}}, which is dedicated to representation theory of the Poincare group. These representations are obtained by the method of [[w:induced representation|''induced representations'']] or, in physics parlance, the method of the [[w:little group|''little group'']], pioneered by Wigner in 1939 for this type of group and put on firm mathematical footing by [[w:George Mackey|''George Mackey'']] in the fifties.&lt;/ref&gt;

The requirement of Lorentz invariance takes on perhaps its most dramatic effect in [[w:string theory|''string theory'']]. ''Classical'' relativistic strings can be handled in the Lagrangian framework by using the [[w:Nambu&amp;ndash;Goto action|''Nambu&amp;ndash;Goto action'']].&lt;ref&gt;{{harvnb|Zwiebach|2004|loc=Section 6.4.}}&lt;/ref&gt; This results in a relativistically invariant theory in any spacetime dimension.&lt;ref&gt;{{harvnb|Zwiebach|2004|loc=Chapter 7.}}&lt;/ref&gt; But as it turns out, the theory of [[w:Open string (physics)|''open'']] and [[w:closed string|''closed'']] [[w:bosonic string theory|''bosonic strings'']] (the simplest string theory) is impossible to quantize in such a way that the Lorentz group is represented on the space of states (a [[w:Hilbert space|''Hilbert space'']]) unless the dimension of spacetime is 26.&lt;ref&gt;{{harvnb|Zwiebach|2004|loc=Section 12.5.}}&lt;/ref&gt; The corresponding result for [[w:superstring theory|''superstring theory'']] is again deduced demanding Lorentz invariance, but now with [[w:supersymmetry|''supersymmetry'']]. In these theories the [[w:Poincaré algebra|''Poincaré algebra'']] is replaced by a [[w:supersymmetry algebra|''supersymmetry algebra'']] which is a [[w:graded Lie algebra|{{math|''Z''&lt;sub&gt;2&lt;/sub&gt;}}-''graded Lie algebra'']] extending the Poincaré algebra. The structure of such an algebra is to a large degree fixed by the demands of Lorentz invariance. In particular, the fermionic operators (grade {{math|1}}) belong to a (0, &amp;frac12;) or (&amp;frac12;, 0) representation space of the (ordinary) Lorentz Lie algebra.&lt;ref name=&quot;Weinberg 2000 loc=Section 25.2&quot;&gt;{{harvnb|Weinberg|2000|loc=Section 25.2.}}&lt;/ref&gt; The only possible dimension of spacetime in such theories is 10.&lt;ref&gt;{{harvnb|Zwiebach|2004|loc=Last paragraph, section 12.6.}}&lt;/ref&gt;

== Finite-dimensional representations ==
Representation theory of groups in general, and Lie groups in particular, is a very rich subject. The full Lorentz group is no exception. The Lorentz group has some properties that makes it &quot;agreeable&quot; and others that make it &quot;not very agreeable&quot; within the context of representation theory. The group is [[w:simple (abstract algebra)|''simple'']] and thus [[w:Semisimple Lie algebra|''semisimple'']], but is not [[w:Connected set|''connected'']], and none of its components are [[w:simply connected space|''simply connected'']]. Perhaps most importantly, the Lorentz group is not [[w:compact space|''compact'']].&lt;ref&gt;These facts can be found in most introductory mathematics and physics texts. See e.g. {{harvtxt|Rossmann|2002}}, {{harvtxt|Hall|2015}} and {{harvtxt|Tung|1985}}.&lt;/ref&gt;

For finite-dimensional representations, the presence of semisimplicity means that the Lorentz group can be dealt with the same way as other semisimple groups using a well-developed theory. In addition, all representations are built from the [[w:Irreducible representation|''irreducible'']] ones, since the Lie algebra possesses the ''complete reducibility property''.&lt;ref group=nb&gt;{{harvtxt|Hall|2015|loc=Section 4.4.}}
{{Paragraph break}}
One says that a group has the ''complete reducibility property'' if every representation decomposes as a direct sum of irreducible representations.&lt;/ref&gt;&lt;ref&gt;{{harvtxt|Hall|2015|loc=Theorem 4.34 and following discussion.}}&lt;/ref&gt; But, the non-compactness of the Lorentz group, in combination with lack of simple connectedness, cannot be dealt with in all the aspects as in the simple framework that applies to simply connected, compact groups. Non-compactness implies, for a connected simple Lie group, that no nontrivial finite-dimensional [[w:Unitary representation|''unitary'']] representations exist.&lt;ref name=&quot;Wigner 1939&quot;&gt;{{harvnb|Wigner|1939}}&lt;/ref&gt; Lack of simple connectedness gives rise to [[w:spin representation|''spin representations'']] of the group.&lt;ref&gt;{{harvnb|Hall|2015|loc=Appendix D2.}}&lt;/ref&gt; The non-connectedness means that, for representations of the full Lorentz group, one has to deal with [[w:T-symmetry|''time reversal'']] and [[w:Parity (physics)|''space inversion'']] separately.&lt;ref&gt;{{harvnb|Greiner|Reinhardt|1996}}&lt;/ref&gt;&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Section 2.6 and Chapter 5.}}&lt;/ref&gt;

=== History ===
The development of the finite-dimensional representation theory of the Lorentz group mostly follows that of the subject in general. Lie theory originated with [[w:Sophus Lie|''Sophus Lie'']] in 1873.&lt;ref&gt;{{harvnb|Lie|1888}}, 1890, 1893&lt;/ref&gt; By 1888 the [[w:list of simple lie groups|''classification of simple Lie algebras'']] was essentially completed by [[w:Wilhelm Killing|''Wilhelm Killing'']].&lt;ref&gt;{{harvnb|Killing|1888}}&lt;/ref&gt; In 1913 the [[w:Cartan's theorem|''theorem of highest weight'']] for representations of simple Lie algebras, the path that will be followed here, was completed by [[w:Élie Cartan|''Élie Cartan'']].&lt;ref&gt;{{harvnb|Cartan|1913}}&lt;/ref&gt; [[w:Richard Brauer|''Richard Brauer'']] was 1935–38 largely responsible for the development of the [[w:Weyl-Brauer matrices|''Weyl-Brauer matrices'']] describing how spin representations of the Lorentz Lie algebra can be embedded in [[w:Clifford algebra|''Clifford algebras'']].&lt;ref&gt;{{harvnb|Brauer|Weyl|1935}} ''Spinors in n dimensions''.&lt;/ref&gt; The Lorentz group has also historically received special attention in representation theory, see [[w:Representation theory of the Lorentz group#infinite-dimensional unitary representations#history|''History of infinite-dimensional unitary representations'']] below, due to its exceptional importance in physics. Mathematicians [[w:Hermann Weyl|''Hermann Weyl'']]&lt;ref&gt;{{harvnb|Weyl|1931}} ''The Theory of Groups and Quantum Mechanics''.&lt;/ref&gt;&lt;ref&gt;{{harvnb|Weyl|1939}} ''The Classical Groups. Their Invariants and Representations''.&lt;/ref&gt; and [[w:Harish-Chandra|''Harish-Chandra'']]&lt;ref&gt;{{harvnb|Harish-Chandra|1947}} ''Infinite irreducible representations of the Lorentz group''.&lt;/ref&gt; and physicists [[w:Eugene Wigner|''Eugene Wigner'']]&lt;ref&gt;{{harvnb|Wigner|1939}} ''On unitary representations of the inhomogeneous Lorentz group''.&lt;/ref&gt; and [[w:Valentine Bargmann|''Valentine Bargmann'']]&lt;ref&gt;{{harvnb|Bargmann|1947}} ''Irreducible unitary representations of the Lorenz group''.&lt;/ref&gt;&lt;ref&gt;Bargmann was also a [[w:mathematician|''mathematician'']]. He worked as [[w:Albert Einstein|''Albert Einsteins'']] assistant at the [[w:Institute for Advanced Study|''Institute for Advanced Study'']] in Princeton.&lt;/ref&gt; made substantial contributions both to general representation theory and in particular to the Lorentz group.&lt;ref&gt;{{harvnb|Bargmann|Wigner|1948}} ''Group theoretical discussion of relativistic wave equations''.&lt;/ref&gt; Physicist [[w:Paul Dirac|''Paul Dirac'']] was perhaps the first to manifestly knit everything together in a practical application of major lasting importance with the [[w:Dirac equation|''Dirac equation'']] in 1928.&lt;ref&gt;{{harvnb|Dirac|1928}}&lt;/ref&gt;

=== Strategy ===
Classification of the finite-dimensional irreducible representations generally consists of two steps. The first step is to ''hypothesize'' the existence of representations. One assumes heuristically that all representations that ''a priori'' could exist, do exist. One investigates the properties of these hypothetical representations, primarily using the Lie algebra.&lt;ref&gt;{{harvnb|Hall|2003|loc=Introduction to chapter 7.}}&lt;/ref&gt; The goal of this study is twofold. First, some of these hypothetical representations may not exist. The goal in this situation is to show that their existence would imply a falsehood such as {{math|0 {{=}} 1}}. If this can be done, then the initial hypothesis that the representation existed must be false, and one can therefore exclude these hypothetical representations from later studies. Second, one can better understand the representations that do exist. These representations must have enough structure to manifest the symmetries of the group action, but describing this structure may not be easy. Before a classification has been completed, it is unclear which representations fall into the first class and which fall into the second.

If this first step of the classification is successful, it results in a tentative classification of the possible representations. This is often a short list. Each list entry is a single representation or a family of related representations, and ideally, the entry gives requirements so specific that they can be met by at most a single representation. The second step consists of explicit construction of the representations on this list. If successful, it justifies the existence hypotheses made in the first step. The results of investigations performed in the first step provide hints about how to construct the representations, i.e. construction of a vector space {{mvar|V}} and a specified Lie algebra action on {{mvar|V}}, since most of the properties they must have are then known.

For finite-dimensional irreducible representations of finite-dimensional semisimple Lie algebras the general result is Cartan's [[w:Cartan's theorem|''theorem of highest weight'']].&lt;ref&gt;{{harvtxt|Hall|2015|loc=Theorems 9.4 and 9.5.}}&lt;/ref&gt; It provides a classification of the irreducible representations in terms of the [[w:Weight (representation theory)|''weights'']] of the Lie algebra.

For some semisimple Lie algebras, especially non-compact ones, it is easier to proceed indirectly via [[w:Unitarian trick|''Weyl's unitarian trick'']] instead of applying Cartan's theorem directly. In the present case of {{math|'''so'''(3; 1)}} one sets up a chain of isomorphisms between Lie algebras and other correspondences preserving irreducible representations, so that the representations may be obtained from representations of {{math|SU(2) ⊗ SU(2)}}. See equation {{EquationNote|(A1)}}and references around it. It is essential here that {{math|SU(2)}} is compact, since then the irreducible representations of {{math|SU(2) ⊗ SU(2)}} are simply tensor products of irreducible representations of {{math|SU(2)}}, that can all be obtained from the irreducible representations of {{math|'''su'''(2)}}.&lt;ref group=nb&gt;The latter are all unitary, or can be made unitary, see footnote in [[#Non-unitarity|non-unitarity]]. This is probably the origin of the name of the trick.&lt;/ref&gt;

Then the classification part. Cartan's theorem is applied to {{math|'''su'''(2)}} (together with knowledge of its highest weights) and one obtains a classification of the representations of {{math|'''so'''(3; 1)}} via {{EquationNote|(A1)}}. An explicit construction of the representations of {{math|SL(2, ℂ)}} is then given (which is not much more difficult to obtain than the more basic {{math|'''su'''(2)}} representations), thus completing the task with the {{math|(''m'', ''n'')}} representations of {{math|'''so'''(3; 1)}} as the final result.

Representative matrices may be obtained by choice of basis in the representation space. An [[#Explicit formula for matrix elements|explicit formula for matrix elements]] is presented and some [[#Common representations|common representations]] are listed.

The [[w:Lie correspondence|''Lie correspondence'']] is subsequently employed for obtaining group representations of the connected component of the Lorentz group, {{math|SO(3, 1)&lt;sup&gt;+&lt;/sup&gt;}}. This is effected by taking the [[w:matrix exponential|''matrix exponential'']] of the matrices of the Lie algebra representation, a topic which is investigated in some depth. A subtlety arises due to the (in physics parlance) [[#Fundamental group|doubly connected]] nature of {{math|SO(3, 1)&lt;sup&gt;+&lt;/sup&gt;}}. This results in the [[w:projective representation|''projective representations'']] or two-value representations that are actually [[w:spin representation|''spin representations'']] of the covering group {{math|SL(2, ℂ)}}.

The Lie correspondence gives results only for the connected component of the groups, and thus the components of the full Lorentz that contain the operations of [[w:T-symmetry|''time reversal'']] and [[w:Parity (physics)|''space inversion'']] are treated separately, mostly from physical considerations, by defining representatives for the space inversion and time reversal matrices.

===The Lie algebra===
[[File:Wilhelm Karl Joseph Killing.jpeg|upright|thumb|left|[[w:Wilhelm Killing|''Wilhelm Killing'']], Independent discoverer of [[w:Lie algebra|''Lie algebras'']]. The simple Lie algebras were first classified by him in 1888.]]
According to the general [[w:representation theory|''representation theory'']] of Lie groups, one first looks for the representations of the [[w:complexification|''complexification'']], {{math|'''so'''(3; 1)&lt;sub&gt;'''C'''&lt;/sub&gt;}} of the Lie algebra {{math|'''so'''(3; 1)}} of the Lorentz group.  A convenient basis for {{math|'''so'''(3; 1)}} is given by the three [[#Explicit formulas|generators]] {{math|''J''&lt;sub&gt;''i''&lt;/sub&gt;}} of [[w:rotation|''rotations'']] and the three generators {{math|''K''&lt;sub&gt;''i''&lt;/sub&gt;}} of [[w:Lorentz boost|''boosts'']]. They are explicitly given in [[#Conventions and Lie algebra bases|conventions and Lie algebra bases]].

Now [[w:Complexification (Lie group)|''complexify'']] the Lie algebra, and then change basis to the components of&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Equations 5.6.7&amp;ndash;8.}}&lt;/ref&gt;

:&lt;math&gt;\mathbf{A} = \frac{\mathbf{J} + i \mathbf{K}}{2}\,,\quad \mathbf{B} = \frac{\mathbf{J} - i \mathbf{K}}{2}\,.&lt;/math&gt;

One may verify that the components of {{math|'''A''' {{=}} (''A''&lt;sub&gt;1&lt;/sub&gt;, ''A''&lt;sub&gt;2&lt;/sub&gt;, ''A''&lt;sub&gt;3&lt;/sub&gt;)}} and {{math|'''B''' {{=}} (''B''&lt;sub&gt;1&lt;/sub&gt;, ''B''&lt;sub&gt;2&lt;/sub&gt;, ''B''&lt;sub&gt;3&lt;/sub&gt;)}} separately satisfy the [[w:commutation relation|''commutation relations'']] of the Lie algebra [[w:su(2)|'''''su'''(2)'']] and moreover that they commute with each other,&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Equations 5.6.9&amp;ndash;11.}}&lt;/ref&gt;

:&lt;math&gt;\left[A_i ,A_j\right] = i\varepsilon_{ijk}A_k\,,\quad \left[B_i ,B_j\right] = i\varepsilon_{ijk}B_k\,,\quad \left[A_i ,B_j\right] = 0\,,&lt;/math&gt;

where {{math|''i'', ''j'', ''k''}} are indices which each take values {{math|1, 2, 3}}, and {{math|''ε&lt;sub&gt;ijk&lt;/sub&gt;''}} is the three-dimensional [[w:Levi-Civita symbol|''Levi-Civita symbol'']]. Let {{math|'''A'''&lt;sub&gt;'''C'''&lt;/sub&gt;}} and {{math|'''B'''&lt;sub&gt;'''C'''&lt;/sub&gt;}} denote the complex [[w:linear span|''linear span'']] of {{math|'''A'''}} and {{math|'''B'''}} respectively.

One has the isomorphisms&lt;ref name=&quot;Hall 2003&quot;&gt;{{harvnb|Hall|2003|loc=Chapter 6.}}&lt;/ref&gt;&lt;ref group=nb&gt;{{harvnb|Knapp|2001}} The rather mysterious looking third isomorphism is proved in chapter 2, paragraph 4.&lt;/ref&gt;
{{NumBlk|::|&lt;math&gt;\mathfrak{so}(3; 1) \hookrightarrow \mathfrak{so}(3; 1)_C \approx \mathbf{A}_C \oplus \mathbf{B}_C \approx \mathfrak{su}(2)_C \oplus \mathfrak{su}(2)_C \approx \mathfrak{sl}(2, C) \oplus \mathfrak{sl}(2, C) \approx \mathfrak{sl}(2, C) \oplus i\mathfrak{sl}(2, C) = \mathfrak{sl}(2, C)_C \hookleftarrow \mathfrak{sl}(2, C) \,,&lt;/math&gt;|{{EquationRef|A1}}}}

where '''sl'''(2, '''C''') is the complexification of '''su'''(2) ≈ {{math|'''A'''}} ≈ {{math|'''B'''}}.

The utility of these isomorphisms comes from the fact that all irreducible [[w:Representation theory of SU(2)|''representations of '''su'''(2)'']] are known. Every irreducible representation of '''su'''(2) is isomorphic to one of the [[w:highest weight representation|''highest weight representations'']]. Moreover, there is a one-to-one correspondence between linear representations of '''su'''(2) and complex linear representations of '''sl'''(2, '''C''').&lt;ref&gt;{{harvnb|Hall|2003|loc=Chapter 4.}}&lt;/ref&gt;

====The unitarian trick====
[[File:Hermann Weyl ETH-Bib Portr 00890.jpg|upright|thumb|right|[[w:Hermann Weyl|''Hermann Weyl'']], inventor the [[w:unitarian trick|''unitarian trick'']]. There are several concepts and formulas in representation theory named after Weyl, e.g. the [[w:Weyl group|''Weyl group'']] and the [[w:Weyl character formula|''Weyl character formula'']].&lt;br&gt;Photo courtesy of [http://www.e-pics.ethz.ch/link.jsp?catalog=ETHBIB.Bildarchiv&amp;id=Portr_00890&amp;view=info&amp;c=ba ETH-Bibliothek Zürich, Bildarchiv] ]]

In {{EquationNote|A1|(A1)}}, all isomorphisms are {{math|'''C'''}}-linear (the last is just a defining equality). The most important part of the manipulations below is that the {{math|'''R'''}}-linear (irreducible) representations of a (real or complex) Lie algebra are in one-to-one correspondence with {{math|'''C'''}}-linear (irreducible) representation of its complexification.&lt;ref&gt;{{harvnb|Rossmann|2002|loc=Section 6.5.}}&lt;/ref&gt; With this in mind, it is seen that the {{math|'''R'''}}-linear representations of the '''real forms''' of the far left, {{math|'''so'''(3; 1)}}, and the far right, {{math|'''sl'''(2, '''C''')}}, in {{EquationNote|A1|(A1)}}can be obtained from the {{math|'''C'''}}-linear representations of {{math|'''sl'''(2, '''C''') ⊕ '''sl'''(2, '''C''')}}.

The manipulations to obtain representations of a non-compact algebra (here '''so'''(3; 1)), and subsequently the non-compact group itself, from qualitative knowledge about unitary representations of a compact group (here {{math|SU(2)}}) is a variant of Weyl's so-called [[w:unitarian trick|''unitarian trick'']]. The trick specialized to {{math|SL(2, '''C''')}} can be summarized concisely.&lt;ref name=&quot;Knapp 2001&quot;&gt;{{harvnb|Knapp|2001|loc=Section 2.3.}}&lt;/ref&gt;

Let {{math|''V''}} be a finite-dimensional complex vector space. The following statements are equivalent, in the sense that if ''one'' of them holds, then there is a uniquely determined (modulo choice of basis for {{math|''V''}}) ''corresponding'' representation (either via given Lie algebra isomorphisms, or via complexification of Lie algebras per above, or via restriction to real forms, or via the exponential mapping (to be introduced), or, finally, via a standard mechanism (also to be introduced) for obtaining Lie algebra representations given group representations) of the appropriate type for the other groups and Lie algebras:
* There is a representation of {{math|SL(2, '''R''')}} on {{math|''V''}}.
* There is a representation of {{math|SU(2)}} on {{math|''V''}}.
* There is a holomorphic representation of {{math|SL(2, '''C''')}} on {{math|''V''}}.
* There is a representation of {{math|'''sl'''(2, '''R''')}} on {{math|''V''}}.
* There is a representation of {{math|'''su'''(2)}} on {{math|''V''}}.
* There is a complex linear representation of {{math|'''sl'''(2, '''C''')}} on {{math|''V''}}.
If one representation is irreducible, then all of them are. In this list, direct products (groups) or direct sums (Lie algebras) may be introduced (if done consistently). The ''essence'' of the trick is that the starting point in the above list is immaterial. Both qualitative knowledge (like existence theorems for one item on the list) and concrete realizations for one item on the list will translate and propagate, respectively, to the others.

Now, the representations of {{math|'''sl'''(2, '''C''') ⊕ '''sl'''(2, '''C''')}}, which is the Lie algebra of {{math|SL(2, '''C''') × SL(2, '''C''')}}, are supposed to be irreducible. This means that they must be tensor products of complex linear representations of {{math|'''sl'''(2, '''C''')}}, as can be seen by restriction to the subgroup {{math|SU(2) × SU(2) ⊂ SL(2, '''C''') × SL(2, '''C''')}}, a compact group to which the [[w:Peter–Weyl theorem|''Peter–Weyl theorem'']] applies.&lt;ref name=&quot;ReferenceC&quot;&gt;{{harvnb|Knapp|2001}}&lt;/ref&gt; The irreducible unitary representations of {{math|SU(2) × SU(2)}} are precisely the tensor products of irreducible unitary representations of {{math|SU(2)}}. These stand in one-to-one correspondence with the [[w:Holomorphic function|''holomorphic'']] representations of {{math|SL(2, '''C''') × SL(2, '''C''')}}&lt;ref name=&quot;ReferenceC&quot;/&gt; and these, in turn, are in one-to-one correspondence with the complex linear representations of {{math|'''sl'''(2, '''C''') ⊕ '''sl'''(2, '''C''')}} because {{math|SL(2, '''C''') × SL(2, '''C''')}} is simply connected.&lt;ref name=&quot;ReferenceC&quot;/&gt;

For {{math|'''sl'''(2, '''C''')}}, there exists the highest weight representations (obtainable, via the trick, from the corresponding {{math|'''su'''(2)}}-representations), here indexed by {{math|μ}} for {{math|1=μ = 0, 1, … }}. The tensor products of two complex linear factors then form the irreducible complex linear representations of {{math|'''sl'''(2, '''C''') ⊕ '''sl'''(2, '''C''')}}. For reference, if {{math|(''π''&lt;sub&gt;1&lt;/sub&gt;, ''U'')}} and {{math|(''π''&lt;sub&gt;2&lt;/sub&gt;, ''V'')}} are representations of a Lie algebra {{math|'''g'''}}, then their tensor product {{math|(''π''&lt;sub&gt;1&lt;/sub&gt; ⊗ ''π''&lt;sub&gt;2&lt;/sub&gt;, ''U'' ⊗ ''V'')}} is given by either of&lt;ref&gt;{{harvnb|Hall|2015|loc=Definition 4.20 and following remarks}}.&lt;/ref&gt;&lt;ref group=nb&gt;Tensor products of representations, {{math|''π''&lt;sub&gt;''g''&lt;/sub&gt; ⊗ ''π''&lt;sub&gt;''h''&lt;/sub&gt;}} of {{math|'''g''' ⊕ '''h'''}} can, when both factors come from the same Lie algebra {{math|1=('''h''' = '''g''')}}, either be thought of as a representation of {{math|'''g'''}} or {{math|'''g''' ⊕ '''g'''}}.&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\pi_1\otimes\pi_2(X) = \pi_1(X) \otimes \mathrm{Id}_V + \mathrm{Id}_U \otimes \pi_2(X), \quad X \in \mathbf{g}, \qquad \pi_1\otimes\pi_2(X, Y) = \pi_1(X) \otimes \mathrm{Id}_V + \mathrm{Id}_U \otimes \pi_2(Y), \quad (X,Y) \in \mathbf{g} \oplus \mathbf{g},&lt;/math&gt;|{{EquationRef|A0}}}}

where {{math|Id}} is the identity operator. Here, the latter interpretation is intended. The not necessarily complex linear representations of {{math|'''sl'''(2, '''C''')}} come using another variant of the unitarian trick as is shown in the last Lie algebra isomorphism in {{EquationNote|(A1)}}.

==== The representations ====
The representations for all Lie algebras and groups involved in the unitarian trick can now be obtained. The real linear representations for {{math|'''sl'''(2, '''C''')}} and {{math|'''so'''(3; 1)}} follow here assuming the complex linear representations of {{math|'''sl'''(2, '''C''')}} are known. Explicit realizations and group representations are given later.

===== sl(2, C) =====
The complex linear representations of the complexification of {{math|'''sl'''(2, '''C''')}}, {{math|'''sl'''(2, '''C''')&lt;sub&gt;'''C'''&lt;/sub&gt;}}, obtained via isomorphisms in {{EquationNote|(A1)}}, stand in one-to-one correspondence with the real linear representations of {{math|'''sl'''(2, '''C''')}}.&lt;ref name=&quot;ReferenceC&quot;/&gt; The set of all, ''at least real linear'', irreducible representations of {{math|'''sl'''(2, '''C''')}} are thus indexed by a pair {{math|(μ, ν)}}. The complex linear ones, corresponding precisely to the complexification of the real linear {{math|'''su'''(2)}} representations, are of the form {{math|(μ, 0)}}, while the conjugate linear ones are the {{math|(0, ν)}}.&lt;ref name=&quot;ReferenceC&quot;/&gt; All others are real linear only. The linearity properties follow from the canonical injection, the far right in {{EquationNote|(A1)}}, of {{math|'''sl'''(2, '''C''')}} into its complexification. Representations on the form {{math|(ν, ν)}} or {{math|(μ, ν) ⊕ (ν, μ)}} are given by ''real'' matrices (the latter is not irreducible). Explicitly, the real linear {{math|(μ, ν)}}-representations of {{math|'''sl'''(2, '''C''')}} are

&lt;math&gt;\phi_{\mu, \nu}(X) = \phi_\mu \otimes \overline{\phi}_\nu(X) = \phi_\mu(X) \otimes \mathrm{Id}_{\nu + 1} + \mathrm{Id}_{\mu + 1} \otimes \overline{\phi_\nu(X)}, \quad X \in \mathfrak{sl}(2, \mathbb{C})&lt;/math&gt;

where {{math|Φ&lt;sub&gt;μ&lt;/sub&gt;, μ {{=}} 0,1, &amp;hellip;}} are the complex linear irreducible representations of {{math|'''sl'''(2, '''C''')}} and {{math|{{overline|Φ}}&lt;sub&gt;ν&lt;/sub&gt;, ν {{=}} 0,1, &amp;hellip;}} their complex conjugate representations. Here the tensor product is interpreted in the former sense of {{EquationNote|(A0)}}. These representations are [[#Concrete realization|concretely realized]] below.

===== so(3; 1) =====
Via the displayed isomorphisms in {{EquationNote|(A1)}}and knowledge of the complex linear irreducible representations of {{math|sl(2, '''C''') ⊕ sl(2, '''C''')}}, upon solving for {{math|'''J'''}} and {{math|'''K'''}}, all irreducible representations of {{math|'''so'''(3; 1)}}&lt;sub&gt;'''C'''&lt;/sub&gt;, and, by restriction, those of {{math|'''so'''(3; 1)}} are known. It's worth noting that the representations of {{math|'''so'''(3; 1)}} obtained this way are real linear (and not complex or conjugate linear) because the algebra is not closed upon conjugation, but they are still irreducible.&lt;ref name=&quot;Hall 2003&quot;/&gt; Since {{math|'''so'''(3; 1)}} is [[w:semisimple Lie algebra|''semisimple'']],&lt;ref name=&quot;Hall 2003&quot;/&gt; all its representations can be built up as [[w:direct sum|''direct sums'']] of the irreducible ones.

Thus the finite dimensional irreducible representations of the Lorentz algebra are classified by an ordered pair of half-integers {{math|1=''m'' = μ/2}} and {{math|1=''n'' = ν/2}}, conventionally written as one of
:&lt;math&gt;(m,n) \equiv D^{(m,n)} \equiv \pi_{m,n}.&lt;/math&gt;

The notation {{math|''D''&lt;sup&gt;(''m'',''n'')&lt;/sup&gt;}} is usually reserved for the group representations. Let {{math|''π''&lt;sub&gt;(''m'', ''n'')&lt;/sub&gt; : '''so'''(3; 1) → '''gl'''(''V'')}}, where {{mvar|V}} is a vector space, denote the irreducible representations of {{math|'''so'''(3; 1)}} according to this classification. These are, up to a [[w:Matrix similarity|''similarity transformation'']], uniquely given by&lt;ref group=nb&gt;Combine {{harvtxt|Weinberg|2002|loc=Equations 5.6.7-8, 5.6.14-15}} with {{harvtxt|Hall|2015|loc=Proposition 4.18}} about Lie algebra representations of group tensor product representations.&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{align}
\pi_{m,n}(J_i) &amp; = 1_{(2m+1)}\otimes J^{(n)}_i + J^{(m)}_i\otimes 1_{(2n+1)}\\
\pi_{m,n}(K_i) &amp; = i(1_{(2m+1)}\otimes J^{(n)}_i - J^{(m)}_i \otimes 1_{(2n+1)}),
\end{align}&lt;/math&gt;|{{EquationRef|A2}}}}

where the {{math|'''J'''&lt;sup&gt;(''n'')&lt;/sup&gt; {{=}} (''J''&lt;sup&gt;(''n'')&lt;/sup&gt;&lt;sub&gt;1&lt;/sub&gt;, ''J''&lt;sup&gt;(''n'')&lt;/sup&gt;&lt;sub&gt;2&lt;/sub&gt;, ''J''&lt;sup&gt;(''n'')&lt;/sup&gt;&lt;sub&gt;3&lt;/sub&gt;)}} are the {{math|(2''n'' + 1)}}-dimensional irreducible spin {{mvar|n}} [[Representation theory of SU(2)|representations of {{math|'''so'''(3)}}]] ≈ {{math|'''su'''(2)}} and {{math|1&lt;sub&gt;''n''&lt;/sub&gt;}} is the {{mvar|n}}-dimensional [[w:Identity matrix|''unit matrix'']].

====== Explicit formula for matrix elements ======
Let {{math|''π''&lt;sub&gt;(''m'', ''n'')&lt;/sub&gt; : '''so'''(3; 1) → '''gl'''(''V'')}}, where {{mvar|V}} is a vector space, denote the irreducible representations of {{math|'''so'''(3; 1)}} according to the {{math|(''m'',  ''n'')}} classification. In components, with {{math|−''m'' ≤ ''a'', ''a′'' ≤ ''m''}}, {{math|−''n'' ≤ ''b'', ''b′'' ≤ ''n''}}, the representations are given by&lt;ref name=&quot;Weinberg 2002 Chapter 2&quot;&gt;{{harvnb|Weinberg|2002|loc=Section 5.6.}} The equations follow from equations 5.6.7-8 and 5.6.14-15.&lt;/ref&gt;

:&lt;math&gt;\begin{align}
(\pi_{m,n}(J_i))_{a'b' , ab} &amp;= \delta_{b'b}(J_i^{(m)})_{a'a} + \delta_{a'a}(J_i^{(n)})_{b'b},\\
(\pi_{m,n}(K_i))_{a'b' , ab} &amp;= i(\delta_{a'a}(J_i^{(n)})_{b'b} - \delta_{b'b}(J_i^{(m)})_{a'a}),
\end{align}&lt;/math&gt;

where {{math|δ}} is the [[w:Kronecker delta|''Kronecker delta'']] and the {{math|''J''&lt;sub&gt;''i''&lt;/sub&gt;&lt;sup&gt;(''n'')&lt;/sup&gt;}} are the {{math|(2''n'' + 1)}}-dimensional irreducible representations of {{math|'''so'''(3)}}, also termed [[w:Rotation group SO(3)#A note on Lie algebra|''spin matrices'']] or ''angular momentum matrices''. These are explicitly given as&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Equations 5.6.16-17.}}&lt;/ref&gt;

:&lt;math&gt;\begin{align}
\left(J_1^{(j)}\right)_{a'a} &amp;= \frac{1}{2}\left(\sqrt{(j - a)(j + a + 1)}\delta_{a',a + 1} + \sqrt{(j + a)(j - a + 1)}\delta_{a',a - 1}\right),\\
\left(J_2^{(j)}\right)_{a'a} &amp;= \frac{1}{2i}\left(\sqrt{(j - a)(j + a + 1)}\delta_{a',a + 1} - \sqrt{(j + a)(j - a + 1)}\delta_{a',a - 1}\right),\\
\left(J_3^{(j)}\right)_{a'a} &amp;= a\delta_{a'a}.
\end{align}&lt;/math&gt;

====== Common representations ======
{| class=&quot;wikitable&quot; align=left valign=center style=&quot;margin-right:3em&quot;
 |
 ! {{mvar|m}}=0
 ! &amp;frac12;
 ! 1
 |-
 !{{mvar|n}}=0
 | style=&quot;padding:8px; color:#CC00FF&quot; |'''scalar'''
 |&lt;div style=&quot;color:#CC00FF&quot;&gt;Weyl spinor&lt;/div&gt;bispinor
 | style=&quot;padding:8px&quot; |&lt;div style=&quot;color:#CC00FF&quot;&gt;self-dual 2-form&lt;/div&gt;2-form field
 |-
 !&amp;frac12;
 | style=&quot;color:#CC00FF&quot; |Weyl spinor&lt;br&gt;(right-handed)
 | style=&quot;color:#CC00FF&quot; align=center |'''4-vector'''
 |Rarita–Schwinger&lt;br&gt;field
 |-
 !1
 | style=&quot;padding:8px; color:#CC00FF&quot; |anti-self-dual&lt;br&gt;2-form
 |
 | style=&quot;padding:8px; color:#CC00FF&quot; |'''traceless'''&lt;br&gt;
'''symmetric tensor'''
 |-
 | colspan=4 style=&quot;padding-left:20px; font-size:80%&quot; |
&lt;span style=&quot;color:#CC00FF;&quot;&gt;Purple&lt;/span&gt;: {{math|(''m'', ''n'')}} complex irreducible representations &amp;nbsp;
Black: {{math|(''m'', ''n'') ⊕ (''n'', ''m'')}}&lt;br&gt;
&lt;span style=&quot;color:#CC00FF;&quot;&gt;'''Bold'''&lt;/span&gt;: {{math|(''m'', ''m'')}}
 |}
Since for any irreducible representation for which {{math|''m'' ≠ ''n''}} it is essential to operate over the field of [[w:complex numbers|''complex numbers'']], the [[w:direct sum of representations|''direct sum of representations'']] {{math|(''m'', ''n'')}} and {{math|(''n'', ''m'')}} has a particular relevance to physics, since it permits to use [[w:linear operator|''linear operators'']] over [[w:real numbers|''real numbers'']]. 
* (0, 0) is the Lorentz [[w:scalar (mathematics)|''scalar'']]&lt;!-- do not forget here is mathematics! --&gt; representation.  This representation is carried by relativistic [[w:scalar field|''scalar field'']] theories.
* (&amp;frac12;, 0) is the left-handed [[w:Weyl spinor|''Weyl spinor'']] and (0, &amp;frac12;) is the right-handed Weyl spinor representation. Fermionic [[w:supersymmetry|''supersymmetry'']] generators transform under one of these representations.&lt;ref name=&quot;Weinberg 2000 loc=Section 25.2&quot;/&gt;&lt;!-- where is a Majorana spinor? --&gt;
* (&amp;frac12;, 0) ⊕ (0, &amp;frac12;) is the [[w:bispinor|''bispinor'']] representation. (See also [[w:Dirac spinor|''Dirac spinor'']] and [[#Weyl spinors and bispinors|Weyl spinors and bispinors]] below.)
* (&amp;frac12;, &amp;frac12;) is the [[w:four-vector|''four-vector'']] representation.  The [[w:four-momentum|''four-momentum'']] of a [[w:particle|''particle'']] (either massless or [[w:invariant mass|''massive'']]) transforms under this representation.
* (1, 0) is the [[w:Hodge dual#Four dimensions|''self-dual 2-form field'']] representation and (0, 1) is the anti-self-dual 2-form field representation.
* (1, 0) ⊕ (0, 1) is the [[w:adjoint representation|''adjoint representation'']] and the representation of a [[w:P-symmetry|''parity'']]-invariant [[w:2-form|''2-form'']] field (a.k.a. [[w:curvature form|''curvature form'']]). The [[w:electromagnetic field tensor|''electromagnetic field tensor'']] transforms under this representation.
* (1, &amp;frac12;) ⊕ (&amp;frac12;, 1) is the [[w:Rarita–Schwinger equation|''Rarita–Schwinger'']] field representation.
* (1, 1) is the spin&amp;nbsp;2 representation of a traceless [[w:symmetric bilinear form|''symmetric'']] [[w:tensor field|''tensor field'']].&lt;ref group=nb&gt;The &quot;traceless&quot; property can be expressed as {{math|1=''S''&lt;sub&gt;''αβ''&lt;/sub&gt;[[w:metric tensor (general relativity)|''''g'''']]&lt;sup&gt;''αβ''&lt;/sup&gt; = 0}}, or {{math|1=''S''&lt;sub&gt;''α''&lt;/sub&gt;&lt;sup&gt;''α''&lt;/sup&gt; = 0}}, or {{math|1=''S''&lt;sup&gt;''αβ''&lt;/sup&gt;''g''&lt;sub&gt;''αβ''&lt;/sub&gt; = 0}} depending on the presentation of the field: covariant, mixed, and contravariant respectively.&lt;/ref&gt; A physical example is the traceless part of the [[w:energy-momentum tensor|''energy-momentum tensor'']] {{math|''T''&lt;sup&gt;''μν''&lt;/sup&gt;}}.&lt;ref name=&quot;Tung 1985&quot;&gt;{{harvnb|Tung|1985}}&lt;/ref&gt;&lt;ref group=nb&gt;This doesn't necessarily come symmetric directly from the Lagrangian by using [[w:Noether's theorem|''Noether's theorem'']], but it can be symmetrized as the [[w:Belinfante–Rosenfeld stress–energy tensor|''Belinfante–Rosenfeld stress–energy tensor'']].&lt;/ref&gt;
* ({{sfrac|3|2}}, 0) ⊕ (0, {{sfrac|3|2}}) would be the symmetry of the hypothesized [[w:gravitino|''gravitino'']].&lt;ref group=nb&gt;This is provided parity is a symmetry. Else there would be two flavors, ({{sfrac|3|2}}, 0) and (0, {{sfrac|3|2}}) in analogy with [[w:neutrino|''neutrinos'']].&lt;/ref&gt; It can be obtained from the (1, &amp;frac12;) ⊕ (&amp;frac12;, 1)-representation.&lt;ref&gt;{{Harvnb|Weinberg|2002}} See footnote on p. 232.&lt;/ref&gt;

=== The group ===
The approach in this section is based on theorems that, in turn, are based on the fundamental [[w:Lie correspondence|''Lie correspondence'']].&lt;ref&gt;{{harvnb|Lie|1888}}&lt;/ref&gt; The Lie correspondence is in essence a dictionary between connected Lie groups and Lie algebras.&lt;ref&gt;{{harvnb|Rossmann|2002|loc=Section 2.5.}}&lt;/ref&gt; The link between them is the [[w:exponential map (Lie theory)|''exponential mapping'']] from the Lie algebra to the Lie group, denoted {{math|exp:'''g''' → '''G'''}}. It is one-to-one in a neighborhood of the identity.

==== The Lie correspondence ====
[[File:Sophus Lie.jpg|upright|thumb|right|[[Sophus Lie]], the originator of [[Lie group theory|Lie theory]]. The theory of [[manifolds]] was not discovered in Lie's time, so he worked ''locally'' with subsets of {{math|'''R'''&lt;sup&gt;''n''&lt;/sup&gt;}}. The structure would today be called a '''local group'''.]]
The Lie correspondence and some results based on it needed here and below are stated for reference. If {{mvar|G}} denotes a Lie group and {{math|'''g'''}} a Lie algebra, let {{math|Γ('''g''')}} denote the group [[w:Generating set of a group|''generated'']] by {{math|exp('''g''')}}, the image of the Lie algebra under the exponential mapping,&lt;ref group=nb&gt;The exponential mapping need not be onto and the image is in those cases not a group, see e.g. [[#Non-surjectiveness of exponential mapping|non-surjectiveness of exponential mapping for SL(2, C)]] below. Therefore one takes all finite products of elements in the image in order to obtain a group, which necessarily must be closed under multiplication.&lt;/ref&gt; and let {{math|''L''(''G'')}} denote the Lie algebra of {{math|''G''}}. The Lie correspondence reads in modern language as follows:
*There is a one-to-one correspondence between connected and [[w:simply connected|''simply connected'']] Lie groups {{math|''G''}} and Lie algebras {{math|'''g'''}} under which {{math|'''g'''}} corresponds to {{math|''L''(''G'')}} and {{math|''G''}} to {{math|Γ('''g''')}}. Equivalently, {{math|Γ(''L''(''G'')) {{=}} ''G''}} and {{math|''L''(Γ('''g''')) {{=}} '''g'''}}.&lt;ref&gt;{{harvnb|Rossmann|2002}} Theorem 1, Paragraph 2.5.&lt;/ref&gt; {{EquationRef|(Lie)}}
A ''linear'' Lie group is one that has at least one faithful finite-dimensional representation.&lt;ref group=nb&gt;This is not always the case. For example, the [[w:universal covering group|''universal covering group'']] of the linear Lie group [[w:SL(2, R)|''SL(2, R)'']] is not linear. See {{harvtxt|Hall|2015|loc=Proposition 5.16.}} A [[w:Quotient group|''quotient'']] of a matrix Lie group need not be linear. This is e.g. the case for the quotient of the [[w:Heisenberg group|''Heisenberg group'']] by a discrete subgroup of its [[w:Center (group theory)|''center'']]. See {{harvtxt|Hall|2015|loc=Section 4.8.}} However, if {{mvar|G}} is a compact Lie group, it is representable as a matrix Lie group. This is a consequence of the [[w:Peter&amp;ndash;Weyl theorem|''Peter&amp;ndash;Weyl theorem'']]. See {{harvtxt|Rossmann|2002|loc=Section 6.2.}}&lt;/ref&gt;&lt;ref group=nb&gt;It's a rather deep fact that all finite-dimensional Lie algebras are linear, meaning that they are all Lie subalgebras of the Lie algebra of matrices. This is the content of [[w:Ado's theorem|''Ado's theorem'']]. See {{harvtxt|Hall|2015|loc=Section 5.10.}}&lt;/ref&gt;  The following are some corollaries that will be used in the sequel:
:*A connected linear Lie group {{math|''G''}} is abelian if and only if {{math|'''g'''}} is abelian.&lt;ref name=&quot;Rossmann 2002&quot;&gt;{{harvnb|Rossmann|2002}} Proposition 3, Paragraph 2.5.&lt;/ref&gt; {{EquationRef|(Lie i)}}
:*A connected subgroup {{math|''H''}} with Lie algebra {{math|'''h'''}} of a connected linear Lie group {{math|''G''}} is normal if and only if {{math|'''h''' ⊂ '''g'''}} is an ideal.&lt;ref name=&quot;Rossmann 2002&quot;/&gt; {{EquationRef|(Lie ii)}}
:*If {{math|''G'', ''H''}} are linear Lie groups with Lie algebras {{math|'''g''', '''h'''}} and {{math|Π:''G'' → ''H''}} is a group homomorphism, then {{math|π:'''g''' → '''h'''}}, its pushforward at the identity, is a Lie algebra homomorphism and {{math|Π(''e''&lt;sup&gt;''iX''&lt;/sup&gt;) {{=}} ''e''&lt;sup&gt;''i''π(''X'')&lt;/sup&gt;}} for every {{math|''X'' ∈ '''g'''}}.&lt;ref&gt;{{harvnb|Rossmann|2002}} Theorem 1, Paragraph 2.6.&lt;/ref&gt; {{EquationRef|(Lie iii)}}

==== Lie algebra representations from group representations ====
Using the above theorem it is always possible to pass from a representation of a Lie group {{math|''G''}} to a representation of its Lie algebra {{math|'''g'''}}. If {{math|Π : ''G'' → GL(''V'')}} is a group representation for some vector space {{math|''V''}}, then its [[w:Pushforward (differential)|''pushforward'']] (differential) at the identity, or '''Lie map''', {{math|''π'' : '''g''' → End ''V''}} is a Lie algebra representation. It is explicitly computed using&lt;ref group=nb&gt;{{harvnb|Hall|2003}}, Equation 2.16. Due to the physicist conventions, the formula here differs with a factor of {{math|i}} in the exponent.&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\pi(X) = \frac{d}{dt}\Pi(e^{itX})|_{t = 0}, \quad X \in \mathfrak{g}.&lt;/math&gt;|{{EquationRef|G6}}}}

This, of course, holds for the Lorentz group in particular, but not all Lie algebra representations arise this way because their corresponding group representations may not exist as proper representations, i.e. they are projective, see below.

==== Group representations from Lie algebra representations ====
[[File:Lorentz group commutative diagram 2.svg|300px|right|thumb|Here {{mvar|V}} is a finite-dimensional vector space, {{math|GL(''V'')}} is the set of all invertible linear transformations on {{mvar|V}} and {{math|'''gl'''(''V'')}} is its Lie algebra. The maps {{mvar|π}} and {{math|Π}} are Lie algebra and group representations respectively, and {{math|exp}} is the exponential mapping. The diagram commutes only up to a sign if {{math|Π}} is projective.]]
Given a {{math|'''so'''(3; 1)}} representation, one may try to construct a representation of {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}, the [[w:identity component|''identity component'']] of the Lorentz group, by using the exponential mapping. Since {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} is a [[w:matrix Lie group|''matrix Lie group'']], the exponential mapping is simply the [[w:matrix exponential|''matrix exponential'']]. If {{mvar|X}} is an element of '''so'''(3; 1) in the standard representation, then

{{NumBlk|:|&lt;math&gt;\Lambda = e^{iX} \equiv \sum_{n=0}^{\infty} \frac{(iX)^n}{n!}&lt;/math&gt;|{{EquationRef|G1}}}}

is a Lorentz transformation by general properties of Lie algebras. Motivated by this and the Lie correspondence theorem stated above, let {{math|''π'' : '''so'''(3; 1) → '''gl'''(''V'')}} for some vector space {{mvar|V}} be a representation and tentatively define a representation {{math|Π}} of {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} by first setting

{{NumBlk|:|&lt;math&gt;\Pi_U(e^{iX}) = e^{i\pi(X)}, \quad X \in \mathfrak{so}(3; 1).&lt;/math&gt;|{{EquationRef|G2}}}}

The subscript {{math|''U''}} indicates a small open set containing the identity. Its precise meaning is defined below. There are at least two potential problems with this definition. The first is that it is not obvious that this yields a [[w:group homomorphism|''group homomorphism'']], or even a well defined map at all (local existence). The second problem is that for a given {{math|''g'' ∈ ''U'' ⊂ SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} there may not be exactly one {{math|''X'' ∈ '''so'''(3; 1)}} such that {{math|1=''g'' = ''e''&lt;sup&gt;''iX''&lt;/sup&gt;}} (local uniqueness). The soundness of the tentative definition {{EquationNote|(G2)}}is shown in several steps below:
#{{math|Π&lt;sub&gt;''U''&lt;/sub&gt;}} is a local homomorphism.
#{{math|Π(''g'')}} defined along a path using properties of {{math|Π&lt;sub&gt;''U''&lt;/sub&gt;}} is a global homomorphism.
#The exponential mapping {{math|exp:'''so'''(3; 1) → SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} is surjective.
#{{math|Π(''g'')}} defined along a path coincides with {{math|Π&lt;sub&gt;''U''&lt;/sub&gt;(''g'')}} with {{math|''U'' {{=}} SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}.

===== Local existence and uniqueness =====
A theorem&lt;ref&gt;{{harvnb|Hall|2015|loc=Corollary 3.44.}}&lt;/ref&gt; based on the [[w:inverse function theorem|''inverse function theorem'']] states that the map {{math|exp : '''so'''(3; 1) → SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} is one-to-one for {{math|''X''}} small enough {{EquationRef|(A)}}. This makes the map well-defined. The qualitative form of the [[w:Baker–Campbell–Hausdorff formula|''Baker–Campbell–Hausdorff formula'']] then guarantees that it ''is'' a group homomorphism, still for {{math|X}} small enough {{EquationRef|(B)}}. Let {{math|''U'' ⊂ SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} denote image under the exponential mapping of the open set in {{math|'''so'''(3; 1)}} where conditions {{EquationNote|(A)}}and {{EquationNote|(B)}}both hold. Let {{math|''g'', ''h'' ∈ ''U'', ''g'' {{=}} ''e''&lt;sup&gt;''X''&lt;/sup&gt;, ''h'' {{=}} ''e''&lt;sup&gt;''Y''&lt;/sup&gt;}}, then&lt;ref&gt;{{harvnb|Hall|2015|loc=Equations 5.1-4.}}&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{align}\Pi_U(gh) &amp;= \Pi_U(e^Xe^Y) = e^{i\pi \log(e^Xe^Y)} &amp;\text{(by definition and by }\textbf{(A)}\text{)}\\
&amp;= e^{i\pi(X + Y + \frac{1}{2}[X, Y] + \frac{1}{12}[X, [X, Y]] -\frac{1}{12}[Y, [X, Y]] + \ldots)} &amp;\text{(by Baker–Campbell–Hausdorff and }\textbf{(B)}\text{)}\\
&amp;= e^{i\pi(X) + i\pi(Y) + \frac{1}{2}[i\pi(X), i\pi(Y)] + \frac{1}{12}[i\pi(X), [i\pi(X), i\pi(Y)]] -\frac{1}{12}[i\pi(Y), [i\pi(X), i\pi(Y)]] + \ldots)} &amp;\text{(since }\pi \text{ is a Lie algebra homomorphism)}\\
&amp;= e^{\log (e^{i\pi(X)}e^{i\pi(Y)})} &amp;\text{(by Baker–Campbell–Hausdorff and }\textbf{(B)}\text{ again)}\\
&amp;= e^{\log(\Pi_U(g)\Pi_U(h))} = \Pi_U(g)\Pi_U(h) &amp;\text{(by definition and by }\textbf{(A)}\text{)}\\
\end{align}&lt;/math&gt;|{{EquationRef|G7}}}}

This shows that the map {{math|Π&lt;sub&gt;''U''&lt;/sub&gt;}} is a well-defined group homomorphism on {{math|''U''}}.

===== Global existence and uniqueness =====
Technically, formula {{EquationNote|(G2)}}is used to ''define'' {{math|Π}} near the identity. For other elements {{math|''g'' ∉ ''U''}} one chooses a [[w:path (topology)|''path'']] from the identity to {{mvar|g}} and defines {{math|Π}} along that path by partitioning it finely enough so that formula {{EquationNote|(G2)}}can be used again on the resulting factors in the partition. In detail, one sets&lt;ref&gt;{{harvnb|Hall|2015|loc=Step 1 of proof of theorem 5.10.}}&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;g = g_n = (g_n g_{n-1}^{-1})(g_{n-1}g_{n-2}^{-1})\cdots(g_{2}g_{1}^{-1})(g_{1}g_{0}), \qquad \Pi(g) \equiv \Pi_U(g_{n}g_{n-1}^{-1})\Pi_U(g_{n-1}g_{n-2}^{-1})\cdots\Pi_U(g_{2}g_{1}^{-1})\Pi_U(g_{1}g_{0}), \quad g_0 = 1&lt;/math&gt;|{{EquationRef|G3}}}}

where the {{mvar|g&lt;sub&gt;i&lt;/sub&gt;}} are on the path and the factors on the far right are uniquely defined by {{EquationNote|(G2)}}provided that all {{math|''g''&lt;sub&gt;''i''&lt;/sub&gt; ''g''&lt;sub&gt;''i''+1&lt;/sub&gt;&lt;sup&gt;−1&lt;/sup&gt; ∈ ''U''}} and, for all conceivable pairs {{math|''h'',''k''}} of points on the path between {{math|''g''&lt;sub&gt;''i''&lt;/sub&gt;}} and {{math|''g''&lt;sub&gt;''i+1''&lt;/sub&gt;}}, {{math|''hk''&lt;sup&gt;−1&lt;/sup&gt; ∈ ''U''}} as well. For each {{math|i}} take, by the inverse function theorem, the unique {{mvar|X&lt;sub&gt;i&lt;/sub&gt;}} such that {{math| exp(''X''&lt;sub&gt;''i''&lt;/sub&gt;)}} = {{math|''g''&lt;sub&gt;''i''&lt;/sub&gt;''g''&lt;sub&gt;''i''−1&lt;/sub&gt;&lt;sup&gt;−1&lt;/sup&gt;}} and obtain
{{NumBlk|:|&lt;math&gt;\Pi(g) = \Pi_U(e^{iX_n})\Pi_U(e^{iX_{n - 1}})\cdots\Pi_U(e^{iX_2})\Pi_U(e^{iX_1}) = e^{i\pi(X_n)}e^{i\pi(X_{n - 1})}\cdots e^{i\pi(X_2)}e^{i\pi(X_1)}.&lt;/math&gt;|{{EquationRef|G4}}}}

By [[w:compactness|''compactness'']] of the path there is an {{mvar|n}} large enough so that {{math|Π(''g'')}} is well defined, possibly depending on the partition and/or the path, whether {{math|''g''}} is close to the identity or not.

====== Partition independence ======
It turns out that the result is always independent of the partitioning of the path.&lt;ref&gt;{{harvnb|Hall|2015|loc=Step 2 of proof of theorem 5.10.}}&lt;/ref&gt; To demonstrate the independence of a chosen path, one employs the Baker–Campbell–Hausdorff formula. It shows that {{math|Π&lt;sub&gt;''U''&lt;/sub&gt;}} is a group homomorphism for elements in {{math|''U''}}.

To see this, first fix a partitioning used in {{EquationNote|(G3)}}. Then insert a new point {{math|h}} somewhere on the path, say 
:&lt;math&gt;g = \cdots(g_{i + 1}h^{-1})(hg_i^{-1})\cdots, \qquad \cdots \Pi_U(g_{i + 1}h^{-1})\Pi_U(hg_i^{-1})\cdots.&lt;/math&gt;
But 
:&lt;math&gt; \cdots \Pi_U(g_{i + 1}h^{-1})\Pi_U(hg_i^{-1})\cdots = \cdots\Pi_U(g_{i + 1}h^{-1}hg_i^{-1})\cdots = \cdots\Pi_U(g_{i + 1}g_i^{-1})\cdots&lt;/math&gt;
as a consequence of the Baker–Campbell–Hausdorff formula and the conditions on the original partitioning. Thus, adding a point on the path has no effect on the definition of {{math|Π(''g'')}}.

Then, for any two given partitions of a given path, they have common '''refinement''', their union. This refinement can be reached from any of the two partitionings by, one-by-one, adding points from the other partition. No individual addition changes the definition of {{math|Π(''g'')}}, hence, since there are finitely many points in each partition, the value of {{math|Π(''g'')}} must have been the same for the two partitionings to begin with.

====== Path independence ======
For [[w:simply connected|''simply connected'']] groups, the construction will be independent of the path as well, yielding a well defined representation.&lt;ref&gt;{{harvnb|Hall|2015|loc=Step 3 of proof of theorem 5.10.}}&lt;/ref&gt; In that case formula {{EquationNote|(G2)}}can unambiguously be used directly. Simply connected spaces have the property that any two paths can be continuously deformed into each other. Any such deformation is called a [[w:homotopy|''homotopy'']] and is usually chosen as a continuous function {{math|''H''}} from the unit square {{math|{''s'',''t'' ∈ '''R''': 0 ≤ ''s'', ''t'' ≤ 1} }} into the group. For {{math|''s'' {{=}} 0}} the image is one of the paths, for {{math|''s'' {{=}} 1}} the other, for intermediate {{math|''s''}}, an intermediate path results, but endpoints are kept fixed.

One deforms the path, a little bit at a time, using the previous result, the independence of partitioning. Each consecutive deformation is so small that two consecutive deformed paths can be partitioned using ''the same partition points''. Thus two consecutive deformed paths yield the same value for {{math|Π(''g'')}}. But any two ''pairs'' of consecutive deformations need not have the same choice partition points, so the actual path laid out in the group as one progresses through the deformation does indeed change.

Using compactness arguments, in a finite number of steps, the original ({{math|''s'' {{=}} 0}}) path is deformed into the other ({{math|''s'' {{=}} 1}}) without affecting the value of {{math|Π(''g'')}}.&lt;ref&gt;{{harvnb|Hall|2015|loc=Step 3 of proof of theorem 5.10. gives a detailed account.}}&lt;/ref&gt;

====== Global homomorphism ======
The map {{math|Π&lt;sub&gt;''U''&lt;/sub&gt;}} is, by the Baker-Campbell-Hausdorff formula, a local homomorphism. To show that {{math|Π}} is a global homomorphism, consider two elements {{math|''g'', ''h'' ∈ SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}. Lay out paths {{math|''p''&lt;sub&gt;''g''&lt;/sub&gt;, ''p''&lt;sub&gt;''h''&lt;/sub&gt;}} from the identity to them and define a path {{math|''p''&lt;sub&gt;''gh''&lt;/sub&gt;}} going along {{math|''p''&lt;sub&gt;''g''&lt;/sub&gt;(2''t'')}} for {{math|0 ≤ ''t'' ≤ &amp;frac12;}} and along {{math|''p''&lt;sub&gt;''g''&lt;/sub&gt; · ''p''&lt;sub&gt;''h''&lt;/sub&gt;(2''t'' - 1)}} for {{math|&amp;frac12; ≤ ''t'' ≤ 1}}. This is a path from the identity to {{math|''gh''}}. Select adequate partitionings for {{math|''p''&lt;sub&gt;''g''&lt;/sub&gt;, ''p''&lt;sub&gt;''h''&lt;/sub&gt;}}. This corresponds to a choice of &quot;times&quot; {{math|''t''&lt;sub&gt;0&lt;/sub&gt;, ''t''&lt;sub&gt;1&lt;/sub&gt;, ...''t''&lt;sub&gt;m&lt;/sub&gt;}} and {{math|''s''&lt;sub&gt;0&lt;/sub&gt;, ''s''&lt;sub&gt;1&lt;/sub&gt;, ...''s''&lt;sub&gt;n&lt;/sub&gt;}}. Divide the first set with 2 and divide the second set with 2 and add &amp;frac12; and so obtain a new (adequate) set of &quot;times&quot; to be used for {{math|''p''&lt;sub&gt;''gh''&lt;/sub&gt;}}. Direct computation shows that, with these partitionings (and hence all partitionings), {{math|Π(''gh'') {{=}} Π(''g'')Π(''h'')}}.&lt;ref&gt;{{harvnb|Hall|2015|loc=Theorem 5.6.}}&lt;/ref&gt;

===== Surjectiveness of exponential mapping =====
From a practical point of view, it is important that formula {{EquationNote|(G2)}}can be used for all elements of the group. The Lie correspondence theorem above guarantees that {{EquationNote|(G2)}}holds for all {{math|''X'' ∈ '''so'''(3; 1)}}, but provides no guarantee that all {{math|''g'' ∈ SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} are in the image of {{math|exp:'''so'''(3; 1) → SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}. For general Lie groups, this is not the case, especially not for non-compact groups, as for example for {{math|SL(2, '''C''')}}, the universal covering group of {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}. It will be treated in this respect below.

But {{math|exp: '''so'''(3; 1) → SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} ''is'' surjective. One way to see this is to make use of the isomorphism {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt; ≈ PGL(2, '''C''')}}, the latter being the [[w:Möbius group|''Möbius group'']]. It is a quotient of {{math|GL(''n'', '''C''')}} (see the linked article). Let {{math|p:GL(''n'', '''C''') → PGL(2, '''C''')}} denote the quotient map. Now {{math|exp:'''gl'''(''n'', '''C''') → GL(''n'', '''C''')}} is onto.&lt;ref&gt;{{harvnb|Hall|2015|loc=Theorem 2.10.}}&lt;/ref&gt; Apply the Lie correspondence theorem with {{math|π}} being the differential at the identity of {{math|''p''}}. Then for all {{math|''X'' ∈ '''gl'''(''n'', '''C''')}} {{math|''p''(e&lt;sup&gt;''iX''&lt;/sup&gt;) {{=}} e&lt;sup&gt;iπ(''X'')&lt;/sup&gt;}}. Since the left hand side is surjective (both {{math|exp}} and {{math|''p''}} are), the right hand side is surjective and hence {{math|exp:'''pgl'''(2, '''C''') → PGL(2, '''C''')}} is surjective.&lt;ref&gt;{{harvnb|Bourbaki|1998|p=424.}}&lt;/ref&gt; Finally, recycle the argument once more, but now with the known isomorphism between {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} and {{math|PGL(2, '''C''')}} to find that {{math|exp}} is onto for the connected component of the Lorentz group.

===== Consistency =====
From the way {{math|Π(''g'')}} has been defined for elements far from the identity, it not immediately clear that formula {{EquationNote|(G2)}}holds for all elements of {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}, i.e. that one can take {{math|''U'' {{=}} ''G''}} in {{EquationNote|(G2)}}. But, in summary,
*{{math|Π}} is a uniquely constructed homomorphism.
*Using {{EquationNote|(G6)}}with {{math|Π}} as defined here, then one ends up with the {{math|π}} one started with since {{math|Π}} was defined that way near the identity, and {{EquationNote|(G6)}}depends only on an arbitrarily small neighborhood of the identity. 
*{{math|exp: '''so'''(3; 1) → SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} is surjective.

Hence {{EquationNote|(G2)}}holds everywhere.&lt;ref&gt;{{harvnb|Hall|2003|loc=Step 5 in proof of theorem 3.7.}}&lt;/ref&gt; One finally unconditionally writes

{{NumBlk|:|&lt;math&gt;\Pi(e^{iX}) = e^{i\pi(X)}, \quad X \in \mathfrak{so}(3; 1).&lt;/math&gt;|{{EquationRef|G6}}}}

==== Fundamental group ====
The above construction relies on simple connectedness. The result needs modifications for non-simply connected groups per below. To exhibit the [[w:fundamental group|''fundamental group'']] of {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}, one may consider first the topology of its [[#The covering group|covering group {{math|SL(2, ℂ)}}]]. By the [[w:polar decomposition theorem|''polar decomposition theorem'']], any matrix {{math|''λ'' ∈ SL(2, C)}} may be ''uniquely'' expressed as&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Section 2.7 p.88.}}&lt;/ref&gt;

:&lt;math&gt;\lambda = ue^h, \det u = 1, \operatorname{tr} v = 0,&lt;/math&gt;

where {{mvar|u}} is [[w:Unitary matrix|''unitary'']] with [[w:determinant|''determinant'']] one, hence in {{math|SU(2)}}, and {{mvar|h}} is [[w:Hermitian matrix|''Hermitian'']] with [[w:Trace (linear algebra)|''trace'']] zero. The [[w:Trace (linear algebra)|''trace'']] and [[w:determinant|''determinant'']] conditions imply&lt;ref name=&quot;Weinberg 2002 loc=Section 2.7&quot;&gt;{{harvnb|Weinberg|2002|loc=Section 2.7.}}&lt;/ref&gt;

:&lt;math&gt;h = \left(\begin{matrix}c&amp;a-ib\\a+ibc&amp;-c\end{matrix}\right), \quad u = \left(\begin{matrix}d+ie&amp;f+ig\\-f+ig&amp;d-ie\end{matrix}\right), \quad d^2 + e^2 + f^2 + g^2 = 1,&lt;/math&gt;

with {{math|(''a'', ''b'', ''c'') ∈ ℝ&lt;sup&gt;3&lt;/sup&gt;}} unconstrained and {{math|(''d'', ''e'', ''f'', ''g'') ∈ ℝ&lt;sup&gt;4&lt;/sup&gt;}} constrained to the [[w:3-sphere|''3-sphere'']] {{math|''S''&lt;sup&gt;3&lt;/sup&gt;}}. It follows that the manifestly continuous one-to-one map {{math|ℝ&lt;sup&gt;3&lt;/sup&gt; × ''S''&lt;sup&gt;3&lt;/sup&gt; → SL(2, ℂ); (''r'', ''s'') ↦ ''u''(''s'')''e''&lt;sup&gt;''h''(''r'')&lt;/sup&gt;}} is a [[w:homeomorphism|''homeomorphism'']] (hence preserves the fundamental group). Since {{math|ℝ&lt;sup&gt;''n''&lt;/sup&gt;}} is simply connected for all {{mvar|n}} and {{math|''S''&lt;sup&gt;''n''&lt;/sup&gt;}} is simply connected for {{math|''n'' &gt; 1}} and since simple connectedness is preserved under cartesian products, it follows that {{math|SL(2, ℂ)}} is simply connected. Now, {{math|SO(3; 1) ≈ SL(2, ℂ)/&lt;nowiki&gt;{&lt;/nowiki&gt;''I'', −''I''&lt;nowiki&gt;}&lt;/nowiki&gt;}}, where {{math|&lt;nowiki&gt;{&lt;/nowiki&gt;''I'', −''I''&lt;nowiki&gt;}&lt;/nowiki&gt; }} is the center of {{math|SL(2, ℂ)}}. Identifying {{math|''λ''}} and {{math|−''λ''}} amounts to identifying {{math|''u''}} with {{math|−''u''}}, which in turn amounts to identifying [[w:antipodal point|''antipodal points'']] on {{math|''S''&lt;sup&gt;3&lt;/sup&gt;}}. Thus topologically,&lt;ref name=&quot;Weinberg 2002 loc=Section 2.7&quot;/&gt;

:&lt;math&gt;SO(3; 1) \approx \mathbb R^3 \times  S^3/Z_2,&lt;/math&gt;

where last factor is not simply connected: Geometrically, it is easy to see (for visualization purposes, replace {{math|''S''&lt;sup&gt;3&lt;/sup&gt;}} by {{math|''S''&lt;sup&gt;2&lt;/sup&gt;}}) that a path from {{math|''u''}} to {{math|−''u''}} in {{math|SU(2) ≈ ''S''&lt;sup&gt;3&lt;/sup&gt;}} ''is'' a loop in {{math|''S''&lt;sup&gt;3&lt;/sup&gt;/''Z''&lt;sub&gt;2&lt;/sub&gt;}} since {{math|''u''}} and {{math|−''u''}} are antipodal points, and that it is not contractible to a point. But a path from {{math|''u''}} to {{math|−''u''}}, thence to {{math|''u''}} again, a loop in {{math|'''S'''&lt;sup&gt;3&lt;/sup&gt;}} and a ''double loop'' (considering {{math|''p''(''ue''&lt;sup&gt;''h''&lt;/sup&gt;) {{=}} ''p''(−''ue''&lt;sup&gt;''h''&lt;/sup&gt;)}}, where {{mvar|p}} is the covering map {{math|SL(2, ℂ) → SL(3; 1)}}) in {{math|''S''&lt;sup&gt;3&lt;/sup&gt;/''Z''&lt;sub&gt;2&lt;/sub&gt;}} that ''is'' contractible to a point (continuously move away from {{math|−''u''}} &quot;upstairs&quot; in {{math|''S''&lt;sup&gt;3&lt;/sup&gt;}} and shrink the path there to the point {{math|''u''}}).&lt;ref name=&quot;Weinberg 2002 loc=Section 2.7&quot;/&gt; Thus {{math|''π''&lt;sub&gt;1&lt;/sub&gt;(SO(3; 1))}} is a two-element group with two equivalence classes of loops as its elements &amp;ndash; or put more simply, {{math|SO(3; 1)}} is ''doubly connected''.

==== Projective representations ====
For a group that is connected but not simply connected, such as {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}, the result ''may'' depend on the [[w:homotopy class|''homotopy class'']] of the chosen path.&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Appendix B, Chapter 2.}}&lt;/ref&gt; The result, when using {{EquationNote|(G2)}}, will then depend on ''which'' {{mvar|X}} in the Lie algebra is used to obtain the representative matrix for {{mvar|g}}.

Since {{math|π&lt;sub&gt;1&lt;/sub&gt;(SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;)}} per above has two elements, not all representations of the Lie algebra will yield representations of the group, but some will instead yield [[w:projective representation|''projective representations'']].&lt;ref&gt;{{harvnb|Hall|2015|loc=Appendix C.3.}}&lt;/ref&gt;&lt;ref group=nb&gt;One should note that the terminology differs between mathematics and physics. In the linked article term projective representation has a slightly different meaning than in physics, where a projective representation is though of as a local section (a local inverse) of the [[w:covering map|''covering map'']] from the covering group onto the group being covered, composed with a proper representation of the covering group. Since this can be done (locally) continuously in two ways in the case at hand as explained below, one also speaks of a double-valued or two-valued representation.&lt;/ref&gt; Once these conclusions have been reached, and once one knows whether a representation is projective, there is no need to be concerned about paths and partitions. Formula {{EquationNote|(G2)}}applies to all group elements and all representations, including the projective ones.

For the Lorentz group, the {{math|(''m'', ''n'')}}-representation is projective when {{math|''m'' + ''n''}} is a half-integer. See the section [[#spinors|spinors]].

For a projective representation {{math|Π}} of {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}, it holds that&lt;ref name=&quot;Weinberg 2002 loc=Section 2.7&quot;/&gt;

{{NumBlk|:|&lt;math&gt;
[\Pi(\Lambda_1)\Pi(\Lambda_2)\Pi^{-1}(\Lambda_1\Lambda_2)]^2 = 1\Rightarrow \Pi(\Lambda_1\Lambda_2) = \pm \Pi(\Lambda_1)\Pi(\Lambda_2), \qquad \Lambda_1,\Lambda_2 \in \mathrm{SO}(3; 1)   ~,
&lt;/math&gt;|{{EquationRef|G5}}}}

since any loop in {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} traversed twice, due to the double connectedness, is [[w:contractible|''contractible'']] to a point, so that its homotopy class is that of a constant map. It follows that {{math|Π}} is a double-valued function. One cannot consistently chose a sign to obtain a continuous representation of all of {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}, but this is possible ''locally'' around any point.&lt;ref name=&quot;Wigner 1939&quot;/&gt;

=== The covering group ===
Consider {{math|'''sl'''(2, '''C''')}} as a ''real'' Lie algebra with basis

:&lt;math&gt;(\frac{1}{\sqrt{2}}\sigma_1, \frac{1}{\sqrt{2}}\sigma_2, \frac{1}{\sqrt{2}}\sigma_3, \frac{i}{\sqrt{2}}\sigma_1, \frac{i}{\sqrt{2}}\sigma_2, \frac{i}{\sqrt{2}}\sigma_3)\equiv(j_1, j_2, j_3, k_1, k_2, k_3),&lt;/math&gt;
 
where the sigmas are the [[w:Pauli matrices|''Pauli matrices'']]. From the relations

{{NumBlk|:|&lt;math&gt;[\sigma_i, \sigma_j] = 2i\epsilon_{ijk}\sigma_k&lt;/math&gt;|{{EquationRef|J1}}}}

one obtains

{{NumBlk|:|&lt;math&gt;[j_i, j_j] = i\epsilon_{ijk}j_k, \quad [j_i, k_j] = i\epsilon_{ijk}k_k, \quad [k_i, k_j] = -i\epsilon_{ijk}j_k,&lt;/math&gt;|{{EquationRef|J2}}}}

which are exactly on the form of the {{math|3}}-dimensional version of the commutation relations for {{math|'''so'''(3; 1)}} (see [[#Conventions and Lie algebra bases|conventions and Lie algebra bases]] below). Thus, one may map {{math|''J''&lt;sub&gt;''i''&lt;/sub&gt;  ↔  ''j''&lt;sub&gt;''i''&lt;/sub&gt;}}, {{math|''K''&lt;sub&gt;''i''&lt;/sub&gt; ↔ ''k''&lt;sub&gt;''i''&lt;/sub&gt;}}, and extend by linearity to obtain an isomorphism. Since {{math|SL(2, '''C''')}} is simply connected, it is the [[w:universal covering group|''universal covering group'']] of {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}.

==== A geometric view ====
[[File:Wigner.jpg|upright|thumb|left|[[Eugene Wigner|E.P. Wigner]] investigated the Lorentz group in depth and is known for the [[Bargmann-Wigner equations]]. The realization of the covering group given here is from his 1939 paper.]]

Let {{math|π&lt;sub&gt;''g''&lt;/sub&gt;}} denote the set of path homotopy classes {{math|[''p''&lt;sub&gt;''g''&lt;/sub&gt;]}} of paths {{math|''p''&lt;sub&gt;''g''&lt;/sub&gt;(''t''), 0 ≤ ''t'' ≤ 1}}, from {{math|1 ∈ SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} to {{math|''g'' ∈ SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} and define the set
{{NumBlk|:|&lt;math&gt;G = \{(g,[p_g]): g\in \mathrm{SO}(3; 1)^+,[p_g]\in \pi_g\} &lt;/math&gt;|{{EquationRef|C1}}}}
and endow it with the multiplication operation
{{NumBlk|:|&lt;math&gt;(g_1,[p_1])(g_2,[p_2]) = (g_1g_2,[p_{12}]),\quad g_1,g_2\in \mathrm{SO}(3; 1)^+,\quad [p_1]\in\pi_{g_1}, [p_2]\in \pi_{g_2}, [p_{12}]\in \pi_{g_{12}},\quad p_{12}(t) = p_1(t)\cdot p_2(t)~.&lt;/math&gt;|{{EquationRef|C2}}}}
The dot on the far right denotes [[w:Fundamental group#Definition|''path multiplication'']].

With this multiplication, {{mvar|''G''}} is a [[w:group (mathematics)|''group'']] and {{math|''G'' ≈ SL(2, '''C''')}},&lt;ref&gt;{{harvnb|Wigner|1939|p=27.}}&lt;/ref&gt; the universal covering group of {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}. By the above construction, there is, since each {{math|π&lt;sub&gt;''g''&lt;/sub&gt;}} has two elements, a [[w:covering map|''2:1 covering map'']] {{math|''p'' : ''G'' → SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} and an isomorphism {{math|''G'' ≈ SL(2, '''C''')}}. According to [[w:covering group|''covering group'']] theory, the Lie algebras {{math|'''so'''(3; 1)}}, {{math|'''sl'''(2, '''C''')}} and {{math|'''g'''}} of {{math|''G''}} are all isomorphic. The covering map {{math|''p'':''G'' → SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} is simply given by {{math|''p''(''g'',[''p''&lt;sub&gt;''g''&lt;/sub&gt;]) {{=}} ''g''}}.

==== An algebraic view ====
For an algebraic view of the universal covering group, let {{math|SL(2, '''C''')}} act on the set of all Hermitian {{gaps|2|×|2}} matrices {{math|'''h'''}} by the operation&lt;ref name=&quot;Weinberg 2002 loc=Section 2.7&quot;/&gt;
{{NumBlk|:|&lt;math&gt;\mathbf{P}(A): \mathbf{h} \rightarrow \mathbf{h}; \quad X \rightarrow A^\dagger XA, \quad X \in \mathbf{h}, A \in \mathrm{SL}(2, C).&lt;/math&gt;|{{EquationRef|C3}}}}

Since {{math|''X'' ∈ '''h'''}} is Hermitian, {{math|''A''&lt;sup&gt;†&lt;/sup&gt;''XA''}} is again Hermitian because {{math|1=(''A''&lt;sup&gt;†&lt;/sup&gt;''XA'')&lt;sup&gt;†&lt;/sup&gt; = ''A''&lt;sup&gt;†&lt;/sup&gt;''X''&lt;sup&gt;†&lt;/sup&gt;''A''&lt;sup&gt;††&lt;/sup&gt; = ''A''&lt;sup&gt;†&lt;/sup&gt;''XA''}}, and also {{math|1=''A''&lt;sup&gt;†&lt;/sup&gt;(''αX'' + ''βY'')''A'' = ''αA''&lt;sup&gt;†&lt;/sup&gt;''XA'' + ''βA''&lt;sup&gt;†&lt;/sup&gt;''YA''}}, so the action is linear as well. An element of {{math|'''h'''}} may generally be written in the form
{{NumBlk|:|&lt;math&gt;X = \bigl(\begin{smallmatrix}
\xi_4 + \xi_3&amp;\xi_1 + i\xi_2\\ \xi_1 - i\xi_2&amp;\xi_4 - \xi_3\\
\end{smallmatrix}\bigr) &lt;/math&gt;|{{EquationRef|C4}}}}
for {{mvar|ξ&lt;sub&gt;i&lt;/sub&gt;}} real, showing that {{math|'''h'''}} is a 4-dimensional real vector space. Moreover, {{math|1=(''AB'')&lt;sup&gt;†&lt;/sup&gt;''X''(''AB'') = ''B''&lt;sup&gt;†&lt;/sup&gt;''A''&lt;sup&gt;†&lt;/sup&gt;''XAB''}} meaning that {{math|'''P'''}} is a group homomorphism into {{math|GL('''h''') ⊂ End '''h'''}}. Thus {{math|'''P''' : SL(2, '''C''') → GL ('''h''')}} is a 4-dimensional representation of {{math|SL(2, '''C''')}}. Its kernel must in particular take the identity matrix to itself, {{math|''A''&lt;sup&gt;†&lt;/sup&gt;''IA'' {{=}} ''A''&lt;sup&gt;†&lt;/sup&gt;''A'' {{=}} ''I'' ⇒ ''A''&lt;sup&gt;†&lt;/sup&gt; {{=}} ''A''&lt;sup&gt;−1&lt;/sup&gt;}}. Thus {{math|''AX'' {{=}} ''XA''}} for {{mvar|A}} in the kernel so, by [[w:Schur's lemma|''Schur's lemma'']],&lt;ref group=nb&gt;In particular, {{math|''A''}} commutes with the [[w:Pauli matrices|''Pauli matrices'']], hence with all of {{math|SU(2)}} making Schur's lemma applicable.&lt;/ref&gt; {{mvar|A}} is a multiple of the identity, which must be {{math|±''I''}} since {{math|det ''A'' {{=}} 1}}.&lt;ref name=Gelfand_1&gt;{{harvnb|Gelfand|Minlos|Shapiro|1963}} This construction of the covering group is treated in paragraph 4, section 1, chapter 1 in Part II.&lt;/ref&gt; Now map {{math|'''h'''}} to spacetime {{math|'''R'''&lt;sup&gt;4&lt;/sup&gt;}} endowed with the Lorentz metric, [[w:Minkowski space|''Minkowski space'']], via 
{{NumBlk|:|&lt;math&gt;X = (\xi_1,\xi_2,\xi_3,\xi_4) \leftrightarrow \overrightarrow{(\xi_1,\xi_2,\xi_3,\xi_4)} = (x,y,z,t) = \overrightarrow{X}.&lt;/math&gt;|{{EquationRef|C5}}}}
The action of {{math|'''P'''(''A'')}} on {{math|'''h'''}} preserves determinants since {{math|1=det(''A''&lt;sup&gt;†&lt;/sup&gt;''XA'') = (det ''A'')(det ''A''&lt;sup&gt;†&lt;/sup&gt;)(det ''X'') = det ''X''}}. The induced representation {{math|'''p'''}} of {{math|SL(2, '''C''')}} on {{math|'''R'''&lt;sup&gt;4&lt;/sup&gt;}}, via the above isomorphism, given by
{{NumBlk|:|:&lt;math&gt;\mathbf{p}(A)\overrightarrow{X} = \overrightarrow{AXA^\dagger}&lt;/math&gt;|{{EquationRef|C6}}}}
will preserve the Lorentz inner product since 
:{{math|1=−det ''X'' = ''ξ''&lt;sub&gt;1&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt; + ''ξ''&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt; + ''ξ''&lt;sub&gt;3&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt; − ''ξ''&lt;sub&gt;4&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt; = ''x''&lt;sup&gt;2&lt;/sup&gt; + ''y''&lt;sup&gt;2&lt;/sup&gt; + ''z''&lt;sup&gt;2&lt;/sup&gt; − ''t''&lt;sup&gt;2&lt;/sup&gt;}}. 
This means that {{math|'''p'''(A)}} belongs to the full Lorentz group {{math|SO(3; 1)}}. By the [[w:main theorem of connectedness|''main theorem of connectedness'']], since {{math|SL(2, '''C''')}} is connected, its image under {{math|'''p'''}} in {{math|SO(3; 1)}} is connected as well, and hence is contained in {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}.

It can be shown that the [[w:Pushforward (differential)|''Lie map'']] of {{math|'''p''' : SL(2, '''C''') → SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}, {{math|''π'' : '''sl'''(2, '''C''') → '''so'''(3; 1)}} is a Lie algebra isomorphism (its kernel is {{math|{∅}&lt;nowiki/&gt;}}&lt;ref group = nb&gt;The kernel of a Lie algebra homomorphism is an [[w:ideal (Lie algebra)|''ideal'']], hence a subspace. Since {{math|'''p'''}} is {{math|2:1}} and both {{math|SL(2, '''C''')}} and {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} are {{nowrap|{{math|6}}-dimensional}}, the kernel must be {{nowrap|{{math|0}}-dimensional}}, hence {{math|{∅}&lt;nowiki/&gt;}}.&lt;/ref&gt; and must therefore be an isomorphism for dimensional reasons). The map {{math|'''P'''}} is also onto.&lt;ref group=nb&gt;

The exponential map is one-to-one in a neighborhood of the identity in {{math|SL(2, '''C''')}}, hence the composition {{math|exp ∘ ''σ'' ∘ log:SL(2, '''C''') → SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}, where {{mvar|σ}} is the Lie algebra isomorphism, is onto an open neighborhood {{math|''U'' ⊂ SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} containing the identity. Such a neighborhood generates the connected component.&lt;/ref&gt;

Thus {{math|SL(2, '''C''')}}, since it is simply connected, is the universal covering group of {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}, isomorphic to the group {{math|''G''}} of above.

==== Representations of {{math|SL(2, C)}} and {{math|sl(2, C)}} ====
The complex linear representations of {{math|'''sl'''(2, '''C''')}} and {{math|SL(2, '''C''')}} are more straightforward to obtain than the {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} representations. If {{math|π&lt;sub&gt;μ&lt;/sub&gt;}} is a representation of {{math|'''su'''(2)}} with highest weight {{math|μ}}, then the complexification of {{math|π&lt;sub&gt;μ&lt;/sub&gt;}} is a complex linear representation of {{math|'''sl'''(2, '''C''')}}. All complex linear representation of {{math|'''sl'''(2, '''C''')}} are of this form. The [[w:Holomorphic function|''holomorphic'']] group representations (meaning the corresponding Lie algebra representation is complex linear) are obtained by exponentiation. By simple connectedness of {{math|SL(2, '''C''')}}, this always yields a representation of the group as opposed to in the {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} case. The real linear representations of {{math|'''sl'''(2, '''C''')}} are exactly the {{math|(μ, ν)}}-representations presented earlier. They can be exponentiated too. The {{math|(μ, 0)}}-representations are complex linear and are (isomorphic to) the highest weight-representations. These are usually indexed with only one integer.

It is also possible to obtain representations of {{math|SL(2, '''C''')}} directly. This will be done below. Then, using the unitarian trick, going the other way, one finds {{math|'''sl'''(2, '''C''')}}-,{{math|SU(2)}}-,{{math|'''su'''(2)}}-,{{math|SL(2, '''R''')}}-, and {{math|'''sl'''(2, '''R''')}}-representations as well as {{math|'''so'''(3; 1)}}-representations (via {{EquationNote|(A1)}}) and, possibly projective, {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}-representations (via projection from {{math|SL(2, '''C''')}}, see below, or exponentiation).

The mathematics convention is used in this section for convenience. Lie algebra elements differ by a factor of {{math|''i''}} and there is no factor of {{math|''i''}} in the exponential mapping compared to the physics convention used elsewhere. Let the basis of {{math|'''sl'''(2, '''C''')}} be&lt;ref&gt;{{harvnb|Hall|2015|loc=First displayed equations in section 4.6.}}&lt;/ref&gt;
{{NumBlk|:|&lt;math&gt;
H = \bigl(\begin{smallmatrix} 1&amp;0\\ 0&amp;-1\\ \end{smallmatrix}\bigr), \quad 
X = \bigl(\begin{smallmatrix} 0&amp;1\\ 0&amp;0\\ \end{smallmatrix}\bigr), \quad
Y = \bigl(\begin{smallmatrix} 0&amp;0\\ 1&amp;0\\ \end{smallmatrix}\bigr).
&lt;/math&gt;|{{EquationRef|S1}}}}
This choice of basis, and the notation, is standard in the mathematical literature.

===== Concrete realization =====
The irreducible holomorphic {{math|(''n'' + 1)}}-dimensional representations of {{math|SL(2, '''C''')}}, {{math|''n'' ≥ 0}}, can be realized on a set of functions {{math|ℙ&lt;sup&gt;2&lt;/sup&gt;&lt;sub&gt;''n''&lt;/sub&gt; {{=}} {''P'':'''C'''&lt;sup&gt;2&lt;/sup&gt; → '''C'''}&lt;nowiki/&gt;}} where each {{math|''P'' ∈ ℙ&lt;sup&gt;2&lt;/sup&gt;&lt;sub&gt;''n''&lt;/sub&gt;}} is a [[w:homogeneous polynomial|''homogeneous polynomial'']] of [[w:Degree of a polynomial|''degree'']] {{math|''n''}} in 2 variables.&lt;ref&gt;{{harvnb|Hall|2015|loc=Example 4.10.}}&lt;/ref&gt;&lt;ref name=&quot;harvnb|Knapp|2001&quot;&gt;{{harvnb|Knapp|2001|loc=Chapter 2.}}&lt;/ref&gt; The elements of {{math|ℙ&lt;sup&gt;2&lt;/sup&gt;&lt;sub&gt;''n''&lt;/sub&gt;}} appears as {{math|''P''(''z''&lt;sub&gt;1&lt;/sub&gt;,&amp;nbsp;''z''&lt;sub&gt;2&lt;/sub&gt;) {{=}} ''c''&lt;sub&gt;''n''&lt;/sub&gt;''z''&lt;sub&gt;1&lt;/sub&gt;&lt;sup&gt;n&lt;/sup&gt; + ''c''&lt;sub&gt;''n''−1&lt;/sub&gt;''z''&lt;sub&gt;1&lt;/sub&gt;&lt;sup&gt;''n''−1&lt;/sup&gt;''z''&lt;sub&gt;2&lt;/sub&gt; + ... + ''c''&lt;sub&gt;''n''&lt;/sub&gt;''z''&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;''n''&lt;/sup&gt;}}. The action of {{math|SL(2, '''C''')}} is given by&lt;ref&gt;{{harvnb|Knapp|2001}} Equation 2.1.&lt;/ref&gt;&lt;ref&gt;{{harvnb|Hall|2015|loc=Equation 4.2.}}&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;[\Phi_n(g)]P(z) = \biggl[\Phi\biggl(\begin{matrix} a&amp;b\\ c&amp;d\\ \end{matrix}\biggr)P\biggr] \biggl(\begin{matrix} z_1\\ z_2\\ \end{matrix}\biggr) = 
P\biggl( \biggl(\begin{matrix} a&amp;b\\ c&amp;d\\ \end{matrix}\biggr)^{-1} \biggl(\begin{matrix} z_1\\ z_2\\ \end{matrix}\biggr) \biggr), \quad P \in \mathbb{P}^2_n.&lt;/math&gt;|{{EquationRef|S2}}}}

The associated {{math|'''sl'''(2, '''C''')}}-action is, using {{EquationNote|(G6)}}and the definition above, given by&lt;ref&gt;{{harvnb|Hall|2015|loc=Third equation after 4.3.}}&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\phi_n(X) = \left .\frac{d}{dt}\Phi(e^tX)\right|_{t = 0} \Rightarrow \phi_n(X)P(z) = \left .\frac{d}{dt}P(e^{-tX}z)\right|_{t = 0},\quad z = \biggl(\begin{matrix} z_1\\ z_2\\ \end{matrix}\biggr).&lt;/math&gt;|{{EquationRef|S3}}}}

Defining {{math|''z''(''t'') {{=}} e&lt;sup&gt;−''tX''&lt;/sup&gt;''z'' {{=}} (''z''&lt;sub&gt;1&lt;/sub&gt;(''t''), ''z''&lt;sub&gt;2&lt;/sub&gt;(''t''))&lt;sup&gt;T&lt;/sup&gt;}} and using the [[w:chain rule|''chain rule'']] one finds&lt;ref&gt;{{harvnb|Hall|2015|loc=Equation 4.4.}}&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\phi_n(X)P = \left .\frac{\partial P}{\partial z_1}\frac{dz_1}{dt}\right|_{t = 0} + \left .\frac{\partial P}{\partial z_2}\frac{dz_2}{dt}\right|_{t = 0} =
-\frac{\partial P}{\partial z_1}(X_{11}z_1 + X_{12}z_2) - \frac{\partial P}{\partial z_2}(X_{21}z_1 + X_{22}z_2).&lt;/math&gt;|{{EquationRef|S4}}}}

The basis elements of {{math|'''sl'''(2, '''C''')}} are then represented by&lt;ref&gt;{{harvnb|Hall|2015|loc=Equation before 4.5.}}&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\phi_n(H) = -z_1\frac{\partial}{\partial z_1} + z_2\frac{\partial}{\partial z_2}, \quad \phi_n(X) = -z_2\frac{\partial}{\partial z_1}, \quad \phi_n(Y) = -z_1\frac{\partial}{\partial z_2}.&lt;/math&gt;|{{EquationRef|S5}}}}

on the space {{math|''P'' ∈ ℙ&lt;sup&gt;2&lt;/sup&gt;&lt;sub&gt;''n''&lt;/sub&gt;}} (all {{math|''n''}}). By employing the unitarian trick one obtains representations for {{math|SU(2)}}, {{math|'''su'''(2)}}, {{math|SL(2, '''R'''}}, and {{math|'''sl'''(2, '''R''')}}, all are obtained by restriction of either {{EquationNote|(S2)}}or {{EquationNote|(S4)}}. They are formally identical to {{EquationNote|(S2)}}or {{EquationNote|(S4)}}. With a choice of basis for {{math|''P'' ∈ ℙ&lt;sup&gt;2&lt;/sup&gt;&lt;sub&gt;''n''&lt;/sub&gt;}}, all these representations become matrix groups or matrix Lie algebras.

The {{math|(μ, ν)}}-representations are realized on a space of polynomials {{math|ℙ&lt;sup&gt;2&lt;/sup&gt;&lt;sub&gt;''μν''&lt;/sub&gt;}} in {{math|z&lt;sub&gt;1&lt;/sub&gt;, {{overline|z}}&lt;sub&gt;1&lt;/sub&gt;, z&lt;sub&gt;2&lt;/sub&gt;, {{overline|z}}&lt;sub&gt;2&lt;/sub&gt;}}, homogeneous of degree {{math|μ}} in {{math|z&lt;sub&gt;1&lt;/sub&gt;, z&lt;sub&gt;2&lt;/sub&gt;}} and homogeneous of degree {{math|ν}} in {{math|{{overline|z}}&lt;sub&gt;1&lt;/sub&gt;, {{overline|z}}&lt;sub&gt;2&lt;/sub&gt;}}.&lt;ref name=&quot;harvnb|Knapp|2001&quot;/&gt; The representations are given by&lt;ref&gt;{{harvnb|Knapp|2001}} Equation 2.4.&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;[\Phi_{\mu,\nu}(g)]P(z) = \biggl[\Phi\biggl(\begin{matrix} a&amp;b\\ c&amp;d\\ \end{matrix}\biggr)P\biggr] \biggl(\begin{matrix} z_1\\ z_2\\ \end{matrix}\biggr) = 
P\biggl( \biggl(\begin{matrix} a&amp;b\\ c&amp;d\\ \end{matrix}\biggr)^{-1} \biggl(\begin{matrix} z_1\\ z_2\\ \end{matrix}\biggr) \biggr), \quad P \in \mathbb{P}^2_{\mu,\nu}.&lt;/math&gt;|{{EquationRef|S6}}}}

By carrying out the same steps as above, one finds

{{NumBlk|:|&lt;math&gt;\phi_{\mu,\nu}(X)P  = -\frac{\partial P}{\partial z_1}(X_{11}z_1 + X_{12}z_2) - \frac{\partial P}{\partial z_2}(X_{21}z_1 + X_{22}z_2)
-\frac{\partial P}{\partial \overline{z_1}}(\overline{X_{11}}\overline{z_1} + \overline{X_{12}}\overline{z_2})
-\frac{\partial P}{\partial \overline{z_2}}(\overline{X_{21}}\overline{z_1} + \overline{X_{22}}\overline{z_2}), \quad X \in \mathfrak{sl}(2, \mathbb{C}),&lt;/math&gt;|{{EquationRef|S7}}}}

from which the expressions

{{NumBlk|:|&lt;math&gt;\phi_{\mu,\nu}(H) = -z_1\frac{\partial}{\partial z_1} + z_2\frac{\partial}{\partial z_2}
-\overline{z_1}\frac{\partial}{\partial \overline{z_1}} + \overline{z_2}\frac{\partial}{\partial \overline{z_2}}, \quad
\phi_{\mu,\nu}(X) = -z_2\frac{\partial}{\partial z_1} - \overline{z_2}\frac{\partial}{\partial \overline{z_1}}, \quad
\phi_{\mu,\nu}(Y) = -z_1\frac{\partial}{\partial z_2} - \overline{z_1}\frac{\partial}{\partial \overline{z_2}}
&lt;/math&gt;|{{EquationRef|S8}}}}

for the basis elements follow.

===== Non-surjectiveness of exponential mapping =====
[[File:Commutative diagram SO(3, 1) latex.svg|300px|thumb|left|This diagram shows the web of maps discussed in the text. Here {{Math|''V''}} is a finite-dimensional vector space carrying representations of {{math|'''sl'''(2, '''C''')}}, {{math|'''so'''(3; 1)}}, {{math|SL(2, '''C''')}}, {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}, {{math|exp}} is the exponential mapping, {{math|'''p'''}} is the covering map from {{math|SL(2, '''C''')}} onto {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} and {{math|σ}} is the Lie algebra isomorphism induced by it.

The maps {{math|Π, π}} and the two {{math|Φ}} are representations. the picture is only partially true when {{math|Π}} is projective.]]

Unlike in the case {{math|exp: '''so'''(3; 1) → SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}, the exponential mapping {{math|exp: '''sl'''(2, '''C''') → SL(2, '''C''')}} is not onto.&lt;ref&gt;{{harvnb|Rossmann|2002|loc=Section 2.1.}}&lt;/ref&gt; The [[w:conjugacy class|''conjugacy class'']]es of {{math|SL(2, '''C''')}} are represented by the matrices&lt;ref&gt;{{harvnb|Rossmann|2002|loc=Example 4, section 2.1.}}&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;a(\alpha) = \bigl(\begin{smallmatrix} \alpha&amp;0\\ 0&amp;\alpha^{-1}\\ \end{smallmatrix}\bigr) = 
e^{\operatorname{log}(\alpha)H},\alpha \neq 0, \quad
p = \bigl(\begin{smallmatrix} 1&amp;1\\ 0&amp;1\\ \end{smallmatrix}\bigr) = e^X, \quad 
q = \bigl(\begin{smallmatrix} -1&amp;1\\ 0&amp;-1\\ \end{smallmatrix}\bigr),&lt;/math&gt;|{{EquationRef|S6}}}}

but there is no element {{math|''Q''}} in {{math|'''sl'''(2, '''C''')}} such that {{math|''q'' {{=}} exp(''Q'')}}.&lt;ref group=nb&gt;{{harvnb|Rossmann|2002}} From Example 4 in section 2.1 : This can be seen as follows. The matrix ''q'' has eigenvalues {{math|{-1, -1} }}, but it is not [[w:Diagonalizable matrix|''diagonalizable'']]. If {{math|''q'' {{=}} exp(''Q'')}}, then {{math|''Q''}} has eigenvalues {{math|λ, −λ}} with {{math|λ {{=}} ''iπ + 2π''ik}} for some {{math|''k''}} because the tracelessness of {{math|'''sl'''(2, '''C''')}}-matrices forces them to be negatives of each other. But then {{math|''Q''}} is diagonalizable, hence {{math|''q''}} is diagonalizable. This is a contradiction.&lt;/ref&gt;

In general, if {{math|''g''}} is an element of a connected Lie group {{math|''G''}} with Lie algebra {{math|'''g'''}}, then&lt;ref&gt;{{harvnb|Hall|2015|loc=Corollary 3.47.}}&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;g = e^{X_1}e^{X_2}\cdots e^{X_n}, \quad X_i \in \mathfrak{g}, 1 \leq i \leq n.&lt;/math&gt;|{{EquationRef|S7}}}}

This follows from the compactness of a path from the identity to {{math|''g''}} and the one-to-one nature of {{math|exp}} near the identity. In the case of the matrix {{math|''q''}}, one may write

{{NumBlk|:|&lt;math&gt;e^{-X}e^{i\pi H} = e^{\bigl(\begin{smallmatrix} 0&amp;-1\\ 0&amp;0\\ \end{smallmatrix}\bigr)}e^{i\pi\bigl(\begin{smallmatrix} 1&amp;0\\ 0&amp;-1\\ \end{smallmatrix}\bigr)} 
= \bigl(\begin{smallmatrix} 1&amp;-1\\ 0&amp;1\\ \end{smallmatrix}\bigr)\bigl(\begin{smallmatrix} -1&amp;0\\ 0&amp;-1\\ \end{smallmatrix}\bigr) = 
\bigl(\begin{smallmatrix} -1&amp;1\\ 0&amp;-1\\ \end{smallmatrix}\bigr) = q.&lt;/math&gt;|{{EquationRef|S8}}}}

The [[w:kernel (algebra)#Group homomorphism|''kernel'']] of the [[w:covering map|''covering map'']] {{math|'''p''':''SL''(2, '''C''') → '''SO'''(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} of above is {{math|''N'' {{=}} {''I'', −''I''}&lt;nowiki/&gt;}}, a [[w:normal subgroup|''normal subgroup'']] of {{math|SL(2, '''C''')&lt;sup&gt;+&lt;/sup&gt;}}. The composition {{math|'''p''' ∘ exp: '''sl'''(2, '''C''') → SO(3; 1)}} ''is'' onto. If a matrix {{math|''a''}} is not in the image of {{math|exp}}, then there is a matrix {{math|''b''}} [[w:Equivalence relation#Abstract algebra#Group theory|''equivalent'']] to it with respect to {{math|'''p'''}}, meaning {{math|'''p'''(''b'') {{=}} '''p'''(''a'')}}, that is in the image of {{math|exp}}. The condition for equivalence is {{math|''a''&lt;sup&gt;−1&lt;/sup&gt;''b'' ∈ ''N''}}.&lt;ref&gt;{{harvnb|Hall|2003|loc=Appendix A.}}&lt;/ref&gt; In the case of the matrix {{math|''q''}}, one may solve for {{math|''p''}} in the equation {{math|''p''&lt;sup&gt;−1&lt;/sup&gt;''q'' {{=}} -''I'' ∈ ''N''}}. One finds

{{NumBlk|:|&lt;math&gt;p = \bigl(\begin{smallmatrix} 1&amp;-1\\ 0&amp;1\\ \end{smallmatrix}\bigr) = e^{-X}, \quad 
p^{-1}q = \bigl(\begin{smallmatrix} 1&amp;1\\ 0&amp;1\\ \end{smallmatrix}\bigr)\bigl(\begin{smallmatrix} -1&amp;1\\ 0&amp;-1\\ \end{smallmatrix}\bigr) = 
\bigl(\begin{smallmatrix} -1&amp;0\\ 0&amp;-1\\ \end{smallmatrix}\bigr) = -I.&lt;/math&gt;|{{EquationRef|S9}}}}

As a corollary, since the covering map {{math|'''p'''}} is a homomorphism,the mapping version of the Lie correspondence {{EquationNote|(G6)}}can be used to provide a proof of the surjectiveness of {{math|exp}} for {{math|'''so'''(3; 1)}}. Let {{math|σ}} denote the isomorphism between {{math|'''sl'''(2, '''C''')}} and {{math|'''so'''(3; 1)}}. Refer to the commutative diagram. One has {{math|'''p''' ∘ exp: '''sl'''(2, '''C''') → SO(3; 1) {{=}} exp ∘ σ}} for all {{math|''X'' ∈ '''sl'''(2, '''C''')}}. Since {{math|'''p''' ∘ exp}} is onto, {{math|exp ∘ σ}} is onto, and hence {{math|exp: '''so'''(3; 1) → SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} is onto as well.

===== {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}-representations from {{math|SL(2, C)}}-representations =====
By the [[w:first isomorphism theorem|''first isomorphism theorem'']], a representation {{math|(Φ, ''V'')}} of {{math|SL(2, '''C''')}} descends to a representation {{math|(Π, ''V'')}} of {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} if and only if {{math|ker '''p''' ⊂ ker Φ}}. Refer to the commutative diagram. If this condition holds, then both elements in the fiber {{math|'''p'''&lt;sup&gt;−1&lt;/sup&gt;(''g''), ''g'' ∈ SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} will be mapped by {{math|Φ}} to the same representative, and the expression {{math|Φ('''p'''&lt;sup&gt;−1&lt;/sup&gt;(''g''))}} makes sense. One may thus define {{math|Π: SO(3; 1)&lt;sup&gt;+&lt;/sup&gt; → GL(''V''), Π(''g'') {{=}} Φ('''p'''&lt;sup&gt;−1&lt;/sup&gt;(''g''))}}. In particular, if {{math|Π}} is [[w:Faithful representation|''faithful'']], i.e. having kernel = {{math|''I''}}, then there is no corresponding proper representation of {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}, but there is a projective one as was shown in a previous section, corresponding to the two possible choices of representative in each fiber {{math|'''p'''&lt;sup&gt;−1&lt;/sup&gt;(''g'')}}.

Lie algebra representations of {{math|'''so'''(3; 1)}} are obtained from {{math|'''sl'''(2, '''C''')}}-representations simply by composition with {{math|σ&lt;sup&gt;−1&lt;/sup&gt;}}.

===== {{math|SL(2, C)}}-representations from {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}-representations =====
{{math|SL(2, '''C''')}}-representations can be obtained from non-projective {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}-representations by composition with the projection map {{math|'''p'''}}. These are always representations since they are compositions of group homomorphisms. Such a representation is never faithful because {{math|Ker '''p''' {{=}} {''I'', −''I''}&lt;nowiki/&gt;}}. If the {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}-representation is projective, then the resulting {{math|SL(2, '''C''')}}-representation would be projective as well. Instead, the isomorphism {{math|σ:'''so'''(3; 1) → '''sl'''(3, '''C''')}} can be employed, composed with {{math|exp:'''sl'''(2, '''C''') → SL(2, '''C''')}}. This is always a non-projective representation.

=== Properties of the (''m'', ''n'') representations ===
The {{math|(''m'', ''n'')}} representations are irreducible, and they are the only irreducible representations.&lt;ref name=&quot;ReferenceC&quot;&gt;{{harvnb|Knapp|2001}}&lt;/ref&gt;
*Irreducibility follows from the unitarian trick&lt;ref name=&quot;Knapp 2001&quot;/&gt; and that a representation {{math|Π}} of {{math|SU(2) × SU(2)}} is irreducible if and only if {{math|Π {{=}} Π&lt;sub&gt;μ&lt;/sub&gt; ⊗ Π&lt;sub&gt;ν&lt;/sub&gt;}},&lt;ref group=nb&gt;{{harvnb|Rossmann|2002|loc=Proposition 10, paragraph 6.3.}} This is easiest proved using [[w:character theory|''character theory'']].&lt;/ref&gt; where {{math|Π&lt;sub&gt;μ&lt;/sub&gt;, Π&lt;sub&gt;ν&lt;/sub&gt;}} are irreducible representations of {{math|SU(2)}}.
*Uniqueness follows from that the {{math|Π&lt;sub&gt;''m''&lt;/sub&gt;}} are the only irreducible representations of {{math|SU(2)}}, which is one of the conclusions of the theorem of the highest weight.&lt;ref&gt;{{harvnb|Hall|2015|loc=Theorems 9.4&amp;ndash;5.}}&lt;/ref&gt;

==== Dimension ====
The {{math|(''m'', ''n'')}} representations are {{math|(2''m'' + 1)(2''n'' + 1)}}-dimensional.&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Chapter 5.}}&lt;/ref&gt; It follows from the Weyl dimension formula. For a Lie algebra {{math|'''g'''}} it reads&lt;ref&gt;{{harvnb|Hall|2015|loc= Theorem 10.18.}}&lt;/ref&gt;

&lt;math&gt;\operatorname{dim}\pi_\mu = \frac{\Pi_{\alpha \in R^+} \langle\alpha, \mu + \delta \rangle}{\Pi_{\alpha \in R^+} \langle\alpha, \delta \rangle},&lt;/math&gt;

where {{math|''R''&lt;sup&gt;+&lt;/sup&gt;}} is the set of positive roots and {{math|δ}} is half the sum of the positive roots. The inner product {{math|&lt;⋅,⋅&gt;}} is that of the Lie algebra {{math|'''g'''}}, invariant under the action of the Weyl group on {{math|'''h''' ⊂ '''g'''}}, the [[w:Cartan subalgebra|''Cartan subalgebra'']]. The roots (really elements of {{math|'''h'''*)}} are via this inner product identified with elements of {{math|'''h'''}}. For {{math|'''sl'''(2, '''C''')}}, the formula reduces to {{math|dim ''π''&lt;sub&gt;''μ''&lt;/sub&gt; {{=}} ''μ'' + 1 {{=}} 2''m'' + 1}}.&lt;ref&gt;{{harvnb|Hall|2003|p=235.}}&lt;/ref&gt; By taking tensor products, the result follows.

A quicker approach is, of course, to simply count the dimensions in any concrete realization, such as the one given in [[#representations of SL(2, C) and sl(2, C)|representations of {{math|SL(2,''' C''')}} and {{math|'''sl'''(2, ''' C''')}}]].

==== Faithfulness ====
If a representation {{math|Π}} of a Lie group {{math|''G''}} is not faithful, then {{math|''N'' {{=}} ker Π}} is a nontrivial normal subgroup because {{math|Π(''n'') {{=}} ''I'' ⇒ Π(''gng''&lt;sup&gt;−1&lt;/sup&gt;) {{=}} Π(''g'')Π(''n'')Π(''g'')&lt;sup&gt;−1&lt;/sup&gt; {{=}} Π(''g'')Π(''g'')&lt;sup&gt;−1&lt;/sup&gt; {{=}} ''I''}}. There are three relevant cases.
#{{math|''N''}} is non-discrete and [[w:abelian group|''abelian'']].
#{{math|''N''}} is non-discrete and non-abelian.
#{{math|''N''}} is discrete. In this case {{math|''N'' ⊂ ''Z''}}, where {{math|''Z''}} is the center of {{math|''G''}}.&lt;ref group=nb&gt;Any discrete normal subgroup of a [[w:path connected|''path connected'']] group {{math|''G''}} is contained in the center {{math|''Z''}} of {{math|''G''}}.
{{Paragraph break}}
{{harvnb|Hall|2015|loc=Exercise 11, chapter 1.}}&lt;/ref&gt;
In the case of {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}, the first case is excluded since {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} is semi-simple.&lt;ref group=nb&gt;A semisimple Lie group does not have any non-discrete normal [[w:abelian group|''abelian subgroups'']]. This can be taken as the definition of semisimplicity.&lt;/ref&gt; The second case (and the first case) is excluded because {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} is simple.&lt;ref group=nb&gt;A simple group does not have any non-discrete normal subgroups.&lt;/ref&gt; For the third case, {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} is isomorphic to the quotient {{math|SL(2, '''C''')/{''I'', −''I''}&lt;nowiki/&gt;}}. But {{math|{''I'', −''I''}&lt;nowiki/&gt;}} is the center of {{math|SL(2, '''C''')}}. It follows that the center of {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} is trivial, and this excludes the third case. The conclusion is that every representation {{math|Π:SO(3; 1)&lt;sup&gt;+&lt;/sup&gt; → GL(''V'')}} and every projective representation {{math|Π:SO(3; 1)&lt;sup&gt;+&lt;/sup&gt; → PGL(''W'')}} for {{math|''V'', ''W''}} finite-dimensional vector spaces are faithful.

By using the fundamental Lie correspondence, the statements and the reasoning above translate directly to Lie algebras with (abelian) nontrivial non-discrete normal subgroups replaced by (one-dimensional) nontrivial ideals in the Lie algebra,&lt;ref&gt;{{harvnb|Rossmann|2002}} Propositions 3 and 6 paragraph 2.5.&lt;/ref&gt; and the center of {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} replaced by the center of {{math|'''sl'''(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}. The center of any semisimple Lie algebra is trivial&lt;ref&gt;{{harvnb|Hall|2003}} See exercise 1, Chapter 6.&lt;/ref&gt; and {{math|'''so'''(3; 1)}} is semi-simple and simple, and hence has no non-trivial ideals.

A related fact is that if the corresponding representation of {{math|SL(2, ℂ)}} is faithful, then the representation is projective. Conversely, if the representation is non-projective, then the corresponding {{math|SL(2, ℂ)}} representation is not faithful, but is {{math|2:1}}.

==== Non-unitarity ====
The {{math|(''m'', ''n'')}} Lie algebra representation is not [[w:Hermitian|''Hermitian'']]. Accordingly, the corresponding (projective) representation of the group is never [[w:unitary matrix|''unitary'']].&lt;ref group=nb&gt;By contrast, there is a trick, also called Weyl's unitarian trick, but unrelated to the unitarian trick of above showing that all finite-dimensional representations are, or can be made, unitary. If {{math|(Π, ''V'')}} is a finite-dimensional representation of a ''compact'' Lie group {{mvar|G}} and if {{math|(·, ·)}} is any [[w:inner product|''inner product'']] on {{mvar|V}}, define a new inner product {{math|(·, ·)&lt;sub&gt;Π&lt;/sub&gt;}} by {{math|(''x'', ''y'')&lt;sub&gt;Π&lt;/sub&gt; {{=}} ∫&lt;sub&gt;''G''&lt;/sub&gt;(Π(''g'')''x'', Π(''g'')''y'' ''dμ''(''g'')}}, where {{mvar|μ}} is [[w:Haar measure|''Haar measure'']] on {{mvar|G}}. Then {{math|Π}} is unitary with respect to {{math|(·, ·)&lt;sub&gt;Π&lt;/sub&gt;}}. See {{harvtxt|Hall|2015|loc=Theorem 4.28.}}
{{Paragraph break}}
Another consequence is that every compact Lie group has the ''complete reducibility property'', meaning that all its finite-dimensional representations decompose as a direct sum of ''irreducible'' representations. {{harvtxt|Hall|2015|loc=Definition 4.24., Theorem 4.28.}}
{{Paragraph break}}
It is also true that there are no infinite-dimensional ''irreducible'' unitary representations of compact Lie groups, stated, but not proved in {{harvtxt|Greiner|Müller|1994|loc=Section 15.2.}}.&lt;/ref&gt; This is due to the non-compactness of the Lorentz group. In fact, a connected simple non-compact Lie group cannot have ''any'' nontrivial unitary finite-dimensional representations.&lt;ref name=&quot;Wigner 1939&quot;/&gt; There is a topological proof of this.&lt;ref&gt;{{harvnb|Bekaert|Boulanger|2006}} p.4.&lt;/ref&gt; Let {{math|''U'':''G'' → GL(''V'')}}, where {{math|''V''}} is finite-dimensional, be a continuous unitary representation of the non-compact connected simple Lie group {{math|''G''}}. Then {{math|''U''(''G'') ⊂ U(''V'') ⊂ GL(''V'')}} where {{math|U(''V'')}} is the compact subgroup of {{math|GL(''V'')}} consisting of unitary transformations of {{math|''V''}}. The [[w:Kernel (algebra)|''kernel'']], {{math|ker ''U''}}, of {{math|''U''}} is a [[w:normal subgroup|''normal subgroup'']] of {{math|''G''}}. Since {{math|''G''}} is simple, {{math|ker ''U''}} is either all of {{math|''G''}}, in which case {{math|''U''}} is trivial, or {{math|ker ''U''}} is trivial, in which case {{math|''U''}} is [[w:Faithful representation|''faithful'']]. In the latter case {{math|''U''}} is a [[w:diffeomorphism|''diffeomorphism'']] onto its image,&lt;ref&gt;{{harvnb|Hall|2003}} Proposition 1.20.&lt;/ref&gt; {{math|''U''(''G'') ≈ ''G''.}}, and {{math|''U''(''G'')}} is Lie group. This would mean that {{math|''U''(''G'')}} is an [[w:Embedded submanifold|''embedded'']] non-compact Lie subgroup of the compact group {{math|U(''V'')}}. This is impossible with the subspace topology on {{math|''U''(''G'') ⊂ U(''V'')}} since all ''embedded'' Lie subgroups of a Lie group are closed&lt;ref&gt;{{harvnb|Lee|2003|loc=Theorem 8.30.}}&lt;/ref&gt; If {{math|''U''(''G'')}} were closed, it would be compact,&lt;ref group=nb&gt;{{harvnb|Lee|2003}} Lemma A.17 (c). Closed subsets of compact sets are compact.&lt;/ref&gt; and then {{math|''G''}} would be compact,&lt;ref group=nb&gt;{{harvnb|Lee|2003}} Lemma A.17 (a). If {{math|''f'':''X'' → ''Y''}} is continuous, {{math|''X''}} is compact, then {{math|''f''(''X'')}} is compact.&lt;/ref&gt; contrary to assumption.&lt;ref group=nb&gt;The non-unitarity is a vital ingredient in the proof of the [[w:Coleman–Mandula theorem|''Coleman–Mandula theorem'']], which has the implication that, contrary to in non-relativistic theories, there can exist no ''ordinary'' symmetry relating particles of different spin. See {{harvtxt|Weinberg|2000|loch=Chapter 24.}}&lt;/ref&gt;

In the case of the Lorentz group, this can also be seen directly from the definitions. The representations of {{math|'''A'''}} and {{math|'''B'''}} used in the construction are Hermitian. This means that {{math|'''J'''}} is Hermitian, but {{math|'''K'''}} is [[w:anti-Hermitian|''anti-Hermitian'']].&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Section 5.6, p. 231.}}&lt;/ref&gt; The non-unitarity is not a problem in quantum field theory, since the objects of concern are not required to have a Lorentz-invariant positive definite norm.&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Section 5.6.}}&lt;/ref&gt;

==== Restriction to SO(3) ====
The {{math|(''m'', ''n'')}} representation is, however, unitary when restricted to the rotation subgroup {{math|SO(3)}}, but these representations are not irreducible as representations of SO(3). A [[w:Clebsch–Gordan coefficients|''Clebsch–Gordan decomposition'']] can be applied showing that an {{math|(''m'', ''n'')}} representation have {{math|SO(3)}}-invariant subspaces of highest weight (spin) {{math|''m'' + ''n'', ''m'' + ''n'' − 1, … , &amp;#124;''m'' − ''n''&amp;#124;}},&lt;ref&gt;{{harvnb|Weinberg|2002|p=231.}}&lt;/ref&gt; where each possible highest weight (spin) occurs exactly once. A weight subspace of highest weight (spin) {{math|''j''}} is {{math|(2''j'' + 1)}}-dimensional. So for example, the (&amp;frac12;, &amp;frac12;) representation has spin&amp;nbsp;1 and spin&amp;nbsp;0 subspaces of dimension 3 and 1 respectively.

Since the [[w:angular momentum|''angular momentum'']] operator is given by {{math|'''J''' {{=}} '''A''' + '''B'''}}, the highest spin in quantum mechanics of the rotation sub-representation will be {{math|(''m'' + ''n'')ℏ}} and the &quot;usual&quot; rules of addition of angular momenta and the formalism of [[w:3-j symbol|''3-j symbols'']], [[w:6-j symbol|''6-j symbols'']], etc. applies.&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Sections 2.5, 5.7.}}&lt;/ref&gt;

==== Spinors ====
It is the {{math|SO(3)}}-invariant subspaces of the irreducible representations that determine whether a representation has spin. From the above paragraph, it is seen that the {{math|(''m'', ''n'')}} representation has spin if {{math|''m'' + ''n''}} is half-integral. The simplest are {{math|( &amp;frac12;, 0)}} and {{math|(0,  &amp;frac12;)}}, the Weyl-spinors of dimension {{math|2}}. Then, for example, {{math|(0, &lt;sup&gt;3&lt;/sup&gt;&amp;frasl;&lt;sub&gt;2&lt;/sub&gt;)}} and {{math|(1, &amp;frac12;)}} are a spin representations of dimensions {{math|2&lt;sup&gt;3&lt;/sup&gt;&amp;frasl;&lt;sub&gt;2&lt;/sub&gt; + 1 {{=}} 4}} and {{math|(2 + 1)(2&amp;frac12; + 1) {{=}} 6}} respectively. Note that, according to the above paragraph, there are subspaces with spin both {{math|&lt;sup&gt;3&lt;/sup&gt;&amp;frasl;&lt;sub&gt;2&lt;/sub&gt;}} and {{math|&amp;frac12;}} in the last two cases, so these representations cannot likely represent a ''single'' physical particle which must be well-behaved under {{math|SO(3)}}. It cannot be ruled out in general, however, that representations with multiple {{math|SO(3)}} subrepresentations with different spin can represent physical particles with well-defined spin. It may be that there is a suitable relativistic wave equation that projects out ''unphysical components'', leaving only a single spin.&lt;ref&gt;{{harvnb|Tung|1985|loc=Section 10.5.}}&lt;/ref&gt;

Construction of pure spin {{math|&lt;sup&gt;''n''&lt;/sup&gt;&amp;frasl;&lt;sub&gt;2&lt;/sub&gt;}} representations for any {{math|''n''}} (under {{math|SO(3)}}) from the irreducible representations involves taking tensor products of the Dirac-representation with a non-spin representation, extraction of a suitable subspace, and finally imposing differential constraints.&lt;ref&gt;{{harvnb|Weinberg|2002}} This is outlined (very briefly) on page 232, hardly more than a footnote.&lt;/ref&gt;

==== Dual representations ====
[[File:Root system A1xA1.svg|150px|thumb|right| The [[root system]] {{math|''A''&lt;sub&gt;1&lt;/sub&gt; × ''A''&lt;sub&gt;1&lt;/sub&gt;}} of {{math|'''sl'''(2, '''C''') ⊕ '''sl'''(2, '''C''')}}.]]
To see if the [[w:dual representation|''dual representation'']] of an irreducible representation is [[w:Representation theory#Equivariant maps and isomorphisms|''isomorphic'']] to the original representation one can consider the following theorems:
#The set of [[w:Weight (representation theory)|''weights'']] of the [[w:dual representation|''dual representation'']] of an irreducible representation of a semisimple Lie algebra is, including multiplicities, the negative of the set of weights for the original representation.&lt;ref&gt;{{harvnb|Hall|2003|loc=Proposition 7.39.}}&lt;/ref&gt; 
#Two irreducible representations are isomorphic if and only if they have the same [[w:highest weight|''highest weight'']].&lt;ref group=nb&gt;This is one of the conclusions of [[w:Cartan's theorem|''Cartan's theorem'']], the theorem of the highest weight.{{Paragraph break}}{{harvtxt|Hall|2015|loc=Theorems 9.4&amp;ndash;5.}}&lt;/ref&gt;
#For each semisimple Lie algebra there exists a unique element {{math|w&lt;sub&gt;0&lt;/sub&gt;}} of the [[w:Weyl group|''Weyl group'']] such that if {{math|μ}} is a dominant integral weight, then {{math|w&lt;sub&gt;0&lt;/sub&gt; ⋅ (−μ)}} is again a dominant integral weight.&lt;ref name=&quot;Hall 2003 loc=Theorem 7.40&quot;&gt;{{harvnb|Hall|2003|loc=Theorem 7.40.}}&lt;/ref&gt;
#If {{math|π&lt;sub&gt;μ&lt;sub&gt;0&lt;/sub&gt;&lt;/sub&gt;}} is an irreducible representation with highest weight {{math|μ&lt;sub&gt;0&lt;/sub&gt;}}, then {{math|π*&lt;sub&gt;μ&lt;sub&gt;0&lt;/sub&gt;&lt;/sub&gt;}} has highest weight {{math|w&lt;sub&gt;0&lt;/sub&gt; ⋅ (−μ)}}.&lt;ref name=&quot;Hall 2003 loc=Theorem 7.40&quot;/&gt;
Here, the elements of the Weyl group are considered as orthogonal transformations, acting by matrix multiplication, on the real vector space of [[w:root system|''roots'']]. One sees that if {{math|−''I''}} is an element of the [[w:Weyl group|''Weyl group'']] of a semisimple Lie algebra, then {{math|w&lt;sub&gt;0&lt;/sub&gt; {{=}} −''I''}}. In the case of {{math|'''sl'''(2, '''C''')}}, the Weyl group is {{math|''W'' {{=}} {''I'', −''I''}&lt;nowiki/&gt;}}.&lt;ref&gt;{{harvnb|Hall|2003|loc=Section 6.6.}}&lt;/ref&gt; It follows that each {{math|π&lt;sub&gt;μ&lt;/sub&gt;, μ {{=}} 0, 1, &amp;hellip;}} is isomorphic to its dual {{math|π&lt;sub&gt;μ&lt;/sub&gt;*}}. The root system of {{math|'''sl'''(2, '''C''') ⊕ '''sl'''(2, '''C''')}} is shown in the figure to the right.&lt;ref group=nb&gt;{{harvnb|Hall|2015|loc=Section 8.2}} The root system is the union of two copies of {{math|''A''&lt;sub&gt;1&lt;/sub&gt;}}, where each copy resides in its own dimensions in the embedding vector space.&lt;/ref&gt; The Weyl group is generated by {{math|{w&lt;sub&gt;γ&lt;/sub&gt;}&lt;nowiki/&gt;}} where {{math|w&lt;sub&gt;γ&lt;/sub&gt;}} is reflection in the plane orthogonal to {{math|γ}} as {{math|γ}} ranges over all roots.&lt;ref group=nb&gt;{{harvnb|Rossmann|2002}} This definition is equivalent to the definition in terms of the connected Lie group whose Lie algebra is the Lie algebra of the root system under consideration.&lt;/ref&gt; One sees that {{math|w&lt;sub&gt;α&lt;/sub&gt; ⋅w&lt;sub&gt;β&lt;/sub&gt; {{=}} −''I''}} so {{math|−''I'' ∈ ''W''}}. Then using the fact that if {{math|π, σ}} are Lie algebra representations and {{math|π ≈ σ}}, then {{math|Π ≈ Σ}}.&lt;ref&gt;{{harvnb|Hall|2003|loc=Second item in proposition 4.5.}}&lt;/ref&gt; The conclusion for {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} is
:&lt;math&gt;\pi_{m, n}^{*} \cong \pi_{m, n}, \quad \Pi_{m, n}^{*} \cong \Pi_{m, n}, \quad 2m, 2n \in \mathbb{N}. &lt;/math&gt;

==== Complex conjugate representations ====
If {{math|π}} is a representation of a Lie algebra, then {{math|{{overline|π}}}} is a representation, where the bar denotes entry-wise complex conjugation in the representative matrices. This follows from that complex conjugation commutes with addition and multiplication.&lt;ref&gt;{{harvnb|Hall|2003|p=219.}}&lt;/ref&gt; In general, every irreducible representation {{math|π}} of {{math|'''sl'''(''n'', '''C''')}} can be written uniquely as {{math|π {{=}} π&lt;sup&gt;+&lt;/sup&gt; + π&lt;sup&gt;−&lt;/sup&gt;}}, where&lt;ref&gt;{{harvnb|Rossmann|2002|loc=Exercise 3 in paragraph 6.5.}}&lt;/ref&gt;

:&lt;math&gt;\pi^{\pm}(X) = \frac{1}{2}(\pi(X) \pm i\pi(i^{-1}X)),&lt;/math&gt;

with {{math|π&lt;sup&gt;+&lt;/sup&gt;}} holomorphic (complex linear) and {{math|π&lt;sup&gt;−&lt;/sup&gt;}} '''anti-holomorphic''' (conjugate linear). For {{math|'''sl'''(2, '''C''')}}, since {{math|π&lt;sub&gt;μ&lt;/sub&gt;}} is holomorphic, {{math|{{overline|π}}&lt;sub&gt;μ&lt;/sub&gt;}} is anti-holomorphic. Direct examination of the explicit expressions for {{math|π&lt;sub&gt;μ, 0&lt;/sub&gt;}} and {{math|π&lt;sub&gt;0, ν&lt;/sub&gt;}} in equation {{EquationNote|(S8)}}below shows that they are holomorphic and anti-holomorphic respectively. Closer examination of the expression {{EquationNote|(S8)}}also allows for identification of {{math|π&lt;sup&gt;+&lt;/sup&gt;}} and {{math|π&lt;sup&gt;−&lt;/sup&gt;}} for {{math|π&lt;sub&gt;μ, ν&lt;/sub&gt;}} as {{math|π&lt;sup&gt;+&lt;/sup&gt;&lt;sub&gt;μ, ν&lt;/sub&gt; {{=}} π&lt;sub&gt;μ&lt;/sub&gt;&lt;sup&gt;⊕&lt;sub&gt;ν + 1&lt;/sub&gt;&lt;/sup&gt;}} and {{math|π&lt;sup&gt;−&lt;/sup&gt;&lt;sub&gt;μ, ν&lt;/sub&gt; {{=}} {{overline|π}}&lt;sub&gt;ν&lt;/sub&gt;&lt;sup&gt;⊕&lt;sub&gt;μ + 1&lt;/sub&gt;&lt;/sup&gt;}}.

Using the above identities (interpreted as pointwise addition of functions), for {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} yields

:&lt;math&gt;\overline{\pi_{m, n}} = \overline{\pi_{m, n}^+ + \pi_{m, n}^-} = 
\overline{\pi_m^{\oplus_{2n + 1}}} + \overline{\overline{\pi_n}^{\oplus_{2m + 1}}} = 
\pi_n^{\oplus_{2m + 1}} + \overline{\pi_m}^{\oplus_{2n + 1}} = \pi_{n, m}^+ + \pi_{n, m}^- = 
\pi_{n, m}, \quad \overline{\Pi_{m, n}} = \Pi_{n, m}, \quad 2m, 2n \in \mathbb{N}, &lt;/math&gt;

where the statement for the group representations follow from {{math|exp({{overline|''X''}})}} = {{math|{{overline|exp(''X'')}}}}. It follows that the irreducible representations {{math|(''m'', ''n'')}} have real matrix representatives if and only if {{math|''m'' {{=}} ''n''}}. Reducible representations on the form {{math|(''m'', ''n'') ⊕ (''n'', ''m'')}} have real matrices too.

=== Induced representations on the Clifford algebra and the Dirac spinor representation ===
[[File:Richard Brauer.jpg|upright|thumb|right|[[Richard Brauer]] and wife Ilse 1970. Brauer generalized the [[spin representation]]s of Lie algebras sitting inside [[Clifford algebra]]s to spin higher than {{sfrac|1|2}}.{{Paragraph break}}Photo courtesy of MFO.]]
In general representation theory, if {{math|(''π'', ''V'')}} is a representation of a Lie algebra '''g''', then there is an associated representation of '''g''' on {{math|[[w:Endomorphism|''End'']] ''V''}}, also denoted {{mvar|π}}, given by

{{NumBlk|:|&lt;math&gt;\pi(X)(A) = [\pi(X),A], \quad A\in \operatorname{End}V,\ X\in\mathfrak{g}.&lt;/math&gt;|{{EquationRef|I1}}}}

Likewise, a representation {{math|(Π, ''V'')}} of a group {{mvar|G}} yields a representation {{math|Π}} on {{math|End  ''V''}} of {{mvar|G}}, still denoted {{math|Π}}, given by&lt;ref&gt;{{harvnb|Hall|2003}} See appendix D.3&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\Pi(g)(A) = \Pi(g)A\Pi(g)^{-1}, \quad A\in \operatorname{End}V,\ g\in G.&lt;/math&gt;|{{EquationRef|I2}}}}

Applying this to the Lorentz group, if {{math|(Π, ''V'')}} is a projective representation, then direct calculation using ([[#equation G4|G4]]) shows that the induced representation on {{math|End ''V''}} is, in fact, a proper representation, i.e. a representation without phase factors.

In quantum mechanics this means that if {{math|(''π'', ''H'')}} or {{math|(Π, ''H'')}} is a representation acting on some Hilbert space {{math|''H''}}, then the corresponding induced representation acts on the set of linear operators on {{math|''H''}}. As an example, the induced representation of the projective spin {{math|(&amp;frac12;, 0) ⊕ (0, &amp;frac12;)}} representation on {{math|End(''H'')}} is the non-projective 4-vector ({&amp;frac12;, &amp;frac12;) representation.&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Equation 5.4.8.}}&lt;/ref&gt;

For simplicity, consider now only the &quot;discrete part&quot; of {{math|End ''H''}}, that is, given a basis for {{math|''H''}}, the set of constant matrices of various dimension, including possibly infinite dimensions. A general element of the full {{math|End ''H''}} is the sum of tensor products of a matrix from the simplified {{math|End ''H''}} and an operator from the left out part. The left out part consists of functions of spacetime, differential and integral operators and the like. See [[w:Dirac operator|''Dirac operator'']] for an illustrative example. Also left out are operators corresponding to other degrees of freedom not related to spacetime, such as gauge degrees of freedom in gauge theories.

The induced 4-vector representation of above on this simplified {{math|End ''H''}} has an invariant 4-dimensional subspace that is spanned by the four [[w:gamma matrices|''gamma matrices'']].&lt;ref name=&quot;Weinberg 2002 loc=Section 5.4&quot;&gt;{{harvnb|Weinberg|2002|loc=Section 5.4.}}&lt;/ref&gt; (Note the different metric convention in the linked article.) In a corresponding way, the complete Clifford [[w:spacetime algebra|''algebra of spacetime'']], {{math|Cℓ&lt;sub&gt;3,1&lt;/sub&gt;('''R''')}}, whose complexification is {{math|M&lt;sub&gt;4&lt;/sub&gt;('''C''')}}, generated by the gamma matrices decomposes as a direct sum of [[w:representation space|''representation spaces'']] of a '''scalar''' irreducible representation (irrep), the {{math|(0, 0)}}, a '''pseudoscalar''' irrep, also the {{math|(0, 0)}}, but with parity inversion eigenvalue −1, see the [[#The full Lorentz group|next section]] below, the already mentioned '''vector''' irrep, {{math|(&amp;frac12;, &amp;frac12;)}}, a '''pseudovector''' irrep, {{math|(&amp;frac12;, &amp;frac12;)}} with parity inversion eigenvalue +1 (not −1), and a '''tensor''' irrep, {{math|(1, 0) ⊕ (0, 1)}}.&lt;ref&gt;{{harvnb|Weinberg|2002|pp=215&amp;ndash;216.}}&lt;/ref&gt; The dimensions add up to {{math|1=1 + 1 + 4 + 4 + 6 = 16}}. In other words,

{{NumBlk|:|&lt;math&gt;Cl_{3,1}(\mathbb{R}) = (0,0) \oplus (\frac{1}{2}, \frac{1}{2}) \oplus [(1, 0) \oplus (0, 1)] \oplus (\frac{1}{2}, \frac{1}{2})_p \oplus (0, 0)_p,&lt;/math&gt;|{{EquationRef|I3}}}}

where, as is [[w:Abuse of notation|''customary'']], a representation is confused with its representation space. This is, in fact, a reasonably convenient way to show that the algebra spanned by the gammas is 16-dimensional.&lt;ref name=&quot;ReferenceA&quot;&gt;{{harvnb|Weinberg|2002}} Section 5.4.&lt;/ref&gt;

==== The {{math|(&amp;frac12;, 0) ⊕ (0, &amp;frac12;)}} spin representation ====

The six-dimensional representation space of the tensor {{math|(1, 0) ⊕ (0, 1)}}-representation inside {{math|Cℓ&lt;sub&gt;3,1&lt;/sub&gt;('''R''')}} has two roles. In particular, letting&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Equation 5.4.6.}}&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\sigma^{\mu\nu} = -\frac{i}{4}[\gamma^\mu, \gamma^\nu],&lt;/math&gt;|{{EquationRef|I4}}}}

where {{math|{γ&lt;sup&gt;μ&lt;/sup&gt; ∈ Cℓ&lt;sub&gt;3,1&lt;/sub&gt;('''R'''): μ {{=}} 0,1,2,3} }} are the gamma matrices, the {{math|{σ&lt;sup&gt;μν&lt;/sup&gt; ∈ Cℓ&lt;sub&gt;3,1&lt;/sub&gt;('''R''')} }}, only 6 of which are non-zero due to antisymmetry of the bracket, span the tensor representation space. Moreover, they have the commutation relations of the Lorentz Lie algebra,&lt;ref name=&quot;ReferenceA&quot;/&gt;

{{NumBlk|:|&lt;math&gt;[\sigma^{\mu\nu},\sigma^{\rho\tau}] = i(\eta^{\tau\mu}\sigma^{\rho\nu} + \eta^{\nu\tau}\sigma^{\mu\rho} - \eta^{\rho\mu}\sigma^{\tau\nu} -\eta^{\nu\rho}\sigma^{\mu\tau}),&lt;/math&gt;|{{EquationRef|I5}}}}

and hence constitute a representation (in addition to being a representation space) sitting inside {{math|Cℓ&lt;sub&gt;3,1&lt;/sub&gt;('''R''')}}, the {{math|(&amp;frac12;, 0) ⊕ (0, &amp;frac12;)}} spin representation. For details, see [[w:bispinor|''bispinor'']] and [[w:Dirac algebra|''Dirac algebra'']].

The conclusion is that every element of the complexified {{math|Cℓ&lt;sub&gt;3,1&lt;/sub&gt;('''R''')}} in {{math|End ''H''}} (i.e. every complex {{gaps|4|×|4}} matrix) has well defined Lorentz transformation properties. In addition, it has a spin-representation of the Lorentz Lie algebra, which upon exponentiation becomes a spin representation of the group, acting on {{math|'''C'''&lt;sup&gt;4&lt;/sup&gt;}}, making it a space of bispinors.

=== Reducible representations ===
There is a multitude of other representations that can be deduced from the irreducible ones, such as those obtained in a standard manner by taking direct sums, tensor products, and quotients of the irreducible representations. Other methods of obtaining representations include the restriction of a representation of a larger group containing the Lorentz group, e.g. {{math|GL(''n'', ℝ)}}. These representations are in general not irreducible, and are not discussed here. It is to be noted though that the Lorenz group and its Lie algebra have the ''complete reducibility property''. This means that every representation reduces to a direct sum of irreducible representations.

=== Space inversion and time reversal ===

The (possibly projective) {{math|(''m'', ''n'')}} representation is irreducible as a representation {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}, the identity component of the Lorentz group, in physics terminology the [[w:orthochronous|''proper orthochronous'']] Lorentz group. If {{math|1=''m'' = ''n''}} it can be extended to a representation of all of {{math|O(3; 1)}}, the full Lorentz group, including [[w:P-symmetry|''space parity inversion'']] and [[w:T-symmetry|''time reversal'']]. The representations {{math|(''m'', ''n'') ⊕ (''n'', ''m'')}} can be extended likewise.&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Section 5.7, pp. 232&amp;ndash;233.}}&lt;/ref&gt;

====Space parity inversion====
For space parity inversion, one considers the [[w:adjoint representation|''adjoint action'']] {{math|Ad&lt;sub&gt;''P''&lt;/sub&gt;}} of {{math|''P'' ∈ SO(3; 1)}} on {{math|'''so'''(3; 1)}}, where {{math|''P''}} is the standard representative of space parity inversion, {{math|1=''P'' = diag(1, −1, −1, −1)}}, given by

{{NumBlk|:|&lt;math&gt;\mathrm{Ad}_P(J_i) = PJ_iP^{-1} = J_i, \qquad \mathrm{Ad}_P(K_i) = PK_iP^{-1} = -K_i.&lt;/math&gt;|{{EquationRef|F1}}}}

It is these properties of {{math|'''K'''}} and {{math|'''J'''}} under {{mvar|P}} that motivate the terms ''vector'' for {{math|'''K'''}} and [[w:pseudovector|''pseudovector'']] or ''axial vector'' for {{math|'''J'''}}. In a similar way, if {{math|π}} is any representation of {{math|'''so'''(3; 1)}} and {{math|Π}} is its associated group representation, then {{math|Π(SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;)}} acts on the representation of {{math|π}} by the adjoint action, {{math|π(''X'') ↦ Π(''g'') π(''X'') Π(''g'')&lt;sup&gt;−1&lt;/sup&gt;}} for {{math|''X'' ∈ '''so'''(3; 1)}}, {{math|g ∈ SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}. If {{math|''P''}} is to be included in {{math|Π}}, then consistency with {{EquationNote|F1|(F1)}}requires that

{{NumBlk|:|&lt;math&gt;\Pi(P)\pi(B_i)\Pi(P)^{-1} = \pi(A_i)&lt;/math&gt;|{{EquationRef|F2}}}}

holds, where {{math|'''A'''}} and {{math|'''B'''}} are defined as in the first section. This can hold only if {{math|''A''&lt;sub&gt;''i''&lt;/sub&gt;}} and {{math|''B''&lt;sub&gt;''i''&lt;/sub&gt;}} have the same dimensions, i.e. only if {{math|1=''m'' = ''n''}}. When {{math|''m'' ≠ ''n''}} then {{math|(''m'', ''n'') ⊕ (''n'', ''m'')}} can be extended to an irreducible representation of {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}, the orthocronous Lorentz group. The parity reversal representative {{math|Π(''P'')}} does not come automatically with the general construction of the {{math|(''m'', ''n'')}} representations. It must be specified separately. The matrix {{math|1=[[w:Gamma matrices|''β'' = ''i'' ''γ''&lt;sup&gt;0&lt;/sup&gt;]]}} (or a multiple of modulus −1 times it) may be used in the {{math|(&amp;frac12;, 0) ⊕ (0, &amp;frac12;)}}&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Section 5.7, p. 233.}}&lt;/ref&gt; representation.

If parity is included with a minus sign (the {{math|1×1}} matrix {{math|[−1]}}) in the {{math|(0,0)}} representation, it is called a [[w:pseudoscalar|''pseudoscalar'']] representation.

==== Time reversal ====
[[w:T-symmetry|''Time reversal'']] {{math|1=''T'' = diag(−1, 1, 1, 1)}}, acts similarly on {{math|'''so'''(3; 1)}} by&lt;ref&gt;{{harvnb|Weinberg|2002}} Equation 2.6.5.&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\mathrm{Ad}_T(J_i) = TJ_iT^{-1} = -J_i,  \qquad \mathrm{Ad}_T(K_i) = TK_iT^{-1} = K_i.&lt;/math&gt;|{{EquationRef|F3}}}}

By explicitly including a representative for {{math|''T''}}, as well as one for {{math|''P''}}, one obtains a representation of the full Lorentz group {{math|SO(3; 1)}}. A subtle problem appears however in application to physics, in particular quantum mechanics. When considering the full [[w:Representation theory of the Poincaré group|''Poincaré group'']], four more generators, the {{math|''P''&lt;sup&gt;μ&lt;/sup&gt;}}, in addition to the {{math|''J''&lt;sup&gt;''i''&lt;/sup&gt;}} and {{math|''K''&lt;sup&gt;''i''&lt;/sup&gt;}} generate the group. These are interpreted as generators of translations. The time-component {{math|''P''&lt;sup&gt;0&lt;/sup&gt;}} is the Hamiltonian {{math|''H''}}. The operator {{math|''T''}} satisfies the relation&lt;ref&gt;{{harvnb|Weinberg|2002}} Equation following 2.6.6.&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\mathrm{Ad}_{T}(iH) = TiHT^{-1} = -iH&lt;/math&gt;|{{EquationRef|F4}}}}

in analogy to the relations above with {{math|'''so'''(3; 1)}} replaced by the full [[w:Representation theory of the Poincare group|''Poincaré algebra'']]. By just cancelling the {{math|''i''}}'s, the result {{math|1=''THT''&lt;sup&gt;−1&lt;/sup&gt; = −''H''}} would imply that for every state {{math|Ψ}} with positive energy {{math|''E''}} in a Hilbert space of quantum states with time-reversal invariance, there would be a state {{math|Π(''T''&lt;sup&gt;−1&lt;/sup&gt;)Ψ}} with negative energy {{math|−''E''}}. Such states do not exist. The operator {{math|Π(''T'')}} is therefore chosen [[w:antilinear|''antilinear'']] and [[w:antiunitary|''antiunitary'']], so that it [[w:anticommutator|''anticommutes'']] with {{math|''i''}}, resulting in {{math|1=''THT''&lt;sup&gt;−1&lt;/sup&gt;}} = {{math|+''H''}}, and its action on Hilbert space likewise becomes antilinear and antiunitary.&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Section 2.6.}}&lt;/ref&gt; It may be expressed as the composition of [[w:complex conjugation|''complex conjugation'']] with multiplication by a unitary matrix.&lt;ref&gt;For a detailed discussion of the spin 0, {{sfrac|1|2}} and 1 cases, see {{harvnb|Greiner|Reinhardt|1996}}.&lt;/ref&gt; This is mathematically sound, see [[w:Wigner's theorem|''Wigner's theorem'']], but if one is very strict with terminology, {{math|Π}} is not a ''representation''.

When constructing theories such as [[w:quantum electrodynamics|''QED'']] which is invariant under space parity and time reversal, Dirac spinors may be used, while theories that do not, such as the [[w:electroweak force|''electroweak force'']], must be formulated in terms of Weyl spinors. The Dirac representation, {{nowrap|(&amp;frac12;, 0) ⊕ (0, &amp;frac12;)}}, is usually taken to include both space parity and time inversions. Without space parity inversion, it is not an irreducible representation.

The third discrete symmetry entering in the [[w:CPT theorem|''CPT theorem'']] along with {{math|''P''}} and {{math|''T''}}, [[w:charge conjugation symmetry|''charge conjugation symmetry'']] {{math|''C''}}, has nothing directly to do with Lorentz invariance.&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Chapter 3.}}&lt;/ref&gt;

== Action on function spaces ==
In the classification of the irreducible finite-dimensional representations of above it was never specified precisely ''how'' a representative of a group or Lie algebra element acts on vectors in the representation space. The action can be anything as long as it is linear. The point silently adopted was that after a choice of basis in the representation space, everything becomes matrices anyway.

If {{mvar|V}} is a vector space of functions of a finite number of variables {{mvar|n}}, then the action on a scalar function {{math|''f'' ∈ ''V''}} given by
{{NumBlk|:|&lt;math&gt;(\Pi(g)f)(x) = f(\Pi_x(g)^{-1}x),\qquad x \in \mathbb{R}^n, f \in V&lt;/math&gt;|{{EquationRef|H1}}}}
produces another function {{math|Π''f'' ∈ ''V''}}. Here {{math|Π&lt;sub&gt;''x''&lt;/sub&gt;}} is an {{mvar|n}}-dimensional representation, and {{math|Π}} is a possibly infinite-dimensional representation. A special case of this construction is when {{mvar|V}} is a space of functions defined on the group {{mvar|G}} itself, viewed as a {{mvar|n}}-dimensional [[w:manifold|''manifold'']] embedded in {{math|'''R'''&lt;sup&gt;''n''&lt;/sup&gt;}}.&lt;ref&gt;{{harvnb|Rossmann|2002}} See section 6.1 for more examples, both finite-dimensional and infinite-dimensional.&lt;/ref&gt; This is the setting in which the [[w:Peter–Weyl theorem|''Peter–Weyl theorem'']] and the [[w:Borel–Weil theorem|''Borel–Weil theorem'']] are formulated. The former demonstrates the existence of a Fourier decomposition of functions on a compact group into [[w:character (mathematics)#Character of a representation|''characters'']] of finite-dimensional representations.&lt;ref name=&quot;ReferenceC&quot;/&gt; The completeness of the characters in this sense can thus be used to prove the existence of the highest weight representations.&lt;ref&gt;{{harvnb|Hall|2015|loc=Theorem 12.6.}}&lt;/ref&gt; The latter theorem, providing more explicit representations, makes use of the [[w:unitarian trick|''unitarian trick'']] to yield representations of complex non-compact groups, e.g. {{math|SL(2, '''C''')}}; in the present case, there is a one-to-one correspondence between representations of {{math|SU(2)}} and [[w:Analyticity of holomorphic functions|''holomorphic'']] representations of {{math|SL(2, '''C''')}}. (A group representation is called holomorphic if its corresponding Lie algebra representation is complex linear.) This theorem too can be used to demonstrate the existence of the highest weight representations.&lt;ref&gt;{{harvnb|Hall|2003|loc=Chapter 7.}}&lt;/ref&gt;

=== Euclidean rotations ===
::Main articles: [[w:Rotation group SO(3)|''Rotation group SO(3)'']], [[w:Spherical harmonics|''Spherical harmonics'']]
The subgroup {{math|SO(3)}} of three-dimensional Euclidean rotations has an infinite-dimensional representation on the Hilbert space {{math|1=''L''&lt;sup&gt;2&lt;/sup&gt;('''S'''&lt;sup&gt;2&lt;/sup&gt;) = span{''Y''&lt;sup&gt;''ℓ''&lt;/sup&gt;&lt;sub&gt;''m''&lt;/sub&gt;, ''ℓ'' ∈ '''N'''&lt;sup&gt;+&lt;/sup&gt;, −''ℓ'' ≤ ''m'' ≤ ''ℓ'' }}}, where the {{math|''Y''&lt;sup&gt;''ℓ''&lt;/sup&gt;&lt;sub&gt;''m''&lt;/sub&gt;}} are [[w:spherical harmonics|''spherical harmonics'']]. Its elements are square integrable complex-valued functions&lt;ref group=nb&gt;The elements of {{math|''L''&lt;sup&gt;2&lt;/sup&gt;('''S'''&lt;sup&gt;2&lt;/sup&gt;)}} are actually equivalence classes of functions. two functions are declared equivalent if they differ merely on a set of [[w:measure zero|''measure zero'']]. The integral is the Lebesgue integral in order to obtain a ''complete'' inner product space.&lt;/ref&gt; on the sphere. The inner product on this space is given by
{{NumBlk|:|&lt;math&gt;\langle f,g\rangle = \int_{\mathbb{S}^2}\overline{f}gd\Omega = \int_0^{2\pi}\int_0^{\pi}\overline{f}g \sin\theta d\theta d\varphi.&lt;/math&gt;|{{EquationRef|H1}}}}

If {{mvar|f}} is an arbitrary square integrable function defined on the unit sphere {{math|'''S'''&lt;sup&gt;2&lt;/sup&gt;}}, then it can be expressed as&lt;ref name=&quot;Gelfand_M_S&quot;&gt;{{harvnb|Gelfand|Minlos|Shapiro|1963}}&lt;/ref&gt;
{{NumBlk|:|&lt;math&gt;|f\rangle = \sum_{l = 1}^\infty\sum_{m = -l}^{m = l} |Y_m^l\rangle\langle Y_m^l|f\rangle, \quad f(\theta, \varphi) = \sum_{l = 1}^\infty\sum_{m = -l}^{m = l}f_{lm}Y^l_m(\theta, \varphi),&lt;/math&gt;|{{EquationRef|H2}}}}
where the expansion coefficients are given by

{{NumBlk|:|&lt;math&gt;f_{lm} = \langle Y_m^l, f \rangle = \int_{\mathbb{S}^2}\overline{{Y^l_m}}fd\Omega = \int_0^{2\pi}\int_0^\pi \overline{{Y_m^l}}(\theta, \varphi)f(\theta, \varphi)\sin \theta d\theta d\varphi.&lt;/math&gt;|{{EquationRef|H3}}}}

The Lorentz group action restricts to that of {{math|SO(3)}} and is expressed as

{{NumBlk|:|&lt;math&gt;(\Pi(R)f)(\theta(x), \varphi(x)) = \sum_{l = 1}^\infty\sum_{m = -l}^{m = l}\sum_{m' = -l}^{m' = l}D^{(l)}_{mm'}(R)f_{lm'}Y^l_m(\theta(R^{-1}x), \varphi(R^{-1}x)), \qquad R \in \mathrm{SO}(3), \quad x \in \mathbb{S}^2.&lt;/math&gt;|{{EquationRef|H4}}}}

This action is unitary, meaning that
{{NumBlk|:|&lt;math&gt;\langle \Pi(R)f,\Pi(R)g\rangle = \langle f,g\rangle \qquad \forall f,g \in \mathbb{S}^2, \quad\forall R \in \mathrm{SO}(3).&lt;/math&gt;|{{EquationRef|H5}}}}
The {{math|''D''&lt;sup&gt;(''ℓ'')&lt;/sup&gt;}} can be obtained from the {{math|''D''&lt;sup&gt;(''m'', ''n'')&lt;/sup&gt;}} of above using [[w:Clebsch–Gordan coefficients|''Clebsch–Gordan decomposition'']], but they are more easily directly expressed as an exponential of an odd-dimensional {{math|'''su'''(2)}}-representation (the 3-dimensional one is exactly {{math|'''so'''(3)}}).&lt;ref&gt;In ''Quantum Mechanics - non-relativistic theory'' by [[w:Course of Theoretical Physics|''Landau and Lifshitz'']] the lowest order {{math|''D''}} are calculated analytically.&lt;/ref&gt;&lt;ref&gt;{{harvnb|Curtright|Fairlie|Zachos|2014}} A formula for {{math|''D''&lt;sup&gt;(''ℓ'')&lt;/sup&gt;}} valid for all ''ℓ'' is given.&lt;/ref&gt; In this case the space {{math|''L''&lt;sup&gt;2&lt;/sup&gt;('''S'''&lt;sup&gt;2&lt;/sup&gt;)}} decomposes neatly into an infinite direct sum of irreducible odd finite-dimensional representations {{math|''V''&lt;sub&gt;2''i'' + 1&lt;/sub&gt;, ''i'' {{=}} 0, 1, &amp;hellip;}} according to&lt;ref&gt;{{harvnb|Hall|2003}} Section 4.3.5.&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;L^2(\mathbb{S}^2) = \sum_{i = 0}^{\infty} V_{2i + 1} \equiv \bigoplus_{i=0}^\infty \operatorname{span}\{Y_m^{2i+1}\}.&lt;/math&gt;|{{EquationRef|H6}}}}

This is characteristic of infinite-dimensional unitary representations of {{math|SO(3)}}. If {{mvar|Π}} is an infinite-dimensional unitary representation on a [[w:separable space|''separable'']]&lt;ref group=nb&gt;A Hilbert space is separable if and only if it has a countable basis. All separable Hilbert spaces are isomorphic.&lt;/ref&gt; Hilbert space, then it decomposes as a direct sum of finite-dimensional unitary representations.&lt;ref name=Gelfand_M_S/&gt; Such a representation is thus never irreducible. All irreducible finite-dimensional representations {{math|(Π, ''V'')}} can be made unitary by an appropriate choice of inner product,&lt;ref name=Gelfand_M_S/&gt;
:&lt;math&gt;\langle f, g\rangle_U \equiv \int_{\mathrm{SO}(3)}\langle \Pi(R)f, \Pi(R)g\rangle dg = \frac{1}{8\pi^2}\int_0^{2\pi}\int_0^{\pi}\int_0^{2\pi} \langle \Pi(R)f, \Pi(R)g\rangle \sin \theta d\varphi d\theta d\psi, \quad f,g \in V,&lt;/math&gt;
where the integral is the unique invariant integral over {{math|SO(3)}} normalized to {{math|1}}, here expressed using the [[w:Euler angles|''Euler angles'']] parametrization. The inner product inside the integral is any inner product on {{math|''V''}}.

=== The Möbius group ===
::Main article: [[w:Möbius transformation|''Möbius transformation'']], [[w:Lorentz group#Relation to the Möbius group|''Lorentz group#Relation to the Möbius group'']]
The identity component of the Lorentz group is isomorphic to the [[w:Möbius transformation|''Möbius group'']] {{math|M}}. This group can be thought of as [[w:conformal mapping|''conformal mappings'']] of either the [[w:complex plane|''complex plane'']] or, via [[w:stereographic projection|''stereographic projection'']], the [[w:Riemann sphere|''Riemann sphere'']]. In this way, the Lorentz group itself can be thought of as acting conformally on the complex plane or on the Riemann sphere.

In the plane, a Möbius transformation characterized by the complex numbers {{math|''a'', ''b'', ''c'', ''d''}} acts on the plane according to&lt;ref&gt;{{harvnb|Churchill|Brown|2014|loc=Chapter 8 pp. 307-310.}}&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;f(z) = \frac{a z + b}{c z + d}, \qquad ad - bc \neq 0&lt;/math&gt;.|{{EquationRef|M1}}}}

and can be represented by complex matrices

{{NumBlk|:|&lt;math&gt;\Pi_f = \begin{pmatrix} A &amp; B \\ C &amp; D \end{pmatrix} = \lambda \begin{pmatrix} a &amp; b \\ c &amp; d \end{pmatrix}, \qquad \lambda \in \mathbb C-\{0\},\, \operatorname{det} \Pi_f = 1,&lt;/math&gt;|{{EquationRef|M2}}}}

since multiplication by a nonzero complex scalar does not change {{mvar|f}}. These are elements of {{math|SL(2,  ℂ)}} and are unique up to a sign (since {{math|±Π&lt;sub&gt;''f''&lt;/sub&gt;}} give the same {{mvar|f}}), hence {{math|M ≈ SL(2,  ℂ)/{I,  −I} ≈ SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}}.

=== The Riemann P-functions ===
::Main article: [[w:Riemann's differential equation|''Riemann's differential equation'']]
The [[w:Riemann's differential equation|''Riemann P-functions'']], solutions of Riemann's differential equation, are an example of a set of functions that transform among themselves under the action of the Lorentz group. The Riemann P-functions are expressed as&lt;ref&gt;{{cite journal|last1=Gonzalez|first1=P. A.|last2=Vasquez|first2=Y.|title=Dirac Quasinormal Modes of New Type Black Holes in New Massive Gravity|year=2014|journal= Eur. Phys. J. C |publisher=Springer|location=Berlin·Heidelberg|issn=1434-6044|volume=74:2969|arxiv=1404.5371v2|doi=10.1140/epjc/s10052-014-2969-1|url=http://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2969-1|page=3}}&lt;/ref&gt;

{{NumBlk|
:|&lt;math&gt;w(z)=P  \left\{ \begin{matrix} a &amp; b &amp; c &amp; \; \\ 
\alpha &amp; \beta &amp; \gamma &amp; z \\
\alpha' &amp; \beta' &amp; \gamma' &amp; \;
\end{matrix} \right\} = \left(\frac{z-a}{z-b}\right)^\alpha 
\left(\frac{z-c}{z-b}\right)^\gamma
P \left\{ \begin{matrix} 0 &amp; \infty &amp; 1 &amp; \; \\ 
0 &amp; \alpha+\beta+\gamma &amp; 0 &amp; \;\frac{(z-a)(c-b)}{(z-b)(c-a)} \\
\alpha'-\alpha &amp; \alpha+\beta'+\gamma &amp; \gamma'-\gamma &amp; \;
\end{matrix} \right\},
&lt;/math&gt;|{{EquationRef|T1}}}}

where the {{math|''a'',  ''b'',  ''c'',  ''α'',  ''β'',  ''γ'',  ''α&amp;prime;'',  ''β&amp;prime;'',  ''γ&amp;prime;''}} are complex constants. The P-function on the right hand side can be expressed using standard [[w:hypergeometric functions|''hypergeometric functions'']]. The connection is&lt;ref&gt;{{harvnb|Abramowitz|Stegun|1965|loc=Equation 15.6.5.}}&lt;/ref&gt;

{{NumBlk|
:|&lt;math&gt; P  \left\{ \begin{matrix} 0 &amp; \infty &amp; 1 &amp; \; \\ 
0 &amp; a &amp; 0 &amp; z \\
1-c &amp; b &amp; c-a-b &amp; \;
\end{matrix} \right\} = \;_2F_1(a,b;c;z).&lt;/math&gt;|{{EquationRef|T2}}}}

The set of constants {{math|0, ∞, 1}} in the upper row on the left hand side are the [[w:regular singular point|''regular singular points'']] of the [[w:Hypergeometric function#The hypergeometric differential equation|''Gauss' hypergeometric equation'']].&lt;ref&gt;{{harvnb|Simmons|1972|loc=Sections 30, 31.}}&lt;/ref&gt; Its ''exponents'', i. e. solutions of the [[w:indicial equation|''indicial equation'']], for expansion around the singular point {{math|0}} are {{math|0}} and {{math|1 − ''c''}} ,corresponding to the two linearly independent solutions,&lt;ref group = nb&gt;See {{harvtxt|Simmons|1972|loc=Section 30.}} for precise conditions under which two [[w:Frobenius method|''Frobenius method'']] yields two linearly independent solutions. If the exponents do not differ by an integer, this is always the case.&lt;/ref&gt; and for expansion around the singular point {{math|1}} they are {{math|0}} and {{math|''c'' − ''a'' − ''b''}}.&lt;ref&gt;{{harvnb|Simmons|1972|loc=Sections 30.}}&lt;/ref&gt; Similarly, the exponents for {{math|∞}} are {{mvar|a}} and {{mvar|b}} for the two solutions.&lt;ref&gt;{{harvnb|Simmons|1972|loc=Section 31.}}&lt;/ref&gt;

One has thus

{{NumBlk|
:|&lt;math&gt;w(z)=
\left(\frac{z-a}{z-b}\right)^\alpha 
\left(\frac{z-c}{z-b}\right)^\gamma
\;_2F_1 \left(
\alpha+\beta +\gamma, 
\alpha+\beta'+\gamma; 
1+\alpha-\alpha';
\frac{(z-a)(c-b)}{(z-b)(c-a)} \right) 
,&lt;/math&gt;|{{EquationRef|T3}}}}

where the condition (sometimes called Riemann's identity)&lt;ref&gt;{{harvnb|Simmons|1972|loc=Equation 11 in appendix E, chapter 5.}}&lt;/ref&gt;

:&lt;math&gt;\alpha+\alpha'+\beta+\beta'+\gamma+\gamma'=1&lt;/math&gt;

on the exponents of the solutions of Riemann's differential equation has been used to define {{math|''γ''&amp;prime;}}.

The first set of constants on the left hand side in {{EquationNote|(T1)}}, {{math|''a'', ''b'', ''c''}} denotes the regular singular points of Riemann's differential equation. The second set, {{math|''α'', ''β'', ''γ''}}, are the corresponding exponents at {{math|''a'', ''b'', ''c''}} for one of the two linearly independent solutions, and, accordingly, {{math|''α&amp;prime;'', ''β&amp;prime;'', ''γ&amp;prime;''}} are exponents at {{math|''a'', ''b'', ''c''}} for the second solution.

Define an action of the Lorentz group on the set of all Riemann P-functions by first setting

{{NumBlk|:|&lt;math&gt;u(\Lambda)(z)=\frac{Az+B}{Cz+D},&lt;/math&gt;|{{EquationRef|T4}}}}

where {{math|''A'',  ''B'',  ''C'',  ''D''}} are the entries in

{{NumBlk|:|&lt;math&gt;\lambda = \begin{pmatrix} A &amp; B \\ C &amp; D \end{pmatrix} \in SL(2, \mathbb C),&lt;/math&gt;|{{EquationRef|T5}}}}

for {{math|Λ {{=}} ''p''(''λ'') ∈ SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} a Lorentz transformation.

Define

{{NumBlk|:|&lt;math&gt;[\Pi(\Lambda) P](z) = P[u(\Lambda)(z)],&lt;/math&gt;|{{EquationRef|T6}}}}

where {{mvar|P}} is a Riemann P-function. The resulting function is again a Riemann P-function. The effect of the Mobius transformation of the argument is that of shifting the [[w:pole (complex analysis)|''poles'']] to new locations, hence changing the critical points, but there is no change in the exponents of the differential equation the new function satisfies. The new function is expressed as

{{NumBlk|
:|&lt;math&gt;[\Pi(\Lambda) P](u) = P \left\{ \begin{matrix} 
\eta &amp; \zeta &amp; \theta &amp; \; \\ 
\alpha &amp; \beta &amp; \gamma &amp; u \\
\alpha' &amp; \beta' &amp; \gamma' &amp; \;
\end{matrix} \right\},&lt;/math&gt;
|{{EquationRef|T6}}}}

where

{{NumBlk|
:|&lt;math&gt;\eta=\frac{Aa+B}{Ca+D}
\quad \text{ and } \quad
\zeta=\frac{Ab+B}{Cb+D}
\quad \text{ and } \quad
\theta=\frac{Ac+B}{Cc+D}.&lt;/math&gt;
|{{EquationRef|T7}}}}

==Infinite-dimensional unitary representations==&lt;!--caution: internal #-link --&gt;

===History===
The Lorentz group {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} and its double cover {{math|SL(2, '''C''')}} also have infinite dimensional unitary representations, studied independently by {{harvtxt|Bargmann|1947}}, {{harvtxt|Gelfand|Naimark|1947}} and {{harvtxt|Harish-Chandra|1947}} at the instigation of [[w:Paul Dirac|''Paul Dirac'']]. This trail of development begun with {{harvtxt|Dirac|1936}} where he devised matrices {{math|'''U'''}} and {{math|'''B'''}} necessary for description of higher spin (compare [[w:Dirac matrices|''Dirac matrices'']]), elaborated upon by {{harvtxt|Fierz|1939}}, see also {{harvtxt|Fierz|Pauli|1939}}, and proposed precursors of the [[w:Bargmann-Wigner equations|''Bargmann-Wigner equations'']]. In {{harvtxt|Dirac|1945}} he proposed a concrete infinite-dimensional representation space whose elements were called '''expansors''' as a generalization of tensors. These ideas were incorporated by Harish&amp;ndash;Chandra and expanded with '''expinors''' as an infinite-dimensional generalization of spinors in his 1947 paper.

The [[w:Plancherel formula|''Plancherel formula'']] for these groups was first obtained by Gelfand and Naimark through involved calculations. The treatment was subsequently considerably simplified by {{harvtxt|Harish-Chandra|1951}} and {{harvtxt|Gelfand|Graev|1953}}, based on an analogue for {{math|SL(2, '''C''')}} of the integration formula of [[w:Hermann Weyl|''Hermann Weyl'']] for [[w:compact Lie group|''compact Lie groups'']]. Elementary accounts of this approach can be found in {{harvtxt|Rühl|1970}} and {{harvtxt|Knapp|2001}}.

The theory of [[w:zonal spherical function|''spherical functions'']] for the Lorentz group, required for [[w:harmonic analysis|''harmonic analysis'']] on the 3-dimensional [[w:hyperboloid#Relation to the sphere|''unit quasi-sphere'']] in Minkowski space, or equivalently 3-dimensional [[w:hyperbolic space|''hyperbolic space'']], is considerably easier than the general theory. It only involves representations from the spherical [[w:principal series|''principal series'']] and can be treated directly, because in radial coordinates the [[w:Laplacian|''Laplacian'']] on the hyperboloid is equivalent to the Laplacian on {{math|'''R'''}}. This theory is discussed in {{harvtxt|Takahashi|1963}}, {{harvtxt|Helgason|1968}}, {{harvtxt|Helgason|2000}} and the posthumous text of {{harvtxt|Jorgenson|Lang|2008}}.

===Principal series===

The '''principal series''', or '''unitary principal series''', are the unitary representations [[w:induced representation|''induced'']] from the one-dimensional representations of the lower triangular subgroup&amp;nbsp;{{mvar|B}} of {{math|1=''G'' = SL(2, '''C''')}}. Since the one-dimensional representations of {{mvar|B}} correspond to the representations of the diagonal matrices, with non-zero complex entries {{mvar|z}} and {{math|''z''&lt;sup&gt;−1&lt;/sup&gt;}}, they thus have the form

:&lt;math&gt;\chi_{\nu,k}\begin{pmatrix}z&amp; 0\\ c&amp; z^{-1}\end{pmatrix}=r^{i\nu} e^{ik\theta},&lt;/math&gt;

for {{mvar|k}} an integer, {{mvar|ν}} real and with {{mvar|1=''z'' = ''re''&lt;sup&gt;''iθ''&lt;/sup&gt;}}. The representations are [[w:irreducible representation|''irreducible'']]; the only repetitions, i.e. isomorphisms of representations, occur when {{mvar|k}} is replaced by {{math|−''k''}}.  By definition the representations are realized on L&lt;sup&gt;2&lt;/sup&gt; sections of [[w:line bundle|''line bundles'']] on {{math|1=''G''/''B'' = S&lt;sup&gt;2&lt;/sup&gt;}}, which is isomorphic to the [[w:Riemann sphere|''Riemann sphere'']]. When {{math|1=''k'' = 0}}, these representations constitute the so-called '''spherical principal series'''.

The restriction of a principal series to the maximal compact subgroup {{math|1=''K'' = SU(2)}} of&amp;nbsp;{{mvar|G}} can also be realized as an induced representation of&amp;nbsp;{{mvar|K}} using the identification {{math|1=''G'' / ''B'' = ''K'' / ''T''}}, where {{math|1=''T'' = ''B'' ∩ ''K''}} is the [[w:maximal torus|''maximal torus'']] in&amp;nbsp;{{mvar|K}} consisting of diagonal matrices with {{math|1={{!}} ''z'' {{!}} = 1}}. It is the representation induced from the 1-dimensional representation {{math|''z''&lt;sup&gt;k&lt;/sup&gt; ''T''}}, and is independent of&amp;nbsp;{{mvar|ν}}. By [[w:Frobenius reciprocity|''Frobenius reciprocity'']], on&amp;nbsp;{{mvar|K}} they decompose as a direct sum of the irreducible representations of&amp;nbsp;{{mvar|K}} with dimensions {{math|1=&amp;#124;''k''&amp;#124; + 2''m'' + 1}} with {{mvar|m}} a non-negative integer.

Using the identification between the Riemann sphere minus a point and {{math|'''C'''}}, the principal series can be defined directly on {{math|L&lt;sup&gt;2&lt;/sup&gt;('''C''')}} by the formula&lt;ref&gt;{{harvnb|Gelfand|Graev|Pyatetskii-Shapiro|1969}}&lt;/ref&gt;

:&lt;math&gt;\pi_{\nu,k}\begin{pmatrix}a&amp; b\\ c&amp; d\end{pmatrix}^{-1}f(z)=|cz+d|^{-2-i\nu} \left({cz+d\over |cz+d|}\right)^{-k}f\left({az+b\over cz+d}\right).&lt;/math&gt;

Irreducibility can be checked in a variety of ways:

* The representation is already irreducible on&amp;nbsp;{{mvar|B}}. This can be seen directly, but is also a special case of general results on irreducibility of induced representations due to [[w:François Bruhat|''François Bruhat'']] and [[w:George Mackey|''George Mackey'']], relying on the [[w:Bruhat decomposition|''Bruhat decomposition'']] {{math|1=''G'' = ''B'' ∪ ''B'' ''s'' ''B''}} where {{mvar|s}} is the [[w:Weyl group|''Weyl group'']] element&lt;ref&gt;{{harvnb|Knapp|2001|loc=Chapter II.}}&lt;/ref&gt;

:&lt;math&gt;\begin{pmatrix}0&amp; -1\\ 1&amp; 0\end{pmatrix}&lt;/math&gt;.

* The action of the Lie algebra &lt;math&gt;\mathfrak{g}&lt;/math&gt; of&amp;nbsp;{{mvar|G}} can be computed on the algebraic direct sum of the irreducible subspaces of&amp;nbsp;{{mvar|K}} can be computed explicitly and the it can be verified directly that the lowest-dimensional subspace generates this direct sum as a &lt;math&gt;\mathfrak{g}&lt;/math&gt;-module.&lt;ref name=&quot;Harish-Chandra 1947&quot;&gt;{{harvnb|Harish-Chandra|1947}}&lt;/ref&gt;&lt;ref name=&quot;Taylor 1986&quot;&gt;{{harvnb|Taylor|1986}}&lt;/ref&gt;

===Complementary series===

The for {{math|0 &lt; t &lt; 2}}, the complementary series is defined on [[w:L2 space|{{math|L&lt;sup&gt;2&lt;/sup&gt;}} ''functions'']] {{mvar|f}} on {{math|'''C'''}} for the inner product&lt;ref&gt;{{harvnb|Knapp|2001}} Chapter 2. Equation 2.12.&lt;/ref&gt;

:&lt;math&gt; (f,g)=\int \int {f(z) \overline{g(w)}\, dz\, dw\over |z-w|^{2-t}}.&lt;/math&gt;

with the action given by&lt;ref&gt;{{harvnb|Bargmann|1947}}&lt;/ref&gt;&lt;ref&gt;{{harvnb|Gelfand|Graev|1953}}&lt;/ref&gt;

:&lt;math&gt;\pi_{t}\begin{pmatrix}a&amp; b\\ c&amp; d\end{pmatrix}^{-1}f(z)=|cz+d|^{-2-t} f\left({az+b\over cz+d}\right).&lt;/math&gt;

The representations in the complementary series are irreducible and pairwise non-isomorphic. As a representation of&amp;nbsp;{{mvar|K}}, each is isomorphic to the Hilbert space direct sum of all the odd dimensional irreducible representations of {{math|1=''K'' = SU(2)}}.  Irreducibility can be proved by analyzing the action of &lt;math&gt;\mathfrak{g}&lt;/math&gt; on the 
algebraic sum of these subspaces&lt;ref name=&quot;Harish-Chandra 1947&quot;/&gt;&lt;ref name=&quot;Taylor 1986&quot;/&gt; or directly without using the Lie algebra.&lt;ref&gt;{{harvnb|Gelfand|Naimark|1947}}&lt;/ref&gt;&lt;ref&gt;{{harvnb|Takahashi|1963|page=343.}}&lt;/ref&gt;

===Plancherel theorem===
The only irreducible unitary representations of {{math|SL(2, '''C''')}} are the principal series, the complementary series and the trivial representation.
Since {{math|−''I''}} acts as {{math|(−1)&lt;sup&gt;''k''&lt;/sup&gt;}} on the principal series and trivially on the remainder, these will give all the irreducible unitary representations of the Lorentz group, provided {{mvar|k}} is taken to be even.

To decompose the left regular representation of&amp;nbsp;{{mvar|G}} on {{math|L&lt;sup&gt;2&lt;/sup&gt;(''G'')}}, only the principal series are required. This immediately yields the decomposition on the subrepresentations {{math|L&lt;sup&gt;2&lt;/sup&gt;(''G''/±''I'')}}, the left regular representation of the Lorentz group, and {{math|L&lt;sup&gt;2&lt;/sup&gt;(''G''/''K'')}},
the regular representation on 3-dimensional hyperbolic space. (The former only involves principal series representations with ''k'' even and the latter only those with {{math|1=''k'' = 0}}.)

The left and right regular representation {{mvar|λ}} and {{mvar|ρ}} are defined on {{math|L&lt;sup&gt;2&lt;/sup&gt;(''G'')}} by

:&lt;math&gt;\lambda(g)f(x)=f(g^{-1}x),\,\,\rho(g)f(x)=f(xg).&lt;/math&gt;

Now if {{mvar|f}} is an element of {{math|C&lt;sub&gt;c&lt;/sub&gt;(''G'')}}, the operator {{math|''π''&lt;sub&gt;''&amp;nu;'',''k''&lt;/sub&gt;(''f'')}} defined by

:&lt;math&gt;\pi_{\nu,k}(f)=\int_G f(g)\pi(g)\, dg&lt;/math&gt;

is [[w:Hilbert–Schmidt operator|''Hilbert–Schmidt'']]. Define a Hilbert space&amp;nbsp;{{mvar|H}} by

:&lt;math&gt; H=\bigoplus_{k\ge 0} HS(L^2(C)) \otimes L^2(R, c_k(\nu^2 + k^2)^{1/2} d\nu),&lt;/math&gt;

where

:&lt;math&gt;c_0=1/4\pi^{3/2}, \,\, c_k=1/(2\pi)^{3/2}\,\,(k\ne 0)&lt;/math&gt;

and {{math|HS(L&lt;sup&gt;2&lt;/sup&gt;('''C'''))}} denotes the Hilbert space of Hilbert–Schmidt operators on {{math|L&lt;sup&gt;2&lt;/sup&gt;('''C''')}}.&lt;ref group=nb&gt;Note that for a Hilbert space&amp;nbsp;{{mvar|H}}, {{math|HS(''H'')}} may be identified canonically with the Hilbert space tensor product of {{mvar|H}} and its conjugate space.&lt;/ref&gt;  Then the map&amp;nbsp;{{mvar|U}} defined on {{math|C&lt;sub&gt;c&lt;/sub&gt;(''G'')}} by

:&lt;math&gt;U(f)(\nu,k) = \pi_{\nu,k}(f)&lt;/math&gt;

extends to a unitary of {{math|L&lt;sup&gt;2&lt;/sup&gt;(''G'')}} onto {{mvar|H}}.

The map&amp;nbsp;{{mvar|U}} satisfies the intertwining property

:&lt;math&gt; U(\lambda(x)\rho(y)f)(\nu,k) = \pi_{\nu,k}(x)^{-1} \pi_{\nu,k}(f)\pi_{\nu,k}(y).&lt;/math&gt;

If {{math|''f''&lt;sub&gt;1&lt;/sub&gt;}}, {{math|''f''&lt;sub&gt;2&lt;/sub&gt;}} are in {{math|C&lt;sub&gt;c&lt;/sub&gt;(''G'')}} then by unitarity

:&lt;math&gt; (f_1,f_2) = \sum_{k\ge 0} c_k^2 \int_{-\infty}^\infty {\rm Tr}(\pi_{\nu,k}(f_1)\pi_{\nu,k}(f_2)^*) (\nu^2 +k^2) \, d\nu.&lt;/math&gt;

Thus if {{math|1=''f'' = ''f''&lt;sub&gt;1&lt;/sub&gt; ∗ ''f''&lt;sub&gt;2&lt;/sub&gt;*&lt;!-- these asterisks are different! do not touch anything unless you can think about it, please --&gt;}} denotes the [[w:convolution|''convolution'']] of {{math|''f''&lt;sub&gt;1&lt;/sub&gt;}} and {{math|''f''&lt;sub&gt;2&lt;/sub&gt;*}}, and
&lt;math&gt;f_2^*(g)=\overline{f_2(g^{-1})}&lt;/math&gt;, then

:&lt;math&gt; f(1) = \sum_{k\ge 0} c_k^2 \int_{-\infty}^\infty  {\rm Tr}(\pi_{\nu,k}(f)) (\nu^2+k^2)\, d\nu.&lt;/math&gt;

The last two displayed formulas are usually referred to as the [[w:Plancherel formula|''Plancherel formula'']] and the [[w:inverse Fourier transform|''Fourier inversion'']] formula respectively.
The Plancherel formula extends to all {{math|''f''&lt;sub&gt;''i''&lt;/sub&gt;}} in {{math|L&lt;sub&gt;2&lt;/sub&gt;(''G'')}}. By a theorem of [[w:Jacques Dixmier|''Jacques Dixmier'']] and [[w:Paul Malliavin|''Paul Malliavin'']],
every function&amp;nbsp;{{mvar|f}} in &lt;math&gt;C^\infty_c(G)&lt;/math&gt; is a finite sum of convolutions of similar functions, the inversion formula holds for such {{mvar|f}}.
It can be extended to much wider classes of functions satisfying mild differentiability conditions.&lt;ref name=&quot;ReferenceC&quot;/&gt;

=== Classification ===
The strategy followed in the classification of the irreducible infinite-dimensional representations is, in analogy to the finite-dimensional case, to ''assume'' they exist, and to investigate their properties. Thus first assume that an irreducible [[w:Strong operator topology|''strongly continuous'']] infinite-dimensional representation {{math|Π&lt;sub&gt;''H''&lt;/sub&gt;}} on a Hilbert space {{mvar|H}} of {{math|SO(3; 1)&lt;sup&gt;+&lt;/sup&gt;}} is at hand.&lt;ref&gt;{{harvnb|Folland|2015|loc=Section 3.1.}}&lt;/ref&gt; Since {{math|SO(3)}} is a subgroup, {{math|Π&lt;sub&gt;''H''&lt;/sub&gt;}} is a representation of it as well. Each irreducible subrepresentation of {{math|SO(3)}} is finite-dimensional, and the {{math|SO(3)}} representation is reducible into a direct sum of irreducible finite-dimensional unitary representations of {{math|SO(3)}} if {{math|Π&lt;sub&gt;''H''&lt;/sub&gt;}} is unitary.&lt;ref&gt;{{harvnb|Folland|2015|loc=Theorem 5.2.}}&lt;/ref&gt;

The steps are the following:&lt;ref&gt;{{harvnb|Tung|1985|loc=Section 10.3.3.}}&lt;/ref&gt;
#Chose a suitable basis of common eigenvectors of {{math|''J''&lt;sup&gt;2&lt;/sup&gt;}} and {{math|''J''&lt;sub&gt;3&lt;/sub&gt;}}.
#Compute matrix elements of {{math|''J''&lt;sub&gt;1&lt;/sub&gt;, ''J''&lt;sub&gt;2&lt;/sub&gt;, ''J''&lt;sub&gt;3&lt;/sub&gt;}} and {{math|''K''&lt;sub&gt;1&lt;/sub&gt;, ''K''&lt;sub&gt;2&lt;/sub&gt;, ''K''&lt;sub&gt;3&lt;/sub&gt;}}.
#Enforce Lie algebra commutation relations.
#Require unitarity together with orthonormality of the basis.&lt;ref group=nb&gt;If one instead demands finite-dimensionality, one ends up with the {{math|(''m'', ''n'')}} representations, see {{harvtxt|Tung|1985|loc=Problem 10.8.}} If one demands neither, then one obtains a broader classification of ''all'' irreducible representations, including the finite-dimensional and the unitary ones. This approach is taken by in {{harvtxt|Harish-Chandra|1947}}.&lt;/ref&gt;

==== Step 1 ====
One may suitably choose a basis and label the basis vectors by

:&lt;math&gt;\left |j_0\, j_1;j\, m\right\rangle.&lt;/math&gt;

If this was a ''finite-dimensional'' representation, then {{math|''j''&lt;sub&gt;0&lt;/sub&gt;}} would correspond the lowest occurring eigenvalue {{math|''j''(''j'' + 1)}} of {{math|''J''&lt;sup&gt;2&lt;/sup&gt;}} in the representation, equal to {{math|{{!}}''m'' − ''n''{{!}}}}, and {{math|''j''&lt;sub&gt;1&lt;/sub&gt;}} would correspond to the highest occurring eigenvalue, equal to {{math|''m'' + ''n''}}. In the infinite-dimensional case, {{math|''j''&lt;sub&gt;0&lt;/sub&gt; ≥ 0}} retains this meaning, but {{math|''j''&lt;sub&gt;1&lt;/sub&gt;}} does not.&lt;ref name=&quot;Tung 1985&quot;/&gt; One assumes for simplicity that a given {{mvar|j}} occurs at most once in a given representation (this is the case for finite-dimensional representations), and it can be shown&lt;ref&gt;{{harvnb|Harish-Chandra|1947|loc=Footnote p. 374.}}&lt;/ref&gt; that the assumption is possible to avoid (with a slightly more complicated calculation) with the same results.

==== Step 2 ====
The next step is to compute the matrix elements of the operators {{math|''J''&lt;sub&gt;1&lt;/sub&gt;, ''J''&lt;sub&gt;2&lt;/sub&gt;, ''J''&lt;sub&gt;3&lt;/sub&gt;}} and {{math|''K''&lt;sub&gt;1&lt;/sub&gt;, ''K''&lt;sub&gt;2&lt;/sub&gt;, ''K''&lt;sub&gt;3&lt;/sub&gt;}} forming the basis of the Lie algebra of {{math|'''so'''(3; 1)}}. The matrix elements of

:&lt;math&gt;J_\pm = J_1 \pm iJ_2, J_3&lt;/math&gt;

(here one is operating in the comlpexified Lie algebra) are known from the representation theory of the rotation group, and are given by&lt;ref&gt;{{harvnb|Tung|1985|loc=Equations 7.3-13, 7.3-14.}}&lt;/ref&gt;&lt;ref&gt;{{harvnb|Harish-Chandra|1947|loc=Equation 8.}}&lt;/ref&gt;

:&lt;math&gt;\left \langle j\, m\right|J_+ \left |j\, m-1\right\rangle = \left \langle j\, m-1\right|J_- \left |j\, m\right\rangle = \sqrt{(j+m)(j-m+1)}, \quad \left \langle j, m\right|J_3 \left |j\, m\right\rangle = m,&lt;/math&gt;

where the labels {{math|''j''&lt;sub&gt;0&lt;/sub&gt;}} and {{math|''j''&lt;sub&gt;1&lt;/sub&gt;}} have been dropped since they are the same for all basis vectors in the representation.

Due to the commutation relations

:&lt;math&gt;[J_i,K_j] = i\epsilon_{ijk}K_k,&lt;/math&gt;

the triple {{math|(''K''&lt;sub&gt;''i''&lt;/sub&gt;, ''K''&lt;sub&gt;''i''&lt;/sub&gt;, ''K''&lt;sub&gt;''i''&lt;/sub&gt;) ≡ '''''K'''''}} is a [[w:Tensor operator#Vector operators|''vector operator'']]&lt;ref&gt;{{harvnb|Hall|2015|loc=Proposition C.7.}}&lt;/ref&gt; and the [[w:Wigner&amp;ndash;Eckart theorem|''Wigner&amp;ndash;Eckart theorem'']]&lt;ref&gt;{{harvnb|Hall|2015|loc=Appendix C.2.}}&lt;/ref&gt; applies for computation of matrix elements between the states represented by the chosen basis.&lt;ref&gt;{{harvnb|Tung|1985|loc=Step II section 10.2.}}&lt;/ref&gt; The matrix elements of

:&lt;math&gt;\begin{align} K^{(1)}_0 &amp;= K_3\\
K^{(1)}_{\pm 1} &amp;= \mp\frac{1}{\sqrt 2}(K_1 \pm iK_2),\end{align}&lt;/math&gt;

where the superscript {{math|(1)}} signifies that the defined quantities are the components of a [[w:spherical tensor operator|''spherical tensor operator'']] of rank {{math|''k'' {{=}} 1}} (which explains the factor &amp;radic;&lt;span style=&quot;text-decoration: overline&quot;&gt;2&lt;/span&gt; as well) and the subscripts {{math|0, ±1}} are referred to as {{mvar|q}} in formulas below, are given by&lt;ref&gt;{{harvnb|Tung|1985|loc=Equations 10.3-5. Tung's notation for Clebsch&amp;ndash;Gordan coefficients differ from the one used here.}}&lt;/ref&gt;

:&lt;math&gt;\begin{align}\left\langle j'\,m'\right|K^{(1)}_0\left|j\,m\right\rangle &amp;= \langle j' \, m' \,k = 1 \,q = 0 | j \, m \rangle \langle j \| K^{(1)} \| j'\rangle\\
\left\langle j' m'\right|K^{(1)}_{\pm 1}\left|j\,m\right\rangle &amp;= \langle j' \, m' \, k= 1 \,q = \pm 1 | j \, m \rangle \langle j \| K^{(1)} \| j'\rangle.
\end{align}&lt;/math&gt;

Here the first factors on the right hand sides are [[w:Clebsch&amp;ndash;Gordan coefficient|''Clebsch&amp;ndash;Gordan coefficients'']] for coupling {{math|''j''&amp;prime;}} with {{mvar|k}} to get {{mvar|j}}. The second factors are the ''reduced matrix elements''. They do not depend on {{math|''m'', ''m&amp;prime;''}} or {{mvar|q}}, but depend on {{math|''j'', ''j&amp;prime;''}} and, of course, {{math|'''''K'''''}}. For a complete list of non-vanishing equations, see {{harvtxt|Harish-Chandra|1947|p=375}}.

==== Step 3 ====
The next step is to demand that the Lie algebra relations hold, i.e. that

:&lt;math&gt;[K_\pm, K_3] = \pm J_\pm, \quad [K_+, K_-] = -2J_3.&lt;/math&gt;

This results in a set of equations&lt;ref&gt;{{harvnb|Tung|1985|loc=Equation VII-3.}}&lt;/ref&gt; for which the solutions are&lt;ref&gt;{{harvnb|Tung|1985|loc=Equations 10.3-5, 7, 8.}}&lt;/ref&gt;
:&lt;math&gt;\begin{align}
\langle j \| K^{(1)} \| j\rangle = i\frac{j_1j_0}{\sqrt{j(j+1)}},\\
\langle j \| K^{(1)} \| j-1\rangle = -B_j\xi_j\sqrt{j(2j-1)},\\
\langle j-1 \| K^{(1)} \| j\rangle = B_j\xi_j^{-1}\sqrt{j(2j+1)},\\
 \end{align}&lt;/math&gt;

where

:&lt;math&gt;B_j = \sqrt{\frac{(j^2 - j_0^2)(j^2 - j_1^2)}{j^2(4j^2 - 1)}},&lt;/math&gt;

and

:&lt;math&gt;\begin{align}j_0 &amp;= 0, \frac{1}{2}, 1, \ldots,\\
j_1 &amp;\in \mathbb C,\\
\xi_j&amp;\in \mathbb C.\\
\end{align}&lt;/math&gt;

==== Step 4 ====
The imposition of the requirement of unitarity of the corresponding representation of the ''group'' restricts the possible values for the arbitrary complex numbers {{math|''j''&lt;sub&gt;0&lt;/sub&gt;}} and {{math|''ξ''&lt;sub&gt;''j''&lt;/sub&gt;}}. Unitarity of the group representation translates to the requirement of the Lie algebra representatives being Hermitian, meaning

:&lt;math&gt;\begin{align}K_\pm^\dagger &amp;= K_\mp,\\ K_3^\dagger &amp;= K_3.\end{align}&lt;/math&gt;

This translates to&lt;ref&gt;{{harvnb|Tung|1985|loc=Equation VII-9.}}&lt;/ref&gt;

:&lt;math&gt;\begin{align}
\langle j \| K^{(1)} \| j\rangle &amp;= \overline{\langle j \| K^{(1)} \| j\rangle},\\
\langle j \| K^{(1)} \| j-1\rangle &amp;= -\overline{\langle j-1 \| K^{(1)} \| j\rangle},\\
\end{align}&lt;/math&gt;

leading to&lt;ref&gt;{{harvnb|Tung|1985|loc=Equations VII-10, 11.}}&lt;/ref&gt;

:&lt;math&gt;\begin{align}j_0(j_1 + \overline{j_1}) &amp;= 0,\\
|B_j|(|\xi_j|^2 - e^{-2i\beta_j}) &amp;= 0,\end{align}&lt;/math&gt;
where {{math|''β''&lt;sub&gt;''j''&lt;/sub&gt;}} is the angle of {{math|''B''&lt;sub&gt;''j''&lt;/sub&gt;}} on polar form. For {{math|{{!}}''B''&lt;sub&gt;''j''&lt;/sub&gt;{{!}} ≠ 0}} one has &lt;math&gt;|\zeta_j|^2=1&lt;/math&gt;, and {{math|''ξ''&lt;sub&gt;''j''&lt;/sub&gt; {{=}} 1}} is chosen by convention. There are two possible cases. The first with {{math|''j''&lt;sub&gt;1&lt;/sub&gt; + {{overline|''j''&lt;sub&gt;1&lt;/sub&gt;}} {{=}} 0}} gives, with {{math|''j''&lt;sub&gt;1&lt;/sub&gt; {{=}} − ''iν''}}, {{mvar|ν}} real,&lt;ref&gt;{{harvnb|Tung|1985|loc=Equations VII-12.}}&lt;/ref&gt;

:&lt;math&gt;\begin{align}\langle j \| K^{(1)} \| j\rangle &amp;= \frac{\nu j_0}{j(j+1)},\\
B_j &amp;= \sqrt{\frac{(j^2 - j_0^2)(j^2 + \nu^2)}{4j^2 - 1}}.
\end{align}&lt;/math&gt;

This is principal series and the elements may be denoted {{math|(''j''&lt;sub&gt;0&lt;/sub&gt;, ''ν''), 2''j''&lt;sub&gt;0&lt;/sub&gt; ∈ ℕ, ''ν'' ∈ ℝ}}. For the other possibility, {{math|''j''&lt;sub&gt;0&lt;/sub&gt; {{=}} 0}}, one has&lt;ref&gt;{{harvnb|Tung|1985|loc=Equations VII-13.}}&lt;/ref&gt;

:&lt;math&gt;\begin{align}\langle j \| K^{(1)} \| j\rangle &amp;= 0,\\
B_j &amp;= \sqrt{\frac{(j^2 - \nu^2)}{4j^2 - 1}}.
\end{align}&lt;/math&gt;

One needs to require that &lt;math&gt;B_j^2&lt;/math&gt; is real and positive for {{math|''j'' {{=}} 1, 2, ...}} (because {{math|''B''&lt;sub&gt;0&lt;/sub&gt; {{=}} B&lt;sub&gt;''j''&lt;sub&gt;0&lt;/sub&gt;&lt;/sub&gt;}}), leading to {{math|−1 ≤ ''ν'' ≤ 1}}. This is complementary series and its elements may be denoted {{math|(0, ''ν''), −1 ≤ ''ν'' ≤ 1}}.

This shows that the representations of above are ''all'' infinite-dimensional irreducible unitary representations.

== Explicit formulas ==

=== Conventions and Lie algebra bases ===
The metric of choice is given by {{math|η}} = {{math|diag(−1, 1, 1, 1)}}, and the physics convention for Lie algebras and the exponential mapping is used. These choices are arbitrary, but once they are made, fixed. One possible choice of [[w:basis (linear algebra)|''basis'']] for the Lie algebra is, in the 4-vector representation, given by

:&lt;math&gt;\begin{align}
J_1 &amp;= J^{23} = -J^{32} = i\biggl(\begin{smallmatrix}
0&amp;0&amp;0&amp;0\\ 0&amp;0&amp;0&amp;0\\ 0&amp;0&amp;0&amp;-1\\ 0&amp;0&amp;1&amp;0\\
\end{smallmatrix}\biggr),\\
J_2 &amp;= J^{31} = -J^{13} = i\biggl(\begin{smallmatrix}
0&amp;0&amp;0&amp;0\\ 0&amp;0&amp;0&amp;1\\ 0&amp;0&amp;0&amp;0\\ 0&amp;-1&amp;0&amp;0\\
\end{smallmatrix}\biggr),\\
J_3 &amp;= J^{12} = -J^{21} = i\biggl(\begin{smallmatrix}
0&amp;0&amp;0&amp;0\\ 0&amp;0&amp;-1&amp;0\\ 0&amp;1&amp;0&amp;0\\ 0&amp;0&amp;0&amp;0\\
\end{smallmatrix}\biggr),\\
K_1 &amp;= J^{01} = J^{10} = i\biggl(\begin{smallmatrix}
0&amp;1&amp;0&amp;0\\ 1&amp;0&amp;0&amp;0\\ 0&amp;0&amp;0&amp;0\\ 0&amp;0&amp;0&amp;0\\
\end{smallmatrix}\biggr),\\
K_2 &amp;= J^{02} = J^{20} = i\biggl(\begin{smallmatrix}
0&amp;0&amp;1&amp;0\\ 0&amp;0&amp;0&amp;0\\ 1&amp;0&amp;0&amp;0\\ 0&amp;0&amp;0&amp;0\\
\end{smallmatrix}\biggr),\\
K_3 &amp;= J^{03} = J^{30} = i\biggl(\begin{smallmatrix}
0&amp;0&amp;0&amp;1\\ 0&amp;0&amp;0&amp;0\\ 0&amp;0&amp;0&amp;0\\ 1&amp;0&amp;0&amp;0\\
\end{smallmatrix}\biggr).
\end{align}&lt;/math&gt;

The commutation relations of the Lie algebra '''so'''(3; 1) are&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Equation 2.4.12.}}&lt;/ref&gt;

:&lt;math&gt;[J^{\mu\nu},J^{\rho\sigma}] = i(\eta^{\sigma\mu}J^{\rho\nu} + \eta^{\nu\sigma}J^{\mu\rho} - \eta^{\rho\mu}J^{\sigma\nu} -\eta^{\nu\rho}J^{\mu\sigma}).&lt;/math&gt;

In three-dimensional notation, these are&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Equations 2.4.18-2.4.20.}}&lt;/ref&gt;

:&lt;math&gt;[J_i,J_j] = i\epsilon_{ijk}J_k, \quad [J_i,K_j] = i\epsilon_{ijk}K_k, \quad [K_i,K_j] = -i\epsilon_{ijk}J_k.&lt;/math&gt;

The choice of basis above satisfies the relations, but other choices are possible. The multiple use of the symbol {{mvar|J}} above and in the sequel should be observed.

=== Weyl spinors and bispinors ===
[[File:Paul Dirac, 1933.jpg|upright|thumb|right|Solutions to the [[Dirac equation]] transform under the {{math|({{sfrac|1|2}}, 0) ⊕ (0, {{sfrac|1|2}})}}-representation. Dirac discovered the [[gamma matrices]] in his search for a relativistically invariant equation, then already known to mathematicians.&lt;ref name=&quot;Weinberg 2002 loc=Section 5.4&quot;/&gt;]]
By taking, in turn, {{math|''m'' {{=}} {{sfrac|1|2}}}}, {{math|''n'' {{=}} 0}} and {{math|''m'' {{=}} 0}}, {{math|''n'' {{=}} {{sfrac|1|2}}}} and by setting

:&lt;math&gt;J_i^{(\frac{1}{2})} = \frac{1}{2}\sigma_i&lt;/math&gt;

in the general expression {{EquationNote|G1|(G1)}}, and by using the trivial relations {{nowrap|1&lt;sub&gt;1&lt;/sub&gt; {{=}} 1}} and {{math|''J''&lt;sup&gt;(0)&lt;/sup&gt; {{=}} 0}}, one obtains

{{NumBlk|:|&lt;math&gt;\begin{align}
\pi_{(\frac{1}{2},0)}(J_i) &amp; = \frac{1}{2}(\sigma_i\otimes 1_{(1)} + 1_{(2)}\otimes J^{(0)}_i) = \frac{1}{2}\sigma_i\quad\pi_{(\frac{1}{2},0)}(K_i) = \frac{i}{2}(1_{(2)}\otimes J^{(0)}_i - \sigma_i \otimes 1_{(1)}) = -\frac{i}{2}\sigma_i,\\
\pi_{(0,\frac{1}{2})}(J_i) &amp; = \frac{1}{2}(J^{(0)}_i\otimes 1_{(2)} + 1_{(1)}\otimes \sigma_i) = \frac{1}{2}\sigma_i\quad\pi_{(0,\frac{1}{2})}(K_i) = \frac{i}{2}(1_{(1)}\otimes\sigma_i  - J^{(0)}_i \otimes 1_{(2)}) = +\frac{i}{2}\sigma_i.
\end{align}&lt;/math&gt;|{{EquationRef|W1}}}}

These are the left-handed and right-handed [[w:Weyl spinor|''Weyl spinor'']] representations. They act by matrix multiplication on 2-dimensional [[w:complex coordinate space|''complex vector spaces'']] (with a choice of basis) {{math|''V''&lt;sub&gt;L&lt;/sub&gt;}} and {{math|''V''&lt;sub&gt;R&lt;/sub&gt;}}, whose elements {{math|Ψ&lt;sub&gt;L&lt;/sub&gt;}} and {{math|Ψ&lt;sub&gt;R&lt;/sub&gt;}} are called left- and right-handed Weyl spinors respectively. Given {{math|(''π''({{sfrac|1|2}},0), ''V''&lt;sub&gt;L&lt;/sub&gt;)}} and {{math|(''π''(0,{{sfrac|1|2}}), ''V''&lt;sub&gt;R&lt;/sub&gt;)}} one may form their direct sum as representations,&lt;ref&gt;{{harvnb|Weinberg|2002|loc=Equations 5.4.19, 5.4.20.}}&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{align}
\pi_{(\frac{1}{2},0) \oplus (0,\frac{1}{2})}(J_i) &amp;= \frac{1}{2}\biggl(\begin{matrix}
\sigma_i&amp;0\\ 0&amp;\sigma_i\\
\end{matrix}\biggr),\\
\pi_{(\frac{1}{2},0) \oplus (0,\frac{1}{2})}(K_i) &amp;= \frac{i}{2}
\biggl(\begin{matrix}
\sigma_i&amp;0\\ 0&amp;-\sigma_i\\
\end{matrix}\biggr)\\
\end{align}.&lt;/math&gt;|{{EquationRef|D1}}}}

This is, up to a similarity transformation, the {{math|({{sfrac|1|2}},0) ⊕ (0,{{sfrac|1|2}})}} [[w:Dirac spinor|''Dirac spinor'']] representation of {{math|'''so'''(3; 1)}}. It acts on the 4-component elements {{math|(Ψ&lt;sub&gt;L&lt;/sub&gt;, Ψ&lt;sub&gt;R&lt;/sub&gt;)}} of {{math|(''V''&lt;sub&gt;L&lt;/sub&gt; ⊕ ''V''&lt;sub&gt;R&lt;/sub&gt;)}}, called [[w:bispinor|''bispinors'']], by matrix multiplication. The representation may be obtained in a more general and basis independent way using [[w:Clifford algebra|''Clifford algebras'']]. These expressions for bispinors and Weyl spinors all extend by linearity of Lie algebras and representations to all of {{math|'''so'''(3; 1)}}. Expressions for the group representations are obtained by exponentiation.

== See also ==
* [[Representation theory]]
* [[w:Bargmann–Wigner equations|''Bargmann–Wigner equations'']]
* [[w:Center of mass (relativistic)|''Center of mass (relativistic)'']]
* [[w:Dirac algebra|''Dirac algebra'']]
* [[w:Gamma matrices|''Gamma matrices'']]
* [[w:Lorentz group|''Lorentz group'']]
* [[w:Möbius transformation|''Möbius transformation'']]
* [[w:Poincaré group|''Poincaré group'']]
* [[w:Representation theory of the Poincaré group|''Representation theory of the Poincaré group'']]
* [[w:Symmetry in quantum mechanics|''Symmetry in quantum mechanics'']]
* [[w:Wigner's classification|''Wigner's classification'']]

==Remarks==
{{reflist|group=nb}}

==Notes==
{{reflist|30em}}

==Freely available online references==
*{{cite arXiv|last = Bekaert|last2 = Boulanger|first = X.|first2 = N.|year = 2006|title = The unitary representations of the Poincare group in any spacetime dimension|eprint = hep-th/0611263}} Expanded version of the lectures presented at the second Modave summer school in mathematical physics (Belgium, August 2006).
*{{citation|last = Curtright|first = T L |last2=Fairlie|first2= D B|last3=Zachos|first3 = C K|year = 2014|title = A compact formula for rotations as spin matrix polynomials| journal =SIGMA| volume=10| page=084|doi=10.3842/SIGMA.2014.084|authorlink1=Thomas Curtright|authorlink2=David Fairlie|authorlink3=Cosmas Zachos|arxiv = 1402.3541 |bibcode = 2014SIGMA..10..084C }} Group elements of SU(2) are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group.

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[[w:Category:Representation theory of Lie groups|''Category:Representation theory of Lie groups'']]
[[w:Category:Special relativity|''Category:Special relativity'']]
[[w:Category:Quantum mechanics|''Category:Quantum mechanics'']]
[[w:Category:Hendrik Lorentz|''Category:Hendrik Lorentz'']]</text>
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    <title>Quantum gravity (Planck)</title>
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      <text bytes="63208" sha1="ngzy03i1dkuj9zxeu0anpt6dmi3tsnv" xml:space="preserve">'''Simulating gravitational and atomic orbits via n-body rotating particle-particle orbital pairs at the Planck scale'''

The following describes a geometrical method for simulating gravitational orbits and atomic orbitals via an n-body network of rotating individual particle-particle orbital pairs &lt;ref&gt;Macleod, Malcolm J.; {{Cite journal |title=Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs |journal=RG |date=Dec 2024 | doi=10.13140/RG.2.2.11378.00961/1}}&lt;/ref&gt;. Although the simulation is dimensionless (the only physical constant used is the [[w:fine structure constant |fine structure constant alpha]]), it can translate via the [[w:Planck_units |Planck units]] for comparisons with real world orbits. The orbits generated by this dimensionless geometrical approach can be formulated, and despite not using Newtonian physics these formulas demonstrate consistency; for example the derived formulas for radius R, period T and (M + m) will reduce [[v:Quantum_gravity_(Planck)#Kepler's_formula_=_G |Kepler's formula]] to '''G'''. Likewise the atomic orbital shells naturally quantize according to pi without relying on built-in postulates.

[[File:Gravitational-regular-3body-orbit.gif|thumb|right|640px|A regular 3-body orbit. The simulation begins with the start (x, y) co-ordinates of each point. No other parameters are required. r0=2*α; x1=1789.5722983; y1=0; x2=cos(pi*2/3)*r0; y2=sin(pi*2/3)*r0; x3=cos(pi*2/3)*r0; y3=sin(pi*2/3)*r0]]
:&lt;math&gt;\frac{4 \pi^2 R^3}{(M+m) T} = \frac{l_p c^2}{m_P} = G&lt;/math&gt;


For simulating gravity, orbiting objects ''A'', ''B'', ''C''... are sub-divided into discrete points, each point can be represented as 1 unit of [[w:Planck mass |Planck mass]] ''m''&lt;sub&gt;P&lt;/sub&gt; (for example, a 1kg satellite would be divided into 1kg/''m''&lt;sub&gt;P&lt;/sub&gt; = 45940509 points). Each point in object ''A'' then forms an orbital pair with every point in objects ''B'', ''C''..., resulting in a universe-wide, n-body network of rotating point-to-point orbital pairs . 

Each orbital pair rotates 1 unit of length per unit of time, when these orbital pair rotations are summed and mapped over time, gravitational orbits emerge between the objects ''A'', ''B'', ''C''... 

The base simulation requires only the start position (''x'', ''y'' coordinates) of each point, as it maps only rotations of the points within their respective orbital pairs then information regarding the macro objects ''A'', ''B'', ''C''...; momentum, center of mass, barycenter etc ... is not required (each orbital is calculated independently of all other orbitals). 


For simulating electron transition within the atom, the electron is assigned as a single mass point, the nucleus as multiple points clustered together (a 2-body orbit), and an incoming 'photon' is added to the orbital radius in a series of discrete steps (rather than a single 'jump' between orbital shells). As the electron continues to orbit the nucleus during this transition phase, the electron path traces a [[w:hyperbolic spiral |hyperbolic spiral]]. Although we are mapping the electron transition as a gravitational orbit on a 2-D plane, periodically the transition spiral angles converge to give an integer orbital radius (360°=4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r''), a radial quantization (as a function of pi and so of geometrical origin) naturally emerges. Furthermore, the transition steps between these radius can then be used to solve the transition frequency, replicating the Bohr model.  In this context the Bohr model is a gravitational model, and thus is not superseded by the [[w:Schrodinger equation |Schrodinger wave equation]], but rather is complementary to this equation (they each measure different aspects of the transition).


[[File:complex-orbit-pts26-r17-1-7-1.gif|thumb|right|640px|By selecting the start co-ordinates on a 2-D plane for each point accordingly, we can 'design' the required orbits. The 26 points orbit each other resulting in 325 point-point orbitals.]]
=== Theory ===
In the simulation, particles are treated as an electric wave-state to (Planck) mass point-state oscillation, the wave-state as the duration of particle frequency in Planck time units, the point-state duration as 1 unit of Planck time (as a point, this state can be assigned mapping coordinates), the particle itself is a continuous oscillation between these 2 states (i.e.: the particle is not a fixed entity). For example, an electron has a frequency (wave-state duration) = 10&lt;sup&gt;23&lt;/sup&gt; units of Planck time followed by the mass state (1 unit of Planck time). The background to this oscillation is given in the [[v:Electron (mathematical) |mathematical electron]] model.

If the electron '''has (is)''' mass (1 unit of Planck mass) for 1 unit of Planck time, and then '''no''' mass for 10&lt;sup&gt;23&lt;/sup&gt; units of Planck time (the wave-state), then in order for a (hypothetical) object composed only of electrons to '''have (be)''' 1 unit of Planck mass at every unit of Planck time, the object will require 10&lt;sup&gt;23&lt;/sup&gt; electrons. This is because orbital rotation occurs at each unit of Planck time and so the simulation requires this object to have a unit of Planck mass at each unit of Planck time (i.e.: on average there will always be 1 electron in the mass point state). We would then measure the mass of this object as 1 Planck mass (the measured mass of an object reflects the average number of units of Planck mass per unit of Planck time). For the simulation program, this Planck mass object can now be defined as a point (it will have point co-ordinates at each unit of Planck time and so can be mapped). As the simulation is dividing the mass of objects into these Planck mass size points and then rotating these points around each other as point-to-point orbital pairs, then by definition gravity is a mass to mass interaction. 

Nevertheless, although this is a mass-point to mass-point rotation, and so referred to here as a point-point orbital, it is still a particle to particle orbital, albeit the particles are both in the mass state. We can also map individual particle to particle orbitals albeit as gravitational orbits, the H atom is a well-researched particle-to-particle orbital pair (an electron orbiting a proton) and so can be used as reference. To map orbital transitions between energy levels, the simulation uses the [[v:Quantum_gravity_(Planck)#Photon_orbital_model |photon-orbital model]], in which the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase. The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton). 

Orbital momentum derives from this orbital radius, it is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus. Furthermore, orbital transition (between orbitals) occurs between the orbital radius and the photon, the electron has a passive role. Transition (the electron path) follows a specific [[v:Fine-structure_constant_(spiral) |hyperbolic spiral]] for which the angle component periodically converges to give integer radius where ''r'' = Bohr radius; at 360° radius =4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r''. As these spiral angles (360°, 360+120°, 360+180°, 360+216° ...) are linked directly to pi, and as the electron is following a semi-classical gravitational orbit, this particular quantization has a geometrical origin.

Although the simulation is not optimized for atomic orbitals (the nucleus is treated simply as a cluster of points), the transition period '''t''' measured between these integer radius can be used to solve the transition frequencies '''f''' via the general formula &lt;math&gt;f/c = t \lambda_H/(n_f^2-n_i^2)&lt;/math&gt;. 
 
In summary, both gravitational and atomic orbitals reflect the same particle-to-particle orbital pairing, the distinction being the state of the particles; gravitational orbitals are mass to mass whereas atomic orbitals are predominately wave to wave. There are not 2 separate forces used by the simulation, instead particles are treated as oscillations between the 2 states (electric wave and mass point). The gravity-mass Bohr model can then be seen as complementary to the electric-wave Schrödinger equation.


=== N-body orbitals ===
[[File:8body-27orbital-gravitational-orbit.gif|thumb|right|640px|8-body (8 mass points, 28 orbitals), the resulting orbit is a function of the start positions of each point]]

The simulation universe is a 4-axis hypersphere expanding in increments &lt;ref&gt;Macleod, Malcolm; {{Cite journal |title=2. Programming cosmic microwave background for Planck unit Simulation Hypothesis modelling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.31308.16004/7 }}&lt;/ref&gt; with 3-axis (the [[v:Black-hole_(Planck)  |hypersphere surface]]) projected onto an (''x'', ''y'') plane with the ''z'' axis as the simulation timeline (the expansion axis). Each point is assigned start (''x'', ''y'', ''z'' = 0) co-ordinates and forms pairs with all other points, resulting in a universe-wide n-body network of point-point orbital pairs. The barycenter for each orbital pairing is its center, the points located at each orbital 'pole'.

The simulation itself is dimensionless, simply rotating circles. To translate to dimensioned gravitational or atomic orbits, we can use the Planck units ([[w:Planck mass |Planck mass m&lt;sub&gt;P&lt;/sub&gt;]], [[w:Planck length |Planck length l&lt;sub&gt;p&lt;/sub&gt;]], [[w:Planck time |Planck time t&lt;sub&gt;p&lt;/sub&gt;]]), such that the simulation increments in discrete steps (each step assigned as 1 unit of Planck time), during each step (for each unit of Planck time), the orbitals rotate 1 unit of (Planck) length (at velocity ''c'' = ''l''&lt;sub&gt;p&lt;/sub&gt;/''t''&lt;sub&gt;p&lt;/sub&gt;) in hyper-sphere co-ordinates. These rotations are then all summed and averaged to give new point co-ordinates. As this occurs for every point before the next increment to the simulation clock (the next unit of Planck time), the orbits can be updated in 'real time' (simulation time) on a serial processor. 


Orbital pair rotation on the (''x'', ''y'') plane occurs in discrete steps according to an angle '''β'''  as defined by the orbital pair radius (the atomic orbital '''β''' has an additional alpha term). 

:&lt;math&gt;\beta_{gravity} = \frac{1}{r_{ij} r_{orbital} \sqrt{r_{orbital}}}&lt;/math&gt; 

:&lt;math&gt;\beta_{atomic} = \frac{1}{\sqrt{2\alpha} r_{orbital} \sqrt{r_{orbital}}}&lt;/math&gt;


As the simulation treats each (point-point) orbital independently (independent of all other orbitals), no information regarding the points (other than their initial start coordinates) is required by the simulation.

Although orbital and so point rotation occurs at ''c'', the [[v:Relativity (Planck) |hyper-sphere expansion]] &lt;ref&gt;Macleod, Malcolm; {{Cite journal |title=1. Programming relativity for Planck scale Simulation Hypothesis modeling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.18574.00326/3 }}&lt;/ref&gt; is equidistant and so `invisible' to the observer. Instead observers (being constrained to 3D space) will register these 4-axis orbits (in hyper-sphere co-ordinates) as a circular motion on a 2-D plane (in 3-D space). An apparent [[w:Time_dilation |time dilation]] effect emerges as a consequence.


[[File:4body-orbital-3x10x-gravitational-orbit.gif|thumb|right|640px|Symmetrical 4 body orbit; (3 center mass points, 1 orbiting point, 6 orbital pairs). Note that all points orbit each other.]]
==== 2 body orbits ('''x, y''' plane) ====
For simple 2-body orbits, to reduce computation only 1 point is assigned as the orbiting point and the remaining points are assigned as the central mass. For example the ratio of earth mass to moon mass is 81:1 and so we can simulate this orbit accordingly. However we note that the only actual distinction between a 2-body orbit and a complex orbit being that the central mass points are assigned ('''x, y''') co-ordinates relatively close to each other, and the orbiting point is assigned ('''x, y''') co-ordinates distant from the central points (this becomes the orbital radius) ... this is because the simulation treats all points equally, the center points also orbiting each other according to their orbital radius, for the simulation itself there is no difference between simple 2-body and complex n-body orbits. 

The [[w:Schwarzschild radius |Schwarzschild radius]] formula in Planck units

:&lt;math&gt;r_s = \frac{2 l_p M}{m_P}&lt;/math&gt;

As the simulation itself is dimensionless, we can remove the dimensioned length component &lt;math&gt;2 l_p&lt;/math&gt;, and as each point is analogous to 1 unit of Planck mass &lt;math&gt;m_P&lt;/math&gt;, then the Schwarzschild radius for the simulation becomes the number of central mass points. We then assign ('''x, y''') co-ordinates (to the central mass points) within a circle radius &lt;math&gt;r_s&lt;/math&gt; = number of central points = total points - 1 (the orbiting point).


After every orbital has rotated 1 length unit (anti-clockwise in these examples), the new co-ordinates for each rotation per point are then averaged and summed, the process then repeats. After 1 complete orbit (return to the start position by the orbiting point), the period '''t''' (as the number of increments to the simulation clock) and the ('''x, y''') plane orbit length '''l''' (distance as measured on the 2-D plane) are noted. 

'''Key''':

1. '''i''' = r&lt;sub&gt;s&lt;/sub&gt;; the number of center mass points (the orbited object). 

2. '''j''' = total number of points, as here there is only 1 orbiting point; '''j''' = '''i''' + 1

3. '''k&lt;sub&gt;r&lt;/sub&gt;''' is a mass to radius co-efficient in the form &lt;math&gt;j_{max} = (k_r i + 1)&lt;/math&gt;. This function defines orbital radius in terms of the central mass Schwarzschild radius (&lt;math&gt;i&lt;/math&gt;) and the orbiting point (1), thus quantizing the radius. When &lt;math&gt;k_r&lt;/math&gt; = 1 then &lt;math&gt;j_{max} = j&lt;/math&gt;, and the radius is at a minimum giving an analogue gravitational principal quantum number &lt;math&gt;n_g = j_{max}/j&lt;/math&gt;.

4. '''x, y''' = start co-ordinates for each point (on a 2-D plane), '''z''' = 0. 

5. '''r&lt;sub&gt;α&lt;/sub&gt;''' = a radius constant, here r&lt;sub&gt;α&lt;/sub&gt; = sqrt(2α) = 16.55512; where alpha = inverse [[w:fine structure constant |fine structure constant]] = 137.035 999 084 (CODATA 2018). This constant adapts the simulation specifically to gravitational and atomic orbitals.

:&lt;math&gt;r_{\alpha} = \sqrt{2\alpha} &lt;/math&gt;

:&lt;math&gt;r_{orbit} = {r_{\alpha}}^2 \;*\; r_{wavelength} &lt;/math&gt;

6. Rotation angle β
:&lt;math&gt;\beta_{orbital} = \frac{1}{r_{ij} r_{orbital} \sqrt{r_{orbital}}}&lt;/math&gt; 

:&lt;math&gt;r_{ij} = \sqrt{\frac{2 j}{i}}&lt;/math&gt; (for each gravitational orbital in the simulation)

:&lt;math&gt;r_{ij} = \sqrt{2 \alpha}&lt;/math&gt; (for each atomic orbital in the simulation)


==== Orbital formulas (2-D plane)====

:&lt;math&gt;j = i +  1&lt;/math&gt;

:&lt;math&gt;r_{orbit} = 2 \alpha 2 \frac{(k_r i + 1)^2}{i^2}&lt;/math&gt;, orbital radius (center mass to point)

:&lt;math&gt;r_{ij} = \sqrt{\frac{i}{j}}&lt;/math&gt; (averaged for each orbit)

:&lt;math&gt;t_{orbit} = \frac{2\pi}{ \beta_{orbit}} = 16 \pi {\alpha}^{3/2} \frac{{(k_r i + 1)}^3}{i^{5/2} j^{1/2}}&lt;/math&gt;, orbiting point period

:&lt;math&gt;r_{barycenter} = \frac{r_{orbit}}{j}&lt;/math&gt;

:&lt;math&gt;l_{orbit} = 2 \pi (r_{orbit} - r_{barycenter})&lt;/math&gt;, distance travelled by orbiting point

:&lt;math&gt;v_{orbit} = \sqrt{\frac{i}{r_{orbit}j}}&lt;/math&gt;, orbiting point velocity  


Examples (dimensionless). The simulation parameters agree closely with the calculated parameters:

a)
:source code Gravitational-Orbital-simulation-2body.py &lt;ref&gt;https://codingthecosmos.com/files/Gravitational-Orbital-simulation-2body.py Gravitational-Orbital-simulation-2body.py&lt;/ref&gt;&lt;ref&gt;https://codingthecosmos.com/#coding simulation source codes&lt;/ref&gt;
:points = 8 (1 orbiting point and 7 center mass points) 
:i = 7, j = 8
:k&lt;sub&gt;r&lt;/sub&gt; = 32
:&lt;math&gt;\sqrt{2j/i}&lt;/math&gt; = 1.511858


Calculated
:calculated orbit period = 2504836149.00059
:calculated orbit radius = 566322.241497
:calculated orbit length = 3113519.13854
:calculated orbit barycenter = 70790.280187, 0
:n&lt;sub&gt;g&lt;/sub&gt; = (k&lt;sub&gt;r&lt;/sub&gt; i + 1)/j = 28.125 


Simulation
 simulation orbit period = 2504839696   (simulated/calculated = 1.000001416)    
 simulation orbit length = 3113519.129787637 (1.0000000028)             
 simulation orbit barycenter; x = 70790.28092,  y = 0.000732
 simulation orbit radius = 566322.240887


b)
:points = 82 (1 orbiting point and 81 center mass points)
:i = 81, j = 82
:k&lt;sub&gt;r&lt;/sub&gt; = 14
:&lt;math&gt;\sqrt{2j/i}&lt;/math&gt; = 1.422916


Calculated
:calculated orbit period = 220490545.8705734
:calculated orbit radius = 107625.788776
:calculated orbit length = 667986.0335558
:calculated orbit barycenter = 1312.5096192, 0
:n&lt;sub&gt;g&lt;/sub&gt; = (k&lt;sub&gt;r&lt;/sub&gt; i + 1)/j ratio = 13.84 


Simulation
 simulation orbit period = 222295442   (1.0081858)    
 simulation orbit length = 667954.5848732             
 simulation orbit barycenter; x = 1309.983113,  y = 19.760475


===== Earth moon orbit===== 
The earth to moon mass ratio approximates 81:1 and so can be simulated as a 2-body orbit with the moon as a single orbiting point as in the above example. Here we use the orbital parameters to determine the value for the mass to radius coefficient k&lt;sub&gt;r&lt;/sub&gt;. Planck length &lt;math&gt;l_p&lt;/math&gt;, Planck mass &lt;math&gt;m_P&lt;/math&gt; and &lt;math&gt;c&lt;/math&gt; are used to convert between the dimensionless formulas and dimensioned SI units.

Reference values

:&lt;math&gt;M&lt;/math&gt; = 5.9722 x 10&lt;sup&gt;24&lt;/sup&gt;kg (earth)

:&lt;math&gt;m&lt;/math&gt; = 7.346 x 10&lt;sup&gt;22&lt;/sup&gt;kg (moon)

:&lt;math&gt;T_{orbit}&lt;/math&gt; = 27.321661*86400 = 2360591.51s


To simplify, we assume a circular orbit which then gives us this radius

:&lt;math&gt;R_{orbit} = (\frac{G (M+m) T_{orbit}^2}{4 \pi^2})^{(1/3)}&lt;/math&gt; = 384714027m 

:&lt;math&gt;G = \frac{l_p c^2}{m_P}&lt;/math&gt; = 0.66725e-10 


The mass ratio

:&lt;math&gt;i = \frac{M}{m}&lt;/math&gt; = 81.298666, j = i + 1


We then find a value for &lt;math&gt;k_r&lt;/math&gt; using T&lt;sub&gt;orbit&lt;/sub&gt; as our reference (reversing the orbit period equation). 

:&lt;math&gt;T_o = T_{orbit} \frac{m_P}{M} \frac{c}{l_p} = 16 \pi {\alpha}^{3/2} \frac{(k_r i + 1)^3}{i^{5/2} j^{1/2}}&lt;/math&gt; (dimensionless orbital period) 

:&lt;math&gt;k_r = \frac{1}{i} {(\frac{T_o i^{5/2} j^{1/2}}{16 \pi {\alpha}^{3/2}})}^{(1/3)} - \frac{1}{i}&lt;/math&gt; = 12581.4468


Dimensionless solutions

:&lt;math&gt;r_{orbit} = 2 \alpha 2 \frac{(k_r i + 1)^2}{i^2}&lt;/math&gt; = 86767420100 

:&lt;math&gt;t_{orbit} = 16 \pi {\alpha}^{3/2} \frac{{(k_r i + 1)}^3}{i^{5/2} j^{1/2}}&lt;/math&gt; = 0.159610040233 x 10&lt;sup&gt;18&lt;/sup&gt;

:&lt;math&gt;r_{barycenter} = \frac{r_{orbit}}{j}&lt;/math&gt; = 1054299229.62

:&lt;math&gt;l_{orbit} = 2 \pi (r_{orbit} - r_{barycenter})&lt;/math&gt; = 538551421685

:&lt;math&gt;v = \sqrt{\frac{i}{r_{orbit}j}}&lt;/math&gt; = 0.33741701 x 10&lt;sup&gt;-5&lt;/sup&gt;


Converting back to dimensioned values

:&lt;math&gt;R = r_{orbit} l_p \frac{M}{m_P} = R_{orbit}&lt;/math&gt; = 384714027m 

:&lt;math&gt;T = t_{orbit} \frac{l_p}{c} \frac{M}{m_P} = T_{orbit}&lt;/math&gt; = 2360591.51s

:&lt;math&gt;B = \frac{R}{j}&lt;/math&gt; = 4674608.301m (barycenter)

:&lt;math&gt;L = 2\pi (R - B)&lt;/math&gt; = 2387858091.51m (distance moon travelled around the barycenter)

:&lt;math&gt;V = c \sqrt{\frac{i}{r_{orbit}j}}&lt;/math&gt; = 1011.551m/s (velocity of the moon around the barycenter)


If we expand the velocity term

:&lt;math&gt;v_{orbit} = c \sqrt{\frac{i}{r_{orbit}j}}&lt;/math&gt;

:&lt;math&gt;v_{orbit}^3 = \frac{G M}{T_{orbit}} 2\pi \frac{i^2}{j^2}&lt;/math&gt;


Note: The [[w:standard gravitational parameter | standard gravitational parameter]] ''μ'' is the product of the gravitational constant ''G'' and the mass ''M'' of that body. For several objects in the Solar System, the value of ''μ'' is known to greater accuracy than either ''G'' or ''M''.

:&lt;math&gt;\mu_{earth}&lt;/math&gt; = 3.986004418(8)e14
:&lt;math&gt;\mu_{moon}&lt;/math&gt; = 4.9048695(9)e12
:&lt;math&gt;i = \frac{\mu_{earth}}{\mu_{moon}}&lt;/math&gt; = 81.2662685

:&lt;math&gt;k_r = \frac{c}{2 \sqrt{\alpha}} {(\frac{T_{orbit}}{2 \pi \mu_{earth}})}^{1/3} \frac{(i+1)}{i}^{1/6} - \frac{1}{i}&lt;/math&gt; = 12580.3462

:&lt;math&gt;t_{orbit}&lt;/math&gt; = 0.15956776936 x 10&lt;sup&gt;18&lt;/sup&gt;

:&lt;math&gt;r_{orbit}&lt;/math&gt; = 86752239934 

 
===== Kepler's formula = G=====
Kepler's formula reduces to G

:&lt;math&gt;R = 2 \alpha 2 (\frac{k_r i + 1}{i})^2 l_p \frac{M}{m_P}&lt;/math&gt;

:&lt;math&gt;T = 16 \pi {\alpha}^{3/2} \frac{{(k_r i + 1)}^3}{i^{5/2} (i+1)^{1/2}} \frac{l_p}{c} \frac{M}{m_P}&lt;/math&gt;

:&lt;math&gt;M+m = M (\frac{i+1}{i})&lt;/math&gt;

:&lt;math&gt;\frac{4 \pi^2 R^3}{(M+m) T} = \frac{l_p c^2}{m_P} = G&lt;/math&gt;


==== Orbital vs. Newton ====
A 3-body orbit is compared with the equivalent orbit using Newtonian dynamics. The start positions are the same
 r0=2*α; x1=3490.3069; y1=0; x2=cos(pi*2/3)*r0; y2=sin(pi*2/3)*r0; x3=cos(pi*2/3)*r0; y3=sin(pi*2/3)*r0
The m1 orbit (orbiting point) was used to determine the optimal G for the Newtonian orbit (G = 0.4956). The initial velocities for the Newtonian orbit were estimated by taken the difference between the first 2 orbital points (x1, y1), (x2, y2), (x3, y3) 
 1,    3490.306899038,    0.000000000,    -137.035999177,    237.353313041,     -137.035999177,    -237.353313041
 2,    3490.306898988,    0.013513518,    -137.055178780,    237.346554801,     -137.016819523,    -237.360068318

Period of orbit (&lt;math&gt;k_r&lt;/math&gt; = 2.19006)
:&lt;math&gt;t_{calc}&lt;/math&gt; = 1122034 
:&lt;math&gt;t_{orbital}&lt;/math&gt; = 1121397
:&lt;math&gt;t_{newton}&lt;/math&gt; = 1125633

# Code for calculating Newtonian orbit; Newton-vs-Orbital_Newton.py &lt;ref&gt;https://codingthecosmos.com/files/Newton-vs-Orbital_Newton.py Newton-vs-Orbital_Newton.py&lt;/ref&gt;
# Orbital data file; data-3b-long.txt&lt;ref&gt;https://codingthecosmos.com/files/data-3b-long.txt  data-3b-long.txt&lt;/ref&gt;
# Comparison Orbital data; Newton-vs-Orbital_Orbital.py &lt;ref&gt;https://codingthecosmos.com/files/Newton-vs-Orbital_Orbital.py  Newton-vs-Orbital_.py&lt;/ref&gt;


Qu for AI. These plots are of a 3-body gravitational orbit using 2 different simulation methods; Orbital and Newtonian. 
Analyze the plots and then compare the 2 orbital methods. (note: access to links may require to login, chatgpt does not save when text has images).

1. Analysis by Qwen&lt;ref&gt;https://chat.qwen.ai/s/c09d84e6-88e4-4c5a-b543-ca0c3315df23?fev=0.0.85 Qwen on orbital vs Newtonian for a 3-body orbit&lt;/ref&gt;
:Orbital Method  
  **Key Strength:** Demonstrates superior long-term orbital stability, maintaining circular orbits for all bodies (m1, m2, m3) without significant precession or drift.  
  **Methodological Insight:** Likely employs a higher-order or adaptive-step integration scheme, or incorporates corrections to preserve angular momentum and energy. This aligns with its ability to sustain symmetry and minimize cumulative errors over time.  
  **Probable Applications:**  
    - High-precision astrophysical simulations (e.g., exoplanet dynamics, binary star systems).  
    - Long-term stability studies (e.g., solar system evolution, galactic dynamics).  
  **Symmetry:**   
    - Maintains near-perfect symmetry in both $ x_2 - x_3 $ and $ y_2 + y_3 $ deviations (see symmetry check plot), indicating robust conservation of system invariants.  
    - **Methodological Advantage:** Likely incorporates explicit constraints or higher-order force calculations to enforce symmetry, reducing numerical artifacts. 

:Newtonian Method  
  **Key Strength:** Simplicity and computational efficiency via the leapfrog integrator, which conserves phase-space volume and energy over short timescales.  
  **Limitations:** Accumulates numerical errors in long-term simulations, leading to orbital precession (e.g., m2/m3 drifting into elliptical orbits) and asymmetry (notably in the y-direction).  
  **Probable Applications:**  
    - Rapid prototyping or educational demonstrations of gravitational systems.  
    - Short-term simulations where exact energy conservation is less critical (e.g., collisional stellar dynamics).  
  **Symmetry:** 
    - Exhibits growing asymmetry in $ y_2 + y_3 $ deviations, suggesting energy leakage and phase errors.  
    - **Root Cause:** The leapfrog integrator, while symplectic, is a second-order method prone to secular drift in angular momentum and energy over long integration periods.  


2. Analysis by Claude&lt;ref&gt;https://claude.ai/share/5b0f07aa-c32b-460f-827c-3d6e4ee2c80f Claude on orbital vs Newtonian for a 3-body orbit&lt;/ref&gt;
:Orbital Method Advantages:
     Superior long-term stability for predicting general orbital structure
     Better preservation of symmetry properties
     More consistent orbital parameters over extended timeframes
     Potentially more efficient for extremely long simulations where qualitative behavior is more important than exact positions
:Newtonian Method Advantages:
     More physically complete treatment of gravitational interactions
     Better representation of subtle dynamical evolution
     Allows natural emergence of secular effects
     More suitable for studying the intrinsic chaos of three-body systems


==== Gravitational coupling constant ====
In the above, the points were assigned a mass as a theoretical unit of Planck mass. Conventionally, the [[w:Gravitational coupling constant | Gravitational coupling constant]] ''α&lt;sub&gt;G&lt;/sub&gt;'' characterizes the gravitational attraction between a given pair of elementary particles in terms of a particle (i.e.: electron) mass to Planck mass ratio;

:&lt;math&gt;\alpha_G = \frac{G m_e^2}{\hbar c} = (\frac{m_e}{m_P})(\frac{m_e}{m_P}) = 1.75... x10^{-45}&lt;/math&gt;

For the purposes of this simulation, particles are treated as an oscillation between an electric wave-state (duration particle frequency) and a mass point-state (duration 1 unit of Planck time). This inverse α&lt;sub&gt;G&lt;/sub&gt; then represents the probability that any 2 electrons will be in the mass point-state at any unit of Planck time ([[v:Electron_(mathematical) |wave-mass oscillation at the Planck scale]] &lt;ref&gt;Macleod, M.J. {{Cite journal |title= Programming Planck units from a mathematical electron; a Simulation Hypothesis |journal=Eur. Phys. J. Plus |volume=113 |pages=278 |date=22 March 2018 | doi=10.1140/epjp/i2018-12094-x }}&lt;/ref&gt;). 

:&lt;math&gt;{\alpha_G}^{-1} = \frac{m_P^2}{m_e^2} = 0.57... x10^{45}&lt;/math&gt;

As mass is not treated as a constant property of the particle, measured particle mass becomes the averaged frequency of discrete point mass at the Planck level. If 2 dice are thrown simultaneously and a win is 2 'sixes', then approximately every (1/6)x(1/6) = (1/36) = 36 throws (frequency) of the dice will result in a win. Likewise, the inverse of α&lt;sub&gt;G&lt;/sub&gt; is the frequency of occurrence of the mass point-state between the 2 electrons. As 1 second requires 10&lt;sup&gt;42&lt;/sup&gt; units of Planck time (&lt;math&gt;t_p = 10^{-42}s&lt;/math&gt;), this occurs about once every 3 minutes.

:&lt;math&gt;\frac{{\alpha_G}^{-1}}{t_p}&lt;/math&gt;

Gravity now has a similar magnitude to the strong force (at this, the Planck level), albeit this interaction occurs seldom (only once every 3 minutes between 2 electrons), and so when averaged over time (the macro level), gravity appears weak.  

If particles oscillate between an electric wave state to Planck mass (for 1 unit of Planck-time) point-state, then at any discrete unit of Planck time, a number of particles will simultaneously be in the mass point-state. If an assigned point contains only electrons, and as the frequency of the electron = f&lt;sub&gt;e&lt;/sub&gt;, then the point will require 10&lt;sup&gt;23&lt;/sup&gt; electrons so that, on average for each unit of Planck time there will be 1 electron in the mass point state, and so the point will have a mass equal to Planck mass (i.e.: experience continuous gravity at every unit of Planck time).

:&lt;math&gt;f_e = \frac{m_P}{m_e} = 10^{23}&lt;/math&gt;

For example a 1kg satellite orbits the earth, for any given unit of Planck time, satellite (B) will have &lt;math&gt;1kg/m_P = 45940509&lt;/math&gt; particles in the point-state. The earth (A) will have &lt;math&gt;5.9738 \;x10^{24} kg/m_P = 0.274 \;x10^{33}&lt;/math&gt; particles in the point-state, and so the earth-satellite coupling constant becomes the number of rotating orbital pairs (at unit of Planck time) between earth and the satellite;

:&lt;math&gt;N_{orbitals} =  (\frac{m_A}{m_P})(\frac{m_B}{m_P})  = 0.1261\; x10^{41}&lt;/math&gt;
 

Examples:

:&lt;math&gt;i = \frac{M_{earth}}{m_P} = 0.27444  \;x10^{33}&lt;/math&gt; (earth as the center mass)

:&lt;math&gt;i 2 l_p = 0.00887&lt;/math&gt; (earth Schwarzschild radius)

:&lt;math&gt;s = \frac{1kg}{m_P} = 45940509&lt;/math&gt; (1kg orbiting satellite)

:&lt;math&gt;j = N_{orbitals} = i*s = 0.1261  \;x10^{41}&lt;/math&gt;


1) 1kg satellite at earth surface orbit 

:&lt;math&gt;r_{o} = 6371000 km&lt;/math&gt; (earth surface)

:&lt;math&gt;j_{max} = \frac{j}{r_\alpha}\sqrt{\frac{r_{o}}{i l_p}} = 0.288645\;x10^{44}&lt;/math&gt;

:&lt;math&gt;n_g = \frac{j_{max}}{j} = 2289.41&lt;/math&gt;

:&lt;math&gt;r = r_{\alpha}^2 n_g^2 i l_p = r_{o} &lt;/math&gt;

:&lt;math&gt;v = \frac{c}{n_g r_{\alpha}} = 7909.7924&lt;/math&gt; m/s

:&lt;math&gt;t = 2 \pi \frac{r_{outer}}{v_{outer}} = 5060.8374&lt;/math&gt; s 


2) 1kg satellite at a synchronous orbit radius

:&lt;math&gt;r_o = 42164.17 km&lt;/math&gt; 

:&lt;math&gt;j_{max} =  \frac{j}{r_\alpha} \sqrt{\frac{r_{o}}{i l_p}}  = 0.74256\;x10^{44}&lt;/math&gt;

:&lt;math&gt;n_g = \frac{j_{max}}{j} = 5889.674&lt;/math&gt;

:&lt;math&gt;r = r_{\alpha}^2 n_g^2 i l_p = r_{o} &lt;/math&gt;

:&lt;math&gt;v = \frac{c}{n_g r_{\alpha}} = 3074.66&lt;/math&gt; m/s

:&lt;math&gt;t = 2 \pi \frac{r_{outer}}{v_{outer}} = 86164.09165&lt;/math&gt; s 


3) The energy required to lift a 1 kg satellite into geosynchronous orbit is the difference between the energy of each of the 2 orbits (geosynchronous and earth). 

:&lt;math&gt;E_{orbital} = \frac{h c}{2 \pi r_{6371}} - \frac{h c}{2 \pi r_{42164}} = 0.412 x10^{-32}J&lt;/math&gt; (energy per orbital)

:&lt;math&gt;N_{orbitals} = \frac{M_{earth}m_{satellite}}{m_P^2} = 0.126 x10^{41}&lt;/math&gt; (number of orbitals)

:&lt;math&gt;E_{total} = E_{orbital} N_{orbitals} = 53 MJ/kg&lt;/math&gt;


4) The orbital angular momentum of the planets derived from the angular momentum of the respective orbital pairs.

:&lt;math&gt;N_{sun} = \frac{M_{sun}}{m_P} &lt;/math&gt;

:&lt;math&gt;N_{planet} = \frac{M_{planet}}{m_P} &lt;/math&gt;

:&lt;math&gt;N_{orbitals} = N_{sun}N_{planet} &lt;/math&gt;

:&lt;math&gt;n_g = \sqrt{\frac{R_{radius} m_P}{2 \alpha l_p M_{sun}}} &lt;/math&gt;

:&lt;math&gt;L_{oam} = 2\pi \frac{M r^2}{T} = N_{orbitals} n_g\frac{h}{2\pi} \sqrt{2 \alpha},\;\frac{kg m^2}{s} &lt;/math&gt;

The orbital angular momentum of the planets;

 mercury = .9153 x10&lt;sup&gt;39&lt;/sup&gt;  
 venus    = .1844 x10&lt;sup&gt;41&lt;/sup&gt;  
 earth    = .2662 x10&lt;sup&gt;41&lt;/sup&gt; 
 mars     = .3530 x10&lt;sup&gt;40&lt;/sup&gt; 
 jupiter   = .1929 x10&lt;sup&gt;44&lt;/sup&gt;   
 pluto   = .365 x10&lt;sup&gt;39&lt;/sup&gt;   


Orbital angular momentum combined with orbit velocity cancels ''n&lt;sub&gt;g&lt;/sub&gt;'' giving an orbit constant. Adding momentum to an orbit will therefore result in a greater distance of separation and a corresponding reduction in orbit velocity accordingly.

:&lt;math&gt;L_{oam}v_g = N_{orbitals} \frac{h c}{2\pi},\;\frac{kg m^3}{s^2} &lt;/math&gt;


==== Precession ====
Precession is a change in the orientation of the rotational axis of a rotating body. The first of three tests to establish observational evidence for the theory of general relativity, as proposed by Albert Einstein in 1915, concerned the &quot;anomalous&quot; precession of the[[w:Tests_of_general_relativity | perihelion of Mercury]]. 


Schwarzschild radius &lt;math&gt;\lambda_{sun} = i 2 l_p&lt;/math&gt; = 2953.25m

from
:&lt;math&gt;r_{o} = 2 \alpha {n_g}^2 i l_p = \alpha {n_g}^2 \lambda_{sun}&lt;/math&gt;

:&lt;math&gt;n_g = \frac{j_{max}}{j}&lt;/math&gt;


semi-minor axis: &lt;math&gt;b = \alpha l^2 \lambda_{sun}&lt;/math&gt;

semi-major axis: &lt;math&gt;a = \alpha n^2 \lambda_{sun}&lt;/math&gt;

radius of curvature :&lt;math&gt;L = \frac{b^2}{a} = \frac{a l^4 \lambda_{sun}}{n^2}&lt;/math&gt;

:&lt;math&gt;\frac{3 \lambda_{sun}}{2 L} = \frac{3 \alpha n^2}{2 \alpha^2 l^4} = \frac{3 n^2}{2 \alpha l^4}&lt;/math&gt;


The conversion factor from radians to arcseconds

2π = 1296000 arcseconds


&lt;math&gt;T_{earth}&lt;/math&gt; = 365.25 days

arcseconds per 100 years (drift) 

drift = &lt;math&gt;\frac{3 n^2}{2 \alpha l^4} * 1296000 * \frac{100 T_{earth}}{T_{planet}}&lt;/math&gt; 


By inserting the above formulas for ''a'' and ''b'', the standard formula for precession reduces to &lt;ref&gt;https://codingthecosmos.com/ai_pdf/Chat-Orbital-Precession-in-Planck-Units.pdf  ChatGPT precession -pdf&lt;/ref&gt; (note: all 3 tests in terms of α, ''n'', ''l'' are shown here &lt;ref&gt;https://chatgpt.com/share/684d6375-238c-8012-b1b8-905c80daaf07 ChatGPT 3 tests of relativity -online&lt;/ref&gt;&lt;ref&gt;https://codingthecosmos.com/ai_pdf/Chat_3tests-of-general-relativity.pdf ChatGPT 3 tests of relativity -pdf&lt;/ref&gt;);

&lt;math&gt;\frac{6\pi G M}{a (1-e^2) c^2} = \frac{3 \pi n^2}{\alpha l^4} = \frac{3 n^2}{2 \alpha l^4} * 1296000&lt;/math&gt;


 Mercury (eccentricity = 0.205630)
 T = 87.9691 days
 a = 57909050 km (''n'' = 378.2734) 
 b = 56671523 km (''l'' = 374.2096)
 drift = 42.98

 Venus (eccentricity = 0.006772) 
 T = 224.701 days
 a = 108208000 km (''n'' = 517.085) 
 b = 108205519 km (''l'' = 517.079)
 drift = 8.6247

 Earth (eccentricity = 0.0167)
 T = 365.25 days
 a = 149598000 km (''n'' = 607.989) 
 b = 149577138 km (''l'' = 607.946)
 drift = 3.8388

 Mars (eccentricity = 0.0934)
 T = 686.980 days
 a = 227939366 km (''n'' = 750.485) 
 b = 226942967 km (''l'' = 748.843)
 drift = 1.351


[[File:Gravitational-potential-energy-8body-1-2.gif|thumb|right|640px|8-body circular orbit plus 1-body with opposing orbitals 1:2]]
==== Orbital alignment ====
Orbital trajectory is a measure of alignment of the orbitals. In the above examples, all orbitals rotate in the same direction = aligned. If all orbitals are unaligned the object will appear to 'fall' = straight line orbit (source code &lt;ref&gt;https://codingthecosmos.com/files/ Gravitational-orbitals-2body-elliptical-05-2025.c&lt;/ref&gt;).

In this example, for comparison, onto an 8-body orbit (blue circle orbiting the center mass green circle), is imposed a single point (yellow dot) with a ratio of 1 orbital (anti-clockwise around the center mass) to 2 orbitals (clockwise around the center mass) giving an elliptical orbit.

The change in orbit velocity (acceleration towards the center and deceleration from the center) derives automatically from the change in the orbital radius (there is no barycenter). 

The orbital drift (as determined where the blue and yellow meet) is due to orbiting points rotating around each other. Note: if all orbitals are aligned only along the orbital path, as in this simulation, then the semi-major axis = orbital radius.


AI comparisons between angular momentum (standard physics) and orbital alignment; conclusions and links; 

1. Grok: The classical model uses angular momentum to shape orbits, while the orbital pair model uses alignment, yet both parameters function analogously: maximum values yield circular orbits, zero values yield linear paths, and intermediate values yield ellipses. This functional similarity, supported by a possible mathematical mapping (e.g., heff∝∣S∣h), demonstrates their mutual consistency in describing orbital dynamics. &lt;ref&gt;https://x.com/i/grok/share/RJ1ap3xqJsKr0N71JlKoXdnj5 Grok angular momentum and orbital alignment&lt;/ref&gt; 

2. Chat: Thus angular momentum in the usual sense and orbital-pair alignment at the Planck scale are two faces of the same coin: they both track exactly the net rotational “pull” around the center, and this single scalar L fully determines whether you orbit in a circle, ellipse, or straight line.&lt;ref&gt;https://chatgpt.com/share/6858a28b-18f8-8012-9ccc-72c46711cc08 
 Chat angular momentum and orbital alignment&lt;/ref&gt; 

3. Qwen: Key Equation .... This bridges classical angular momentum and the alignment framework, proving their consistency &lt;ref&gt;https://chat.qwen.ai/s/f92dc700-9b68-4943-8da4-3c575d10168f Qwen angular momentum and orbital alignment&lt;/ref&gt;

4. DeepSeek: The classical angular momentum model and the Planck-scale orbital alignment model describe the same physical reality at different scales ... This consistency arises because the Planck-scale model quantizes angular momentum, preserving its vector nature and conservation—proving it is not a replacement but a discrete reformulation of classical principles &lt;ref&gt;https://codingthecosmos.com/ai_pdf/DeepSeek_Angular-Momentum-vs-Alignment.pdf DeepSeek angular momentum and orbital alignment&lt;/ref&gt;.


[[File:orbit-points32-orbitals496-clumping-over-time.gif|thumb|right|640px|32 mass points (496 orbitals) begin with random co-ordinates, after 2&lt;sup&gt;32&lt;/sup&gt; steps they have clumped to form 1 large mass and 2 orbiting masses.]]
==== Freely moving points ====
The simulation calculates each point as if freely moving in space, and so is useful with 'dust' clouds where the freedom of movement is not restricted.   

In this animation, 32 mass points begin with random co-ordinates (the only input parameter here are the start (''x'', ''y'') coordinates of each point). We then fast-forward 2&lt;sup&gt;32&lt;/sup&gt; steps to see that the points have now clumped to form 1 larger mass and 2 orbiting masses. The larger center mass is then zoomed in on to show the component points are still orbiting each other, there are still 32 freely orbiting points, only the proximity between them has changed, they have formed ''planets''. 


[[File:relativistic-quantum-gravity-orbitals-codingthecosmos.png|thumb|right|480px|Illustration of B's cylindrical orbit relative to A's time-line axis]]
==== Hyper-sphere orbit ====
{{main|Relativity (Planck)}}

Each point moves 1 unit of (Planck) length per 1 unit of (Planck) time in '''x, y, z''' (hyper-sphere) co-ordinates, the simulation 4-axis hyper-sphere universe expanding in uniform (Planck) steps (the simulation clock-rate) as the origin of the speed of light, and so (hyper-sphere) time and velocity are constants. Particles are pulled along by this expansion, the expansion as the origin of motion, and so all objects, including orbiting objects, travel at, and only at, the speed of light in these hyper-sphere co-ordinates &lt;ref&gt;Macleod, Malcolm; {{Cite journal |title=1. Programming relativity for Planck unit Simulation Hypothesis modelling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.18574.00326/3 }}&lt;/ref&gt;. Time becomes [[v:God_(programmer)#Universe_time-line |time-line]]. 

While ''B'' (satellite) has a circular orbit period on a 2-axis plane (the horizontal axis representing 3-D space) around ''A'' (planet), it also follows a cylindrical orbit (from B&lt;sup&gt;1&lt;/sup&gt; to B&lt;sup&gt;11&lt;/sup&gt;) around the ''A'' time-line (vertical expansion) axis ('''t&lt;sub&gt;d&lt;/sub&gt;''') in hyper-sphere co-ordinates. ''A'' is moving with the universe expansion (along the time-line axis) at (''v = c''), but is stationary in 3-D space (''v'' = 0). ''B'' is orbiting ''A'' at (''v = c''), but the time-line axis motion is equivalent (and so `invisible') to both ''A'' and ''B'', as a result the orbital period and velocity measures will be defined in terms of 3-D space co-ordinates by observers on ''A'' and ''B''. 


For object '''B'''

:&lt;math&gt;t_d = t \sqrt{1 - v_{outer}^2}&lt;/math&gt; 

For object '''A'''

:&lt;math&gt;t_d = t \sqrt{1 - v_{inner}^2}&lt;/math&gt;


=== Atomic orbitals ===
[[File:H-orbit-transitions-n1-n2-n3-n1.gif|thumb|right|640px|fig 5. H atom orbital transitions from n1-n2, n2-n3, n3-n1 via 2 photon capture, photons expand/contract the orbital radius. The spiral pattern emerges because the electron is continuously pulled in an anti-clockwise direction by the rotating orbital.]]
In the atom we find individual particle to particle orbitals, and as such the atomic orbital is principally a wave-state orbital (during the orbit the electron is predominately in the electric wave-state). The wave-state is defined by a wave-function, we can however map (assign co-ordinates to) the mass point-states and so follow the electron orbit, for example, in 1 orbit at the lowest energy level in the H atom, the electron will oscillate between wave-state to point-state approximately &lt;math&gt;2\pi4\alpha^2&lt;/math&gt; = 471960 times, and so a plot of the electron as a circular obit around the nucleus will be the sum of 471960 'dots'. 

This permits us to treat the electron orbit around the nucleus as a simple 2-body gravitational orbit with the electron as the orbiting point. Although this (gravitational orbit) approach can only map the electron-as-mass point-state (and so offers no direct information regarding the electron as a wave), during electron transition between ''n''-shell orbitals we find the electron follows a [[v:Fine-structure_constant_(spiral) |hyperbolic spiral]] which can be used to derive the transition frequencies, this is significant because periodically the spiral angle components converge giving integer radius values (360°=4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r'').

As these spiral angles (360°, 360+120°, 360+180°, 360+216° ...) are linked directly to pi via this spiral geometry, we may ask if quantization of the atom has a geometrical origin. &lt;ref&gt;Macleod, Malcolm J.; {{Cite journal |title=Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs |journal=RG |date=Dec 2024 | doi=10.13140/RG.2.2.11378.00961}}&lt;/ref&gt;.


==== Theory ====
{{see|Fine-structure_constant_(spiral)}}
=====Hyperbolic spiral=====
[[File:Hyperbol-spiral-1.svg|thumb|right|320px|Hyperbolic spiral]]
A [[w:hyperbolic spiral |hyperbolic spiral]] is a type of [[w:spiral|spiral]] with a pitch angle that increases with distance from its center. As this curve widens (radius '''r''' increases), it approaches an [[w:asymptotic line|asymptotic line]] (the '''y'''-axis) with the limit set by a scaling factor '''a''' (as '''r''' approaches infinity, the '''y''' axis approaches '''a'''). 

For the particular spiral that the electron transition path maps, periodically the spiral angles converge to give integer radius, the general form for this type of spiral (beginning at the outer limit ranging inwards); 

:&lt;math&gt;x = a^2 \frac{cos(\varphi)}{\varphi^2},\; y = a^2 \frac{sin(\varphi)}{\varphi^2},\;0 &lt; \varphi &lt; 4\pi&lt;/math&gt;

:radius = &lt;math&gt;\sqrt(x^2 + y^2) r&lt;/math&gt; 

:&lt;math&gt;\varphi = (2)\pi, \; 4r&lt;/math&gt; (360°)
:&lt;math&gt;\varphi = (4/3)\pi,\; 9r&lt;/math&gt; (240°)
:&lt;math&gt;\varphi = (1)\pi, \;  16r&lt;/math&gt; (180°)
:&lt;math&gt;\varphi = (4/5)\pi, \; 25r&lt;/math&gt; (144°)
:&lt;math&gt;\varphi = (2/3)\pi, \; 36r&lt;/math&gt; (120°)


[[File:Bohr atom model (mul).svg|thumb|right|320px|Electron at different ''n'' level orbitals]]
=====Principal quantum number '''n'''=====
The H atom has 1 proton and 1 electron orbiting the proton, in the [[w:Bohr model |Bohr model]] (which approximates a gravitational orbit), the electron can be found at select radius ([[w:Bohr radius |the Bohr radius]]) from the proton (nucleus), these radius represent the permitted energy levels (orbital regions) at which the electron may orbit the proton. Electron transition (to a higher energy level) occurs when an incoming photon provides the required energy (momentum). Conversely emission of a photon will result in electron transition to a lower energy level.

The [[w:principal quantum number |principal quantum number ''n'']] denotes the energy level for each orbital. As ''n'' increases, the electron is at a higher energy level and is therefore less tightly bound to the nucleus (as ''n'' increases, the electron orbit is further from the nucleus). Each shell can accommodate up to ''n''&lt;sup&gt;2&lt;/sup&gt; (1, 4, 9, 16 ... ) electrons. Accounting for two states of spin this becomes 2''n''&lt;sup&gt;2&lt;/sup&gt; electrons. As these energy levels are fixed according to this integer ''n'', the orbitals may be said to be quantized.


=====(Bohr) orbital=====
The basic orbital radius has 2 components, dimensionless (the [[w:fine structure constant|fine structure constant alpha]]) and dimensioned (electron + proton wavelength);

wavelength = &lt;math&gt;\lambda_H = \lambda_p + \lambda_e&lt;/math&gt;

radius = &lt;math&gt;r_{orbital} = 2\alpha n^2 (\lambda_H)&lt;/math&gt;

As a mass point, the electron orbits the proton at a fixed radius (the Bohr radius) in a series of steps (the duration of each step corresponds to the wavelength component). The distance travelled per step (per wave-point oscillation) equates to the distance between mass point states and is the inverse of the radius

[[File:atomic-orbital-rotation-step.png|thumb|right|208px|electron (blue dot) moving 1 step anti-clockwise along the alpha orbital circumference]]
length = &lt;math&gt;l_{orbital} = \frac{1}{r_{orbital}}&lt;/math&gt;

Duration = 1 step per wavelength and so velocity

velocity = &lt;math&gt;v_{orbital} = \frac{1}{2\alpha n}&lt;/math&gt;


Giving period of orbit

period = &lt;math&gt;t_{orbital} = \frac{2\pi r_{orbital}} {v_{orbital}} = 2\pi 2\alpha 2\alpha n^3 \lambda_H&lt;/math&gt; 
 

As we are not mapping the wavelength component, a base (reference) orbital (''n''=1)

:&lt;math&gt;t_{ref} = 2\pi 4\alpha^2&lt;/math&gt; = 471964.356...  


The angle of rotation depends on the orbital radius

:&lt;math&gt;\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}\sqrt{2\alpha}}&lt;/math&gt;


===== Photon orbital model =====
The electron can jump between ''n'' energy levels via the absorption or emission of a photon. In the Photon-orbital model&lt;ref&gt;Macleod, Malcolm J.; {{Cite journal |title=Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs |journal=RG |date=Dec 2024 | doi=10.13140/RG.2.2.11378.00961}}&lt;/ref&gt;, the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase such that &lt;math&gt;orbital \;radius + photon = zero&lt;/math&gt; (cancel).

The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton), as such it is the orbital radius that absorbs or emits the photon during transition, in the process the orbital radius is extended or reduced (until the photon is completely absorbed/emitted). The electron itself has a `passive' role in the transition phase. It is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus (orbital momentum comes from the orbital radius), and this rotation continues during the transition phase resulting in the electron following a spiral path.


The photon is actually 2 photons as per the Rydberg formula (denoted initial and final).

:&lt;math&gt;\lambda_{photon} = R.(\frac{1}{n_i^2}-\frac{1}{n_f^2}) = \frac{R}{n_i^2}-\frac{R}{n_f^2}&lt;/math&gt;

:&lt;math&gt;\lambda_{photon} = (+\lambda_i) - (+\lambda_f)&lt;/math&gt;


The wavelength of the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon corresponds to the wavelength of the orbital radius. The (+&lt;math&gt;\lambda_i&lt;/math&gt;) will then delete the orbital radius as described above (''orbital'' + ''photon'' = ''zero''), however the (-&lt;math&gt;\lambda_f&lt;/math&gt;), because of the Rydberg minus term, will have the same phase as the orbital radius and so conversely will increase the orbital radius. And so for the duration of the (+&lt;math&gt;\lambda_i&lt;/math&gt;) photon wavelength, the orbital radius does not change as the 2 photons cancel each other;

:&lt;math&gt;r_{orbital} = r_{orbital} + (\lambda_i - \lambda_f)&lt;/math&gt;


However, the (&lt;math&gt;\lambda_f&lt;/math&gt;) has the longer wavelength, and so after the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon has been absorbed, and for the remaining duration of this (&lt;math&gt;\lambda_f&lt;/math&gt;) photon wavelength, the orbital radius will be extended until the (&lt;math&gt;\lambda_f&lt;/math&gt;) is also absorbed. For example, the electron is at the ''n'' = 1 orbital. To jump from an initial &lt;math&gt;n_i = 1&lt;/math&gt; orbital to a final &lt;math&gt;n_f = 2&lt;/math&gt; orbital, first the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon is absorbed (&lt;math&gt;\lambda_i + \lambda_{orbital} = zero&lt;/math&gt; which corresponds to 1 complete ''n'' = 1 orbit by the electron, the '''orbital phase'''), then the remaining (&lt;math&gt;\lambda_f&lt;/math&gt;) photon continues until it too is absorbed (the '''transition phase''').

:&lt;math&gt;t_{ref} \sim 2\pi 4\alpha^2 &lt;/math&gt;

:&lt;math&gt;\lambda_i = 1t_{ref}&lt;/math&gt;

:&lt;math&gt;\lambda_f = 4t_{ref}&lt;/math&gt; (''n'' = 2)


After the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon is absorbed, the (&lt;math&gt;\lambda_f&lt;/math&gt;) photon still has &lt;math&gt;\lambda_f = (n_f^2 - n_i^2)t_{ref} = 3 t_{ref}&lt;/math&gt; steps remaining until it too is absorbed.


[[File:atomic-orbital-transition-alpha-steps.png|thumb|right|277px|orbital transition during orbital rotation]]
This process does not occur as a single `jump' between energy levels by the electron, but rather absorption/emission of the photon takes place in discrete steps, each step corresponds to a unit of &lt;math&gt;r_{incr}&lt;/math&gt; (both photon and orbital radius may be considered as constructs from multiple units of this geometry);

:&lt;math&gt;r_{incr} = -\frac{1}{2 \pi 2\alpha r_{wavelength}}&lt;/math&gt;


In summary; the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon, which has the same wavelength as the orbital radius, deletes the orbital radius in steps &lt;math&gt;r = r_{orbital}&lt;/math&gt;

:&lt;math&gt;r = r + r_{incr}&lt;/math&gt; 
://&lt;math&gt;\lambda_i&lt;/math&gt; photon


Conversely, because of its minus term, the (&lt;math&gt;\lambda_f&lt;/math&gt;) photon will simultaneously extend the orbital radius accordingly;

WHILE (&lt;math&gt;r &lt; 4 r_{orbital}&lt;/math&gt;) 

:&lt;math&gt;r = r - r_{incr}&lt;/math&gt; 
://&lt;math&gt;\lambda_f&lt;/math&gt; photon


The model assumes orbits also follow along a [[Quantum_gravity_(Planck)#Hyper-sphere_orbit|timeline ''z''-axis]]

:&lt;math&gt;t_{orbital} = t_{ref} \sqrt{1 - \frac{1}{(v_{orbital})^2}}&lt;/math&gt; 

The orbital phase has a fixed radius, however at the transition phase this needs to be calculated for each discrete step as the orbital velocity depends on the radius;

:&lt;math&gt;t_{transition} = t_{ref} \sqrt{1 - \frac{1}{(v_{transition})^2}}&lt;/math&gt; 


====AI analysis====
AI was used to condense the derivation (summarize the method and give conclusions)&lt;ref&gt;https://codingthecosmos.com/ AI model analysis&lt;/ref&gt;. This presumes familiarity with the Bohr model. 


Chat GPT chatgpt.com/share/67ce62fc-8bf8-8012-8622-37a7a4fae6d6 &lt;ref&gt;https://chatgpt.com/share/67ce62fc-8bf8-8012-8622-37a7a4fae6d6 
 Chat GPT n-shell spiral analysis&lt;/ref&gt;
:Conclusion : the stability of the n orbital shells (with &lt;math&gt;m = n^2&lt;/math&gt;) arises because at these levels the spiral’s radial and angular increments are in perfect resonance. The electron’s semi-classical motion then traces a repeating, closed path. This geometrical resonance, which directly involves π and the circular nature of the motion, naturally leads to the quantized orbital shells without invoking the usual quantum-mechanical postulates.


Deepseek deepseek-spiral.pdf &lt;ref&gt;https://codingthecosmos.com/ai_pdf/Deepseek-spiral-03-2025.pdf Deepseek n-shell spiral analysis pdf&lt;/ref&gt;
:Conclusion : The levels &lt;math&gt;m = n^2&lt;/math&gt; are stable because the spiral’s geometry enforces a self-consistent synchronization between radial growth and angular displacement. This resonance prevents dispersion, creating discrete orbital shells without invoking quantum mechanics. The quantization of m into squares is a geometric constraint ensuring constructive interference in the spiral’s path, much like how integer multiples of wavelengths stabilize standing waves. The stability of &lt;math&gt;m = n^2&lt;/math&gt; levels arises from geometric resonance in the spiral, where radial expansion and angular progression harmonize to form closed or self-reinforcing paths. This ensures discrete, stable orbital shells.


Qwen chat.qwen.ai/s/9fe132a6-91d7-4ec6-8c82-ebf2e1b2b422 &lt;ref&gt;https://chat.qwen.ai/s/9fe132a6-91d7-4ec6-8c82-ebf2e1b2b422 Qwen n-shell spiral analysis&lt;/ref&gt;
:Conclusion : The stability of &lt;math&gt;m = n^2&lt;/math&gt; orbits is a geometric necessity . The spiral’s radial and angular evolution align to create resonant, non-decaying paths at these specific radii. These orbits are &quot;stable&quot; because they satisfy the minimal condition for periodicity and constructive interference, ensuring the electron’s motion remains bounded without invoking quantum theory.  


Claude claude.ai/share/355e21e4-d623-4810-962a-fb1892c2ef3f &lt;ref&gt;https://claude.ai/share/355e21e4-d623-4810-962a-fb1892c2ef3f Claude n-shell spiral analysis&lt;/ref&gt;&lt;ref&gt;https://codingthecosmos.com/ai_pdf/Claude-spiral-03-2025.pdf Claude n-shell spiral analysis pdf&lt;/ref&gt;
:Conclusion : The stability of orbits at m = n² emerges from the geometric properties of the spiral itself, specifically from resonances in the relationship between angular rotation and radial expansion. This geometric perspective provides a semi-classical explanation for why certain orbital levels are preferred without invoking quantum mechanical principles.


Grok x.com/i/grok/share/2ERWIbPFpB0wlCsaqcGErGVX2 &lt;ref&gt;https://x.com/i/grok/share/2ERWIbPFpB0wlCsaqcGErGVX2 Grok n-shell spiral analysis&lt;/ref&gt;&lt;ref&gt;https://codingthecosmos.com/ai_pdf/Grok-spiral-03-2025.pdf Grok n-shell spiral analysis pdf&lt;/ref&gt;
:Conclusion : The integer ( n ) introduces a quantization effect, and at these points, the spiral’s geometry—through the number of steps and the resulting angular position—creates a resonant or balanced configuration. This allows the electron, moving semi-classically, to occupy stable orbital shells, not because of quantum wavefunctions, but due to the inherent structure and symmetry of the spiral’s path at these discrete, integer-squared intervals.


==== Simulation ====
The simulation treats the atomic orbital as a 2-body gravitational orbit with the electron (single point) orbiting a central mass - the nucleus. The nucleus is a set of individual points (also orbiting each other) and not a static mass (static entity). The difference between gravitational (between macro objects) and atomic (gravitational) orbits is only in the angle of rotation &lt;math&gt;\beta&lt;/math&gt;' which has an additional &lt;math&gt;r_{\alpha}&lt;/math&gt; term included as the atomic orbital wavelength component is dominated by the particle wave-state (the mass-state is treated as a point), and so velocity along the 2-D (gravitational) plane (we are only mapping the radial component of the orbital) will decrease proportionately.

:&lt;math&gt;\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}} \sqrt{2\alpha}}&lt;/math&gt; 

 # Source code for atomic orbital transitions Transition_spiral.py &lt;ref&gt;https://codingthecosmos.com/files/Transition_spiral.py Transition_spiral.py&lt;/ref&gt;

[[File:Alpha-hyperbolic-spiral.gif|thumb|right|640px|Bohr radius during ionization, as the H atom electron reaches each ''n'' level, it completes 1 orbit (for illustration) then continues outward (actual velocity will become slower as radius increases according to angle β)]]
=====Spiral angle=====
In this example (see simulation ''n''=2 to ''n''=7), for an idealized Rydberg atom (a nucleus of point size, infinite mass and disregarding wavelength), the electron transition starts at the initial (''n''&lt;sub&gt;i&lt;/sub&gt; = 1) orbital

:&lt;math&gt;\varphi = 0, \;r_{orbital} = 2\alpha&lt;/math&gt; 

:&lt;math&gt;x = r_{orbital},\; y = 0&lt;/math&gt; 


For each step during transition, setting t = step number (FOR t = 1 TO ...), we can calculate the radius ''r'' and &lt;math&gt;n_f^2&lt;/math&gt; at each step.

:&lt;math&gt;r = r_{orbital} + \frac{t}{2\pi 2\alpha}&lt;/math&gt; (number of increments ''t'' of &lt;math&gt;r_{incr}&lt;/math&gt;)

:&lt;math&gt;\varphi = \varphi + \beta&lt;/math&gt;

:&lt;math&gt;n_f^2 = 1 + \frac{t}{2\pi 4\alpha^2}&lt;/math&gt; (&lt;math&gt;n_f^2&lt;/math&gt; as a function of ''t'') 


The spiral angle and &lt;math&gt;n_f^2&lt;/math&gt; are interchangeable

:&lt;math&gt;\varphi =4 \pi \frac{(n_f^2 - n_f)}{n_f^2}&lt;/math&gt; (&lt;math&gt;\varphi&lt;/math&gt; at any &lt;math&gt;n_f^2&lt;/math&gt;) 


We can then re-write (&lt;math&gt;n_f&lt;/math&gt; is only an integer at prescribed spiral angles);

:&lt;math&gt;\beta = \frac{1}{{r_{orbital}}^2 n_f^3}&lt;/math&gt;


Giving integer values at these spiral angles

:&lt;math&gt;\varphi = (2)\pi, \; r = 4 r_{orbital}&lt;/math&gt; (360°)
:&lt;math&gt;\varphi = (8/3)\pi,\; r = 9 r_{orbital}&lt;/math&gt; (360+120°)
:&lt;math&gt;\varphi = (3)\pi, \;  r = 16 r_{orbital}&lt;/math&gt; (360+180°)
:&lt;math&gt;\varphi = (16/5)\pi, \;  r = 25 r_{orbital}&lt;/math&gt; (360+216°)
:&lt;math&gt;\varphi = (10/3)\pi, \;  r = 36 r_{orbital}&lt;/math&gt; (360+240°)
:&lt;math&gt;\varphi = (7/4)\pi, \;  r = 49 r_{orbital}&lt;/math&gt; 
:&lt;math&gt;\varphi = (7/2)\pi, \;  r = 64 r_{orbital}&lt;/math&gt; (360+270°)


===== Rydberg atom =====
At the ''n'' = 1 orbital, 1 complete rotation becomes (the dimensionless terms are measured on a 2-D plane);

:&lt;math&gt;t_{ref} = \frac{2\pi r_{orbital}}{v_{orbital}} = 2\pi 2\alpha 2\alpha&lt;/math&gt; 

:&lt;math&gt;1t_{ref}&lt;/math&gt; = 471964.3563... 

:&lt;math&gt;4t_{ref}&lt;/math&gt; = 1887857.4255...

:&lt;math&gt;9t_{ref}&lt;/math&gt; = 4247679.2074...

:&lt;math&gt;16t_{ref}&lt;/math&gt; = 7551429.7021...


===== H atom =====
Experimental values for H(1s-ns) transitions (''n'' the [[w:principal quantum number |principal quantum number]]).

H(1s-2s) =  2466 061 413 187.035 kHz &lt;ref&gt;http://www2.mpq.mpg.de/~haensch/pdf/Improved%20Measurement%20of%20the%20Hydrogen%201S-2S%20Transition%20Frequency.pdf&lt;/ref&gt;  

H(1s-3s) =  2922 743 278 665.79 kHz &lt;ref&gt;https://pubmed.ncbi.nlm.nih.gov/33243883/&lt;/ref&gt;  

H(1s-4s) =  3082 581 563 822.63 kHz &lt;ref&gt;https://codata.org/&lt;/ref&gt;   

H(1s-∞s) =  3288 086 857 127.60 kHz &lt;ref&gt;https://codata.org/ (109678.77174307cm-1)&lt;/ref&gt; (''n'' = ∞)


R = 10973731.568157 &lt;ref&gt;https://codata.org/ (mean)&lt;/ref&gt; ([[w:Rydberg constant |Rydberg constant]])

α =137.035999177 (inverse fine structure constant &lt;ref&gt;https://codata.org/ (mean)&lt;/ref&gt;


The wavelength of the H atom, for simplification the respective particle wavelengths are presumed constant irrespective of the vicinity of the electron to the proton. 

&lt;math&gt;r_{wavelength} = \lambda_H = \frac{2c}{\lambda_e + \lambda_p}&lt;/math&gt;


Dividing (dimensioned) wavelength (&lt;math&gt;r_{wavelength}&lt;/math&gt;) by the (dimensioned) transition frequency returns a dimensionless number (the alpha component of the photon). 

:&lt;math&gt;h_{(1s-ns)} = \lambda_H \frac{(n^2 - 1)}{H(1s-ns)}&lt;/math&gt;

&lt;math&gt;h_{(1s-2s)}&lt;/math&gt; = 1887839.82626...

&lt;math&gt;h_{(1s-3s)}&lt;/math&gt; = 4247634.04874...

&lt;math&gt;h_{(1s-4s)}&lt;/math&gt; = 7551347.55306...


===== Simulation atom =====
The following example simulates an electron transition, the electron begins at radius &lt;math&gt;r = r_{orbital}&lt;/math&gt; and makes a 360° rotation at orbital radius (the orbital phase) and then moves in incremental steps to higher orbitals (the transition phase) mapping a hyperbolic spiral path (red line) in the process (photon orbital model). 

The period &lt;math&gt;t_{sim}&lt;/math&gt; and length &lt;math&gt;l_{sim}&lt;/math&gt; are measured at integer &lt;math&gt;n^2 r&lt;/math&gt; (''n'' = 1, 2, 3...) radius. For a Rydberg atom, these radius correspond precisely to the electron path at the [[v:Fine-structure_constant_(spiral) |(hyperbolic) spiral]] angles; (360°(''1r''), 360°(''4r''), 360+120°(''9r''); 360+180°(''16r''), 360+216°(''25r''), 360+240°(''36r'') ...) (the angles converge to give integer values at these radius), and we find that as the simulation nucleus mass increases, the integer radius values approach these angles (table 2.). The period &lt;math&gt;t_{sim}&lt;/math&gt; can then be used to calculate the transition frequencies.


In this example, the nucleus = 249 mass points (start ''x'', ''y'' co-ordinates close to 0, 0), the electron = 1 mass point (at radius ''x'' = ''r'', ''y'' = 0), with ''t''&lt;sub&gt;sim&lt;/sub&gt; = period and ''l''&lt;sub&gt;sim&lt;/sub&gt; = distance travelled by the electron (&lt;math&gt;l_{orbital} = l_{sim}&lt;/math&gt; at ''n''=1), the radius coefficient ''r''&lt;sub&gt;n&lt;/sub&gt; = radius divided by &lt;math&gt;r_{orbital}&lt;/math&gt;. As this is a gravitational orbit, although the nucleus comprises 249 points clumped close together, these points are independent of each other (they also rotate around each other), and so the `nucleus' size and shape is not static (the simulation is not optimised for a nucleus). Table 1. gives the relative values and the ''x'', ''y'' co-ordinates for the electron, nucleus center and barycenter.
[[File:H-atom-electron-transition-nucleus-plot.gif|thumb|right|640px|H atom electron transition spiral plotting the nucleus and barycenter as the electron transitions from n=1 to n=8]]


:&lt;math&gt;j_{atom} = 250&lt;/math&gt; (atomic mass)

:&lt;math&gt;i_{nucleus} = j_{atom} -1 = 249&lt;/math&gt; (relative nucleus mass)

:&lt;math&gt;r_{wavelength} = 2 (\frac{j_{atom}}{i_{nucleus}})^2&lt;/math&gt; = 2.0160965

:&lt;math&gt;r_{orbital} = 2 \alpha \;*\; r_{wavelength} &lt;/math&gt; (radius) = 552.5556

:&lt;math&gt;t_n = \frac{t_{sim}}{r_{wavelength}}&lt;/math&gt; 

:&lt;math&gt;l_n = \frac{l_{sim}}{l_{orbital}} - l_{orbital}&lt;/math&gt; 

:&lt;math&gt;r_b = r_{sim} - \frac{r_{sim}}{j_{atom}}&lt;/math&gt; 

:&lt;math&gt;r_n = \frac{r_b}{r_{orbital}}&lt;/math&gt; 


{| class=&quot;wikitable&quot;
|+table 1. Electron transition (mass = 250; ''r''&lt;sub&gt;n&lt;/sub&gt;= 1 to 5)
! ''r''&lt;sub&gt;n&lt;/sub&gt;
! ''t''&lt;sub&gt;sim&lt;/sub&gt;
! ''l''&lt;sub&gt;n&lt;/sub&gt;
! angle
! ''x'', ''y'' (electron) 
! ''x'', ''y'' (nucleus) 
! ''x'', ''y'' (barycenter) 
|-1
| 1
| 471957.072
| 0.9999897
| 360°
| 550.334,  0.0036
| -2.2102,  -0.00002 
| -0.00004,  -0.00001
|-
| 4
| 1887867.293
| 2.000012
| 359.952489°
| 2202.8558,  0.0001
| -7.9565,  -1.9475
| 0.8868,  -1.9397
|-
| 9 
| 4247689.502 
| 4.000014
| 119.92712° 
| -2473.180,  4296.283 
| 13.558,  -10.325  
| 3.611,  6.901
|-
| 16 
| 7551439.538 
| 6.000014 
| 179.91669°
| -8815.254,  12.818 
| 25.636,  13.303
| -9.728,  13.301
|-
| 25 
| 11799118.905 
| 8.000014  
| 215.9122°
| -11158.64,  -8081.13
| 16.580,  39.083
| -28.118,  6.602
|}


Comparison of the spiral angle at ''r''&lt;sub&gt;n&lt;/sub&gt; = 4 (360°), 9 (360+120°), 16 (360+180°) with different mass (''m'' = 64, 128, 250, 500, Rydberg). For the proton:electron mass ratio; ''m'' = 1836.15267...

{| class=&quot;wikitable&quot;
|+ table 2. Spiral angle at &lt;math&gt;r_n&lt;/math&gt; = 4, 9, 16 
! mass 
! ''r''&lt;sub&gt;n&lt;/sub&gt; = 4 
! ''r''&lt;sub&gt;n&lt;/sub&gt; = 9
! ''r''&lt;sub&gt;n&lt;/sub&gt; = 16
|-
| ''m'' = 64
| 359.80318°
| 119.70323°
| 179.66239°
|-
| ''m'' = 128
| 359.90394°
| 119.85415°
| 179.83377°
|-
| ''m'' = 250
| 359.95249°
| 119.92711°
| 179.91669°
|-
| ''m'' = 500
| 359.97706°
| 119.96501° 
| 
|-
| Rydberg
| 360°
| 360+120°
| 360+180°
|}


=== Geometrically coded universe === 
* [[Simulation_hypothesis_(Planck)]]: A geometrical Planck scale simulation universe
* [[Electron_(mathematical)]]: Mathematical electron from Planck units
* [[Planck_units_(geometrical)]]: Planck units as geometrical forms
* [[Physical_constant_(anomaly)]]: Anomalies in the physical constants 
* [[Quantum_gravity_(Planck)]]: Gravity at the Planck scale 
* [[Fine-structure_constant_(spiral)]]:  Quantization via pi 
* [[Relativity_(Planck)]]: 4-axis hypersphere as origin of motion
* [[Black-hole_(Planck)]]: CMB and Planck units
* [[Sqrt_Planck_momentum]]: Link between charge and mass


== External links ==
* [https://codingthecosmos.com/ Planck scale modelling using geometrical objects] 
* [https://www.amazon.com/Our-Mathematical-Universe-Ultimate-Reality/dp/0307599809 Our Mathematical Universe: My Quest for the Ultimate Nature of Reality] -Max Tegmark (Book)
* [https://link.springer.com/article/10.1134/S0202289308020011/ Dirac-Kerr-Newman black-hole electron] -Alexander Burinskii (article)
* [https://plato.stanford.edu/entries/pythagoras/ Pythagoras &quot;all is number&quot;] - Stanford University
* [[w:Mathematical universe hypothesis | Mathematical universe hypothesis]]
* [[w:Philosophy of mathematics | Philosophy of mathematics]]
* [[w:Philosophy of physics | Philosophy of physics]]
* [[w:Platonism | Platonism]] 


==References==
{{Reflist}}


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      <text bytes="171215" sha1="jwdajecv7j6517begy5xysa9u7xt8fn" xml:space="preserve">[[File:Leslie Ward - Vanity Fair, Newspapermen, ^Algy^, The Hon Algernon Henry Bourke, Januray 20, 1898 - B1979.14.521 - Yale Center for British Art.jpg|thumb|Hon. Algernon Bourke, ''Vanity Fair'', 1898]]

==Also Known As==
* Family name: Bourke [pronounced ''burk'']&lt;ref name=&quot;:62&quot;&gt;{{Cite journal|date=2024-05-07|title=Earl of Mayo|url=https://en.wikipedia.org/w/index.php?title=Earl_of_Mayo&amp;oldid=1222668659|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Earl_of_Mayo.&lt;/ref&gt;
* The Hon. Algernon Bourke
** Button Bourke&lt;ref&gt;&quot;A Tory 'Reformer' at the India Office.&quot; ''India'' 10 November 1911, Friday: 4 [of 12], Col. 1b [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004300/19111110/007/0004#. Print: same title, p. 228.&lt;/ref&gt;
** Algy Bourke
* Mrs. Guendoline Bourke
** Guendoline&lt;ref name=&quot;:1&quot; /&gt; [The National Portrait Gallery, London, uses this spelling for Lafayette's portrait of Bourke in costume for the ball,&lt;ref name=&quot;:23&quot; /&gt; as does Russell Harris in the Lafayette Archive, held by the Victoria and Albert&lt;ref name=&quot;:25&quot;&gt;Harris, Russell. &quot;Hon Mrs Algernon Bourke, née Guendoline Irene Emily Sloane-Stanley (1870-1967), as Salammbô.&quot; ''In Calm Prose'' 2011. http://www.rvondeh.dircon.co.uk/incalmprose/.&lt;/ref&gt;.]
** Gwendolen&lt;ref&gt;General Register Office. ''England and Wales Civil Registration Indexes''. London, England: General Register Office. FreeBMD. ''England &amp; Wales, Civil Registration Marriage Index, 1837-1915''[database on-line]. Provo, UT, USA: Ancestry.com Operations Inc, 2006.&lt;/ref&gt;{{rp|Marriage Index}} &lt;ref name=&quot;:15&quot; /&gt;{{rp|''Morning Post'' article about her name}} &lt;ref&gt;General Register Office. ''England and Wales Civil Registration Indexes''. London, England: General Register Office. FreeBMD. ''England &amp; Wales, Civil Registration Marriage Index, 1837-1915''[database on-line]. Provo, UT, USA: Ancestry.com Operations Inc, 2006.&lt;/ref&gt;{{rp|Electoral Register}}
** Gwendoline&lt;ref name=&quot;:14&quot;&gt;City of Westminster Archives Centre; London, England; ''Westminster Church of England Parish Registers''; Reference: ''SPWP/PR/1/2''. Ancestry.com. ''Westminster, London, England, Church of England Births and Baptisms, 1813-1919'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2020.&lt;/ref&gt;{{rp|Births and Baptisms}}
* Shelley Bontein and Emilie Sloane-Stanley Bontein
* See also the [[Social Victorians/People/Mayo|page for the Earl of Mayo]], the Hon. Algernon Bourke's father and then brother, and other Bourkes

== Overview ==

=== Algernon Bourke ===
Although the Hon. Algernon Henry Bourke was born in Dublin in 1854 and came from a family whose title is in the Peerage of Ireland,&lt;ref name=&quot;:6&quot;&gt;1911 England Census.&lt;/ref&gt; he seems to have spent much of his adult life generally in England and especially in London. He was &quot;a noted fisherman.&quot;&lt;ref&gt;&quot;London Correspondence.&quot; ''Freeman's Journal'' 21 December 1897, Tuesday: 5 [of 8], Col. 5c [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000056/18971221/027/0005. Same print title, n.p.&lt;/ref&gt;

Because he was the son of the [[Social Victorians/People/Mayo|Earl of Mayo]], perhaps, or perhaps because he was so involved in projects that got reported on, he was mentioned a great deal in the newspapers, but after his bankruptcy, he seems to have receded in prominence, in part because he was living outside of the U.K., and apparently separately from his wife, Guendoline Bourke.

Bourke ran as the Conservative candidate for Parliament from Clapham (population, c. 70,000) in 1885, a race he did not win. As a candidate he is described like this:&lt;blockquote&gt;Acted as a newspaper correspondent during the Zulu war. Subsequently Poor-law inspector in the West of Ireland. &quot;A loyal supporter of Church and State.&quot; Desires to reduce the School Board expenditure, and revive trade; and is opposed to Mr. Chamberlain's &quot;police of hasty and experimental reform.&quot;&lt;ref&gt;&quot;Clapham (70,000).&quot; ''South London Chronicle'' 17 October 1885, Saturday: 5 [of 8], Col. 5a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000443/18851017/113/0005. Print title ''South London Chronicle and Southwark and Lambeth Ensign'', p. 5.&lt;/ref&gt;&lt;/blockquote&gt;The London ''Weekly Dispatch'' says he is &quot;a dashing and unscrupulous young Tory.&quot;&lt;ref&gt;&quot;The Political Campaign in London.&quot; ''Weekly Dispatch'' (London) 15 November 1885, Sunday: 9 [of 16], Col. 3c [of 4]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003358/18851115/069/0009. Same print title and p.&lt;/ref&gt;

&quot;Algy&quot; Bourke was &quot;Man of the Day&quot; (No. DCCII [522) for ''Vanity Fair'' in 1898, caricatured by Leslie Ward (above right):&lt;blockquote&gt;Son of the great and murdered Lord Mayo, he is contemporary with the outbreak of the Crimean War, he is a Member of the London Stock Exchange, he has a beautiful wife and a daughter, and, being a very fashionable young man, he was once refused as their Member by the worthy electors of Clapham. He was an Eton boy, of course: and less naturally he went to Cambridge; where he was made President of the Beefsteak, the Amateur Dramatic, the Athenaeum, the True Blue, and the Hibernian Clubs. When he came down he tried journalism and went to Zululand as a ''Daily Telegraph'' ‘‘special”; after which he was improved into an Inspector of Workhouses [2, Col. 2c – 3, Col. 1a] in Ireland: which may account for his proficiency as a caterer. For seven years he worked under the late Mr. Chinnery on ''The Times'': being popularly supposed to look after that journal's morals. He is a good man of business, and a great organiser who has made White's Club pay even if it be less “smart&quot; than it was. He has done much for Willis’s since he took it in hand; he did well with his Battersea venture, and he thinks that he only failed with the Summer Club in Kensington Gardens because people would not go to the wrong side of the Park. Moreover, he runs a Club at Brighton, and he is Chairman of the Grand Hotel at Monte Carlo: whither he once organised a cheap trip. Altogether he is a veritable Clubman, and a very successful arranger of amusements, associations, and restaurants.

He is a popular fellow who is known to all of us; and though he is a little inclined to be quarrelsome, no one can get much the better of him. He is also a quick grasper of facts and a good talker. His favourite sports are fishing and the organising of associations for the introduction of salmon to the Thames. By way of being an art critic, he has made an interesting collection of engravings of the members of White’s Club from its foundation; but his friends say that he is not a well-dressed man. He has also written a history of White’s, and he is now writing one of Brooks's Club. He is a genial person, who looks as if the world agreed with him well.

He is an aquisition [sic] to a house party; and they call him “Algy.”&lt;ref&gt;&quot;Men of the Day.&quot; — No. DCCII [522]. The Hon. Algernon Henry Bourke.&quot; ''Vanity Fair'' 20 January 1898, Thursday: 2 [of 4], Col. 2c – 3, Col. 3a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/9900020/18980120/010/0002 and https://www.britishnewspaperarchive.co.uk/viewer/BL/9900020/18980120/005/0003. Same print title, pp. 41–42. Portrait is full page, on p. 1.&lt;/ref&gt;&lt;/blockquote&gt;The Hon. Algernon Bourke and Mr. Algernon Bourke, depending on the newspaper article, were the same person. Calling him Mr. Bourke in the newspapers, especially when considered as a businessman or (potential) member of Parliament, does not rule out the son of an earl, who would normally be accorded the honorific of ''Honorable''.

=== Guendoline Sloane-Stanley Bourke ===

Mrs. Guendoline Bourke exhibited at dog shows successfully and was a [[Social Victorians/Timeline/1900s#Society Sportswomen|noted deerstalker]] and &quot;an appreciative listener to good music.&quot;&lt;ref&gt;&quot;Vanity Fair.&quot; ''Lady of the House'' 15 June 1899, Thursday: 4 [of 44], Col. 2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004836/18990615/019/0004.&lt;/ref&gt; Her personal beauty is often mentioned in reports, and ''The World'' says she was &quot;a magnificent woman.&quot;&lt;ref&gt;&quot;Beauties of To-Day. From the ''World''.&quot; ''Clifton Society'' 24 June 1897, Thursday: 14 [of 16], Col. 2c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002164/18970624/066/0014. Same print and p.&lt;/ref&gt; She is the first listed in the ''Graphic''&lt;nowiki/&gt;'s 1891 &quot;Leading Ladies of Society&quot;:&lt;blockquote&gt;The Hon. Mrs. Algernon Bourke is a daughter (Gwendoline Irene Emily) of the late Hans Sloane Stanley, Esq., of Poultons, Southampton, and 49, Cadogan Square, S.W. She married, on December 15th, 1887, the Hon. Algernon Bourke, third son of the sixth Earl of Mayo, Governor-General of India (who was assassinated in 1872), and nephew of Lord Connemara, Governor of Madras. Mr. Bourke is a member of the London Stock Exchange, and resides at 33, Cadogan Terrace, S.W.&lt;ref&gt;&quot;Leading Ladies of Society.&quot; The Graphic 28 March 1891, Saturday: 6 [of 28], Col. 2c [of 3]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/9000057/18910328/019/0006. Print: same title, p. 346.&lt;/ref&gt;&lt;/blockquote&gt;She attended many social events without her husband, especially into the 20th century, usually with an appreciative description of what she wore. She was a sponsor of Irish art needlework as well. Unlike her husband's, Guendoline's social status seems to have risen as time passed, and she appears in stories associated with the Princess of Wales, and then later with Queen Alexandra.

=== The Sloane-Stanley Family ===
Guendoline's family consisted of a younger brother, Cyril Sloane-Stanley, as well as her parents, Hans Sloane-Stanley and Emilie Edwards Sloane-Stanley. Exactly one year after she and Algernon Bourke married, Hans Sloane-Stanley died (in 1888), leaving an estate worth £33,704 7s. 5d.&lt;ref name=&quot;:17&quot;&gt;Principal Probate Registry; London, England; ''Calendar of the Grants of Probate and Letters of Administration made in the Probate Registries of the High Court of Justice in England''. Ancestry.com. ''England &amp; Wales, National Probate Calendar (Index of Wills and Administrations), 1858-1995'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2010.&lt;/ref&gt; (1888, 321) Her mother remarried almost exactly a year after that, to James Shelly Bontein. Bontein's father had been Gentleman Usher and Clerk of the Robes to Queen Victoria.&lt;ref name=&quot;:18&quot;&gt;&quot;Marriages.&quot; &quot;Births, Marriages, and Deaths.&quot; ''Belfast News-Letter'' 6 December 1889, Friday: 1 [of 8], Col. 1a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000038/18891206/001/0001. Same print title and p.&lt;/ref&gt;

Shortly after his death ''Truth'' described Guendoline and Cyril's father Hans Sloane-Stanley:&lt;blockquote&gt;The death of Mr. Sloane Stanley, of Paultons Park, is much regretted in South Hants, as he was one of the most popular landlords in the county, and was greatly esteemed. Mr. Sloane Stanley was well known in yachting circles, and for many years he was Commodore of the Royal Southern Yacht Club, and owned the schooner ''Star of the West''. He was one of the very few owners who continued to keep up the old custom of giving his crew a laying-up supper at the close of each season. There were great festivities at Paultons only a few months ago, when Miss Sloane Stanley was married to Mr. Algernon Bourke.&lt;ref&gt;&quot;Entre Nous.&quot; ''Truth'' 27 December 1888, Thursday: 6 [of 48], Col. 2b [of 2]. ''British Newspaper Archive'' [https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18881227/023/0006# https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18881227/023/0006]. Same print title, p. 1136.&lt;/ref&gt;&lt;/blockquote&gt;When he died in 1944, Cyril Sloane-Stanley's estate was quite a bit larger than his father's had been 50 years before. The probate was divided between what was limited to &quot;settled land&quot; and what was &quot;save and except settled land.&quot; What was not settled land totalled £356,114 12s. 10d. and went to John Everett, company director; the Hon. Elwyn Villiers Rhys, captain, H.M. army; and William Adam de Geijer, retired captain, H.M. army.&lt;ref name=&quot;:17&quot; /&gt; (1944, 430) His daughter Lavender was married to John Everett, and Diane was married to Elwyn Villiers Rhys. What was settled land totalled £168,975 and went to William Adam de Geijer, retired captain, H.M. army, and George Lawrence Stewart, solicitor.&lt;ref name=&quot;:17&quot; /&gt; (1944, 430)

The Sloane-Stanleys descend from Hans Sloane (1660–1753), whose 71,000-item collections &quot;provid[ed] the foundation of the British Museum, the British Library, and the Natural History Museum, London.&quot;&lt;ref name=&quot;:19&quot;&gt;{{Cite journal|date=2025-01-07|title=Hans Sloane|url=https://en.wikipedia.org/wiki/Hans_Sloane|journal=Wikipedia https://en.wikipedia.org/wiki/Hans_Sloane|language=en|via=}}&lt;/ref&gt; Much of this Hans Sloane's wealth came from his medical practice in Jamaica, where he went as physician to the Governor General of Jamaica, the 2nd Duke of Albemarle, and where he married &quot;a wealthy heiress of sugar plantations&quot; worked by enslaved Jamaicans.&lt;ref name=&quot;:19&quot; /&gt; His great-nephew, Hans Sloane, inherited Paultons, near Romsey, &quot;and in recognition of this he adopted the additional surname of Stanley in 1821.&quot;&lt;ref&gt;{{Cite journal|date=2023-10-06|title=Hans Sloane (MP)|url=https://en.wikipedia.org/wiki/Hans_Sloane_(MP)|journal=Wikipedia https://en.wikipedia.org/wiki/Hans_Sloane_(MP)|language=en}}&lt;/ref&gt;

== Acquaintances, Friends and Enemies ==

=== Algernon Bourke ===

* Best man at [[Social Victorians/Timeline/1887#Wedding of Algernon Bourke and Gwendolen Sloane Stanley|his wedding]]: the Hon. Michael Sandys
* [[Social Victorians/People/Montrose|Marcus Henry Milner]], &quot;one of the zealous assistants of that well-known firm of stockbrokers, Messrs. Bourke and Sandys&quot;&lt;ref name=&quot;:8&quot;&gt;&quot;Metropolitan Notes.&quot; ''Nottingham Evening Post'' 31 July 1888, Tuesday: 4 [of 4], Col. 2a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000321/18880731/025/0004.&lt;/ref&gt;
* Caroline, Duchess of Montrose — her &quot;legal advisor&quot; on the day of her marriage to Marcus Henry Milner&lt;ref&gt;&quot;Metropolitan Notes.&quot; ''Nottingham Evening Post'' 31 July 1888, Tuesday: 4 [of 4], Col. 1b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000321/18880731/025/0004.&lt;/ref&gt;
* Wilfred Scawen Blunt
* Alexander Meyrick Broadley&lt;ref name=&quot;:24&quot;&gt;Broadley, Alexander Meyrick. ''How We Defended Arábi and His Friends, a Story of Egypt and the Egyptians''. London: Chapman and Hall, 1884. Google Books: [https://www.google.com/books/edition/How_We_Defended_Ar%C3%A1bi_and_His_Friends/75YRzFDvaO4C?hl=en&amp;gbpv=1&amp;bsq=Bourke https://www.google.com/books/edition/How_We_Defended_Ar%C3%A1bi_and_His_Friends/75YRzFDvaO4C].&lt;/ref&gt; (2) [Blunt, Broadley and Bourke were mutual friends; Blunt and Bourke were cousins.]
* &quot;Natty Rothschild,&quot; according to Blunt
* Brett, who was Hartington's secretary in the early 80s.&lt;ref&gt;Blunt, Wilfred Scawen. ''Gordon at Khartoum, Being a Personal Narrative of Events in Continuation of &quot;A Secret History of the English Occupation of Egypt''.&quot; London: Stephen Swift, 1911.&lt;/ref&gt;

=== Guendoline Bourke ===

* Bridesmaids at [[Social Victorians/Timeline/1887#Wedding of Algernon Bourke and Gwendolen Sloane Stanley|her wedding]]: Lady Florence Bourke, Miss Nora Bourke, Miss Edwards, and Miss Ewart
* Lord and Lady Alington, Belvedere House, Scarborough
* [[Social Victorians/People/William James|Evelyn James]] 

== Organizations ==

=== Guendoline Bourke ===

* Member, the Ladies Committee for the [[Social Victorians/London Clubs#Prince's Skating Club|Prince's Skating Club]], which also included [[Social Victorians/People/Princess Louise|Princess Louise]] (Duchess of Argyll), the [[Social Victorians/People/Portland|Duchess of Portland]], [[Social Victorians/People/Londonderry|Lady Londonderry]], [[Social Victorians/People/Campbell|Lady Archibald Campbell]], [[Social Victorians/People/Ribblesdale|Lady Ribblesdale]], and [[Social Victorians/People/Asquith|Mrs. Asquith]]&lt;ref name=&quot;:11&quot;&gt;&quot;What the 'World' Says.&quot; ''Northwich Guardian'' 01 November 1902, Saturday: 6 [of 8], Col. 8a [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001975/19021101/134/0006. Print title: The ''Guardian'', p. 6.&lt;/ref&gt; (in 1902, at least)

=== Algernon Bourke ===
* [[Social Victorians/Schools#Eton|Eton]]
* Cambridge University, Trinity College, 1873, Michaelmas term&lt;ref name=&quot;:7&quot;&gt;Cambridge University Alumni, 1261–1900. Via Ancestry.&lt;/ref&gt;
* Conservative Party
* 1879: Appointed a Poor Law Inspector in Ireland, Relief of Distress Act
* 1881: Partner, with 2 uncles, in Brunton, Bourke, and Co.&lt;ref&gt;&quot;From Our London Correspondent.&quot; ''Manchester Courier'' 24 August 1881, Wednesday: 5 [of 8], Col. 4a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000206/18810824/030/0005. Print: ''Manchester Courier and Lancaster General Advertiser'', p. 5.&lt;/ref&gt; (one of the [[Social Victorians/British Aristocracy#Sons of Peers on the Stock Exchange|sons of peers on the Stock Exchange]])
* 1885: Office of the 7th Surrey Rifles Regiment&lt;ref&gt;&quot;7th Surrey Rifles.&quot; ''South London Press'' 08 August 1885, Saturday: 12 [of 16], Col. 4a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000213/18850808/165/0012. Print p. 12.&lt;/ref&gt;
* 1886: Battersea Friendly Angling Society&lt;ref&gt;&quot;Battersea Friendly Angling Society.&quot; ''Fishing Gazette'' 17 April 1886, Saturday: 6 [of 20], Col. 2a [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002553/18860417/030/0006. Same print title, p. 218.&lt;/ref&gt;
* 27 February 1886: one of the Vice Presidents of the [[Social Victorians/London Clubs#Bolingbroke Reading-Room and Institute|Bolingbroke Reading-Room and Institute]]
* Special Correspondent of The ''Times'' for the Zulu War, accompanying Lord Chelmsford
* Head, Messrs. Bourke and Sandys, &quot;that well-known firm of stockbrokers&quot;&lt;ref name=&quot;:8&quot; /&gt; ( – 1901 [at least])
* White's gentleman's club, St. James's,&lt;ref&gt;{{Cite journal|date=2024-10-09|title=White's|url=https://en.wikipedia.org/wiki/White's|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/White%27s.&lt;/ref&gt; Manager (1897)&lt;ref&gt;&quot;Side Lights on Drinking.&quot; ''Waterford Standard'' 28 April 1897, Wednesday: 3 [of 4], Col. 7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001678/18970428/053/0003.&lt;/ref&gt;
* Willis's Rooms (described in 1895):&lt;blockquote&gt;... the Hon. Algernon Burke [sic], son of the 6th Earl of Mayo, has turned the place into a smart restaurant where choice dinners are served and eaten while a stringed band discourses music. Willis's Rooms are now the favourite dining place for ladies who have no club of their own, or for gentlemen who are debarred by rules from inviting ladies to one of their own clubs. The same gentleman runs a hotel in Brighton, and has promoted several clubs. He has a special faculty for organising places of the kind, without which such projects end in failure.&lt;ref&gt;&quot;Lenten Dullness.&quot; ''Cheltenham Looker-On'' 23 March 1895, Saturday: 11 [of 24], Col. 2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000226/18950323/004/0011. Print p. 275.&lt;/ref&gt;&lt;/blockquote&gt;
*The [[Social Victorians/London Clubs#Pelican|Pelican Club]], known for its boxing (1891)

==== Boards of Directors ====
*1883: One of the directors, the Franco-English Tunisian Esparto Fibre Supply Company, Ltd.&lt;ref&gt;''Money Market Review'', 20 Jan 1883 (Vol 46): 124.&lt;/ref&gt;
*1891: One of the founders, the Discount Banking Company, Ltd., which says Algernon Bourke is a director of District Messenger Services and News Company, Ltd.&lt;ref&gt;&quot;Public Company.&quot; ''Nottingham Journal'' 31 October 1891, Saturday: 4 [of 8], Col. 8a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001896/18911031/099/0004. Print title: ''The Nottingham Daily Express'', p. 4.&lt;/ref&gt;
*1894: One of the directors, the Frozen Lake, Ltd., with Admiral Maxse, Lord [[Social Victorians/People/Beresford|Marcus Beresford]], [[Social Victorians/People/Williams|Hwfa Williams]]&lt;ref&gt;&quot;The Frozen Lake, Limited.&quot; ''St James's Gazette'' 08 June 1894, Friday: 15 [of 16], Col. 4a [of 4]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001485/18940608/085/0015. Print p. 15.&lt;/ref&gt;&lt;blockquote&gt;London is to have new amusement this winter, for which Mr Algernon Bourke, who has taught us that it is possible to eat as well in St. James’s as on the Boulevards, and Mr Hwfa Williams, of Sandown fame, are jointly responsible. The &quot;Frozen Lake,&quot; under which title a real ice-skating rink is about to be constructed under their auspices, will no doubt be gladly welcomed by all skaters, and the venture is likely to prove a success.&lt;ref&gt;&quot;Society Gossip.&quot; ''Weston-super-Mare Gazette, and General Advertiser'' 6 June 1894, Wednesday: 4 [of 4], Col. 4b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001444/18940606/044/0004. Print title: ''Weston-super-Mare Gazette'', p. 4.&lt;/ref&gt;&lt;/blockquote&gt;

==== Committees ====

*Member, General Committee, [[Social Victorians/London Clubs#Baths|the Baths Club]] (1892)
*Member, Men's Committee of the [[Social Victorians/London Clubs#Prince's Skating Club|Prince's Skating Club]], which also included Lord Edward Cecil, Lord Redesdale, Mr. [[Social Victorians/People/Lyttelton|Alfred Lyttelton]], Sir Edgar Vincent, Sir William Hart Dyke, and Mr. [[Social Victorians/People/Grenfell|W. H. Grenfell]]&lt;ref name=&quot;:11&quot; /&gt; (1902, at least)
*[[Social Victorians/Timeline/1896#25 March 1896, Wednesday|The Sala Memorial Fund]], member of the committee (from 25 March 1896)
* Member of an &quot;influential committee&quot; headed by the Lord Mayor &quot;to restore salmon to the Thames&quot; (June 1899)&lt;ref&gt;&quot;Salmon in the Thames.&quot; ''Berks and Oxon Advertiser'' 30 June 1899, Friday: 5 [of 8], Col. 4a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002298/18990630/079/0005. Print n.p.&lt;/ref&gt;

== Timeline ==

=== 1870s ===
'''1872 February 8''', Richard Bourke, 6th Earl of Mayo was assassinated while inspecting a &quot;convict settlement at Port Blair in the Andaman Islands ... by Sher Ali Afridi, a former Afghan soldier.&quot;&lt;ref&gt;{{Cite journal|date=2024-12-01|title=Richard Bourke, 6th Earl of Mayo|url=https://en.wikipedia.org/wiki/Richard_Bourke,_6th_Earl_of_Mayo|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Richard_Bourke,_6th_Earl_of_Mayo.&lt;/ref&gt; The Hon. Algernon's brother Dermot became the 7th Earl at 19 years old.

'''1876 November 24, Friday''', the Hon. Algernon Bourke was one of 6 men (2 students, one of whom was Bourke; 2 doctors; a tutor and another man) from Cambridge who gave evidence as witnesses in an inquest about the death from falling off a horse of a student.&lt;ref&gt;&quot;The Fatal Accident to a Sheffield Student at Cambridge.&quot; ''Sheffield Independent'' 25 November 1876, Saturday: 7 [of 12], Col. 5a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000181/18761125/040/0007. Print title: ''Sheffield and Rotherham Independent'', n. p.&lt;/ref&gt;

'''1879 December 27, Saturday – 29, Monday''', Algernon Bourke was in Kilrush as a Local Government Board Inspector:&lt;blockquote&gt;Among many distinguished visitors at the Vandeleur Arms Hotel, Kilrush this week was the Hon. Algernon Bourke Local Government Board Inspector who arrived on Saturday, and sojourned there until 2 o'clock on Monday, when the honourable gentleman left by Steamer tor Limerick.&lt;ref&gt;&quot;Fashionable Intelligence.&quot; ''Kilrush Herald and Kilkee Gazette'' 01 January 1880, Thursday: 2 [of 5], Col. 3a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003186/18800101/011/0002. Print title ''Kilrush Herald'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;

=== 1880s ===
'''4 February 1880, Wednesday''', Hon. Algernon Bourke attended the [[Social Victorians/Timeline/1880#Grand Ball at Palmerstown House Hosted by the Earl of Mayo|grand ball at Palmerstown House hosted by the Earl of Mayo]]. 

'''1880 March 30, Tuesday''', Algernon Bourke was working in the judicial system in Newcastle, County Limerick, possibly as Poorhouse Inspector:&lt;blockquote&gt;A sworn enquiry was held to-day at the Workhorse, Newcastle West, by the Hon Algernon Bourke, L.G.I., to enquire into charges preferred by Dr. Pierce, Medical Office, against Dr. O'Shaughnesay. The enquiry was adjourned till Thursday next. Mr Moran, sol., Rathkeale, was engaged for Dr. O'Shaughnessy.&lt;ref&gt;&quot;Sworn Enquiry.&quot; &quot;Limerick County. Newcastle West Intelligence.&quot; ''Bassett's Chronicle'' 31 March 1880, Wednesday: 3 [of 4], Col. 3b–c [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003471/18800331/044/0003.  Print title ''Bassett's Daily Chronicle'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;'''1880 April 17, Saturday''', in-jokes dominate this report mentioning Algernon Bourke in the context of the Kildare and National Hunt races in Dublin:&lt;blockquote&gt;And in mopy Upper Mount-street, where young Algernon Bourke, of the Onety-oneth, had promised to call for, and afterwards spin down to the races in his mail phaeton, the Blake girls; and in fastidious Fitzwilliam-place, and exclusive &quot;Murryan-squeer,&quot; from which dashing army men, in their neatly-appointed, well horsed drags were to &quot;tool&quot; down sweet young Dublin lasses of the ''crême d la crême'' [sic], many an anxious forecasting of the weather was taken, lest by an unpropitious shower that last triumph of Mrs. Manning, or the Forrests, or Miss Sedford, or any of the ''grandes dames de la mode'' should be rendered as worthless as a Confederate &quot;greenback.&quot; But by ten o'clock all doubts were happily set aside, and up struck the lovely April day in all its spring-time glory and then the road, oh, the road!&lt;ref&gt;&quot;To Punchestown and Back by the Old Road.&quot; ''Illustrated Sporting and Dramatic News'' 17 April 1880, Saturday: 6 [of 24], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001857/18800417/013/0006. Same print title, p. 102.&lt;/ref&gt;&lt;/blockquote&gt;'''1881 May 10, Tuesday''', Algernon Bourke attended the [[Social Victorians/Timeline/1881#1881 May 10, Tuesday|wedding of Marion Lascelles, eldest daughter of the Hon. Egremont W. Lascelles, brother of the Earl of Harewood, and Lieutenant Henry Dent Brocklehurst, of the Second Life Guards, nephew of Mr. Philip Brocklehurst, of Swithamley Park, Macclesfield]]. His gift was an &quot;old enamelled watch set in pearls.&quot;&lt;ref&gt;&quot;Nuptial Rejoicings at Middlethorpe Manor. Marriage of Miss Lascelles and Lieut. Brocklehurst.&quot; ''Yorkshire Gazette'' 14 May 1881, Saturday: 9 [of 12], Cols. 3a–4a [of 6]. ''British Newspaper Archive''https://www.britishnewspaperarchive.co.uk/viewer/bl/0000266/18810514/057/0009. Print same title and p.&lt;/ref&gt;

'''1881 May 23, Monday, 2:00 p.m.''', Algernon Bourke is listed among the Honourables at the [[Social Victorians/Timeline/1881#Queen's Levee at St. James's Palace|Queen's Levee at St. James's Palace]].

'''1881 July 14, Thursday afternoon, beginning about 2 p.m.''', Algernon Bourke was invited to a Garden Party at Marlborough House hosted by [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]]. Members of the family of the [[Social Victorians/People/Mayo|Earl of Mayo]] were also among the 1,500 or so invited guests.

'''1881 July 22, Friday''', Algernon Bourke was invited to an [[Social Victorians/Timeline/1881#22 July 1881, Friday|evening party at Marlborough House hosted by the Prince and Princess of Wales]].

'''1881 September 17, Saturday''', Algernon Bourke was reported among the company at Doncaster during race week.&lt;ref&gt;&quot;List of the Company.&quot; ''York Herald'' 17 September 1881, Saturday: 8 [of 16], Col. 4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000499/18810917/183/0008. Same print title and p.&lt;/ref&gt;

'''1881 November 22, Tuesday''', Algernon Bourke was sued in Dublin by Henry Naylor because he &quot;had declined to pay&quot; for a £35 piano.&lt;ref&gt;&quot;Henry Naylor v. the Hon. Algernon Bourke.&quot; &quot;Exchequer Division.&quot; &quot;High Court of Justice.&quot; ''Belfast Morning News'' 23 November 1881, Wednesday: 3 [of 4], Col. 8a [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000428/18811123/015/0003. Same print title, n.p.&lt;/ref&gt;

'''1881 December 8, Thursday''', Algernon Bourke was part of a [[Social Victorians/Timeline/1881#Battue at Palmerstown|battue at Palmerstown]], when the group bagged 172 pheasants, hares and rabbits.

'''1882 March 7, Tuesday''', Algernon Bourke attended the [[Social Victorians/Timeline/1882#7 March 1882, Tuesday|fashionable wedding of Reginald Chandos-Pole and Violet Denison]].

'''1882 March 15, Wednesday''', Algernon Bourke attended [[Social Victorians/Timeline/1882#The Marchioness of Salisbury's Assembly|the Marchioness of Salisbury's first reception of the season]].

'''1882 July 13, Thursday''', Algernon Bourke was invited to the [[Social Victorians/1882-07-13 Marlborough House Garden Party|Garden Party at Marlborough House for Queen Victoria]] hosted by [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]]. The more than 1,000 people invited also included a number of people from the family of the [[Social Victorians/People/Mayo|Earl of Mayo]].

'''1882 September 19''', Alexander M. Broadley in Tunis received a letter from Algernon Bourke asking him to provide legal counsel to defend prisoner-of war Arábi in Egypt, with Wilfred Scawen Blunt, which he did.&lt;ref name=&quot;:24&quot; /&gt; (2)

'''1882 September 28, Saturday''', Algernon Bourke attended the [[Social Victorians/Timeline/1882#The Wedding of John M'Donald and Georgiana Lambart|wedding of John M'Donald and Georgiana Lambart]].

'''1883 March 21, Wednesday''', the Evening Irish Times announced that Algernon Bourke &quot;has arrived at Kingstown from England.&quot;&lt;ref&gt;&quot;Court and Fashion.&quot; ''Evening Irish Times'' 21 March 1883, Wednesday: 7 [of 8], Col. 5a [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003464/18830321/086/0007. Same print title and p.&lt;/ref&gt;

'''23 July 1883, Monday, noon''', the Hon. Algernon Bourke was invited to a [[Social Victorians/Timeline/1883#Garden Party at Marlborough House, at Noon|garden party at Marlborough House]] hosted by the Prince and Princess of Wales.

'''31 October 1883, Wednesday''', Algernon Bourke attended the wedding of [[Social Victorians/Timeline/1883#Wedding of Lady Cecelia Hay and Captain George Webbe|Lady Cecelia Hay and Captain George Webbe]].&lt;p&gt;

'''1884 February 16, Saturday''', Algernon Bourke attended [[Social Victorians/Timeline/1884#16 February 1884, Saturday|the funeral of Thomas Chenery, editor of the ''Times'']].

'''1884 April 4, Saturday''', Algernon Bourke was (may have been?) one of the [[Social Victorians/Timeline/1884#5 April 1884, Saturday|&quot;Supporters of the Pall&quot; at the funeral]] of [[Social Victorians/People/Leopold|Prince Leopold George Duncan Albert, Duke of Albany]] at St. George's, Windsor.

'''1884 April 26, Saturday''', Algernon Bourke attended a [[Social Victorians/Timeline/1884#26 April 1884, Saturday|dinner party at the Lord Mayor's Mansion House for conservatives to meet Sir Stafford Northcote]].

'''1884 May 3, Saturday''', the &quot;Rochester Conservatives&quot; announced that they would &quot;bring forward the Hon. Algernon Bourke, brother of Lord Mayo, as their second candidate,&quot;&lt;ref&gt;&quot;Election Intelligence.&quot; ''Yorkshire Gazette'' 03 May 1884, Saturday: 4 [of 12], Col. 6a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000266/18840503/011/0004.&lt;/ref&gt; but because he would not be the first candidate, Bourke declined.&lt;ref&gt;&quot;Rochester.&quot; London ''Daily Chronicle'' 09 May 1884, Friday: 3 [of 8], Col. 8b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005049/18840509/049/0003.&lt;/ref&gt;

'''1884 June 18, Wednesday''', Mr. Algernon Bourke was on a committee to watch a [[Social Victorians/Timeline/1884#18 June 1884, Wednesday|Mr. Bishop's &quot;thought-reading&quot; experiment]], which was based on a challenge by Henry Labourchere made the year before. This &quot;experiment&quot; took place before a fashionable audience.

'''1884 July 25, Friday, afternoon''', the Hon. Algernon Bourke was invited to a [[Social Victorians/Timeline/1884#Garden Party at Marlborough House hosted by the Prince and Princess of Wales|Garden Party at Marlborough House hosted by the Prince and Princess of Wales]].

'''1885 January 22, Thursday''', Algernon Bourke's gift to [[Social Victorians/Timeline/1885#Wedding of George Buckle and Alicia Payn|George Buckle and Alicia Payn for their wedding]] was an antique cabinet.

'''1885 July 7, Tuesday''', Algernon Bourke attended [[Social Victorians/Timeline/1885#7 July 1885, Tuesday|Eva Bourke's wedding to Windham Wyndham-Quin]] at St. Mary Abbots, Kensington.

'''1885 July 13, Monday''', Algernon Bouurke was at Victoria Station as part of the [[Social Victorians/Timeline/1885#Arrival of Lord Wolseley in London from Egypt|crowd greeting Lord Wolseley on his return from Egypt]].

'''1885 July 24, Friday''', the Hon. Algernon Bourke was invited to a [[Social Victorians/1885-07-24 Marlborough House Ball|ball at Marlborough House]] hosted by the Prince and Princess of Wales.

'''1885 September 26, Saturday''', Algernon Bourke took part in the [[Social Victorians/Timeline/1885#26 September 1885, Saturday|Ealing Conservative Club fete and meeting]] supporting Salisbury's government and condemning &quot;the dictates of one man&quot; — Gladstone — for Gordon's death.

'''1885 October 3, Saturday''', the Hon. Algernon Bourke was named as the Conservative candidate for Clapham in the Battersea and Clapham borough after the Redistribution Bill determined the electoral districts for South London.&lt;ref&gt;&quot;South London Candidates.&quot; ''South London Press'' 03 October 1885, Saturday: 9 [of 16], Col. 5b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000213/18851003/096/0009. Print p. 9.&lt;/ref&gt; On Sunday 15 November 1885 the London ''Weekly Dispatch'' supported Moulton, the Liberal candidate, who ultimately won the election:&lt;blockquote&gt;

Though a successful lawyer, Mr. Moulton is much more than that. He is a thorough and independent student of political science, who may be trusted to do good service to the Liberal cause with brain as well as with tongue. It will be matter for hearty congratulation if he defeats the Hon. Algernon Henry Bourke, who is a dashing and unscrupulous young Tory, and a nephew of the well-known politician with the same surname.&lt;ref&gt;&quot;The Political Campaign in London. VI. — The South-West Divisions.&quot; ''Weekly Dispatch'' (London) 15 November 1885, Sunday: 9 [of 16], Col. 3c [of 4]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003358/18851115/069/0009. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

On Saturday 21 November 1885 the ''South London Press'' reported on posters for Bourke's candidacy:&lt;blockquote&gt;

The Hon. Algernon Bourke, Conservative candidate for Clapham, has a very industrious billsticker, who pastes up his patron’s bills in every possible place where they can be seen to advantage. It is unfortunate, however, that choosing the flank wall of an auctioneer’s the modern &quot;Sam Slap&quot; has produced some curious combinations, such as — &quot;Vote for Bourke,&quot; &quot;Now on View;&quot; &quot;Electors of Clapham, Vote for Mr. Bourke, and&quot; &quot;Be Sold Without Reserve;&quot; &quot;Mr, Bourke will&quot; &quot;Advance Money to&quot; &quot;the Electors of Clapham;&quot; &quot;Great Conservative Meeting. The British Constitution will be&quot; &quot;Offered for Sale this Evening,&quot; &amp;c.&lt;ref&gt;&quot;Pick-up Notes.&quot; ''South London Press'' 21 November 1885, Saturday: 10 [of 16], Col. 1b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000213/18851121/155/0010. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

'''1885 November 3, Tuesday, 11:00 a.m.''', Algernon Bourke attended the [[Social Victorians/Mayo-Ponsonby Wedding 1885-11-03|wedding of his brother, Dermot, 7th  Earl of Mayo and Geraldine Ponsonby]]. He gave them 2 Sheraton secretaires.

'''1886 January 5, Tuesday, late''', the Hon. Algernon Bourke attended the [[Social Victorians/Timeline/1886#Twelfth Night|Twelfth Night celebration at the Drury Lane theatre]].

'''1886 March 13, Saturday evening''', an Hon. Mr. Bourke attended a [[Social Victorians/1886-03-13 Reception at the French Embassy|reception at the French Embassy]], possibly Algernon Bourke or possibly [[Social Victorians/People/Mayo|one of his brothers]].

'''1886 July 10, Saturday''', Hon. Algernon Bourke was invited to a [[Social Victorians/Timeline/1886#Garden Party at Marlborough House Given to the Queen|garden party at Marlborough House given to the Queen]]. Guendoline Sloane Stanley is not mentioned but Mr. and Mrs. Hans Sloane Stanley are, as are Mr. and Mrs. F. Sloane Stanley.&lt;p&gt;

'''1886 July 21, Wednesday''', Algernon Bourke was invited to the [[Social Victorians/1886-07-21 Marlborough House Ball|Ball at Marlborough House]], as were a [[Social Victorians/People/Bourke#The Sloane-Stanleys 2|Mr. and Mrs. F. Sloane-Stanley]], possibly the parents of Guendoline Sloane-Stanley (if the &quot;F&quot; is a mistake), who married Bourke on 15 December 1887. Guendoline is not mentioned as having been invited.

'''1886 July 27, Tuesday''', Algernon Bourke attended a service honoring a memorial at St. Paul's for his father, who had been assassinated.&lt;ref&gt;&quot;Memorial to the Late Earl of Mayo.&quot; ''Northern Whig'' 28 July 1886, Wednesday: 6 [of 8], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000434/18860728/143/0006. Print p. 6.&lt;/ref&gt;

'''1886 September 2, Thursday''', Mr. Algernon Bourke was part of a group of mostly aristocratic men taking part in [[Social Victorians/Timeline/1886#Augustus Harris's A Run of Luck|a &quot;trial-rehearsal&quot; as part of Augustus Harris's production]] ''A Run of Luck'', about sports.

'''1886 October 2, Saturday''', the Duke of Beaufort and the Hon. Algernon Bourke arrived in Yougal: &quot;His grace has taken a residence at Lismore for a few weeks, to enjoy some salmon fishing on the Blackwater before the close of the season.&quot;&lt;ref&gt;&quot;Chippenham.&quot; ''Wilts and Gloucestershire Standard'' 02 October 1886, Saturday: 8 [of 8], Col. 6a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001955/18861002/142/0008. Print p. 8.&lt;/ref&gt;

'''1886 October 11, Monday''', Algernon Bourke may have been taking part in a [[Social Victorians/Timeline/1886#Performance of Run of Luck|performance of ''Run of Luck'' at the Drury Lane]].

'''1886 October 23, Saturday''', Algernon Bourke was [[Social Victorians/Timeline/1886#Party at Wemyss Castle, Fife|staying at Wemyss Castle, Fife]].

'''1886 December 30, Thursday''', Algernon Bourke was back in London and attending the [[Social Victorians/Timeline/1886#Augustus Harris's The Forty Thieves|&quot;Forty Thieves&quot; pantomime at the Drury Lane Theatre]].

'''1887 January 5, Wednesday''', the Hon. Algernon Bourke was one of the chief mourners at the [[Social Victorians/Timeline/1887#Funeral of Lady Margaret Harriett Bourke|funeral of Lady Margaret Harriett Bourke]].

'''1887 March 1, 2:00 p.m.''', Algernon Bourke is listed among the Messieurs attending the [[Social Victorians/Timeline/1887#Queen's Levee at St. James's Palace|Queen's Levee at St. James's Palace]].&lt;p&gt;

'''1887 May''', a &quot;signalling incident&quot; in 1907 [[Social Victorians/Timeline/1887#May 1887|caused the Waterford ''Evening News'' to recall a similar event]] that had occurred 20 years earlier, in which Algernon Bourke, as special correspondent for the ''Times'', caused the incident to be publicized:&lt;blockquote&gt;During the manoeuvres in connection with the 1887 Jubilee of Queen Victoria a signal was observed going up from [[Social Victorians/People/Beresford|Lord Charles [Beresford]]]'s ship. It was a message to his wife, Lady Beresford, to the effect that, as he should be late for dinner, she was not to wait. Beyond the hilarity this domestic signal evoked, nothing more would have been heard of it, but Mr. Algernon Bourke (Lord Mayo's brother) was acting as special correspondent for the &quot;Times,&quot; and that paper the next morning contained a full and humorous report of the incident. Then there was trouble.&lt;ref&gt;&quot;Signalling Incident.&quot; ''Evening News'' (Waterford) 13 November 1907, Wednesday: 1 [of 4], Col. 6c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004557/19071113/021/0001.&lt;/ref&gt;&lt;/blockquote&gt;

'''1887 June 15, Wednesday''', the Hon. Algernon Bourke attended a [[Social Victorians Foreign Office Reception 1887-06-15|reception at the Foreign Office in honor of Queen Victoria's Golden Jubilee]].

'''1887 July 6, Wednesday''', Algernon Bourke was invited to and, presumably, attended the State Ball at Buckingham Palace.&lt;ref&gt;&quot;The State Ball at Buckingham Palace.&quot; ''Morning Post'' 08 July 1887, Friday: 3 [of 8], Col. 5a–6c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18870708/013/0003. Same print title and p.&lt;/ref&gt; (Col. 1c)

'''1887 August 6, Saturday''', the ''Brighton Gazette'' says that the &quot;Hon. Mrs and Mr Algernon Bourke&quot; were staying at the Royal Crescent Hotel in Brighton, but they didn't marry until 15 December 1887.&lt;ref&gt;&quot;Royal Crescent Hotel.&quot; ''Brighton Gazette'' 6 August 1887, Saturday: 3 [of 8], Col. 5c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000938/18870806/047/0003. Print title ''Brighton Gazette and Sussex Telegraph'', p. 3.&lt;/ref&gt; Perhaps an elder relative, because she is mentioned first?

'''1887 November 9, Wednesday''', the ''Hampshire Advertiser County Newspaper'' announced that&lt;blockquote&gt;A marriage is arranged, and will take place early in January, between Mr. Algernon Bourke, third son of the late Earl of Mayo, and Miss Guendolen Sloane Stanley, only daughter of Mr. Hans Sloane Stanley, of Paultons.&lt;ref&gt;&quot;Romsey, Nov. 9.&quot; ''Hampshire Advertiser'' 9 November 1887, Wednesday: 3 [of 4], Col. 7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18871109/034/0003. Print title ''Hampshire Advertiser County Newspaper'', p. 3.&lt;/ref&gt;&lt;/blockquote&gt;Shortly after, the papers announced that the wedding would not take place.

'''1887 December 15, Thursday''', Hon. [[Social Victorians/Timeline/1887#Wedding of Algernon Bourke and Gwendolen Sloane Stanley|Algernon Bourke and Guendoline Stanley were married at St. Paul's]], Knightsbridge, by Bourke's uncle the Hon. and Rev. George Bourke. Only family members attended because of &quot;the recent death of a near relative of the bride.&quot;&lt;ref&gt;&quot;Court Circular.&quot; ''Morning Post'' 16 December 1887, Friday: 5 [of 8], Col. 7c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18871216/066/0005.&lt;/ref&gt; Who the &quot;near relative of the bride&quot; was not in her nuclear family, and perhaps that explains the cancellation of the wedding and then the changing of the wedding date and not some problem in the couple.

'''1888 – 1899 January 1''', the Hon. Algernon Bourke was &quot;proprietor&quot; of [[Social Victorians/London Clubs#White's|White's Club, St. James's Street]].&lt;ref name=&quot;:9&quot;&gt;&quot;The Hon. Algernon Bourke's Affairs.&quot; ''Eastern Morning News'' 19 October 1899, Thursday: 6 [of 8], Col. 7c [of7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001152/18991019/139/0006. Print p. 6.&lt;/ref&gt;

'''1888 January 21, Saturday''', Guendoline Bourke attended the wedding of [[Social Victorians/Timeline/1888#Hamilton-Ewart Wedding|Florence Ewart and Henry Hamilton]].

'''1888 March 7, Wednesday''', assuming that this date is not a week after the actual date, [[Social Victorians/People/Beresford|Lady Charles Beresford]] held a [[Social Victorians/Timeline/1888#1888 March 7, Wednesday|notable and well-attended &quot;at home&quot;]] that Guendoline Bourke attended, reported for being dressed in white and being among the beautiful women present.

'''6 April 1888, Friday''', Algernon and Guendoline Bourke attended the [[Social Victorians/Timeline/1888#6 April 1888, Friday|New Forest United Hunt ball at the New Forest Hall, Lyndhurst]].

'''1888 May 2, Wednesday''', Algernon and Guendoline Bourke attended the [[Social Victorians/Timeline/1888#The Marchioness of Salisbury's Reception|Marchioness of Salisbury's reception]] at the Salisbury home on Arlington-street.

'''1888 May 22, Tuesday''', the Dowager Countess of Mayo presented Guendoline Bourke at the [[Social Victorians/Timeline/1888#Queen's Drawing Room|Queen's drawing-room]] hosted by the Princess of Wales. This is Guendoline Bourke's dress:&lt;blockquote&gt;Empire robe de cour of white satin duchesse, lined with rich pink silk, sufficiently bright to give a beautiful shell-like tint through the satin; tulle underdress, with upper skirt, embroidered with pearl, and caught up in Greek folds with large pink Tosca roses; white satin bodice, with Josephine pink sash tied at side, Headdress, veil and plumes; ornaments, diamonds.&lt;ref&gt;&quot;Dresses at the Drawing-Room.&quot; ''Epsom Journal'' 22 May 1888, Tuesday: 3 [of 6], Col. 5b–c [of 6]. ''British Newspapers Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004837/18880522/034/0003. Print: title ''Local Journal'', p. 3.&lt;/ref&gt;&lt;/blockquote&gt;

Another description:&lt;blockquote&gt;Mrs. Algernon Bourke's train was of white satin lined with pink, which showed through with charmingly shell-like effect. The dress, fashioned after those of the Empire period, was of white satin embroidered with pearls. A very broad sash of pink satin made the waist seem quaintly short, a trying thing to any but the young and tall, both of which qualifications Mrs. Bourke most happily possesses. She carried a lovely posy of La France roses.&lt;ref&gt;&quot;Gossip on Dress.&quot; ''Boston Spa News'' 25 May 1888, Friday: 2 [of 8], Col. 1b–2b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003395/18880525/014/0002. Print title The News, n.p.&lt;/ref&gt; (Col. 1c)&lt;/blockquote&gt;'''1888 June 8, Friday''', Hon. Algernon and Mrs. Bourke attended a [[Social Victorians/Timeline/1888#Dinner and Dance Hosted by Lord and Lady Wimborne at Hamilton House|dinner and dance Hosted by Lord and Lady Wimborne at Hamilton House]] featuring Prince and Princess Christian of Schleswig-Holstein, and for the ball, the King of Sweden and Norway and the Prince and Princess of Wales and their daughters were present.

'''1888 June 19, Tuesday''', Guendoline Bourke was one of the principal guests at the wedding of [[Social Victorians/Timeline/1888#19 June 1888, Tuesday|Captain Philip Green and Miss Mabel Emilie Scott]].

'''1888 July 26''', [[Social Victorians/People/Montrose|Caroline Graham Stirling-Crawford]] (known as Mr. Manton for her horse-breeding and -racing operations) and Marcus Henry Milner married.&lt;ref name=&quot;:12&quot;&gt;&quot;Hon. Caroline Agnes Horsley-Beresford.&quot; {{Cite web|url=https://thepeerage.com/p6863.htm#i68622|title=Person Page|website=thepeerage.com|access-date=2020-11-21}}&lt;/ref&gt; According to the ''Nottingham Evening Post'' of 31 July 1888,&lt;blockquote&gt;LONDON GOSSIP.

(From the ''World''.)

The marriage of &quot;Mr. Manton&quot; was the surprise as well the sensation of last week. Although some wise people noticed a certain amount of youthful ardour in the attentions paid by Mr. Marcus Henry Milner to Caroline Duchess of Montrose at '''Mrs. Oppenheim's ball''', nobody was prepared for the sudden ''dénouement''; '''and it''' were not for the accidental and unseen presence [[Social Victorians/People/Mildmay|a well-known musical amateur]] who had received permission to practice on the organ, the ceremony performed at half-past nine on Thursday morning at St. Andrew's, Fulham, by the Rev. Mr. Propert, would possibly have remained a secret for some time to come. Although the evergreen Duchess attains this year the limit of age prescribed the Psalmist, the bridegroom was only born in 1864. Mr. &quot;Harry&quot; Milner (familiarly known in the City as &quot;Millions&quot;) was one of the zealous assistants of that well-known firm of stockbrokers, Messrs. Bourke and Sandys, and Mr. Algernon Bourke, the head of the house (who, of course, takes a fatherly interest in the match) went down to Fulham to give away the Duchess. The ceremony was followed by a ''partie carrée'' luncheon at the Bristol, and the honeymoon began with a visit to the Jockey Club box at Sandown. Mr. Milner and the Duchess of Montrose have now gone to Newmarket. The marriage causes a curious reshuffling of the cards of affinity. Mr. Milner is now the stepfather of the [[Social Victorians/People/Montrose|Duke of Montrose]], his senior by twelve years; he is also the father-in-law of [[Social Victorians/People/Lady Violet Greville|Lord Greville]], Mr. Murray of Polnaise, and [[Social Victorians/People/Breadalbane|Lord Breadalbane]].&lt;ref name=&quot;:8&quot; /&gt;&lt;/blockquote&gt;

'''1888 December 1st week''', according to &quot;Society Gossip&quot; from the ''World'', the Hon. Algernon Bourke was suffering from malaria, presumably which he caught when he was in South Africa:&lt;blockquote&gt;I am sorry to hear that Mr. Algernon Bourke, who married Miss Sloane-Stanley a short time ago, has been very dangerously ill. Certain complications followed an attack of malarian fever, and last week his mother, the Dowager Lady Mayo, and his brother, Lord Mayo, were hastily summoned to Brighton. Since then a change for the better has taken place, and he is now out of danger.&lt;ref&gt;&quot;Society Gossip. What the ''World'' Says.&quot; ''Hampshire Advertiser'' 08 December 1888, Saturday: 2 [of 8], Col. 5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18881208/037/0002. Print title: ''The Hampshire Advertiser County Newspaper''; print p. 2.&lt;/ref&gt;&lt;/blockquote&gt;

'''1888 December 20, Thursday''', the Sloane-Stanley family, including Guendoline Bourke, attended the [[Social Victorians/Timeline/1888#20 December 1888, Thursday|funeral of Hans Sloane Stanley]]. Algernon Bourke did not attend because he was still too ill.

'''1889 January 22, 2:30 p.m., Tuesday''', Algernon and Guendoline Bourke sent a gift for the [[Social Victorians/Cecil Lambton Wedding 1889 January 22|wedding of Lady Eleanor Lambton and Lord Robert]] Cecil, a pair of antique mirrors.

'''1889 May 18, Saturday''', Algernon Bourke attended the [[Social Victorians/Timeline/1889#18 May 1889, Saturday|opening of the Italian Opera season at Covent Garden]].

'''1889 May 27, Monday, 11 p.m.''', the dancing commenced at [[Social Victorians/Timeline/1889#The Queen's State Ball at Buckingham Palace|the Queen's State Ball at Buckingham Palace]], with both the Hon. Algernon and the Hon. Guendoline Bourke present.

'''1889 June 8, Saturday''', the Hon. Algernon Bourke contributed some art he owned to the collection of the Royal Institute of Painters in Water-Colours' [[Social Victorians/Timeline/1889#8 June 1889, Saturday|exhibition of &quot;the works of the 'English Humourists in Art.'&quot;]]

'''1889 July 2, Tuesday''', Guendoline and Algernon Bourke sat in the Muriettas' box at a [[Social Victorians/Timeline/1889#The Shah at a Covent Garden Opera Performance|gala performance at Covent Garden also attended by the Prince and Princess of Wales, a number of other royals and the Shah]].&lt;p&gt;

'''1889 27 July, Saturday''', Guendoline and Algernon Bourke attended a [[Social Victorians/Timeline/1889#Garden Party Hosted by Mr. and Mrs. Augustus Harris|garden party hosted by Mr. and Mrs. Augustus Harris]], which was attended by a people from the theatre and arts worlds.&lt;p&gt;

'''1889 December 2, Monday''', Guendoline Bourk's mother, Emilie Sloane-Stanley, married James Shelly Bontein:&lt;blockquote&gt;&lt;p&gt;

BONTEIN—STANLEY — December 2, at St. George's, Hanover Square, London, by the Rev. G. S. de Sansmarez, James Shelly, only son of the late James Bontein, Gentleman Usher and Clerk of the Robes to the Queen, to Emilie Josephine, widow of Hans Sloane Stanley, of Paultons.&lt;ref name=&quot;:18&quot; /&gt;&lt;/blockquote&gt;'''1889 December 17, Tuesday''', Hon. Algernon and Mrs. Bourke gave a gift to [[Dangan-Neville Wedding|Lady Violet Nevill for her wedding to Henry Wellesley, Viscount Dangan]] and so were probably in attendance.

=== 1890s ===
'''1890 January 9, Thursday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1890#The York Hunt Ball|fancy-dress Hunt Ball in York]]. She&lt;blockquote&gt;looked a picture in a Gainsborough gown. The white satin skirt was flounced with sable and veiled with ''chiffon'', the setuage of which was left to show without being hemmed up. There was a broad sash of rose-pink silk and each buttonhole was filled round with crimped lisse.&lt;ref&gt;&quot;Our London Letter.&quot; ''Irish Society'' (Dublin) 11 January 1890, Saturday: 17 [of 24], Col. 2a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001939/18900111/042/0017. Same print title, p. 29.&lt;/ref&gt;&lt;/blockquote&gt;'''1890 February''' '''12, Wednesday''', Hon. Algernon and Mrs. Bourke attended [[Social Victorians/Timeline/1890#Lady Constance Leslie's Reception|Lady Constance Leslie's reception]] at her house in Stratford-place.

'''1890 April 9, Wednesday''', Guendoline Bourke attended [[Social Victorians/Timeline/1890#The New Forest United Hunt Ball|the New Forest United Hunt Ball]].

'''1890 June 3, Tuesday''', Guendoline Bourke attended the 2:30 p.m. [[Social Victorians/Timeline/1890#Münster-Hay Wedding|wedding of Count Alexander Münster and Lady Muriel Henrietta Constance Hay]]. She is also listed as having attended a [[Social Victorians/Timeline/1890#Dinner and Concert Hosted by Mrs. Arthur Williams and Ball by Mrs. Menzies|ball hosted by Mrs. J. Menzies (daughter of Mrs. Arthur Wilson)]] that Prince Eddie, the Duke of Clarence and Avondale, also attended, that night.  

'''1890 July 4, Friday, 11 p.m.''', the Hon. Algernon and Guendoline Bourke attended [[Social Victorians/Timeline/1890#The Queen's State Ball at Buckingham Palace|the Queen's State Ball at Buckingham Palace]]. The dancing commenced shortly after 11:00. 

'''1890 July 15, Tuesday''', Hon. Algernon and Mrs. Bourke were invited to a [[Social Victorians/Timeline/1890#Garden Party at Marlborough House to Meet the Queen|garden party at Marlborough House to meet the Queen]]. 

'''1890 July 19, Saturday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1890#Wedding of James Francis Harry St. Clair-Erskine and Violet Aline Vyner|wedding of James Francis Harry St. Clair-Erskine and Violet Aline Vyner]], the two of them giving &quot;four small silver dessert dishes&quot; and Guendoline giving an &quot;enamel and diamond pin.&quot;&lt;ref&gt;&quot;Marriage of Lord Loughborough with Miss Vyner.&quot; ''Fife Free Press'' 26 July 1890, Saturday: 2 [of 8], Col. 1a–2b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001110/18900726/015/0002. Same print title and p.&lt;/ref&gt; (Col. 2b)

'''1890 July 24, Thursday''', Algernon and Guendoline Bourke attended a [[Social Victorians/Timeline/1890#Dinner and Dance Hosted by Lord Alington|dance hosted by Lord Alington]] attended also by the Prince and Princess of Wales and Princesses Victoria and Maud.

'''1890 September 6, Saturday''', the ''Country Gentleman'' (as it was called at the time) reported that &quot;Muckross, the only deer forest in Ireland, it may be said, has this year been rented by Mr. Algernon Bourke, who will next week be joined there for the stalking season by his brother, Lord Mayo.&quot;&lt;ref&gt;&quot;Shooting. Moors, Forests, and Fishings.&quot; ''Sporting Gazette'' 06 September 1890, Saturday: 11 [of 38], Col. 1c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002525/18900906/065/0011. Print: ''Country Gentleman'', p. 1251.&lt;/ref&gt; On 11 October 1890 the ''St. James's Gazette'' says,&lt;blockquote&gt;The Earl of Durham has been staying at Muchross, county Kerry, on a visit to the Hon. A. Bourke, who has rented the celebrated shootings and fishings on that estate for the autumn.&lt;ref&gt;&quot;Court and Society.&quot; ''St James's Gazette'' 11 October 1890, Saturday: 12 [of 16], Col. 1b [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001485/18901011/064/0012. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;'''1890 October 25, Saturday''', the Hon. Algernon and Mrs. Bourke gave a gold-mounted box to [[Social Victorians/Loder De Vere Beauclerk Wedding|Lady Louise De Vere Beauclerk on her wedding to Gerald Loder, M.P.]], so they were probably present at the wedding, or at least the reception. Mrs. Bontein [sic Bontine], Guendoline's mother, gave a silver box, suggesting the relationship was through the women.

'''1890 November 29, 11:30 Saturday morning''', Algernon Bourke's gift for the [[Social Victorians/Dudley-Beckwith Wedding 1890-11-29|wedding of the Hon. Francis Dudley and Miss Forbes Beckwith]] was some cases of a Bordeaux wine: &quot;three dozen Cantenac, 1875 vintage.&quot;&lt;ref&gt;&quot;Marriage of Lord Leigh's Heir. Descriptive Sketch of the Ceremony, and Full List of Guests and Presents.&quot; ''Leamington Spa Courier'' 6 December 1890, Saturday: 6 [of 10], Cols. 1a–4a. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18901206/021/0006. Same print title and p.&lt;/ref&gt;{{rp|Col. 3b}} Guendoline Bourke is not listed as having been invited to the reception, but this list from the ''Leamington Spa Courier'' has some gaps.

'''1890 December 4, Thursday''', Guendoline and Algernon Bourke attended the [[Mure-Portal Wedding 1890-12-04|wedding of Miss Mure and Mr. S. J. Portal]]. Their gift is not recorded.

'''1891 January''', Algernon Bourke took party in a [[Social Victorians/Timeline/1891#Shooting Party in Kallarnet, Totton|shooting party in Kallarnet, Totton]].

'''1891 June 24, Wednesday''', the Hon. Algernon and Mrs. Bourke attended a [[Social Victorians/Timeline/1891#Dinner and Ball Hosted by Lord and Lady Wimborne|dinner and ball Hosted by Lord and Lady Wimborne]] featuring Princess Mary Adelaide, the Duke of Teck, and Princess Victoria.

'''1891 July 9, Thursday''', Algernon and Guendoline Bourke were invited to a [[Social Victorians/1891-07-09 Garden Party|large Garden Party at Marlborough House]] hosted by the [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]] in honor of Queen Victoria and the German Emperor and Empress. The more than 3,000 people invited also included a number of people from the [[Social Victorians/People/Mayo|family of the Earl of Mayo]].

'''1891 July 22, Wednesday''', Guendoline Bourke attended a [[Social Victorians/Timeline/1891#Dinner and Dance at Alington House|dance at the Earl and Countess Alington]]'s that also included the Prince and Princess of Wales.

'''1891 October 22, Thursday''', Hon. and Mrs. Bourke attended at least the reception of the [[Social Victorians/Timeline/1891#Le Strange Astley Wedding|Le Strange—Astley Wedding]], although perhaps the couple is not the Algernon Bourkes.

'''1891 November 22, Sunday''', the London ''Weekly Dispatch'' reports a performance by American &quot;Lady Magnet&quot; Mrs. Abbott, who claimed to be able to lift anybody using only her magnetic properties. An enthusiastic &quot;committee of some fifteen gentlemen presented a written and signed testimonial&quot; supporting Mrs. Abbott, &quot;the Hon. Algernon Bourke, Professor Atkinson, Dr. Hides, and three other doctors who prefer to remain incog., being among the signatories. All the medical gentlemen concerned assured the ''Evening News and Post'' reporter of their complete and unconditional surrender. One of them went so far as to say that he had come with the full determination of disbelieving, but had been quite able to act up to his resolve.&quot;&lt;ref&gt;&quot;The Lady Magnet. Draws Crowds of People Who Divide in Opinion about Her.&quot; ''Weekly Dispatch'' (London) 22 November 1891, Sunday: 16 [of 16], Cols. 3a–4b [of 4]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003358/18911122/203/0016. Print: same title and p.&lt;/ref&gt;

'''1892''', the Hon. Algernon Bourke privately published his ''The History of White's'', the exclusive gentleman's club.

'''1892 January 27, Saturday''', Algernon and Guendoline Bourke attended the very fashionable [[Social Victorians/Timeline/1892#The Wedding of Lord Henry Cavendish Bentinck, M.P., and Lady Olivia Taylour|wedding of Lord Henry Cavendish Bentinck, M.P., and Lady Olivia Taylour]]. Their gift was not noted in the list.

'''1892 February''' '''10, Wednesday''', Guendoline Bourke attended the [[Social Victorians/1892-02-10 Alington Leigh Wedding|very fashionable wedding of Henry, Lord Alington and Evelyn Henriette Leigh]] [[Social Victorians/1892-02-10 Alington Leigh Wedding|in St. Paul's, Knightsbridge]]

'''1892 April''' '''10, Wednesday, about 2:30 p.m.''', Guendoline Bourke attended [[Social Victorians/1892-02-10 Alington Leigh Wedding|the very fashionable wedding between Henry Sturt, Lord Alington and Evelyn Leigh]]. Her gift was a &quot;tortoiseshell and gold heart-shaped tray.&quot;&lt;ref name=&quot;:02&quot;&gt;&quot;Lord Alington to Miss Leigh.&quot; ''Gentlewoman'' 20 February 1892, Saturday: 21 [of 46], Cols. 1a–3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18920220/092/0021. Same print title, p. 237.&lt;/ref&gt; (Col. 3a) 

'''1892 June 25, Saturday''', the ''Gentlewoman''&lt;nowiki/&gt;'s &quot;Overheard by the Little Bird&quot; says &quot;That pretty Mrs. Algernon Bourke has been staying here, but returned to England in time for Ascot.&quot;&lt;ref&gt;Little Bird, The. &quot;Overheard by the Little Bird.&quot; ''Gentlewoman'' 25 June 1892, Saturday: 32 [of 60], Col. 3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18920625/157/0032. Same print title, p. 860.&lt;/ref&gt; 

'''1892 December 13, Tuesday''', the ''Gentlewoman'' says Guendoline Bourke is lovely in its coverage of [[Social Victorians/Timeline/1892#Wedding of Miss Eleanor M. Ewart and Captain Guy Withington|Eleanor M. Ewart and Captain Guy Withington's wedding]]. 

'''1892 December 22, Thursday''', Algernon Bourke attended the [[Social Victorians/Timeline/1892#22 December 1892, Thursday|monthly meeting of the Zoological Society in Hanover-square]].&lt;p&gt;

'''1893 February 11, Tuesday''', Algernon Bourke opened Willis's Restaurant:&lt;blockquote&gt;Mr. Algernon Bourke has in his time done many things, and has generally done them well. His recently published history of White's Club is now a standard work. White's Club itself was a few years ago in its agony when Mr. Bourke stepped in and gave it a renewed lease of life. Under Mr. Bourke's auspices &quot;Willis's Restaurant&quot; opened its doors to the public on Tuesday last in a portion of the premises formerly so well known as Willis's Rooms. This new venture is to rival the Amphitryon in the matter of cuisine and wines; but it is not, like the Amphitryon, a club, but open to the public generally. Besides the restaurant proper, there are several ''cabinets particuliers'', and these are decorated with the very best of taste, and contain some fine portraits of the Georges.&lt;ref&gt;&quot;Marmaduke.&quot; &quot;Letter from the Linkman.&quot; ''Truth'' 20 April 1893, Thursday: 25 [of 56], Col. 1a [of 2]. ''British Newspaper Archive'' [https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18930420/075/0025# https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18930420/075/0025]. Print p. 855.&lt;/ref&gt;&lt;/blockquote&gt;

'''1893 February 7, Tuesday''', Guendoline Bourke attended [[Social Victorians/Timeline/1893#1893 February 7, Tuesday|the reception after Lady Emily Cadogan's wedding]].

'''1893 February 20, Monday''', the Hon. Algernon Bourke is listed as having attended the [[Social Victorians/Timeline/1893#Queen's Levee at St. James's Palace|Queen's Levee at St. James's Palace]] held by the Prince of Wales; because wives generally are not listed, it seems likely Guendoline Bourke attended as well.

'''1893 February 28, Tuesday, 3:00 p.m.''', Guendoline Bourke attended a [[Social Victorians/Queens Drawing Room 1893-02-28|Queen's Drawing Room at Buckingham Palace]].&lt;p&gt;

'''1893 March 22, Wednesday''', Guendoline Bourke attended [[Social Victorians/Timeline/1893#22 March 1893, Wednesday|Lady Wimborne's reception]].

'''1893 April 1, Saturday''', Algernon Bourke published a letter to the editor of the ''Times'', reprinted in the ''Kildare Observer'', arguing against Gladstone's Home Rule bill on the grounds that Ireland would not be able to take out a loan on its own behalf because of its obligations to the U.K., including what was called its share of the national debt.&lt;ref&gt;&quot;Irish Unionist Alliance.&quot; ''Kildare Observer and Eastern Counties Advertiser'' 01 April 1893, Saturday: 6 [of 8], Col. 4c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001870/18930401/062/0006. Print: The ''Kildare Observer'', n.p.&lt;/ref&gt;

'''1893 May 13, Saturday''', Algernon Bourke was seen at [[Social Victorians/Timeline/1893#13 May 1893, Saturday|exhibitions of art and furniture for sale by Christie's and on display by Lord Clifden]].

'''1893 July 13, Thursday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1893#The Countess of Listowel's Garden Party|Countess of Listowel's Garden Party]] [[Social Victorians/Timeline/1893#The Countess of Listowel's Garden Party|at her residence, Kingston House, Princes-gate]], accompanied by Miss Adeane.

'''1893 July 14, Friday''', Guendoline Bourke attended [[Social Victorians/Sandown Races 1893-07-14|the races at Sandown]] wearing a dark-blue-and-white outfit and black hat that got described in the newspaper.

'''1893 August 1, Tuesday – August 4, Friday''', Guendoline Bourke, at least, was at [[Social Victorians/Timeline/1893#1 August 1893, Tuesday – 4 August 1893, Friday|the Goodwood races]], mentioned in the ''Gentlewoman'' for her beauty, although none of the dresses were noted.

'''1893 November 4–11, Wednesday–Saturday''', Guendoline Bourke was at a [[Social Victorians/Timeline/1893#Ralph and Mary Sneyd Hosted a Shooting Party|shooting party at Keele Hall hosted by Ralph and Mary Sneyd]].

'''1893 November 30, Thursday''', with Sir Walter Gilbey the Hon. Algernon Bourke &quot;assisted&quot; in &quot;forming [a] collection&quot; of engravings by George Morland that was exhibited at Messrs. J. and W. Vokins’s, Great Portland-street.&lt;ref&gt;&quot;The George Morland Exhibition at Vokins's.&quot; ''Sporting Life'' 30 November 1893, Thursday: 4 [of 4], Col. 4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000893/18931130/058/0004.&lt;/ref&gt;

'''1893 December 14, Thursday, afternoon''', Guendoline Bourke attended the [[Social Victorians/1893-12-14 Wedding Adele Grant and George, 7th Earl of Essex|wedding of American Adele Grant and George, 7th Earl of Essex]] and gave a &quot;pearl and gold box.&quot;&lt;ref name=&quot;:22&quot;&gt;&quot;Wedding of the Earl of Essex.&quot; ''Herts Advertiser'' 16 December 1893, Saturday; 8 [of 8], Col. 1a–4b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000415/18931216/125/0008. Print title: ''The Herts Advertiser and St Albans Times'', p. 8.&lt;/ref&gt;{{rp|Col. 3c}} Mr. and Mrs. Shelley Bontein also attended, and Mrs. Bontein gave a &quot;green leather bag and purse, with coronet and monogram in gold.&quot;&lt;ref name=&quot;:22&quot; /&gt;{{rp|3b}}

'''1894 January 27, Saturday''', Psyche in &quot;The Social Peepshow&quot; in the ''Gentlewoman'' reported on a [[Social Victorians/Timeline/1894#27 January 1894, Saturday|ball hosted by Lord and Lady Dunraven at Adare Manor]] that Guendoline Bourke attended.

'''1894 January 31, Wednesday''', Algernon and Guendoline Bourke, who was dressed more stylishly than most, attended the [[Social Victorians/Timeline/1894#Also 31 January 1894, Wednesday|Kildare Hunt Ball]] hosted by Dermot, [[Social Victorians/People/Mayo|Earl of Mayo]] and Geraldine, Countess of Mayo.

'''1894 February 24, Saturday''', ''The Field'' reported on a series of tennis matches; Algernon Bourke attended the one played at the Prince's Club.&lt;ref&gt;&quot;Tennis.&quot; ''Field'' 24 February 1894, Saturday: 39 [of 72], Col. 1c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002446/18940224/349/0039. Print title ''The Field, The Country Gentleman's Newspaper'', p. 249.&lt;/ref&gt;

'''1894 March 31, Saturday''', Psyche, in the &quot;Social Peepshow&quot; column in the ''Gentlewoman'', says that &quot;Mr. Algernon Bourke has still further embellished Willis's restaurant hard by [the St. James's Theatre], by the addition of some valuable old tapestry that lately came to the hammer at Christie's.&quot;&lt;ref&gt;Psyche. &quot;The Social Peepshow.&quot; ''Gentlewoman'' 31 March 1894, Saturday: 16 [of 56], Col. 2b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18940331/081/0016. Same print title, p. 408.&lt;/ref&gt;

'''1894 April 13, Friday''', Guendoline Bourke set sail on the [[Social Victorians/Timeline/1894#P. and O. Line S.S. Rome for Gibraltar|P. and O. Line ''S.S. Rome'' for Gibraltar]] along with her stepfather, Mr. Shelley Bontein, and her brother, Mr. Sloane Stanley.

'''31 May 1894, Thursday''', the Hon. Algernon and Mrs. Bourke attended the [[Social Victorians/Timeline/1894#Reception at Devonshire House|Duchess of Devonshire's reception at Devonshire House]].&lt;p&gt;

'''1894 June 18, Monday''', the London ''Echo'' reported that Algernon Bourke was [[Social Victorians/London Clubs#Brooks'|writing a history of Brooks' Club]].&lt;p&gt;

'''1894 June 20, Wednesday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1894#Princess Louise, Marchioness of Lorne Opened the Annual Sale of the Scottish Home Industries|Annual Sale of the Scottish Home Industries]]; her outfit was described in the article in ''Lady's Pictorial''.

'''1894 August 2, Thursday''', the column &quot;Overheard by the Little Bird&quot; says, &quot;At Willis' [restaurant] — 'What a smart cotillon Mr. and Mrs. Algernon Bourke gave on Thursday evening.&quot;&lt;ref&gt;Bird, The Little. &quot;Overheard by the Little Bird.&quot; ''Gentlewoman'' 04 August 1894, Saturday: 30 [of 56], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18940804/148/0030. Print title same, p. 144.&lt;/ref&gt; Willis's Restaurant, King-street, St. James's, was a restaurant Algernon Bourke opened in 1893.&lt;p&gt;

'''1894 September 7, Saturday''', Algernon and Guendoline Bourke were at a [[Social Victorians/Timeline/1894#7 September 1894, Saturday|shooting party at Witley]], which had been loaned to one of his brothers by William Ward, 2nd Earl of Dudley.&lt;p&gt;

'''1894 October 22, Thursday''', Algernon and Guendoline Bourke attended the [[Social Victorians/Timeline/1894#Wedding of Lord Connemara and Mrs. Coleman|luncheon after the wedding of Lord Connemara and Mrs. Coleman]].

'''1894 November 3, Saturday''', Psyche, in &quot;The Social Peepshow&quot; for the Gentlewoman, reported that Guendoline Bourke had been [[Social Victorians/Timeline/1894#3 November 1894, Saturday|seen shopping in London]].

'''1895 January 5, Saturday, 2:00 p.m.''', Algernon and Guendoline Bourke gave an old mother-of-pearl workbox to [[Wolverton-Ward Wedding 1895-01-05|Lady Edith Ward for her wedding to Frederick Glyn, Lord Wolverton]] and presumably attended the wedding and reception afterwards.&lt;p&gt;

'''1895 February 23, Saturday''', the Hon. Algernon Bourke attended the [[Social Victorians/Timeline/1895#23 February 1895, Saturday|fashionable wedding of Laurence Currie and Edith Sibyl Mary Finch]]. Guendoline Bourke is not listed as having attended, but she is not noted as absent, either. Daphne Bourke was born on 5 April 1895, probably explaining Guendoline's absence.

'''1895 March 24, Sunday – 30 March, Saturday''', Algernon Bourke was [[Social Victorians/Timeline/1895#24, Sunday – 30 March 1895, Saturday|enjoying the sunny weather in Brighton]].

'''1895 April 27, Saturday''', Algernon Bourke attended the [[Social Victorians/Timeline/1895#1895 April 27, Saturday|wedding of Norah Bourke and Henry E. A. Lindsay]]. Again, Guendoline Bourke is not listed as having attended. Daphne Bourke was born on 5 April 1895, and Psyche, writing the &quot;Social Peepshow&quot; column in the Gentlewoman, says,&lt;blockquote&gt;

I regret to hear of the serious illness of Mrs. Algernon Bourke, whose first child was born a fortnight ago. It is feared that the attack is of the nature of typhoid, but happily the patient's strength keeps up. Mrs. Bourke is at her mother's house in Clarges-street.&lt;ref&gt;Psyche. &quot;The Social Peepshow.&quot; ''Gentlewoman'' 27 April 1895, Saturday: 28 [of 84], Col. 2c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18950427/147/0028. Same print title, p. 506.&lt;/ref&gt;&lt;/blockquote&gt;

'''1895 July 13, Saturday''', Algernon Bourke donated 10s. to the ''Daily Telegraph'' National Shilling Testimonial to W. G. Grace.&lt;ref&gt;&quot;''Daily Telegraph'' National Shilling Testimonial to W. G. Grace.&quot; ''Daily Telegraph &amp; Courier'' (London) 13 July 1895, Saturday: 7 [of 12], Col. 7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/18950713/079/0007. Print: ''Daily Telegraph'', p. 7.&lt;/ref&gt;

'''1895 August 24, Saturday''', &quot;Marmaduke&quot; in the ''Graphic'' says that Algernon Bourke &quot;opened a cyclists' club in Chelsea.&quot;&lt;ref&gt;&quot;Marmaduke.&quot; &quot;Court and Club.&quot; The ''Graphic'' 24 August 1895, Saturday: 11 [of 32], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/9000057/18950824/017/0011. Print p. 223.&lt;/ref&gt;

'''1895 October''', the Hon. Algernon Bourke [[Social Victorians/Timeline/1900s#24 October 1902, Friday|opened the Prince's ice-skating rink for the]] season.if the newspapers were right that 1902 was the 7th season. He also was planning a bicycling club for Kensington Gardens to open the following season.&lt;ref&gt;Mackenzie, Ethel Morell (Miss). &quot;Pins and Needles.&quot; ''Hull Daily News'' 12 October 1895, Saturday: 24 [of 40], Col. 1a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003443/18951012/024/0024. Print title: ''Hull News Supplement'', p. 1[6? 8?].&lt;/ref&gt;

'''1895 October 7, Monday''', the Hon. Algernon and Mrs. Bourke attended the [[Social Victorians/Timeline/1895#Adeane-Cator Wedding|Maud Adeane–John Cator wedding]].

'''1895 December 11, Wednesday''', Guendoline and Algernon Bourke attended a [[Social Victorians/Timeline/1895#Sneyd Party to Meet the Duke of Coburg|shooting party at the Sneyds' to meet the Duke of Coburg]].

'''1895 December 18, Wednesday''', Algernon and Guendoline Bourke attended the [[Social Victorians/Timeline/1895#Wedding of Lady Albreda Fitzwilliam and the Hon. Charles Bourke|wedding of Lady Albreda Fitzwilliam and the Hon. Charles Bourke]]. Their gift is not noted in the newspaper account.

'''1896 March 17, Tuesday''', Algernon Bourke attended the [[Social Victorians/Timeline/1896#17 March 1896, Tuesday|annual dinner of the Cymmrodorion, or the Honourable Society of Cymmrodorion]], a society for Welsh culture and history.

'''1896 April 21, Monday''', Mr. and Mrs. A. Bourke sent a gift — a &quot;box for miniature&quot; — for [[Social Victorians/Timeline/1896#Monday, 1896 April 27|the wedding of Lady Angela St. Clair Erskine and James Stewart Forbes]].

'''1896 May 21, Thursday''', the Hon. and Mrs. Algernon Bourke attended [[Social Victorians/Timeline/1896#Mrs. C. H. Wilson's Ball|Mrs. C. H. Wilson's ball in Grosvenor-square, London]].

'''1896 May 26, Tuesday,  through 28 May, Thursday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1896#Coming of Age of Mr Sloane Stanley|3-day celebration in honor of the coming of age of her brother, Cyril Sloane Stanley]].

'''1896 June 15, Monday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1896#Dinner and Dance Hosted by the Countess of Huntingdon|dance hosted by the Earl and Countess of Huntingdon]] after their dinner party.

'''1896 July 13, Monday''', Algernon Bourke (listed among the &quot;Honourables&quot;) and Mrs. A. Bourke (Listed among the &quot;Honourable Ladies&quot;) were invited to the [[Social Victorians/Timeline/1896#Queen's Garden Party at Buckingham Palace|Queen's Garden Party at Buckingham Palace]].

'''1896 June 29, Monday''', the Hon. Mrs. Algernon Bourke attended the [[Social Victorians/Cadogan-Scott Wedding 1896-06-29|wedding and reception of Lady Sophie Cadogan and Sir Samuel Scott]]. Algernon Bourke published a letter to the editor of the ''Daily Telegraph'' about White's Club — and thus Bourke's — &quot;[[Social Victorians/London Clubs#Summer Club|Summer Club]]&quot; in Kensington Park, the subject of a little controversy.

'''1896 July 21, Tuesday''', the Hon. Algernon and Mrs. Bourke attended a [[Social Victorians/Timeline/1896#Dinner Hosted by Sir Horace and Lady Farquhar|dinner hosted by Sir Horace and Lady Farquhar in Grosvenor-square]].

'''1896 August 5, Wednesday''', Algernon and Guendoline Bourke attended at the [[Social Victorians/Timeline/1896#5 August 1896|wedding of the Hon. Terence Bourke and Miss Eveline Haines]] and gave the bride an &quot;enamel muff chain.&quot;&lt;p&gt;

'''1896 August 10, Monday''', the Morning Leader reported that the Hon. Algernon Bourke, for the Foreign Office, received Li Hung Chang at St. Paul's:&lt;blockquote&gt;At St. Paul's Li Hung was received by Field-Marshal Simmons, Colonel Lane, the Hon. Algernon Bourke, of the Foreign Office (who made the necessary arrangements for the visit) and Canon Newbolt, on behalf of the Dean and Chapter. A crowd greeted Li with a cheer as he drove up in Lord Lonsdale’s striking equipage, and his Excellency was carried up the steps in an invalid chair by two stalwart constables. He walked through the centre door with his suite, and was immediately conducted by Canon Newbolt to General Gordon’s tomb in the north aisle, where a detachment of boys from the Gordon Home received him as a guard of honor.

Li inspected the monument with marked interest, and drew the attention of his suite to the remarkable likeness to the dead hero. He laid a handsome wreath of royal purple asters, lilies, maidenhair fern, and laurel, tied with a broad band of purple silk, on the tomb.

The visit was not one of inspection of the building, but on passing the middle aisle the interpreter called the attention of His Excellency to the exquisite architecture and decoration of the chancel. Li shook hands in hearty English fashion with Canon Newbolt and the other gentlemen who had received him, and, assisted by his two sons, walked down the steps to his carriage. He returned with his suite to Carlton House-terrace by way of St. Paul’s Churchyard, Cannon-st., Queen Victoria-st., and the Embankment.&lt;ref&gt;&quot;At St. Paul's.&quot; ''Morning Leader'' 10 August 1896, Monday: 7 [of 12], Col. 2b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004833/18960810/134/0007. Print p. 7.&lt;/ref&gt;&lt;/blockquote&gt;

'''1896 August 19, Wednesday''', the ''Edinburgh Evening News'' reported on the catering that White's Club and Mr Algernon Bourke arranged for the visiting Li Hung Chang:&lt;blockquote&gt;It is probably not generally known (says the &quot;Chef&quot;) that Mr Algernon Bourke, manager of White's Club, London, has undertaken to the whole of the catering for our illustrious visitor front the Flowery Land. Li Hung Chang has five native cooks in his retinue, and the greatest good fellowship exists between them and their English ''confreres'', although considerable difficulty is experienced in conversation in understanding one another's meaning. There are between 40 and and 50 to cater for daily, besides a staff about 30; that Mr Lemaire finds his time fully occupied. The dishes for his Excellency are varied and miscellaneous, and from 14 to 20 courses are served at each meal. The bills of fare contain such items as bird's-nest soup, pigs' kidneys stewed in cream, boiled ducks and green ginger, sharks' fins, shrimps and prawns stewed with leeks and muscatel grapes, fat pork saute with peas and kidney beans. The meal usually winds with fruit and sponge cake, and freshly-picked green tea as liqueur.&lt;ref&gt;&quot;Li Hung Chang's Diet.&quot; ''Edinburgh Evening News'' 19 August 1896, Wednesday: 3 [of 4], Col. 8b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000452/18960819/057/0003.&lt;/ref&gt;&lt;/blockquote&gt;

'''1896 November 6, Friday''', both Algernon and Guendoline Bourke were on the committee for the [[Social Victorians/London Clubs#Prince's Skating Club|Prince's Club ice-skating rink]], which [[Social Victorians/Timeline/1896#Opening of the Prince's Club Skating Rink|opened on this day]].

'''1896 November 22, week of''', Mrs. Algernon Bourke was part of a [[Social Victorians/Timeline/1896#Shooting Party at the Charles Wilsons' Warter Priory, Yorkshire|shooting party at the Charles Wilsons' Warter Priory, Yorkshire]].&lt;p&gt;

'''1896 November 25, Wednesday''', Mr. and Mrs. Algernon Bouke attended [[Social Victorians/Timeline/1896#Lord and Lady Burton Hosted a Party for Derby Day|Lord and Lady Burton's party for Derby Day]].&lt;p&gt;

'''1896 December 4, Friday''', the Orleans Club at Brighton was robbed:&lt;blockquote&gt;The old building of the Orleans Club at Brighton, which opens its new club house at 33, Brunswick-terrace to-day, was the scene of a very ingenious burglary during the small hours of yesterday morning. The greater portion of the club property had already been removed to the new premises, but Mr Algernon Bourke, his private secretary, and some of the officials of the club, still occupied bed-rooms at the house in the King’s-road. The corner shop of the street front is occupied by Mr. Marx, a jeweller in a large way of business, and upon his manager arriving at nine o'clock he discovered that the place had been entered through hole in the ceiling, and a great part of a very valuable stock of jewelry extracted. An examination of the morning rooms of the club, which runs over Mr. Marx's establishment reveal a singularly neat  specimen of the burglar's art. A piece of the flooring about 15in square had been removed by a series of holes bored side by side with a centre-bit, at a spot where access to the lofty shop was rendered easy by a tall showcase which stood convemently near. A massive iron girder had been avoided by a quarter of an inch, and this circumstance and the general finish of the operation point to an artist in his profession, who had acquired an intimate knowledge of the premises. The club doors were all found locked yesterday morning, and the means of egress adopted by the thief are at present a mystery.&lt;ref&gt;&quot;Burglary at Brighton.&quot; ''Daily Telegraph &amp; Courier'' (London) 05 December 1896, Saturday: 5 [of 12], Col. 7a [of 7]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/18961205/090/0005. Print title: ''Daily Telegraph''; p. 5.&lt;/ref&gt;&lt;/blockquote&gt;

'''1896 December 10, Thursday''', Guendoline Bourke was present to help staff a stall at the [[Social Victorians/Timeline/1896#10 December 1896, Thursday|Irish Industries Exhibition and Sale, Brighton]].

'''1896 December 31, Thursday''', Guendoline Bourke hosted a New Year's Eve dance:&lt;blockquote&gt;Mrs. Algernon Bourke gave a highly satisfactory and enjoyable dance on Thursday night, when the old year was danced out and the new one danced in. Most of the silver gilters at present in to len were to the fore.&lt;ref&gt;&quot;The Man about Town.&quot; ''Sporting Gazette'' 02 January 1897, Saturday: 7 [of 34], Col. 3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002525/18970102/041/0007. Print title ''The County Gentleman'', p. 7.&lt;/ref&gt;&lt;/blockquote&gt;

'''1897 January 9, Saturday''', Psyche in &quot;The Social Peepshow&quot; says that Algernon Bourke's &quot;cheerful countenance was quite in keeping with the [Christmas] season,&quot; seen in London.&lt;ref&gt;Psyche. &quot;The Social Peepshow.&quot; ''Gentlewoman'' 9 January 1897, Saturday: 22 [of 56], Col. 2a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970109/097/0022. Same print title, p. 40.&lt;/ref&gt; 

'''1897 January 13, Wednesday – 18, Monday''', Algernon and Guendoline Bourke were guests of the [[Social Victorians/Timeline/1897#The Warwickshire Hunt Club Ball|house party associated with the Warwickshire Hunt Ball]] at [[Social Victorians/People/Warwick|Warwick Castle]].

'''1897 January 30, Saturday''', Guendoline Bourke was reported to have been out shopping in London: &quot;Another charming figure was that of Mrs. Algernon Bourke all in chinchilla, with something of pale blue in a smart toque.&quot;&lt;ref&gt;Psyche. &quot;The Social Peepshow.&quot; ''Gentlewoman'' 30 January 1897, Saturday: 20 [of 59]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970130/107/0020. Same print title, p. 134.&lt;/ref&gt;

'''1897 May 31, Monday''', Hon. Algernon and Mrs. Bourke were present at a [[Social Victorians/Timeline/1897#House Party at Warwick Castle|House Party at Warwick Castle]] hosted by the Earl and Countess of Warwick.

'''1897 June 2, Wednesday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1897#Reception at the Foreign Office|reception at the Foreign Office]].

'''1897 June 12, Saturday''', the ''Gentlewoman'' reported on Guendoline Bourke's dress and hat at the [[Social Victorians/Timeline/1897#The Duchess of Albany's Bazaar at the Imperial Institute|Duchess of Albany's Bazaar at the Imperial Institute]]. 

'''1897 June 19, Saturday''', Psyche in &quot;The Social Peepshow&quot; column in the ''Gentlewoman'' writes that Guendoline Bourke was seen driving in London, &quot;in blue, ... looking as usual very handsome.&quot;&lt;ref&gt;Psyche. &quot;The Social Peepshow.&quot; ''Gentlewoman'' 19 June 1897, Saturday: 28 [of 108], Col. 2b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970619/159/0028. Same print title, p. 848.&lt;/ref&gt;

'''1897 June 28, Monday''', Algernon and Guendoline Bourke were invited to the [[Social Victorians/Diamond Jubilee Garden Party|Garden Party at Buckingham Palace]], the final official event of the London Diamond Jubilee celebrations. Members of the family of the [[Social Victorians/People/Mayo|Earl of Mayo]] were also among the 5,000–6,000 people invited.

'''1897 July 2, Friday''', the Hon. A. and Mrs. A. Bourke and Mr. and Mrs. Bourke attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House.

'''1897 July 8, Thursday, 11:00 p.m.''', Hon. Algernon and Guendoline Bourke were present at [[Social Victorians/Timeline/1890#Queen's State Ball at Buckingham Palace|the Queen's State Ball at Buckingham Palace]]. The dancing commenced shortly after 11:00 p.m.

'''1897 July 11–16, week of''', a dog of Guendoline Bourke's won a prize at the [[Social Victorians/Timeline/1897#The Ladies' Kennel Association show in the Royal Botanic Gardens in Regent's Park|Ladies' Kennel Association show in the Royal Botanic Gardens in Regent's Park]]. 

'''1897 July 23, Friday''', both the Hon. Algernon Bourke and Guendoline Bourke attended the [[Social Victorians/Timeline/1897#Bourke-Curzon Cricket Match at the Queen's Club|Bourke-Curzon cricket match at the Queen's Club]], which Algernon Bourke's team lost. 

'''1897 July 23 — or July 30, Friday''', Guendoline Bourke attended [[Social Victorians/Timeline/1897#Lady Burton's party at Chesterfield House|Lady Burton's party at Chesterfield House]]. &lt;blockquote&gt;Far the prettiest women in the room were Lady Henry Bentinck (who looked perfectly lovely in pale yellow, with a Iong blue sash; and Mrs. Algernon Bourke, who was as smart as possible in pink, with pink and white ruchings on her sleeves and a tall pink feather in her hair.&lt;ref&gt;&quot;Lady Burton's Party at Chesterfield House.&quot; ''Belper &amp; Alfreton Chronicle'' 30 July 1897, Friday: 7 [of 8], Col. 1c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004151/18970730/162/0007. Print title: ''Belper and Alfreton Chronicle''; n.p.&lt;/ref&gt;&lt;/blockquote&gt;

'''1897 August 2, Monday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1897#Warwick House Party for the Easton Lodge Cricket Week Games|Earl and Countess of Warwick's house party for Easton Lodge cricket week]].

'''1897 August 2, Monday''', Mrs. Algernon Bourke was listed as among [[Social Victorians/Timeline/1897#The Most Beautiful Women in England|the most beautiful women in England]] in an article from ''Vanity Fair'' that was reprinted elsewhere.

'''1897 September 25, Saturday''', according to the ''Pall Mall Gazette'',&lt;blockquote&gt;The [[Social Victorians/People/Mayo|Dowager-Countess of Mayo]] is staying with her son, the Hon. Algernon Bourke, at Bramnber, near Brighton.&lt;ref&gt;&quot;Pall Mall Gazette Office.&quot; ''Pall Mall Gazette'' 25 September 1897, Saturday: 8 [of 10], Col. 2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18970925/023/0008. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;'''1897 October 2, Saturday''', &quot;Yenatrix&quot; in &quot;Kennel Column&quot; in the ''Gentlewoman'' reported that Guendoline Bourke had joined the Ladies' Kennel Association.&lt;ref&gt;Yenatrix. &quot;Kennel Column.&quot; ''Gentlewoman'' 02 October 1897, Saturday: 39 [of 61], Col. 2a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18971002/182/0039. Same print title, p. 434.&lt;/ref&gt;

'''1897 October 9, Saturday''', Algernon and Guendoline Bourke were at [[Social Victorians/Timeline/1897#Harrogate|Harrogate, presumably taking the waters and baths]]. Lady May was on her way to visit Algernon Bourke in Brighton:&lt;blockquote&gt;The Earl of Mayo is expected to return from Sweden on Saturday next. Lady Mayo leaves Bournemouth on Sarurday for Brighton, where she will pay a two days' visit to her brother-in-law, the [[Social Victorians/People/Bourke|Hon. Algernon Bourke]]. The Earl and Countess will then return to Palmerstown, their seat in County Kildare.&lt;ref&gt;&quot;Pall Mall Gazette Office.&quot; ''Pall Mall Gazette'' 7 October 1897, Thursday: 8 [of 12], Col. 3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18971007/022/0008. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;&lt;p&gt;

'''1897 October 30, Saturday''', ''Black and White'' published '''J.P.B.'''&lt;nowiki/&gt;'s &quot;The Case of Mrs. Elliott,&quot;&lt;ref name=&quot;:13&quot;&gt;J.P.B. &quot;The Case of Mrs. Elliott.&quot; ''Black &amp; White'' 30 October 1897, Saturday: 12 [of 34], Cols. 1a–2b [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004617/18971030/036/0012. Print title ''Black and White'', p. 542.&lt;/ref&gt; an odd short short story in which the Honourable Algernon Bourke Herriott is &quot;rude to Mrs. Elliott,&quot;&lt;ref name=&quot;:13&quot; /&gt;{{rp|Col. 2b}} presumably having proposed sexual relations while her husband is out. J.P.B. links to the biographical Algernon Bourke's career in the stock market in the description of Mrs. Christine Elliott not even simulating interest in her husband's bicycling: &quot;a soul is a grievous burthen for a stockbroker's wife,&quot;&lt;ref name=&quot;:13&quot; /&gt;{{rp|Col. 2a}} suggesting that Mr. Elliott rather than Algernon Bourke Herriott is the stockbroker. The Hon. Algy&lt;blockquote&gt;was a senior member of several junior clubs. A woman had dubbed him once &quot;a rip with a taste for verses.&quot; The description was severe, but not unwarranted. His was a pretty pagan sensualism, though, singing from a wine palate to Church music. For the rest, he had just imagination enough to despise mediocrity.&lt;ref name=&quot;:13&quot; /&gt;{{rp|Col. 2a}}&lt;/blockquote&gt;

'''1897 November 25–26, Thursday–Friday''', Guendoline Bourke was in Brighton, helping the Countess of Mayo at the [[Social Victorians/Timeline/1897#The Irish Industries' Association Annual Exhibition|bazaar of the Irish Industries' Association]].

'''1897 December 7, Tuesday''', Algernon Bourke attended the [[Social Victorians/Timeline/1897#7 December 1897, Tuesday|7th annual dinner for the Actors' Benevolent Fund]].

'''1897 December 20, Monday, the week of''', Guendoline Bourke's Mousie was the smallest dog in the [[Social Victorians/Timeline/1897#Ladies' Kennel Association's Dog Show|Ladies' Kennel Association show in Earl's Court]].

'''1897 December 30''', Algernon and Guendoline Bourke attended a [[Social Victorians/Timeline/1897#Blenheim Palace Party with Amateur Theatricals|party at Blenheim Palace in which people performed tableaux vivants]] that got reported on, many of whom wearing the costumes from the Duchess of Devonshire's ball. The ''Irish Independent'' said Algernon Bourke was &quot;mainly responsible for the living pictures.&quot;&lt;ref&gt;&quot;Mr Algernon Bourke ....&quot; ''Irish Independent'' 05 January 1898, Wednesday: 6 [of 8], Col. 2c [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001985/18980105/115/0006.&lt;/ref&gt;

'''1898''', Algernon Bourke called a meeting at White's Club about attempting to [[Social Victorians/Timeline/1900s#The Thames Salmon Experiment|restock the Thames with salmon]]. In 1899 he was on a [[Social Victorians/People/Bourke#Committees|committee led by the Lord Mayor about this topic]] as well.

'''1898 February 3, Thursday''',  Algernon Bourke was among [[Social Victorians/Timeline/1898#The Dundee Evening Telegraph Report on People at Monte Carlo|those visiting Monte Carlo according to the Dundee ''Evening Telegraph'']].

'''1898 March 12, Saturday''', ''The World'' reported on Algernon Bourke's upgrading of the Orleans Club at Brighton:&lt;blockquote&gt;

The Orleans Club at Brighton is flourishing exceedingly, and the new buildings which Mr. Algernon Bourke has just had erected at the back of the comfortable mansion at the corner of Lansdowne-place now provide all that was wanting to make the present habitat of the club all that its members desire. The new billiard-room is rapidly approaching completion, and the coffee-room, excellent and spacious now, was open on Saturday night, when every table was occupied by club diners and their guests, all of whom were enthusiastic over the excellence of this latest addition to the comfort of the house. All interested may be congratulated on what is practically new lease of life to the Orleans Club, than which there is no more comfortable place stay within the four seas.&lt;ref&gt;&quot;From '''The World''.'&quot; ''East &amp; South Devon Advertiser'' 12 March 1898, Saturday: 6 pop 8], Col. 2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001639/18980312/132/0006. Print title ''The East and South Devon Advertiser'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;

'''1898 March 30, Wednesday''', Algernon Bourke was charged with assaulting a Mr. Potter, but it is not clear from this account what exactly happened:&lt;blockquote&gt;The Hon. Algernon H. Bourke, of Bramber, was summoned, at the instance of Mr. Walter John Potter, clerk to Mr. G. A. Flowers, solicitor, of Steyning, for assault, on the 30th March. — Mr. J. Edward Dell supported the case, and Mr. J. C. Buckwell defended, and pleaded not guilty. — The evidence was to the effect that Mr. Potter had occasion go to defendant's house on Wednesday last to serve a writ. He was going to drop the letter into [Col. 5c–6a] defendant's pocket when he turned and struck him a violent blow on the chest, making witness stagger backwards. Witness put up his hands to keep his balance, and defendant then struck him violently across the head with a weeding spud. — Richard Reed, who was at work for Mr. Bourke on the date named, and was working in garden at the time of the alleged assault, gave corroborative evidence. — Defendant, in the witness box, made a similar statement. — The magistrates differed as to whether the assault was committed, and dismissed the case.&lt;ref&gt;&quot;Steyning.&quot; ''Sussex Express'' 9 April 1898, Saturday: 2 [of 12], Col. 5c–6a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000654/18980409/036/0002. Print: ''The Sussex Express, Surrey Standard, Weald of the Kent Mail, Hants and County Advertiser'', p. 2.&lt;/ref&gt;&lt;/blockquote&gt;'''1898 April 12, Tuesday''', Algernon Bourke was among [[Social Victorians/Timeline/1898#1898 April 12, Tuesday|those visiting Monte Carlo according to the ''Gentlewoman'']].

'''1898 May 25, Wednesday''', Guendoline Bourke wore pink to [[Social Victorians/1898-05-25 Savoy Dinner Dance Hwfa|Mrs. Hwfa Williams' dinner-dance at the Savoy]].

'''1898 June 7, Tuesday''', the Hon. Algernon and Mrs. A. Bourke were invited to and probably attended the [[Social Victorians/Timeline/1898#7 June 1898, Tuesday|State Ball at Buckingham Palace hosted by the Prince and Princess of Wales]].

'''1898 July 4, Thursday afternoon''', the Hon. Algernon and Mrs. Bourke were invited to and probably attended the [[Social Victorians/Timeline/1898#Garden Party at Marlborough House|Garden Party at Marlborough House given to the Queen and Shah of Persia]].

'''1898 October 29, Saturday''', Algernon Bourke attended a [[Social Victorians/Timeline/1898#Tennis Championship Game at Prince's Club, Knightsbridge|tennis match at Prince's Club, Knightsbridge]].

'''1898 November 22, Tuesday''', Algernon Bourke was present at a [[Social Victorians/Timeline/1898#Shooting Party Hosted by William James|shooting party hosted by Mr. William James]].

'''1898 December 3, Saturday''', Hon. Algernon and Mrs. A. Bourke attended the [[Social Victorians/Timeline/1898#The Funeral of Lady Connemara|funeral of Lady Connemara in Christ Church]], Down street, Piccadilly.&lt;p&gt;

'''1899 January 10, Tuesday''', the Brighton Championship Dog Show opened:&lt;blockquote&gt;Princess of Wales a Winner at the Ladies’ Kennel Club Show.

[Exclusive to &quot;The Leader.&quot;)

The Brighton Championship Dog Show opened in the Dome and Corn Exchange yesterday, and was very well patronised by visitors and exhibitors. Among the latter was H.R.H. the Princess of Wales, who did very well; and others included Princess Sophie Duleep Singh, Countess De Grey, Sir Edgar Boehm, the Hon Mrs. Algernon Bourke, Lady Cathcart, Lady Reid, Mr. Shirley (chairman of the Kennel Club), and the Rev. Hans Hamiiton (president of the Kennel Club).

The entry of bloodhounds is one of the best seen for some time; the Great Danes are another strong lot; deerhounds are a fine entry, all good dogs, and most of the best kennels represented; borzois are another very stylish lot. The bigger dogs are, as usual, in the Corn Exchange and the &quot;toy&quot; dogs in the Dome. To everyone's satsfaction the Princess of Wales carried off two first prizes with Alex in the borzois class.&lt;ref&gt;&quot;Dogs at Brighton.&quot; ''Morning Leader'' 11 January 1899, Wednesday: 8 [of 12], Col. 3b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004833/18990111/142/0008. Print p. 8.&lt;/ref&gt;&lt;/blockquote&gt;

'''1899 January 11, Wednesday''', Guendoline Bourke attended [[Social Victorians/Timeline/1899#11 January 1899, Wednesday|a luncheon at Stanfield-hall, home of Mr. and Mrs. Basil Montogomery, for Princess Henry of Battenberg]], that also included the Countess of Dudley (sister of Mrs. Montgomery), General Oliphant, and the Mayor and Mayoress of Romsey.

'''1899 January 17–18, Tuesday and Wednesday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1899#Ladies' Kennel Association in Brighton|Ladies' Kennel Association in Brighton]], where she showed an Italian greyhound named Brenda.

'''1899 February 7, Tuesday''', Guendoline Bourke was a member of the very high-ranking committee organizing the [[Social Victorians/Timeline/1899#Gordon Memorial College Ball|Gordon Memorial College Ball at the Hotel Cecil on 7 February 1899]]. The committee had been planning for the ball, of course, for at least 3 weeks before.

'''1899 February 22, Wednesday – April''', Guendoline Bourke was part of [[Social Victorians/Timeline/1899#Society in St. Moritz|Society in St. Moritz]]. 1899 March 29, Wednesday, the ''Dundee Advertiser'' says that [[Social Victorians/Timeline/1899#29 March 1899, Wednesday|Cyril Sloane-Stanley was spending part of the winter in St. Moritz]] with his sister Guendoline Bourke.

'''1899 April 7, Friday, probably''', oddly, Algernon and Guendoline Bourke are not reported to have attended the [[Social Victorians/Timeline/1899#Funeral of the Hon. Charles Bourke, C.B.|Funeral of the Hon. Charles Bourke, C.B.]] or even to have sent flowers.

'''1899 April 8, Saturday''', the ''Gentlewoman'' reported that Guendoline Bourke had gone to [[Social Victorians/Timeline/1899#8 April 1899, Saturday|St. Moritz with her brother, Mr. Stanley, who had gotten engaged to Lady Cairns]].

'''1899 April 26, Wednesday''', according to &quot;Local and District News&quot; for Totton, Guendoline Bourke was &quot;ill with influenza in Paris, and Mrs. Shelley Bontein, her mother, has gone out to nurse her.&quot;&lt;ref&gt;&quot;Local and District News. Totton.&quot; ''Hampshire Advertiser'' 26 April 1899, Wednesday: 4 [of 4], Col. 2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18990426/037/0004. Print title ''Hampshire Advertiser County Newspaper'', p. 4.&lt;/ref&gt;

'''1899 June 1, Thursday, or 2, Friday''', the Hon. Algernon and Guendoline Bourke attended the [[Social Victorians/Timeline/1899#Wedding of Roger Cyril Sloane Stanley and Olivia, Countess Cairns|wedding of her brother, Sloane Stanley and Olivia Countess Cairns]] at Holy Trinity Church, Brompton.

'''1899 June 8, Thursday''', Algernon Bourke's money troubles:&lt;blockquote&gt;The Hon. Algernon Bourke, son of the Earl of Mayo, has been appearing before the official receivers in connection with a winding-up order made against Willis’ Restaurant, Limited. The companyf [sic] was formed to acquire the well known restaurant from the Hon. H. A. Bourke. The chairman reminded the creditors that on the last occasion the meeting was adjourned because Mr. Bourke said he thought he would be able in the course of a fortnight to obtain an offer for a sum sufficient to satisfy the creditors and debenture holders. He had received a letter from Mr. Bourke to the effect that he had been unable to complete arrangements. Having looked into the affairs of the company more closely, it appeared to him that Mr. Bourke was legally liable to repay the sum of £5,000 which was advanced to White's Club, and the question would arise whether Mr. Bourke was not also liable to repay the sum of £4,000.&lt;ref&gt;&quot;Mr. Bourke Must Pay.&quot; ''Irish Independent'' 8 June 1899, Thursday: 4 [of 8], Col. 8c [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001985/18990608/118/0004. Print title: ''The Irish Weekly Independent'', p. 4.&lt;/ref&gt;&lt;/blockquote&gt;'''1899 July 1, Saturday''', Algernon Bourke attended a [[Social Victorians/Timeline/1899#1 July 1899, Saturday|meeting in London at the Duke of Westminster's Grosvenor House]] about preserving Killarney as part of the National Trust and seems to have been acting for someone who wanted to purchase the Muckross Estate.

'''1899 July 5, Wednesday''', Guendoline Bourke attended a [[Social Victorians/Timeline/1899#Dinner and Dance at Devonshire House|dance at Devonshire House hosted by the Duke and Duchess of Devonshire]].

'''1899 July 6, Thursday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1899#Joan Wilson and Guy Fairfax's Wedding|wedding of Joan Wilson and Guy Fairfax in St. Mark's, near Grosvenor Square]].

'''1899 July 14, Friday''', Guendoline Bourke attended [[Social Victorians/Timeline/1899#14 July 1899, Friday|Ernest Beckett's dinner party]].

'''1899 July 18, Tuesday''', Guendoline Bourke attended a [[Social Victorians/Timeline/1899#Ludovici Lecture on Impressionism|lecture on Impressionism by Ludovici hosted by the Countess of Mayo]].

'''1899 July 28, Friday''', [[Social Victorians/London Clubs#White's|White's Club]] was no longer under Algernon Bourke's management and was reconstituting itself after the possibility that it would have to close.

'''1889 July 31, Wednesday''', the Hon. Algernon and Mrs. Bourke attended a [[Social Victorians/Timeline/1889#Fete of the Uxbridge Habitation of the Primrose League|Fete of the Uxbridge Habitation of the Primrose League]] at Hillingdon Court and hosted by the Hon. Algernon and Lady Mary Mills.

'''1899 September 9, Saturday''', the ''Eastern Morning News'' includes Algernon Bourke (&quot;St. James's-street, London, club proprietor&quot;) in a list of men &quot;Receiving Orders,&quot; which it is reprinting from the ''London Gazette''.&lt;ref&gt;&quot;Receiving Orders.&quot; ''Eastern Morning'' News 9 September 1899, Saturday: 5 [of 8], Col. 3c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001152/18990909/074/0005. Same print title and p.&lt;/ref&gt;&lt;p&gt;

'''1899 October 19, Thursday''', the Hon. Algernon Bourke had a bankruptcy hearing:&lt;blockquote&gt;The public examination of the Hon. Algernon Bourke was held before Mr Registrar Giffard yesterday, at the London Bankruptcy Court. The debtor, described as proprietor of a St. James's-street club, furnished a statement of affairs showing unsecured debts £13,694 and debts fully secured £12,800, with assets which are estimated at £4,489 [?]. He stated, in reply to the Official Receiver, that he was formerly a member of the Stock Exchange, but had nothing to do with the firm of which he was a member during the last ten years. He severed his connection with the firm in May last, and believed he was indebted to them to the extent of £2,000 or £3,000. He repudiated a claim which they now made for £37,300. In 1889 he became proprietor of White's Club, St. James's-street, and carried it on until January 1st last, when he transferred it to a company called Recreations, Limited. One of the objects of the company was to raise money on debentures. The examination was formally adjourned.&lt;ref name=&quot;:9&quot; /&gt;&lt;/blockquote&gt;

'''1899 October 20, Friday''', the ''Morning Leader'' mentions Bourke's bankruptcy:&lt;blockquote&gt;Mr. Algernon Bourke, whose bankruptcy is much talked about, has been connected with numerous enterprises in clubland. He raised White's from the slough into which it had sunk after the secession of the Prince of Wales. He started the Willis Restaurant, put fresh life into the Orleans Club at Brighton, arranged a big restaurant for the bicyclists in the time of the bicycle parade, and was concerned at first in the smart and short-lived Trafalgar Bicycle Club. At one time his name spelt success. Latterly his luck has left him. He is a brother of Lord Mayo, a son of the peer who was assassinated at the post of duty, and is one of the best known men about town of the day.&lt;ref&gt;&quot;Club, Stage, and Salon.&quot; ''Morning Leader'' 20 October 1899, Friday: 6 [of 12], Col. 5b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004833/18991020/085/0006. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;'''1899 November 8, Wednesday''', the Hon. Algernon Bourke's bankruptcy case came up again:&lt;blockquote&gt;At Bankruptcy Court, yesterday, the case the Hon. Algernon Bourke again came on for hearing before Mr. Registrar Giffard, and the examination was concluded. The debtor has at various times been proprietor of White’s Club, St. James’s-street, and the Orleans’ Club, Brighton, and also of Willis's Restaurant, King-street, St. James's. He attributed his failure to losses sustained by the conversion of White’s Club and the Orleans' Club into limited companies, to the payment of excessive Interest on borrowed money, and other causes. The liabilities amount to £26,590, of which £13,694 are stated to be unsecured, and assets £4,409.&lt;ref&gt;&quot;Affairs of the Hon. A. Bourke.&quot; ''Globe'' 09 November 1899, Thursday: 2 [of 8], Col. 1c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001652/18991109/020/0002. Print p. 2.&lt;/ref&gt;&lt;/blockquote&gt;

'''1899 December 23, Saturday''', &quot;Mr. Algernon Bourke has departed for a tour in Africa, being at present the guest of his brother in Tunis.&quot;&lt;ref&gt;&quot;The Society Pages.&quot; ''Walsall Advertiser'' 23 December 1899, Saturday: 7 [of 8], Col. 7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001028/18991223/143/0007. Print p. 7.&lt;/ref&gt;

'''1899 December 29, Friday''', Guendoline Bourke was at the [[Social Victorians/Timeline/1899#Christmas Party Hosted by the Duke and Duchess of Marlborough|Christmas Party Hosted by the Duke and Duchess of Marlborough]].&lt;p&gt;

'''1899 December 31''', the San Francisco newspaper ''The Wave'' wrote the following about London society:&lt;blockquote&gt;The most prominent untitled people in London may be said to be Mr. and Mrs. [[Social Victorians/People/Williams|Hwfa Williams]], Mr. and Mrs. [[Social Victorians/People/Grenfell|Willie Grenfell]] and Mr. Algy Bourke. That they are passing rich, goes without saying, and that they entertain lavishly, understood — for to be untitled, prominent and successful, argues wealth, hospitality and cleverness.&lt;ref&gt;&quot;London.&quot; The (San Francisco) ''Wave'' 14 January 1899 (Vol. XIX, No. 2): 14. ''The Internet Archive'' https://archive.org/details/wave19unse/page/n20/mode/1up.&lt;/ref&gt;&lt;/blockquote&gt;

=== 1900s ===

'''1900 February 15, Thursday''', Daphne Bourke, the four-year-old daughter of the Hon. Algernon and Mrs. Bourke was a bridesmaid in the [[Social Victorians/Wilson Chesterfield Wedding 1900-02-15|wedding of Enid Wilson and the Earl of Chesterfield]].&lt;ref&gt;&quot;London Day by Day.&quot; ''Daily Telegraph'' 15 February 1900, Thursday: 8 [of 12], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/19000215/175/0008. Name in British Newspaper Archive: ''Daily Telegraph &amp; Courier'' (London). Print p. 8.&lt;/ref&gt; Guendoline Bourke, &quot;who was in grey, wore a chinchilla toque with violets.&quot;&lt;ref&gt;&quot;Society. Entertainments, Balls, &amp;c.&quot; ''The Queen'' 24 February 1900, Saturday: 40 [of 76], Col. 1a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/19000224/235/0040. Print: ''The Queen, The Lady's Newspaper'', p. 308.&lt;/ref&gt;

'''1900 March 10, Saturday''', the ''Weekly Irish Times'' reprinted society gossip from ''The World'':&lt;blockquote&gt;Mrs. Algernon Bourke, who has been staying with her husband's uncle, old Connemara, during Mr. Algernon Bourke's absence abroad, has taken a new house near Portman square, and will be settling there before Easter.&lt;ref&gt;&quot;Society Gossip.&quot; ''Weekly Irish Times'' 10 March 1900, Saturday: 17 [of 20], Col. 1b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001684/19000310/116/0017. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;'''1900 July''' '''17, Tuesday''', Guendoline Bourke took part in the [[Social Victorians/Timeline/1900s#17 July 1900, Tuesday|Children's Fete in support of the National Society for the Prevention of Cruelty to Children]] on the grounds of the Royal Botanic Society. Daphe was 5 at this time, so it seems logical that she would have been there, too.

'''1900 July 30, Monday''', Guendoline Bourke attended [[Social Victorians/Timeline/1900s#Barber of Seville at Covent Garden|''The Barber of Seville'' at Covent Garden]].

'''1890 August 6, Friday''', &quot;[[Social Victorians/Timeline/1890#Beautiful Women|Beautiful Women]],&quot; an article in ''Vanity Fair'' that was reprinted elsewhere, mentions Guendoline Bourke (&quot;Lady Algernon Bourke&quot;) as one of the most beautiful women in England.

'''1900 August 11, Saturday''', Guendoline Bourke got&lt;blockquote&gt;the pretty little Yorkshire String, an especially tiny mite, weighing only 2&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;lb, and carrying a very promising coat, ... at the Aquarium Show.&lt;ref&gt;&quot;The Witchampton Kennel.&quot; &quot;Ladies Kennels.&quot; ''Ladies' Field'' 11 August 1900, Saturday: 16 [of 60], Col. 2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0006043/19000811/043/0016. Print title same, p. 390.&lt;/ref&gt;&lt;/blockquote&gt;&lt;p&gt;

'''1900 September 16''', the Hon. Algernon Bourke became the heir presumptive to the Earldom of Mayo when his older brother Captain Hon. Sir Maurice Archibald Bourke died.&lt;p&gt;

'''1900 October 06, Saturday''', the ''Weekly Irish Times'' says that Mr. Algernon Bourke, now heir presumptive to the earldom of Mayo, &quot;has been for some months lately staying with Mr. Terence Bourke in Morocco.&quot;&lt;ref&gt;&quot;Society Gossip.&quot; ''Weekly Irish Times'' 06 October 1900, Saturday: 14 [of 20], Col. 3b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001684/19001006/121/0014. Print p. 14.&lt;/ref&gt;&lt;p&gt;

'''1901 May 30, Thursday''', the Hon. Mrs. Algernon Bourke attended the fashionable [[Social Victorians/Timeline/1900s#1901 May 30, Thursday|Ladies' Kennel Association Dog Show at the Botanic Garden]].

'''1901 July 2, Tuesday''', Guendoline Bourke — &quot;pretty Mrs. Algernon Bourke, in a mauve gown and and purple tulle toque&quot; — attended a children's party at the Botanic Gardens hosted by the Earl and Countess of Kilmorey.&lt;ref&gt;&quot;The Earl of Kilmorey, K.P.&quot; ''Gentlewoman'' 13 July 1901: Saturday, 50 [of 84], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010713/237/0050. Print: title the same, p. 60.&lt;/ref&gt;

'''1901 July 4, Thursday''', Guendoline Bourke — dressed &quot;in pale grey, with her pretty little girl,&quot; 6-year-old Daphne — attended a [[Social Victorians/Timeline/1900s#The Countess of Yarborough's Children's Party|children's party hosted by the Countess of Yarborough]].&lt;ref&gt;&quot;The Countess of Yarborough ....&quot; ''Gentlewoman'' 13 July 1901, Saturday: 76 [of 84], Col. 2b, 3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010713/381/0076. Print p. xxxvi.&lt;/ref&gt;{{rp|Col. 3a}}

'''1901 July 4–6, Thursday–Saturday''', Guendoline Bourke helped staff the Perthshire stall&lt;ref&gt;&quot;The Great County Sale.&quot; ''Gentlewoman'' 29 June 1901, Saturday: 43 [of 72], Col. 3a [of 3]. ''British Newspaper Archive'' [https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010629/223/0043# https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010629/223/0043]. Same print title, pp. 679.&lt;/ref&gt; at the [[Social Victorians/Timeline/1900s#The Great County Sale|Great County Sale in the Imperial Gardens of the Earl's Court Exhibition]].

'''1901 July 20, Saturday''', the ''Gentlewoman'' published the Hon. Mrs. Algernon Bourke's portrait (identified with &quot;Perthshire&quot;) in its 3rd series of &quot;The Great County Sale at Earl's Court. Portraits of Stallholders.&quot;&lt;ref&gt;&quot;The Great County Sale at Earl's Court. Portraits of Stallholders.&quot; ''Gentlewoman'' 20 July 1901, Saturday: 31 [of 60], Col. 4b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010720/141/0031. Print n.p.&lt;/ref&gt; Their daughter Daphne appears in the portrait as well.

'''1901 July 23, Tuesday''', an &quot;Hon. Mrs. Bourke&quot; was in the [[Social Victorians/Timeline/1900s#Lord and Lady Algernon Gordon Lennox|party &quot;entertained by Lord and Lady Algernon Gordon Lennox]].&quot;&lt;p&gt;

'''1901 September 12, Thursday''', Mrs. Guendoline Bourke wanted her name listed as Mrs. Algernon Bourke in the Electoral Register, apparently a frequent complaint:&lt;blockquote&gt;Mr. Underhill, the Conservative agent, mentioned to the Revising Barrister (Mr. William F. Webster) that the name of the Hon. Mrs. Guendoline Bourke was on the list in respect of the house, 75, Gloucester-place. The lady had written to him to say that she was the Hon. Mrs. Algernon Bourke and that she wished that name to appear on the register. In reply to the Revising Barrister, Mr. Underhill said that “Algernon” was the name of the lady’s husband.

Mr. Cooke, the rate-collector, said that Mrs. Bourke had asked to be addressed Mrs. Algernon Bourke, but that the Town Clerk thought the address was not a correct one. The lady signed her cheques Guendoline.”

Mr. Underhill said the agents frequently had indignant letters from ladies because they were not addressed by their husband’s Christian name.

The Revising Barrister — lf a lady gave me the name of Mrs. John Smith I should say I had not got the voter’s name. The name Guendoline must remain.&lt;ref name=&quot;:15&quot;&gt;&quot;Ladies’ Names.&quot; ''Morning Post'' 12 September 1901, Thursday: 7 [of 10], Col. 3a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/19010912/130/0007. Print p. 7.&lt;/ref&gt;&lt;/blockquote&gt;

'''1901 October 26, Friday''', Algernon Bourke was on the Men's Committee of the [[Social Victorians/London Clubs#Prince's Club Ice-skating Rink|Prince's Club Ice-skating Rink]], which had [[Social Victorians/Timeline/1900s#The Prince's Club Ice-skating Rink Opening|its official opening on his day]]. 

'''1902 January''', Algernon Bourke is mentioned in [[Social Victorians/Schools#&quot;More of My Contemporaries at School.&quot;|reminiscences of Eton written by the &quot;Earl of X&quot;]] as being among those in the &quot;world of letters,&quot; and whose brother, later the Earl of Mayo, the Earl of X did not like. 

'''1902 January 25, Saturday''', Mrs. Algernon Bourke gave a box to Lady Helen Stewart-Vane-Tempest in honor of [[Social Victorians/Stewart-Stavordale Wedding 1902-01-25|Lady Helen's wedding to Giles Fox-Strangways, Lord Stavordale]]. 

'''1902 April 26, Saturday''', Mrs. A. Bourke is listed as being at the Norfolk Hotel in Brighton.&lt;ref&gt;&quot;Guide to Visitors at Hotels and Boarding Houses.&quot; ''Brighton Gazette'' 26 April 1902, Saturday: 3 [of 8], Col. 4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000938/19020426/116/0003. Same print title and p.&lt;/ref&gt; 

'''1902 May, End of''', Guendoline Bourke attended a [[Social Victorians/Timeline/1900s#End of May 1902|party at Blenheim Palace hosted by the Duke and Duchess of Marlborough]]. 

'''1902 June 11, Monday''', the Hon. Mrs. Algernon Bourke had a dog entered in the [[Social Victorians/Timeline/1900s#Ladies' Kennel Association Show|Ladies' Kennel Association competitions in the Botanic Gardens]]. 

'''1902 September 4, Thursday''', the ''Daily Express'' reported that &quot;Mrs. Algernon Bourke is staying with Lord and Lady Alington at Scarborough.&quot;&lt;ref&gt;&quot;Onlooker.&quot; &quot;My Social Diary.&quot; &quot;Where People Are.&quot; ''Daily Express'' 04 September 1902, Thursday: 5 [of 8], Col. 1b? [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004848/19020904/099/0005. Print p. 4, Col. 7b [of 7].&lt;/ref&gt;

'''1902 September 22, Monday''', Guendoline Bourke was a guest at the [[Social Victorians/Timeline/1900s#Earl and Countess of Mar and Kellie's House Party|large house party hosted by the Earl and Countess of Mar and Kellie]].

'''1902 October 24, Friday''', the Hon. Algernon Bourke [[Social Victorians/Timeline/1900s#Annual Opening of the Prince's Ice-skating Rink|opened the Prince's ice-skating rink for the season]], which he had been doing since 1895.

'''1902 October 25, Saturday''', Algernon Bourke was bequeathed £500 by his uncle [[Social Victorians/People/Mayo|Robert Bourke]], who had died 3 September 1902.&lt;ref&gt;&quot;Will of Lord Connemara.&quot; ''Kildare Observer and Eastern Counties Advertiser'' 25 October 1902, Saturday: 2 [of 8], Col. 4b–c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001870/19021025/037/0002. Print title the ''Kildare Observer'', n.p.&lt;/ref&gt;&lt;p&gt;

'''1902 October 31, Friday''', the [[Social Victorians/Timeline/1900s#Annual Opening of the Prince's Ice-skating Rink|7th opening of the Prince's Skating Club]]. Guendoline Bourke was on the Women's Committee and Algernon Bourke was on the Men's.&lt;p&gt;

'''1902 November 8, Friday, beginning, perhaps''', Guendoline Bourke was part of the [[Social Victorians/Timeline/1900s#8 November 1902, Saturday|Earl and Countess of Warwick's shooting party at Easton Lodge]].&lt;p&gt;

'''1902 December 9, Tuesday''', Guendoline Bourke attended [[Social Victorians/Timeline/1900s#9 December 1902, Tuesday|Lady Eva Wyndham-Quin's &quot;at home,&quot; held at the Welch Industrial depot]] for the sale Welsh-made Christmas gifts and cards. Bourke wore &quot;a fur coat and a black picture hat.&quot;&lt;ref&gt;&quot;A Lady Correspondent.&quot; &quot;Society in London.&quot; ''South Wales Daily News'' 11 December 1902, Thursday: 4 [of 8], Col. 5a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000919/19021211/082/0004. Print p. 4.&lt;/ref&gt;

'''1903 February 6, Friday''', Hon. Mrs A. Bourke was present at a [[Social Victorians/Timeline/1900s#Dinner Party Hosted by Lord Lieutenant of Ireland and the Countess of Dudley|dinner party Hosted by Lord Lieutenant of Ireland and the Countess of Dudley]]. &lt;p&gt;

'''1903 February 9, Monday''', Guendoline Bourke was present at a [[Social Victorians/Timeline/1900s#Dinner Party Hosted by Lord Lieutenant of Ireland and the Countess of Dudley|house party at Dublin Castle hosted by the Lord Lieutenant and Countess of Dudley that began the Viceregal season]].

'''1903 March 17, Tuesday''', Guendoline Bourke staffed a booth at a [[Social Victorians/Timeline/1900s#1903 March 17, Tuesday|sale of the Irish Industries Association]] on St. Patrick's Day with [[Social Victorians/People/Mayo|Lady Mayo]], [[Social Victorians/People/Dudley|Georgina Lady Dudley]] and [[Social Victorians/People/Beresford|Miss Beresford]]. A number of other aristocratic women were also present at the sale in other booths, including [[Social Victorians/People/Londonderry|Lady Londonderry]] and [[Social Victorians/People/Lucan|Lady Lucan]].

'''1903 June 19, Friday''', Guendoline Bourke was invited to the [[Social Victorians/Timeline/1900s#Grand Ball in the Waterloo Chamber at Windsor Castle|grand ball at Windsor Castle]], the end of the Ascot-week festivities.

'''1903 June 23, Tuesday''', Guendoline and Daphne Bourke were invited to a [[Social Victorians/Timeline/1900s#1903 June 23, Tuesday|children's party at Buckingham Palace for Prince Eddie's birthday]].

'''1903 July 10, Friday, or so''', Guendoline Bourke attended a [[Social Victorians/Timeline/1900s#Party Hosted by the Duke and Duchess of Marlborough|party hosted by the Duke and Duchess of Marlborough]].

'''1904 May 17, Tuesday''', Guendoline Bourke had agreed to let Daphne appear in the tableaux vivants arranged by Sir Philip Burne-Jones for the [[Social Victorians/Timeline/1900s#Countess Cadogan's Great Bazaar|Countess of Cadogan's great bazaar]]. Some mothers had had to decline because of the outbreaks of measles and chicken pox.&lt;p&gt;

'''1904 June 30, Thursday''', Guendoline and Daphne Bourke attended another birthday party for Prince Eddie at Buckingham Palace, and the ''Gentlewoman'' says, &quot;No prettier little girl was to be seen that day than little Miss Daphne Bourke, the daughter of the Hon. Mrs. Algernon Bourke, with her wonderful Irish eyes and colouring, her pretty white frock being relieved with a rose pink sash.&quot;&lt;ref&gt;&quot;Prince Eddie's Birthday.&quot; ''Gentlewoman'' 02 July 1904, Saturday: 68 [of 92]. Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19040702/360/0068. Print: title the same, p. 42.&lt;/ref&gt;&lt;p&gt;

'''1904 September 15, Thursday''', according to what was at the time called the ''Irish Daily Independent and Nation'', Algernon Bourke was living in Venice and not in the UK at this point:&lt;blockquote&gt;Algernon Bourke, who usually lives in Venice, has spent some time in England during the present summer, and has now gone on a fishing expedition to Sweden, accompanied by his brother, Lord Mayo. Lady Mayo has been staying meanwhile in Ireland, and has had a visit from her mother, Lady Maria Ponsonby, who is a sister of Lend Obventry.&lt;ref name=&quot;:10&quot;&gt;&quot;Society Notes.&quot; ''Irish Independent'' 15 September 1904, Thursday: 4 [of 8], Col. 5b [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001986/19040915/131/0004. Print title: ''Irish Daily Independent and Nation'', p. 4.&lt;/ref&gt;&lt;/blockquote&gt;

'''1904 October 22, Saturday''', the ''Gentlewoman'' reported that &quot;Mrs. Algernon Bourke is paying a visit to Venice, which Mr. Bourke has made his headquarters for several years past, as he is connected with some very artistic stone and marble works situated near the Grand Canal.&quot;&lt;ref&gt;&quot;The Social Peepshow.&quot; ''Gentlewoman'' 22 October 1904, Saturday: 24 [of 6ths 8], Col. 2c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19041022/112/0024. Print title same, p. 672.&lt;/ref&gt;

'''1905 February 17, Friday''', the Dundee ''Evening Post'' reported that Algernon Bourke &quot;set up a shop in Venice for the sale of art treasures and old furniture.&quot;&lt;ref&gt;&quot;Social News.&quot; Dundee ''Evening Post'' 17 February 1905, Friday: 6 [of 6], Col. 7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000582/19050217/105/0006. Print p. 6.&lt;/ref&gt;

'''1905 April 26, Wednesday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1900s#New Forest United Hunt Ball|New Forest United Hunt Ball]], as did her brother Captain R. C. H. Sloane Stanley and his wife Olivia Countess Cairns.&lt;p&gt;

'''1905 June 5, Monday''', Algernon Bourke wrote to the ''Times'' from Venice that &quot;The Venetian wits have suggested a motto for Admiral Togo, Togo Tenga Tutto (Togo takes the lot).&quot;&lt;ref&gt;&quot;Mr. Algernon Bourke.&quot; ''Hull Daily Mail'' 08 June 1905, Thursday: 2 [of 6], Col. 6a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000324/19050608/008/0002. Print title ''Daily Mail'', p. 6.&lt;/ref&gt;&lt;p&gt;

'''1905, last week of July''', Guendoline Bourke and daughter Daphne Bourke — who was 10 years old — attended [[Social Victorians/Timeline/1900s#Last week of July, 1905|Lady Cadogan's children's party at Chelsea House]]. Daphne was &quot;One of loveliest little girls present.&quot;&lt;ref&gt;&quot;Court and Social News.&quot; ''Belfast News-Letter'' 01 August 1905, Tuesday: 7 [of 10], Col. 6b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000038/19050801/157/0007. Print p. 7.&lt;/ref&gt;&lt;p&gt;

'''1906 March 9, Friday''', Guendoline Bourke was a reference for Mr. Frances Burgess, who taught piano, singing, voice production, organ and music theory. Burgess was &quot;Organist and Choirmaster of St. Columbs', North Kensington, Director of the Plainsong and Medieval Music Society's Choir, etc., etc.&quot;&lt;ref name=&quot;:21&quot;&gt;&quot;Mr. Francis Burgess.&quot; ''Kilburn Times'' 9 March 1906, Friday: 3 [of 8], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001813/19060309/086/0003. Print title: ''Kilburn Times Hampstead and North-western Press'', p. 3.&lt;/ref&gt;&lt;p&gt;

'''1906 December 10, Monday''', Guendoline Bourke was seen in the tea room, possibly with Lady Grosvenor, at [[Social Victorians/Timeline/1900s#1906 December 10, Monday|Lady Dudley's sale of Irish needlework]].&lt;p&gt;

'''1907 May''', a &quot;naval signalling incident&quot; [[Social Victorians/Timeline/1887#May 1887|caused the Waterford ''Evening News'' to recall a similar event]] that had occurred 20 years earlier, in which Algernon Bourke, as special correspondent for the ''Times'', publicized [[Social Victorians/People/Beresford|Lord Charles Beresford]]'s use of his ship's signalling capabilities to send a message to his wife about being late for dinner:&lt;blockquote&gt;

The naval signalling incident is still in the air. It is expected that the matter will not be threshed out until Emperor William leaves England. A story of a former signalling incident in which [[Social Victorians/People/Beresford|Lord Charles Beresford]] was concerned is going the rounds at the moment.&lt;/blockquote&gt;

'''1907 August 24, Saturday''', Algernon Bourke was present at [[Social Victorians/Timeline/1900s#Polo Week at Eaton Hall, Duke and Duchess of Westminster|Polo Week at Eaton Hall, hosted by the Duke and Duchess of Westminster]].

'''1908 July 30, Thursday''', Guendoline Bourke was at [[Social Victorians/Timeline/1900s#Glorious Goodwood. Cup Day and Dresses.|Cup Day at the Goodwood races]], wearing salmon-pink with a matching hat.

'''1909 April 20, Tuesday''', Lady Rosemary Cairns — daughter of Olivia Sloan-Stanley, Countess Cairns and Cyril Sloane-Stanley — and Wyndham Portal were [[Social Victorians/Timeline/1900s#20 April 1909, Tuesday|married in St. Margaret's, Westminster]]. Lavender and Diane Sloane-Stanley were bridesmaids.&lt;p&gt;

'''1909 May 22, Saturday''', Algernon Bourke appears to have been living in Pisa. A columnist for the ''Queen'' reported on the Royal School of Art Needlework:&lt;blockquote&gt;Lady Leconfield [?] was there, also her sister-in-law, the [[Social Victorians/People/Mayo|Dowager Lady Mayo]], only just back from her winter on the Continent, when she spent most of the time at Pisa, where her son Mr Algernon Bourke has also been staying. The latter is a great connoisseur as regards [art?] notably in what is really good in the way of old Italian sculpture and carving. He and his handsome wife have a place near to Putney, and this winter again Mr Bourke, as the result of his Italian travels, has been sending home such relics of the old Italian palace gardens as as stone and marble carved vases, garden seats, and what-not of the kind — not all for himself and his own gardens by any means, I fancy; but his friends, relying on his knowledge in such matters, get him when abroad to choose for [them?] the adornment of their English terraces and gardens.&lt;ref&gt;&quot;My Social Diary.&quot; The ''Queen'' 22 May 1909, Saturday: 31 [of 86], Col. 1b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/19090522/203/0031. Print p. 871.&lt;/ref&gt;&lt;/blockquote&gt;'''1909 September''', the Hon. Algernon Bourke was among the [[Social Victorians/Timeline/1900s#Visitors in Venice from the U.K.|many visitors from &quot;England&quot; in Venice]] in September.

=== 1910s ===
'''1910 April 20, Wednesday''', the ''Tatler'' printed an &quot;open letter&quot; to Geraldine, Countess of Mayo, as part of its &quot;The Searchlight in Society&quot; series and mentioned Algernon Bourke, saying he had been keeping &quot;a curiosity shop at Venice&quot;:&lt;blockquote&gt;The Bourkes have brains, and a good example is afforded by Mr. Algernon Bourke, next brother to Lord Mayo and heir-presumptive to the title. He is a good-looking man who used to be known as Buttons Bourke, and he married well, as his wife was the rich and pretty Miss Guendolen Sloane Stanley. He may be described as a &quot;Jack of all trades,&quot; but it is not I who will say that he is a master of none. He was once in the Stock Exchange, then he took White's Club in hand and restored it to much of its former prestige. After that he dabbled in smart hotels and restaurants, and the last thing I heard of him was that he kept a curiosity shop at Venice.&lt;ref&gt;Candida. &quot;The Searchlight in Society. Our Open Letter. No. CII. The Countess of Mayo.&quot; The ''Tatler'' 20 April 1910, Wednesday: 18 [of 42], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001852/19100420/023/0018. Print title same, p. 72.&lt;/ref&gt;&lt;/blockquote&gt;

'''1911 November 21, Tuesday''', Guendoline Bourke assisted the [[Social Victorians/Timeline/1910s#21 November 1911, Tuesday|Duchess of Marlborough at her at-home]] that included a sale of work by the wives of prisoners.&lt;p&gt;

'''1912 September 27, Friday''', Guendoline and Daphne Bourke were visiting Mr. and Mrs. Shelley Bontein, her mother and stepfather.&lt;ref&gt;&quot;From 'The World.'&quot; ''Berks and Oxon Advertiser'' 27 September 1912, Friday: 2 [of 8], Col. 4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002298/19120927/014/0002. Same print title, n.p.&lt;/ref&gt;&lt;p&gt;

'''1913 April 23, Wednesday''', the Irish Independent reported that Guendoline and Daphne Bourke had arrived in London for the season:&lt;blockquote&gt;&lt;p&gt;

The Hon. Mrs. Algernon Bourke and Miss Bourke have arrived for the season at 75 Gloucester place, Portman square, London.&lt;ref&gt;&quot;Social and Personal.&quot; ''Irish Independent'' 23 April 1913, Wednesday: 4 [of 10], Col. 5b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001715/19130423/081/0004. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;&lt;p&gt;

'''1913 May 7, Wednesday''', Guendoline Bourke presented her daughter Daphne Bourke at court:&lt;blockquote&gt;Mrs. Algernon Bourke presented her daughter, and wore blue and gold broché with a gold lace train.&lt;ref&gt;&quot;Social and Personal.&quot; London ''Daily Chronicle'' 08 May 1913, Thursday: 6 [of 12], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005049/19130508/120/0006. Print p. 6.&lt;/ref&gt;&lt;/blockquote&gt;

The ''Pall Mall Gazette'' has a description of Daphne Bourke's dress, but what exactly &quot;chiffon [[Social Victorians/Terminology#Hoops|paniers]]&quot; means in 1913 is not clear:&lt;blockquote&gt;Court dressmakers appear to have surpassed all previous records in their efforts to make the dresses for to-night’s Court as beautiful as possible. Noticeable among these is the dainty presentation gown to be worn by Miss Bourke, who will be presented by her mother, the Hon. Mrs. Algernon Bourke. This has a skirt of soft white satin draped with chiffon [[Social Victorians/Terminology#Hoops|paniers]] and a bodice veiled with chiffon and trimmed with diamanté and crystal embroidery. Miss Bourke’s train, gracefully hung from the shoulders, is of white satin lined with pale rose pink chiffon and embroidered with crystal and diamanté.&lt;ref&gt;&quot;Fashion Day by Day. Lovely Gowns for To-night's Court.&quot; ''Pall Mall Gazette'' 07 May 1913, Wednesday: 13 [of 18], Col. 1a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/19130507/199/0013. Print n.p.&lt;/ref&gt;&lt;/blockquote&gt;The ''London Evening Standard'' describes Guendoline and Daphne Bourke the same way except with differences in editing:&lt;blockquote&gt;Miss Bourke: Presented by her mother, the Hon. Mrs. Algernon Bourke. Dainty presentation gown of white satin, the skirt draped with chiffon paniers, bodice veiled chiffon and trimmed with diamanté and crystal embroidery. Train gracefully hung from shoulder of white satin embroidered with crystal and diamanté, lined with pale rose pink chiffon.&lt;ref&gt;&quot;Some of the Dresses.&quot; &quot;The King and Queen. Third Court. Most Brilliant of the Year.&quot; ''London Evening Standard'' 08 May 1913, Thursday: 11 [of 18], Col. 4b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/19130508/237/0011. Print title ''The Standard'', p. 11.&lt;/ref&gt;&lt;/blockquote&gt;

According to the ''Lady's Pictorial'', Daphne Bourke's dress was designed and constructed by [[Social Victorians/People/Dressmakers and Costumiers#Messrs Russell and Allen|Messrs. Russell and Allen]], Old Bond-street, W., and the description is identical (except for a couple of commas).&lt;ref&gt;&quot;Their Majesties' Court.&quot; ''Lady's Pictorial'' 17 May 1913, Saturday: 35 [of 64], Col. 2c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005980/19130517/296/0035. Same print title, p. 787.&lt;/ref&gt;

'''1914 May 11, Monday''', Guendoline and Daphne Bourke attended a [[Social Victorians/Timeline/1910s#Dance at the Ritz Hosted by Mrs. George Marjoribanks|dance at the Ritz hosted by Mrs. George Marjoribanks]].

'''1915 January 1, Friday''', Algernon Bourke is listed as being on the Executive Committee of the [[Social Victorians/Timeline/1910s#1915 January 1, Friday|National Food Fund, publicized by the ''Conservative and Unionist Women's Franchise Review'']].

'''1916 August 25, Friday''', Daphne Bourke's and John Fortescue's engagement was announced:&lt;blockquote&gt;A most attractive prospective bride (says the &quot;Star&quot;) is Mr. and Mrs. Algernon Bourke's only daughter, Miss Daphne Bourke, whose engagement has just taken place to Mr. Fortescue, of the Coldstream Guards. Miss Bourke is tall, dark, and very beautiful; and Mr. Fortescue is one of the family of Boconoc, Cornwall, and Dropmore, Maidenhead. At the latter place the two families have been neighbours, for Mr. and Mrs. Algernon Bourke have a charming country residence at Taplow, while Dropmore is famous for its magnificent gardens.&lt;ref&gt;&quot;Personalia.&quot; ''Uxbridge &amp; W. Drayton Gazette'' 25 August 1916, Friday: 4 [of 8], Col. 5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002285/19160825/043/0004. Print title ''The Advertiser'', p. 4.&lt;/ref&gt;&lt;/blockquote&gt;&lt;p&gt;'''1917 June 7, Thursday''', Daphne Bourke and John Grenville Fortescue [[Social Victorians/Timeline/1910s#7 June 1917, Thursday|married in the Coldstream Guards' chapel]].

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==
According to both the ''Morning Post'' and the ''Times'', the Hon. Algernon Bourke was among the Suite of Men in the [[Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts#&quot;Oriental&quot; Procession|&quot;Oriental&quot; procession]] at the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]].&lt;ref name=&quot;:2&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;&lt;ref name=&quot;:3&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt; Based on the people they were dressed as, Guendoline Bourke was probably in this procession but it seems unlikely that Algernone Bourke was.

[[File:Guendoline-Irene-Emily-Bourke-ne-Sloane-Stanley-as-Salammb.jpg|thumb|alt=Black-and-white photograph of a standing woman richly dressed in an historical costume with a headdress and a very large fan|Hon. '''Guendoline''' Bourke as Salammbô. ©National Portrait Gallery, London.]]
=== Hon. Guendoline Bourke ===
[[File:Alfons Mucha - 1896 - Salammbô.jpg|thumb|left|alt=Highly stylized orange-and-yellow painting of a bare-chested woman with a man playing a harp at her feet|Alfons Mucha's 1896 ''Salammbô''.]]
Lafayette's portrait (right) of &quot;Guendoline Irene Emily Bourke (née Sloane-Stanley) as Salammbô&quot; in costume is photogravure #128 in the '''Album''' presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:4&quot;&gt;&quot;Devonshire House Fancy Dress Ball (1897): photogravures by Walker &amp; Boutall after various photographers.&quot; 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.&lt;/ref&gt; The printing on the portrait says, &quot;The Hon. Mrs. Algernon Bourke as Salammbo.&quot;&lt;ref name=&quot;:23&quot;&gt;&quot;Mrs. Algernon Bourke as Salammbo.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158491/Guendoline-Irene-Emily-Bourke-ne-Sloane-Stanley-as-Salammb.&lt;/ref&gt;

The Lafayette Archive has 2 additional poses from the same session on 5 July 1897 as the one chosen for the Album:

* Same image as the Album photograph but higher resolution than the one the National Portrait Gallery, London, gives permission to post (Neg. No. GP [L] [http://lafayette.org.uk/bou1369.html 1369]).
* Standing with fan behind head, includes close-up of skirt fabric and left hand (Neg. No. GP [L] [http://lafayette.org.uk/bou1368-444.html 1368-444]).
* Reclining on pillows and furs, includes close-up of face and headdress (Neg. No. GP [L] [http://lafayette.org.uk/bou1368-442.html 1368-442]).

==== Newspaper Accounts ====
The Hon. Mrs. A. Bourke was dressed as Salambo in the Oriental procession&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; in a costume made by [[Social Victorians/People/Dressmakers and Costumiers#Mrs. Mason|Mrs. Mason]], although Russell Harris says [[Social Victorians/People/Dressmakers and Costumiers#Mr. Charles Alias|Charles Alias]] was the &quot;Costume supplier.&quot;&lt;ref&gt;{{Cite web|url=http://lafayette.org.uk/bou1369.html|title=Hon Mrs Algernon Bourke (1870-1967), née Guendoline Irene Emily Sloane-Stanley by Lafayette 1897|website=lafayette.org.uk|access-date=2025-07-05}}&lt;/ref&gt;

Besides the two that mention her — the ''Morning Post'' and the ''Times'' — only two describe her costume, the London ''Evening Standard'' and the ''Gentlewoman'':
* &quot;Mrs. A. Bourke, as an Egyptian Princess, with the Salambo coiffure, wore a flowing gown of white and silver gauze covered with embroidery of lotus flowers. The top of the gown was ornamented with old green satin embroidered with blue turquoise and gold, and studded with rubies. The train was of old green broché with sides of orange and gold embroidery, and from the ceinture depended long bullion fringe and an embroidered ibis.&quot;&lt;ref&gt;“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.&lt;/ref&gt;{{rp|p. 3, Col. 3b}}
* &quot;(Egyptian Princess), drapery gown of white and silver gauze, covered with embroidery of lotus flowers; the top of gown appliqué with old green satin embroidered blue turquoise and gold, studded rubies; train of old green broché.&quot;&lt;ref&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 40, Col. 3a}}

==== Commentary ====

*The heavy headdress is likely attached to a wig and further stabilized by something under Bourke's chin and by the filet around her forehead. The flowers appear to have jewels at the center. What looks like a neck treatment is probably part of the structure holding the headpiece in place. The placement of the flowers is clearly a copy of the headdress worn by Rose Caron in the 1892 Paris production of the Reyer opera. Made by Eugène Lacoste, this textile, metal and pearl ''coiffe'' still exists (https://www.gazette-drouot.com/article/bijoux-d-opera/54941) and is held by the Bibliothèque Nationale de France.&lt;ref&gt;Merle, Sandrine. &quot;Les bijoux d’Opéra en pleine lumière au Palais Garnier.&quot; ''La Gazette Druout'' 4 July 2025 (No. 26). Retrieved July 2025

https://www.gazette-drouot.com/article/bijoux-d-opera/54941.&lt;/ref&gt; It seems to have been constructed from metal for the gold framework with the leaf design, beads attached to hanging metal frameworks (the blue-green beads look like they're covered with fabric or wrapped with thread) and flowers that may have been made from stiffened pinked fabric (likely organdy) whose color may have darkened in the century that has passed.
*Bourke's posture as she leans in the reclining odalisque pose shows she is wearing a corset.
*The translucent overskirt looks like an apron attached under the bust and hanging from the bodice.
*Russell Harris says the large fan looks more Egyptian rather than Carthaginian.&lt;ref name=&quot;:25&quot; /&gt; 
*The fullness of the skirt is very controlled.
*Her shoes are slippers decorated with real or artificial jewels.

==== Salammbô ====
Salammbô is the fictitious protagonist in Gustave Flaubert's 1862 novel ''Salammbô'', set during the Roman war against Carthage.&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|date=2024-04-29|title=Salammbô|url=https://en.wikipedia.org/w/index.php?title=Salammb%C3%B4&amp;oldid=1221352216|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Salammb%C3%B4.&lt;/ref&gt; Salammbô is a Carthaginian priestess of the lunar goddess Tanit. Matho, a Roman mercenary, breaks into Tanit's temple and steals her sacred veil — the spiritual guardian of Carthage. Salammbô sneaks into the enemy encampment to steal the veil back. She meets Matho in his tent, and &quot;believing each other to be divine apparitions,&quot; they make love,&lt;ref name=&quot;:5&quot; /&gt; although it is also a defilement. Salammbô succceds in getting the veil back, but Matho is tortured and executed, which causes her to die of shock, the effect of both having touched the veil. The plot of the opera is not identical to that of the novel.

What Guendoline Bourke saw in Salammbo as representative of herself is difficult to know, unless the costume itself appealed to her. A production of Ernest Reyer's opera ''Salammbô'', based on Flaubert's novel and published in Paris in 1890, opened at the Paris Opéra on 16 May 1892,&lt;ref&gt;{{Cite journal|date=2024-04-11|title=Ernest Reyer|url=https://en.wikipedia.org/w/index.php?title=Ernest_Reyer&amp;oldid=1218353215|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Ernest_Reyer.&lt;/ref&gt; starring Madame Rose Caron, with Mademoiselle Lucienne Bréval performing when Caron was on vacation.&lt;ref&gt;Jullienn, Adolphe. &quot;Mademoiselle Lucienne Bréval de L'Académie Nationale de Musique [or de l'Opéra in the Table of Contents].&quot; ''Le Théatre'' April 1898 (No. 4). Google Books https://www.google.com/books/edition/_/_oxRAQAAMAAJ. Pp. 8–10.&lt;/ref&gt;{{rp|8, Col. 2c}} This production was widely reviewed and discussed in the papers in the UK, and its production design was notable, especially Caron's costumes, the sets and [[Social Victorians/People/Bourke#Scale of the Production of Salammbo|the very scale of the production]]. Bourke or her costumier clearly saw the opera or images of the performers or its posters, certainly influencing the design of her costume.

* Rose Caron in her Salammbo costume is here: https://www.gettyimages.com/detail/news-photo/rose-caron-french-soprano-in-costume-in-the-title-role-of-news-photo/1439485238.
* A headshot of Bréval in costume is here: https://books.google.com/books/content?id=_oxRAQAAMAAJ&amp;pg=RA3-PP7&amp;img=1&amp;zoom=3&amp;hl=en&amp;bul=1&amp;sig=ACfU3U2Gv8Os_rEmx2gM9SakJkYLJ9hW7g&amp;ci=6%2C1%2C988%2C1371&amp;edge=0.)
* &quot;Salammbo's hair [was] powdered with a violet dust when she first appeared before the eyes of Matho.&quot;&lt;ref&gt;&quot;Salome.&quot; ''Pall Mall Gazette'' 27 February 1893, Monday: 3 [of 8]. Col. 2b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18930227/010/0003. Same print title and p.&lt;/ref&gt;

Salammbo figured in paintings, sculptures and illustrations of editions of Flaubert's novel before Ernest Reyer's 1890 opera. Translations and illustrated editions of Flaubert's novel had come out steadily beginning in the 1880s. Alfons Mucha's 1896 lithograph of Salammbô (above left) was published the year before the ball. Salammbo is often depicted as nude and highly sexual or sexualized (kissing a huge snake, for example, that she holds aloft). Reyer's opera was first produced in 1890 in Brussels. Both Modest Mussorgsky and Sergei Rachmaninoff had attempted but not completed operas based on the novel as well.&lt;ref name=&quot;:5&quot; /&gt; 


Guendoline Bourke's costume and her social life as reported in the newspapers do not suggest that she was a big risk-taker like, for example, the eccentric la Comtesse de Castiglione, who appeared at a ball in a Salammbo costume in 1886, 4 years after Flaubert's novel was first published. In 1889 the ''Edinburgh Evening News'' exaggerates her nudity and doesn't describe the rush in the ballroom to see her but does address the lingering memory:&lt;blockquote&gt;The late Countess Castiglione, whose death in Paris is recorded yesterday, made her first appearance at the Imperial Court in 1866, where her extraordinary beauty made a great impression on Napoleon III., and eventually led to the Empress Eugenie’s undertaking an unexpected and much-talked-of visit to Scotland. The Countess had a face and complexion which would have enchanted Rubens, and her lovely golden hair touched her feet. Even at the present day Paris has not forgotten her costume, or rather absence of costume as Salammbo, in which character she figured at a certain memorable ball at the Tuileries, wearing her hair, her jewels, and very little else. The Empress Eugenie, when she was presented to her thus lightly arrayed, declared that she must be cold, and insisted upon her there and then donning a mantle. Mme. de Castiglione was never again invited to an entertainment over which the Empress Eugenie presided.&lt;ref&gt;&quot;A Countess’ Queer Ball Costume.&quot; ''Edinburgh Evening News'' 2 December 1899, Saturday: 2 [of 6], Col. 7b [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000452/18991202/024/0002. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;Given how widely this incident was discussed at the time of the death of la Comtesse in 1889, Guendoline Bourke might easily have known about it. But she was developing relationships with people like the Princess of Wales, and what Countess Castigiolone did does not sound at all like her. 

===== Scale of the Production of ''Salammbô'' =====

* &quot;In Reyer's opera of 'Salammbo,' lately produced at the Grand Opera in Paris, there were 1,400 persons on the stage in the last act.&quot;&lt;ref&gt;&quot;Facts and Fancies.&quot; ''Louth and North Lincolnshire Advertiser'' 9 July 1892, Saturday: 3 [of 8], Col. 6c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000313/18920709/038/0003. Same print title and p.&lt;/ref&gt;
* &quot;the battle scene in [''Salammbo''] requires no less than 3000 square yards of 'decorative surface' [probably canvas]. This establishes a record, the next largest surface being that of the salles des fetes in 'Don Giovanni.'&quot;&lt;ref&gt;&quot;A French paper gives interesting details....&quot; ''Sevenoaks Chronicle and Kentish Advertiser'' 26 August 1892, Friday: 2 [of 8], Col. 3c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001067/18920826/032/0002. Same print title, n.p.

&lt;/ref&gt;

===== Influence of the Production of ''Salammbo'' =====
Rose Caron's productions were influential, including for the costumes she wore. The 1892 ''Lohengrin'' she starred in was the source of the costumes worn by [[Social Victorians/People/Stonor#Julia Caroline Stonor, Marquise of Hautpoul|Julia Stonor, Marquise of Hautpoul]] and her brother, [[Social Victorians/People/Stonor#Hon. Harry Stonor|Hon. Harry Stonor]].

Women's clothing was influenced by the costumes in the opera, particularly those worn by Rose Caron. One color of intense red was called Salammbo. A bonnet was named the Salammbô:&lt;blockquote&gt;About the smartest thing in bonnets for ordinary complimentary mourning is called the Salammbô, and is copied from a head-dress worn by a leading artiste at one of the Paris theatres. It is made of jet, and has a rose on each side of the front from the centres of which rise two black ospreys.&lt;ref&gt;Mantalini, Miss. &quot;The Shows in the London Shops. With Mems. about Millinery.&quot; ''Pall Mall Budget'' 29 December 1892, Thursday: 22 [of 40], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005967/18921229/092/0022. Same print title, p. 1928.&lt;/ref&gt;&lt;/blockquote&gt;In a long illustrated article describing the wedding of Princess Marie of Edinburgh, the ''Lady's Pictorial'' provides a sketch of &quot;a very pretty [hat] (No. 4) of brown mirror velvet trimmed with mink and a brown velvet bow in front '''with Salammbo''' '&lt;nowiki/&gt;''fantaisie''&lt;nowiki/&gt;'&quot; that is among the bride's millinery.&lt;ref&gt;&quot;The Marriage of H.R.H. Princess Marie of Edinburgh and H.R.H. Ferdinand Crown Prince of Roumania.&quot; ''Lady's Pictorial'' 14 January 1893, Saturday: 40 [of 76], Col. 3c [of 3]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0005980/18930114/064/0040. Same print title, p. 56.&lt;/ref&gt;&lt;p&gt;

Shoes appeared:&lt;blockquote&gt;At Mrs Merritt's, Savile-street, the stock is particularly attractive, there being so many new styles in shoes this season. One of the latest designs is the Salammbo Shoe, glace kid, with one strap, a jet buckle, and very low French heels. This shoe is especially designed for tender feet, as it is very light in weight.&lt;ref&gt;&quot;House and Home. Local Letter for Women Reader [sic], (By Our Lady Contributor).&quot; ''Hull Daily Mail'' 22 July 1897, Thursday: 5 [of 6], Col. 1b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000324/18970722/069/0005. Same print title, n.p.&lt;/ref&gt;&lt;/blockquote&gt;Patterns for making the Tunique Romaine and Corsage Salammbo were for sale just a few months after the opening:&lt;blockquote&gt;Some of the leading fashionable novelties described in ''Le Follet de Paris'' are almost ahead of the season, but they look so well that it will not be long before our provincial dressmakers have them. A revival and modification of the ancient tunic is one item which is transforming the modern gowns of tailor-built tweeds into long clinging draperies, of simple cut but ineffable grace. We have had the Russian blouse with us now for the last couple of months. Now the reign of Tunique Romaine and Corsage Salambo is upon us.

... [including the address to send off for the patterns]

A very successful novelty is the ''corsage'' “Salammbo.” In reality, it is more of a blouse and short tunic than a ''corsage'', as there is no attempt at shaping to the figure. In [sic] consists, indeed, of two straight pieces of material cut round on the shoulders, where the back and front are fastened together by clasps. There is no arm-hole, and the two pieces meet at the waist under the arm, and then hang open on to the skirt. There being no dart, the waist is as wide as the shoulders; the fullness is drawn to the centre under a ''ceinture Russe'', or of oxydised silver. The outlines are trimmed with ''galon'' or some similar garniture. The &quot;Salammbo” ''guimpe'' or ''corsage'' are made of flannel or ''mousseline de laine'' of bright colour, and are worn with fitting bodices or skirts of serge, or woollen of dark colour. They are very effective, and nothing can be easier to make, while their addition to a frock constitutes a separate costume.

The fitting bodices worn under the ''guimpes'' or ''robes'' &quot;Salammbo&quot; are very simply made; being round-waisted, they are without side pieces, and only require a seam under each arm; one in the centre of the back, and one or two darts in front, according to the figure. The skirt is mounted on a round waistband, and the ''ceinture'' worn over this gives the bodice and skirt the effect of a princess dress.&lt;ref&gt;&quot;A Womans Ceilidh.&quot; ''Oban Times and Argyllshire Advertiser'' 3 September 1892, Saturday: 3 [of 8], Col. 6a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000462/18920903/078/0003. Print title: ''The Oban Times'', p. 3.&lt;/ref&gt;&lt;/blockquote&gt;Stationery even before the opera opened in Paris:&lt;blockquote&gt;The last fad in fancy stationery is the carte Salammbo, a delightfully smooth surface for writing upon, the envelopes are very small, square, and of the wallet make; the paper folds over once to fit. The newest shades are rose pink, pale English blue, apple green, and the evergreen heliotrope.&lt;ref&gt;&quot;Fashions of the Month.&quot; ''Nottinghamshire Guardian'' 27 February 1892, Saturday: 7 [of 8], Col. 2b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000176/18920227/059/0007. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;[[File:Algernon Henry Bourke Vanity Fair 20 January 1898.jpg|thumb|alt=Old colored drawing of an elegant elderly man dressed in a 19th-century tuxedo with a cloak, top hat and formal pointed shoes with bows, standing facing 1/4 to his right|''Algy'' — Algernon Henry Bourke — by &quot;Spy,&quot; ''Vanity Fair'' 20 January 1898]]

=== Hon. Algernon Bourke ===
[[File:Hon-Algernon-Henry-Bourke-as-Izaak-Walton.jpg|thumb|left|alt=Black-and-white photograph of a man richly dressed in an historical costume sitting in a fireplace that does not have a fire and holding a tankard|Hon. Algernon Henry Bourke as Izaak Walton. ©National Portrait Gallery, London.]]
'''Lafayette's portrait''' (left) of &quot;Hon. Algernon Henry Bourke as Izaak Walton&quot; in costume is photogravure #129 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:4&quot; /&gt; The printing on the portrait says, &quot;The Hon. Algernon Bourke as Izaak Walton.&quot;&lt;ref&gt;&quot;Hon. Algernon Bourke as Izaak Walton.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158492/Hon-Algernon-Henry-Bourke-as-Izaak-Walton.&lt;/ref&gt;

This portrait is amazing and unusual: Algernon Bourke is not using a photographer's set with theatrical flats and props, certainly not one used by anyone else at the ball itself. Isaak Walton (baptised 21 September 1593 – 15 December 1683) wrote ''The Compleat Angler''.&lt;ref&gt;{{Cite journal|date=2021-09-15|title=Izaak Walton|url=https://en.wikipedia.org/w/index.php?title=Izaak_Walton&amp;oldid=1044447858|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Izaak_Walton.&lt;/ref&gt; A cottage Walton lived in and willed to the people of Stafford was photographed in 1888, suggesting that its relationship to Walton was known in 1897, raising a question about whether Bourke could have used the fireplace in the cottage for his portrait. (This same cottage still exists, as the [https://www.staffordbc.gov.uk/izaak-waltons-cottage Isaak Walton Cottage] museum.)

A caricature portrait (right) of the Hon. Algernon Bourke, called &quot;Algy,&quot; by Leslie Ward (&quot;Spy&quot;) was published in the 20 January 1898 issue of ''Vanity Fair'' as Number 702 in its &quot;Men of the Day&quot; series,&lt;ref&gt;{{Cite journal|date=2024-01-14|title=List of Vanity Fair (British magazine) caricatures (1895–1899)|url=https://en.wikipedia.org/w/index.php?title=List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899)&amp;oldid=1195518024|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899).&lt;/ref&gt; giving an indication of what he looked like out of costume.

=== Mr. and Mrs. Bourke ===
The ''Times'' made a distinction between the Hon. Mr. and Mrs. A. Bourke and Mr. and Mrs. Bourke, including both in the article.&lt;ref name=&quot;:3&quot; /&gt; Occasionally this same article mentions the same people more than once in different contexts and parts of the article, so they may be the same couple. (See [[Social Victorians/People/Bourke#Notes and Question|Notes and Question]] #2, below.)

== Demographics ==

=== The Bourkes ===
*Nationality: Anglo-Irish&lt;ref&gt;{{Cite journal|date=2020-11-14|title=Richard Bourke, 6th Earl of Mayo|url=https://en.wikipedia.org/w/index.php?title=Richard_Bourke,_6th_Earl_of_Mayo&amp;oldid=988654078|journal=Wikipedia|language=en}}&lt;/ref&gt;
*Occupation: journalist. 1895: restaurant, hotel and club owner and manager&lt;ref&gt;''Cheltenham Looker-On'', 23 March 1895. Via Ancestry but taken from the BNA.&lt;/ref&gt;

==== Residences ====
*Ireland: 1873: Palmerston House, Straffan, Co. Kildare.&lt;ref name=&quot;:7&quot; /&gt; Not Co. Mayo?
*1888–1891: 33 Cadogan Terrace, S.W., Kensington and Chelsea, a dwelling house&lt;ref&gt;Kensington and Chelsea, London, England, Electoral Registers, 1889–1970, Register of Voters, 1891.&lt;/ref&gt;
*1894: 181 Pavilion Road, Kensington and Chelsea&lt;ref&gt;Kensington and Chelsea, London, England, Electoral Registers, 1889–1970. Register of Voters, 1894. Via Ancestry.&lt;/ref&gt;
*1900: 181 Pavilion Road, Kensington and Chelsea&lt;ref&gt;Kensington and Chelsea, London, England, Electoral Registers, 1889–1970. Register of Voters, 1900. Via Ancestry.&lt;/ref&gt;
*1904: Algernon Bourke was &quot;usually liv[ing] in Venice&quot;&lt;ref name=&quot;:10&quot; /&gt;
*1906: 75, Gloucester-place, W.&lt;ref name=&quot;:21&quot; /&gt;
*Guendoline Bourke
*1911: 1911 Fulham, London&lt;ref name=&quot;:6&quot; /&gt;
*20 Eaton Square, S.W. (in 1897)&lt;ref name=&quot;:0&quot;&gt;{{Cite book|url=https://books.google.com/books?id=Pl0oAAAAYAAJ|title=Who's who|date=1897|publisher=A. &amp; C. Black|language=en}} 712, Col. 1b.&lt;/ref&gt; (London home of the [[Social Victorians/People/Mayo|Earl of Mayo]])

=== The Sloane-Stanleys ===

==== Residences ====

* 1871: Chester Street, St George Hanover Square (Census), with 5 servants, including a cook and a footman.&lt;ref name=&quot;:16&quot;&gt;The National Archives; Kew, London, England; ''1871 England Census''; Class: ''RG10''; Piece: ''104''; Folio: ''21''; Page: ''37''; GSU roll: ''838763''. Ancestry.com. ''1871 England Census'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations Inc, 2004.&lt;/ref&gt;
* 1881–1885&lt;ref&gt;''UK, City and County Directories, 1600s-1900s''. Ancestry.com. ''UK, City and County Directories, 1766 - 1946'' [database on-line]. Provo, UT, USA: Ancestry.com Operations, Inc., 2013.&lt;/ref&gt; [at least]: 14 Halkin Street, W., St. Georges, 14 servants, including a governess, a house steward, an under butler, a footman and a cook.&lt;ref&gt;''Census Returns of England and Wales, 1881''. Kew, Surrey, England: The National Archives of the UK (TNA): Public Record Office (PRO), 1881. Class: ''RG11''; Piece: ''98''; Folio: ''66''; Page: ''37''; GSU roll: ''1341022''. Ancestry.com and The Church of Jesus Christ of Latter-day Saints. ''1881 England Census'' [database on-line]. Provo, UT, USA: Ancestry.com Operations Inc, 2004.&lt;/ref&gt;
* 1888: 49, Cadogan-square, St. Luke, Chelsea&lt;ref&gt;Ancestry.com. ''London, England, Overseer Returns, 1863-1894'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2013.&lt;/ref&gt;
* 1899, Roger Cyril Sloane-Stanley: 4 Down St., St George, Hanover Square&lt;ref&gt;London Metropolitan Archives; London, England; ''Electoral Registers''. Ancestry.com. ''London, England, Electoral Registers, 1832-1965'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2010.&lt;/ref&gt;
* 1911, Roger Cyril Sloane-Stanley: Paultons, Ower, Romsey

== Family ==
*Hon. Algernon Henry Bourke (31 December 1854 – 7 April 1922)&lt;ref&gt;&quot;Hon. Algernon Henry Bourke.&quot; {{Cite web|url=https://www.thepeerage.com/p29657.htm#i296561|title=Person Page|website=www.thepeerage.com|access-date=2020-12-10}}&lt;/ref&gt;
*Guendoline Irene Emily Sloane-Stanley Bourke (c. 1869 – 30 December 1967)&lt;ref name=&quot;:1&quot;&gt;&quot;Guendoline Irene Emily Stanley.&quot; {{Cite web|url=https://www.thepeerage.com/p51525.htm#i515247|title=Person Page|website=www.thepeerage.com|access-date=2020-12-10}}&lt;/ref&gt;
#Daphne Marjory Bourke (5 April 1895 – 22 May 1962)

=== Relations ===
*Hon. Algernon Henry Bourke (the 3rd son of the [[Social Victorians/People/Mayo|6th Earl of Mayo]]) was the older brother of Lady Florence Bourke.&lt;ref name=&quot;:0&quot; /&gt;
*Wilfred Blunt was a cousin of Algernon Bourke: Bourke's mother's &quot;mother was one of the Blunts of Crabbet Park, Sussex, which makes them kinswomen of Mr. Alfred Scawen Blunt, poet, Egyptophil and counsel for Arabi Pasha in his trial.&quot;&lt;ref&gt;&quot;From ''Truth''.&quot; ''Mid-Lothian Journal'' 23 August 1912, Friday: 8 [of 8], 2c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002721/19120823/147/0008. Print title and p. same.&lt;/ref&gt;
*Lord Alfred Douglas was a cousin of Algernon Bourke.

==== Other Bourkes ====
*Hubert Edward Madden Bourke (after 1925, Bourke-Borrowes)&lt;ref&gt;&quot;Hubert Edward Madden Bourke-Borrowes.&quot; {{Cite web|url=https://www.thepeerage.com/p52401.htm#i524004|title=Person Page|website=www.thepeerage.com|access-date=2021-08-25}} https://www.thepeerage.com/p52401.htm#i524004.&lt;/ref&gt;
*Lady Eva Constance Aline Bourke, who married [[Social Victorians/People/Dunraven|Windham Henry Wyndham-Quin]] on 7 July 1885;&lt;ref&gt;&quot;Lady Eva Constance Aline Bourke.&quot; {{Cite web|url=https://www.thepeerage.com/p2575.htm#i25747|title=Person Page|website=www.thepeerage.com|access-date=2020-12-02}} https://www.thepeerage.com/p2575.htm#i25747.&lt;/ref&gt; he became 5th Earl of Dunraven and Mount-Earl on 14 June 1926.

=== The Sloane-Stanleys ===
* Emilie Josephine S Stanley ( 21 December 1848 [baptism]&lt;ref&gt;London Metropolitan Archives; &quot;London, England, UK&quot; ; ''London Church of England Parish Registers''; Reference Number: ''P87/Tri/001''. Ancestry.com. ''London, England, Church of England Births and Baptisms, 1813-1923'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2010.&lt;/ref&gt;  – October 1945)
* Hans T Sloane Stanley (11 May 1840 [baptism]&lt;ref&gt;Ancestry.com. ''England, Select Births and Christenings, 1538-1975'' [database on-line]. Provo, UT, USA: Ancestry.com Operations, Inc., 2014.&lt;/ref&gt; – 15 December 1888&lt;ref&gt;Ancestry.com. ''UK and Ireland, Find a Grave® Index, 1300s-Current'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2012.&lt;/ref&gt;)
* James Shell[e?]y Bontein ()
*# Gwendoline&lt;ref name=&quot;:14&quot; /&gt; Irene Emily G Stanley (c. 1870&lt;ref name=&quot;:16&quot; /&gt; – )
*# '''Roger Cyril Hans Sloane Stanley''' (29 April 1875&lt;ref&gt;The National Archives; Kew, Surrey, England; ''WO 42 War Office: Officers' Birth Certificates, Wills and Personal Papers 1755-1908''; Reference: ''WO 42/72''. Ancestry.com. ''UK, Officers' Birth Certificates, Wills and Personal Papers, 1755-1908'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2023.&lt;/ref&gt; – 18 November 1944&lt;ref&gt;''Find a Grave''. Find a Grave®. http://www.findagrave.com/cgi-bin/fg.cgi. Ancestry.com. ''UK and Ireland, Find a Grave® Index, 1300s-Current'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2012.&lt;/ref&gt;)


* Olivia Elizabeth Berens, Countess Cairns&lt;ref&gt;The National Archives of the UK (TNA); Kew, Surrey, England; ''Census Returns of England and Wales, 1911''. Ancestry.com. ''1911 England Census'' [database on-line]. Provo, UT, USA: Ancestry.com Operations, Inc., 2011.&lt;/ref&gt; (c. 1871 – 20 June 1951&lt;ref&gt;&quot;Olivia Elizabeth Berens.&quot; Person Page 3908; person #39077. ''The Peerage: A Genealogical Survey of the Peerage of Britain as well as the Royal Families of Europe''. https://www.thepeerage.com/p3908.htm#i39077.

&lt;/ref&gt;)
* Arthur William Cairns, 2nd Earl Cairns (21 December 1861 – 14 January 1890)&lt;ref name=&quot;:20&quot;&gt;&quot;Arthur William Cairns, 2nd Earl Cairns.&quot; Person Page 3908; Person #39076. ''The Peerage: A Genealogical Survey of the Peerage of Britain as well as the Royal Families of Europe''. https://www.thepeerage.com/p3908.htm#i39076.&lt;/ref&gt;
*# Lady Louise Rosemary Kathleen Virginia Cairns (10 March 1889 – 17 May 1962)&lt;ref name=&quot;:20&quot; /&gt;
* Roger Cyril Hans Sloane Stanley (1875 – 18 November 1944)
*# Lavender Elizabeth (20 May 1900 [baptism]&lt;ref&gt;Hampshire Archives and Local Studies; Winchester, England, UK; ''Anglican Parish Registers''; Reference: ''35M76/PR3''. Ancestry.com. ''Hampshire, England, Church of England Baptisms, 1813-1921''[database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2023.&lt;/ref&gt; – )
*# Diane Sloane Stanley (c. 1905 – )


* Lavender Elizabeth (20 May 1900 [baptism] – )
* John Everett ()


* Diane Sloane Stanley (c. 1905 – )
* Elwyn Villiers Rhys ()

== Writings, Memoirs, Biographies, Papers ==

=== Writings ===
* Bourke, the Hon. Algernon. ''The History of White's''. London: Algernon Bourke [privately published], 1892.
* Bourke, the Hon. Algernon, ed., &quot;with a brief Memoir.&quot; ''Correspondence of Mr Joseph Jekyll with His Sister-in-Law, Lady Gertrude Sloane Stanley, 1818–1838''. John Murray, 1893.
* Bourke, the Hon. Algernon, ed. ''Correspondence of Mr Joseph Jekyll''. John Murray, 1894.

=== Papers ===
* Where are the papers for the Earl of Mayo family? Are Algernon and Guendoline Bourke's papers with them?

== Notes and Questions ==
#The portrait of Algernon Bourke in costume as Isaac Walton is really an amazing portrait with a very interesting setting, far more specific than any of the other Lafayette portraits of these people in their costumes. Where was it shot? Lafayette is given credit, but it's not one of his usual backdrops. If this portrait was taken the night of the ball, then this fireplace was in Devonshire House; if not, then whose fireplace is it?
#The ''Times'' lists Hon. A. Bourke (at 325) and Hon. Mrs. A. Bourke (at 236) as members of a the &quot;Oriental&quot; procession, Mr. and Mrs. A. Bourke (in the general list of attendees), and then a small distance down Mr. and Mrs. Bourke (now at 511 and 512, respectively). This last couple with no honorifics is also mentioned in the report in the London ''Evening Standard'', which means the Hon. Mrs. A. Bourke, so the ''Times'' may have repeated the Bourkes, who otherwise are not obviously anyone recognizable. If they are not the Hon. Mr. and Mrs. A. Bourke, then they are unidentified. It seems likely that they are the same, however, as the newspapers were not perfectly consistent in naming people with their honorifics, even in a single story, especially a very long and detailed one in which people could be named more than once.
#Three slightly difficult-to-identify men were among the Suite of Men in the [[Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts#&quot;Oriental&quot; Procession|&quot;Oriental&quot; procession]]: [[Social Victorians/People/Halifax|Gordon Wood]], [[Social Victorians/People/Portman|Arthur B. Portman]] and [[Social Victorians/People/Sarah Spencer-Churchill Wilson|Wilfred Wilson]]. The identification of Gordon Wood and Wilfred Wilson is high because of contemporary newspaper accounts. The Hon. Algernon Bourke, who was also in the Suite of Men, is not difficult to identify at all. Arthur Portman appears in a number of similar newspaper accounts, but none of them mentions his family of origin.
#[http://thepeerage.com The Peerage] has no other Algernon Bourkes.
#The Hon Algernon Bourke is #235 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who were present]]; the Hon. Guendoline Bourke is #236; a Mr. Bourke is #703; a Mrs. Bourke is #704.
#Hans Stanley-Sloane's estate was £33,704 7s. 5d. in the final probate in December 1889,&lt;ref&gt;Principal Probate Registry; London, England; ''Calendar of the Grants of Probate and Letters of Administration made in the Probate Registries of the High Court of Justice in England''. Ancestry.com. ''England &amp; Wales, National Probate Calendar (Index of Wills and Administrations), 1858-1995'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2010.&lt;/ref&gt; which might lead his widow to consider remarrying.
== Footnotes ==
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      <text bytes="172215" sha1="jjapwfjg4ar9rqbjdu2fei8acrlp8p0" xml:space="preserve">Algernon and Guendoline Bourke were &quot;satirized&quot; by [[Social Victorians/People/Oscar Wilde|Oscar Wilde]] in his 1895 ''The Importance of Being Earnest''.&lt;ref&gt;Fahey, David M. ''The Politics of Drink in England, from Gladstone to Lloyd George''. Cambridge Scholars Publishing, Lady Stephenson Library, Newcastle upon Tyne, 2022: p. 21, n. 34. ''Google Books'' (retrieved July 2025) https://www.google.com/books/edition/The_Politics_of_Drink_in_England_from_Gl/8HRjEAAAQBAJ.&lt;/ref&gt; Algernon Bourke was a cousin of Lord Alfred Douglas, Wilde's Bosie, in 1898 attempting to facilitate a reconciliation between Bosie and his father.&lt;ref name=&quot;:26&quot;&gt;Ellmann, Richard. ''Osar Wilde''. Alfred A. Knopf, 1988.&lt;/ref&gt; (568)[[File:Leslie Ward - Vanity Fair, Newspapermen, ^Algy^, The Hon Algernon Henry Bourke, Januray 20, 1898 - B1979.14.521 - Yale Center for British Art.jpg|thumb|Hon. Algernon Bourke, ''Vanity Fair'', 1898]]

==Also Known As==
* Family name: Bourke [pronounced ''burk'']&lt;ref name=&quot;:62&quot;&gt;{{Cite journal|date=2024-05-07|title=Earl of Mayo|url=https://en.wikipedia.org/w/index.php?title=Earl_of_Mayo&amp;oldid=1222668659|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Earl_of_Mayo.&lt;/ref&gt;
* The Hon. Algernon Bourke
** Button Bourke&lt;ref&gt;&quot;A Tory 'Reformer' at the India Office.&quot; ''India'' 10 November 1911, Friday: 4 [of 12], Col. 1b [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004300/19111110/007/0004#. Print: same title, p. 228.&lt;/ref&gt;
** Algy Bourke
* Mrs. Guendoline Bourke
** Guendoline&lt;ref name=&quot;:1&quot; /&gt; [The National Portrait Gallery, London, uses this spelling for Lafayette's portrait of Bourke in costume for the ball,&lt;ref name=&quot;:23&quot; /&gt; as does Russell Harris in the Lafayette Archive, held by the Victoria and Albert&lt;ref name=&quot;:25&quot;&gt;Harris, Russell. &quot;Hon Mrs Algernon Bourke, née Guendoline Irene Emily Sloane-Stanley (1870-1967), as Salammbô.&quot; ''In Calm Prose'' 2011. http://www.rvondeh.dircon.co.uk/incalmprose/.&lt;/ref&gt;.]
** Gwendolen&lt;ref&gt;General Register Office. ''England and Wales Civil Registration Indexes''. London, England: General Register Office. FreeBMD. ''England &amp; Wales, Civil Registration Marriage Index, 1837-1915''[database on-line]. Provo, UT, USA: Ancestry.com Operations Inc, 2006.&lt;/ref&gt;{{rp|Marriage Index}} &lt;ref name=&quot;:15&quot; /&gt;{{rp|''Morning Post'' article about her name}} &lt;ref&gt;General Register Office. ''England and Wales Civil Registration Indexes''. London, England: General Register Office. FreeBMD. ''England &amp; Wales, Civil Registration Marriage Index, 1837-1915''[database on-line]. Provo, UT, USA: Ancestry.com Operations Inc, 2006.&lt;/ref&gt;{{rp|Electoral Register}}
** Gwendoline&lt;ref name=&quot;:14&quot;&gt;City of Westminster Archives Centre; London, England; ''Westminster Church of England Parish Registers''; Reference: ''SPWP/PR/1/2''. Ancestry.com. ''Westminster, London, England, Church of England Births and Baptisms, 1813-1919'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2020.&lt;/ref&gt;{{rp|Births and Baptisms}}
* Shelley Bontein and Emilie Sloane-Stanley Bontein
* See also the [[Social Victorians/People/Mayo|page for the Earl of Mayo]], the Hon. Algernon Bourke's father and then brother, and other Bourkes

== Overview ==

=== Algernon Bourke ===
Although the Hon. Algernon Henry Bourke was born in Dublin in 1854 and came from a family whose title is in the Peerage of Ireland,&lt;ref name=&quot;:6&quot;&gt;1911 England Census.&lt;/ref&gt; he seems to have spent much of his adult life generally in England and especially in London. He was &quot;a noted fisherman.&quot;&lt;ref&gt;&quot;London Correspondence.&quot; ''Freeman's Journal'' 21 December 1897, Tuesday: 5 [of 8], Col. 5c [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000056/18971221/027/0005. Same print title, n.p.&lt;/ref&gt;

Because he was the son of the [[Social Victorians/People/Mayo|Earl of Mayo]], perhaps, or perhaps because he was so involved in projects that got reported on, he was mentioned a great deal in the newspapers, but after his bankruptcy, he seems to have receded in prominence, in part because he was living outside of the U.K., and apparently separately from his wife, Guendoline Bourke.

Bourke ran as the Conservative candidate for Parliament from Clapham (population, c. 70,000) in 1885, a race he did not win. As a candidate he is described like this:&lt;blockquote&gt;Acted as a newspaper correspondent during the Zulu war. Subsequently Poor-law inspector in the West of Ireland. &quot;A loyal supporter of Church and State.&quot; Desires to reduce the School Board expenditure, and revive trade; and is opposed to Mr. Chamberlain's &quot;police of hasty and experimental reform.&quot;&lt;ref&gt;&quot;Clapham (70,000).&quot; ''South London Chronicle'' 17 October 1885, Saturday: 5 [of 8], Col. 5a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000443/18851017/113/0005. Print title ''South London Chronicle and Southwark and Lambeth Ensign'', p. 5.&lt;/ref&gt;&lt;/blockquote&gt;The London ''Weekly Dispatch'' says he is &quot;a dashing and unscrupulous young Tory.&quot;&lt;ref&gt;&quot;The Political Campaign in London.&quot; ''Weekly Dispatch'' (London) 15 November 1885, Sunday: 9 [of 16], Col. 3c [of 4]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003358/18851115/069/0009. Same print title and p.&lt;/ref&gt;

&quot;Algy&quot; Bourke was &quot;Man of the Day&quot; (No. DCCII [522) for ''Vanity Fair'' in 1898, caricatured by Leslie Ward (above right):&lt;blockquote&gt;Son of the great and murdered Lord Mayo, he is contemporary with the outbreak of the Crimean War, he is a Member of the London Stock Exchange, he has a beautiful wife and a daughter, and, being a very fashionable young man, he was once refused as their Member by the worthy electors of Clapham. He was an Eton boy, of course: and less naturally he went to Cambridge; where he was made President of the Beefsteak, the Amateur Dramatic, the Athenaeum, the True Blue, and the Hibernian Clubs. When he came down he tried journalism and went to Zululand as a ''Daily Telegraph'' ‘‘special”; after which he was improved into an Inspector of Workhouses [2, Col. 2c – 3, Col. 1a] in Ireland: which may account for his proficiency as a caterer. For seven years he worked under the late Mr. Chinnery on ''The Times'': being popularly supposed to look after that journal's morals. He is a good man of business, and a great organiser who has made White's Club pay even if it be less “smart&quot; than it was. He has done much for Willis’s since he took it in hand; he did well with his Battersea venture, and he thinks that he only failed with the Summer Club in Kensington Gardens because people would not go to the wrong side of the Park. Moreover, he runs a Club at Brighton, and he is Chairman of the Grand Hotel at Monte Carlo: whither he once organised a cheap trip. Altogether he is a veritable Clubman, and a very successful arranger of amusements, associations, and restaurants.

He is a popular fellow who is known to all of us; and though he is a little inclined to be quarrelsome, no one can get much the better of him. He is also a quick grasper of facts and a good talker. His favourite sports are fishing and the organising of associations for the introduction of salmon to the Thames. By way of being an art critic, he has made an interesting collection of engravings of the members of White’s Club from its foundation; but his friends say that he is not a well-dressed man. He has also written a history of White’s, and he is now writing one of Brooks's Club. He is a genial person, who looks as if the world agreed with him well.

He is an aquisition [sic] to a house party; and they call him “Algy.”&lt;ref&gt;&quot;Men of the Day.&quot; — No. DCCII [522]. The Hon. Algernon Henry Bourke.&quot; ''Vanity Fair'' 20 January 1898, Thursday: 2 [of 4], Col. 2c – 3, Col. 3a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/9900020/18980120/010/0002 and https://www.britishnewspaperarchive.co.uk/viewer/BL/9900020/18980120/005/0003. Same print title, pp. 41–42. Portrait is full page, on p. 1.&lt;/ref&gt;&lt;/blockquote&gt;The Hon. Algernon Bourke and Mr. Algernon Bourke, depending on the newspaper article, were the same person. Calling him Mr. Bourke in the newspapers, especially when considered as a businessman or (potential) member of Parliament, does not rule out the son of an earl, who would normally be accorded the honorific of ''Honorable''.

=== Guendoline Sloane-Stanley Bourke ===

Mrs. Guendoline Bourke exhibited at dog shows successfully and was a [[Social Victorians/Timeline/1900s#Society Sportswomen|noted deerstalker]] and &quot;an appreciative listener to good music.&quot;&lt;ref&gt;&quot;Vanity Fair.&quot; ''Lady of the House'' 15 June 1899, Thursday: 4 [of 44], Col. 2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004836/18990615/019/0004.&lt;/ref&gt; Her personal beauty is often mentioned in reports, and ''The World'' says she was &quot;a magnificent woman.&quot;&lt;ref&gt;&quot;Beauties of To-Day. From the ''World''.&quot; ''Clifton Society'' 24 June 1897, Thursday: 14 [of 16], Col. 2c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002164/18970624/066/0014. Same print and p.&lt;/ref&gt; She is the first listed in the ''Graphic''&lt;nowiki/&gt;'s 1891 &quot;Leading Ladies of Society&quot;:&lt;blockquote&gt;The Hon. Mrs. Algernon Bourke is a daughter (Gwendoline Irene Emily) of the late Hans Sloane Stanley, Esq., of Poultons, Southampton, and 49, Cadogan Square, S.W. She married, on December 15th, 1887, the Hon. Algernon Bourke, third son of the sixth Earl of Mayo, Governor-General of India (who was assassinated in 1872), and nephew of Lord Connemara, Governor of Madras. Mr. Bourke is a member of the London Stock Exchange, and resides at 33, Cadogan Terrace, S.W.&lt;ref&gt;&quot;Leading Ladies of Society.&quot; The Graphic 28 March 1891, Saturday: 6 [of 28], Col. 2c [of 3]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/9000057/18910328/019/0006. Print: same title, p. 346.&lt;/ref&gt;&lt;/blockquote&gt;She attended many social events without her husband, especially into the 20th century, usually with an appreciative description of what she wore. She was a sponsor of Irish art needlework as well. Unlike her husband's, Guendoline's social status seems to have risen as time passed, and she appears in stories associated with the Princess of Wales, and then later with Queen Alexandra.

=== The Sloane-Stanley Family ===
Guendoline's family consisted of a younger brother, Cyril Sloane-Stanley, as well as her parents, Hans Sloane-Stanley and Emilie Edwards Sloane-Stanley. Exactly one year after she and Algernon Bourke married, Hans Sloane-Stanley died (in 1888), leaving an estate worth £33,704 7s. 5d.&lt;ref name=&quot;:17&quot;&gt;Principal Probate Registry; London, England; ''Calendar of the Grants of Probate and Letters of Administration made in the Probate Registries of the High Court of Justice in England''. Ancestry.com. ''England &amp; Wales, National Probate Calendar (Index of Wills and Administrations), 1858-1995'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2010.&lt;/ref&gt; (1888, 321) Her mother remarried almost exactly a year after that, to James Shelly Bontein. Bontein's father had been Gentleman Usher and Clerk of the Robes to Queen Victoria.&lt;ref name=&quot;:18&quot;&gt;&quot;Marriages.&quot; &quot;Births, Marriages, and Deaths.&quot; ''Belfast News-Letter'' 6 December 1889, Friday: 1 [of 8], Col. 1a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000038/18891206/001/0001. Same print title and p.&lt;/ref&gt;

Shortly after his death ''Truth'' described Guendoline and Cyril's father Hans Sloane-Stanley:&lt;blockquote&gt;The death of Mr. Sloane Stanley, of Paultons Park, is much regretted in South Hants, as he was one of the most popular landlords in the county, and was greatly esteemed. Mr. Sloane Stanley was well known in yachting circles, and for many years he was Commodore of the Royal Southern Yacht Club, and owned the schooner ''Star of the West''. He was one of the very few owners who continued to keep up the old custom of giving his crew a laying-up supper at the close of each season. There were great festivities at Paultons only a few months ago, when Miss Sloane Stanley was married to Mr. Algernon Bourke.&lt;ref&gt;&quot;Entre Nous.&quot; ''Truth'' 27 December 1888, Thursday: 6 [of 48], Col. 2b [of 2]. ''British Newspaper Archive'' [https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18881227/023/0006# https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18881227/023/0006]. Same print title, p. 1136.&lt;/ref&gt;&lt;/blockquote&gt;When he died in 1944, Cyril Sloane-Stanley's estate was quite a bit larger than his father's had been 50 years before. The probate was divided between what was limited to &quot;settled land&quot; and what was &quot;save and except settled land.&quot; What was not settled land totalled £356,114 12s. 10d. and went to John Everett, company director; the Hon. Elwyn Villiers Rhys, captain, H.M. army; and William Adam de Geijer, retired captain, H.M. army.&lt;ref name=&quot;:17&quot; /&gt; (1944, 430) His daughter Lavender was married to John Everett, and Diane was married to Elwyn Villiers Rhys. What was settled land totalled £168,975 and went to William Adam de Geijer, retired captain, H.M. army, and George Lawrence Stewart, solicitor.&lt;ref name=&quot;:17&quot; /&gt; (1944, 430)

The Sloane-Stanleys descend from Hans Sloane (1660–1753), whose 71,000-item collections &quot;provid[ed] the foundation of the British Museum, the British Library, and the Natural History Museum, London.&quot;&lt;ref name=&quot;:19&quot;&gt;{{Cite journal|date=2025-01-07|title=Hans Sloane|url=https://en.wikipedia.org/wiki/Hans_Sloane|journal=Wikipedia https://en.wikipedia.org/wiki/Hans_Sloane|language=en|via=}}&lt;/ref&gt; Much of this Hans Sloane's wealth came from his medical practice in Jamaica, where he went as physician to the Governor General of Jamaica, the 2nd Duke of Albemarle, and where he married &quot;a wealthy heiress of sugar plantations&quot; worked by enslaved Jamaicans.&lt;ref name=&quot;:19&quot; /&gt; His great-nephew, Hans Sloane, inherited Paultons, near Romsey, &quot;and in recognition of this he adopted the additional surname of Stanley in 1821.&quot;&lt;ref&gt;{{Cite journal|date=2023-10-06|title=Hans Sloane (MP)|url=https://en.wikipedia.org/wiki/Hans_Sloane_(MP)|journal=Wikipedia https://en.wikipedia.org/wiki/Hans_Sloane_(MP)|language=en}}&lt;/ref&gt;

== Acquaintances, Friends and Enemies ==

=== Algernon Bourke ===

* Best man at [[Social Victorians/Timeline/1887#Wedding of Algernon Bourke and Gwendolen Sloane Stanley|his wedding]]: the Hon. Michael Sandys
* [[Social Victorians/People/Montrose|Marcus Henry Milner]], &quot;one of the zealous assistants of that well-known firm of stockbrokers, Messrs. Bourke and Sandys&quot;&lt;ref name=&quot;:8&quot;&gt;&quot;Metropolitan Notes.&quot; ''Nottingham Evening Post'' 31 July 1888, Tuesday: 4 [of 4], Col. 2a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000321/18880731/025/0004.&lt;/ref&gt;
* Caroline, Duchess of Montrose — her &quot;legal advisor&quot; on the day of her marriage to Marcus Henry Milner&lt;ref&gt;&quot;Metropolitan Notes.&quot; ''Nottingham Evening Post'' 31 July 1888, Tuesday: 4 [of 4], Col. 1b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000321/18880731/025/0004.&lt;/ref&gt;
* Wilfred Scawen Blunt
* Alexander Meyrick Broadley&lt;ref name=&quot;:24&quot;&gt;Broadley, Alexander Meyrick. ''How We Defended Arábi and His Friends, a Story of Egypt and the Egyptians''. London: Chapman and Hall, 1884. Google Books: [https://www.google.com/books/edition/How_We_Defended_Ar%C3%A1bi_and_His_Friends/75YRzFDvaO4C?hl=en&amp;gbpv=1&amp;bsq=Bourke https://www.google.com/books/edition/How_We_Defended_Ar%C3%A1bi_and_His_Friends/75YRzFDvaO4C].&lt;/ref&gt; (2) [Blunt, Broadley and Bourke were mutual friends; Blunt and Bourke were cousins.]
* &quot;Natty Rothschild,&quot; according to Blunt
* Brett, who was Hartington's secretary in the early 80s.&lt;ref&gt;Blunt, Wilfred Scawen. ''Gordon at Khartoum, Being a Personal Narrative of Events in Continuation of &quot;A Secret History of the English Occupation of Egypt''.&quot; London: Stephen Swift, 1911.&lt;/ref&gt;

=== Guendoline Bourke ===

* Bridesmaids at [[Social Victorians/Timeline/1887#Wedding of Algernon Bourke and Gwendolen Sloane Stanley|her wedding]]: Lady Florence Bourke, Miss Nora Bourke, Miss Edwards, and Miss Ewart
* Lord and Lady Alington, Belvedere House, Scarborough
* [[Social Victorians/People/William James|Evelyn James]] 

== Organizations ==

=== Guendoline Bourke ===

* Member, the Ladies Committee for the [[Social Victorians/London Clubs#Prince's Skating Club|Prince's Skating Club]], which also included [[Social Victorians/People/Princess Louise|Princess Louise]] (Duchess of Argyll), the [[Social Victorians/People/Portland|Duchess of Portland]], [[Social Victorians/People/Londonderry|Lady Londonderry]], [[Social Victorians/People/Campbell|Lady Archibald Campbell]], [[Social Victorians/People/Ribblesdale|Lady Ribblesdale]], and [[Social Victorians/People/Asquith|Mrs. Asquith]]&lt;ref name=&quot;:11&quot;&gt;&quot;What the 'World' Says.&quot; ''Northwich Guardian'' 01 November 1902, Saturday: 6 [of 8], Col. 8a [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001975/19021101/134/0006. Print title: The ''Guardian'', p. 6.&lt;/ref&gt; (in 1902, at least)

=== Algernon Bourke ===
* [[Social Victorians/Schools#Eton|Eton]]
* Cambridge University, Trinity College, 1873, Michaelmas term&lt;ref name=&quot;:7&quot;&gt;Cambridge University Alumni, 1261–1900. Via Ancestry.&lt;/ref&gt;
* Conservative Party
* 1879: Appointed a Poor Law Inspector in Ireland, Relief of Distress Act
* 1881: Partner, with 2 uncles, in Brunton, Bourke, and Co.&lt;ref&gt;&quot;From Our London Correspondent.&quot; ''Manchester Courier'' 24 August 1881, Wednesday: 5 [of 8], Col. 4a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000206/18810824/030/0005. Print: ''Manchester Courier and Lancaster General Advertiser'', p. 5.&lt;/ref&gt; (one of the [[Social Victorians/British Aristocracy#Sons of Peers on the Stock Exchange|sons of peers on the Stock Exchange]])
* 1885: Office of the 7th Surrey Rifles Regiment&lt;ref&gt;&quot;7th Surrey Rifles.&quot; ''South London Press'' 08 August 1885, Saturday: 12 [of 16], Col. 4a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000213/18850808/165/0012. Print p. 12.&lt;/ref&gt;
* 1886: Battersea Friendly Angling Society&lt;ref&gt;&quot;Battersea Friendly Angling Society.&quot; ''Fishing Gazette'' 17 April 1886, Saturday: 6 [of 20], Col. 2a [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002553/18860417/030/0006. Same print title, p. 218.&lt;/ref&gt;
* 27 February 1886: one of the Vice Presidents of the [[Social Victorians/London Clubs#Bolingbroke Reading-Room and Institute|Bolingbroke Reading-Room and Institute]]
* Special Correspondent of The ''Times'' for the Zulu War, accompanying Lord Chelmsford
* Head, Messrs. Bourke and Sandys, &quot;that well-known firm of stockbrokers&quot;&lt;ref name=&quot;:8&quot; /&gt; ( – 1901 [at least])
* White's gentleman's club, St. James's,&lt;ref&gt;{{Cite journal|date=2024-10-09|title=White's|url=https://en.wikipedia.org/wiki/White's|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/White%27s.&lt;/ref&gt; Manager (1897)&lt;ref&gt;&quot;Side Lights on Drinking.&quot; ''Waterford Standard'' 28 April 1897, Wednesday: 3 [of 4], Col. 7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001678/18970428/053/0003.&lt;/ref&gt;
* Willis's Rooms (described in 1895):&lt;blockquote&gt;... the Hon. Algernon Burke [sic], son of the 6th Earl of Mayo, has turned the place into a smart restaurant where choice dinners are served and eaten while a stringed band discourses music. Willis's Rooms are now the favourite dining place for ladies who have no club of their own, or for gentlemen who are debarred by rules from inviting ladies to one of their own clubs. The same gentleman runs a hotel in Brighton, and has promoted several clubs. He has a special faculty for organising places of the kind, without which such projects end in failure.&lt;ref&gt;&quot;Lenten Dullness.&quot; ''Cheltenham Looker-On'' 23 March 1895, Saturday: 11 [of 24], Col. 2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000226/18950323/004/0011. Print p. 275.&lt;/ref&gt;&lt;/blockquote&gt;
*The [[Social Victorians/London Clubs#Pelican|Pelican Club]], known for its boxing (1891)

==== Boards of Directors ====
*1883: One of the directors, the Franco-English Tunisian Esparto Fibre Supply Company, Ltd.&lt;ref&gt;''Money Market Review'', 20 Jan 1883 (Vol 46): 124.&lt;/ref&gt;
*1891: One of the founders, the Discount Banking Company, Ltd., which says Algernon Bourke is a director of District Messenger Services and News Company, Ltd.&lt;ref&gt;&quot;Public Company.&quot; ''Nottingham Journal'' 31 October 1891, Saturday: 4 [of 8], Col. 8a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001896/18911031/099/0004. Print title: ''The Nottingham Daily Express'', p. 4.&lt;/ref&gt;
*1894: One of the directors, the Frozen Lake, Ltd., with Admiral Maxse, Lord [[Social Victorians/People/Beresford|Marcus Beresford]], [[Social Victorians/People/Williams|Hwfa Williams]]&lt;ref&gt;&quot;The Frozen Lake, Limited.&quot; ''St James's Gazette'' 08 June 1894, Friday: 15 [of 16], Col. 4a [of 4]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001485/18940608/085/0015. Print p. 15.&lt;/ref&gt;&lt;blockquote&gt;London is to have new amusement this winter, for which Mr Algernon Bourke, who has taught us that it is possible to eat as well in St. James’s as on the Boulevards, and Mr Hwfa Williams, of Sandown fame, are jointly responsible. The &quot;Frozen Lake,&quot; under which title a real ice-skating rink is about to be constructed under their auspices, will no doubt be gladly welcomed by all skaters, and the venture is likely to prove a success.&lt;ref&gt;&quot;Society Gossip.&quot; ''Weston-super-Mare Gazette, and General Advertiser'' 6 June 1894, Wednesday: 4 [of 4], Col. 4b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001444/18940606/044/0004. Print title: ''Weston-super-Mare Gazette'', p. 4.&lt;/ref&gt;&lt;/blockquote&gt;

==== Committees ====

*Member, General Committee, [[Social Victorians/London Clubs#Baths|the Baths Club]] (1892)
*Member, Men's Committee of the [[Social Victorians/London Clubs#Prince's Skating Club|Prince's Skating Club]], which also included Lord Edward Cecil, Lord Redesdale, Mr. [[Social Victorians/People/Lyttelton|Alfred Lyttelton]], Sir Edgar Vincent, Sir William Hart Dyke, and Mr. [[Social Victorians/People/Grenfell|W. H. Grenfell]]&lt;ref name=&quot;:11&quot; /&gt; (1902, at least)
*[[Social Victorians/Timeline/1896#25 March 1896, Wednesday|The Sala Memorial Fund]], member of the committee (from 25 March 1896)
* Member of an &quot;influential committee&quot; headed by the Lord Mayor &quot;to restore salmon to the Thames&quot; (June 1899)&lt;ref&gt;&quot;Salmon in the Thames.&quot; ''Berks and Oxon Advertiser'' 30 June 1899, Friday: 5 [of 8], Col. 4a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002298/18990630/079/0005. Print n.p.&lt;/ref&gt;

== Timeline ==

=== 1870s ===
'''1872 February 8''', Richard Bourke, 6th Earl of Mayo was assassinated while inspecting a &quot;convict settlement at Port Blair in the Andaman Islands ... by Sher Ali Afridi, a former Afghan soldier.&quot;&lt;ref&gt;{{Cite journal|date=2024-12-01|title=Richard Bourke, 6th Earl of Mayo|url=https://en.wikipedia.org/wiki/Richard_Bourke,_6th_Earl_of_Mayo|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Richard_Bourke,_6th_Earl_of_Mayo.&lt;/ref&gt; The Hon. Algernon's brother Dermot became the 7th Earl at 19 years old.

'''1876 November 24, Friday''', the Hon. Algernon Bourke was one of 6 men (2 students, one of whom was Bourke; 2 doctors; a tutor and another man) from Cambridge who gave evidence as witnesses in an inquest about the death from falling off a horse of a student.&lt;ref&gt;&quot;The Fatal Accident to a Sheffield Student at Cambridge.&quot; ''Sheffield Independent'' 25 November 1876, Saturday: 7 [of 12], Col. 5a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000181/18761125/040/0007. Print title: ''Sheffield and Rotherham Independent'', n. p.&lt;/ref&gt;

'''1879 December 27, Saturday – 29, Monday''', Algernon Bourke was in Kilrush as a Local Government Board Inspector:&lt;blockquote&gt;Among many distinguished visitors at the Vandeleur Arms Hotel, Kilrush this week was the Hon. Algernon Bourke Local Government Board Inspector who arrived on Saturday, and sojourned there until 2 o'clock on Monday, when the honourable gentleman left by Steamer tor Limerick.&lt;ref&gt;&quot;Fashionable Intelligence.&quot; ''Kilrush Herald and Kilkee Gazette'' 01 January 1880, Thursday: 2 [of 5], Col. 3a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003186/18800101/011/0002. Print title ''Kilrush Herald'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;

=== 1880s ===
'''4 February 1880, Wednesday''', Hon. Algernon Bourke attended the [[Social Victorians/Timeline/1880#Grand Ball at Palmerstown House Hosted by the Earl of Mayo|grand ball at Palmerstown House hosted by the Earl of Mayo]]. 

'''1880 March 30, Tuesday''', Algernon Bourke was working in the judicial system in Newcastle, County Limerick, possibly as Poorhouse Inspector:&lt;blockquote&gt;A sworn enquiry was held to-day at the Workhorse, Newcastle West, by the Hon Algernon Bourke, L.G.I., to enquire into charges preferred by Dr. Pierce, Medical Office, against Dr. O'Shaughnesay. The enquiry was adjourned till Thursday next. Mr Moran, sol., Rathkeale, was engaged for Dr. O'Shaughnessy.&lt;ref&gt;&quot;Sworn Enquiry.&quot; &quot;Limerick County. Newcastle West Intelligence.&quot; ''Bassett's Chronicle'' 31 March 1880, Wednesday: 3 [of 4], Col. 3b–c [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003471/18800331/044/0003.  Print title ''Bassett's Daily Chronicle'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;'''1880 April 17, Saturday''', in-jokes dominate this report mentioning Algernon Bourke in the context of the Kildare and National Hunt races in Dublin:&lt;blockquote&gt;And in mopy Upper Mount-street, where young Algernon Bourke, of the Onety-oneth, had promised to call for, and afterwards spin down to the races in his mail phaeton, the Blake girls; and in fastidious Fitzwilliam-place, and exclusive &quot;Murryan-squeer,&quot; from which dashing army men, in their neatly-appointed, well horsed drags were to &quot;tool&quot; down sweet young Dublin lasses of the ''crême d la crême'' [sic], many an anxious forecasting of the weather was taken, lest by an unpropitious shower that last triumph of Mrs. Manning, or the Forrests, or Miss Sedford, or any of the ''grandes dames de la mode'' should be rendered as worthless as a Confederate &quot;greenback.&quot; But by ten o'clock all doubts were happily set aside, and up struck the lovely April day in all its spring-time glory and then the road, oh, the road!&lt;ref&gt;&quot;To Punchestown and Back by the Old Road.&quot; ''Illustrated Sporting and Dramatic News'' 17 April 1880, Saturday: 6 [of 24], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001857/18800417/013/0006. Same print title, p. 102.&lt;/ref&gt;&lt;/blockquote&gt;'''1881 May 10, Tuesday''', Algernon Bourke attended the [[Social Victorians/Timeline/1881#1881 May 10, Tuesday|wedding of Marion Lascelles, eldest daughter of the Hon. Egremont W. Lascelles, brother of the Earl of Harewood, and Lieutenant Henry Dent Brocklehurst, of the Second Life Guards, nephew of Mr. Philip Brocklehurst, of Swithamley Park, Macclesfield]]. His gift was an &quot;old enamelled watch set in pearls.&quot;&lt;ref&gt;&quot;Nuptial Rejoicings at Middlethorpe Manor. Marriage of Miss Lascelles and Lieut. Brocklehurst.&quot; ''Yorkshire Gazette'' 14 May 1881, Saturday: 9 [of 12], Cols. 3a–4a [of 6]. ''British Newspaper Archive''https://www.britishnewspaperarchive.co.uk/viewer/bl/0000266/18810514/057/0009. Print same title and p.&lt;/ref&gt;

'''1881 May 23, Monday, 2:00 p.m.''', Algernon Bourke is listed among the Honourables at the [[Social Victorians/Timeline/1881#Queen's Levee at St. James's Palace|Queen's Levee at St. James's Palace]].

'''1881 July 14, Thursday afternoon, beginning about 2 p.m.''', Algernon Bourke was invited to a Garden Party at Marlborough House hosted by [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]]. Members of the family of the [[Social Victorians/People/Mayo|Earl of Mayo]] were also among the 1,500 or so invited guests.

'''1881 July 22, Friday''', Algernon Bourke was invited to an [[Social Victorians/Timeline/1881#22 July 1881, Friday|evening party at Marlborough House hosted by the Prince and Princess of Wales]].

'''1881 September 17, Saturday''', Algernon Bourke was reported among the company at Doncaster during race week.&lt;ref&gt;&quot;List of the Company.&quot; ''York Herald'' 17 September 1881, Saturday: 8 [of 16], Col. 4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000499/18810917/183/0008. Same print title and p.&lt;/ref&gt;

'''1881 November 22, Tuesday''', Algernon Bourke was sued in Dublin by Henry Naylor because he &quot;had declined to pay&quot; for a £35 piano.&lt;ref&gt;&quot;Henry Naylor v. the Hon. Algernon Bourke.&quot; &quot;Exchequer Division.&quot; &quot;High Court of Justice.&quot; ''Belfast Morning News'' 23 November 1881, Wednesday: 3 [of 4], Col. 8a [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000428/18811123/015/0003. Same print title, n.p.&lt;/ref&gt;

'''1881 December 8, Thursday''', Algernon Bourke was part of a [[Social Victorians/Timeline/1881#Battue at Palmerstown|battue at Palmerstown]], when the group bagged 172 pheasants, hares and rabbits.

'''1882 March 7, Tuesday''', Algernon Bourke attended the [[Social Victorians/Timeline/1882#7 March 1882, Tuesday|fashionable wedding of Reginald Chandos-Pole and Violet Denison]].

'''1882 March 15, Wednesday''', Algernon Bourke attended [[Social Victorians/Timeline/1882#The Marchioness of Salisbury's Assembly|the Marchioness of Salisbury's first reception of the season]].

'''1882 July 13, Thursday''', Algernon Bourke was invited to the [[Social Victorians/1882-07-13 Marlborough House Garden Party|Garden Party at Marlborough House for Queen Victoria]] hosted by [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]]. The more than 1,000 people invited also included a number of people from the family of the [[Social Victorians/People/Mayo|Earl of Mayo]].

'''1882 September 19''', Alexander M. Broadley in Tunis received a letter from Algernon Bourke asking him to provide legal counsel to defend prisoner-of war Arábi in Egypt, with Wilfred Scawen Blunt, which he did.&lt;ref name=&quot;:24&quot; /&gt; (2)

'''1882 September 28, Saturday''', Algernon Bourke attended the [[Social Victorians/Timeline/1882#The Wedding of John M'Donald and Georgiana Lambart|wedding of John M'Donald and Georgiana Lambart]].

'''1883 March 21, Wednesday''', the Evening Irish Times announced that Algernon Bourke &quot;has arrived at Kingstown from England.&quot;&lt;ref&gt;&quot;Court and Fashion.&quot; ''Evening Irish Times'' 21 March 1883, Wednesday: 7 [of 8], Col. 5a [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003464/18830321/086/0007. Same print title and p.&lt;/ref&gt;

'''23 July 1883, Monday, noon''', the Hon. Algernon Bourke was invited to a [[Social Victorians/Timeline/1883#Garden Party at Marlborough House, at Noon|garden party at Marlborough House]] hosted by the Prince and Princess of Wales.

'''31 October 1883, Wednesday''', Algernon Bourke attended the wedding of [[Social Victorians/Timeline/1883#Wedding of Lady Cecelia Hay and Captain George Webbe|Lady Cecelia Hay and Captain George Webbe]].&lt;p&gt;

'''1884 February 16, Saturday''', Algernon Bourke attended [[Social Victorians/Timeline/1884#16 February 1884, Saturday|the funeral of Thomas Chenery, editor of the ''Times'']].

'''1884 April 4, Saturday''', Algernon Bourke was (may have been?) one of the [[Social Victorians/Timeline/1884#5 April 1884, Saturday|&quot;Supporters of the Pall&quot; at the funeral]] of [[Social Victorians/People/Leopold|Prince Leopold George Duncan Albert, Duke of Albany]] at St. George's, Windsor.

'''1884 April 26, Saturday''', Algernon Bourke attended a [[Social Victorians/Timeline/1884#26 April 1884, Saturday|dinner party at the Lord Mayor's Mansion House for conservatives to meet Sir Stafford Northcote]].

'''1884 May 3, Saturday''', the &quot;Rochester Conservatives&quot; announced that they would &quot;bring forward the Hon. Algernon Bourke, brother of Lord Mayo, as their second candidate,&quot;&lt;ref&gt;&quot;Election Intelligence.&quot; ''Yorkshire Gazette'' 03 May 1884, Saturday: 4 [of 12], Col. 6a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000266/18840503/011/0004.&lt;/ref&gt; but because he would not be the first candidate, Bourke declined.&lt;ref&gt;&quot;Rochester.&quot; London ''Daily Chronicle'' 09 May 1884, Friday: 3 [of 8], Col. 8b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005049/18840509/049/0003.&lt;/ref&gt;

'''1884 June 18, Wednesday''', Mr. Algernon Bourke was on a committee to watch a [[Social Victorians/Timeline/1884#18 June 1884, Wednesday|Mr. Bishop's &quot;thought-reading&quot; experiment]], which was based on a challenge by Henry Labourchere made the year before. This &quot;experiment&quot; took place before a fashionable audience.

'''1884 July 25, Friday, afternoon''', the Hon. Algernon Bourke was invited to a [[Social Victorians/Timeline/1884#Garden Party at Marlborough House hosted by the Prince and Princess of Wales|Garden Party at Marlborough House hosted by the Prince and Princess of Wales]].

'''1885 January 22, Thursday''', Algernon Bourke's gift to [[Social Victorians/Timeline/1885#Wedding of George Buckle and Alicia Payn|George Buckle and Alicia Payn for their wedding]] was an antique cabinet.

'''1885 July 7, Tuesday''', Algernon Bourke attended [[Social Victorians/Timeline/1885#7 July 1885, Tuesday|Eva Bourke's wedding to Windham Wyndham-Quin]] at St. Mary Abbots, Kensington.

'''1885 July 13, Monday''', Algernon Bouurke was at Victoria Station as part of the [[Social Victorians/Timeline/1885#Arrival of Lord Wolseley in London from Egypt|crowd greeting Lord Wolseley on his return from Egypt]].

'''1885 July 24, Friday''', the Hon. Algernon Bourke was invited to a [[Social Victorians/1885-07-24 Marlborough House Ball|ball at Marlborough House]] hosted by the Prince and Princess of Wales.

'''1885 September 26, Saturday''', Algernon Bourke took part in the [[Social Victorians/Timeline/1885#26 September 1885, Saturday|Ealing Conservative Club fete and meeting]] supporting Salisbury's government and condemning &quot;the dictates of one man&quot; — Gladstone — for Gordon's death.

'''1885 October 3, Saturday''', the Hon. Algernon Bourke was named as the Conservative candidate for Clapham in the Battersea and Clapham borough after the Redistribution Bill determined the electoral districts for South London.&lt;ref&gt;&quot;South London Candidates.&quot; ''South London Press'' 03 October 1885, Saturday: 9 [of 16], Col. 5b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000213/18851003/096/0009. Print p. 9.&lt;/ref&gt; On Sunday 15 November 1885 the London ''Weekly Dispatch'' supported Moulton, the Liberal candidate, who ultimately won the election:&lt;blockquote&gt;

Though a successful lawyer, Mr. Moulton is much more than that. He is a thorough and independent student of political science, who may be trusted to do good service to the Liberal cause with brain as well as with tongue. It will be matter for hearty congratulation if he defeats the Hon. Algernon Henry Bourke, who is a dashing and unscrupulous young Tory, and a nephew of the well-known politician with the same surname.&lt;ref&gt;&quot;The Political Campaign in London. VI. — The South-West Divisions.&quot; ''Weekly Dispatch'' (London) 15 November 1885, Sunday: 9 [of 16], Col. 3c [of 4]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003358/18851115/069/0009. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

On Saturday 21 November 1885 the ''South London Press'' reported on posters for Bourke's candidacy:&lt;blockquote&gt;

The Hon. Algernon Bourke, Conservative candidate for Clapham, has a very industrious billsticker, who pastes up his patron’s bills in every possible place where they can be seen to advantage. It is unfortunate, however, that choosing the flank wall of an auctioneer’s the modern &quot;Sam Slap&quot; has produced some curious combinations, such as — &quot;Vote for Bourke,&quot; &quot;Now on View;&quot; &quot;Electors of Clapham, Vote for Mr. Bourke, and&quot; &quot;Be Sold Without Reserve;&quot; &quot;Mr, Bourke will&quot; &quot;Advance Money to&quot; &quot;the Electors of Clapham;&quot; &quot;Great Conservative Meeting. The British Constitution will be&quot; &quot;Offered for Sale this Evening,&quot; &amp;c.&lt;ref&gt;&quot;Pick-up Notes.&quot; ''South London Press'' 21 November 1885, Saturday: 10 [of 16], Col. 1b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000213/18851121/155/0010. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

'''1885 November 3, Tuesday, 11:00 a.m.''', Algernon Bourke attended the [[Social Victorians/Mayo-Ponsonby Wedding 1885-11-03|wedding of his brother, Dermot, 7th  Earl of Mayo and Geraldine Ponsonby]]. He gave them 2 Sheraton secretaires.

'''1886 January 5, Tuesday, late''', the Hon. Algernon Bourke attended the [[Social Victorians/Timeline/1886#Twelfth Night|Twelfth Night celebration at the Drury Lane theatre]].

'''1886 March 13, Saturday evening''', an Hon. Mr. Bourke attended a [[Social Victorians/1886-03-13 Reception at the French Embassy|reception at the French Embassy]], possibly Algernon Bourke or possibly [[Social Victorians/People/Mayo|one of his brothers]].

'''1886 July 10, Saturday''', Hon. Algernon Bourke was invited to a [[Social Victorians/Timeline/1886#Garden Party at Marlborough House Given to the Queen|garden party at Marlborough House given to the Queen]]. Guendoline Sloane Stanley is not mentioned but Mr. and Mrs. Hans Sloane Stanley are, as are Mr. and Mrs. F. Sloane Stanley.&lt;p&gt;

'''1886 July 21, Wednesday''', Algernon Bourke was invited to the [[Social Victorians/1886-07-21 Marlborough House Ball|Ball at Marlborough House]], as were a [[Social Victorians/People/Bourke#The Sloane-Stanleys 2|Mr. and Mrs. F. Sloane-Stanley]], possibly the parents of Guendoline Sloane-Stanley (if the &quot;F&quot; is a mistake), who married Bourke on 15 December 1887. Guendoline is not mentioned as having been invited.

'''1886 July 27, Tuesday''', Algernon Bourke attended a service honoring a memorial at St. Paul's for his father, who had been assassinated.&lt;ref&gt;&quot;Memorial to the Late Earl of Mayo.&quot; ''Northern Whig'' 28 July 1886, Wednesday: 6 [of 8], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000434/18860728/143/0006. Print p. 6.&lt;/ref&gt;

'''1886 September 2, Thursday''', Mr. Algernon Bourke was part of a group of mostly aristocratic men taking part in [[Social Victorians/Timeline/1886#Augustus Harris's A Run of Luck|a &quot;trial-rehearsal&quot; as part of Augustus Harris's production]] ''A Run of Luck'', about sports.

'''1886 October 2, Saturday''', the Duke of Beaufort and the Hon. Algernon Bourke arrived in Yougal: &quot;His grace has taken a residence at Lismore for a few weeks, to enjoy some salmon fishing on the Blackwater before the close of the season.&quot;&lt;ref&gt;&quot;Chippenham.&quot; ''Wilts and Gloucestershire Standard'' 02 October 1886, Saturday: 8 [of 8], Col. 6a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001955/18861002/142/0008. Print p. 8.&lt;/ref&gt;

'''1886 October 11, Monday''', Algernon Bourke may have been taking part in a [[Social Victorians/Timeline/1886#Performance of Run of Luck|performance of ''Run of Luck'' at the Drury Lane]].

'''1886 October 23, Saturday''', Algernon Bourke was [[Social Victorians/Timeline/1886#Party at Wemyss Castle, Fife|staying at Wemyss Castle, Fife]].

'''1886 December 30, Thursday''', Algernon Bourke was back in London and attending the [[Social Victorians/Timeline/1886#Augustus Harris's The Forty Thieves|&quot;Forty Thieves&quot; pantomime at the Drury Lane Theatre]].

'''1887 January 5, Wednesday''', the Hon. Algernon Bourke was one of the chief mourners at the [[Social Victorians/Timeline/1887#Funeral of Lady Margaret Harriett Bourke|funeral of Lady Margaret Harriett Bourke]].

'''1887 March 1, 2:00 p.m.''', Algernon Bourke is listed among the Messieurs attending the [[Social Victorians/Timeline/1887#Queen's Levee at St. James's Palace|Queen's Levee at St. James's Palace]].&lt;p&gt;

'''1887 May''', a &quot;signalling incident&quot; in 1907 [[Social Victorians/Timeline/1887#May 1887|caused the Waterford ''Evening News'' to recall a similar event]] that had occurred 20 years earlier, in which Algernon Bourke, as special correspondent for the ''Times'', caused the incident to be publicized:&lt;blockquote&gt;During the manoeuvres in connection with the 1887 Jubilee of Queen Victoria a signal was observed going up from [[Social Victorians/People/Beresford|Lord Charles [Beresford]]]'s ship. It was a message to his wife, Lady Beresford, to the effect that, as he should be late for dinner, she was not to wait. Beyond the hilarity this domestic signal evoked, nothing more would have been heard of it, but Mr. Algernon Bourke (Lord Mayo's brother) was acting as special correspondent for the &quot;Times,&quot; and that paper the next morning contained a full and humorous report of the incident. Then there was trouble.&lt;ref&gt;&quot;Signalling Incident.&quot; ''Evening News'' (Waterford) 13 November 1907, Wednesday: 1 [of 4], Col. 6c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004557/19071113/021/0001.&lt;/ref&gt;&lt;/blockquote&gt;

'''1887 June 15, Wednesday''', the Hon. Algernon Bourke attended a [[Social Victorians Foreign Office Reception 1887-06-15|reception at the Foreign Office in honor of Queen Victoria's Golden Jubilee]].

'''1887 July 6, Wednesday''', Algernon Bourke was invited to and, presumably, attended the State Ball at Buckingham Palace.&lt;ref&gt;&quot;The State Ball at Buckingham Palace.&quot; ''Morning Post'' 08 July 1887, Friday: 3 [of 8], Col. 5a–6c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18870708/013/0003. Same print title and p.&lt;/ref&gt; (Col. 1c)

'''1887 August 6, Saturday''', the ''Brighton Gazette'' says that the &quot;Hon. Mrs and Mr Algernon Bourke&quot; were staying at the Royal Crescent Hotel in Brighton, but they didn't marry until 15 December 1887.&lt;ref&gt;&quot;Royal Crescent Hotel.&quot; ''Brighton Gazette'' 6 August 1887, Saturday: 3 [of 8], Col. 5c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000938/18870806/047/0003. Print title ''Brighton Gazette and Sussex Telegraph'', p. 3.&lt;/ref&gt; Perhaps an elder relative, because she is mentioned first?

'''1887 November 9, Wednesday''', the ''Hampshire Advertiser County Newspaper'' announced that&lt;blockquote&gt;A marriage is arranged, and will take place early in January, between Mr. Algernon Bourke, third son of the late Earl of Mayo, and Miss Guendolen Sloane Stanley, only daughter of Mr. Hans Sloane Stanley, of Paultons.&lt;ref&gt;&quot;Romsey, Nov. 9.&quot; ''Hampshire Advertiser'' 9 November 1887, Wednesday: 3 [of 4], Col. 7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18871109/034/0003. Print title ''Hampshire Advertiser County Newspaper'', p. 3.&lt;/ref&gt;&lt;/blockquote&gt;Shortly after, the papers announced that the wedding would not take place.

'''1887 December 15, Thursday''', Hon. [[Social Victorians/Timeline/1887#Wedding of Algernon Bourke and Gwendolen Sloane Stanley|Algernon Bourke and Guendoline Stanley were married at St. Paul's]], Knightsbridge, by Bourke's uncle the Hon. and Rev. George Bourke. Only family members attended because of &quot;the recent death of a near relative of the bride.&quot;&lt;ref&gt;&quot;Court Circular.&quot; ''Morning Post'' 16 December 1887, Friday: 5 [of 8], Col. 7c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18871216/066/0005.&lt;/ref&gt; Who the &quot;near relative of the bride&quot; was not in her nuclear family, and perhaps that explains the cancellation of the wedding and then the changing of the wedding date and not some problem in the couple.

'''1888 – 1899 January 1''', the Hon. Algernon Bourke was &quot;proprietor&quot; of [[Social Victorians/London Clubs#White's|White's Club, St. James's Street]].&lt;ref name=&quot;:9&quot;&gt;&quot;The Hon. Algernon Bourke's Affairs.&quot; ''Eastern Morning News'' 19 October 1899, Thursday: 6 [of 8], Col. 7c [of7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001152/18991019/139/0006. Print p. 6.&lt;/ref&gt;

'''1888 January 21, Saturday''', Guendoline Bourke attended the wedding of [[Social Victorians/Timeline/1888#Hamilton-Ewart Wedding|Florence Ewart and Henry Hamilton]].

'''1888 March 7, Wednesday''', assuming that this date is not a week after the actual date, [[Social Victorians/People/Beresford|Lady Charles Beresford]] held a [[Social Victorians/Timeline/1888#1888 March 7, Wednesday|notable and well-attended &quot;at home&quot;]] that Guendoline Bourke attended, reported for being dressed in white and being among the beautiful women present.

'''6 April 1888, Friday''', Algernon and Guendoline Bourke attended the [[Social Victorians/Timeline/1888#6 April 1888, Friday|New Forest United Hunt ball at the New Forest Hall, Lyndhurst]].

'''1888 May 2, Wednesday''', Algernon and Guendoline Bourke attended the [[Social Victorians/Timeline/1888#The Marchioness of Salisbury's Reception|Marchioness of Salisbury's reception]] at the Salisbury home on Arlington-street.

'''1888 May 22, Tuesday''', the Dowager Countess of Mayo presented Guendoline Bourke at the [[Social Victorians/Timeline/1888#Queen's Drawing Room|Queen's drawing-room]] hosted by the Princess of Wales. This is Guendoline Bourke's dress:&lt;blockquote&gt;Empire robe de cour of white satin duchesse, lined with rich pink silk, sufficiently bright to give a beautiful shell-like tint through the satin; tulle underdress, with upper skirt, embroidered with pearl, and caught up in Greek folds with large pink Tosca roses; white satin bodice, with Josephine pink sash tied at side, Headdress, veil and plumes; ornaments, diamonds.&lt;ref&gt;&quot;Dresses at the Drawing-Room.&quot; ''Epsom Journal'' 22 May 1888, Tuesday: 3 [of 6], Col. 5b–c [of 6]. ''British Newspapers Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004837/18880522/034/0003. Print: title ''Local Journal'', p. 3.&lt;/ref&gt;&lt;/blockquote&gt;

Another description:&lt;blockquote&gt;Mrs. Algernon Bourke's train was of white satin lined with pink, which showed through with charmingly shell-like effect. The dress, fashioned after those of the Empire period, was of white satin embroidered with pearls. A very broad sash of pink satin made the waist seem quaintly short, a trying thing to any but the young and tall, both of which qualifications Mrs. Bourke most happily possesses. She carried a lovely posy of La France roses.&lt;ref&gt;&quot;Gossip on Dress.&quot; ''Boston Spa News'' 25 May 1888, Friday: 2 [of 8], Col. 1b–2b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003395/18880525/014/0002. Print title The News, n.p.&lt;/ref&gt; (Col. 1c)&lt;/blockquote&gt;'''1888 June 8, Friday''', Hon. Algernon and Mrs. Bourke attended a [[Social Victorians/Timeline/1888#Dinner and Dance Hosted by Lord and Lady Wimborne at Hamilton House|dinner and dance Hosted by Lord and Lady Wimborne at Hamilton House]] featuring Prince and Princess Christian of Schleswig-Holstein, and for the ball, the King of Sweden and Norway and the Prince and Princess of Wales and their daughters were present.

'''1888 June 19, Tuesday''', Guendoline Bourke was one of the principal guests at the wedding of [[Social Victorians/Timeline/1888#19 June 1888, Tuesday|Captain Philip Green and Miss Mabel Emilie Scott]].

'''1888 July 26''', [[Social Victorians/People/Montrose|Caroline Graham Stirling-Crawford]] (known as Mr. Manton for her horse-breeding and -racing operations) and Marcus Henry Milner married.&lt;ref name=&quot;:12&quot;&gt;&quot;Hon. Caroline Agnes Horsley-Beresford.&quot; {{Cite web|url=https://thepeerage.com/p6863.htm#i68622|title=Person Page|website=thepeerage.com|access-date=2020-11-21}}&lt;/ref&gt; According to the ''Nottingham Evening Post'' of 31 July 1888,&lt;blockquote&gt;LONDON GOSSIP.

(From the ''World''.)

The marriage of &quot;Mr. Manton&quot; was the surprise as well the sensation of last week. Although some wise people noticed a certain amount of youthful ardour in the attentions paid by Mr. Marcus Henry Milner to Caroline Duchess of Montrose at '''Mrs. Oppenheim's ball''', nobody was prepared for the sudden ''dénouement''; '''and it''' were not for the accidental and unseen presence [[Social Victorians/People/Mildmay|a well-known musical amateur]] who had received permission to practice on the organ, the ceremony performed at half-past nine on Thursday morning at St. Andrew's, Fulham, by the Rev. Mr. Propert, would possibly have remained a secret for some time to come. Although the evergreen Duchess attains this year the limit of age prescribed the Psalmist, the bridegroom was only born in 1864. Mr. &quot;Harry&quot; Milner (familiarly known in the City as &quot;Millions&quot;) was one of the zealous assistants of that well-known firm of stockbrokers, Messrs. Bourke and Sandys, and Mr. Algernon Bourke, the head of the house (who, of course, takes a fatherly interest in the match) went down to Fulham to give away the Duchess. The ceremony was followed by a ''partie carrée'' luncheon at the Bristol, and the honeymoon began with a visit to the Jockey Club box at Sandown. Mr. Milner and the Duchess of Montrose have now gone to Newmarket. The marriage causes a curious reshuffling of the cards of affinity. Mr. Milner is now the stepfather of the [[Social Victorians/People/Montrose|Duke of Montrose]], his senior by twelve years; he is also the father-in-law of [[Social Victorians/People/Lady Violet Greville|Lord Greville]], Mr. Murray of Polnaise, and [[Social Victorians/People/Breadalbane|Lord Breadalbane]].&lt;ref name=&quot;:8&quot; /&gt;&lt;/blockquote&gt;

'''1888 December 1st week''', according to &quot;Society Gossip&quot; from the ''World'', the Hon. Algernon Bourke was suffering from malaria, presumably which he caught when he was in South Africa:&lt;blockquote&gt;I am sorry to hear that Mr. Algernon Bourke, who married Miss Sloane-Stanley a short time ago, has been very dangerously ill. Certain complications followed an attack of malarian fever, and last week his mother, the Dowager Lady Mayo, and his brother, Lord Mayo, were hastily summoned to Brighton. Since then a change for the better has taken place, and he is now out of danger.&lt;ref&gt;&quot;Society Gossip. What the ''World'' Says.&quot; ''Hampshire Advertiser'' 08 December 1888, Saturday: 2 [of 8], Col. 5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18881208/037/0002. Print title: ''The Hampshire Advertiser County Newspaper''; print p. 2.&lt;/ref&gt;&lt;/blockquote&gt;

'''1888 December 20, Thursday''', the Sloane-Stanley family, including Guendoline Bourke, attended the [[Social Victorians/Timeline/1888#20 December 1888, Thursday|funeral of Hans Sloane Stanley]]. Algernon Bourke did not attend because he was still too ill.

'''1889 January 22, 2:30 p.m., Tuesday''', Algernon and Guendoline Bourke sent a gift for the [[Social Victorians/Cecil Lambton Wedding 1889 January 22|wedding of Lady Eleanor Lambton and Lord Robert]] Cecil, a pair of antique mirrors.

'''1889 May 18, Saturday''', Algernon Bourke attended the [[Social Victorians/Timeline/1889#18 May 1889, Saturday|opening of the Italian Opera season at Covent Garden]].

'''1889 May 27, Monday, 11 p.m.''', the dancing commenced at [[Social Victorians/Timeline/1889#The Queen's State Ball at Buckingham Palace|the Queen's State Ball at Buckingham Palace]], with both the Hon. Algernon and the Hon. Guendoline Bourke present.

'''1889 June 8, Saturday''', the Hon. Algernon Bourke contributed some art he owned to the collection of the Royal Institute of Painters in Water-Colours' [[Social Victorians/Timeline/1889#8 June 1889, Saturday|exhibition of &quot;the works of the 'English Humourists in Art.'&quot;]]

'''1889 July 2, Tuesday''', Guendoline and Algernon Bourke sat in the Muriettas' box at a [[Social Victorians/Timeline/1889#The Shah at a Covent Garden Opera Performance|gala performance at Covent Garden also attended by the Prince and Princess of Wales, a number of other royals and the Shah]].&lt;p&gt;

'''1889 27 July, Saturday''', Guendoline and Algernon Bourke attended a [[Social Victorians/Timeline/1889#Garden Party Hosted by Mr. and Mrs. Augustus Harris|garden party hosted by Mr. and Mrs. Augustus Harris]], which was attended by a people from the theatre and arts worlds.&lt;p&gt;

'''1889 December 2, Monday''', Guendoline Bourk's mother, Emilie Sloane-Stanley, married James Shelly Bontein:&lt;blockquote&gt;&lt;p&gt;

BONTEIN—STANLEY — December 2, at St. George's, Hanover Square, London, by the Rev. G. S. de Sansmarez, James Shelly, only son of the late James Bontein, Gentleman Usher and Clerk of the Robes to the Queen, to Emilie Josephine, widow of Hans Sloane Stanley, of Paultons.&lt;ref name=&quot;:18&quot; /&gt;&lt;/blockquote&gt;'''1889 December 17, Tuesday''', Hon. Algernon and Mrs. Bourke gave a gift to [[Dangan-Neville Wedding|Lady Violet Nevill for her wedding to Henry Wellesley, Viscount Dangan]] and so were probably in attendance.

=== 1890s ===
'''1890 January 9, Thursday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1890#The York Hunt Ball|fancy-dress Hunt Ball in York]]. She&lt;blockquote&gt;looked a picture in a Gainsborough gown. The white satin skirt was flounced with sable and veiled with ''chiffon'', the setuage of which was left to show without being hemmed up. There was a broad sash of rose-pink silk and each buttonhole was filled round with crimped lisse.&lt;ref&gt;&quot;Our London Letter.&quot; ''Irish Society'' (Dublin) 11 January 1890, Saturday: 17 [of 24], Col. 2a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001939/18900111/042/0017. Same print title, p. 29.&lt;/ref&gt;&lt;/blockquote&gt;'''1890 February''' '''12, Wednesday''', Hon. Algernon and Mrs. Bourke attended [[Social Victorians/Timeline/1890#Lady Constance Leslie's Reception|Lady Constance Leslie's reception]] at her house in Stratford-place.

'''1890 April 9, Wednesday''', Guendoline Bourke attended [[Social Victorians/Timeline/1890#The New Forest United Hunt Ball|the New Forest United Hunt Ball]].

'''1890 June 3, Tuesday''', Guendoline Bourke attended the 2:30 p.m. [[Social Victorians/Timeline/1890#Münster-Hay Wedding|wedding of Count Alexander Münster and Lady Muriel Henrietta Constance Hay]]. She is also listed as having attended a [[Social Victorians/Timeline/1890#Dinner and Concert Hosted by Mrs. Arthur Williams and Ball by Mrs. Menzies|ball hosted by Mrs. J. Menzies (daughter of Mrs. Arthur Wilson)]] that Prince Eddie, the Duke of Clarence and Avondale, also attended, that night.  

'''1890 July 4, Friday, 11 p.m.''', the Hon. Algernon and Guendoline Bourke attended [[Social Victorians/Timeline/1890#The Queen's State Ball at Buckingham Palace|the Queen's State Ball at Buckingham Palace]]. The dancing commenced shortly after 11:00. 

'''1890 July 15, Tuesday''', Hon. Algernon and Mrs. Bourke were invited to a [[Social Victorians/Timeline/1890#Garden Party at Marlborough House to Meet the Queen|garden party at Marlborough House to meet the Queen]]. 

'''1890 July 19, Saturday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1890#Wedding of James Francis Harry St. Clair-Erskine and Violet Aline Vyner|wedding of James Francis Harry St. Clair-Erskine and Violet Aline Vyner]], the two of them giving &quot;four small silver dessert dishes&quot; and Guendoline giving an &quot;enamel and diamond pin.&quot;&lt;ref&gt;&quot;Marriage of Lord Loughborough with Miss Vyner.&quot; ''Fife Free Press'' 26 July 1890, Saturday: 2 [of 8], Col. 1a–2b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001110/18900726/015/0002. Same print title and p.&lt;/ref&gt; (Col. 2b)

'''1890 July 24, Thursday''', Algernon and Guendoline Bourke attended a [[Social Victorians/Timeline/1890#Dinner and Dance Hosted by Lord Alington|dance hosted by Lord Alington]] attended also by the Prince and Princess of Wales and Princesses Victoria and Maud.

'''1890 September 6, Saturday''', the ''Country Gentleman'' (as it was called at the time) reported that &quot;Muckross, the only deer forest in Ireland, it may be said, has this year been rented by Mr. Algernon Bourke, who will next week be joined there for the stalking season by his brother, Lord Mayo.&quot;&lt;ref&gt;&quot;Shooting. Moors, Forests, and Fishings.&quot; ''Sporting Gazette'' 06 September 1890, Saturday: 11 [of 38], Col. 1c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002525/18900906/065/0011. Print: ''Country Gentleman'', p. 1251.&lt;/ref&gt; On 11 October 1890 the ''St. James's Gazette'' says,&lt;blockquote&gt;The Earl of Durham has been staying at Muchross, county Kerry, on a visit to the Hon. A. Bourke, who has rented the celebrated shootings and fishings on that estate for the autumn.&lt;ref&gt;&quot;Court and Society.&quot; ''St James's Gazette'' 11 October 1890, Saturday: 12 [of 16], Col. 1b [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001485/18901011/064/0012. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;'''1890 October 25, Saturday''', the Hon. Algernon and Mrs. Bourke gave a gold-mounted box to [[Social Victorians/Loder De Vere Beauclerk Wedding|Lady Louise De Vere Beauclerk on her wedding to Gerald Loder, M.P.]], so they were probably present at the wedding, or at least the reception. Mrs. Bontein [sic Bontine], Guendoline's mother, gave a silver box, suggesting the relationship was through the women.

'''1890 November 29, 11:30 Saturday morning''', Algernon Bourke's gift for the [[Social Victorians/Dudley-Beckwith Wedding 1890-11-29|wedding of the Hon. Francis Dudley and Miss Forbes Beckwith]] was some cases of a Bordeaux wine: &quot;three dozen Cantenac, 1875 vintage.&quot;&lt;ref&gt;&quot;Marriage of Lord Leigh's Heir. Descriptive Sketch of the Ceremony, and Full List of Guests and Presents.&quot; ''Leamington Spa Courier'' 6 December 1890, Saturday: 6 [of 10], Cols. 1a–4a. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18901206/021/0006. Same print title and p.&lt;/ref&gt;{{rp|Col. 3b}} Guendoline Bourke is not listed as having been invited to the reception, but this list from the ''Leamington Spa Courier'' has some gaps.

'''1890 December 4, Thursday''', Guendoline and Algernon Bourke attended the [[Mure-Portal Wedding 1890-12-04|wedding of Miss Mure and Mr. S. J. Portal]]. Their gift is not recorded.

'''1891 January''', Algernon Bourke took party in a [[Social Victorians/Timeline/1891#Shooting Party in Kallarnet, Totton|shooting party in Kallarnet, Totton]].

'''1891 June 24, Wednesday''', the Hon. Algernon and Mrs. Bourke attended a [[Social Victorians/Timeline/1891#Dinner and Ball Hosted by Lord and Lady Wimborne|dinner and ball Hosted by Lord and Lady Wimborne]] featuring Princess Mary Adelaide, the Duke of Teck, and Princess Victoria.

'''1891 July 9, Thursday''', Algernon and Guendoline Bourke were invited to a [[Social Victorians/1891-07-09 Garden Party|large Garden Party at Marlborough House]] hosted by the [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]] in honor of Queen Victoria and the German Emperor and Empress. The more than 3,000 people invited also included a number of people from the [[Social Victorians/People/Mayo|family of the Earl of Mayo]].

'''1891 July 22, Wednesday''', Guendoline Bourke attended a [[Social Victorians/Timeline/1891#Dinner and Dance at Alington House|dance at the Earl and Countess Alington]]'s that also included the Prince and Princess of Wales.

'''1891 October 22, Thursday''', Hon. and Mrs. Bourke attended at least the reception of the [[Social Victorians/Timeline/1891#Le Strange Astley Wedding|Le Strange—Astley Wedding]], although perhaps the couple is not the Algernon Bourkes.

'''1891 November 22, Sunday''', the London ''Weekly Dispatch'' reports a performance by American &quot;Lady Magnet&quot; Mrs. Abbott, who claimed to be able to lift anybody using only her magnetic properties. An enthusiastic &quot;committee of some fifteen gentlemen presented a written and signed testimonial&quot; supporting Mrs. Abbott, &quot;the Hon. Algernon Bourke, Professor Atkinson, Dr. Hides, and three other doctors who prefer to remain incog., being among the signatories. All the medical gentlemen concerned assured the ''Evening News and Post'' reporter of their complete and unconditional surrender. One of them went so far as to say that he had come with the full determination of disbelieving, but had been quite able to act up to his resolve.&quot;&lt;ref&gt;&quot;The Lady Magnet. Draws Crowds of People Who Divide in Opinion about Her.&quot; ''Weekly Dispatch'' (London) 22 November 1891, Sunday: 16 [of 16], Cols. 3a–4b [of 4]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003358/18911122/203/0016. Print: same title and p.&lt;/ref&gt;

'''1892''', the Hon. Algernon Bourke privately published his ''The History of White's'', the exclusive gentleman's club.

'''1892 January 27, Saturday''', Algernon and Guendoline Bourke attended the very fashionable [[Social Victorians/Timeline/1892#The Wedding of Lord Henry Cavendish Bentinck, M.P., and Lady Olivia Taylour|wedding of Lord Henry Cavendish Bentinck, M.P., and Lady Olivia Taylour]]. Their gift was not noted in the list.

'''1892 February''' '''10, Wednesday''', Guendoline Bourke attended the [[Social Victorians/1892-02-10 Alington Leigh Wedding|very fashionable wedding of Henry, Lord Alington and Evelyn Henriette Leigh]] [[Social Victorians/1892-02-10 Alington Leigh Wedding|in St. Paul's, Knightsbridge]]

'''1892 April''' '''10, Wednesday, about 2:30 p.m.''', Guendoline Bourke attended [[Social Victorians/1892-02-10 Alington Leigh Wedding|the very fashionable wedding between Henry Sturt, Lord Alington and Evelyn Leigh]]. Her gift was a &quot;tortoiseshell and gold heart-shaped tray.&quot;&lt;ref name=&quot;:02&quot;&gt;&quot;Lord Alington to Miss Leigh.&quot; ''Gentlewoman'' 20 February 1892, Saturday: 21 [of 46], Cols. 1a–3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18920220/092/0021. Same print title, p. 237.&lt;/ref&gt; (Col. 3a) 

'''1892 June 25, Saturday''', the ''Gentlewoman''&lt;nowiki/&gt;'s &quot;Overheard by the Little Bird&quot; says &quot;That pretty Mrs. Algernon Bourke has been staying here, but returned to England in time for Ascot.&quot;&lt;ref&gt;Little Bird, The. &quot;Overheard by the Little Bird.&quot; ''Gentlewoman'' 25 June 1892, Saturday: 32 [of 60], Col. 3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18920625/157/0032. Same print title, p. 860.&lt;/ref&gt; 

'''1892 December 13, Tuesday''', the ''Gentlewoman'' says Guendoline Bourke is lovely in its coverage of [[Social Victorians/Timeline/1892#Wedding of Miss Eleanor M. Ewart and Captain Guy Withington|Eleanor M. Ewart and Captain Guy Withington's wedding]]. 

'''1892 December 22, Thursday''', Algernon Bourke attended the [[Social Victorians/Timeline/1892#22 December 1892, Thursday|monthly meeting of the Zoological Society in Hanover-square]].&lt;p&gt;

'''1893 February 11, Tuesday''', Algernon Bourke opened Willis's Restaurant:&lt;blockquote&gt;Mr. Algernon Bourke has in his time done many things, and has generally done them well. His recently published history of White's Club is now a standard work. White's Club itself was a few years ago in its agony when Mr. Bourke stepped in and gave it a renewed lease of life. Under Mr. Bourke's auspices &quot;Willis's Restaurant&quot; opened its doors to the public on Tuesday last in a portion of the premises formerly so well known as Willis's Rooms. This new venture is to rival the Amphitryon in the matter of cuisine and wines; but it is not, like the Amphitryon, a club, but open to the public generally. Besides the restaurant proper, there are several ''cabinets particuliers'', and these are decorated with the very best of taste, and contain some fine portraits of the Georges.&lt;ref&gt;&quot;Marmaduke.&quot; &quot;Letter from the Linkman.&quot; ''Truth'' 20 April 1893, Thursday: 25 [of 56], Col. 1a [of 2]. ''British Newspaper Archive'' [https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18930420/075/0025# https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18930420/075/0025]. Print p. 855.&lt;/ref&gt;&lt;/blockquote&gt;

'''1893 February 7, Tuesday''', Guendoline Bourke attended [[Social Victorians/Timeline/1893#1893 February 7, Tuesday|the reception after Lady Emily Cadogan's wedding]].

'''1893 February 20, Monday''', the Hon. Algernon Bourke is listed as having attended the [[Social Victorians/Timeline/1893#Queen's Levee at St. James's Palace|Queen's Levee at St. James's Palace]] held by the Prince of Wales; because wives generally are not listed, it seems likely Guendoline Bourke attended as well.

'''1893 February 28, Tuesday, 3:00 p.m.''', Guendoline Bourke attended a [[Social Victorians/Queens Drawing Room 1893-02-28|Queen's Drawing Room at Buckingham Palace]].&lt;p&gt;

'''1893 March 22, Wednesday''', Guendoline Bourke attended [[Social Victorians/Timeline/1893#22 March 1893, Wednesday|Lady Wimborne's reception]].

'''1893 April 1, Saturday''', Algernon Bourke published a letter to the editor of the ''Times'', reprinted in the ''Kildare Observer'', arguing against Gladstone's Home Rule bill on the grounds that Ireland would not be able to take out a loan on its own behalf because of its obligations to the U.K., including what was called its share of the national debt.&lt;ref&gt;&quot;Irish Unionist Alliance.&quot; ''Kildare Observer and Eastern Counties Advertiser'' 01 April 1893, Saturday: 6 [of 8], Col. 4c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001870/18930401/062/0006. Print: The ''Kildare Observer'', n.p.&lt;/ref&gt;

'''1893 May 13, Saturday''', Algernon Bourke was seen at [[Social Victorians/Timeline/1893#13 May 1893, Saturday|exhibitions of art and furniture for sale by Christie's and on display by Lord Clifden]].

'''1893 July 13, Thursday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1893#The Countess of Listowel's Garden Party|Countess of Listowel's Garden Party]] [[Social Victorians/Timeline/1893#The Countess of Listowel's Garden Party|at her residence, Kingston House, Princes-gate]], accompanied by Miss Adeane.

'''1893 July 14, Friday''', Guendoline Bourke attended [[Social Victorians/Sandown Races 1893-07-14|the races at Sandown]] wearing a dark-blue-and-white outfit and black hat that got described in the newspaper.

'''1893 August 1, Tuesday – August 4, Friday''', Guendoline Bourke, at least, was at [[Social Victorians/Timeline/1893#1 August 1893, Tuesday – 4 August 1893, Friday|the Goodwood races]], mentioned in the ''Gentlewoman'' for her beauty, although none of the dresses were noted.

'''1893 November 4–11, Wednesday–Saturday''', Guendoline Bourke was at a [[Social Victorians/Timeline/1893#Ralph and Mary Sneyd Hosted a Shooting Party|shooting party at Keele Hall hosted by Ralph and Mary Sneyd]].

'''1893 November 30, Thursday''', with Sir Walter Gilbey the Hon. Algernon Bourke &quot;assisted&quot; in &quot;forming [a] collection&quot; of engravings by George Morland that was exhibited at Messrs. J. and W. Vokins’s, Great Portland-street.&lt;ref&gt;&quot;The George Morland Exhibition at Vokins's.&quot; ''Sporting Life'' 30 November 1893, Thursday: 4 [of 4], Col. 4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000893/18931130/058/0004.&lt;/ref&gt;

'''1893 December 14, Thursday, afternoon''', Guendoline Bourke attended the [[Social Victorians/1893-12-14 Wedding Adele Grant and George, 7th Earl of Essex|wedding of American Adele Grant and George, 7th Earl of Essex]] and gave a &quot;pearl and gold box.&quot;&lt;ref name=&quot;:22&quot;&gt;&quot;Wedding of the Earl of Essex.&quot; ''Herts Advertiser'' 16 December 1893, Saturday; 8 [of 8], Col. 1a–4b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000415/18931216/125/0008. Print title: ''The Herts Advertiser and St Albans Times'', p. 8.&lt;/ref&gt;{{rp|Col. 3c}} Mr. and Mrs. Shelley Bontein also attended, and Mrs. Bontein gave a &quot;green leather bag and purse, with coronet and monogram in gold.&quot;&lt;ref name=&quot;:22&quot; /&gt;{{rp|3b}}

'''1894 January 27, Saturday''', Psyche in &quot;The Social Peepshow&quot; in the ''Gentlewoman'' reported on a [[Social Victorians/Timeline/1894#27 January 1894, Saturday|ball hosted by Lord and Lady Dunraven at Adare Manor]] that Guendoline Bourke attended.

'''1894 January 31, Wednesday''', Algernon and Guendoline Bourke, who was dressed more stylishly than most, attended the [[Social Victorians/Timeline/1894#Also 31 January 1894, Wednesday|Kildare Hunt Ball]] hosted by Dermot, [[Social Victorians/People/Mayo|Earl of Mayo]] and Geraldine, Countess of Mayo.

'''1894 February 24, Saturday''', ''The Field'' reported on a series of tennis matches; Algernon Bourke attended the one played at the Prince's Club.&lt;ref&gt;&quot;Tennis.&quot; ''Field'' 24 February 1894, Saturday: 39 [of 72], Col. 1c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002446/18940224/349/0039. Print title ''The Field, The Country Gentleman's Newspaper'', p. 249.&lt;/ref&gt;

'''1894 March 31, Saturday''', Psyche, in the &quot;Social Peepshow&quot; column in the ''Gentlewoman'', says that &quot;Mr. Algernon Bourke has still further embellished Willis's restaurant hard by [the St. James's Theatre], by the addition of some valuable old tapestry that lately came to the hammer at Christie's.&quot;&lt;ref&gt;Psyche. &quot;The Social Peepshow.&quot; ''Gentlewoman'' 31 March 1894, Saturday: 16 [of 56], Col. 2b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18940331/081/0016. Same print title, p. 408.&lt;/ref&gt;

'''1894 April 13, Friday''', Guendoline Bourke set sail on the [[Social Victorians/Timeline/1894#P. and O. Line S.S. Rome for Gibraltar|P. and O. Line ''S.S. Rome'' for Gibraltar]] along with her stepfather, Mr. Shelley Bontein, and her brother, Mr. Sloane Stanley.

'''31 May 1894, Thursday''', the Hon. Algernon and Mrs. Bourke attended the [[Social Victorians/Timeline/1894#Reception at Devonshire House|Duchess of Devonshire's reception at Devonshire House]].&lt;p&gt;

'''1894 June 18, Monday''', the London ''Echo'' reported that Algernon Bourke was [[Social Victorians/London Clubs#Brooks'|writing a history of Brooks' Club]].&lt;p&gt;

'''1894 June 20, Wednesday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1894#Princess Louise, Marchioness of Lorne Opened the Annual Sale of the Scottish Home Industries|Annual Sale of the Scottish Home Industries]]; her outfit was described in the article in ''Lady's Pictorial''.

'''1894 August 2, Thursday''', the column &quot;Overheard by the Little Bird&quot; says, &quot;At Willis' [restaurant] — 'What a smart cotillon Mr. and Mrs. Algernon Bourke gave on Thursday evening.&quot;&lt;ref&gt;Bird, The Little. &quot;Overheard by the Little Bird.&quot; ''Gentlewoman'' 04 August 1894, Saturday: 30 [of 56], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18940804/148/0030. Print title same, p. 144.&lt;/ref&gt; Willis's Restaurant, King-street, St. James's, was a restaurant Algernon Bourke opened in 1893.&lt;p&gt;

'''1894 September 7, Saturday''', Algernon and Guendoline Bourke were at a [[Social Victorians/Timeline/1894#7 September 1894, Saturday|shooting party at Witley]], which had been loaned to one of his brothers by William Ward, 2nd Earl of Dudley.&lt;p&gt;

'''1894 October 22, Thursday''', Algernon and Guendoline Bourke attended the [[Social Victorians/Timeline/1894#Wedding of Lord Connemara and Mrs. Coleman|luncheon after the wedding of Lord Connemara and Mrs. Coleman]].

'''1894 November 3, Saturday''', Psyche, in &quot;The Social Peepshow&quot; for the Gentlewoman, reported that Guendoline Bourke had been [[Social Victorians/Timeline/1894#3 November 1894, Saturday|seen shopping in London]].

'''1895 January 5, Saturday, 2:00 p.m.''', Algernon and Guendoline Bourke gave an old mother-of-pearl workbox to [[Wolverton-Ward Wedding 1895-01-05|Lady Edith Ward for her wedding to Frederick Glyn, Lord Wolverton]] and presumably attended the wedding and reception afterwards.&lt;p&gt;

'''1895 February 23, Saturday''', the Hon. Algernon Bourke attended the [[Social Victorians/Timeline/1895#23 February 1895, Saturday|fashionable wedding of Laurence Currie and Edith Sibyl Mary Finch]]. Guendoline Bourke is not listed as having attended, but she is not noted as absent, either. Daphne Bourke was born on 5 April 1895, probably explaining Guendoline's absence.

'''1895 March 24, Sunday – 30 March, Saturday''', Algernon Bourke was [[Social Victorians/Timeline/1895#24, Sunday – 30 March 1895, Saturday|enjoying the sunny weather in Brighton]].

'''1895 April 27, Saturday''', Algernon Bourke attended the [[Social Victorians/Timeline/1895#1895 April 27, Saturday|wedding of Norah Bourke and Henry E. A. Lindsay]]. Again, Guendoline Bourke is not listed as having attended. Daphne Bourke was born on 5 April 1895, and Psyche, writing the &quot;Social Peepshow&quot; column in the Gentlewoman, says,&lt;blockquote&gt;

I regret to hear of the serious illness of Mrs. Algernon Bourke, whose first child was born a fortnight ago. It is feared that the attack is of the nature of typhoid, but happily the patient's strength keeps up. Mrs. Bourke is at her mother's house in Clarges-street.&lt;ref&gt;Psyche. &quot;The Social Peepshow.&quot; ''Gentlewoman'' 27 April 1895, Saturday: 28 [of 84], Col. 2c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18950427/147/0028. Same print title, p. 506.&lt;/ref&gt;&lt;/blockquote&gt;

'''1895 July 13, Saturday''', Algernon Bourke donated 10s. to the ''Daily Telegraph'' National Shilling Testimonial to W. G. Grace.&lt;ref&gt;&quot;''Daily Telegraph'' National Shilling Testimonial to W. G. Grace.&quot; ''Daily Telegraph &amp; Courier'' (London) 13 July 1895, Saturday: 7 [of 12], Col. 7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/18950713/079/0007. Print: ''Daily Telegraph'', p. 7.&lt;/ref&gt;

'''1895 August 24, Saturday''', &quot;Marmaduke&quot; in the ''Graphic'' says that Algernon Bourke &quot;opened a cyclists' club in Chelsea.&quot;&lt;ref&gt;&quot;Marmaduke.&quot; &quot;Court and Club.&quot; The ''Graphic'' 24 August 1895, Saturday: 11 [of 32], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/9000057/18950824/017/0011. Print p. 223.&lt;/ref&gt;

'''1895 October''', the Hon. Algernon Bourke [[Social Victorians/Timeline/1900s#24 October 1902, Friday|opened the Prince's ice-skating rink for the]] season.if the newspapers were right that 1902 was the 7th season. He also was planning a bicycling club for Kensington Gardens to open the following season.&lt;ref&gt;Mackenzie, Ethel Morell (Miss). &quot;Pins and Needles.&quot; ''Hull Daily News'' 12 October 1895, Saturday: 24 [of 40], Col. 1a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003443/18951012/024/0024. Print title: ''Hull News Supplement'', p. 1[6? 8?].&lt;/ref&gt;

'''1895 October 7, Monday''', the Hon. Algernon and Mrs. Bourke attended the [[Social Victorians/Timeline/1895#Adeane-Cator Wedding|Maud Adeane–John Cator wedding]].

'''1895 December 11, Wednesday''', Guendoline and Algernon Bourke attended a [[Social Victorians/Timeline/1895#Sneyd Party to Meet the Duke of Coburg|shooting party at the Sneyds' to meet the Duke of Coburg]].

'''1895 December 18, Wednesday''', Algernon and Guendoline Bourke attended the [[Social Victorians/Timeline/1895#Wedding of Lady Albreda Fitzwilliam and the Hon. Charles Bourke|wedding of Lady Albreda Fitzwilliam and the Hon. Charles Bourke]]. Their gift is not noted in the newspaper account.

'''1896 March 17, Tuesday''', Algernon Bourke attended the [[Social Victorians/Timeline/1896#17 March 1896, Tuesday|annual dinner of the Cymmrodorion, or the Honourable Society of Cymmrodorion]], a society for Welsh culture and history.

'''1896 April 21, Monday''', Mr. and Mrs. A. Bourke sent a gift — a &quot;box for miniature&quot; — for [[Social Victorians/Timeline/1896#Monday, 1896 April 27|the wedding of Lady Angela St. Clair Erskine and James Stewart Forbes]].

'''1896 May 21, Thursday''', the Hon. and Mrs. Algernon Bourke attended [[Social Victorians/Timeline/1896#Mrs. C. H. Wilson's Ball|Mrs. C. H. Wilson's ball in Grosvenor-square, London]].

'''1896 May 26, Tuesday,  through 28 May, Thursday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1896#Coming of Age of Mr Sloane Stanley|3-day celebration in honor of the coming of age of her brother, Cyril Sloane Stanley]].

'''1896 June 15, Monday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1896#Dinner and Dance Hosted by the Countess of Huntingdon|dance hosted by the Earl and Countess of Huntingdon]] after their dinner party.

'''1896 July 13, Monday''', Algernon Bourke (listed among the &quot;Honourables&quot;) and Mrs. A. Bourke (Listed among the &quot;Honourable Ladies&quot;) were invited to the [[Social Victorians/Timeline/1896#Queen's Garden Party at Buckingham Palace|Queen's Garden Party at Buckingham Palace]].

'''1896 June 29, Monday''', the Hon. Mrs. Algernon Bourke attended the [[Social Victorians/Cadogan-Scott Wedding 1896-06-29|wedding and reception of Lady Sophie Cadogan and Sir Samuel Scott]]. Algernon Bourke published a letter to the editor of the ''Daily Telegraph'' about White's Club — and thus Bourke's — &quot;[[Social Victorians/London Clubs#Summer Club|Summer Club]]&quot; in Kensington Park, the subject of a little controversy.

'''1896 July 21, Tuesday''', the Hon. Algernon and Mrs. Bourke attended a [[Social Victorians/Timeline/1896#Dinner Hosted by Sir Horace and Lady Farquhar|dinner hosted by Sir Horace and Lady Farquhar in Grosvenor-square]].

'''1896 August 5, Wednesday''', Algernon and Guendoline Bourke attended at the [[Social Victorians/Timeline/1896#5 August 1896|wedding of the Hon. Terence Bourke and Miss Eveline Haines]] and gave the bride an &quot;enamel muff chain.&quot;&lt;p&gt;

'''1896 August 10, Monday''', the Morning Leader reported that the Hon. Algernon Bourke, for the Foreign Office, received Li Hung Chang at St. Paul's:&lt;blockquote&gt;At St. Paul's Li Hung was received by Field-Marshal Simmons, Colonel Lane, the Hon. Algernon Bourke, of the Foreign Office (who made the necessary arrangements for the visit) and Canon Newbolt, on behalf of the Dean and Chapter. A crowd greeted Li with a cheer as he drove up in Lord Lonsdale’s striking equipage, and his Excellency was carried up the steps in an invalid chair by two stalwart constables. He walked through the centre door with his suite, and was immediately conducted by Canon Newbolt to General Gordon’s tomb in the north aisle, where a detachment of boys from the Gordon Home received him as a guard of honor.

Li inspected the monument with marked interest, and drew the attention of his suite to the remarkable likeness to the dead hero. He laid a handsome wreath of royal purple asters, lilies, maidenhair fern, and laurel, tied with a broad band of purple silk, on the tomb.

The visit was not one of inspection of the building, but on passing the middle aisle the interpreter called the attention of His Excellency to the exquisite architecture and decoration of the chancel. Li shook hands in hearty English fashion with Canon Newbolt and the other gentlemen who had received him, and, assisted by his two sons, walked down the steps to his carriage. He returned with his suite to Carlton House-terrace by way of St. Paul’s Churchyard, Cannon-st., Queen Victoria-st., and the Embankment.&lt;ref&gt;&quot;At St. Paul's.&quot; ''Morning Leader'' 10 August 1896, Monday: 7 [of 12], Col. 2b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004833/18960810/134/0007. Print p. 7.&lt;/ref&gt;&lt;/blockquote&gt;

'''1896 August 19, Wednesday''', the ''Edinburgh Evening News'' reported on the catering that White's Club and Mr Algernon Bourke arranged for the visiting Li Hung Chang:&lt;blockquote&gt;It is probably not generally known (says the &quot;Chef&quot;) that Mr Algernon Bourke, manager of White's Club, London, has undertaken to the whole of the catering for our illustrious visitor front the Flowery Land. Li Hung Chang has five native cooks in his retinue, and the greatest good fellowship exists between them and their English ''confreres'', although considerable difficulty is experienced in conversation in understanding one another's meaning. There are between 40 and and 50 to cater for daily, besides a staff about 30; that Mr Lemaire finds his time fully occupied. The dishes for his Excellency are varied and miscellaneous, and from 14 to 20 courses are served at each meal. The bills of fare contain such items as bird's-nest soup, pigs' kidneys stewed in cream, boiled ducks and green ginger, sharks' fins, shrimps and prawns stewed with leeks and muscatel grapes, fat pork saute with peas and kidney beans. The meal usually winds with fruit and sponge cake, and freshly-picked green tea as liqueur.&lt;ref&gt;&quot;Li Hung Chang's Diet.&quot; ''Edinburgh Evening News'' 19 August 1896, Wednesday: 3 [of 4], Col. 8b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000452/18960819/057/0003.&lt;/ref&gt;&lt;/blockquote&gt;

'''1896 November 6, Friday''', both Algernon and Guendoline Bourke were on the committee for the [[Social Victorians/London Clubs#Prince's Skating Club|Prince's Club ice-skating rink]], which [[Social Victorians/Timeline/1896#Opening of the Prince's Club Skating Rink|opened on this day]].

'''1896 November 22, week of''', Mrs. Algernon Bourke was part of a [[Social Victorians/Timeline/1896#Shooting Party at the Charles Wilsons' Warter Priory, Yorkshire|shooting party at the Charles Wilsons' Warter Priory, Yorkshire]].&lt;p&gt;

'''1896 November 25, Wednesday''', Mr. and Mrs. Algernon Bouke attended [[Social Victorians/Timeline/1896#Lord and Lady Burton Hosted a Party for Derby Day|Lord and Lady Burton's party for Derby Day]].&lt;p&gt;

'''1896 December 4, Friday''', the Orleans Club at Brighton was robbed:&lt;blockquote&gt;The old building of the Orleans Club at Brighton, which opens its new club house at 33, Brunswick-terrace to-day, was the scene of a very ingenious burglary during the small hours of yesterday morning. The greater portion of the club property had already been removed to the new premises, but Mr Algernon Bourke, his private secretary, and some of the officials of the club, still occupied bed-rooms at the house in the King’s-road. The corner shop of the street front is occupied by Mr. Marx, a jeweller in a large way of business, and upon his manager arriving at nine o'clock he discovered that the place had been entered through hole in the ceiling, and a great part of a very valuable stock of jewelry extracted. An examination of the morning rooms of the club, which runs over Mr. Marx's establishment reveal a singularly neat  specimen of the burglar's art. A piece of the flooring about 15in square had been removed by a series of holes bored side by side with a centre-bit, at a spot where access to the lofty shop was rendered easy by a tall showcase which stood convemently near. A massive iron girder had been avoided by a quarter of an inch, and this circumstance and the general finish of the operation point to an artist in his profession, who had acquired an intimate knowledge of the premises. The club doors were all found locked yesterday morning, and the means of egress adopted by the thief are at present a mystery.&lt;ref&gt;&quot;Burglary at Brighton.&quot; ''Daily Telegraph &amp; Courier'' (London) 05 December 1896, Saturday: 5 [of 12], Col. 7a [of 7]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/18961205/090/0005. Print title: ''Daily Telegraph''; p. 5.&lt;/ref&gt;&lt;/blockquote&gt;

'''1896 December 10, Thursday''', Guendoline Bourke was present to help staff a stall at the [[Social Victorians/Timeline/1896#10 December 1896, Thursday|Irish Industries Exhibition and Sale, Brighton]].

'''1896 December 31, Thursday''', Guendoline Bourke hosted a New Year's Eve dance:&lt;blockquote&gt;Mrs. Algernon Bourke gave a highly satisfactory and enjoyable dance on Thursday night, when the old year was danced out and the new one danced in. Most of the silver gilters at present in to len were to the fore.&lt;ref&gt;&quot;The Man about Town.&quot; ''Sporting Gazette'' 02 January 1897, Saturday: 7 [of 34], Col. 3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002525/18970102/041/0007. Print title ''The County Gentleman'', p. 7.&lt;/ref&gt;&lt;/blockquote&gt;

'''1897 January 9, Saturday''', Psyche in &quot;The Social Peepshow&quot; says that Algernon Bourke's &quot;cheerful countenance was quite in keeping with the [Christmas] season,&quot; seen in London.&lt;ref&gt;Psyche. &quot;The Social Peepshow.&quot; ''Gentlewoman'' 9 January 1897, Saturday: 22 [of 56], Col. 2a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970109/097/0022. Same print title, p. 40.&lt;/ref&gt; 

'''1897 January 13, Wednesday – 18, Monday''', Algernon and Guendoline Bourke were guests of the [[Social Victorians/Timeline/1897#The Warwickshire Hunt Club Ball|house party associated with the Warwickshire Hunt Ball]] at [[Social Victorians/People/Warwick|Warwick Castle]].

'''1897 January 30, Saturday''', Guendoline Bourke was reported to have been out shopping in London: &quot;Another charming figure was that of Mrs. Algernon Bourke all in chinchilla, with something of pale blue in a smart toque.&quot;&lt;ref&gt;Psyche. &quot;The Social Peepshow.&quot; ''Gentlewoman'' 30 January 1897, Saturday: 20 [of 59]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970130/107/0020. Same print title, p. 134.&lt;/ref&gt;

'''1897 May 31, Monday''', Hon. Algernon and Mrs. Bourke were present at a [[Social Victorians/Timeline/1897#House Party at Warwick Castle|House Party at Warwick Castle]] hosted by the Earl and Countess of Warwick.

'''1897 June 2, Wednesday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1897#Reception at the Foreign Office|reception at the Foreign Office]].

'''1897 June 12, Saturday''', the ''Gentlewoman'' reported on Guendoline Bourke's dress and hat at the [[Social Victorians/Timeline/1897#The Duchess of Albany's Bazaar at the Imperial Institute|Duchess of Albany's Bazaar at the Imperial Institute]]. 

'''1897 June 19, Saturday''', Psyche in &quot;The Social Peepshow&quot; column in the ''Gentlewoman'' writes that Guendoline Bourke was seen driving in London, &quot;in blue, ... looking as usual very handsome.&quot;&lt;ref&gt;Psyche. &quot;The Social Peepshow.&quot; ''Gentlewoman'' 19 June 1897, Saturday: 28 [of 108], Col. 2b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970619/159/0028. Same print title, p. 848.&lt;/ref&gt;

'''1897 June 28, Monday''', Algernon and Guendoline Bourke were invited to the [[Social Victorians/Diamond Jubilee Garden Party|Garden Party at Buckingham Palace]], the final official event of the London Diamond Jubilee celebrations. Members of the family of the [[Social Victorians/People/Mayo|Earl of Mayo]] were also among the 5,000–6,000 people invited.

'''1897 July 2, Friday''', the Hon. A. and Mrs. A. Bourke and Mr. and Mrs. Bourke attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House.

'''1897 July 8, Thursday, 11:00 p.m.''', Hon. Algernon and Guendoline Bourke were present at [[Social Victorians/Timeline/1890#Queen's State Ball at Buckingham Palace|the Queen's State Ball at Buckingham Palace]]. The dancing commenced shortly after 11:00 p.m.

'''1897 July 11–16, week of''', a dog of Guendoline Bourke's won a prize at the [[Social Victorians/Timeline/1897#The Ladies' Kennel Association show in the Royal Botanic Gardens in Regent's Park|Ladies' Kennel Association show in the Royal Botanic Gardens in Regent's Park]]. 

'''1897 July 23, Friday''', both the Hon. Algernon Bourke and Guendoline Bourke attended the [[Social Victorians/Timeline/1897#Bourke-Curzon Cricket Match at the Queen's Club|Bourke-Curzon cricket match at the Queen's Club]], which Algernon Bourke's team lost. 

'''1897 July 23 — or July 30, Friday''', Guendoline Bourke attended [[Social Victorians/Timeline/1897#Lady Burton's party at Chesterfield House|Lady Burton's party at Chesterfield House]]. &lt;blockquote&gt;Far the prettiest women in the room were Lady Henry Bentinck (who looked perfectly lovely in pale yellow, with a Iong blue sash; and Mrs. Algernon Bourke, who was as smart as possible in pink, with pink and white ruchings on her sleeves and a tall pink feather in her hair.&lt;ref&gt;&quot;Lady Burton's Party at Chesterfield House.&quot; ''Belper &amp; Alfreton Chronicle'' 30 July 1897, Friday: 7 [of 8], Col. 1c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004151/18970730/162/0007. Print title: ''Belper and Alfreton Chronicle''; n.p.&lt;/ref&gt;&lt;/blockquote&gt;

'''1897 August 2, Monday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1897#Warwick House Party for the Easton Lodge Cricket Week Games|Earl and Countess of Warwick's house party for Easton Lodge cricket week]].

'''1897 August 2, Monday''', Mrs. Algernon Bourke was listed as among [[Social Victorians/Timeline/1897#The Most Beautiful Women in England|the most beautiful women in England]] in an article from ''Vanity Fair'' that was reprinted elsewhere.

'''1897 September 25, Saturday''', according to the ''Pall Mall Gazette'',&lt;blockquote&gt;The [[Social Victorians/People/Mayo|Dowager-Countess of Mayo]] is staying with her son, the Hon. Algernon Bourke, at Bramnber, near Brighton.&lt;ref&gt;&quot;Pall Mall Gazette Office.&quot; ''Pall Mall Gazette'' 25 September 1897, Saturday: 8 [of 10], Col. 2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18970925/023/0008. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;'''1897 October 2, Saturday''', &quot;Yenatrix&quot; in &quot;Kennel Column&quot; in the ''Gentlewoman'' reported that Guendoline Bourke had joined the Ladies' Kennel Association.&lt;ref&gt;Yenatrix. &quot;Kennel Column.&quot; ''Gentlewoman'' 02 October 1897, Saturday: 39 [of 61], Col. 2a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18971002/182/0039. Same print title, p. 434.&lt;/ref&gt;

'''1897 October 9, Saturday''', Algernon and Guendoline Bourke were at [[Social Victorians/Timeline/1897#Harrogate|Harrogate, presumably taking the waters and baths]]. Lady May was on her way to visit Algernon Bourke in Brighton:&lt;blockquote&gt;The Earl of Mayo is expected to return from Sweden on Saturday next. Lady Mayo leaves Bournemouth on Sarurday for Brighton, where she will pay a two days' visit to her brother-in-law, the [[Social Victorians/People/Bourke|Hon. Algernon Bourke]]. The Earl and Countess will then return to Palmerstown, their seat in County Kildare.&lt;ref&gt;&quot;Pall Mall Gazette Office.&quot; ''Pall Mall Gazette'' 7 October 1897, Thursday: 8 [of 12], Col. 3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18971007/022/0008. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;&lt;p&gt;

'''1897 October 30, Saturday''', ''Black and White'' published '''J.P.B.'''&lt;nowiki/&gt;'s &quot;The Case of Mrs. Elliott,&quot;&lt;ref name=&quot;:13&quot;&gt;J.P.B. &quot;The Case of Mrs. Elliott.&quot; ''Black &amp; White'' 30 October 1897, Saturday: 12 [of 34], Cols. 1a–2b [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004617/18971030/036/0012. Print title ''Black and White'', p. 542.&lt;/ref&gt; an odd short short story in which the Honourable Algernon Bourke Herriott is &quot;rude to Mrs. Elliott,&quot;&lt;ref name=&quot;:13&quot; /&gt;{{rp|Col. 2b}} presumably having proposed sexual relations while her husband is out. J.P.B. links to the biographical Algernon Bourke's career in the stock market in the description of Mrs. Christine Elliott not even simulating interest in her husband's bicycling: &quot;a soul is a grievous burthen for a stockbroker's wife,&quot;&lt;ref name=&quot;:13&quot; /&gt;{{rp|Col. 2a}} suggesting that Mr. Elliott rather than Algernon Bourke Herriott is the stockbroker. The Hon. Algy&lt;blockquote&gt;was a senior member of several junior clubs. A woman had dubbed him once &quot;a rip with a taste for verses.&quot; The description was severe, but not unwarranted. His was a pretty pagan sensualism, though, singing from a wine palate to Church music. For the rest, he had just imagination enough to despise mediocrity.&lt;ref name=&quot;:13&quot; /&gt;{{rp|Col. 2a}}&lt;/blockquote&gt;

'''1897 November 25–26, Thursday–Friday''', Guendoline Bourke was in Brighton, helping the Countess of Mayo at the [[Social Victorians/Timeline/1897#The Irish Industries' Association Annual Exhibition|bazaar of the Irish Industries' Association]].

'''1897 December 7, Tuesday''', Algernon Bourke attended the [[Social Victorians/Timeline/1897#7 December 1897, Tuesday|7th annual dinner for the Actors' Benevolent Fund]].

'''1897 December 20, Monday, the week of''', Guendoline Bourke's Mousie was the smallest dog in the [[Social Victorians/Timeline/1897#Ladies' Kennel Association's Dog Show|Ladies' Kennel Association show in Earl's Court]].

'''1897 December 30''', Algernon and Guendoline Bourke attended a [[Social Victorians/Timeline/1897#Blenheim Palace Party with Amateur Theatricals|party at Blenheim Palace in which people performed tableaux vivants]] that got reported on, many of whom wearing the costumes from the Duchess of Devonshire's ball. The ''Irish Independent'' said Algernon Bourke was &quot;mainly responsible for the living pictures.&quot;&lt;ref&gt;&quot;Mr Algernon Bourke ....&quot; ''Irish Independent'' 05 January 1898, Wednesday: 6 [of 8], Col. 2c [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001985/18980105/115/0006.&lt;/ref&gt;

'''1898''', Algernon Bourke called a meeting at White's Club about attempting to [[Social Victorians/Timeline/1900s#The Thames Salmon Experiment|restock the Thames with salmon]]. In 1899 he was on a [[Social Victorians/People/Bourke#Committees|committee led by the Lord Mayor about this topic]] as well.

'''1898 February 3, Thursday''',  Algernon Bourke was among [[Social Victorians/Timeline/1898#The Dundee Evening Telegraph Report on People at Monte Carlo|those visiting Monte Carlo according to the Dundee ''Evening Telegraph'']].

'''1898 March 12, Saturday''', ''The World'' reported on Algernon Bourke's upgrading of the Orleans Club at Brighton:&lt;blockquote&gt;

The Orleans Club at Brighton is flourishing exceedingly, and the new buildings which Mr. Algernon Bourke has just had erected at the back of the comfortable mansion at the corner of Lansdowne-place now provide all that was wanting to make the present habitat of the club all that its members desire. The new billiard-room is rapidly approaching completion, and the coffee-room, excellent and spacious now, was open on Saturday night, when every table was occupied by club diners and their guests, all of whom were enthusiastic over the excellence of this latest addition to the comfort of the house. All interested may be congratulated on what is practically new lease of life to the Orleans Club, than which there is no more comfortable place stay within the four seas.&lt;ref&gt;&quot;From '''The World''.'&quot; ''East &amp; South Devon Advertiser'' 12 March 1898, Saturday: 6 pop 8], Col. 2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001639/18980312/132/0006. Print title ''The East and South Devon Advertiser'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;

'''1898 March 30, Wednesday''', Algernon Bourke was charged with assaulting a Mr. Potter, but it is not clear from this account what exactly happened:&lt;blockquote&gt;The Hon. Algernon H. Bourke, of Bramber, was summoned, at the instance of Mr. Walter John Potter, clerk to Mr. G. A. Flowers, solicitor, of Steyning, for assault, on the 30th March. — Mr. J. Edward Dell supported the case, and Mr. J. C. Buckwell defended, and pleaded not guilty. — The evidence was to the effect that Mr. Potter had occasion go to defendant's house on Wednesday last to serve a writ. He was going to drop the letter into [Col. 5c–6a] defendant's pocket when he turned and struck him a violent blow on the chest, making witness stagger backwards. Witness put up his hands to keep his balance, and defendant then struck him violently across the head with a weeding spud. — Richard Reed, who was at work for Mr. Bourke on the date named, and was working in garden at the time of the alleged assault, gave corroborative evidence. — Defendant, in the witness box, made a similar statement. — The magistrates differed as to whether the assault was committed, and dismissed the case.&lt;ref&gt;&quot;Steyning.&quot; ''Sussex Express'' 9 April 1898, Saturday: 2 [of 12], Col. 5c–6a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000654/18980409/036/0002. Print: ''The Sussex Express, Surrey Standard, Weald of the Kent Mail, Hants and County Advertiser'', p. 2.&lt;/ref&gt;&lt;/blockquote&gt;'''1898 April 12, Tuesday''', Algernon Bourke was among [[Social Victorians/Timeline/1898#1898 April 12, Tuesday|those visiting Monte Carlo according to the ''Gentlewoman'']].

'''1898 May 25, Wednesday''', Guendoline Bourke wore pink to [[Social Victorians/1898-05-25 Savoy Dinner Dance Hwfa|Mrs. Hwfa Williams' dinner-dance at the Savoy]].

'''1898 June 7, Tuesday''', the Hon. Algernon and Mrs. A. Bourke were invited to and probably attended the [[Social Victorians/Timeline/1898#7 June 1898, Tuesday|State Ball at Buckingham Palace hosted by the Prince and Princess of Wales]].

'''1898 July 4, Thursday afternoon''', the Hon. Algernon and Mrs. Bourke were invited to and probably attended the [[Social Victorians/Timeline/1898#Garden Party at Marlborough House|Garden Party at Marlborough House given to the Queen and Shah of Persia]].

'''1898 October 29, Saturday''', Algernon Bourke attended a [[Social Victorians/Timeline/1898#Tennis Championship Game at Prince's Club, Knightsbridge|tennis match at Prince's Club, Knightsbridge]].

'''1898 November''', Algernon Bourke received a letter from Lord Alfred Douglas and attempted to help Douglas and his father, the Marquis of Queensberry, to reconcile. Bourke met with the Marquis and sent a positive reply to Douglas the next day.&lt;ref name=&quot;:26&quot; /&gt; (568–569)

'''1898 November 22, Tuesday''', Algernon Bourke was present at a [[Social Victorians/Timeline/1898#Shooting Party Hosted by William James|shooting party hosted by Mr. William James]].

'''1898 December 3, Saturday''', Hon. Algernon and Mrs. A. Bourke attended the [[Social Victorians/Timeline/1898#The Funeral of Lady Connemara|funeral of Lady Connemara in Christ Church]], Down street, Piccadilly.&lt;p&gt;

'''1899 January 10, Tuesday''', the Brighton Championship Dog Show opened:&lt;blockquote&gt;Princess of Wales a Winner at the Ladies’ Kennel Club Show.

[Exclusive to &quot;The Leader.&quot;)

The Brighton Championship Dog Show opened in the Dome and Corn Exchange yesterday, and was very well patronised by visitors and exhibitors. Among the latter was H.R.H. the Princess of Wales, who did very well; and others included Princess Sophie Duleep Singh, Countess De Grey, Sir Edgar Boehm, the Hon Mrs. Algernon Bourke, Lady Cathcart, Lady Reid, Mr. Shirley (chairman of the Kennel Club), and the Rev. Hans Hamiiton (president of the Kennel Club).

The entry of bloodhounds is one of the best seen for some time; the Great Danes are another strong lot; deerhounds are a fine entry, all good dogs, and most of the best kennels represented; borzois are another very stylish lot. The bigger dogs are, as usual, in the Corn Exchange and the &quot;toy&quot; dogs in the Dome. To everyone's satsfaction the Princess of Wales carried off two first prizes with Alex in the borzois class.&lt;ref&gt;&quot;Dogs at Brighton.&quot; ''Morning Leader'' 11 January 1899, Wednesday: 8 [of 12], Col. 3b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004833/18990111/142/0008. Print p. 8.&lt;/ref&gt;&lt;/blockquote&gt;

'''1899 January 11, Wednesday''', Guendoline Bourke attended [[Social Victorians/Timeline/1899#11 January 1899, Wednesday|a luncheon at Stanfield-hall, home of Mr. and Mrs. Basil Montogomery, for Princess Henry of Battenberg]], that also included the Countess of Dudley (sister of Mrs. Montgomery), General Oliphant, and the Mayor and Mayoress of Romsey.

'''1899 January 17–18, Tuesday and Wednesday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1899#Ladies' Kennel Association in Brighton|Ladies' Kennel Association in Brighton]], where she showed an Italian greyhound named Brenda.

'''1899 February 7, Tuesday''', Guendoline Bourke was a member of the very high-ranking committee organizing the [[Social Victorians/Timeline/1899#Gordon Memorial College Ball|Gordon Memorial College Ball at the Hotel Cecil on 7 February 1899]]. The committee had been planning for the ball, of course, for at least 3 weeks before.

'''1899 February 22, Wednesday – April''', Guendoline Bourke was part of [[Social Victorians/Timeline/1899#Society in St. Moritz|Society in St. Moritz]]. 1899 March 29, Wednesday, the ''Dundee Advertiser'' says that [[Social Victorians/Timeline/1899#29 March 1899, Wednesday|Cyril Sloane-Stanley was spending part of the winter in St. Moritz]] with his sister Guendoline Bourke.

'''1899 April 7, Friday, probably''', oddly, Algernon and Guendoline Bourke are not reported to have attended the [[Social Victorians/Timeline/1899#Funeral of the Hon. Charles Bourke, C.B.|Funeral of the Hon. Charles Bourke, C.B.]] or even to have sent flowers.

'''1899 April 8, Saturday''', the ''Gentlewoman'' reported that Guendoline Bourke had gone to [[Social Victorians/Timeline/1899#8 April 1899, Saturday|St. Moritz with her brother, Mr. Stanley, who had gotten engaged to Lady Cairns]].

'''1899 April 26, Wednesday''', according to &quot;Local and District News&quot; for Totton, Guendoline Bourke was &quot;ill with influenza in Paris, and Mrs. Shelley Bontein, her mother, has gone out to nurse her.&quot;&lt;ref&gt;&quot;Local and District News. Totton.&quot; ''Hampshire Advertiser'' 26 April 1899, Wednesday: 4 [of 4], Col. 2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18990426/037/0004. Print title ''Hampshire Advertiser County Newspaper'', p. 4.&lt;/ref&gt;

'''1899 June 1, Thursday, or 2, Friday''', the Hon. Algernon and Guendoline Bourke attended the [[Social Victorians/Timeline/1899#Wedding of Roger Cyril Sloane Stanley and Olivia, Countess Cairns|wedding of her brother, Sloane Stanley and Olivia Countess Cairns]] at Holy Trinity Church, Brompton.

'''1899 June 8, Thursday''', Algernon Bourke's money troubles:&lt;blockquote&gt;The Hon. Algernon Bourke, son of the Earl of Mayo, has been appearing before the official receivers in connection with a winding-up order made against Willis’ Restaurant, Limited. The companyf [sic] was formed to acquire the well known restaurant from the Hon. H. A. Bourke. The chairman reminded the creditors that on the last occasion the meeting was adjourned because Mr. Bourke said he thought he would be able in the course of a fortnight to obtain an offer for a sum sufficient to satisfy the creditors and debenture holders. He had received a letter from Mr. Bourke to the effect that he had been unable to complete arrangements. Having looked into the affairs of the company more closely, it appeared to him that Mr. Bourke was legally liable to repay the sum of £5,000 which was advanced to White's Club, and the question would arise whether Mr. Bourke was not also liable to repay the sum of £4,000.&lt;ref&gt;&quot;Mr. Bourke Must Pay.&quot; ''Irish Independent'' 8 June 1899, Thursday: 4 [of 8], Col. 8c [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001985/18990608/118/0004. Print title: ''The Irish Weekly Independent'', p. 4.&lt;/ref&gt;&lt;/blockquote&gt;'''1899 July 1, Saturday''', Algernon Bourke attended a [[Social Victorians/Timeline/1899#1 July 1899, Saturday|meeting in London at the Duke of Westminster's Grosvenor House]] about preserving Killarney as part of the National Trust and seems to have been acting for someone who wanted to purchase the Muckross Estate.

'''1899 July 5, Wednesday''', Guendoline Bourke attended a [[Social Victorians/Timeline/1899#Dinner and Dance at Devonshire House|dance at Devonshire House hosted by the Duke and Duchess of Devonshire]].

'''1899 July 6, Thursday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1899#Joan Wilson and Guy Fairfax's Wedding|wedding of Joan Wilson and Guy Fairfax in St. Mark's, near Grosvenor Square]].

'''1899 July 14, Friday''', Guendoline Bourke attended [[Social Victorians/Timeline/1899#14 July 1899, Friday|Ernest Beckett's dinner party]].

'''1899 July 18, Tuesday''', Guendoline Bourke attended a [[Social Victorians/Timeline/1899#Ludovici Lecture on Impressionism|lecture on Impressionism by Ludovici hosted by the Countess of Mayo]].

'''1899 July 28, Friday''', [[Social Victorians/London Clubs#White's|White's Club]] was no longer under Algernon Bourke's management and was reconstituting itself after the possibility that it would have to close.

'''1889 July 31, Wednesday''', the Hon. Algernon and Mrs. Bourke attended a [[Social Victorians/Timeline/1889#Fete of the Uxbridge Habitation of the Primrose League|Fete of the Uxbridge Habitation of the Primrose League]] at Hillingdon Court and hosted by the Hon. Algernon and Lady Mary Mills.

'''1899 September 9, Saturday''', the ''Eastern Morning News'' includes Algernon Bourke (&quot;St. James's-street, London, club proprietor&quot;) in a list of men &quot;Receiving Orders,&quot; which it is reprinting from the ''London Gazette''.&lt;ref&gt;&quot;Receiving Orders.&quot; ''Eastern Morning'' News 9 September 1899, Saturday: 5 [of 8], Col. 3c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001152/18990909/074/0005. Same print title and p.&lt;/ref&gt;&lt;p&gt;

'''1899 October 19, Thursday''', the Hon. Algernon Bourke had a bankruptcy hearing:&lt;blockquote&gt;The public examination of the Hon. Algernon Bourke was held before Mr Registrar Giffard yesterday, at the London Bankruptcy Court. The debtor, described as proprietor of a St. James's-street club, furnished a statement of affairs showing unsecured debts £13,694 and debts fully secured £12,800, with assets which are estimated at £4,489 [?]. He stated, in reply to the Official Receiver, that he was formerly a member of the Stock Exchange, but had nothing to do with the firm of which he was a member during the last ten years. He severed his connection with the firm in May last, and believed he was indebted to them to the extent of £2,000 or £3,000. He repudiated a claim which they now made for £37,300. In 1889 he became proprietor of White's Club, St. James's-street, and carried it on until January 1st last, when he transferred it to a company called Recreations, Limited. One of the objects of the company was to raise money on debentures. The examination was formally adjourned.&lt;ref name=&quot;:9&quot; /&gt;&lt;/blockquote&gt;

'''1899 October 20, Friday''', the ''Morning Leader'' mentions Bourke's bankruptcy:&lt;blockquote&gt;Mr. Algernon Bourke, whose bankruptcy is much talked about, has been connected with numerous enterprises in clubland. He raised White's from the slough into which it had sunk after the secession of the Prince of Wales. He started the Willis Restaurant, put fresh life into the Orleans Club at Brighton, arranged a big restaurant for the bicyclists in the time of the bicycle parade, and was concerned at first in the smart and short-lived Trafalgar Bicycle Club. At one time his name spelt success. Latterly his luck has left him. He is a brother of Lord Mayo, a son of the peer who was assassinated at the post of duty, and is one of the best known men about town of the day.&lt;ref&gt;&quot;Club, Stage, and Salon.&quot; ''Morning Leader'' 20 October 1899, Friday: 6 [of 12], Col. 5b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004833/18991020/085/0006. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;'''1899 November 8, Wednesday''', the Hon. Algernon Bourke's bankruptcy case came up again:&lt;blockquote&gt;At Bankruptcy Court, yesterday, the case the Hon. Algernon Bourke again came on for hearing before Mr. Registrar Giffard, and the examination was concluded. The debtor has at various times been proprietor of White’s Club, St. James’s-street, and the Orleans’ Club, Brighton, and also of Willis's Restaurant, King-street, St. James's. He attributed his failure to losses sustained by the conversion of White’s Club and the Orleans' Club into limited companies, to the payment of excessive Interest on borrowed money, and other causes. The liabilities amount to £26,590, of which £13,694 are stated to be unsecured, and assets £4,409.&lt;ref&gt;&quot;Affairs of the Hon. A. Bourke.&quot; ''Globe'' 09 November 1899, Thursday: 2 [of 8], Col. 1c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001652/18991109/020/0002. Print p. 2.&lt;/ref&gt;&lt;/blockquote&gt;

'''1899 December 23, Saturday''', &quot;Mr. Algernon Bourke has departed for a tour in Africa, being at present the guest of his brother in Tunis.&quot;&lt;ref&gt;&quot;The Society Pages.&quot; ''Walsall Advertiser'' 23 December 1899, Saturday: 7 [of 8], Col. 7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001028/18991223/143/0007. Print p. 7.&lt;/ref&gt;

'''1899 December 29, Friday''', Guendoline Bourke was at the [[Social Victorians/Timeline/1899#Christmas Party Hosted by the Duke and Duchess of Marlborough|Christmas Party Hosted by the Duke and Duchess of Marlborough]].&lt;p&gt;

'''1899 December 31''', the San Francisco newspaper ''The Wave'' wrote the following about London society:&lt;blockquote&gt;The most prominent untitled people in London may be said to be Mr. and Mrs. [[Social Victorians/People/Williams|Hwfa Williams]], Mr. and Mrs. [[Social Victorians/People/Grenfell|Willie Grenfell]] and Mr. Algy Bourke. That they are passing rich, goes without saying, and that they entertain lavishly, understood — for to be untitled, prominent and successful, argues wealth, hospitality and cleverness.&lt;ref&gt;&quot;London.&quot; The (San Francisco) ''Wave'' 14 January 1899 (Vol. XIX, No. 2): 14. ''The Internet Archive'' https://archive.org/details/wave19unse/page/n20/mode/1up.&lt;/ref&gt;&lt;/blockquote&gt;

=== 1900s ===

'''1900 February 15, Thursday''', Daphne Bourke, the four-year-old daughter of the Hon. Algernon and Mrs. Bourke was a bridesmaid in the [[Social Victorians/Wilson Chesterfield Wedding 1900-02-15|wedding of Enid Wilson and the Earl of Chesterfield]].&lt;ref&gt;&quot;London Day by Day.&quot; ''Daily Telegraph'' 15 February 1900, Thursday: 8 [of 12], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/19000215/175/0008. Name in British Newspaper Archive: ''Daily Telegraph &amp; Courier'' (London). Print p. 8.&lt;/ref&gt; Guendoline Bourke, &quot;who was in grey, wore a chinchilla toque with violets.&quot;&lt;ref&gt;&quot;Society. Entertainments, Balls, &amp;c.&quot; ''The Queen'' 24 February 1900, Saturday: 40 [of 76], Col. 1a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/19000224/235/0040. Print: ''The Queen, The Lady's Newspaper'', p. 308.&lt;/ref&gt;

'''1900 March 10, Saturday''', the ''Weekly Irish Times'' reprinted society gossip from ''The World'':&lt;blockquote&gt;Mrs. Algernon Bourke, who has been staying with her husband's uncle, old Connemara, during Mr. Algernon Bourke's absence abroad, has taken a new house near Portman square, and will be settling there before Easter.&lt;ref&gt;&quot;Society Gossip.&quot; ''Weekly Irish Times'' 10 March 1900, Saturday: 17 [of 20], Col. 1b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001684/19000310/116/0017. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;'''1900 July''' '''17, Tuesday''', Guendoline Bourke took part in the [[Social Victorians/Timeline/1900s#17 July 1900, Tuesday|Children's Fete in support of the National Society for the Prevention of Cruelty to Children]] on the grounds of the Royal Botanic Society. Daphe was 5 at this time, so it seems logical that she would have been there, too.

'''1900 July 30, Monday''', Guendoline Bourke attended [[Social Victorians/Timeline/1900s#Barber of Seville at Covent Garden|''The Barber of Seville'' at Covent Garden]].

'''1890 August 6, Friday''', &quot;[[Social Victorians/Timeline/1890#Beautiful Women|Beautiful Women]],&quot; an article in ''Vanity Fair'' that was reprinted elsewhere, mentions Guendoline Bourke (&quot;Lady Algernon Bourke&quot;) as one of the most beautiful women in England.

'''1900 August 11, Saturday''', Guendoline Bourke got&lt;blockquote&gt;the pretty little Yorkshire String, an especially tiny mite, weighing only 2&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;lb, and carrying a very promising coat, ... at the Aquarium Show.&lt;ref&gt;&quot;The Witchampton Kennel.&quot; &quot;Ladies Kennels.&quot; ''Ladies' Field'' 11 August 1900, Saturday: 16 [of 60], Col. 2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0006043/19000811/043/0016. Print title same, p. 390.&lt;/ref&gt;&lt;/blockquote&gt;&lt;p&gt;

'''1900 September 16''', the Hon. Algernon Bourke became the heir presumptive to the Earldom of Mayo when his older brother Captain Hon. Sir Maurice Archibald Bourke died.&lt;p&gt;

'''1900 October 06, Saturday''', the ''Weekly Irish Times'' says that Mr. Algernon Bourke, now heir presumptive to the earldom of Mayo, &quot;has been for some months lately staying with Mr. Terence Bourke in Morocco.&quot;&lt;ref&gt;&quot;Society Gossip.&quot; ''Weekly Irish Times'' 06 October 1900, Saturday: 14 [of 20], Col. 3b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001684/19001006/121/0014. Print p. 14.&lt;/ref&gt;&lt;p&gt;

'''1901 May 30, Thursday''', the Hon. Mrs. Algernon Bourke attended the fashionable [[Social Victorians/Timeline/1900s#1901 May 30, Thursday|Ladies' Kennel Association Dog Show at the Botanic Garden]].

'''1901 July 2, Tuesday''', Guendoline Bourke — &quot;pretty Mrs. Algernon Bourke, in a mauve gown and and purple tulle toque&quot; — attended a children's party at the Botanic Gardens hosted by the Earl and Countess of Kilmorey.&lt;ref&gt;&quot;The Earl of Kilmorey, K.P.&quot; ''Gentlewoman'' 13 July 1901: Saturday, 50 [of 84], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010713/237/0050. Print: title the same, p. 60.&lt;/ref&gt;

'''1901 July 4, Thursday''', Guendoline Bourke — dressed &quot;in pale grey, with her pretty little girl,&quot; 6-year-old Daphne — attended a [[Social Victorians/Timeline/1900s#The Countess of Yarborough's Children's Party|children's party hosted by the Countess of Yarborough]].&lt;ref&gt;&quot;The Countess of Yarborough ....&quot; ''Gentlewoman'' 13 July 1901, Saturday: 76 [of 84], Col. 2b, 3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010713/381/0076. Print p. xxxvi.&lt;/ref&gt;{{rp|Col. 3a}}

'''1901 July 4–6, Thursday–Saturday''', Guendoline Bourke helped staff the Perthshire stall&lt;ref&gt;&quot;The Great County Sale.&quot; ''Gentlewoman'' 29 June 1901, Saturday: 43 [of 72], Col. 3a [of 3]. ''British Newspaper Archive'' [https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010629/223/0043# https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010629/223/0043]. Same print title, pp. 679.&lt;/ref&gt; at the [[Social Victorians/Timeline/1900s#The Great County Sale|Great County Sale in the Imperial Gardens of the Earl's Court Exhibition]].

'''1901 July 20, Saturday''', the ''Gentlewoman'' published the Hon. Mrs. Algernon Bourke's portrait (identified with &quot;Perthshire&quot;) in its 3rd series of &quot;The Great County Sale at Earl's Court. Portraits of Stallholders.&quot;&lt;ref&gt;&quot;The Great County Sale at Earl's Court. Portraits of Stallholders.&quot; ''Gentlewoman'' 20 July 1901, Saturday: 31 [of 60], Col. 4b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010720/141/0031. Print n.p.&lt;/ref&gt; Their daughter Daphne appears in the portrait as well.

'''1901 July 23, Tuesday''', an &quot;Hon. Mrs. Bourke&quot; was in the [[Social Victorians/Timeline/1900s#Lord and Lady Algernon Gordon Lennox|party &quot;entertained by Lord and Lady Algernon Gordon Lennox]].&quot;&lt;p&gt;

'''1901 September 12, Thursday''', Mrs. Guendoline Bourke wanted her name listed as Mrs. Algernon Bourke in the Electoral Register, apparently a frequent complaint:&lt;blockquote&gt;Mr. Underhill, the Conservative agent, mentioned to the Revising Barrister (Mr. William F. Webster) that the name of the Hon. Mrs. Guendoline Bourke was on the list in respect of the house, 75, Gloucester-place. The lady had written to him to say that she was the Hon. Mrs. Algernon Bourke and that she wished that name to appear on the register. In reply to the Revising Barrister, Mr. Underhill said that “Algernon” was the name of the lady’s husband.

Mr. Cooke, the rate-collector, said that Mrs. Bourke had asked to be addressed Mrs. Algernon Bourke, but that the Town Clerk thought the address was not a correct one. The lady signed her cheques Guendoline.”

Mr. Underhill said the agents frequently had indignant letters from ladies because they were not addressed by their husband’s Christian name.

The Revising Barrister — lf a lady gave me the name of Mrs. John Smith I should say I had not got the voter’s name. The name Guendoline must remain.&lt;ref name=&quot;:15&quot;&gt;&quot;Ladies’ Names.&quot; ''Morning Post'' 12 September 1901, Thursday: 7 [of 10], Col. 3a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/19010912/130/0007. Print p. 7.&lt;/ref&gt;&lt;/blockquote&gt;

'''1901 October 26, Friday''', Algernon Bourke was on the Men's Committee of the [[Social Victorians/London Clubs#Prince's Club Ice-skating Rink|Prince's Club Ice-skating Rink]], which had [[Social Victorians/Timeline/1900s#The Prince's Club Ice-skating Rink Opening|its official opening on his day]]. 

'''1902 January''', Algernon Bourke is mentioned in [[Social Victorians/Schools#&quot;More of My Contemporaries at School.&quot;|reminiscences of Eton written by the &quot;Earl of X&quot;]] as being among those in the &quot;world of letters,&quot; and whose brother, later the Earl of Mayo, the Earl of X did not like. 

'''1902 January 25, Saturday''', Mrs. Algernon Bourke gave a box to Lady Helen Stewart-Vane-Tempest in honor of [[Social Victorians/Stewart-Stavordale Wedding 1902-01-25|Lady Helen's wedding to Giles Fox-Strangways, Lord Stavordale]]. 

'''1902 April 26, Saturday''', Mrs. A. Bourke is listed as being at the Norfolk Hotel in Brighton.&lt;ref&gt;&quot;Guide to Visitors at Hotels and Boarding Houses.&quot; ''Brighton Gazette'' 26 April 1902, Saturday: 3 [of 8], Col. 4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000938/19020426/116/0003. Same print title and p.&lt;/ref&gt; 

'''1902 May, End of''', Guendoline Bourke attended a [[Social Victorians/Timeline/1900s#End of May 1902|party at Blenheim Palace hosted by the Duke and Duchess of Marlborough]]. 

'''1902 June 11, Monday''', the Hon. Mrs. Algernon Bourke had a dog entered in the [[Social Victorians/Timeline/1900s#Ladies' Kennel Association Show|Ladies' Kennel Association competitions in the Botanic Gardens]]. 

'''1902 September 4, Thursday''', the ''Daily Express'' reported that &quot;Mrs. Algernon Bourke is staying with Lord and Lady Alington at Scarborough.&quot;&lt;ref&gt;&quot;Onlooker.&quot; &quot;My Social Diary.&quot; &quot;Where People Are.&quot; ''Daily Express'' 04 September 1902, Thursday: 5 [of 8], Col. 1b? [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004848/19020904/099/0005. Print p. 4, Col. 7b [of 7].&lt;/ref&gt;

'''1902 September 22, Monday''', Guendoline Bourke was a guest at the [[Social Victorians/Timeline/1900s#Earl and Countess of Mar and Kellie's House Party|large house party hosted by the Earl and Countess of Mar and Kellie]].

'''1902 October 24, Friday''', the Hon. Algernon Bourke [[Social Victorians/Timeline/1900s#Annual Opening of the Prince's Ice-skating Rink|opened the Prince's ice-skating rink for the season]], which he had been doing since 1895.

'''1902 October 25, Saturday''', Algernon Bourke was bequeathed £500 by his uncle [[Social Victorians/People/Mayo|Robert Bourke]], who had died 3 September 1902.&lt;ref&gt;&quot;Will of Lord Connemara.&quot; ''Kildare Observer and Eastern Counties Advertiser'' 25 October 1902, Saturday: 2 [of 8], Col. 4b–c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001870/19021025/037/0002. Print title the ''Kildare Observer'', n.p.&lt;/ref&gt;&lt;p&gt;

'''1902 October 31, Friday''', the [[Social Victorians/Timeline/1900s#Annual Opening of the Prince's Ice-skating Rink|7th opening of the Prince's Skating Club]]. Guendoline Bourke was on the Women's Committee and Algernon Bourke was on the Men's.&lt;p&gt;

'''1902 November 8, Friday, beginning, perhaps''', Guendoline Bourke was part of the [[Social Victorians/Timeline/1900s#8 November 1902, Saturday|Earl and Countess of Warwick's shooting party at Easton Lodge]].&lt;p&gt;

'''1902 December 9, Tuesday''', Guendoline Bourke attended [[Social Victorians/Timeline/1900s#9 December 1902, Tuesday|Lady Eva Wyndham-Quin's &quot;at home,&quot; held at the Welch Industrial depot]] for the sale Welsh-made Christmas gifts and cards. Bourke wore &quot;a fur coat and a black picture hat.&quot;&lt;ref&gt;&quot;A Lady Correspondent.&quot; &quot;Society in London.&quot; ''South Wales Daily News'' 11 December 1902, Thursday: 4 [of 8], Col. 5a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000919/19021211/082/0004. Print p. 4.&lt;/ref&gt;

'''1903 February 6, Friday''', Hon. Mrs A. Bourke was present at a [[Social Victorians/Timeline/1900s#Dinner Party Hosted by Lord Lieutenant of Ireland and the Countess of Dudley|dinner party Hosted by Lord Lieutenant of Ireland and the Countess of Dudley]]. &lt;p&gt;

'''1903 February 9, Monday''', Guendoline Bourke was present at a [[Social Victorians/Timeline/1900s#Dinner Party Hosted by Lord Lieutenant of Ireland and the Countess of Dudley|house party at Dublin Castle hosted by the Lord Lieutenant and Countess of Dudley that began the Viceregal season]].

'''1903 March 17, Tuesday''', Guendoline Bourke staffed a booth at a [[Social Victorians/Timeline/1900s#1903 March 17, Tuesday|sale of the Irish Industries Association]] on St. Patrick's Day with [[Social Victorians/People/Mayo|Lady Mayo]], [[Social Victorians/People/Dudley|Georgina Lady Dudley]] and [[Social Victorians/People/Beresford|Miss Beresford]]. A number of other aristocratic women were also present at the sale in other booths, including [[Social Victorians/People/Londonderry|Lady Londonderry]] and [[Social Victorians/People/Lucan|Lady Lucan]].

'''1903 June 19, Friday''', Guendoline Bourke was invited to the [[Social Victorians/Timeline/1900s#Grand Ball in the Waterloo Chamber at Windsor Castle|grand ball at Windsor Castle]], the end of the Ascot-week festivities.

'''1903 June 23, Tuesday''', Guendoline and Daphne Bourke were invited to a [[Social Victorians/Timeline/1900s#1903 June 23, Tuesday|children's party at Buckingham Palace for Prince Eddie's birthday]].

'''1903 July 10, Friday, or so''', Guendoline Bourke attended a [[Social Victorians/Timeline/1900s#Party Hosted by the Duke and Duchess of Marlborough|party hosted by the Duke and Duchess of Marlborough]].

'''1904 May 17, Tuesday''', Guendoline Bourke had agreed to let Daphne appear in the tableaux vivants arranged by Sir Philip Burne-Jones for the [[Social Victorians/Timeline/1900s#Countess Cadogan's Great Bazaar|Countess of Cadogan's great bazaar]]. Some mothers had had to decline because of the outbreaks of measles and chicken pox.&lt;p&gt;

'''1904 June 30, Thursday''', Guendoline and Daphne Bourke attended another birthday party for Prince Eddie at Buckingham Palace, and the ''Gentlewoman'' says, &quot;No prettier little girl was to be seen that day than little Miss Daphne Bourke, the daughter of the Hon. Mrs. Algernon Bourke, with her wonderful Irish eyes and colouring, her pretty white frock being relieved with a rose pink sash.&quot;&lt;ref&gt;&quot;Prince Eddie's Birthday.&quot; ''Gentlewoman'' 02 July 1904, Saturday: 68 [of 92]. Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19040702/360/0068. Print: title the same, p. 42.&lt;/ref&gt;&lt;p&gt;

'''1904 September 15, Thursday''', according to what was at the time called the ''Irish Daily Independent and Nation'', Algernon Bourke was living in Venice and not in the UK at this point:&lt;blockquote&gt;Algernon Bourke, who usually lives in Venice, has spent some time in England during the present summer, and has now gone on a fishing expedition to Sweden, accompanied by his brother, Lord Mayo. Lady Mayo has been staying meanwhile in Ireland, and has had a visit from her mother, Lady Maria Ponsonby, who is a sister of Lend Obventry.&lt;ref name=&quot;:10&quot;&gt;&quot;Society Notes.&quot; ''Irish Independent'' 15 September 1904, Thursday: 4 [of 8], Col. 5b [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001986/19040915/131/0004. Print title: ''Irish Daily Independent and Nation'', p. 4.&lt;/ref&gt;&lt;/blockquote&gt;

'''1904 October 22, Saturday''', the ''Gentlewoman'' reported that &quot;Mrs. Algernon Bourke is paying a visit to Venice, which Mr. Bourke has made his headquarters for several years past, as he is connected with some very artistic stone and marble works situated near the Grand Canal.&quot;&lt;ref&gt;&quot;The Social Peepshow.&quot; ''Gentlewoman'' 22 October 1904, Saturday: 24 [of 6ths 8], Col. 2c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19041022/112/0024. Print title same, p. 672.&lt;/ref&gt;

'''1905 February 17, Friday''', the Dundee ''Evening Post'' reported that Algernon Bourke &quot;set up a shop in Venice for the sale of art treasures and old furniture.&quot;&lt;ref&gt;&quot;Social News.&quot; Dundee ''Evening Post'' 17 February 1905, Friday: 6 [of 6], Col. 7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000582/19050217/105/0006. Print p. 6.&lt;/ref&gt;

'''1905 April 26, Wednesday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1900s#New Forest United Hunt Ball|New Forest United Hunt Ball]], as did her brother Captain R. C. H. Sloane Stanley and his wife Olivia Countess Cairns.&lt;p&gt;

'''1905 June 5, Monday''', Algernon Bourke wrote to the ''Times'' from Venice that &quot;The Venetian wits have suggested a motto for Admiral Togo, Togo Tenga Tutto (Togo takes the lot).&quot;&lt;ref&gt;&quot;Mr. Algernon Bourke.&quot; ''Hull Daily Mail'' 08 June 1905, Thursday: 2 [of 6], Col. 6a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000324/19050608/008/0002. Print title ''Daily Mail'', p. 6.&lt;/ref&gt;&lt;p&gt;

'''1905, last week of July''', Guendoline Bourke and daughter Daphne Bourke — who was 10 years old — attended [[Social Victorians/Timeline/1900s#Last week of July, 1905|Lady Cadogan's children's party at Chelsea House]]. Daphne was &quot;One of loveliest little girls present.&quot;&lt;ref&gt;&quot;Court and Social News.&quot; ''Belfast News-Letter'' 01 August 1905, Tuesday: 7 [of 10], Col. 6b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000038/19050801/157/0007. Print p. 7.&lt;/ref&gt;&lt;p&gt;

'''1906 March 9, Friday''', Guendoline Bourke was a reference for Mr. Frances Burgess, who taught piano, singing, voice production, organ and music theory. Burgess was &quot;Organist and Choirmaster of St. Columbs', North Kensington, Director of the Plainsong and Medieval Music Society's Choir, etc., etc.&quot;&lt;ref name=&quot;:21&quot;&gt;&quot;Mr. Francis Burgess.&quot; ''Kilburn Times'' 9 March 1906, Friday: 3 [of 8], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001813/19060309/086/0003. Print title: ''Kilburn Times Hampstead and North-western Press'', p. 3.&lt;/ref&gt;&lt;p&gt;

'''1906 December 10, Monday''', Guendoline Bourke was seen in the tea room, possibly with Lady Grosvenor, at [[Social Victorians/Timeline/1900s#1906 December 10, Monday|Lady Dudley's sale of Irish needlework]].&lt;p&gt;

'''1907 May''', a &quot;naval signalling incident&quot; [[Social Victorians/Timeline/1887#May 1887|caused the Waterford ''Evening News'' to recall a similar event]] that had occurred 20 years earlier, in which Algernon Bourke, as special correspondent for the ''Times'', publicized [[Social Victorians/People/Beresford|Lord Charles Beresford]]'s use of his ship's signalling capabilities to send a message to his wife about being late for dinner:&lt;blockquote&gt;

The naval signalling incident is still in the air. It is expected that the matter will not be threshed out until Emperor William leaves England. A story of a former signalling incident in which [[Social Victorians/People/Beresford|Lord Charles Beresford]] was concerned is going the rounds at the moment.&lt;/blockquote&gt;

'''1907 August 24, Saturday''', Algernon Bourke was present at [[Social Victorians/Timeline/1900s#Polo Week at Eaton Hall, Duke and Duchess of Westminster|Polo Week at Eaton Hall, hosted by the Duke and Duchess of Westminster]].

'''1908 July 30, Thursday''', Guendoline Bourke was at [[Social Victorians/Timeline/1900s#Glorious Goodwood. Cup Day and Dresses.|Cup Day at the Goodwood races]], wearing salmon-pink with a matching hat.

'''1909 April 20, Tuesday''', Lady Rosemary Cairns — daughter of Olivia Sloan-Stanley, Countess Cairns and Cyril Sloane-Stanley — and Wyndham Portal were [[Social Victorians/Timeline/1900s#20 April 1909, Tuesday|married in St. Margaret's, Westminster]]. Lavender and Diane Sloane-Stanley were bridesmaids.&lt;p&gt;

'''1909 May 22, Saturday''', Algernon Bourke appears to have been living in Pisa. A columnist for the ''Queen'' reported on the Royal School of Art Needlework:&lt;blockquote&gt;Lady Leconfield [?] was there, also her sister-in-law, the [[Social Victorians/People/Mayo|Dowager Lady Mayo]], only just back from her winter on the Continent, when she spent most of the time at Pisa, where her son Mr Algernon Bourke has also been staying. The latter is a great connoisseur as regards [art?] notably in what is really good in the way of old Italian sculpture and carving. He and his handsome wife have a place near to Putney, and this winter again Mr Bourke, as the result of his Italian travels, has been sending home such relics of the old Italian palace gardens as as stone and marble carved vases, garden seats, and what-not of the kind — not all for himself and his own gardens by any means, I fancy; but his friends, relying on his knowledge in such matters, get him when abroad to choose for [them?] the adornment of their English terraces and gardens.&lt;ref&gt;&quot;My Social Diary.&quot; The ''Queen'' 22 May 1909, Saturday: 31 [of 86], Col. 1b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/19090522/203/0031. Print p. 871.&lt;/ref&gt;&lt;/blockquote&gt;'''1909 September''', the Hon. Algernon Bourke was among the [[Social Victorians/Timeline/1900s#Visitors in Venice from the U.K.|many visitors from &quot;England&quot; in Venice]] in September.

=== 1910s ===
'''1910 April 20, Wednesday''', the ''Tatler'' printed an &quot;open letter&quot; to Geraldine, Countess of Mayo, as part of its &quot;The Searchlight in Society&quot; series and mentioned Algernon Bourke, saying he had been keeping &quot;a curiosity shop at Venice&quot;:&lt;blockquote&gt;The Bourkes have brains, and a good example is afforded by Mr. Algernon Bourke, next brother to Lord Mayo and heir-presumptive to the title. He is a good-looking man who used to be known as Buttons Bourke, and he married well, as his wife was the rich and pretty Miss Guendolen Sloane Stanley. He may be described as a &quot;Jack of all trades,&quot; but it is not I who will say that he is a master of none. He was once in the Stock Exchange, then he took White's Club in hand and restored it to much of its former prestige. After that he dabbled in smart hotels and restaurants, and the last thing I heard of him was that he kept a curiosity shop at Venice.&lt;ref&gt;Candida. &quot;The Searchlight in Society. Our Open Letter. No. CII. The Countess of Mayo.&quot; The ''Tatler'' 20 April 1910, Wednesday: 18 [of 42], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001852/19100420/023/0018. Print title same, p. 72.&lt;/ref&gt;&lt;/blockquote&gt;

'''1911 November 21, Tuesday''', Guendoline Bourke assisted the [[Social Victorians/Timeline/1910s#21 November 1911, Tuesday|Duchess of Marlborough at her at-home]] that included a sale of work by the wives of prisoners.&lt;p&gt;

'''1912 September 27, Friday''', Guendoline and Daphne Bourke were visiting Mr. and Mrs. Shelley Bontein, her mother and stepfather.&lt;ref&gt;&quot;From 'The World.'&quot; ''Berks and Oxon Advertiser'' 27 September 1912, Friday: 2 [of 8], Col. 4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002298/19120927/014/0002. Same print title, n.p.&lt;/ref&gt;&lt;p&gt;

'''1913 April 23, Wednesday''', the Irish Independent reported that Guendoline and Daphne Bourke had arrived in London for the season:&lt;blockquote&gt;&lt;p&gt;

The Hon. Mrs. Algernon Bourke and Miss Bourke have arrived for the season at 75 Gloucester place, Portman square, London.&lt;ref&gt;&quot;Social and Personal.&quot; ''Irish Independent'' 23 April 1913, Wednesday: 4 [of 10], Col. 5b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001715/19130423/081/0004. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;&lt;p&gt;

'''1913 May 7, Wednesday''', Guendoline Bourke presented her daughter Daphne Bourke at court:&lt;blockquote&gt;Mrs. Algernon Bourke presented her daughter, and wore blue and gold broché with a gold lace train.&lt;ref&gt;&quot;Social and Personal.&quot; London ''Daily Chronicle'' 08 May 1913, Thursday: 6 [of 12], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005049/19130508/120/0006. Print p. 6.&lt;/ref&gt;&lt;/blockquote&gt;

The ''Pall Mall Gazette'' has a description of Daphne Bourke's dress, but what exactly &quot;chiffon [[Social Victorians/Terminology#Hoops|paniers]]&quot; means in 1913 is not clear:&lt;blockquote&gt;Court dressmakers appear to have surpassed all previous records in their efforts to make the dresses for to-night’s Court as beautiful as possible. Noticeable among these is the dainty presentation gown to be worn by Miss Bourke, who will be presented by her mother, the Hon. Mrs. Algernon Bourke. This has a skirt of soft white satin draped with chiffon [[Social Victorians/Terminology#Hoops|paniers]] and a bodice veiled with chiffon and trimmed with diamanté and crystal embroidery. Miss Bourke’s train, gracefully hung from the shoulders, is of white satin lined with pale rose pink chiffon and embroidered with crystal and diamanté.&lt;ref&gt;&quot;Fashion Day by Day. Lovely Gowns for To-night's Court.&quot; ''Pall Mall Gazette'' 07 May 1913, Wednesday: 13 [of 18], Col. 1a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/19130507/199/0013. Print n.p.&lt;/ref&gt;&lt;/blockquote&gt;The ''London Evening Standard'' describes Guendoline and Daphne Bourke the same way except with differences in editing:&lt;blockquote&gt;Miss Bourke: Presented by her mother, the Hon. Mrs. Algernon Bourke. Dainty presentation gown of white satin, the skirt draped with chiffon paniers, bodice veiled chiffon and trimmed with diamanté and crystal embroidery. Train gracefully hung from shoulder of white satin embroidered with crystal and diamanté, lined with pale rose pink chiffon.&lt;ref&gt;&quot;Some of the Dresses.&quot; &quot;The King and Queen. Third Court. Most Brilliant of the Year.&quot; ''London Evening Standard'' 08 May 1913, Thursday: 11 [of 18], Col. 4b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/19130508/237/0011. Print title ''The Standard'', p. 11.&lt;/ref&gt;&lt;/blockquote&gt;

According to the ''Lady's Pictorial'', Daphne Bourke's dress was designed and constructed by [[Social Victorians/People/Dressmakers and Costumiers#Messrs Russell and Allen|Messrs. Russell and Allen]], Old Bond-street, W., and the description is identical (except for a couple of commas).&lt;ref&gt;&quot;Their Majesties' Court.&quot; ''Lady's Pictorial'' 17 May 1913, Saturday: 35 [of 64], Col. 2c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005980/19130517/296/0035. Same print title, p. 787.&lt;/ref&gt;

'''1914 May 11, Monday''', Guendoline and Daphne Bourke attended a [[Social Victorians/Timeline/1910s#Dance at the Ritz Hosted by Mrs. George Marjoribanks|dance at the Ritz hosted by Mrs. George Marjoribanks]].

'''1915 January 1, Friday''', Algernon Bourke is listed as being on the Executive Committee of the [[Social Victorians/Timeline/1910s#1915 January 1, Friday|National Food Fund, publicized by the ''Conservative and Unionist Women's Franchise Review'']].

'''1916 August 25, Friday''', Daphne Bourke's and John Fortescue's engagement was announced:&lt;blockquote&gt;A most attractive prospective bride (says the &quot;Star&quot;) is Mr. and Mrs. Algernon Bourke's only daughter, Miss Daphne Bourke, whose engagement has just taken place to Mr. Fortescue, of the Coldstream Guards. Miss Bourke is tall, dark, and very beautiful; and Mr. Fortescue is one of the family of Boconoc, Cornwall, and Dropmore, Maidenhead. At the latter place the two families have been neighbours, for Mr. and Mrs. Algernon Bourke have a charming country residence at Taplow, while Dropmore is famous for its magnificent gardens.&lt;ref&gt;&quot;Personalia.&quot; ''Uxbridge &amp; W. Drayton Gazette'' 25 August 1916, Friday: 4 [of 8], Col. 5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002285/19160825/043/0004. Print title ''The Advertiser'', p. 4.&lt;/ref&gt;&lt;/blockquote&gt;&lt;p&gt;'''1917 June 7, Thursday''', Daphne Bourke and John Grenville Fortescue [[Social Victorians/Timeline/1910s#7 June 1917, Thursday|married in the Coldstream Guards' chapel]].

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==
According to both the ''Morning Post'' and the ''Times'', the Hon. Algernon Bourke was among the Suite of Men in the [[Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts#&quot;Oriental&quot; Procession|&quot;Oriental&quot; procession]] at the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]].&lt;ref name=&quot;:2&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;&lt;ref name=&quot;:3&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt; Based on the people they were dressed as, Guendoline Bourke was probably in this procession but it seems unlikely that Algernone Bourke was.

[[File:Guendoline-Irene-Emily-Bourke-ne-Sloane-Stanley-as-Salammb.jpg|thumb|alt=Black-and-white photograph of a standing woman richly dressed in an historical costume with a headdress and a very large fan|Hon. '''Guendoline''' Bourke as Salammbô. ©National Portrait Gallery, London.]]
=== Hon. Guendoline Bourke ===
[[File:Alfons Mucha - 1896 - Salammbô.jpg|thumb|left|alt=Highly stylized orange-and-yellow painting of a bare-chested woman with a man playing a harp at her feet|Alfons Mucha's 1896 ''Salammbô''.]]
Lafayette's portrait (right) of &quot;Guendoline Irene Emily Bourke (née Sloane-Stanley) as Salammbô&quot; in costume is photogravure #128 in the '''Album''' presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:4&quot;&gt;&quot;Devonshire House Fancy Dress Ball (1897): photogravures by Walker &amp; Boutall after various photographers.&quot; 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.&lt;/ref&gt; The printing on the portrait says, &quot;The Hon. Mrs. Algernon Bourke as Salammbo.&quot;&lt;ref name=&quot;:23&quot;&gt;&quot;Mrs. Algernon Bourke as Salammbo.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158491/Guendoline-Irene-Emily-Bourke-ne-Sloane-Stanley-as-Salammb.&lt;/ref&gt;

The Lafayette Archive has 2 additional poses from the same session on 5 July 1897 as the one chosen for the Album:

* Same image as the Album photograph but higher resolution than the one the National Portrait Gallery, London, gives permission to post (Neg. No. GP [L] [http://lafayette.org.uk/bou1369.html 1369]).
* Standing with fan behind head, includes close-up of skirt fabric and left hand (Neg. No. GP [L] [http://lafayette.org.uk/bou1368-444.html 1368-444]).
* Reclining on pillows and furs, includes close-up of face and headdress (Neg. No. GP [L] [http://lafayette.org.uk/bou1368-442.html 1368-442]).

==== Newspaper Accounts ====
The Hon. Mrs. A. Bourke was dressed as Salambo in the Oriental procession&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; in a costume made by [[Social Victorians/People/Dressmakers and Costumiers#Mrs. Mason|Mrs. Mason]], although Russell Harris says [[Social Victorians/People/Dressmakers and Costumiers#Mr. Charles Alias|Charles Alias]] was the &quot;Costume supplier.&quot;&lt;ref&gt;{{Cite web|url=http://lafayette.org.uk/bou1369.html|title=Hon Mrs Algernon Bourke (1870-1967), née Guendoline Irene Emily Sloane-Stanley by Lafayette 1897|website=lafayette.org.uk|access-date=2025-07-05}}&lt;/ref&gt;

Besides the two that mention her — the ''Morning Post'' and the ''Times'' — only two describe her costume, the London ''Evening Standard'' and the ''Gentlewoman'':
* &quot;Mrs. A. Bourke, as an Egyptian Princess, with the Salambo coiffure, wore a flowing gown of white and silver gauze covered with embroidery of lotus flowers. The top of the gown was ornamented with old green satin embroidered with blue turquoise and gold, and studded with rubies. The train was of old green broché with sides of orange and gold embroidery, and from the ceinture depended long bullion fringe and an embroidered ibis.&quot;&lt;ref&gt;“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.&lt;/ref&gt;{{rp|p. 3, Col. 3b}}
* &quot;(Egyptian Princess), drapery gown of white and silver gauze, covered with embroidery of lotus flowers; the top of gown appliqué with old green satin embroidered blue turquoise and gold, studded rubies; train of old green broché.&quot;&lt;ref&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 40, Col. 3a}}

==== Commentary on Her Costume ====

*The heavy headdress is likely attached to a wig and further stabilized by something under Bourke's chin and by the filet around her forehead. The flowers appear to have jewels at the center. What looks like a neck treatment is probably part of the structure holding the headpiece in place. The placement of the flowers is clearly a copy of the headdress worn by Rose Caron in the 1892 Paris production of the Reyer opera. Made by Eugène Lacoste, this textile, metal and pearl ''coiffe'' still exists (https://www.gazette-drouot.com/article/bijoux-d-opera/54941) and is held by the Bibliothèque Nationale de France.&lt;ref&gt;Merle, Sandrine. &quot;Les bijoux d’Opéra en pleine lumière au Palais Garnier.&quot; ''La Gazette Druout'' 4 July 2025 (No. 26). Retrieved July 2025

https://www.gazette-drouot.com/article/bijoux-d-opera/54941.&lt;/ref&gt; It seems to have been constructed from metal for the gold framework with the leaf design, beads attached to hanging metal frameworks (the blue-green beads look like they're covered with fabric or wrapped with thread) and flowers that may have been made from stiffened pinked fabric (likely organdy) whose color may have darkened in the century that has passed.
*Bourke's posture as she leans in the reclining odalisque pose shows she is wearing a corset.
*The translucent overskirt looks like an apron attached under the bust and hanging from the bodice.
*Russell Harris says the large fan looks more Egyptian rather than Carthaginian.&lt;ref name=&quot;:25&quot; /&gt; 
*The fullness of the skirt is very controlled.
*Her shoes are slippers decorated with real or artificial jewels.

==== Salammbô ====
Salammbô is the fictitious protagonist in Gustave Flaubert's 1862 novel ''Salammbô'', set during the Roman war against Carthage.&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|date=2024-04-29|title=Salammbô|url=https://en.wikipedia.org/w/index.php?title=Salammb%C3%B4&amp;oldid=1221352216|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Salammb%C3%B4.&lt;/ref&gt; Salammbô is a Carthaginian priestess of the lunar goddess Tanit. Matho, a Roman mercenary, breaks into Tanit's temple and steals her sacred veil — the spiritual guardian of Carthage. Salammbô sneaks into the enemy encampment to steal the veil back. She meets Matho in his tent, and &quot;believing each other to be divine apparitions,&quot; they make love,&lt;ref name=&quot;:5&quot; /&gt; although it is also a defilement. Salammbô succceds in getting the veil back, but Matho is tortured and executed, which causes her to die of shock, the effect of both having touched the veil. The plot of the opera is not identical to that of the novel.

What Guendoline Bourke saw in Salammbo as representative of herself is difficult to know, unless the costume itself appealed to her. A production of Ernest Reyer's opera ''Salammbô'', based on Flaubert's novel and published in Paris in 1890, opened at the Paris Opéra on 16 May 1892,&lt;ref&gt;{{Cite journal|date=2024-04-11|title=Ernest Reyer|url=https://en.wikipedia.org/w/index.php?title=Ernest_Reyer&amp;oldid=1218353215|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Ernest_Reyer.&lt;/ref&gt; starring Madame Rose Caron, with Mademoiselle Lucienne Bréval performing when Caron was on vacation.&lt;ref&gt;Jullienn, Adolphe. &quot;Mademoiselle Lucienne Bréval de L'Académie Nationale de Musique [or de l'Opéra in the Table of Contents].&quot; ''Le Théatre'' April 1898 (No. 4). Google Books https://www.google.com/books/edition/_/_oxRAQAAMAAJ. Pp. 8–10.&lt;/ref&gt;{{rp|8, Col. 2c}} This production was widely reviewed and discussed in the papers in the UK, and its production design was notable, especially Caron's costumes, the sets and [[Social Victorians/People/Bourke#Scale of the Production of Salammbo|the very scale of the production]]. Bourke or her costumier clearly saw the opera or images of the performers or its posters, certainly influencing the design of her costume.

* Rose Caron in her Salammbo costume is here: https://www.gettyimages.com/detail/news-photo/rose-caron-french-soprano-in-costume-in-the-title-role-of-news-photo/1439485238.
* A headshot of Bréval in costume is here: https://books.google.com/books/content?id=_oxRAQAAMAAJ&amp;pg=RA3-PP7&amp;img=1&amp;zoom=3&amp;hl=en&amp;bul=1&amp;sig=ACfU3U2Gv8Os_rEmx2gM9SakJkYLJ9hW7g&amp;ci=6%2C1%2C988%2C1371&amp;edge=0.)
* &quot;Salammbo's hair [was] powdered with a violet dust when she first appeared before the eyes of Matho.&quot;&lt;ref&gt;&quot;Salome.&quot; ''Pall Mall Gazette'' 27 February 1893, Monday: 3 [of 8]. Col. 2b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18930227/010/0003. Same print title and p.&lt;/ref&gt;

Salammbo figured in paintings, sculptures and illustrations of editions of Flaubert's novel before Ernest Reyer's 1890 opera. Translations and illustrated editions of Flaubert's novel had come out steadily beginning in the 1880s. Alfons Mucha's 1896 lithograph of Salammbô (above left) was published the year before the ball. Salammbo is often depicted as nude and highly sexual or sexualized (kissing a huge snake, for example, that she holds aloft). Reyer's opera was first produced in 1890 in Brussels. Both Modest Mussorgsky and Sergei Rachmaninoff had attempted but not completed operas based on the novel as well.&lt;ref name=&quot;:5&quot; /&gt; 


Guendoline Bourke's costume and her social life as reported in the newspapers do not suggest that she was a big risk-taker like, for example, the eccentric la Comtesse de Castiglione, who appeared at a ball in a Salammbo costume in 1886, 4 years after Flaubert's novel was first published. In 1889 the ''Edinburgh Evening News'' exaggerates her nudity and doesn't describe the rush in the ballroom to see her but does address the lingering memory:&lt;blockquote&gt;The late Countess Castiglione, whose death in Paris is recorded yesterday, made her first appearance at the Imperial Court in 1866, where her extraordinary beauty made a great impression on Napoleon III., and eventually led to the Empress Eugenie’s undertaking an unexpected and much-talked-of visit to Scotland. The Countess had a face and complexion which would have enchanted Rubens, and her lovely golden hair touched her feet. Even at the present day Paris has not forgotten her costume, or rather absence of costume as Salammbo, in which character she figured at a certain memorable ball at the Tuileries, wearing her hair, her jewels, and very little else. The Empress Eugenie, when she was presented to her thus lightly arrayed, declared that she must be cold, and insisted upon her there and then donning a mantle. Mme. de Castiglione was never again invited to an entertainment over which the Empress Eugenie presided.&lt;ref&gt;&quot;A Countess’ Queer Ball Costume.&quot; ''Edinburgh Evening News'' 2 December 1899, Saturday: 2 [of 6], Col. 7b [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000452/18991202/024/0002. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;Given how widely this incident was discussed at the time of the death of la Comtesse in 1889, Guendoline Bourke might easily have known about it. But she was developing relationships with people like the Princess of Wales, and what Countess Castigiolone did does not sound at all like her. 

===== Scale of the Production of ''Salammbo'' =====

* &quot;In Reyer's opera of 'Salammbo,' lately produced at the Grand Opera in Paris, there were 1,400 persons on the stage in the last act.&quot;&lt;ref&gt;&quot;Facts and Fancies.&quot; ''Louth and North Lincolnshire Advertiser'' 9 July 1892, Saturday: 3 [of 8], Col. 6c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000313/18920709/038/0003. Same print title and p.&lt;/ref&gt;
* &quot;the battle scene in [''Salammbo''] requires no less than 3000 square yards of 'decorative surface' [probably canvas]. This establishes a record, the next largest surface being that of the salles des fetes in 'Don Giovanni.'&quot;&lt;ref&gt;&quot;A French paper gives interesting details....&quot; ''Sevenoaks Chronicle and Kentish Advertiser'' 26 August 1892, Friday: 2 [of 8], Col. 3c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001067/18920826/032/0002. Same print title, n.p.

&lt;/ref&gt;

===== Influence of the Production of ''Salammbo'' =====
Rose Caron's productions were influential, including for the costumes she wore. The 1892 ''Lohengrin'' she starred in was the source of the costumes worn by [[Social Victorians/People/Stonor#Julia Caroline Stonor, Marquise of Hautpoul|Julia Stonor, Marquise of Hautpoul]] and her brother, [[Social Victorians/People/Stonor#Hon. Harry Stonor|Hon. Harry Stonor]].

Women's clothing was influenced by the costumes in the opera, particularly those worn by Rose Caron. One color of intense red was called Salammbo. A bonnet was named the Salammbô:&lt;blockquote&gt;About the smartest thing in bonnets for ordinary complimentary mourning is called the Salammbô, and is copied from a head-dress worn by a leading artiste at one of the Paris theatres. It is made of jet, and has a rose on each side of the front from the centres of which rise two black ospreys.&lt;ref&gt;Mantalini, Miss. &quot;The Shows in the London Shops. With Mems. about Millinery.&quot; ''Pall Mall Budget'' 29 December 1892, Thursday: 22 [of 40], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005967/18921229/092/0022. Same print title, p. 1928.&lt;/ref&gt;&lt;/blockquote&gt;In a long illustrated article describing the wedding of Princess Marie of Edinburgh, the ''Lady's Pictorial'' provides a sketch of &quot;a very pretty [hat] (No. 4) of brown mirror velvet trimmed with mink and a brown velvet bow in front '''with Salammbo''' '&lt;nowiki/&gt;''fantaisie''&lt;nowiki/&gt;'&quot; that is among the bride's millinery.&lt;ref&gt;&quot;The Marriage of H.R.H. Princess Marie of Edinburgh and H.R.H. Ferdinand Crown Prince of Roumania.&quot; ''Lady's Pictorial'' 14 January 1893, Saturday: 40 [of 76], Col. 3c [of 3]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0005980/18930114/064/0040. Same print title, p. 56.&lt;/ref&gt;&lt;p&gt;

Shoes appeared:&lt;blockquote&gt;At Mrs Merritt's, Savile-street, the stock is particularly attractive, there being so many new styles in shoes this season. One of the latest designs is the Salammbo Shoe, glace kid, with one strap, a jet buckle, and very low French heels. This shoe is especially designed for tender feet, as it is very light in weight.&lt;ref&gt;&quot;House and Home. Local Letter for Women Reader [sic], (By Our Lady Contributor).&quot; ''Hull Daily Mail'' 22 July 1897, Thursday: 5 [of 6], Col. 1b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000324/18970722/069/0005. Same print title, n.p.&lt;/ref&gt;&lt;/blockquote&gt;Patterns for making the Tunique Romaine and Corsage Salammbo were for sale just a few months after the opening:&lt;blockquote&gt;Some of the leading fashionable novelties described in ''Le Follet de Paris'' are almost ahead of the season, but they look so well that it will not be long before our provincial dressmakers have them. A revival and modification of the ancient tunic is one item which is transforming the modern gowns of tailor-built tweeds into long clinging draperies, of simple cut but ineffable grace. We have had the Russian blouse with us now for the last couple of months. Now the reign of Tunique Romaine and Corsage Salambo is upon us.

... [including the address to send off for the patterns]

A very successful novelty is the ''corsage'' “Salammbo.” In reality, it is more of a blouse and short tunic than a ''corsage'', as there is no attempt at shaping to the figure. In [sic] consists, indeed, of two straight pieces of material cut round on the shoulders, where the back and front are fastened together by clasps. There is no arm-hole, and the two pieces meet at the waist under the arm, and then hang open on to the skirt. There being no dart, the waist is as wide as the shoulders; the fullness is drawn to the centre under a ''ceinture Russe'', or of oxydised silver. The outlines are trimmed with ''galon'' or some similar garniture. The &quot;Salammbo” ''guimpe'' or ''corsage'' are made of flannel or ''mousseline de laine'' of bright colour, and are worn with fitting bodices or skirts of serge, or woollen of dark colour. They are very effective, and nothing can be easier to make, while their addition to a frock constitutes a separate costume.

The fitting bodices worn under the ''guimpes'' or ''robes'' &quot;Salammbo&quot; are very simply made; being round-waisted, they are without side pieces, and only require a seam under each arm; one in the centre of the back, and one or two darts in front, according to the figure. The skirt is mounted on a round waistband, and the ''ceinture'' worn over this gives the bodice and skirt the effect of a princess dress.&lt;ref&gt;&quot;A Womans Ceilidh.&quot; ''Oban Times and Argyllshire Advertiser'' 3 September 1892, Saturday: 3 [of 8], Col. 6a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000462/18920903/078/0003. Print title: ''The Oban Times'', p. 3.&lt;/ref&gt;&lt;/blockquote&gt;Stationery even before the opera opened in Paris:&lt;blockquote&gt;The last fad in fancy stationery is the carte Salammbo, a delightfully smooth surface for writing upon, the envelopes are very small, square, and of the wallet make; the paper folds over once to fit. The newest shades are rose pink, pale English blue, apple green, and the evergreen heliotrope.&lt;ref&gt;&quot;Fashions of the Month.&quot; ''Nottinghamshire Guardian'' 27 February 1892, Saturday: 7 [of 8], Col. 2b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000176/18920227/059/0007. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;[[File:Algernon Henry Bourke Vanity Fair 20 January 1898.jpg|thumb|alt=Old colored drawing of an elegant elderly man dressed in a 19th-century tuxedo with a cloak, top hat and formal pointed shoes with bows, standing facing 1/4 to his right|''Algy'' — Algernon Henry Bourke — by &quot;Spy,&quot; ''Vanity Fair'' 20 January 1898]]

=== Hon. Algernon Bourke ===
[[File:Hon-Algernon-Henry-Bourke-as-Izaak-Walton.jpg|thumb|left|alt=Black-and-white photograph of a man richly dressed in an historical costume sitting in a fireplace that does not have a fire and holding a tankard|Hon. Algernon Henry Bourke as Izaak Walton. ©National Portrait Gallery, London.]]
'''Lafayette's portrait''' (left) of &quot;Hon. Algernon Henry Bourke as Izaak Walton&quot; in costume is photogravure #129 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:4&quot; /&gt; The printing on the portrait says, &quot;The Hon. Algernon Bourke as Izaak Walton.&quot;&lt;ref&gt;&quot;Hon. Algernon Bourke as Izaak Walton.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158492/Hon-Algernon-Henry-Bourke-as-Izaak-Walton.&lt;/ref&gt;

This portrait is amazing and unusual: Algernon Bourke is not using a photographer's set with theatrical flats and props, certainly not one used by anyone else at the ball itself. Isaak Walton (baptised 21 September 1593 – 15 December 1683) wrote ''The Compleat Angler''.&lt;ref&gt;{{Cite journal|date=2021-09-15|title=Izaak Walton|url=https://en.wikipedia.org/w/index.php?title=Izaak_Walton&amp;oldid=1044447858|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Izaak_Walton.&lt;/ref&gt; A cottage Walton lived in and willed to the people of Stafford was photographed in 1888, suggesting that its relationship to Walton was known in 1897, raising a question about whether Bourke could have used the fireplace in the cottage for his portrait. (This same cottage still exists, as the [https://www.staffordbc.gov.uk/izaak-waltons-cottage Isaak Walton Cottage] museum.)

A caricature portrait (right) of the Hon. Algernon Bourke, called &quot;Algy,&quot; by Leslie Ward (&quot;Spy&quot;) was published in the 20 January 1898 issue of ''Vanity Fair'' as Number 702 in its &quot;Men of the Day&quot; series,&lt;ref&gt;{{Cite journal|date=2024-01-14|title=List of Vanity Fair (British magazine) caricatures (1895–1899)|url=https://en.wikipedia.org/w/index.php?title=List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899)&amp;oldid=1195518024|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899).&lt;/ref&gt; giving an indication of what he looked like out of costume.

=== Mr. and Mrs. Bourke ===
The ''Times'' made a distinction between the Hon. Mr. and Mrs. A. Bourke and Mr. and Mrs. Bourke, including both in the article.&lt;ref name=&quot;:3&quot; /&gt; Occasionally this same article mentions the same people more than once in different contexts and parts of the article, so they may be the same couple. (See [[Social Victorians/People/Bourke#Notes and Question|Notes and Question]] #2, below.)

== Demographics ==

=== The Bourkes ===
*Nationality: Anglo-Irish&lt;ref&gt;{{Cite journal|date=2020-11-14|title=Richard Bourke, 6th Earl of Mayo|url=https://en.wikipedia.org/w/index.php?title=Richard_Bourke,_6th_Earl_of_Mayo&amp;oldid=988654078|journal=Wikipedia|language=en}}&lt;/ref&gt;
*Occupation: journalist. 1895: restaurant, hotel and club owner and manager&lt;ref&gt;''Cheltenham Looker-On'', 23 March 1895. Via Ancestry but taken from the BNA.&lt;/ref&gt;

==== Residences ====
*Ireland: 1873: Palmerston House, Straffan, Co. Kildare.&lt;ref name=&quot;:7&quot; /&gt; Not Co. Mayo?
*1888–1891: 33 Cadogan Terrace, S.W., Kensington and Chelsea, a dwelling house&lt;ref&gt;Kensington and Chelsea, London, England, Electoral Registers, 1889–1970, Register of Voters, 1891.&lt;/ref&gt;
*1894: 181 Pavilion Road, Kensington and Chelsea&lt;ref&gt;Kensington and Chelsea, London, England, Electoral Registers, 1889–1970. Register of Voters, 1894. Via Ancestry.&lt;/ref&gt;
*1900: 181 Pavilion Road, Kensington and Chelsea&lt;ref&gt;Kensington and Chelsea, London, England, Electoral Registers, 1889–1970. Register of Voters, 1900. Via Ancestry.&lt;/ref&gt;
*1904: Algernon Bourke was &quot;usually liv[ing] in Venice&quot;&lt;ref name=&quot;:10&quot; /&gt;
*1906: 75, Gloucester-place, W.&lt;ref name=&quot;:21&quot; /&gt;
*Guendoline Bourke
*1911: 1911 Fulham, London&lt;ref name=&quot;:6&quot; /&gt;
*20 Eaton Square, S.W. (in 1897)&lt;ref name=&quot;:0&quot;&gt;{{Cite book|url=https://books.google.com/books?id=Pl0oAAAAYAAJ|title=Who's who|date=1897|publisher=A. &amp; C. Black|language=en}} 712, Col. 1b.&lt;/ref&gt; (London home of the [[Social Victorians/People/Mayo|Earl of Mayo]])

=== The Sloane-Stanleys ===

==== Residences ====

* 1871: Chester Street, St George Hanover Square (Census), with 5 servants, including a cook and a footman.&lt;ref name=&quot;:16&quot;&gt;The National Archives; Kew, London, England; ''1871 England Census''; Class: ''RG10''; Piece: ''104''; Folio: ''21''; Page: ''37''; GSU roll: ''838763''. Ancestry.com. ''1871 England Census'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations Inc, 2004.&lt;/ref&gt;
* 1881–1885&lt;ref&gt;''UK, City and County Directories, 1600s-1900s''. Ancestry.com. ''UK, City and County Directories, 1766 - 1946'' [database on-line]. Provo, UT, USA: Ancestry.com Operations, Inc., 2013.&lt;/ref&gt; [at least]: 14 Halkin Street, W., St. Georges, 14 servants, including a governess, a house steward, an under butler, a footman and a cook.&lt;ref&gt;''Census Returns of England and Wales, 1881''. Kew, Surrey, England: The National Archives of the UK (TNA): Public Record Office (PRO), 1881. Class: ''RG11''; Piece: ''98''; Folio: ''66''; Page: ''37''; GSU roll: ''1341022''. Ancestry.com and The Church of Jesus Christ of Latter-day Saints. ''1881 England Census'' [database on-line]. Provo, UT, USA: Ancestry.com Operations Inc, 2004.&lt;/ref&gt;
* 1888: 49, Cadogan-square, St. Luke, Chelsea&lt;ref&gt;Ancestry.com. ''London, England, Overseer Returns, 1863-1894'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2013.&lt;/ref&gt;
* 1899, Roger Cyril Sloane-Stanley: 4 Down St., St George, Hanover Square&lt;ref&gt;London Metropolitan Archives; London, England; ''Electoral Registers''. Ancestry.com. ''London, England, Electoral Registers, 1832-1965'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2010.&lt;/ref&gt;
* 1911, Roger Cyril Sloane-Stanley: Paultons, Ower, Romsey

== Family ==
*Hon. Algernon Henry Bourke (31 December 1854 – 7 April 1922)&lt;ref&gt;&quot;Hon. Algernon Henry Bourke.&quot; {{Cite web|url=https://www.thepeerage.com/p29657.htm#i296561|title=Person Page|website=www.thepeerage.com|access-date=2020-12-10}}&lt;/ref&gt;
*Guendoline Irene Emily Sloane-Stanley Bourke (c. 1869 – 30 December 1967)&lt;ref name=&quot;:1&quot;&gt;&quot;Guendoline Irene Emily Stanley.&quot; {{Cite web|url=https://www.thepeerage.com/p51525.htm#i515247|title=Person Page|website=www.thepeerage.com|access-date=2020-12-10}}&lt;/ref&gt;
#Daphne Marjory Bourke (5 April 1895 – 22 May 1962)

=== Relations ===
*Hon. Algernon Henry Bourke (the 3rd son of the [[Social Victorians/People/Mayo|6th Earl of Mayo]]) was the older brother of Lady Florence Bourke.&lt;ref name=&quot;:0&quot; /&gt;
*Wilfred Blunt was a cousin of Algernon Bourke: Bourke's mother's &quot;mother was one of the Blunts of Crabbet Park, Sussex, which makes them kinswomen of Mr. Alfred Scawen Blunt, poet, Egyptophil and counsel for Arabi Pasha in his trial.&quot;&lt;ref&gt;&quot;From ''Truth''.&quot; ''Mid-Lothian Journal'' 23 August 1912, Friday: 8 [of 8], 2c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002721/19120823/147/0008. Print title and p. same.&lt;/ref&gt;
*Lord Alfred Douglas was a cousin of Algernon Bourke.

==== Other Bourkes ====
*Hubert Edward Madden Bourke (after 1925, Bourke-Borrowes)&lt;ref&gt;&quot;Hubert Edward Madden Bourke-Borrowes.&quot; {{Cite web|url=https://www.thepeerage.com/p52401.htm#i524004|title=Person Page|website=www.thepeerage.com|access-date=2021-08-25}} https://www.thepeerage.com/p52401.htm#i524004.&lt;/ref&gt;
*Lady Eva Constance Aline Bourke, who married [[Social Victorians/People/Dunraven|Windham Henry Wyndham-Quin]] on 7 July 1885;&lt;ref&gt;&quot;Lady Eva Constance Aline Bourke.&quot; {{Cite web|url=https://www.thepeerage.com/p2575.htm#i25747|title=Person Page|website=www.thepeerage.com|access-date=2020-12-02}} https://www.thepeerage.com/p2575.htm#i25747.&lt;/ref&gt; he became 5th Earl of Dunraven and Mount-Earl on 14 June 1926.

=== The Sloane-Stanleys ===
* Emilie Josephine S Stanley ( 21 December 1848 [baptism]&lt;ref&gt;London Metropolitan Archives; &quot;London, England, UK&quot; ; ''London Church of England Parish Registers''; Reference Number: ''P87/Tri/001''. Ancestry.com. ''London, England, Church of England Births and Baptisms, 1813-1923'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2010.&lt;/ref&gt;  – October 1945)
* Hans T Sloane Stanley (11 May 1840 [baptism]&lt;ref&gt;Ancestry.com. ''England, Select Births and Christenings, 1538-1975'' [database on-line]. Provo, UT, USA: Ancestry.com Operations, Inc., 2014.&lt;/ref&gt; – 15 December 1888&lt;ref&gt;Ancestry.com. ''UK and Ireland, Find a Grave® Index, 1300s-Current'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2012.&lt;/ref&gt;)
* James Shell[e?]y Bontein ()
*# Gwendoline&lt;ref name=&quot;:14&quot; /&gt; Irene Emily G Stanley (c. 1870&lt;ref name=&quot;:16&quot; /&gt; – )
*# '''Roger Cyril Hans Sloane Stanley''' (29 April 1875&lt;ref&gt;The National Archives; Kew, Surrey, England; ''WO 42 War Office: Officers' Birth Certificates, Wills and Personal Papers 1755-1908''; Reference: ''WO 42/72''. Ancestry.com. ''UK, Officers' Birth Certificates, Wills and Personal Papers, 1755-1908'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2023.&lt;/ref&gt; – 18 November 1944&lt;ref&gt;''Find a Grave''. Find a Grave®. http://www.findagrave.com/cgi-bin/fg.cgi. Ancestry.com. ''UK and Ireland, Find a Grave® Index, 1300s-Current'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2012.&lt;/ref&gt;)


* Olivia Elizabeth Berens, Countess Cairns&lt;ref&gt;The National Archives of the UK (TNA); Kew, Surrey, England; ''Census Returns of England and Wales, 1911''. Ancestry.com. ''1911 England Census'' [database on-line]. Provo, UT, USA: Ancestry.com Operations, Inc., 2011.&lt;/ref&gt; (c. 1871 – 20 June 1951&lt;ref&gt;&quot;Olivia Elizabeth Berens.&quot; Person Page 3908; person #39077. ''The Peerage: A Genealogical Survey of the Peerage of Britain as well as the Royal Families of Europe''. https://www.thepeerage.com/p3908.htm#i39077.

&lt;/ref&gt;)
* Arthur William Cairns, 2nd Earl Cairns (21 December 1861 – 14 January 1890)&lt;ref name=&quot;:20&quot;&gt;&quot;Arthur William Cairns, 2nd Earl Cairns.&quot; Person Page 3908; Person #39076. ''The Peerage: A Genealogical Survey of the Peerage of Britain as well as the Royal Families of Europe''. https://www.thepeerage.com/p3908.htm#i39076.&lt;/ref&gt;
*# Lady Louise Rosemary Kathleen Virginia Cairns (10 March 1889 – 17 May 1962)&lt;ref name=&quot;:20&quot; /&gt;
* Roger Cyril Hans Sloane Stanley (1875 – 18 November 1944)
*# Lavender Elizabeth (20 May 1900 [baptism]&lt;ref&gt;Hampshire Archives and Local Studies; Winchester, England, UK; ''Anglican Parish Registers''; Reference: ''35M76/PR3''. Ancestry.com. ''Hampshire, England, Church of England Baptisms, 1813-1921''[database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2023.&lt;/ref&gt; – )
*# Diane Sloane Stanley (c. 1905 – )


* Lavender Elizabeth (20 May 1900 [baptism] – )
* John Everett ()


* Diane Sloane Stanley (c. 1905 – )
* Elwyn Villiers Rhys ()

== Writings, Memoirs, Biographies, Papers ==

=== Writings ===
* Bourke, the Hon. Algernon. ''The History of White's''. London: Algernon Bourke [privately published], 1892.
* Bourke, the Hon. Algernon, ed., &quot;with a brief Memoir.&quot; ''Correspondence of Mr Joseph Jekyll with His Sister-in-Law, Lady Gertrude Sloane Stanley, 1818–1838''. John Murray, 1893.
* Bourke, the Hon. Algernon, ed. ''Correspondence of Mr Joseph Jekyll''. John Murray, 1894.

=== Papers ===
* Where are the papers for the Earl of Mayo family? Are Algernon and Guendoline Bourke's papers with them?

== Notes and Questions ==
#The portrait of Algernon Bourke in costume as Isaac Walton is really an amazing portrait with a very interesting setting, far more specific than any of the other Lafayette portraits of these people in their costumes. Where was it shot? Lafayette is given credit, but it's not one of his usual backdrops. If this portrait was taken the night of the ball, then this fireplace was in Devonshire House; if not, then whose fireplace is it?
#The ''Times'' lists Hon. A. Bourke (at 325) and Hon. Mrs. A. Bourke (at 236) as members of a the &quot;Oriental&quot; procession, Mr. and Mrs. A. Bourke (in the general list of attendees), and then a small distance down Mr. and Mrs. Bourke (now at 511 and 512, respectively). This last couple with no honorifics is also mentioned in the report in the London ''Evening Standard'', which means the Hon. Mrs. A. Bourke, so the ''Times'' may have repeated the Bourkes, who otherwise are not obviously anyone recognizable. If they are not the Hon. Mr. and Mrs. A. Bourke, then they are unidentified. It seems likely that they are the same, however, as the newspapers were not perfectly consistent in naming people with their honorifics, even in a single story, especially a very long and detailed one in which people could be named more than once.
#Three slightly difficult-to-identify men were among the Suite of Men in the [[Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts#&quot;Oriental&quot; Procession|&quot;Oriental&quot; procession]]: [[Social Victorians/People/Halifax|Gordon Wood]], [[Social Victorians/People/Portman|Arthur B. Portman]] and [[Social Victorians/People/Sarah Spencer-Churchill Wilson|Wilfred Wilson]]. The identification of Gordon Wood and Wilfred Wilson is high because of contemporary newspaper accounts. The Hon. Algernon Bourke, who was also in the Suite of Men, is not difficult to identify at all. Arthur Portman appears in a number of similar newspaper accounts, but none of them mentions his family of origin.
#[http://thepeerage.com The Peerage] has no other Algernon Bourkes.
#The Hon Algernon Bourke is #235 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who were present]]; the Hon. Guendoline Bourke is #236; a Mr. Bourke is #703; a Mrs. Bourke is #704.
#Hans Stanley-Sloane's estate was £33,704 7s. 5d. in the final probate in December 1889,&lt;ref&gt;Principal Probate Registry; London, England; ''Calendar of the Grants of Probate and Letters of Administration made in the Probate Registries of the High Court of Justice in England''. Ancestry.com. ''England &amp; Wales, National Probate Calendar (Index of Wills and Administrations), 1858-1995'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2010.&lt;/ref&gt; which might lead his widow to consider remarrying.
== Footnotes ==
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      <text bytes="172566" sha1="bzex98bafhr5arlnbsg8x0kebvuf8hj" xml:space="preserve">Algernon and Guendoline Bourke were &quot;satirized&quot; by [[Social Victorians/People/Oscar Wilde|Oscar Wilde]] in his 1895 ''The Importance of Being Earnest''.&lt;ref&gt;Fahey, David M. ''The Politics of Drink in England, from Gladstone to Lloyd George''. Cambridge Scholars Publishing, Lady Stephenson Library, Newcastle upon Tyne, 2022: p. 21, n. 34. ''Google Books'' (retrieved July 2025) https://www.google.com/books/edition/The_Politics_of_Drink_in_England_from_Gl/8HRjEAAAQBAJ.&lt;/ref&gt; Algernon Bourke was a cousin of Lord Alfred Douglas, Wilde's Bosie, in 1898 attempting to facilitate a reconciliation between Bosie and his father.&lt;ref name=&quot;:26&quot;&gt;Ellmann, Richard. ''Osar Wilde''. Alfred A. Knopf, 1988.&lt;/ref&gt; (568)[[File:Leslie Ward - Vanity Fair, Newspapermen, ^Algy^, The Hon Algernon Henry Bourke, Januray 20, 1898 - B1979.14.521 - Yale Center for British Art.jpg|thumb|Hon. Algernon Bourke, ''Vanity Fair'', 1898]]

==Also Known As==
* Family name: Bourke [pronounced ''burk'']&lt;ref name=&quot;:62&quot;&gt;{{Cite journal|date=2024-05-07|title=Earl of Mayo|url=https://en.wikipedia.org/w/index.php?title=Earl_of_Mayo&amp;oldid=1222668659|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Earl_of_Mayo.&lt;/ref&gt;
* The Hon. Algernon Bourke
** Button Bourke&lt;ref&gt;&quot;A Tory 'Reformer' at the India Office.&quot; ''India'' 10 November 1911, Friday: 4 [of 12], Col. 1b [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004300/19111110/007/0004#. Print: same title, p. 228.&lt;/ref&gt;
** Algy Bourke
* Mrs. Guendoline Bourke
** Guendoline&lt;ref name=&quot;:1&quot; /&gt; [The National Portrait Gallery, London, uses this spelling for Lafayette's portrait of Bourke in costume for the ball,&lt;ref name=&quot;:23&quot; /&gt; as does Russell Harris in the Lafayette Archive, held by the Victoria and Albert&lt;ref name=&quot;:25&quot;&gt;Harris, Russell. &quot;Hon Mrs Algernon Bourke, née Guendoline Irene Emily Sloane-Stanley (1870-1967), as Salammbô.&quot; ''In Calm Prose'' 2011. http://www.rvondeh.dircon.co.uk/incalmprose/.&lt;/ref&gt;.]
** Gwendolen&lt;ref&gt;General Register Office. ''England and Wales Civil Registration Indexes''. London, England: General Register Office. FreeBMD. ''England &amp; Wales, Civil Registration Marriage Index, 1837-1915''[database on-line]. Provo, UT, USA: Ancestry.com Operations Inc, 2006.&lt;/ref&gt;{{rp|Marriage Index}} &lt;ref name=&quot;:15&quot; /&gt;{{rp|''Morning Post'' article about her name}} &lt;ref&gt;General Register Office. ''England and Wales Civil Registration Indexes''. London, England: General Register Office. FreeBMD. ''England &amp; Wales, Civil Registration Marriage Index, 1837-1915''[database on-line]. Provo, UT, USA: Ancestry.com Operations Inc, 2006.&lt;/ref&gt;{{rp|Electoral Register}}
** Gwendoline&lt;ref name=&quot;:14&quot;&gt;City of Westminster Archives Centre; London, England; ''Westminster Church of England Parish Registers''; Reference: ''SPWP/PR/1/2''. Ancestry.com. ''Westminster, London, England, Church of England Births and Baptisms, 1813-1919'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2020.&lt;/ref&gt;{{rp|Births and Baptisms}}
* Shelley Bontein and Emilie Sloane-Stanley Bontein
* See also the [[Social Victorians/People/Mayo|page for the Earl of Mayo]], the Hon. Algernon Bourke's father and then brother, and other Bourkes

== Overview ==

=== Algernon Bourke ===
Although the Hon. Algernon Henry Bourke was born in Dublin in 1854 and came from a family whose title is in the Peerage of Ireland,&lt;ref name=&quot;:6&quot;&gt;1911 England Census.&lt;/ref&gt; he seems to have spent much of his adult life generally in England and especially in London. He was &quot;a noted fisherman.&quot;&lt;ref&gt;&quot;London Correspondence.&quot; ''Freeman's Journal'' 21 December 1897, Tuesday: 5 [of 8], Col. 5c [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000056/18971221/027/0005. Same print title, n.p.&lt;/ref&gt;

Because he was the son of the [[Social Victorians/People/Mayo|Earl of Mayo]], perhaps, or perhaps because he was so involved in projects that got reported on, he was mentioned a great deal in the newspapers, but after his bankruptcy, he seems to have receded in prominence, in part because he was living outside of the U.K., and apparently separately from his wife, Guendoline Bourke.

Bourke ran as the Conservative candidate for Parliament from Clapham (population, c. 70,000) in 1885, a race he did not win. As a candidate he is described like this:&lt;blockquote&gt;Acted as a newspaper correspondent during the Zulu war. Subsequently Poor-law inspector in the West of Ireland. &quot;A loyal supporter of Church and State.&quot; Desires to reduce the School Board expenditure, and revive trade; and is opposed to Mr. Chamberlain's &quot;police of hasty and experimental reform.&quot;&lt;ref&gt;&quot;Clapham (70,000).&quot; ''South London Chronicle'' 17 October 1885, Saturday: 5 [of 8], Col. 5a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000443/18851017/113/0005. Print title ''South London Chronicle and Southwark and Lambeth Ensign'', p. 5.&lt;/ref&gt;&lt;/blockquote&gt;The London ''Weekly Dispatch'' says he is &quot;a dashing and unscrupulous young Tory.&quot;&lt;ref&gt;&quot;The Political Campaign in London.&quot; ''Weekly Dispatch'' (London) 15 November 1885, Sunday: 9 [of 16], Col. 3c [of 4]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003358/18851115/069/0009. Same print title and p.&lt;/ref&gt;

&quot;Algy&quot; Bourke was &quot;Man of the Day&quot; (No. DCCII [522) for ''Vanity Fair'' in 1898, caricatured by Leslie Ward (above right):&lt;blockquote&gt;Son of the great and murdered Lord Mayo, he is contemporary with the outbreak of the Crimean War, he is a Member of the London Stock Exchange, he has a beautiful wife and a daughter, and, being a very fashionable young man, he was once refused as their Member by the worthy electors of Clapham. He was an Eton boy, of course: and less naturally he went to Cambridge; where he was made President of the Beefsteak, the Amateur Dramatic, the Athenaeum, the True Blue, and the Hibernian Clubs. When he came down he tried journalism and went to Zululand as a ''Daily Telegraph'' ‘‘special”; after which he was improved into an Inspector of Workhouses [2, Col. 2c – 3, Col. 1a] in Ireland: which may account for his proficiency as a caterer. For seven years he worked under the late Mr. Chinnery on ''The Times'': being popularly supposed to look after that journal's morals. He is a good man of business, and a great organiser who has made White's Club pay even if it be less “smart&quot; than it was. He has done much for Willis’s since he took it in hand; he did well with his Battersea venture, and he thinks that he only failed with the Summer Club in Kensington Gardens because people would not go to the wrong side of the Park. Moreover, he runs a Club at Brighton, and he is Chairman of the Grand Hotel at Monte Carlo: whither he once organised a cheap trip. Altogether he is a veritable Clubman, and a very successful arranger of amusements, associations, and restaurants.

He is a popular fellow who is known to all of us; and though he is a little inclined to be quarrelsome, no one can get much the better of him. He is also a quick grasper of facts and a good talker. His favourite sports are fishing and the organising of associations for the introduction of salmon to the Thames. By way of being an art critic, he has made an interesting collection of engravings of the members of White’s Club from its foundation; but his friends say that he is not a well-dressed man. He has also written a history of White’s, and he is now writing one of Brooks's Club. He is a genial person, who looks as if the world agreed with him well.

He is an aquisition [sic] to a house party; and they call him “Algy.”&lt;ref&gt;&quot;Men of the Day.&quot; — No. DCCII [522]. The Hon. Algernon Henry Bourke.&quot; ''Vanity Fair'' 20 January 1898, Thursday: 2 [of 4], Col. 2c – 3, Col. 3a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/9900020/18980120/010/0002 and https://www.britishnewspaperarchive.co.uk/viewer/BL/9900020/18980120/005/0003. Same print title, pp. 41–42. Portrait is full page, on p. 1.&lt;/ref&gt;&lt;/blockquote&gt;The Hon. Algernon Bourke and Mr. Algernon Bourke, depending on the newspaper article, were the same person. Calling him Mr. Bourke in the newspapers, especially when considered as a businessman or (potential) member of Parliament, does not rule out the son of an earl, who would normally be accorded the honorific of ''Honorable''.

=== Guendoline Sloane-Stanley Bourke ===

Mrs. Guendoline Bourke exhibited at dog shows successfully and was a [[Social Victorians/Timeline/1900s#Society Sportswomen|noted deerstalker]] and &quot;an appreciative listener to good music.&quot;&lt;ref&gt;&quot;Vanity Fair.&quot; ''Lady of the House'' 15 June 1899, Thursday: 4 [of 44], Col. 2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004836/18990615/019/0004.&lt;/ref&gt; Her personal beauty is often mentioned in reports, and ''The World'' says she was &quot;a magnificent woman.&quot;&lt;ref&gt;&quot;Beauties of To-Day. From the ''World''.&quot; ''Clifton Society'' 24 June 1897, Thursday: 14 [of 16], Col. 2c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002164/18970624/066/0014. Same print and p.&lt;/ref&gt; She is the first listed in the ''Graphic''&lt;nowiki/&gt;'s 1891 &quot;Leading Ladies of Society&quot;:&lt;blockquote&gt;The Hon. Mrs. Algernon Bourke is a daughter (Gwendoline Irene Emily) of the late Hans Sloane Stanley, Esq., of Poultons, Southampton, and 49, Cadogan Square, S.W. She married, on December 15th, 1887, the Hon. Algernon Bourke, third son of the sixth Earl of Mayo, Governor-General of India (who was assassinated in 1872), and nephew of Lord Connemara, Governor of Madras. Mr. Bourke is a member of the London Stock Exchange, and resides at 33, Cadogan Terrace, S.W.&lt;ref&gt;&quot;Leading Ladies of Society.&quot; The Graphic 28 March 1891, Saturday: 6 [of 28], Col. 2c [of 3]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/9000057/18910328/019/0006. Print: same title, p. 346.&lt;/ref&gt;&lt;/blockquote&gt;She attended many social events without her husband, especially into the 20th century, usually with an appreciative description of what she wore. She was a sponsor of Irish art needlework as well. Unlike her husband's, Guendoline's social status seems to have risen as time passed, and she appears in stories associated with the Princess of Wales, and then later with Queen Alexandra.

=== The Sloane-Stanley Family ===
Guendoline's family consisted of a younger brother, Cyril Sloane-Stanley, as well as her parents, Hans Sloane-Stanley and Emilie Edwards Sloane-Stanley. Exactly one year after she and Algernon Bourke married, Hans Sloane-Stanley died (in 1888), leaving an estate worth £33,704 7s. 5d.&lt;ref name=&quot;:17&quot;&gt;Principal Probate Registry; London, England; ''Calendar of the Grants of Probate and Letters of Administration made in the Probate Registries of the High Court of Justice in England''. Ancestry.com. ''England &amp; Wales, National Probate Calendar (Index of Wills and Administrations), 1858-1995'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2010.&lt;/ref&gt; (1888, 321) Her mother remarried almost exactly a year after that, to James Shelly Bontein. Bontein's father had been Gentleman Usher and Clerk of the Robes to Queen Victoria.&lt;ref name=&quot;:18&quot;&gt;&quot;Marriages.&quot; &quot;Births, Marriages, and Deaths.&quot; ''Belfast News-Letter'' 6 December 1889, Friday: 1 [of 8], Col. 1a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000038/18891206/001/0001. Same print title and p.&lt;/ref&gt;

Shortly after his death ''Truth'' described Guendoline and Cyril's father Hans Sloane-Stanley:&lt;blockquote&gt;The death of Mr. Sloane Stanley, of Paultons Park, is much regretted in South Hants, as he was one of the most popular landlords in the county, and was greatly esteemed. Mr. Sloane Stanley was well known in yachting circles, and for many years he was Commodore of the Royal Southern Yacht Club, and owned the schooner ''Star of the West''. He was one of the very few owners who continued to keep up the old custom of giving his crew a laying-up supper at the close of each season. There were great festivities at Paultons only a few months ago, when Miss Sloane Stanley was married to Mr. Algernon Bourke.&lt;ref&gt;&quot;Entre Nous.&quot; ''Truth'' 27 December 1888, Thursday: 6 [of 48], Col. 2b [of 2]. ''British Newspaper Archive'' [https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18881227/023/0006# https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18881227/023/0006]. Same print title, p. 1136.&lt;/ref&gt;&lt;/blockquote&gt;When he died in 1944, Cyril Sloane-Stanley's estate was quite a bit larger than his father's had been 50 years before. The probate was divided between what was limited to &quot;settled land&quot; and what was &quot;save and except settled land.&quot; What was not settled land totalled £356,114 12s. 10d. and went to John Everett, company director; the Hon. Elwyn Villiers Rhys, captain, H.M. army; and William Adam de Geijer, retired captain, H.M. army.&lt;ref name=&quot;:17&quot; /&gt; (1944, 430) His daughter Lavender was married to John Everett, and Diane was married to Elwyn Villiers Rhys. What was settled land totalled £168,975 and went to William Adam de Geijer, retired captain, H.M. army, and George Lawrence Stewart, solicitor.&lt;ref name=&quot;:17&quot; /&gt; (1944, 430)

The Sloane-Stanleys descend from Hans Sloane (1660–1753), whose 71,000-item collections &quot;provid[ed] the foundation of the British Museum, the British Library, and the Natural History Museum, London.&quot;&lt;ref name=&quot;:19&quot;&gt;{{Cite journal|date=2025-01-07|title=Hans Sloane|url=https://en.wikipedia.org/wiki/Hans_Sloane|journal=Wikipedia https://en.wikipedia.org/wiki/Hans_Sloane|language=en|via=}}&lt;/ref&gt; Much of this Hans Sloane's wealth came from his medical practice in Jamaica, where he went as physician to the Governor General of Jamaica, the 2nd Duke of Albemarle, and where he married &quot;a wealthy heiress of sugar plantations&quot; worked by enslaved Jamaicans.&lt;ref name=&quot;:19&quot; /&gt; His great-nephew, Hans Sloane, inherited Paultons, near Romsey, &quot;and in recognition of this he adopted the additional surname of Stanley in 1821.&quot;&lt;ref&gt;{{Cite journal|date=2023-10-06|title=Hans Sloane (MP)|url=https://en.wikipedia.org/wiki/Hans_Sloane_(MP)|journal=Wikipedia https://en.wikipedia.org/wiki/Hans_Sloane_(MP)|language=en}}&lt;/ref&gt;

== Acquaintances, Friends and Enemies ==

=== Algernon Bourke ===

* Best man at [[Social Victorians/Timeline/1887#Wedding of Algernon Bourke and Gwendolen Sloane Stanley|his wedding]]: the Hon. Michael Sandys
* [[Social Victorians/People/Montrose|Marcus Henry Milner]], &quot;one of the zealous assistants of that well-known firm of stockbrokers, Messrs. Bourke and Sandys&quot;&lt;ref name=&quot;:8&quot;&gt;&quot;Metropolitan Notes.&quot; ''Nottingham Evening Post'' 31 July 1888, Tuesday: 4 [of 4], Col. 2a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000321/18880731/025/0004.&lt;/ref&gt;
* Caroline, Duchess of Montrose — her &quot;legal advisor&quot; on the day of her marriage to Marcus Henry Milner&lt;ref&gt;&quot;Metropolitan Notes.&quot; ''Nottingham Evening Post'' 31 July 1888, Tuesday: 4 [of 4], Col. 1b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000321/18880731/025/0004.&lt;/ref&gt;
* Wilfred Scawen Blunt
* Alexander Meyrick Broadley&lt;ref name=&quot;:24&quot;&gt;Broadley, Alexander Meyrick. ''How We Defended Arábi and His Friends, a Story of Egypt and the Egyptians''. London: Chapman and Hall, 1884. Google Books: [https://www.google.com/books/edition/How_We_Defended_Ar%C3%A1bi_and_His_Friends/75YRzFDvaO4C?hl=en&amp;gbpv=1&amp;bsq=Bourke https://www.google.com/books/edition/How_We_Defended_Ar%C3%A1bi_and_His_Friends/75YRzFDvaO4C].&lt;/ref&gt; (2) [Blunt, Broadley and Bourke were mutual friends; Blunt and Bourke were cousins.]
* &quot;Natty Rothschild,&quot; according to Blunt
* Brett, who was Hartington's secretary in the early 80s.&lt;ref&gt;Blunt, Wilfred Scawen. ''Gordon at Khartoum, Being a Personal Narrative of Events in Continuation of &quot;A Secret History of the English Occupation of Egypt''.&quot; London: Stephen Swift, 1911.&lt;/ref&gt;

=== Guendoline Bourke ===

* Bridesmaids at [[Social Victorians/Timeline/1887#Wedding of Algernon Bourke and Gwendolen Sloane Stanley|her wedding]]: Lady Florence Bourke, Miss Nora Bourke, Miss Edwards, and Miss Ewart
* Lord and Lady Alington, Belvedere House, Scarborough
* [[Social Victorians/People/William James|Evelyn James]] 

== Organizations ==

=== Guendoline Bourke ===

* Member, the Ladies Committee for the [[Social Victorians/London Clubs#Prince's Skating Club|Prince's Skating Club]], which also included [[Social Victorians/People/Princess Louise|Princess Louise]] (Duchess of Argyll), the [[Social Victorians/People/Portland|Duchess of Portland]], [[Social Victorians/People/Londonderry|Lady Londonderry]], [[Social Victorians/People/Campbell|Lady Archibald Campbell]], [[Social Victorians/People/Ribblesdale|Lady Ribblesdale]], and [[Social Victorians/People/Asquith|Mrs. Asquith]]&lt;ref name=&quot;:11&quot;&gt;&quot;What the 'World' Says.&quot; ''Northwich Guardian'' 01 November 1902, Saturday: 6 [of 8], Col. 8a [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001975/19021101/134/0006. Print title: The ''Guardian'', p. 6.&lt;/ref&gt; (in 1902, at least)

=== Algernon Bourke ===
* [[Social Victorians/Schools#Eton|Eton]]
* Cambridge University, Trinity College, 1873, Michaelmas term&lt;ref name=&quot;:7&quot;&gt;Cambridge University Alumni, 1261–1900. Via Ancestry.&lt;/ref&gt;
* Conservative Party
* 1879: Appointed a Poor Law Inspector in Ireland, Relief of Distress Act
* 1881: Partner, with 2 uncles, in Brunton, Bourke, and Co.&lt;ref&gt;&quot;From Our London Correspondent.&quot; ''Manchester Courier'' 24 August 1881, Wednesday: 5 [of 8], Col. 4a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000206/18810824/030/0005. Print: ''Manchester Courier and Lancaster General Advertiser'', p. 5.&lt;/ref&gt; (one of the [[Social Victorians/British Aristocracy#Sons of Peers on the Stock Exchange|sons of peers on the Stock Exchange]])
* 1885: Office of the 7th Surrey Rifles Regiment&lt;ref&gt;&quot;7th Surrey Rifles.&quot; ''South London Press'' 08 August 1885, Saturday: 12 [of 16], Col. 4a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000213/18850808/165/0012. Print p. 12.&lt;/ref&gt;
* 1886: Battersea Friendly Angling Society&lt;ref&gt;&quot;Battersea Friendly Angling Society.&quot; ''Fishing Gazette'' 17 April 1886, Saturday: 6 [of 20], Col. 2a [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002553/18860417/030/0006. Same print title, p. 218.&lt;/ref&gt;
* 27 February 1886: one of the Vice Presidents of the [[Social Victorians/London Clubs#Bolingbroke Reading-Room and Institute|Bolingbroke Reading-Room and Institute]]
* Special Correspondent of The ''Times'' for the Zulu War, accompanying Lord Chelmsford
* Head, Messrs. Bourke and Sandys, &quot;that well-known firm of stockbrokers&quot;&lt;ref name=&quot;:8&quot; /&gt; ( – 1901 [at least])
* White's gentleman's club, St. James's,&lt;ref&gt;{{Cite journal|date=2024-10-09|title=White's|url=https://en.wikipedia.org/wiki/White's|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/White%27s.&lt;/ref&gt; Manager (1897)&lt;ref&gt;&quot;Side Lights on Drinking.&quot; ''Waterford Standard'' 28 April 1897, Wednesday: 3 [of 4], Col. 7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001678/18970428/053/0003.&lt;/ref&gt;
* Willis's Rooms (described in 1895):&lt;blockquote&gt;... the Hon. Algernon Burke [sic], son of the 6th Earl of Mayo, has turned the place into a smart restaurant where choice dinners are served and eaten while a stringed band discourses music. Willis's Rooms are now the favourite dining place for ladies who have no club of their own, or for gentlemen who are debarred by rules from inviting ladies to one of their own clubs. The same gentleman runs a hotel in Brighton, and has promoted several clubs. He has a special faculty for organising places of the kind, without which such projects end in failure.&lt;ref&gt;&quot;Lenten Dullness.&quot; ''Cheltenham Looker-On'' 23 March 1895, Saturday: 11 [of 24], Col. 2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000226/18950323/004/0011. Print p. 275.&lt;/ref&gt;&lt;/blockquote&gt;
*The [[Social Victorians/London Clubs#Pelican|Pelican Club]], known for its boxing (1891)

==== Boards of Directors ====
*1883: One of the directors, the Franco-English Tunisian Esparto Fibre Supply Company, Ltd.&lt;ref&gt;''Money Market Review'', 20 Jan 1883 (Vol 46): 124.&lt;/ref&gt;
*1891: One of the founders, the Discount Banking Company, Ltd., which says Algernon Bourke is a director of District Messenger Services and News Company, Ltd.&lt;ref&gt;&quot;Public Company.&quot; ''Nottingham Journal'' 31 October 1891, Saturday: 4 [of 8], Col. 8a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001896/18911031/099/0004. Print title: ''The Nottingham Daily Express'', p. 4.&lt;/ref&gt;
*1894: One of the directors, the Frozen Lake, Ltd., with Admiral Maxse, Lord [[Social Victorians/People/Beresford|Marcus Beresford]], [[Social Victorians/People/Williams|Hwfa Williams]]&lt;ref&gt;&quot;The Frozen Lake, Limited.&quot; ''St James's Gazette'' 08 June 1894, Friday: 15 [of 16], Col. 4a [of 4]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001485/18940608/085/0015. Print p. 15.&lt;/ref&gt;&lt;blockquote&gt;London is to have new amusement this winter, for which Mr Algernon Bourke, who has taught us that it is possible to eat as well in St. James’s as on the Boulevards, and Mr Hwfa Williams, of Sandown fame, are jointly responsible. The &quot;Frozen Lake,&quot; under which title a real ice-skating rink is about to be constructed under their auspices, will no doubt be gladly welcomed by all skaters, and the venture is likely to prove a success.&lt;ref&gt;&quot;Society Gossip.&quot; ''Weston-super-Mare Gazette, and General Advertiser'' 6 June 1894, Wednesday: 4 [of 4], Col. 4b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001444/18940606/044/0004. Print title: ''Weston-super-Mare Gazette'', p. 4.&lt;/ref&gt;&lt;/blockquote&gt;

==== Committees ====

*Member, General Committee, [[Social Victorians/London Clubs#Baths|the Baths Club]] (1892)
*Member, Men's Committee of the [[Social Victorians/London Clubs#Prince's Skating Club|Prince's Skating Club]], which also included Lord Edward Cecil, Lord Redesdale, Mr. [[Social Victorians/People/Lyttelton|Alfred Lyttelton]], Sir Edgar Vincent, Sir William Hart Dyke, and Mr. [[Social Victorians/People/Grenfell|W. H. Grenfell]]&lt;ref name=&quot;:11&quot; /&gt; (1902, at least)
*[[Social Victorians/Timeline/1896#25 March 1896, Wednesday|The Sala Memorial Fund]], member of the committee (from 25 March 1896)
* Member of an &quot;influential committee&quot; headed by the Lord Mayor &quot;to restore salmon to the Thames&quot; (June 1899)&lt;ref&gt;&quot;Salmon in the Thames.&quot; ''Berks and Oxon Advertiser'' 30 June 1899, Friday: 5 [of 8], Col. 4a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002298/18990630/079/0005. Print n.p.&lt;/ref&gt;

== Timeline ==

=== 1870s ===
'''1872 February 8''', Richard Bourke, 6th Earl of Mayo was assassinated while inspecting a &quot;convict settlement at Port Blair in the Andaman Islands ... by Sher Ali Afridi, a former Afghan soldier.&quot;&lt;ref&gt;{{Cite journal|date=2024-12-01|title=Richard Bourke, 6th Earl of Mayo|url=https://en.wikipedia.org/wiki/Richard_Bourke,_6th_Earl_of_Mayo|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Richard_Bourke,_6th_Earl_of_Mayo.&lt;/ref&gt; The Hon. Algernon's brother Dermot became the 7th Earl at 19 years old.

'''1876 November 24, Friday''', the Hon. Algernon Bourke was one of 6 men (2 students, one of whom was Bourke; 2 doctors; a tutor and another man) from Cambridge who gave evidence as witnesses in an inquest about the death from falling off a horse of a student.&lt;ref&gt;&quot;The Fatal Accident to a Sheffield Student at Cambridge.&quot; ''Sheffield Independent'' 25 November 1876, Saturday: 7 [of 12], Col. 5a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000181/18761125/040/0007. Print title: ''Sheffield and Rotherham Independent'', n. p.&lt;/ref&gt;

'''1879 December 27, Saturday – 29, Monday''', Algernon Bourke was in Kilrush as a Local Government Board Inspector:&lt;blockquote&gt;Among many distinguished visitors at the Vandeleur Arms Hotel, Kilrush this week was the Hon. Algernon Bourke Local Government Board Inspector who arrived on Saturday, and sojourned there until 2 o'clock on Monday, when the honourable gentleman left by Steamer tor Limerick.&lt;ref&gt;&quot;Fashionable Intelligence.&quot; ''Kilrush Herald and Kilkee Gazette'' 01 January 1880, Thursday: 2 [of 5], Col. 3a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003186/18800101/011/0002. Print title ''Kilrush Herald'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;

=== 1880s ===
'''4 February 1880, Wednesday''', Hon. Algernon Bourke attended the [[Social Victorians/Timeline/1880#Grand Ball at Palmerstown House Hosted by the Earl of Mayo|grand ball at Palmerstown House hosted by the Earl of Mayo]]. 

'''1880 March 30, Tuesday''', Algernon Bourke was working in the judicial system in Newcastle, County Limerick, possibly as Poorhouse Inspector:&lt;blockquote&gt;A sworn enquiry was held to-day at the Workhorse, Newcastle West, by the Hon Algernon Bourke, L.G.I., to enquire into charges preferred by Dr. Pierce, Medical Office, against Dr. O'Shaughnesay. The enquiry was adjourned till Thursday next. Mr Moran, sol., Rathkeale, was engaged for Dr. O'Shaughnessy.&lt;ref&gt;&quot;Sworn Enquiry.&quot; &quot;Limerick County. Newcastle West Intelligence.&quot; ''Bassett's Chronicle'' 31 March 1880, Wednesday: 3 [of 4], Col. 3b–c [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003471/18800331/044/0003.  Print title ''Bassett's Daily Chronicle'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;'''1880 April 17, Saturday''', in-jokes dominate this report mentioning Algernon Bourke in the context of the Kildare and National Hunt races in Dublin:&lt;blockquote&gt;And in mopy Upper Mount-street, where young Algernon Bourke, of the Onety-oneth, had promised to call for, and afterwards spin down to the races in his mail phaeton, the Blake girls; and in fastidious Fitzwilliam-place, and exclusive &quot;Murryan-squeer,&quot; from which dashing army men, in their neatly-appointed, well horsed drags were to &quot;tool&quot; down sweet young Dublin lasses of the ''crême d la crême'' [sic], many an anxious forecasting of the weather was taken, lest by an unpropitious shower that last triumph of Mrs. Manning, or the Forrests, or Miss Sedford, or any of the ''grandes dames de la mode'' should be rendered as worthless as a Confederate &quot;greenback.&quot; But by ten o'clock all doubts were happily set aside, and up struck the lovely April day in all its spring-time glory and then the road, oh, the road!&lt;ref&gt;&quot;To Punchestown and Back by the Old Road.&quot; ''Illustrated Sporting and Dramatic News'' 17 April 1880, Saturday: 6 [of 24], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001857/18800417/013/0006. Same print title, p. 102.&lt;/ref&gt;&lt;/blockquote&gt;'''1881 May 10, Tuesday''', Algernon Bourke attended the [[Social Victorians/Timeline/1881#1881 May 10, Tuesday|wedding of Marion Lascelles, eldest daughter of the Hon. Egremont W. Lascelles, brother of the Earl of Harewood, and Lieutenant Henry Dent Brocklehurst, of the Second Life Guards, nephew of Mr. Philip Brocklehurst, of Swithamley Park, Macclesfield]]. His gift was an &quot;old enamelled watch set in pearls.&quot;&lt;ref&gt;&quot;Nuptial Rejoicings at Middlethorpe Manor. Marriage of Miss Lascelles and Lieut. Brocklehurst.&quot; ''Yorkshire Gazette'' 14 May 1881, Saturday: 9 [of 12], Cols. 3a–4a [of 6]. ''British Newspaper Archive''https://www.britishnewspaperarchive.co.uk/viewer/bl/0000266/18810514/057/0009. Print same title and p.&lt;/ref&gt;

'''1881 May 23, Monday, 2:00 p.m.''', Algernon Bourke is listed among the Honourables at the [[Social Victorians/Timeline/1881#Queen's Levee at St. James's Palace|Queen's Levee at St. James's Palace]].

'''1881 July 14, Thursday afternoon, beginning about 2 p.m.''', Algernon Bourke was invited to a Garden Party at Marlborough House hosted by [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]]. Members of the family of the [[Social Victorians/People/Mayo|Earl of Mayo]] were also among the 1,500 or so invited guests.

'''1881 July 22, Friday''', Algernon Bourke was invited to an [[Social Victorians/Timeline/1881#22 July 1881, Friday|evening party at Marlborough House hosted by the Prince and Princess of Wales]].

'''1881 September 17, Saturday''', Algernon Bourke was reported among the company at Doncaster during race week.&lt;ref&gt;&quot;List of the Company.&quot; ''York Herald'' 17 September 1881, Saturday: 8 [of 16], Col. 4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000499/18810917/183/0008. Same print title and p.&lt;/ref&gt;

'''1881 November 22, Tuesday''', Algernon Bourke was sued in Dublin by Henry Naylor because he &quot;had declined to pay&quot; for a £35 piano.&lt;ref&gt;&quot;Henry Naylor v. the Hon. Algernon Bourke.&quot; &quot;Exchequer Division.&quot; &quot;High Court of Justice.&quot; ''Belfast Morning News'' 23 November 1881, Wednesday: 3 [of 4], Col. 8a [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000428/18811123/015/0003. Same print title, n.p.&lt;/ref&gt;

'''1881 December 8, Thursday''', Algernon Bourke was part of a [[Social Victorians/Timeline/1881#Battue at Palmerstown|battue at Palmerstown]], when the group bagged 172 pheasants, hares and rabbits.

'''1882 March 7, Tuesday''', Algernon Bourke attended the [[Social Victorians/Timeline/1882#7 March 1882, Tuesday|fashionable wedding of Reginald Chandos-Pole and Violet Denison]].

'''1882 March 15, Wednesday''', Algernon Bourke attended [[Social Victorians/Timeline/1882#The Marchioness of Salisbury's Assembly|the Marchioness of Salisbury's first reception of the season]].

'''1882 July 13, Thursday''', Algernon Bourke was invited to the [[Social Victorians/1882-07-13 Marlborough House Garden Party|Garden Party at Marlborough House for Queen Victoria]] hosted by [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]]. The more than 1,000 people invited also included a number of people from the family of the [[Social Victorians/People/Mayo|Earl of Mayo]].

'''1882 September 19''', Alexander M. Broadley in Tunis received a letter from Algernon Bourke asking him to provide legal counsel to defend prisoner-of war Arábi in Egypt, with Wilfred Scawen Blunt, which he did.&lt;ref name=&quot;:24&quot; /&gt; (2)

'''1882 September 28, Saturday''', Algernon Bourke attended the [[Social Victorians/Timeline/1882#The Wedding of John M'Donald and Georgiana Lambart|wedding of John M'Donald and Georgiana Lambart]].

'''1883 March 21, Wednesday''', the Evening Irish Times announced that Algernon Bourke &quot;has arrived at Kingstown from England.&quot;&lt;ref&gt;&quot;Court and Fashion.&quot; ''Evening Irish Times'' 21 March 1883, Wednesday: 7 [of 8], Col. 5a [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003464/18830321/086/0007. Same print title and p.&lt;/ref&gt;

'''23 July 1883, Monday, noon''', the Hon. Algernon Bourke was invited to a [[Social Victorians/Timeline/1883#Garden Party at Marlborough House, at Noon|garden party at Marlborough House]] hosted by the Prince and Princess of Wales.

'''31 October 1883, Wednesday''', Algernon Bourke attended the wedding of [[Social Victorians/Timeline/1883#Wedding of Lady Cecelia Hay and Captain George Webbe|Lady Cecelia Hay and Captain George Webbe]].&lt;p&gt;

'''1884 February 16, Saturday''', Algernon Bourke attended [[Social Victorians/Timeline/1884#16 February 1884, Saturday|the funeral of Thomas Chenery, editor of the ''Times'']].

'''1884 April 4, Saturday''', Algernon Bourke was (may have been?) one of the [[Social Victorians/Timeline/1884#5 April 1884, Saturday|&quot;Supporters of the Pall&quot; at the funeral]] of [[Social Victorians/People/Leopold|Prince Leopold George Duncan Albert, Duke of Albany]] at St. George's, Windsor.

'''1884 April 26, Saturday''', Algernon Bourke attended a [[Social Victorians/Timeline/1884#26 April 1884, Saturday|dinner party at the Lord Mayor's Mansion House for conservatives to meet Sir Stafford Northcote]].

'''1884 May 3, Saturday''', the &quot;Rochester Conservatives&quot; announced that they would &quot;bring forward the Hon. Algernon Bourke, brother of Lord Mayo, as their second candidate,&quot;&lt;ref&gt;&quot;Election Intelligence.&quot; ''Yorkshire Gazette'' 03 May 1884, Saturday: 4 [of 12], Col. 6a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000266/18840503/011/0004.&lt;/ref&gt; but because he would not be the first candidate, Bourke declined.&lt;ref&gt;&quot;Rochester.&quot; London ''Daily Chronicle'' 09 May 1884, Friday: 3 [of 8], Col. 8b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005049/18840509/049/0003.&lt;/ref&gt;

'''1884 June 18, Wednesday''', Mr. Algernon Bourke was on a committee to watch a [[Social Victorians/Timeline/1884#18 June 1884, Wednesday|Mr. Bishop's &quot;thought-reading&quot; experiment]], which was based on a challenge by Henry Labourchere made the year before. This &quot;experiment&quot; took place before a fashionable audience.

'''1884 July 25, Friday, afternoon''', the Hon. Algernon Bourke was invited to a [[Social Victorians/Timeline/1884#Garden Party at Marlborough House hosted by the Prince and Princess of Wales|Garden Party at Marlborough House hosted by the Prince and Princess of Wales]].

'''1885 January 22, Thursday''', Algernon Bourke's gift to [[Social Victorians/Timeline/1885#Wedding of George Buckle and Alicia Payn|George Buckle and Alicia Payn for their wedding]] was an antique cabinet.

'''1885 July 7, Tuesday''', Algernon Bourke attended [[Social Victorians/Timeline/1885#7 July 1885, Tuesday|Eva Bourke's wedding to Windham Wyndham-Quin]] at St. Mary Abbots, Kensington.

'''1885 July 13, Monday''', Algernon Bouurke was at Victoria Station as part of the [[Social Victorians/Timeline/1885#Arrival of Lord Wolseley in London from Egypt|crowd greeting Lord Wolseley on his return from Egypt]].

'''1885 July 24, Friday''', the Hon. Algernon Bourke was invited to a [[Social Victorians/1885-07-24 Marlborough House Ball|ball at Marlborough House]] hosted by the Prince and Princess of Wales.

'''1885 September 26, Saturday''', Algernon Bourke took part in the [[Social Victorians/Timeline/1885#26 September 1885, Saturday|Ealing Conservative Club fete and meeting]] supporting Salisbury's government and condemning &quot;the dictates of one man&quot; — Gladstone — for Gordon's death.

'''1885 October 3, Saturday''', the Hon. Algernon Bourke was named as the Conservative candidate for Clapham in the Battersea and Clapham borough after the Redistribution Bill determined the electoral districts for South London.&lt;ref&gt;&quot;South London Candidates.&quot; ''South London Press'' 03 October 1885, Saturday: 9 [of 16], Col. 5b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000213/18851003/096/0009. Print p. 9.&lt;/ref&gt; On Sunday 15 November 1885 the London ''Weekly Dispatch'' supported Moulton, the Liberal candidate, who ultimately won the election:&lt;blockquote&gt;

Though a successful lawyer, Mr. Moulton is much more than that. He is a thorough and independent student of political science, who may be trusted to do good service to the Liberal cause with brain as well as with tongue. It will be matter for hearty congratulation if he defeats the Hon. Algernon Henry Bourke, who is a dashing and unscrupulous young Tory, and a nephew of the well-known politician with the same surname.&lt;ref&gt;&quot;The Political Campaign in London. VI. — The South-West Divisions.&quot; ''Weekly Dispatch'' (London) 15 November 1885, Sunday: 9 [of 16], Col. 3c [of 4]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003358/18851115/069/0009. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

On Saturday 21 November 1885 the ''South London Press'' reported on posters for Bourke's candidacy:&lt;blockquote&gt;

The Hon. Algernon Bourke, Conservative candidate for Clapham, has a very industrious billsticker, who pastes up his patron’s bills in every possible place where they can be seen to advantage. It is unfortunate, however, that choosing the flank wall of an auctioneer’s the modern &quot;Sam Slap&quot; has produced some curious combinations, such as — &quot;Vote for Bourke,&quot; &quot;Now on View;&quot; &quot;Electors of Clapham, Vote for Mr. Bourke, and&quot; &quot;Be Sold Without Reserve;&quot; &quot;Mr, Bourke will&quot; &quot;Advance Money to&quot; &quot;the Electors of Clapham;&quot; &quot;Great Conservative Meeting. The British Constitution will be&quot; &quot;Offered for Sale this Evening,&quot; &amp;c.&lt;ref&gt;&quot;Pick-up Notes.&quot; ''South London Press'' 21 November 1885, Saturday: 10 [of 16], Col. 1b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000213/18851121/155/0010. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

'''1885 November 3, Tuesday, 11:00 a.m.''', Algernon Bourke attended the [[Social Victorians/Mayo-Ponsonby Wedding 1885-11-03|wedding of his brother, Dermot, 7th  Earl of Mayo and Geraldine Ponsonby]]. He gave them 2 Sheraton secretaires.

'''1886 January 5, Tuesday, late''', the Hon. Algernon Bourke attended the [[Social Victorians/Timeline/1886#Twelfth Night|Twelfth Night celebration at the Drury Lane theatre]].

'''1886 March 13, Saturday evening''', an Hon. Mr. Bourke attended a [[Social Victorians/1886-03-13 Reception at the French Embassy|reception at the French Embassy]], possibly Algernon Bourke or possibly [[Social Victorians/People/Mayo|one of his brothers]].

'''1886 July 10, Saturday''', Hon. Algernon Bourke was invited to a [[Social Victorians/Timeline/1886#Garden Party at Marlborough House Given to the Queen|garden party at Marlborough House given to the Queen]]. Guendoline Sloane Stanley is not mentioned but Mr. and Mrs. Hans Sloane Stanley are, as are Mr. and Mrs. F. Sloane Stanley.&lt;p&gt;

'''1886 July 21, Wednesday''', Algernon Bourke was invited to the [[Social Victorians/1886-07-21 Marlborough House Ball|Ball at Marlborough House]], as were a [[Social Victorians/People/Bourke#The Sloane-Stanleys 2|Mr. and Mrs. F. Sloane-Stanley]], possibly the parents of Guendoline Sloane-Stanley (if the &quot;F&quot; is a mistake), who married Bourke on 15 December 1887. Guendoline is not mentioned as having been invited.

'''1886 July 27, Tuesday''', Algernon Bourke attended a service honoring a memorial at St. Paul's for his father, who had been assassinated.&lt;ref&gt;&quot;Memorial to the Late Earl of Mayo.&quot; ''Northern Whig'' 28 July 1886, Wednesday: 6 [of 8], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000434/18860728/143/0006. Print p. 6.&lt;/ref&gt;

'''1886 September 2, Thursday''', Mr. Algernon Bourke was part of a group of mostly aristocratic men taking part in [[Social Victorians/Timeline/1886#Augustus Harris's A Run of Luck|a &quot;trial-rehearsal&quot; as part of Augustus Harris's production]] ''A Run of Luck'', about sports.

'''1886 October 2, Saturday''', the Duke of Beaufort and the Hon. Algernon Bourke arrived in Yougal: &quot;His grace has taken a residence at Lismore for a few weeks, to enjoy some salmon fishing on the Blackwater before the close of the season.&quot;&lt;ref&gt;&quot;Chippenham.&quot; ''Wilts and Gloucestershire Standard'' 02 October 1886, Saturday: 8 [of 8], Col. 6a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001955/18861002/142/0008. Print p. 8.&lt;/ref&gt;

'''1886 October 11, Monday''', Algernon Bourke may have been taking part in a [[Social Victorians/Timeline/1886#Performance of Run of Luck|performance of ''Run of Luck'' at the Drury Lane]].

'''1886 October 23, Saturday''', Algernon Bourke was [[Social Victorians/Timeline/1886#Party at Wemyss Castle, Fife|staying at Wemyss Castle, Fife]].

'''1886 December 30, Thursday''', Algernon Bourke was back in London and attending the [[Social Victorians/Timeline/1886#Augustus Harris's The Forty Thieves|&quot;Forty Thieves&quot; pantomime at the Drury Lane Theatre]].

'''1887 January 5, Wednesday''', the Hon. Algernon Bourke was one of the chief mourners at the [[Social Victorians/Timeline/1887#Funeral of Lady Margaret Harriett Bourke|funeral of Lady Margaret Harriett Bourke]].

'''1887 March 1, 2:00 p.m.''', Algernon Bourke is listed among the Messieurs attending the [[Social Victorians/Timeline/1887#Queen's Levee at St. James's Palace|Queen's Levee at St. James's Palace]].&lt;p&gt;

'''1887 May''', a &quot;signalling incident&quot; in 1907 [[Social Victorians/Timeline/1887#May 1887|caused the Waterford ''Evening News'' to recall a similar event]] that had occurred 20 years earlier, in which Algernon Bourke, as special correspondent for the ''Times'', caused the incident to be publicized:&lt;blockquote&gt;During the manoeuvres in connection with the 1887 Jubilee of Queen Victoria a signal was observed going up from [[Social Victorians/People/Beresford|Lord Charles [Beresford]]]'s ship. It was a message to his wife, Lady Beresford, to the effect that, as he should be late for dinner, she was not to wait. Beyond the hilarity this domestic signal evoked, nothing more would have been heard of it, but Mr. Algernon Bourke (Lord Mayo's brother) was acting as special correspondent for the &quot;Times,&quot; and that paper the next morning contained a full and humorous report of the incident. Then there was trouble.&lt;ref&gt;&quot;Signalling Incident.&quot; ''Evening News'' (Waterford) 13 November 1907, Wednesday: 1 [of 4], Col. 6c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004557/19071113/021/0001.&lt;/ref&gt;&lt;/blockquote&gt;

'''1887 June 15, Wednesday''', the Hon. Algernon Bourke attended a [[Social Victorians Foreign Office Reception 1887-06-15|reception at the Foreign Office in honor of Queen Victoria's Golden Jubilee]].

'''1887 July 6, Wednesday''', Algernon Bourke was invited to and, presumably, attended the State Ball at Buckingham Palace.&lt;ref&gt;&quot;The State Ball at Buckingham Palace.&quot; ''Morning Post'' 08 July 1887, Friday: 3 [of 8], Col. 5a–6c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18870708/013/0003. Same print title and p.&lt;/ref&gt; (Col. 1c)

'''1887 August 6, Saturday''', the ''Brighton Gazette'' says that the &quot;Hon. Mrs and Mr Algernon Bourke&quot; were staying at the Royal Crescent Hotel in Brighton, but they didn't marry until 15 December 1887.&lt;ref&gt;&quot;Royal Crescent Hotel.&quot; ''Brighton Gazette'' 6 August 1887, Saturday: 3 [of 8], Col. 5c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000938/18870806/047/0003. Print title ''Brighton Gazette and Sussex Telegraph'', p. 3.&lt;/ref&gt; Perhaps an elder relative, because she is mentioned first?

'''1887 November 9, Wednesday''', the ''Hampshire Advertiser County Newspaper'' announced that&lt;blockquote&gt;A marriage is arranged, and will take place early in January, between Mr. Algernon Bourke, third son of the late Earl of Mayo, and Miss Guendolen Sloane Stanley, only daughter of Mr. Hans Sloane Stanley, of Paultons.&lt;ref&gt;&quot;Romsey, Nov. 9.&quot; ''Hampshire Advertiser'' 9 November 1887, Wednesday: 3 [of 4], Col. 7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18871109/034/0003. Print title ''Hampshire Advertiser County Newspaper'', p. 3.&lt;/ref&gt;&lt;/blockquote&gt;Shortly after, the papers announced that the wedding would not take place.

'''1887 December 15, Thursday''', Hon. [[Social Victorians/Timeline/1887#Wedding of Algernon Bourke and Gwendolen Sloane Stanley|Algernon Bourke and Guendoline Stanley were married at St. Paul's]], Knightsbridge, by Bourke's uncle the Hon. and Rev. George Bourke. Only family members attended because of &quot;the recent death of a near relative of the bride.&quot;&lt;ref&gt;&quot;Court Circular.&quot; ''Morning Post'' 16 December 1887, Friday: 5 [of 8], Col. 7c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18871216/066/0005.&lt;/ref&gt; Who the &quot;near relative of the bride&quot; was not in her nuclear family, and perhaps that explains the cancellation of the wedding and then the changing of the wedding date and not some problem in the couple.

'''1888 – 1899 January 1''', the Hon. Algernon Bourke was &quot;proprietor&quot; of [[Social Victorians/London Clubs#White's|White's Club, St. James's Street]].&lt;ref name=&quot;:9&quot;&gt;&quot;The Hon. Algernon Bourke's Affairs.&quot; ''Eastern Morning News'' 19 October 1899, Thursday: 6 [of 8], Col. 7c [of7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001152/18991019/139/0006. Print p. 6.&lt;/ref&gt;

'''1888 January 21, Saturday''', Guendoline Bourke attended the wedding of [[Social Victorians/Timeline/1888#Hamilton-Ewart Wedding|Florence Ewart and Henry Hamilton]].

'''1888 March 7, Wednesday''', assuming that this date is not a week after the actual date, [[Social Victorians/People/Beresford|Lady Charles Beresford]] held a [[Social Victorians/Timeline/1888#1888 March 7, Wednesday|notable and well-attended &quot;at home&quot;]] that Guendoline Bourke attended, reported for being dressed in white and being among the beautiful women present.

'''6 April 1888, Friday''', Algernon and Guendoline Bourke attended the [[Social Victorians/Timeline/1888#6 April 1888, Friday|New Forest United Hunt ball at the New Forest Hall, Lyndhurst]].

'''1888 May 2, Wednesday''', Algernon and Guendoline Bourke attended the [[Social Victorians/Timeline/1888#The Marchioness of Salisbury's Reception|Marchioness of Salisbury's reception]] at the Salisbury home on Arlington-street.

'''1888 May 22, Tuesday''', the Dowager Countess of Mayo presented Guendoline Bourke at the [[Social Victorians/Timeline/1888#Queen's Drawing Room|Queen's drawing-room]] hosted by the Princess of Wales. This is Guendoline Bourke's dress:&lt;blockquote&gt;Empire robe de cour of white satin duchesse, lined with rich pink silk, sufficiently bright to give a beautiful shell-like tint through the satin; tulle underdress, with upper skirt, embroidered with pearl, and caught up in Greek folds with large pink Tosca roses; white satin bodice, with Josephine pink sash tied at side, Headdress, veil and plumes; ornaments, diamonds.&lt;ref&gt;&quot;Dresses at the Drawing-Room.&quot; ''Epsom Journal'' 22 May 1888, Tuesday: 3 [of 6], Col. 5b–c [of 6]. ''British Newspapers Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004837/18880522/034/0003. Print: title ''Local Journal'', p. 3.&lt;/ref&gt;&lt;/blockquote&gt;

Another description:&lt;blockquote&gt;Mrs. Algernon Bourke's train was of white satin lined with pink, which showed through with charmingly shell-like effect. The dress, fashioned after those of the Empire period, was of white satin embroidered with pearls. A very broad sash of pink satin made the waist seem quaintly short, a trying thing to any but the young and tall, both of which qualifications Mrs. Bourke most happily possesses. She carried a lovely posy of La France roses.&lt;ref&gt;&quot;Gossip on Dress.&quot; ''Boston Spa News'' 25 May 1888, Friday: 2 [of 8], Col. 1b–2b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003395/18880525/014/0002. Print title The News, n.p.&lt;/ref&gt; (Col. 1c)&lt;/blockquote&gt;'''1888 June 8, Friday''', Hon. Algernon and Mrs. Bourke attended a [[Social Victorians/Timeline/1888#Dinner and Dance Hosted by Lord and Lady Wimborne at Hamilton House|dinner and dance Hosted by Lord and Lady Wimborne at Hamilton House]] featuring Prince and Princess Christian of Schleswig-Holstein, and for the ball, the King of Sweden and Norway and the Prince and Princess of Wales and their daughters were present.

'''1888 June 19, Tuesday''', Guendoline Bourke was one of the principal guests at the wedding of [[Social Victorians/Timeline/1888#19 June 1888, Tuesday|Captain Philip Green and Miss Mabel Emilie Scott]].

'''1888 July 26''', [[Social Victorians/People/Montrose|Caroline Graham Stirling-Crawford]] (known as Mr. Manton for her horse-breeding and -racing operations) and Marcus Henry Milner married.&lt;ref name=&quot;:12&quot;&gt;&quot;Hon. Caroline Agnes Horsley-Beresford.&quot; {{Cite web|url=https://thepeerage.com/p6863.htm#i68622|title=Person Page|website=thepeerage.com|access-date=2020-11-21}}&lt;/ref&gt; According to the ''Nottingham Evening Post'' of 31 July 1888,&lt;blockquote&gt;LONDON GOSSIP.

(From the ''World''.)

The marriage of &quot;Mr. Manton&quot; was the surprise as well the sensation of last week. Although some wise people noticed a certain amount of youthful ardour in the attentions paid by Mr. Marcus Henry Milner to Caroline Duchess of Montrose at '''Mrs. Oppenheim's ball''', nobody was prepared for the sudden ''dénouement''; '''and it''' were not for the accidental and unseen presence [[Social Victorians/People/Mildmay|a well-known musical amateur]] who had received permission to practice on the organ, the ceremony performed at half-past nine on Thursday morning at St. Andrew's, Fulham, by the Rev. Mr. Propert, would possibly have remained a secret for some time to come. Although the evergreen Duchess attains this year the limit of age prescribed the Psalmist, the bridegroom was only born in 1864. Mr. &quot;Harry&quot; Milner (familiarly known in the City as &quot;Millions&quot;) was one of the zealous assistants of that well-known firm of stockbrokers, Messrs. Bourke and Sandys, and Mr. Algernon Bourke, the head of the house (who, of course, takes a fatherly interest in the match) went down to Fulham to give away the Duchess. The ceremony was followed by a ''partie carrée'' luncheon at the Bristol, and the honeymoon began with a visit to the Jockey Club box at Sandown. Mr. Milner and the Duchess of Montrose have now gone to Newmarket. The marriage causes a curious reshuffling of the cards of affinity. Mr. Milner is now the stepfather of the [[Social Victorians/People/Montrose|Duke of Montrose]], his senior by twelve years; he is also the father-in-law of [[Social Victorians/People/Lady Violet Greville|Lord Greville]], Mr. Murray of Polnaise, and [[Social Victorians/People/Breadalbane|Lord Breadalbane]].&lt;ref name=&quot;:8&quot; /&gt;&lt;/blockquote&gt;

'''1888 December 1st week''', according to &quot;Society Gossip&quot; from the ''World'', the Hon. Algernon Bourke was suffering from malaria, presumably which he caught when he was in South Africa:&lt;blockquote&gt;I am sorry to hear that Mr. Algernon Bourke, who married Miss Sloane-Stanley a short time ago, has been very dangerously ill. Certain complications followed an attack of malarian fever, and last week his mother, the Dowager Lady Mayo, and his brother, Lord Mayo, were hastily summoned to Brighton. Since then a change for the better has taken place, and he is now out of danger.&lt;ref&gt;&quot;Society Gossip. What the ''World'' Says.&quot; ''Hampshire Advertiser'' 08 December 1888, Saturday: 2 [of 8], Col. 5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18881208/037/0002. Print title: ''The Hampshire Advertiser County Newspaper''; print p. 2.&lt;/ref&gt;&lt;/blockquote&gt;

'''1888 December 20, Thursday''', the Sloane-Stanley family, including Guendoline Bourke, attended the [[Social Victorians/Timeline/1888#20 December 1888, Thursday|funeral of Hans Sloane Stanley]]. Algernon Bourke did not attend because he was still too ill.

'''1889 January 22, 2:30 p.m., Tuesday''', Algernon and Guendoline Bourke sent a gift for the [[Social Victorians/Cecil Lambton Wedding 1889 January 22|wedding of Lady Eleanor Lambton and Lord Robert]] Cecil, a pair of antique mirrors.

'''1889 May 18, Saturday''', Algernon Bourke attended the [[Social Victorians/Timeline/1889#18 May 1889, Saturday|opening of the Italian Opera season at Covent Garden]].

'''1889 May 27, Monday, 11 p.m.''', the dancing commenced at [[Social Victorians/Timeline/1889#The Queen's State Ball at Buckingham Palace|the Queen's State Ball at Buckingham Palace]], with both the Hon. Algernon and the Hon. Guendoline Bourke present.

'''1889 June 8, Saturday''', the Hon. Algernon Bourke contributed some art he owned to the collection of the Royal Institute of Painters in Water-Colours' [[Social Victorians/Timeline/1889#8 June 1889, Saturday|exhibition of &quot;the works of the 'English Humourists in Art.'&quot;]]

'''1889 July 2, Tuesday''', Guendoline and Algernon Bourke sat in the Muriettas' box at a [[Social Victorians/Timeline/1889#The Shah at a Covent Garden Opera Performance|gala performance at Covent Garden also attended by the Prince and Princess of Wales, a number of other royals and the Shah]].

'''1889 27 July, Saturday''', Guendoline and Algernon Bourke attended a [[Social Victorians/Timeline/1889#Garden Party Hosted by Mr. and Mrs. Augustus Harris|garden party hosted by Mr. and Mrs. Augustus Harris]], which was attended by a people from the theatre and arts worlds.

'''1889 August 15''', Thursday, because of mourning on both sides, only the closest friends and family were invited to the [[Social Victorians/Pleydell-Bouverie Skelmersdale Wedding|wedding of Lady Wilma Pleydell-Bouverie and Lord Skelmersdale]], so she may not have attended, but the Hon. Mrs. Algernon Bourke gave the bride a miniature silver kettle.&lt;p&gt;

'''1889 December 2, Monday''', Guendoline Bourk's mother, Emilie Sloane-Stanley, married James Shelly Bontein:&lt;blockquote&gt;&lt;p&gt;

BONTEIN—STANLEY — December 2, at St. George's, Hanover Square, London, by the Rev. G. S. de Sansmarez, James Shelly, only son of the late James Bontein, Gentleman Usher and Clerk of the Robes to the Queen, to Emilie Josephine, widow of Hans Sloane Stanley, of Paultons.&lt;ref name=&quot;:18&quot; /&gt;&lt;/blockquote&gt;'''1889 December 17, Tuesday''', Hon. Algernon and Mrs. Bourke gave a gift to [[Dangan-Neville Wedding|Lady Violet Nevill for her wedding to Henry Wellesley, Viscount Dangan]] and so were probably in attendance.

=== 1890s ===
'''1890 January 9, Thursday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1890#The York Hunt Ball|fancy-dress Hunt Ball in York]]. She&lt;blockquote&gt;looked a picture in a Gainsborough gown. The white satin skirt was flounced with sable and veiled with ''chiffon'', the setuage of which was left to show without being hemmed up. There was a broad sash of rose-pink silk and each buttonhole was filled round with crimped lisse.&lt;ref&gt;&quot;Our London Letter.&quot; ''Irish Society'' (Dublin) 11 January 1890, Saturday: 17 [of 24], Col. 2a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001939/18900111/042/0017. Same print title, p. 29.&lt;/ref&gt;&lt;/blockquote&gt;'''1890 February''' '''12, Wednesday''', Hon. Algernon and Mrs. Bourke attended [[Social Victorians/Timeline/1890#Lady Constance Leslie's Reception|Lady Constance Leslie's reception]] at her house in Stratford-place.

'''1890 April 9, Wednesday''', Guendoline Bourke attended [[Social Victorians/Timeline/1890#The New Forest United Hunt Ball|the New Forest United Hunt Ball]].

'''1890 June 3, Tuesday''', Guendoline Bourke attended the 2:30 p.m. [[Social Victorians/Timeline/1890#Münster-Hay Wedding|wedding of Count Alexander Münster and Lady Muriel Henrietta Constance Hay]]. She is also listed as having attended a [[Social Victorians/Timeline/1890#Dinner and Concert Hosted by Mrs. Arthur Williams and Ball by Mrs. Menzies|ball hosted by Mrs. J. Menzies (daughter of Mrs. Arthur Wilson)]] that Prince Eddie, the Duke of Clarence and Avondale, also attended, that night.  

'''1890 July 4, Friday, 11 p.m.''', the Hon. Algernon and Guendoline Bourke attended [[Social Victorians/Timeline/1890#The Queen's State Ball at Buckingham Palace|the Queen's State Ball at Buckingham Palace]]. The dancing commenced shortly after 11:00. 

'''1890 July 15, Tuesday''', Hon. Algernon and Mrs. Bourke were invited to a [[Social Victorians/Timeline/1890#Garden Party at Marlborough House to Meet the Queen|garden party at Marlborough House to meet the Queen]]. 

'''1890 July 19, Saturday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1890#Wedding of James Francis Harry St. Clair-Erskine and Violet Aline Vyner|wedding of James Francis Harry St. Clair-Erskine and Violet Aline Vyner]], the two of them giving &quot;four small silver dessert dishes&quot; and Guendoline giving an &quot;enamel and diamond pin.&quot;&lt;ref&gt;&quot;Marriage of Lord Loughborough with Miss Vyner.&quot; ''Fife Free Press'' 26 July 1890, Saturday: 2 [of 8], Col. 1a–2b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001110/18900726/015/0002. Same print title and p.&lt;/ref&gt; (Col. 2b)

'''1890 July 24, Thursday''', Algernon and Guendoline Bourke attended a [[Social Victorians/Timeline/1890#Dinner and Dance Hosted by Lord Alington|dance hosted by Lord Alington]] attended also by the Prince and Princess of Wales and Princesses Victoria and Maud.

'''1890 September 6, Saturday''', the ''Country Gentleman'' (as it was called at the time) reported that &quot;Muckross, the only deer forest in Ireland, it may be said, has this year been rented by Mr. Algernon Bourke, who will next week be joined there for the stalking season by his brother, Lord Mayo.&quot;&lt;ref&gt;&quot;Shooting. Moors, Forests, and Fishings.&quot; ''Sporting Gazette'' 06 September 1890, Saturday: 11 [of 38], Col. 1c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002525/18900906/065/0011. Print: ''Country Gentleman'', p. 1251.&lt;/ref&gt; On 11 October 1890 the ''St. James's Gazette'' says,&lt;blockquote&gt;The Earl of Durham has been staying at Muchross, county Kerry, on a visit to the Hon. A. Bourke, who has rented the celebrated shootings and fishings on that estate for the autumn.&lt;ref&gt;&quot;Court and Society.&quot; ''St James's Gazette'' 11 October 1890, Saturday: 12 [of 16], Col. 1b [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001485/18901011/064/0012. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;'''1890 October 25, Saturday''', the Hon. Algernon and Mrs. Bourke gave a gold-mounted box to [[Social Victorians/Loder De Vere Beauclerk Wedding|Lady Louise De Vere Beauclerk on her wedding to Gerald Loder, M.P.]], so they were probably present at the wedding, or at least the reception. Mrs. Bontein [sic Bontine], Guendoline's mother, gave a silver box, suggesting the relationship was through the women.

'''1890 November 29, 11:30 Saturday morning''', Algernon Bourke's gift for the [[Social Victorians/Dudley-Beckwith Wedding 1890-11-29|wedding of the Hon. Francis Dudley and Miss Forbes Beckwith]] was some cases of a Bordeaux wine: &quot;three dozen Cantenac, 1875 vintage.&quot;&lt;ref&gt;&quot;Marriage of Lord Leigh's Heir. Descriptive Sketch of the Ceremony, and Full List of Guests and Presents.&quot; ''Leamington Spa Courier'' 6 December 1890, Saturday: 6 [of 10], Cols. 1a–4a. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18901206/021/0006. Same print title and p.&lt;/ref&gt;{{rp|Col. 3b}} Guendoline Bourke is not listed as having been invited to the reception, but this list from the ''Leamington Spa Courier'' has some gaps.

'''1890 December 4, Thursday''', Guendoline and Algernon Bourke attended the [[Mure-Portal Wedding 1890-12-04|wedding of Miss Mure and Mr. S. J. Portal]]. Their gift is not recorded.

'''1891 January''', Algernon Bourke took party in a [[Social Victorians/Timeline/1891#Shooting Party in Kallarnet, Totton|shooting party in Kallarnet, Totton]].

'''1891 June 24, Wednesday''', the Hon. Algernon and Mrs. Bourke attended a [[Social Victorians/Timeline/1891#Dinner and Ball Hosted by Lord and Lady Wimborne|dinner and ball Hosted by Lord and Lady Wimborne]] featuring Princess Mary Adelaide, the Duke of Teck, and Princess Victoria.

'''1891 July 9, Thursday''', Algernon and Guendoline Bourke were invited to a [[Social Victorians/1891-07-09 Garden Party|large Garden Party at Marlborough House]] hosted by the [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]] in honor of Queen Victoria and the German Emperor and Empress. The more than 3,000 people invited also included a number of people from the [[Social Victorians/People/Mayo|family of the Earl of Mayo]].

'''1891 July 22, Wednesday''', Guendoline Bourke attended a [[Social Victorians/Timeline/1891#Dinner and Dance at Alington House|dance at the Earl and Countess Alington]]'s that also included the Prince and Princess of Wales.

'''1891 October 22, Thursday''', Hon. and Mrs. Bourke attended at least the reception of the [[Social Victorians/Timeline/1891#Le Strange Astley Wedding|Le Strange—Astley Wedding]], although perhaps the couple is not the Algernon Bourkes.

'''1891 November 22, Sunday''', the London ''Weekly Dispatch'' reports a performance by American &quot;Lady Magnet&quot; Mrs. Abbott, who claimed to be able to lift anybody using only her magnetic properties. An enthusiastic &quot;committee of some fifteen gentlemen presented a written and signed testimonial&quot; supporting Mrs. Abbott, &quot;the Hon. Algernon Bourke, Professor Atkinson, Dr. Hides, and three other doctors who prefer to remain incog., being among the signatories. All the medical gentlemen concerned assured the ''Evening News and Post'' reporter of their complete and unconditional surrender. One of them went so far as to say that he had come with the full determination of disbelieving, but had been quite able to act up to his resolve.&quot;&lt;ref&gt;&quot;The Lady Magnet. Draws Crowds of People Who Divide in Opinion about Her.&quot; ''Weekly Dispatch'' (London) 22 November 1891, Sunday: 16 [of 16], Cols. 3a–4b [of 4]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003358/18911122/203/0016. Print: same title and p.&lt;/ref&gt;

'''1892''', the Hon. Algernon Bourke privately published his ''The History of White's'', the exclusive gentleman's club.

'''1892 January 27, Saturday''', Algernon and Guendoline Bourke attended the very fashionable [[Social Victorians/Timeline/1892#The Wedding of Lord Henry Cavendish Bentinck, M.P., and Lady Olivia Taylour|wedding of Lord Henry Cavendish Bentinck, M.P., and Lady Olivia Taylour]]. Their gift was not noted in the list.

'''1892 February''' '''10, Wednesday''', Guendoline Bourke attended the [[Social Victorians/1892-02-10 Alington Leigh Wedding|very fashionable wedding of Henry, Lord Alington and Evelyn Henriette Leigh]] [[Social Victorians/1892-02-10 Alington Leigh Wedding|in St. Paul's, Knightsbridge]]

'''1892 April''' '''10, Wednesday, about 2:30 p.m.''', Guendoline Bourke attended [[Social Victorians/1892-02-10 Alington Leigh Wedding|the very fashionable wedding between Henry Sturt, Lord Alington and Evelyn Leigh]]. Her gift was a &quot;tortoiseshell and gold heart-shaped tray.&quot;&lt;ref name=&quot;:02&quot;&gt;&quot;Lord Alington to Miss Leigh.&quot; ''Gentlewoman'' 20 February 1892, Saturday: 21 [of 46], Cols. 1a–3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18920220/092/0021. Same print title, p. 237.&lt;/ref&gt; (Col. 3a) 

'''1892 June 25, Saturday''', the ''Gentlewoman''&lt;nowiki/&gt;'s &quot;Overheard by the Little Bird&quot; says &quot;That pretty Mrs. Algernon Bourke has been staying here, but returned to England in time for Ascot.&quot;&lt;ref&gt;Little Bird, The. &quot;Overheard by the Little Bird.&quot; ''Gentlewoman'' 25 June 1892, Saturday: 32 [of 60], Col. 3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18920625/157/0032. Same print title, p. 860.&lt;/ref&gt; 

'''1892 December 13, Tuesday''', the ''Gentlewoman'' says Guendoline Bourke is lovely in its coverage of [[Social Victorians/Timeline/1892#Wedding of Miss Eleanor M. Ewart and Captain Guy Withington|Eleanor M. Ewart and Captain Guy Withington's wedding]]. 

'''1892 December 22, Thursday''', Algernon Bourke attended the [[Social Victorians/Timeline/1892#22 December 1892, Thursday|monthly meeting of the Zoological Society in Hanover-square]].&lt;p&gt;

'''1893 February 11, Tuesday''', Algernon Bourke opened Willis's Restaurant:&lt;blockquote&gt;Mr. Algernon Bourke has in his time done many things, and has generally done them well. His recently published history of White's Club is now a standard work. White's Club itself was a few years ago in its agony when Mr. Bourke stepped in and gave it a renewed lease of life. Under Mr. Bourke's auspices &quot;Willis's Restaurant&quot; opened its doors to the public on Tuesday last in a portion of the premises formerly so well known as Willis's Rooms. This new venture is to rival the Amphitryon in the matter of cuisine and wines; but it is not, like the Amphitryon, a club, but open to the public generally. Besides the restaurant proper, there are several ''cabinets particuliers'', and these are decorated with the very best of taste, and contain some fine portraits of the Georges.&lt;ref&gt;&quot;Marmaduke.&quot; &quot;Letter from the Linkman.&quot; ''Truth'' 20 April 1893, Thursday: 25 [of 56], Col. 1a [of 2]. ''British Newspaper Archive'' [https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18930420/075/0025# https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18930420/075/0025]. Print p. 855.&lt;/ref&gt;&lt;/blockquote&gt;

'''1893 February 7, Tuesday''', Guendoline Bourke attended [[Social Victorians/Timeline/1893#1893 February 7, Tuesday|the reception after Lady Emily Cadogan's wedding]].

'''1893 February 20, Monday''', the Hon. Algernon Bourke is listed as having attended the [[Social Victorians/Timeline/1893#Queen's Levee at St. James's Palace|Queen's Levee at St. James's Palace]] held by the Prince of Wales; because wives generally are not listed, it seems likely Guendoline Bourke attended as well.

'''1893 February 28, Tuesday, 3:00 p.m.''', Guendoline Bourke attended a [[Social Victorians/Queens Drawing Room 1893-02-28|Queen's Drawing Room at Buckingham Palace]].&lt;p&gt;

'''1893 March 22, Wednesday''', Guendoline Bourke attended [[Social Victorians/Timeline/1893#22 March 1893, Wednesday|Lady Wimborne's reception]].

'''1893 April 1, Saturday''', Algernon Bourke published a letter to the editor of the ''Times'', reprinted in the ''Kildare Observer'', arguing against Gladstone's Home Rule bill on the grounds that Ireland would not be able to take out a loan on its own behalf because of its obligations to the U.K., including what was called its share of the national debt.&lt;ref&gt;&quot;Irish Unionist Alliance.&quot; ''Kildare Observer and Eastern Counties Advertiser'' 01 April 1893, Saturday: 6 [of 8], Col. 4c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001870/18930401/062/0006. Print: The ''Kildare Observer'', n.p.&lt;/ref&gt;

'''1893 May 13, Saturday''', Algernon Bourke was seen at [[Social Victorians/Timeline/1893#13 May 1893, Saturday|exhibitions of art and furniture for sale by Christie's and on display by Lord Clifden]].

'''1893 July 13, Thursday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1893#The Countess of Listowel's Garden Party|Countess of Listowel's Garden Party]] [[Social Victorians/Timeline/1893#The Countess of Listowel's Garden Party|at her residence, Kingston House, Princes-gate]], accompanied by Miss Adeane.

'''1893 July 14, Friday''', Guendoline Bourke attended [[Social Victorians/Sandown Races 1893-07-14|the races at Sandown]] wearing a dark-blue-and-white outfit and black hat that got described in the newspaper.

'''1893 August 1, Tuesday – August 4, Friday''', Guendoline Bourke, at least, was at [[Social Victorians/Timeline/1893#1 August 1893, Tuesday – 4 August 1893, Friday|the Goodwood races]], mentioned in the ''Gentlewoman'' for her beauty, although none of the dresses were noted.

'''1893 November 4–11, Wednesday–Saturday''', Guendoline Bourke was at a [[Social Victorians/Timeline/1893#Ralph and Mary Sneyd Hosted a Shooting Party|shooting party at Keele Hall hosted by Ralph and Mary Sneyd]].

'''1893 November 30, Thursday''', with Sir Walter Gilbey the Hon. Algernon Bourke &quot;assisted&quot; in &quot;forming [a] collection&quot; of engravings by George Morland that was exhibited at Messrs. J. and W. Vokins’s, Great Portland-street.&lt;ref&gt;&quot;The George Morland Exhibition at Vokins's.&quot; ''Sporting Life'' 30 November 1893, Thursday: 4 [of 4], Col. 4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000893/18931130/058/0004.&lt;/ref&gt;

'''1893 December 14, Thursday, afternoon''', Guendoline Bourke attended the [[Social Victorians/1893-12-14 Wedding Adele Grant and George, 7th Earl of Essex|wedding of American Adele Grant and George, 7th Earl of Essex]] and gave a &quot;pearl and gold box.&quot;&lt;ref name=&quot;:22&quot;&gt;&quot;Wedding of the Earl of Essex.&quot; ''Herts Advertiser'' 16 December 1893, Saturday; 8 [of 8], Col. 1a–4b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000415/18931216/125/0008. Print title: ''The Herts Advertiser and St Albans Times'', p. 8.&lt;/ref&gt;{{rp|Col. 3c}} Mr. and Mrs. Shelley Bontein also attended, and Mrs. Bontein gave a &quot;green leather bag and purse, with coronet and monogram in gold.&quot;&lt;ref name=&quot;:22&quot; /&gt;{{rp|3b}}

'''1894 January 27, Saturday''', Psyche in &quot;The Social Peepshow&quot; in the ''Gentlewoman'' reported on a [[Social Victorians/Timeline/1894#27 January 1894, Saturday|ball hosted by Lord and Lady Dunraven at Adare Manor]] that Guendoline Bourke attended.

'''1894 January 31, Wednesday''', Algernon and Guendoline Bourke, who was dressed more stylishly than most, attended the [[Social Victorians/Timeline/1894#Also 31 January 1894, Wednesday|Kildare Hunt Ball]] hosted by Dermot, [[Social Victorians/People/Mayo|Earl of Mayo]] and Geraldine, Countess of Mayo.

'''1894 February 24, Saturday''', ''The Field'' reported on a series of tennis matches; Algernon Bourke attended the one played at the Prince's Club.&lt;ref&gt;&quot;Tennis.&quot; ''Field'' 24 February 1894, Saturday: 39 [of 72], Col. 1c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002446/18940224/349/0039. Print title ''The Field, The Country Gentleman's Newspaper'', p. 249.&lt;/ref&gt;

'''1894 March 31, Saturday''', Psyche, in the &quot;Social Peepshow&quot; column in the ''Gentlewoman'', says that &quot;Mr. Algernon Bourke has still further embellished Willis's restaurant hard by [the St. James's Theatre], by the addition of some valuable old tapestry that lately came to the hammer at Christie's.&quot;&lt;ref&gt;Psyche. &quot;The Social Peepshow.&quot; ''Gentlewoman'' 31 March 1894, Saturday: 16 [of 56], Col. 2b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18940331/081/0016. Same print title, p. 408.&lt;/ref&gt;

'''1894 April 13, Friday''', Guendoline Bourke set sail on the [[Social Victorians/Timeline/1894#P. and O. Line S.S. Rome for Gibraltar|P. and O. Line ''S.S. Rome'' for Gibraltar]] along with her stepfather, Mr. Shelley Bontein, and her brother, Mr. Sloane Stanley.

'''31 May 1894, Thursday''', the Hon. Algernon and Mrs. Bourke attended the [[Social Victorians/Timeline/1894#Reception at Devonshire House|Duchess of Devonshire's reception at Devonshire House]].&lt;p&gt;

'''1894 June 18, Monday''', the London ''Echo'' reported that Algernon Bourke was [[Social Victorians/London Clubs#Brooks'|writing a history of Brooks' Club]].&lt;p&gt;

'''1894 June 20, Wednesday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1894#Princess Louise, Marchioness of Lorne Opened the Annual Sale of the Scottish Home Industries|Annual Sale of the Scottish Home Industries]]; her outfit was described in the article in ''Lady's Pictorial''.

'''1894 August 2, Thursday''', the column &quot;Overheard by the Little Bird&quot; says, &quot;At Willis' [restaurant] — 'What a smart cotillon Mr. and Mrs. Algernon Bourke gave on Thursday evening.&quot;&lt;ref&gt;Bird, The Little. &quot;Overheard by the Little Bird.&quot; ''Gentlewoman'' 04 August 1894, Saturday: 30 [of 56], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18940804/148/0030. Print title same, p. 144.&lt;/ref&gt; Willis's Restaurant, King-street, St. James's, was a restaurant Algernon Bourke opened in 1893.&lt;p&gt;

'''1894 September 7, Saturday''', Algernon and Guendoline Bourke were at a [[Social Victorians/Timeline/1894#7 September 1894, Saturday|shooting party at Witley]], which had been loaned to one of his brothers by William Ward, 2nd Earl of Dudley.&lt;p&gt;

'''1894 October 22, Thursday''', Algernon and Guendoline Bourke attended the [[Social Victorians/Timeline/1894#Wedding of Lord Connemara and Mrs. Coleman|luncheon after the wedding of Lord Connemara and Mrs. Coleman]].

'''1894 November 3, Saturday''', Psyche, in &quot;The Social Peepshow&quot; for the Gentlewoman, reported that Guendoline Bourke had been [[Social Victorians/Timeline/1894#3 November 1894, Saturday|seen shopping in London]].

'''1895 January 5, Saturday, 2:00 p.m.''', Algernon and Guendoline Bourke gave an old mother-of-pearl workbox to [[Wolverton-Ward Wedding 1895-01-05|Lady Edith Ward for her wedding to Frederick Glyn, Lord Wolverton]] and presumably attended the wedding and reception afterwards.&lt;p&gt;

'''1895 February 23, Saturday''', the Hon. Algernon Bourke attended the [[Social Victorians/Timeline/1895#23 February 1895, Saturday|fashionable wedding of Laurence Currie and Edith Sibyl Mary Finch]]. Guendoline Bourke is not listed as having attended, but she is not noted as absent, either. Daphne Bourke was born on 5 April 1895, probably explaining Guendoline's absence.

'''1895 March 24, Sunday – 30 March, Saturday''', Algernon Bourke was [[Social Victorians/Timeline/1895#24, Sunday – 30 March 1895, Saturday|enjoying the sunny weather in Brighton]].

'''1895 April 27, Saturday''', Algernon Bourke attended the [[Social Victorians/Timeline/1895#1895 April 27, Saturday|wedding of Norah Bourke and Henry E. A. Lindsay]]. Again, Guendoline Bourke is not listed as having attended. Daphne Bourke was born on 5 April 1895, and Psyche, writing the &quot;Social Peepshow&quot; column in the Gentlewoman, says,&lt;blockquote&gt;

I regret to hear of the serious illness of Mrs. Algernon Bourke, whose first child was born a fortnight ago. It is feared that the attack is of the nature of typhoid, but happily the patient's strength keeps up. Mrs. Bourke is at her mother's house in Clarges-street.&lt;ref&gt;Psyche. &quot;The Social Peepshow.&quot; ''Gentlewoman'' 27 April 1895, Saturday: 28 [of 84], Col. 2c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18950427/147/0028. Same print title, p. 506.&lt;/ref&gt;&lt;/blockquote&gt;

'''1895 July 13, Saturday''', Algernon Bourke donated 10s. to the ''Daily Telegraph'' National Shilling Testimonial to W. G. Grace.&lt;ref&gt;&quot;''Daily Telegraph'' National Shilling Testimonial to W. G. Grace.&quot; ''Daily Telegraph &amp; Courier'' (London) 13 July 1895, Saturday: 7 [of 12], Col. 7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/18950713/079/0007. Print: ''Daily Telegraph'', p. 7.&lt;/ref&gt;

'''1895 August 24, Saturday''', &quot;Marmaduke&quot; in the ''Graphic'' says that Algernon Bourke &quot;opened a cyclists' club in Chelsea.&quot;&lt;ref&gt;&quot;Marmaduke.&quot; &quot;Court and Club.&quot; The ''Graphic'' 24 August 1895, Saturday: 11 [of 32], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/9000057/18950824/017/0011. Print p. 223.&lt;/ref&gt;

'''1895 October''', the Hon. Algernon Bourke [[Social Victorians/Timeline/1900s#24 October 1902, Friday|opened the Prince's ice-skating rink for the]] season.if the newspapers were right that 1902 was the 7th season. He also was planning a bicycling club for Kensington Gardens to open the following season.&lt;ref&gt;Mackenzie, Ethel Morell (Miss). &quot;Pins and Needles.&quot; ''Hull Daily News'' 12 October 1895, Saturday: 24 [of 40], Col. 1a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003443/18951012/024/0024. Print title: ''Hull News Supplement'', p. 1[6? 8?].&lt;/ref&gt;

'''1895 October 7, Monday''', the Hon. Algernon and Mrs. Bourke attended the [[Social Victorians/Timeline/1895#Adeane-Cator Wedding|Maud Adeane–John Cator wedding]].

'''1895 December 11, Wednesday''', Guendoline and Algernon Bourke attended a [[Social Victorians/Timeline/1895#Sneyd Party to Meet the Duke of Coburg|shooting party at the Sneyds' to meet the Duke of Coburg]].

'''1895 December 18, Wednesday''', Algernon and Guendoline Bourke attended the [[Social Victorians/Timeline/1895#Wedding of Lady Albreda Fitzwilliam and the Hon. Charles Bourke|wedding of Lady Albreda Fitzwilliam and the Hon. Charles Bourke]]. Their gift is not noted in the newspaper account.

'''1896 March 17, Tuesday''', Algernon Bourke attended the [[Social Victorians/Timeline/1896#17 March 1896, Tuesday|annual dinner of the Cymmrodorion, or the Honourable Society of Cymmrodorion]], a society for Welsh culture and history.

'''1896 April 21, Monday''', Mr. and Mrs. A. Bourke sent a gift — a &quot;box for miniature&quot; — for [[Social Victorians/Timeline/1896#Monday, 1896 April 27|the wedding of Lady Angela St. Clair Erskine and James Stewart Forbes]].

'''1896 May 21, Thursday''', the Hon. and Mrs. Algernon Bourke attended [[Social Victorians/Timeline/1896#Mrs. C. H. Wilson's Ball|Mrs. C. H. Wilson's ball in Grosvenor-square, London]].

'''1896 May 26, Tuesday,  through 28 May, Thursday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1896#Coming of Age of Mr Sloane Stanley|3-day celebration in honor of the coming of age of her brother, Cyril Sloane Stanley]].

'''1896 June 15, Monday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1896#Dinner and Dance Hosted by the Countess of Huntingdon|dance hosted by the Earl and Countess of Huntingdon]] after their dinner party.

'''1896 July 13, Monday''', Algernon Bourke (listed among the &quot;Honourables&quot;) and Mrs. A. Bourke (Listed among the &quot;Honourable Ladies&quot;) were invited to the [[Social Victorians/Timeline/1896#Queen's Garden Party at Buckingham Palace|Queen's Garden Party at Buckingham Palace]].

'''1896 June 29, Monday''', the Hon. Mrs. Algernon Bourke attended the [[Social Victorians/Cadogan-Scott Wedding 1896-06-29|wedding and reception of Lady Sophie Cadogan and Sir Samuel Scott]]. Algernon Bourke published a letter to the editor of the ''Daily Telegraph'' about White's Club — and thus Bourke's — &quot;[[Social Victorians/London Clubs#Summer Club|Summer Club]]&quot; in Kensington Park, the subject of a little controversy.

'''1896 July 21, Tuesday''', the Hon. Algernon and Mrs. Bourke attended a [[Social Victorians/Timeline/1896#Dinner Hosted by Sir Horace and Lady Farquhar|dinner hosted by Sir Horace and Lady Farquhar in Grosvenor-square]].

'''1896 August 5, Wednesday''', Algernon and Guendoline Bourke attended at the [[Social Victorians/Timeline/1896#5 August 1896|wedding of the Hon. Terence Bourke and Miss Eveline Haines]] and gave the bride an &quot;enamel muff chain.&quot;&lt;p&gt;

'''1896 August 10, Monday''', the Morning Leader reported that the Hon. Algernon Bourke, for the Foreign Office, received Li Hung Chang at St. Paul's:&lt;blockquote&gt;At St. Paul's Li Hung was received by Field-Marshal Simmons, Colonel Lane, the Hon. Algernon Bourke, of the Foreign Office (who made the necessary arrangements for the visit) and Canon Newbolt, on behalf of the Dean and Chapter. A crowd greeted Li with a cheer as he drove up in Lord Lonsdale’s striking equipage, and his Excellency was carried up the steps in an invalid chair by two stalwart constables. He walked through the centre door with his suite, and was immediately conducted by Canon Newbolt to General Gordon’s tomb in the north aisle, where a detachment of boys from the Gordon Home received him as a guard of honor.

Li inspected the monument with marked interest, and drew the attention of his suite to the remarkable likeness to the dead hero. He laid a handsome wreath of royal purple asters, lilies, maidenhair fern, and laurel, tied with a broad band of purple silk, on the tomb.

The visit was not one of inspection of the building, but on passing the middle aisle the interpreter called the attention of His Excellency to the exquisite architecture and decoration of the chancel. Li shook hands in hearty English fashion with Canon Newbolt and the other gentlemen who had received him, and, assisted by his two sons, walked down the steps to his carriage. He returned with his suite to Carlton House-terrace by way of St. Paul’s Churchyard, Cannon-st., Queen Victoria-st., and the Embankment.&lt;ref&gt;&quot;At St. Paul's.&quot; ''Morning Leader'' 10 August 1896, Monday: 7 [of 12], Col. 2b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004833/18960810/134/0007. Print p. 7.&lt;/ref&gt;&lt;/blockquote&gt;

'''1896 August 19, Wednesday''', the ''Edinburgh Evening News'' reported on the catering that White's Club and Mr Algernon Bourke arranged for the visiting Li Hung Chang:&lt;blockquote&gt;It is probably not generally known (says the &quot;Chef&quot;) that Mr Algernon Bourke, manager of White's Club, London, has undertaken to the whole of the catering for our illustrious visitor front the Flowery Land. Li Hung Chang has five native cooks in his retinue, and the greatest good fellowship exists between them and their English ''confreres'', although considerable difficulty is experienced in conversation in understanding one another's meaning. There are between 40 and and 50 to cater for daily, besides a staff about 30; that Mr Lemaire finds his time fully occupied. The dishes for his Excellency are varied and miscellaneous, and from 14 to 20 courses are served at each meal. The bills of fare contain such items as bird's-nest soup, pigs' kidneys stewed in cream, boiled ducks and green ginger, sharks' fins, shrimps and prawns stewed with leeks and muscatel grapes, fat pork saute with peas and kidney beans. The meal usually winds with fruit and sponge cake, and freshly-picked green tea as liqueur.&lt;ref&gt;&quot;Li Hung Chang's Diet.&quot; ''Edinburgh Evening News'' 19 August 1896, Wednesday: 3 [of 4], Col. 8b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000452/18960819/057/0003.&lt;/ref&gt;&lt;/blockquote&gt;

'''1896 November 6, Friday''', both Algernon and Guendoline Bourke were on the committee for the [[Social Victorians/London Clubs#Prince's Skating Club|Prince's Club ice-skating rink]], which [[Social Victorians/Timeline/1896#Opening of the Prince's Club Skating Rink|opened on this day]].

'''1896 November 22, week of''', Mrs. Algernon Bourke was part of a [[Social Victorians/Timeline/1896#Shooting Party at the Charles Wilsons' Warter Priory, Yorkshire|shooting party at the Charles Wilsons' Warter Priory, Yorkshire]].&lt;p&gt;

'''1896 November 25, Wednesday''', Mr. and Mrs. Algernon Bouke attended [[Social Victorians/Timeline/1896#Lord and Lady Burton Hosted a Party for Derby Day|Lord and Lady Burton's party for Derby Day]].&lt;p&gt;

'''1896 December 4, Friday''', the Orleans Club at Brighton was robbed:&lt;blockquote&gt;The old building of the Orleans Club at Brighton, which opens its new club house at 33, Brunswick-terrace to-day, was the scene of a very ingenious burglary during the small hours of yesterday morning. The greater portion of the club property had already been removed to the new premises, but Mr Algernon Bourke, his private secretary, and some of the officials of the club, still occupied bed-rooms at the house in the King’s-road. The corner shop of the street front is occupied by Mr. Marx, a jeweller in a large way of business, and upon his manager arriving at nine o'clock he discovered that the place had been entered through hole in the ceiling, and a great part of a very valuable stock of jewelry extracted. An examination of the morning rooms of the club, which runs over Mr. Marx's establishment reveal a singularly neat  specimen of the burglar's art. A piece of the flooring about 15in square had been removed by a series of holes bored side by side with a centre-bit, at a spot where access to the lofty shop was rendered easy by a tall showcase which stood convemently near. A massive iron girder had been avoided by a quarter of an inch, and this circumstance and the general finish of the operation point to an artist in his profession, who had acquired an intimate knowledge of the premises. The club doors were all found locked yesterday morning, and the means of egress adopted by the thief are at present a mystery.&lt;ref&gt;&quot;Burglary at Brighton.&quot; ''Daily Telegraph &amp; Courier'' (London) 05 December 1896, Saturday: 5 [of 12], Col. 7a [of 7]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/18961205/090/0005. Print title: ''Daily Telegraph''; p. 5.&lt;/ref&gt;&lt;/blockquote&gt;

'''1896 December 10, Thursday''', Guendoline Bourke was present to help staff a stall at the [[Social Victorians/Timeline/1896#10 December 1896, Thursday|Irish Industries Exhibition and Sale, Brighton]].

'''1896 December 31, Thursday''', Guendoline Bourke hosted a New Year's Eve dance:&lt;blockquote&gt;Mrs. Algernon Bourke gave a highly satisfactory and enjoyable dance on Thursday night, when the old year was danced out and the new one danced in. Most of the silver gilters at present in to len were to the fore.&lt;ref&gt;&quot;The Man about Town.&quot; ''Sporting Gazette'' 02 January 1897, Saturday: 7 [of 34], Col. 3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002525/18970102/041/0007. Print title ''The County Gentleman'', p. 7.&lt;/ref&gt;&lt;/blockquote&gt;

'''1897 January 9, Saturday''', Psyche in &quot;The Social Peepshow&quot; says that Algernon Bourke's &quot;cheerful countenance was quite in keeping with the [Christmas] season,&quot; seen in London.&lt;ref&gt;Psyche. &quot;The Social Peepshow.&quot; ''Gentlewoman'' 9 January 1897, Saturday: 22 [of 56], Col. 2a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970109/097/0022. Same print title, p. 40.&lt;/ref&gt; 

'''1897 January 13, Wednesday – 18, Monday''', Algernon and Guendoline Bourke were guests of the [[Social Victorians/Timeline/1897#The Warwickshire Hunt Club Ball|house party associated with the Warwickshire Hunt Ball]] at [[Social Victorians/People/Warwick|Warwick Castle]].

'''1897 January 30, Saturday''', Guendoline Bourke was reported to have been out shopping in London: &quot;Another charming figure was that of Mrs. Algernon Bourke all in chinchilla, with something of pale blue in a smart toque.&quot;&lt;ref&gt;Psyche. &quot;The Social Peepshow.&quot; ''Gentlewoman'' 30 January 1897, Saturday: 20 [of 59]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970130/107/0020. Same print title, p. 134.&lt;/ref&gt;

'''1897 May 31, Monday''', Hon. Algernon and Mrs. Bourke were present at a [[Social Victorians/Timeline/1897#House Party at Warwick Castle|House Party at Warwick Castle]] hosted by the Earl and Countess of Warwick.

'''1897 June 2, Wednesday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1897#Reception at the Foreign Office|reception at the Foreign Office]].

'''1897 June 12, Saturday''', the ''Gentlewoman'' reported on Guendoline Bourke's dress and hat at the [[Social Victorians/Timeline/1897#The Duchess of Albany's Bazaar at the Imperial Institute|Duchess of Albany's Bazaar at the Imperial Institute]]. 

'''1897 June 19, Saturday''', Psyche in &quot;The Social Peepshow&quot; column in the ''Gentlewoman'' writes that Guendoline Bourke was seen driving in London, &quot;in blue, ... looking as usual very handsome.&quot;&lt;ref&gt;Psyche. &quot;The Social Peepshow.&quot; ''Gentlewoman'' 19 June 1897, Saturday: 28 [of 108], Col. 2b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970619/159/0028. Same print title, p. 848.&lt;/ref&gt;

'''1897 June 28, Monday''', Algernon and Guendoline Bourke were invited to the [[Social Victorians/Diamond Jubilee Garden Party|Garden Party at Buckingham Palace]], the final official event of the London Diamond Jubilee celebrations. Members of the family of the [[Social Victorians/People/Mayo|Earl of Mayo]] were also among the 5,000–6,000 people invited.

'''1897 July 2, Friday''', the Hon. A. and Mrs. A. Bourke and Mr. and Mrs. Bourke attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House.

'''1897 July 8, Thursday, 11:00 p.m.''', Hon. Algernon and Guendoline Bourke were present at [[Social Victorians/Timeline/1890#Queen's State Ball at Buckingham Palace|the Queen's State Ball at Buckingham Palace]]. The dancing commenced shortly after 11:00 p.m.

'''1897 July 11–16, week of''', a dog of Guendoline Bourke's won a prize at the [[Social Victorians/Timeline/1897#The Ladies' Kennel Association show in the Royal Botanic Gardens in Regent's Park|Ladies' Kennel Association show in the Royal Botanic Gardens in Regent's Park]]. 

'''1897 July 23, Friday''', both the Hon. Algernon Bourke and Guendoline Bourke attended the [[Social Victorians/Timeline/1897#Bourke-Curzon Cricket Match at the Queen's Club|Bourke-Curzon cricket match at the Queen's Club]], which Algernon Bourke's team lost. 

'''1897 July 23 — or July 30, Friday''', Guendoline Bourke attended [[Social Victorians/Timeline/1897#Lady Burton's party at Chesterfield House|Lady Burton's party at Chesterfield House]]. &lt;blockquote&gt;Far the prettiest women in the room were Lady Henry Bentinck (who looked perfectly lovely in pale yellow, with a Iong blue sash; and Mrs. Algernon Bourke, who was as smart as possible in pink, with pink and white ruchings on her sleeves and a tall pink feather in her hair.&lt;ref&gt;&quot;Lady Burton's Party at Chesterfield House.&quot; ''Belper &amp; Alfreton Chronicle'' 30 July 1897, Friday: 7 [of 8], Col. 1c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004151/18970730/162/0007. Print title: ''Belper and Alfreton Chronicle''; n.p.&lt;/ref&gt;&lt;/blockquote&gt;

'''1897 August 2, Monday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1897#Warwick House Party for the Easton Lodge Cricket Week Games|Earl and Countess of Warwick's house party for Easton Lodge cricket week]].

'''1897 August 2, Monday''', Mrs. Algernon Bourke was listed as among [[Social Victorians/Timeline/1897#The Most Beautiful Women in England|the most beautiful women in England]] in an article from ''Vanity Fair'' that was reprinted elsewhere.

'''1897 September 25, Saturday''', according to the ''Pall Mall Gazette'',&lt;blockquote&gt;The [[Social Victorians/People/Mayo|Dowager-Countess of Mayo]] is staying with her son, the Hon. Algernon Bourke, at Bramnber, near Brighton.&lt;ref&gt;&quot;Pall Mall Gazette Office.&quot; ''Pall Mall Gazette'' 25 September 1897, Saturday: 8 [of 10], Col. 2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18970925/023/0008. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;'''1897 October 2, Saturday''', &quot;Yenatrix&quot; in &quot;Kennel Column&quot; in the ''Gentlewoman'' reported that Guendoline Bourke had joined the Ladies' Kennel Association.&lt;ref&gt;Yenatrix. &quot;Kennel Column.&quot; ''Gentlewoman'' 02 October 1897, Saturday: 39 [of 61], Col. 2a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18971002/182/0039. Same print title, p. 434.&lt;/ref&gt;

'''1897 October 9, Saturday''', Algernon and Guendoline Bourke were at [[Social Victorians/Timeline/1897#Harrogate|Harrogate, presumably taking the waters and baths]]. Lady May was on her way to visit Algernon Bourke in Brighton:&lt;blockquote&gt;The Earl of Mayo is expected to return from Sweden on Saturday next. Lady Mayo leaves Bournemouth on Sarurday for Brighton, where she will pay a two days' visit to her brother-in-law, the [[Social Victorians/People/Bourke|Hon. Algernon Bourke]]. The Earl and Countess will then return to Palmerstown, their seat in County Kildare.&lt;ref&gt;&quot;Pall Mall Gazette Office.&quot; ''Pall Mall Gazette'' 7 October 1897, Thursday: 8 [of 12], Col. 3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18971007/022/0008. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;&lt;p&gt;

'''1897 October 30, Saturday''', ''Black and White'' published '''J.P.B.'''&lt;nowiki/&gt;'s &quot;The Case of Mrs. Elliott,&quot;&lt;ref name=&quot;:13&quot;&gt;J.P.B. &quot;The Case of Mrs. Elliott.&quot; ''Black &amp; White'' 30 October 1897, Saturday: 12 [of 34], Cols. 1a–2b [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004617/18971030/036/0012. Print title ''Black and White'', p. 542.&lt;/ref&gt; an odd short short story in which the Honourable Algernon Bourke Herriott is &quot;rude to Mrs. Elliott,&quot;&lt;ref name=&quot;:13&quot; /&gt;{{rp|Col. 2b}} presumably having proposed sexual relations while her husband is out. J.P.B. links to the biographical Algernon Bourke's career in the stock market in the description of Mrs. Christine Elliott not even simulating interest in her husband's bicycling: &quot;a soul is a grievous burthen for a stockbroker's wife,&quot;&lt;ref name=&quot;:13&quot; /&gt;{{rp|Col. 2a}} suggesting that Mr. Elliott rather than Algernon Bourke Herriott is the stockbroker. The Hon. Algy&lt;blockquote&gt;was a senior member of several junior clubs. A woman had dubbed him once &quot;a rip with a taste for verses.&quot; The description was severe, but not unwarranted. His was a pretty pagan sensualism, though, singing from a wine palate to Church music. For the rest, he had just imagination enough to despise mediocrity.&lt;ref name=&quot;:13&quot; /&gt;{{rp|Col. 2a}}&lt;/blockquote&gt;

'''1897 November 25–26, Thursday–Friday''', Guendoline Bourke was in Brighton, helping the Countess of Mayo at the [[Social Victorians/Timeline/1897#The Irish Industries' Association Annual Exhibition|bazaar of the Irish Industries' Association]].

'''1897 December 7, Tuesday''', Algernon Bourke attended the [[Social Victorians/Timeline/1897#7 December 1897, Tuesday|7th annual dinner for the Actors' Benevolent Fund]].

'''1897 December 20, Monday, the week of''', Guendoline Bourke's Mousie was the smallest dog in the [[Social Victorians/Timeline/1897#Ladies' Kennel Association's Dog Show|Ladies' Kennel Association show in Earl's Court]].

'''1897 December 30''', Algernon and Guendoline Bourke attended a [[Social Victorians/Timeline/1897#Blenheim Palace Party with Amateur Theatricals|party at Blenheim Palace in which people performed tableaux vivants]] that got reported on, many of whom wearing the costumes from the Duchess of Devonshire's ball. The ''Irish Independent'' said Algernon Bourke was &quot;mainly responsible for the living pictures.&quot;&lt;ref&gt;&quot;Mr Algernon Bourke ....&quot; ''Irish Independent'' 05 January 1898, Wednesday: 6 [of 8], Col. 2c [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001985/18980105/115/0006.&lt;/ref&gt;

'''1898''', Algernon Bourke called a meeting at White's Club about attempting to [[Social Victorians/Timeline/1900s#The Thames Salmon Experiment|restock the Thames with salmon]]. In 1899 he was on a [[Social Victorians/People/Bourke#Committees|committee led by the Lord Mayor about this topic]] as well.

'''1898 February 3, Thursday''',  Algernon Bourke was among [[Social Victorians/Timeline/1898#The Dundee Evening Telegraph Report on People at Monte Carlo|those visiting Monte Carlo according to the Dundee ''Evening Telegraph'']].

'''1898 March 12, Saturday''', ''The World'' reported on Algernon Bourke's upgrading of the Orleans Club at Brighton:&lt;blockquote&gt;

The Orleans Club at Brighton is flourishing exceedingly, and the new buildings which Mr. Algernon Bourke has just had erected at the back of the comfortable mansion at the corner of Lansdowne-place now provide all that was wanting to make the present habitat of the club all that its members desire. The new billiard-room is rapidly approaching completion, and the coffee-room, excellent and spacious now, was open on Saturday night, when every table was occupied by club diners and their guests, all of whom were enthusiastic over the excellence of this latest addition to the comfort of the house. All interested may be congratulated on what is practically new lease of life to the Orleans Club, than which there is no more comfortable place stay within the four seas.&lt;ref&gt;&quot;From '''The World''.'&quot; ''East &amp; South Devon Advertiser'' 12 March 1898, Saturday: 6 pop 8], Col. 2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001639/18980312/132/0006. Print title ''The East and South Devon Advertiser'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;

'''1898 March 30, Wednesday''', Algernon Bourke was charged with assaulting a Mr. Potter, but it is not clear from this account what exactly happened:&lt;blockquote&gt;The Hon. Algernon H. Bourke, of Bramber, was summoned, at the instance of Mr. Walter John Potter, clerk to Mr. G. A. Flowers, solicitor, of Steyning, for assault, on the 30th March. — Mr. J. Edward Dell supported the case, and Mr. J. C. Buckwell defended, and pleaded not guilty. — The evidence was to the effect that Mr. Potter had occasion go to defendant's house on Wednesday last to serve a writ. He was going to drop the letter into [Col. 5c–6a] defendant's pocket when he turned and struck him a violent blow on the chest, making witness stagger backwards. Witness put up his hands to keep his balance, and defendant then struck him violently across the head with a weeding spud. — Richard Reed, who was at work for Mr. Bourke on the date named, and was working in garden at the time of the alleged assault, gave corroborative evidence. — Defendant, in the witness box, made a similar statement. — The magistrates differed as to whether the assault was committed, and dismissed the case.&lt;ref&gt;&quot;Steyning.&quot; ''Sussex Express'' 9 April 1898, Saturday: 2 [of 12], Col. 5c–6a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000654/18980409/036/0002. Print: ''The Sussex Express, Surrey Standard, Weald of the Kent Mail, Hants and County Advertiser'', p. 2.&lt;/ref&gt;&lt;/blockquote&gt;'''1898 April 12, Tuesday''', Algernon Bourke was among [[Social Victorians/Timeline/1898#1898 April 12, Tuesday|those visiting Monte Carlo according to the ''Gentlewoman'']].

'''1898 May 25, Wednesday''', Guendoline Bourke wore pink to [[Social Victorians/1898-05-25 Savoy Dinner Dance Hwfa|Mrs. Hwfa Williams' dinner-dance at the Savoy]].

'''1898 June 7, Tuesday''', the Hon. Algernon and Mrs. A. Bourke were invited to and probably attended the [[Social Victorians/Timeline/1898#7 June 1898, Tuesday|State Ball at Buckingham Palace hosted by the Prince and Princess of Wales]].

'''1898 July 4, Thursday afternoon''', the Hon. Algernon and Mrs. Bourke were invited to and probably attended the [[Social Victorians/Timeline/1898#Garden Party at Marlborough House|Garden Party at Marlborough House given to the Queen and Shah of Persia]].

'''1898 October 29, Saturday''', Algernon Bourke attended a [[Social Victorians/Timeline/1898#Tennis Championship Game at Prince's Club, Knightsbridge|tennis match at Prince's Club, Knightsbridge]].

'''1898 November''', Algernon Bourke received a letter from Lord Alfred Douglas and attempted to help Douglas and his father, the Marquis of Queensberry, to reconcile. Bourke met with the Marquis and sent a positive reply to Douglas the next day.&lt;ref name=&quot;:26&quot; /&gt; (568–569)

'''1898 November 22, Tuesday''', Algernon Bourke was present at a [[Social Victorians/Timeline/1898#Shooting Party Hosted by William James|shooting party hosted by Mr. William James]].

'''1898 December 3, Saturday''', Hon. Algernon and Mrs. A. Bourke attended the [[Social Victorians/Timeline/1898#The Funeral of Lady Connemara|funeral of Lady Connemara in Christ Church]], Down street, Piccadilly.&lt;p&gt;

'''1899 January 10, Tuesday''', the Brighton Championship Dog Show opened:&lt;blockquote&gt;Princess of Wales a Winner at the Ladies’ Kennel Club Show.

[Exclusive to &quot;The Leader.&quot;)

The Brighton Championship Dog Show opened in the Dome and Corn Exchange yesterday, and was very well patronised by visitors and exhibitors. Among the latter was H.R.H. the Princess of Wales, who did very well; and others included Princess Sophie Duleep Singh, Countess De Grey, Sir Edgar Boehm, the Hon Mrs. Algernon Bourke, Lady Cathcart, Lady Reid, Mr. Shirley (chairman of the Kennel Club), and the Rev. Hans Hamiiton (president of the Kennel Club).

The entry of bloodhounds is one of the best seen for some time; the Great Danes are another strong lot; deerhounds are a fine entry, all good dogs, and most of the best kennels represented; borzois are another very stylish lot. The bigger dogs are, as usual, in the Corn Exchange and the &quot;toy&quot; dogs in the Dome. To everyone's satsfaction the Princess of Wales carried off two first prizes with Alex in the borzois class.&lt;ref&gt;&quot;Dogs at Brighton.&quot; ''Morning Leader'' 11 January 1899, Wednesday: 8 [of 12], Col. 3b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004833/18990111/142/0008. Print p. 8.&lt;/ref&gt;&lt;/blockquote&gt;

'''1899 January 11, Wednesday''', Guendoline Bourke attended [[Social Victorians/Timeline/1899#11 January 1899, Wednesday|a luncheon at Stanfield-hall, home of Mr. and Mrs. Basil Montogomery, for Princess Henry of Battenberg]], that also included the Countess of Dudley (sister of Mrs. Montgomery), General Oliphant, and the Mayor and Mayoress of Romsey.

'''1899 January 17–18, Tuesday and Wednesday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1899#Ladies' Kennel Association in Brighton|Ladies' Kennel Association in Brighton]], where she showed an Italian greyhound named Brenda.

'''1899 February 7, Tuesday''', Guendoline Bourke was a member of the very high-ranking committee organizing the [[Social Victorians/Timeline/1899#Gordon Memorial College Ball|Gordon Memorial College Ball at the Hotel Cecil on 7 February 1899]]. The committee had been planning for the ball, of course, for at least 3 weeks before.

'''1899 February 22, Wednesday – April''', Guendoline Bourke was part of [[Social Victorians/Timeline/1899#Society in St. Moritz|Society in St. Moritz]]. 1899 March 29, Wednesday, the ''Dundee Advertiser'' says that [[Social Victorians/Timeline/1899#29 March 1899, Wednesday|Cyril Sloane-Stanley was spending part of the winter in St. Moritz]] with his sister Guendoline Bourke.

'''1899 April 7, Friday, probably''', oddly, Algernon and Guendoline Bourke are not reported to have attended the [[Social Victorians/Timeline/1899#Funeral of the Hon. Charles Bourke, C.B.|Funeral of the Hon. Charles Bourke, C.B.]] or even to have sent flowers.

'''1899 April 8, Saturday''', the ''Gentlewoman'' reported that Guendoline Bourke had gone to [[Social Victorians/Timeline/1899#8 April 1899, Saturday|St. Moritz with her brother, Mr. Stanley, who had gotten engaged to Lady Cairns]].

'''1899 April 26, Wednesday''', according to &quot;Local and District News&quot; for Totton, Guendoline Bourke was &quot;ill with influenza in Paris, and Mrs. Shelley Bontein, her mother, has gone out to nurse her.&quot;&lt;ref&gt;&quot;Local and District News. Totton.&quot; ''Hampshire Advertiser'' 26 April 1899, Wednesday: 4 [of 4], Col. 2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18990426/037/0004. Print title ''Hampshire Advertiser County Newspaper'', p. 4.&lt;/ref&gt;

'''1899 June 1, Thursday, or 2, Friday''', the Hon. Algernon and Guendoline Bourke attended the [[Social Victorians/Timeline/1899#Wedding of Roger Cyril Sloane Stanley and Olivia, Countess Cairns|wedding of her brother, Sloane Stanley and Olivia Countess Cairns]] at Holy Trinity Church, Brompton.

'''1899 June 8, Thursday''', Algernon Bourke's money troubles:&lt;blockquote&gt;The Hon. Algernon Bourke, son of the Earl of Mayo, has been appearing before the official receivers in connection with a winding-up order made against Willis’ Restaurant, Limited. The companyf [sic] was formed to acquire the well known restaurant from the Hon. H. A. Bourke. The chairman reminded the creditors that on the last occasion the meeting was adjourned because Mr. Bourke said he thought he would be able in the course of a fortnight to obtain an offer for a sum sufficient to satisfy the creditors and debenture holders. He had received a letter from Mr. Bourke to the effect that he had been unable to complete arrangements. Having looked into the affairs of the company more closely, it appeared to him that Mr. Bourke was legally liable to repay the sum of £5,000 which was advanced to White's Club, and the question would arise whether Mr. Bourke was not also liable to repay the sum of £4,000.&lt;ref&gt;&quot;Mr. Bourke Must Pay.&quot; ''Irish Independent'' 8 June 1899, Thursday: 4 [of 8], Col. 8c [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001985/18990608/118/0004. Print title: ''The Irish Weekly Independent'', p. 4.&lt;/ref&gt;&lt;/blockquote&gt;'''1899 July 1, Saturday''', Algernon Bourke attended a [[Social Victorians/Timeline/1899#1 July 1899, Saturday|meeting in London at the Duke of Westminster's Grosvenor House]] about preserving Killarney as part of the National Trust and seems to have been acting for someone who wanted to purchase the Muckross Estate.

'''1899 July 5, Wednesday''', Guendoline Bourke attended a [[Social Victorians/Timeline/1899#Dinner and Dance at Devonshire House|dance at Devonshire House hosted by the Duke and Duchess of Devonshire]].

'''1899 July 6, Thursday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1899#Joan Wilson and Guy Fairfax's Wedding|wedding of Joan Wilson and Guy Fairfax in St. Mark's, near Grosvenor Square]].

'''1899 July 14, Friday''', Guendoline Bourke attended [[Social Victorians/Timeline/1899#14 July 1899, Friday|Ernest Beckett's dinner party]].

'''1899 July 18, Tuesday''', Guendoline Bourke attended a [[Social Victorians/Timeline/1899#Ludovici Lecture on Impressionism|lecture on Impressionism by Ludovici hosted by the Countess of Mayo]].

'''1899 July 28, Friday''', [[Social Victorians/London Clubs#White's|White's Club]] was no longer under Algernon Bourke's management and was reconstituting itself after the possibility that it would have to close.

'''1889 July 31, Wednesday''', the Hon. Algernon and Mrs. Bourke attended a [[Social Victorians/Timeline/1889#Fete of the Uxbridge Habitation of the Primrose League|Fete of the Uxbridge Habitation of the Primrose League]] at Hillingdon Court and hosted by the Hon. Algernon and Lady Mary Mills.

'''1899 September 9, Saturday''', the ''Eastern Morning News'' includes Algernon Bourke (&quot;St. James's-street, London, club proprietor&quot;) in a list of men &quot;Receiving Orders,&quot; which it is reprinting from the ''London Gazette''.&lt;ref&gt;&quot;Receiving Orders.&quot; ''Eastern Morning'' News 9 September 1899, Saturday: 5 [of 8], Col. 3c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001152/18990909/074/0005. Same print title and p.&lt;/ref&gt;&lt;p&gt;

'''1899 October 19, Thursday''', the Hon. Algernon Bourke had a bankruptcy hearing:&lt;blockquote&gt;The public examination of the Hon. Algernon Bourke was held before Mr Registrar Giffard yesterday, at the London Bankruptcy Court. The debtor, described as proprietor of a St. James's-street club, furnished a statement of affairs showing unsecured debts £13,694 and debts fully secured £12,800, with assets which are estimated at £4,489 [?]. He stated, in reply to the Official Receiver, that he was formerly a member of the Stock Exchange, but had nothing to do with the firm of which he was a member during the last ten years. He severed his connection with the firm in May last, and believed he was indebted to them to the extent of £2,000 or £3,000. He repudiated a claim which they now made for £37,300. In 1889 he became proprietor of White's Club, St. James's-street, and carried it on until January 1st last, when he transferred it to a company called Recreations, Limited. One of the objects of the company was to raise money on debentures. The examination was formally adjourned.&lt;ref name=&quot;:9&quot; /&gt;&lt;/blockquote&gt;

'''1899 October 20, Friday''', the ''Morning Leader'' mentions Bourke's bankruptcy:&lt;blockquote&gt;Mr. Algernon Bourke, whose bankruptcy is much talked about, has been connected with numerous enterprises in clubland. He raised White's from the slough into which it had sunk after the secession of the Prince of Wales. He started the Willis Restaurant, put fresh life into the Orleans Club at Brighton, arranged a big restaurant for the bicyclists in the time of the bicycle parade, and was concerned at first in the smart and short-lived Trafalgar Bicycle Club. At one time his name spelt success. Latterly his luck has left him. He is a brother of Lord Mayo, a son of the peer who was assassinated at the post of duty, and is one of the best known men about town of the day.&lt;ref&gt;&quot;Club, Stage, and Salon.&quot; ''Morning Leader'' 20 October 1899, Friday: 6 [of 12], Col. 5b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004833/18991020/085/0006. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;'''1899 November 8, Wednesday''', the Hon. Algernon Bourke's bankruptcy case came up again:&lt;blockquote&gt;At Bankruptcy Court, yesterday, the case the Hon. Algernon Bourke again came on for hearing before Mr. Registrar Giffard, and the examination was concluded. The debtor has at various times been proprietor of White’s Club, St. James’s-street, and the Orleans’ Club, Brighton, and also of Willis's Restaurant, King-street, St. James's. He attributed his failure to losses sustained by the conversion of White’s Club and the Orleans' Club into limited companies, to the payment of excessive Interest on borrowed money, and other causes. The liabilities amount to £26,590, of which £13,694 are stated to be unsecured, and assets £4,409.&lt;ref&gt;&quot;Affairs of the Hon. A. Bourke.&quot; ''Globe'' 09 November 1899, Thursday: 2 [of 8], Col. 1c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001652/18991109/020/0002. Print p. 2.&lt;/ref&gt;&lt;/blockquote&gt;

'''1899 December 23, Saturday''', &quot;Mr. Algernon Bourke has departed for a tour in Africa, being at present the guest of his brother in Tunis.&quot;&lt;ref&gt;&quot;The Society Pages.&quot; ''Walsall Advertiser'' 23 December 1899, Saturday: 7 [of 8], Col. 7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001028/18991223/143/0007. Print p. 7.&lt;/ref&gt;

'''1899 December 29, Friday''', Guendoline Bourke was at the [[Social Victorians/Timeline/1899#Christmas Party Hosted by the Duke and Duchess of Marlborough|Christmas Party Hosted by the Duke and Duchess of Marlborough]].&lt;p&gt;

'''1899 December 31''', the San Francisco newspaper ''The Wave'' wrote the following about London society:&lt;blockquote&gt;The most prominent untitled people in London may be said to be Mr. and Mrs. [[Social Victorians/People/Williams|Hwfa Williams]], Mr. and Mrs. [[Social Victorians/People/Grenfell|Willie Grenfell]] and Mr. Algy Bourke. That they are passing rich, goes without saying, and that they entertain lavishly, understood — for to be untitled, prominent and successful, argues wealth, hospitality and cleverness.&lt;ref&gt;&quot;London.&quot; The (San Francisco) ''Wave'' 14 January 1899 (Vol. XIX, No. 2): 14. ''The Internet Archive'' https://archive.org/details/wave19unse/page/n20/mode/1up.&lt;/ref&gt;&lt;/blockquote&gt;

=== 1900s ===

'''1900 February 15, Thursday''', Daphne Bourke, the four-year-old daughter of the Hon. Algernon and Mrs. Bourke was a bridesmaid in the [[Social Victorians/Wilson Chesterfield Wedding 1900-02-15|wedding of Enid Wilson and the Earl of Chesterfield]].&lt;ref&gt;&quot;London Day by Day.&quot; ''Daily Telegraph'' 15 February 1900, Thursday: 8 [of 12], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/19000215/175/0008. Name in British Newspaper Archive: ''Daily Telegraph &amp; Courier'' (London). Print p. 8.&lt;/ref&gt; Guendoline Bourke, &quot;who was in grey, wore a chinchilla toque with violets.&quot;&lt;ref&gt;&quot;Society. Entertainments, Balls, &amp;c.&quot; ''The Queen'' 24 February 1900, Saturday: 40 [of 76], Col. 1a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/19000224/235/0040. Print: ''The Queen, The Lady's Newspaper'', p. 308.&lt;/ref&gt;

'''1900 March 10, Saturday''', the ''Weekly Irish Times'' reprinted society gossip from ''The World'':&lt;blockquote&gt;Mrs. Algernon Bourke, who has been staying with her husband's uncle, old Connemara, during Mr. Algernon Bourke's absence abroad, has taken a new house near Portman square, and will be settling there before Easter.&lt;ref&gt;&quot;Society Gossip.&quot; ''Weekly Irish Times'' 10 March 1900, Saturday: 17 [of 20], Col. 1b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001684/19000310/116/0017. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;'''1900 July''' '''17, Tuesday''', Guendoline Bourke took part in the [[Social Victorians/Timeline/1900s#17 July 1900, Tuesday|Children's Fete in support of the National Society for the Prevention of Cruelty to Children]] on the grounds of the Royal Botanic Society. Daphe was 5 at this time, so it seems logical that she would have been there, too.

'''1900 July 30, Monday''', Guendoline Bourke attended [[Social Victorians/Timeline/1900s#Barber of Seville at Covent Garden|''The Barber of Seville'' at Covent Garden]].

'''1890 August 6, Friday''', &quot;[[Social Victorians/Timeline/1890#Beautiful Women|Beautiful Women]],&quot; an article in ''Vanity Fair'' that was reprinted elsewhere, mentions Guendoline Bourke (&quot;Lady Algernon Bourke&quot;) as one of the most beautiful women in England.

'''1900 August 11, Saturday''', Guendoline Bourke got&lt;blockquote&gt;the pretty little Yorkshire String, an especially tiny mite, weighing only 2&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;lb, and carrying a very promising coat, ... at the Aquarium Show.&lt;ref&gt;&quot;The Witchampton Kennel.&quot; &quot;Ladies Kennels.&quot; ''Ladies' Field'' 11 August 1900, Saturday: 16 [of 60], Col. 2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0006043/19000811/043/0016. Print title same, p. 390.&lt;/ref&gt;&lt;/blockquote&gt;&lt;p&gt;

'''1900 September 16''', the Hon. Algernon Bourke became the heir presumptive to the Earldom of Mayo when his older brother Captain Hon. Sir Maurice Archibald Bourke died.&lt;p&gt;

'''1900 October 06, Saturday''', the ''Weekly Irish Times'' says that Mr. Algernon Bourke, now heir presumptive to the earldom of Mayo, &quot;has been for some months lately staying with Mr. Terence Bourke in Morocco.&quot;&lt;ref&gt;&quot;Society Gossip.&quot; ''Weekly Irish Times'' 06 October 1900, Saturday: 14 [of 20], Col. 3b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001684/19001006/121/0014. Print p. 14.&lt;/ref&gt;&lt;p&gt;

'''1901 May 30, Thursday''', the Hon. Mrs. Algernon Bourke attended the fashionable [[Social Victorians/Timeline/1900s#1901 May 30, Thursday|Ladies' Kennel Association Dog Show at the Botanic Garden]].

'''1901 July 2, Tuesday''', Guendoline Bourke — &quot;pretty Mrs. Algernon Bourke, in a mauve gown and and purple tulle toque&quot; — attended a children's party at the Botanic Gardens hosted by the Earl and Countess of Kilmorey.&lt;ref&gt;&quot;The Earl of Kilmorey, K.P.&quot; ''Gentlewoman'' 13 July 1901: Saturday, 50 [of 84], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010713/237/0050. Print: title the same, p. 60.&lt;/ref&gt;

'''1901 July 4, Thursday''', Guendoline Bourke — dressed &quot;in pale grey, with her pretty little girl,&quot; 6-year-old Daphne — attended a [[Social Victorians/Timeline/1900s#The Countess of Yarborough's Children's Party|children's party hosted by the Countess of Yarborough]].&lt;ref&gt;&quot;The Countess of Yarborough ....&quot; ''Gentlewoman'' 13 July 1901, Saturday: 76 [of 84], Col. 2b, 3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010713/381/0076. Print p. xxxvi.&lt;/ref&gt;{{rp|Col. 3a}}

'''1901 July 4–6, Thursday–Saturday''', Guendoline Bourke helped staff the Perthshire stall&lt;ref&gt;&quot;The Great County Sale.&quot; ''Gentlewoman'' 29 June 1901, Saturday: 43 [of 72], Col. 3a [of 3]. ''British Newspaper Archive'' [https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010629/223/0043# https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010629/223/0043]. Same print title, pp. 679.&lt;/ref&gt; at the [[Social Victorians/Timeline/1900s#The Great County Sale|Great County Sale in the Imperial Gardens of the Earl's Court Exhibition]].

'''1901 July 20, Saturday''', the ''Gentlewoman'' published the Hon. Mrs. Algernon Bourke's portrait (identified with &quot;Perthshire&quot;) in its 3rd series of &quot;The Great County Sale at Earl's Court. Portraits of Stallholders.&quot;&lt;ref&gt;&quot;The Great County Sale at Earl's Court. Portraits of Stallholders.&quot; ''Gentlewoman'' 20 July 1901, Saturday: 31 [of 60], Col. 4b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010720/141/0031. Print n.p.&lt;/ref&gt; Their daughter Daphne appears in the portrait as well.

'''1901 July 23, Tuesday''', an &quot;Hon. Mrs. Bourke&quot; was in the [[Social Victorians/Timeline/1900s#Lord and Lady Algernon Gordon Lennox|party &quot;entertained by Lord and Lady Algernon Gordon Lennox]].&quot;&lt;p&gt;

'''1901 September 12, Thursday''', Mrs. Guendoline Bourke wanted her name listed as Mrs. Algernon Bourke in the Electoral Register, apparently a frequent complaint:&lt;blockquote&gt;Mr. Underhill, the Conservative agent, mentioned to the Revising Barrister (Mr. William F. Webster) that the name of the Hon. Mrs. Guendoline Bourke was on the list in respect of the house, 75, Gloucester-place. The lady had written to him to say that she was the Hon. Mrs. Algernon Bourke and that she wished that name to appear on the register. In reply to the Revising Barrister, Mr. Underhill said that “Algernon” was the name of the lady’s husband.

Mr. Cooke, the rate-collector, said that Mrs. Bourke had asked to be addressed Mrs. Algernon Bourke, but that the Town Clerk thought the address was not a correct one. The lady signed her cheques Guendoline.”

Mr. Underhill said the agents frequently had indignant letters from ladies because they were not addressed by their husband’s Christian name.

The Revising Barrister — lf a lady gave me the name of Mrs. John Smith I should say I had not got the voter’s name. The name Guendoline must remain.&lt;ref name=&quot;:15&quot;&gt;&quot;Ladies’ Names.&quot; ''Morning Post'' 12 September 1901, Thursday: 7 [of 10], Col. 3a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/19010912/130/0007. Print p. 7.&lt;/ref&gt;&lt;/blockquote&gt;

'''1901 October 26, Friday''', Algernon Bourke was on the Men's Committee of the [[Social Victorians/London Clubs#Prince's Club Ice-skating Rink|Prince's Club Ice-skating Rink]], which had [[Social Victorians/Timeline/1900s#The Prince's Club Ice-skating Rink Opening|its official opening on his day]]. 

'''1902 January''', Algernon Bourke is mentioned in [[Social Victorians/Schools#&quot;More of My Contemporaries at School.&quot;|reminiscences of Eton written by the &quot;Earl of X&quot;]] as being among those in the &quot;world of letters,&quot; and whose brother, later the Earl of Mayo, the Earl of X did not like. 

'''1902 January 25, Saturday''', Mrs. Algernon Bourke gave a box to Lady Helen Stewart-Vane-Tempest in honor of [[Social Victorians/Stewart-Stavordale Wedding 1902-01-25|Lady Helen's wedding to Giles Fox-Strangways, Lord Stavordale]]. 

'''1902 April 26, Saturday''', Mrs. A. Bourke is listed as being at the Norfolk Hotel in Brighton.&lt;ref&gt;&quot;Guide to Visitors at Hotels and Boarding Houses.&quot; ''Brighton Gazette'' 26 April 1902, Saturday: 3 [of 8], Col. 4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000938/19020426/116/0003. Same print title and p.&lt;/ref&gt; 

'''1902 May, End of''', Guendoline Bourke attended a [[Social Victorians/Timeline/1900s#End of May 1902|party at Blenheim Palace hosted by the Duke and Duchess of Marlborough]]. 

'''1902 June 11, Monday''', the Hon. Mrs. Algernon Bourke had a dog entered in the [[Social Victorians/Timeline/1900s#Ladies' Kennel Association Show|Ladies' Kennel Association competitions in the Botanic Gardens]]. 

'''1902 September 4, Thursday''', the ''Daily Express'' reported that &quot;Mrs. Algernon Bourke is staying with Lord and Lady Alington at Scarborough.&quot;&lt;ref&gt;&quot;Onlooker.&quot; &quot;My Social Diary.&quot; &quot;Where People Are.&quot; ''Daily Express'' 04 September 1902, Thursday: 5 [of 8], Col. 1b? [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004848/19020904/099/0005. Print p. 4, Col. 7b [of 7].&lt;/ref&gt;

'''1902 September 22, Monday''', Guendoline Bourke was a guest at the [[Social Victorians/Timeline/1900s#Earl and Countess of Mar and Kellie's House Party|large house party hosted by the Earl and Countess of Mar and Kellie]].

'''1902 October 24, Friday''', the Hon. Algernon Bourke [[Social Victorians/Timeline/1900s#Annual Opening of the Prince's Ice-skating Rink|opened the Prince's ice-skating rink for the season]], which he had been doing since 1895.

'''1902 October 25, Saturday''', Algernon Bourke was bequeathed £500 by his uncle [[Social Victorians/People/Mayo|Robert Bourke]], who had died 3 September 1902.&lt;ref&gt;&quot;Will of Lord Connemara.&quot; ''Kildare Observer and Eastern Counties Advertiser'' 25 October 1902, Saturday: 2 [of 8], Col. 4b–c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001870/19021025/037/0002. Print title the ''Kildare Observer'', n.p.&lt;/ref&gt;&lt;p&gt;

'''1902 October 31, Friday''', the [[Social Victorians/Timeline/1900s#Annual Opening of the Prince's Ice-skating Rink|7th opening of the Prince's Skating Club]]. Guendoline Bourke was on the Women's Committee and Algernon Bourke was on the Men's.&lt;p&gt;

'''1902 November 8, Friday, beginning, perhaps''', Guendoline Bourke was part of the [[Social Victorians/Timeline/1900s#8 November 1902, Saturday|Earl and Countess of Warwick's shooting party at Easton Lodge]].&lt;p&gt;

'''1902 December 9, Tuesday''', Guendoline Bourke attended [[Social Victorians/Timeline/1900s#9 December 1902, Tuesday|Lady Eva Wyndham-Quin's &quot;at home,&quot; held at the Welch Industrial depot]] for the sale Welsh-made Christmas gifts and cards. Bourke wore &quot;a fur coat and a black picture hat.&quot;&lt;ref&gt;&quot;A Lady Correspondent.&quot; &quot;Society in London.&quot; ''South Wales Daily News'' 11 December 1902, Thursday: 4 [of 8], Col. 5a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000919/19021211/082/0004. Print p. 4.&lt;/ref&gt;

'''1903 February 6, Friday''', Hon. Mrs A. Bourke was present at a [[Social Victorians/Timeline/1900s#Dinner Party Hosted by Lord Lieutenant of Ireland and the Countess of Dudley|dinner party Hosted by Lord Lieutenant of Ireland and the Countess of Dudley]]. &lt;p&gt;

'''1903 February 9, Monday''', Guendoline Bourke was present at a [[Social Victorians/Timeline/1900s#Dinner Party Hosted by Lord Lieutenant of Ireland and the Countess of Dudley|house party at Dublin Castle hosted by the Lord Lieutenant and Countess of Dudley that began the Viceregal season]].

'''1903 March 17, Tuesday''', Guendoline Bourke staffed a booth at a [[Social Victorians/Timeline/1900s#1903 March 17, Tuesday|sale of the Irish Industries Association]] on St. Patrick's Day with [[Social Victorians/People/Mayo|Lady Mayo]], [[Social Victorians/People/Dudley|Georgina Lady Dudley]] and [[Social Victorians/People/Beresford|Miss Beresford]]. A number of other aristocratic women were also present at the sale in other booths, including [[Social Victorians/People/Londonderry|Lady Londonderry]] and [[Social Victorians/People/Lucan|Lady Lucan]].

'''1903 June 19, Friday''', Guendoline Bourke was invited to the [[Social Victorians/Timeline/1900s#Grand Ball in the Waterloo Chamber at Windsor Castle|grand ball at Windsor Castle]], the end of the Ascot-week festivities.

'''1903 June 23, Tuesday''', Guendoline and Daphne Bourke were invited to a [[Social Victorians/Timeline/1900s#1903 June 23, Tuesday|children's party at Buckingham Palace for Prince Eddie's birthday]].

'''1903 July 10, Friday, or so''', Guendoline Bourke attended a [[Social Victorians/Timeline/1900s#Party Hosted by the Duke and Duchess of Marlborough|party hosted by the Duke and Duchess of Marlborough]].

'''1904 May 17, Tuesday''', Guendoline Bourke had agreed to let Daphne appear in the tableaux vivants arranged by Sir Philip Burne-Jones for the [[Social Victorians/Timeline/1900s#Countess Cadogan's Great Bazaar|Countess of Cadogan's great bazaar]]. Some mothers had had to decline because of the outbreaks of measles and chicken pox.&lt;p&gt;

'''1904 June 30, Thursday''', Guendoline and Daphne Bourke attended another birthday party for Prince Eddie at Buckingham Palace, and the ''Gentlewoman'' says, &quot;No prettier little girl was to be seen that day than little Miss Daphne Bourke, the daughter of the Hon. Mrs. Algernon Bourke, with her wonderful Irish eyes and colouring, her pretty white frock being relieved with a rose pink sash.&quot;&lt;ref&gt;&quot;Prince Eddie's Birthday.&quot; ''Gentlewoman'' 02 July 1904, Saturday: 68 [of 92]. Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19040702/360/0068. Print: title the same, p. 42.&lt;/ref&gt;&lt;p&gt;

'''1904 September 15, Thursday''', according to what was at the time called the ''Irish Daily Independent and Nation'', Algernon Bourke was living in Venice and not in the UK at this point:&lt;blockquote&gt;Algernon Bourke, who usually lives in Venice, has spent some time in England during the present summer, and has now gone on a fishing expedition to Sweden, accompanied by his brother, Lord Mayo. Lady Mayo has been staying meanwhile in Ireland, and has had a visit from her mother, Lady Maria Ponsonby, who is a sister of Lend Obventry.&lt;ref name=&quot;:10&quot;&gt;&quot;Society Notes.&quot; ''Irish Independent'' 15 September 1904, Thursday: 4 [of 8], Col. 5b [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001986/19040915/131/0004. Print title: ''Irish Daily Independent and Nation'', p. 4.&lt;/ref&gt;&lt;/blockquote&gt;

'''1904 October 22, Saturday''', the ''Gentlewoman'' reported that &quot;Mrs. Algernon Bourke is paying a visit to Venice, which Mr. Bourke has made his headquarters for several years past, as he is connected with some very artistic stone and marble works situated near the Grand Canal.&quot;&lt;ref&gt;&quot;The Social Peepshow.&quot; ''Gentlewoman'' 22 October 1904, Saturday: 24 [of 6ths 8], Col. 2c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19041022/112/0024. Print title same, p. 672.&lt;/ref&gt;

'''1905 February 17, Friday''', the Dundee ''Evening Post'' reported that Algernon Bourke &quot;set up a shop in Venice for the sale of art treasures and old furniture.&quot;&lt;ref&gt;&quot;Social News.&quot; Dundee ''Evening Post'' 17 February 1905, Friday: 6 [of 6], Col. 7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000582/19050217/105/0006. Print p. 6.&lt;/ref&gt;

'''1905 April 26, Wednesday''', Guendoline Bourke attended the [[Social Victorians/Timeline/1900s#New Forest United Hunt Ball|New Forest United Hunt Ball]], as did her brother Captain R. C. H. Sloane Stanley and his wife Olivia Countess Cairns.&lt;p&gt;

'''1905 June 5, Monday''', Algernon Bourke wrote to the ''Times'' from Venice that &quot;The Venetian wits have suggested a motto for Admiral Togo, Togo Tenga Tutto (Togo takes the lot).&quot;&lt;ref&gt;&quot;Mr. Algernon Bourke.&quot; ''Hull Daily Mail'' 08 June 1905, Thursday: 2 [of 6], Col. 6a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000324/19050608/008/0002. Print title ''Daily Mail'', p. 6.&lt;/ref&gt;&lt;p&gt;

'''1905, last week of July''', Guendoline Bourke and daughter Daphne Bourke — who was 10 years old — attended [[Social Victorians/Timeline/1900s#Last week of July, 1905|Lady Cadogan's children's party at Chelsea House]]. Daphne was &quot;One of loveliest little girls present.&quot;&lt;ref&gt;&quot;Court and Social News.&quot; ''Belfast News-Letter'' 01 August 1905, Tuesday: 7 [of 10], Col. 6b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000038/19050801/157/0007. Print p. 7.&lt;/ref&gt;&lt;p&gt;

'''1906 March 9, Friday''', Guendoline Bourke was a reference for Mr. Frances Burgess, who taught piano, singing, voice production, organ and music theory. Burgess was &quot;Organist and Choirmaster of St. Columbs', North Kensington, Director of the Plainsong and Medieval Music Society's Choir, etc., etc.&quot;&lt;ref name=&quot;:21&quot;&gt;&quot;Mr. Francis Burgess.&quot; ''Kilburn Times'' 9 March 1906, Friday: 3 [of 8], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001813/19060309/086/0003. Print title: ''Kilburn Times Hampstead and North-western Press'', p. 3.&lt;/ref&gt;&lt;p&gt;

'''1906 December 10, Monday''', Guendoline Bourke was seen in the tea room, possibly with Lady Grosvenor, at [[Social Victorians/Timeline/1900s#1906 December 10, Monday|Lady Dudley's sale of Irish needlework]].&lt;p&gt;

'''1907 May''', a &quot;naval signalling incident&quot; [[Social Victorians/Timeline/1887#May 1887|caused the Waterford ''Evening News'' to recall a similar event]] that had occurred 20 years earlier, in which Algernon Bourke, as special correspondent for the ''Times'', publicized [[Social Victorians/People/Beresford|Lord Charles Beresford]]'s use of his ship's signalling capabilities to send a message to his wife about being late for dinner:&lt;blockquote&gt;

The naval signalling incident is still in the air. It is expected that the matter will not be threshed out until Emperor William leaves England. A story of a former signalling incident in which [[Social Victorians/People/Beresford|Lord Charles Beresford]] was concerned is going the rounds at the moment.&lt;/blockquote&gt;

'''1907 August 24, Saturday''', Algernon Bourke was present at [[Social Victorians/Timeline/1900s#Polo Week at Eaton Hall, Duke and Duchess of Westminster|Polo Week at Eaton Hall, hosted by the Duke and Duchess of Westminster]].

'''1908 July 30, Thursday''', Guendoline Bourke was at [[Social Victorians/Timeline/1900s#Glorious Goodwood. Cup Day and Dresses.|Cup Day at the Goodwood races]], wearing salmon-pink with a matching hat.

'''1909 April 20, Tuesday''', Lady Rosemary Cairns — daughter of Olivia Sloan-Stanley, Countess Cairns and Cyril Sloane-Stanley — and Wyndham Portal were [[Social Victorians/Timeline/1900s#20 April 1909, Tuesday|married in St. Margaret's, Westminster]]. Lavender and Diane Sloane-Stanley were bridesmaids.&lt;p&gt;

'''1909 May 22, Saturday''', Algernon Bourke appears to have been living in Pisa. A columnist for the ''Queen'' reported on the Royal School of Art Needlework:&lt;blockquote&gt;Lady Leconfield [?] was there, also her sister-in-law, the [[Social Victorians/People/Mayo|Dowager Lady Mayo]], only just back from her winter on the Continent, when she spent most of the time at Pisa, where her son Mr Algernon Bourke has also been staying. The latter is a great connoisseur as regards [art?] notably in what is really good in the way of old Italian sculpture and carving. He and his handsome wife have a place near to Putney, and this winter again Mr Bourke, as the result of his Italian travels, has been sending home such relics of the old Italian palace gardens as as stone and marble carved vases, garden seats, and what-not of the kind — not all for himself and his own gardens by any means, I fancy; but his friends, relying on his knowledge in such matters, get him when abroad to choose for [them?] the adornment of their English terraces and gardens.&lt;ref&gt;&quot;My Social Diary.&quot; The ''Queen'' 22 May 1909, Saturday: 31 [of 86], Col. 1b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/19090522/203/0031. Print p. 871.&lt;/ref&gt;&lt;/blockquote&gt;'''1909 September''', the Hon. Algernon Bourke was among the [[Social Victorians/Timeline/1900s#Visitors in Venice from the U.K.|many visitors from &quot;England&quot; in Venice]] in September.

=== 1910s ===
'''1910 April 20, Wednesday''', the ''Tatler'' printed an &quot;open letter&quot; to Geraldine, Countess of Mayo, as part of its &quot;The Searchlight in Society&quot; series and mentioned Algernon Bourke, saying he had been keeping &quot;a curiosity shop at Venice&quot;:&lt;blockquote&gt;The Bourkes have brains, and a good example is afforded by Mr. Algernon Bourke, next brother to Lord Mayo and heir-presumptive to the title. He is a good-looking man who used to be known as Buttons Bourke, and he married well, as his wife was the rich and pretty Miss Guendolen Sloane Stanley. He may be described as a &quot;Jack of all trades,&quot; but it is not I who will say that he is a master of none. He was once in the Stock Exchange, then he took White's Club in hand and restored it to much of its former prestige. After that he dabbled in smart hotels and restaurants, and the last thing I heard of him was that he kept a curiosity shop at Venice.&lt;ref&gt;Candida. &quot;The Searchlight in Society. Our Open Letter. No. CII. The Countess of Mayo.&quot; The ''Tatler'' 20 April 1910, Wednesday: 18 [of 42], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001852/19100420/023/0018. Print title same, p. 72.&lt;/ref&gt;&lt;/blockquote&gt;

'''1911 November 21, Tuesday''', Guendoline Bourke assisted the [[Social Victorians/Timeline/1910s#21 November 1911, Tuesday|Duchess of Marlborough at her at-home]] that included a sale of work by the wives of prisoners.&lt;p&gt;

'''1912 September 27, Friday''', Guendoline and Daphne Bourke were visiting Mr. and Mrs. Shelley Bontein, her mother and stepfather.&lt;ref&gt;&quot;From 'The World.'&quot; ''Berks and Oxon Advertiser'' 27 September 1912, Friday: 2 [of 8], Col. 4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002298/19120927/014/0002. Same print title, n.p.&lt;/ref&gt;&lt;p&gt;

'''1913 April 23, Wednesday''', the Irish Independent reported that Guendoline and Daphne Bourke had arrived in London for the season:&lt;blockquote&gt;&lt;p&gt;

The Hon. Mrs. Algernon Bourke and Miss Bourke have arrived for the season at 75 Gloucester place, Portman square, London.&lt;ref&gt;&quot;Social and Personal.&quot; ''Irish Independent'' 23 April 1913, Wednesday: 4 [of 10], Col. 5b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001715/19130423/081/0004. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;&lt;p&gt;

'''1913 May 7, Wednesday''', Guendoline Bourke presented her daughter Daphne Bourke at court:&lt;blockquote&gt;Mrs. Algernon Bourke presented her daughter, and wore blue and gold broché with a gold lace train.&lt;ref&gt;&quot;Social and Personal.&quot; London ''Daily Chronicle'' 08 May 1913, Thursday: 6 [of 12], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005049/19130508/120/0006. Print p. 6.&lt;/ref&gt;&lt;/blockquote&gt;

The ''Pall Mall Gazette'' has a description of Daphne Bourke's dress, but what exactly &quot;chiffon [[Social Victorians/Terminology#Hoops|paniers]]&quot; means in 1913 is not clear:&lt;blockquote&gt;Court dressmakers appear to have surpassed all previous records in their efforts to make the dresses for to-night’s Court as beautiful as possible. Noticeable among these is the dainty presentation gown to be worn by Miss Bourke, who will be presented by her mother, the Hon. Mrs. Algernon Bourke. This has a skirt of soft white satin draped with chiffon [[Social Victorians/Terminology#Hoops|paniers]] and a bodice veiled with chiffon and trimmed with diamanté and crystal embroidery. Miss Bourke’s train, gracefully hung from the shoulders, is of white satin lined with pale rose pink chiffon and embroidered with crystal and diamanté.&lt;ref&gt;&quot;Fashion Day by Day. Lovely Gowns for To-night's Court.&quot; ''Pall Mall Gazette'' 07 May 1913, Wednesday: 13 [of 18], Col. 1a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/19130507/199/0013. Print n.p.&lt;/ref&gt;&lt;/blockquote&gt;The ''London Evening Standard'' describes Guendoline and Daphne Bourke the same way except with differences in editing:&lt;blockquote&gt;Miss Bourke: Presented by her mother, the Hon. Mrs. Algernon Bourke. Dainty presentation gown of white satin, the skirt draped with chiffon paniers, bodice veiled chiffon and trimmed with diamanté and crystal embroidery. Train gracefully hung from shoulder of white satin embroidered with crystal and diamanté, lined with pale rose pink chiffon.&lt;ref&gt;&quot;Some of the Dresses.&quot; &quot;The King and Queen. Third Court. Most Brilliant of the Year.&quot; ''London Evening Standard'' 08 May 1913, Thursday: 11 [of 18], Col. 4b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/19130508/237/0011. Print title ''The Standard'', p. 11.&lt;/ref&gt;&lt;/blockquote&gt;

According to the ''Lady's Pictorial'', Daphne Bourke's dress was designed and constructed by [[Social Victorians/People/Dressmakers and Costumiers#Messrs Russell and Allen|Messrs. Russell and Allen]], Old Bond-street, W., and the description is identical (except for a couple of commas).&lt;ref&gt;&quot;Their Majesties' Court.&quot; ''Lady's Pictorial'' 17 May 1913, Saturday: 35 [of 64], Col. 2c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005980/19130517/296/0035. Same print title, p. 787.&lt;/ref&gt;

'''1914 May 11, Monday''', Guendoline and Daphne Bourke attended a [[Social Victorians/Timeline/1910s#Dance at the Ritz Hosted by Mrs. George Marjoribanks|dance at the Ritz hosted by Mrs. George Marjoribanks]].

'''1915 January 1, Friday''', Algernon Bourke is listed as being on the Executive Committee of the [[Social Victorians/Timeline/1910s#1915 January 1, Friday|National Food Fund, publicized by the ''Conservative and Unionist Women's Franchise Review'']].

'''1916 August 25, Friday''', Daphne Bourke's and John Fortescue's engagement was announced:&lt;blockquote&gt;A most attractive prospective bride (says the &quot;Star&quot;) is Mr. and Mrs. Algernon Bourke's only daughter, Miss Daphne Bourke, whose engagement has just taken place to Mr. Fortescue, of the Coldstream Guards. Miss Bourke is tall, dark, and very beautiful; and Mr. Fortescue is one of the family of Boconoc, Cornwall, and Dropmore, Maidenhead. At the latter place the two families have been neighbours, for Mr. and Mrs. Algernon Bourke have a charming country residence at Taplow, while Dropmore is famous for its magnificent gardens.&lt;ref&gt;&quot;Personalia.&quot; ''Uxbridge &amp; W. Drayton Gazette'' 25 August 1916, Friday: 4 [of 8], Col. 5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002285/19160825/043/0004. Print title ''The Advertiser'', p. 4.&lt;/ref&gt;&lt;/blockquote&gt;&lt;p&gt;'''1917 June 7, Thursday''', Daphne Bourke and John Grenville Fortescue [[Social Victorians/Timeline/1910s#7 June 1917, Thursday|married in the Coldstream Guards' chapel]].

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==
According to both the ''Morning Post'' and the ''Times'', the Hon. Algernon Bourke was among the Suite of Men in the [[Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts#&quot;Oriental&quot; Procession|&quot;Oriental&quot; procession]] at the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]].&lt;ref name=&quot;:2&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;&lt;ref name=&quot;:3&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt; Based on the people they were dressed as, Guendoline Bourke was probably in this procession but it seems unlikely that Algernone Bourke was.

[[File:Guendoline-Irene-Emily-Bourke-ne-Sloane-Stanley-as-Salammb.jpg|thumb|alt=Black-and-white photograph of a standing woman richly dressed in an historical costume with a headdress and a very large fan|Hon. '''Guendoline''' Bourke as Salammbô. ©National Portrait Gallery, London.]]
=== Hon. Guendoline Bourke ===
[[File:Alfons Mucha - 1896 - Salammbô.jpg|thumb|left|alt=Highly stylized orange-and-yellow painting of a bare-chested woman with a man playing a harp at her feet|Alfons Mucha's 1896 ''Salammbô''.]]
Lafayette's portrait (right) of &quot;Guendoline Irene Emily Bourke (née Sloane-Stanley) as Salammbô&quot; in costume is photogravure #128 in the '''Album''' presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:4&quot;&gt;&quot;Devonshire House Fancy Dress Ball (1897): photogravures by Walker &amp; Boutall after various photographers.&quot; 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.&lt;/ref&gt; The printing on the portrait says, &quot;The Hon. Mrs. Algernon Bourke as Salammbo.&quot;&lt;ref name=&quot;:23&quot;&gt;&quot;Mrs. Algernon Bourke as Salammbo.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158491/Guendoline-Irene-Emily-Bourke-ne-Sloane-Stanley-as-Salammb.&lt;/ref&gt;

The Lafayette Archive has 2 additional poses from the same session on 5 July 1897 as the one chosen for the Album:

* Same image as the Album photograph but higher resolution than the one the National Portrait Gallery, London, gives permission to post (Neg. No. GP [L] [http://lafayette.org.uk/bou1369.html 1369]).
* Standing with fan behind head, includes close-up of skirt fabric and left hand (Neg. No. GP [L] [http://lafayette.org.uk/bou1368-444.html 1368-444]).
* Reclining on pillows and furs, includes close-up of face and headdress (Neg. No. GP [L] [http://lafayette.org.uk/bou1368-442.html 1368-442]).

==== Newspaper Accounts ====
The Hon. Mrs. A. Bourke was dressed as Salambo in the Oriental procession&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; in a costume made by [[Social Victorians/People/Dressmakers and Costumiers#Mrs. Mason|Mrs. Mason]], although Russell Harris says [[Social Victorians/People/Dressmakers and Costumiers#Mr. Charles Alias|Charles Alias]] was the &quot;Costume supplier.&quot;&lt;ref&gt;{{Cite web|url=http://lafayette.org.uk/bou1369.html|title=Hon Mrs Algernon Bourke (1870-1967), née Guendoline Irene Emily Sloane-Stanley by Lafayette 1897|website=lafayette.org.uk|access-date=2025-07-05}}&lt;/ref&gt;

Besides the two that mention her — the ''Morning Post'' and the ''Times'' — only two describe her costume, the London ''Evening Standard'' and the ''Gentlewoman'':
* &quot;Mrs. A. Bourke, as an Egyptian Princess, with the Salambo coiffure, wore a flowing gown of white and silver gauze covered with embroidery of lotus flowers. The top of the gown was ornamented with old green satin embroidered with blue turquoise and gold, and studded with rubies. The train was of old green broché with sides of orange and gold embroidery, and from the ceinture depended long bullion fringe and an embroidered ibis.&quot;&lt;ref&gt;“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.&lt;/ref&gt;{{rp|p. 3, Col. 3b}}
* &quot;(Egyptian Princess), drapery gown of white and silver gauze, covered with embroidery of lotus flowers; the top of gown appliqué with old green satin embroidered blue turquoise and gold, studded rubies; train of old green broché.&quot;&lt;ref&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 40, Col. 3a}}

==== Commentary on Her Costume ====

*The heavy headdress is likely attached to a wig and further stabilized by something under Bourke's chin and by the filet around her forehead. The flowers appear to have jewels at the center. What looks like a neck treatment is probably part of the structure holding the headpiece in place. The placement of the flowers is clearly a copy of the headdress worn by Rose Caron in the 1892 Paris production of the Reyer opera. Made by Eugène Lacoste, this textile, metal and pearl ''coiffe'' still exists (https://www.gazette-drouot.com/article/bijoux-d-opera/54941) and is held by the Bibliothèque Nationale de France.&lt;ref&gt;Merle, Sandrine. &quot;Les bijoux d’Opéra en pleine lumière au Palais Garnier.&quot; ''La Gazette Druout'' 4 July 2025 (No. 26). Retrieved July 2025

https://www.gazette-drouot.com/article/bijoux-d-opera/54941.&lt;/ref&gt; It seems to have been constructed from metal for the gold framework with the leaf design, beads attached to hanging metal frameworks (the blue-green beads look like they're covered with fabric or wrapped with thread) and flowers that may have been made from stiffened pinked fabric (likely organdy) whose color may have darkened in the century that has passed.
*Bourke's posture as she leans in the reclining odalisque pose shows she is wearing a corset.
*The translucent overskirt looks like an apron attached under the bust and hanging from the bodice.
*Russell Harris says the large fan looks more Egyptian rather than Carthaginian.&lt;ref name=&quot;:25&quot; /&gt; 
*The fullness of the skirt is very controlled.
*Her shoes are slippers decorated with real or artificial jewels.

==== Salammbô ====
Salammbô is the fictitious protagonist in Gustave Flaubert's 1862 novel ''Salammbô'', set during the Roman war against Carthage.&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|date=2024-04-29|title=Salammbô|url=https://en.wikipedia.org/w/index.php?title=Salammb%C3%B4&amp;oldid=1221352216|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Salammb%C3%B4.&lt;/ref&gt; Salammbô is a Carthaginian priestess of the lunar goddess Tanit. Matho, a Roman mercenary, breaks into Tanit's temple and steals her sacred veil — the spiritual guardian of Carthage. Salammbô sneaks into the enemy encampment to steal the veil back. She meets Matho in his tent, and &quot;believing each other to be divine apparitions,&quot; they make love,&lt;ref name=&quot;:5&quot; /&gt; although it is also a defilement. Salammbô succceds in getting the veil back, but Matho is tortured and executed, which causes her to die of shock, the effect of both having touched the veil. The plot of the opera is not identical to that of the novel.

What Guendoline Bourke saw in Salammbo as representative of herself is difficult to know, unless the costume itself appealed to her. A production of Ernest Reyer's opera ''Salammbô'', based on Flaubert's novel and published in Paris in 1890, opened at the Paris Opéra on 16 May 1892,&lt;ref&gt;{{Cite journal|date=2024-04-11|title=Ernest Reyer|url=https://en.wikipedia.org/w/index.php?title=Ernest_Reyer&amp;oldid=1218353215|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Ernest_Reyer.&lt;/ref&gt; starring Madame Rose Caron, with Mademoiselle Lucienne Bréval performing when Caron was on vacation.&lt;ref&gt;Jullienn, Adolphe. &quot;Mademoiselle Lucienne Bréval de L'Académie Nationale de Musique [or de l'Opéra in the Table of Contents].&quot; ''Le Théatre'' April 1898 (No. 4). Google Books https://www.google.com/books/edition/_/_oxRAQAAMAAJ. Pp. 8–10.&lt;/ref&gt;{{rp|8, Col. 2c}} This production was widely reviewed and discussed in the papers in the UK, and its production design was notable, especially Caron's costumes, the sets and [[Social Victorians/People/Bourke#Scale of the Production of Salammbo|the very scale of the production]]. Bourke or her costumier clearly saw the opera or images of the performers or its posters, certainly influencing the design of her costume.

* Rose Caron in her Salammbo costume is here: https://www.gettyimages.com/detail/news-photo/rose-caron-french-soprano-in-costume-in-the-title-role-of-news-photo/1439485238.
* A headshot of Bréval in costume is here: https://books.google.com/books/content?id=_oxRAQAAMAAJ&amp;pg=RA3-PP7&amp;img=1&amp;zoom=3&amp;hl=en&amp;bul=1&amp;sig=ACfU3U2Gv8Os_rEmx2gM9SakJkYLJ9hW7g&amp;ci=6%2C1%2C988%2C1371&amp;edge=0.)
* &quot;Salammbo's hair [was] powdered with a violet dust when she first appeared before the eyes of Matho.&quot;&lt;ref&gt;&quot;Salome.&quot; ''Pall Mall Gazette'' 27 February 1893, Monday: 3 [of 8]. Col. 2b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18930227/010/0003. Same print title and p.&lt;/ref&gt;

Salammbo figured in paintings, sculptures and illustrations of editions of Flaubert's novel before Ernest Reyer's 1890 opera. Translations and illustrated editions of Flaubert's novel had come out steadily beginning in the 1880s. Alfons Mucha's 1896 lithograph of Salammbô (above left) was published the year before the ball. Salammbo is often depicted as nude and highly sexual or sexualized (kissing a huge snake, for example, that she holds aloft). Reyer's opera was first produced in 1890 in Brussels. Both Modest Mussorgsky and Sergei Rachmaninoff had attempted but not completed operas based on the novel as well.&lt;ref name=&quot;:5&quot; /&gt; 


Guendoline Bourke's costume and her social life as reported in the newspapers do not suggest that she was a big risk-taker like, for example, the eccentric la Comtesse de Castiglione, who appeared at a ball in a Salammbo costume in 1886, 4 years after Flaubert's novel was first published. In 1889 the ''Edinburgh Evening News'' exaggerates her nudity and doesn't describe the rush in the ballroom to see her but does address the lingering memory:&lt;blockquote&gt;The late Countess Castiglione, whose death in Paris is recorded yesterday, made her first appearance at the Imperial Court in 1866, where her extraordinary beauty made a great impression on Napoleon III., and eventually led to the Empress Eugenie’s undertaking an unexpected and much-talked-of visit to Scotland. The Countess had a face and complexion which would have enchanted Rubens, and her lovely golden hair touched her feet. Even at the present day Paris has not forgotten her costume, or rather absence of costume as Salammbo, in which character she figured at a certain memorable ball at the Tuileries, wearing her hair, her jewels, and very little else. The Empress Eugenie, when she was presented to her thus lightly arrayed, declared that she must be cold, and insisted upon her there and then donning a mantle. Mme. de Castiglione was never again invited to an entertainment over which the Empress Eugenie presided.&lt;ref&gt;&quot;A Countess’ Queer Ball Costume.&quot; ''Edinburgh Evening News'' 2 December 1899, Saturday: 2 [of 6], Col. 7b [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000452/18991202/024/0002. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;Given how widely this incident was discussed at the time of the death of la Comtesse in 1889, Guendoline Bourke might easily have known about it. But she was developing relationships with people like the Princess of Wales, and what Countess Castigiolone did does not sound at all like her. 

===== Scale of the Production of ''Salammbo'' =====

* &quot;In Reyer's opera of 'Salammbo,' lately produced at the Grand Opera in Paris, there were 1,400 persons on the stage in the last act.&quot;&lt;ref&gt;&quot;Facts and Fancies.&quot; ''Louth and North Lincolnshire Advertiser'' 9 July 1892, Saturday: 3 [of 8], Col. 6c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000313/18920709/038/0003. Same print title and p.&lt;/ref&gt;
* &quot;the battle scene in [''Salammbo''] requires no less than 3000 square yards of 'decorative surface' [probably canvas]. This establishes a record, the next largest surface being that of the salles des fetes in 'Don Giovanni.'&quot;&lt;ref&gt;&quot;A French paper gives interesting details....&quot; ''Sevenoaks Chronicle and Kentish Advertiser'' 26 August 1892, Friday: 2 [of 8], Col. 3c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001067/18920826/032/0002. Same print title, n.p.

&lt;/ref&gt;

===== Influence of the Production of ''Salammbo'' =====
Rose Caron's productions were influential, including for the costumes she wore. The 1892 ''Lohengrin'' she starred in was the source of the costumes worn by [[Social Victorians/People/Stonor#Julia Caroline Stonor, Marquise of Hautpoul|Julia Stonor, Marquise of Hautpoul]] and her brother, [[Social Victorians/People/Stonor#Hon. Harry Stonor|Hon. Harry Stonor]].

Women's clothing was influenced by the costumes in the opera, particularly those worn by Rose Caron. One color of intense red was called Salammbo. A bonnet was named the Salammbô:&lt;blockquote&gt;About the smartest thing in bonnets for ordinary complimentary mourning is called the Salammbô, and is copied from a head-dress worn by a leading artiste at one of the Paris theatres. It is made of jet, and has a rose on each side of the front from the centres of which rise two black ospreys.&lt;ref&gt;Mantalini, Miss. &quot;The Shows in the London Shops. With Mems. about Millinery.&quot; ''Pall Mall Budget'' 29 December 1892, Thursday: 22 [of 40], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005967/18921229/092/0022. Same print title, p. 1928.&lt;/ref&gt;&lt;/blockquote&gt;In a long illustrated article describing the wedding of Princess Marie of Edinburgh, the ''Lady's Pictorial'' provides a sketch of &quot;a very pretty [hat] (No. 4) of brown mirror velvet trimmed with mink and a brown velvet bow in front '''with Salammbo''' '&lt;nowiki/&gt;''fantaisie''&lt;nowiki/&gt;'&quot; that is among the bride's millinery.&lt;ref&gt;&quot;The Marriage of H.R.H. Princess Marie of Edinburgh and H.R.H. Ferdinand Crown Prince of Roumania.&quot; ''Lady's Pictorial'' 14 January 1893, Saturday: 40 [of 76], Col. 3c [of 3]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0005980/18930114/064/0040. Same print title, p. 56.&lt;/ref&gt;&lt;p&gt;

Shoes appeared:&lt;blockquote&gt;At Mrs Merritt's, Savile-street, the stock is particularly attractive, there being so many new styles in shoes this season. One of the latest designs is the Salammbo Shoe, glace kid, with one strap, a jet buckle, and very low French heels. This shoe is especially designed for tender feet, as it is very light in weight.&lt;ref&gt;&quot;House and Home. Local Letter for Women Reader [sic], (By Our Lady Contributor).&quot; ''Hull Daily Mail'' 22 July 1897, Thursday: 5 [of 6], Col. 1b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000324/18970722/069/0005. Same print title, n.p.&lt;/ref&gt;&lt;/blockquote&gt;Patterns for making the Tunique Romaine and Corsage Salammbo were for sale just a few months after the opening:&lt;blockquote&gt;Some of the leading fashionable novelties described in ''Le Follet de Paris'' are almost ahead of the season, but they look so well that it will not be long before our provincial dressmakers have them. A revival and modification of the ancient tunic is one item which is transforming the modern gowns of tailor-built tweeds into long clinging draperies, of simple cut but ineffable grace. We have had the Russian blouse with us now for the last couple of months. Now the reign of Tunique Romaine and Corsage Salambo is upon us.

... [including the address to send off for the patterns]

A very successful novelty is the ''corsage'' “Salammbo.” In reality, it is more of a blouse and short tunic than a ''corsage'', as there is no attempt at shaping to the figure. In [sic] consists, indeed, of two straight pieces of material cut round on the shoulders, where the back and front are fastened together by clasps. There is no arm-hole, and the two pieces meet at the waist under the arm, and then hang open on to the skirt. There being no dart, the waist is as wide as the shoulders; the fullness is drawn to the centre under a ''ceinture Russe'', or of oxydised silver. The outlines are trimmed with ''galon'' or some similar garniture. The &quot;Salammbo” ''guimpe'' or ''corsage'' are made of flannel or ''mousseline de laine'' of bright colour, and are worn with fitting bodices or skirts of serge, or woollen of dark colour. They are very effective, and nothing can be easier to make, while their addition to a frock constitutes a separate costume.

The fitting bodices worn under the ''guimpes'' or ''robes'' &quot;Salammbo&quot; are very simply made; being round-waisted, they are without side pieces, and only require a seam under each arm; one in the centre of the back, and one or two darts in front, according to the figure. The skirt is mounted on a round waistband, and the ''ceinture'' worn over this gives the bodice and skirt the effect of a princess dress.&lt;ref&gt;&quot;A Womans Ceilidh.&quot; ''Oban Times and Argyllshire Advertiser'' 3 September 1892, Saturday: 3 [of 8], Col. 6a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000462/18920903/078/0003. Print title: ''The Oban Times'', p. 3.&lt;/ref&gt;&lt;/blockquote&gt;Stationery even before the opera opened in Paris:&lt;blockquote&gt;The last fad in fancy stationery is the carte Salammbo, a delightfully smooth surface for writing upon, the envelopes are very small, square, and of the wallet make; the paper folds over once to fit. The newest shades are rose pink, pale English blue, apple green, and the evergreen heliotrope.&lt;ref&gt;&quot;Fashions of the Month.&quot; ''Nottinghamshire Guardian'' 27 February 1892, Saturday: 7 [of 8], Col. 2b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000176/18920227/059/0007. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;[[File:Algernon Henry Bourke Vanity Fair 20 January 1898.jpg|thumb|alt=Old colored drawing of an elegant elderly man dressed in a 19th-century tuxedo with a cloak, top hat and formal pointed shoes with bows, standing facing 1/4 to his right|''Algy'' — Algernon Henry Bourke — by &quot;Spy,&quot; ''Vanity Fair'' 20 January 1898]]

=== Hon. Algernon Bourke ===
[[File:Hon-Algernon-Henry-Bourke-as-Izaak-Walton.jpg|thumb|left|alt=Black-and-white photograph of a man richly dressed in an historical costume sitting in a fireplace that does not have a fire and holding a tankard|Hon. Algernon Henry Bourke as Izaak Walton. ©National Portrait Gallery, London.]]
'''Lafayette's portrait''' (left) of &quot;Hon. Algernon Henry Bourke as Izaak Walton&quot; in costume is photogravure #129 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:4&quot; /&gt; The printing on the portrait says, &quot;The Hon. Algernon Bourke as Izaak Walton.&quot;&lt;ref&gt;&quot;Hon. Algernon Bourke as Izaak Walton.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158492/Hon-Algernon-Henry-Bourke-as-Izaak-Walton.&lt;/ref&gt;

This portrait is amazing and unusual: Algernon Bourke is not using a photographer's set with theatrical flats and props, certainly not one used by anyone else at the ball itself. Isaak Walton (baptised 21 September 1593 – 15 December 1683) wrote ''The Compleat Angler''.&lt;ref&gt;{{Cite journal|date=2021-09-15|title=Izaak Walton|url=https://en.wikipedia.org/w/index.php?title=Izaak_Walton&amp;oldid=1044447858|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Izaak_Walton.&lt;/ref&gt; A cottage Walton lived in and willed to the people of Stafford was photographed in 1888, suggesting that its relationship to Walton was known in 1897, raising a question about whether Bourke could have used the fireplace in the cottage for his portrait. (This same cottage still exists, as the [https://www.staffordbc.gov.uk/izaak-waltons-cottage Isaak Walton Cottage] museum.)

A caricature portrait (right) of the Hon. Algernon Bourke, called &quot;Algy,&quot; by Leslie Ward (&quot;Spy&quot;) was published in the 20 January 1898 issue of ''Vanity Fair'' as Number 702 in its &quot;Men of the Day&quot; series,&lt;ref&gt;{{Cite journal|date=2024-01-14|title=List of Vanity Fair (British magazine) caricatures (1895–1899)|url=https://en.wikipedia.org/w/index.php?title=List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899)&amp;oldid=1195518024|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899).&lt;/ref&gt; giving an indication of what he looked like out of costume.

=== Mr. and Mrs. Bourke ===
The ''Times'' made a distinction between the Hon. Mr. and Mrs. A. Bourke and Mr. and Mrs. Bourke, including both in the article.&lt;ref name=&quot;:3&quot; /&gt; Occasionally this same article mentions the same people more than once in different contexts and parts of the article, so they may be the same couple. (See [[Social Victorians/People/Bourke#Notes and Question|Notes and Question]] #2, below.)

== Demographics ==

=== The Bourkes ===
*Nationality: Anglo-Irish&lt;ref&gt;{{Cite journal|date=2020-11-14|title=Richard Bourke, 6th Earl of Mayo|url=https://en.wikipedia.org/w/index.php?title=Richard_Bourke,_6th_Earl_of_Mayo&amp;oldid=988654078|journal=Wikipedia|language=en}}&lt;/ref&gt;
*Occupation: journalist. 1895: restaurant, hotel and club owner and manager&lt;ref&gt;''Cheltenham Looker-On'', 23 March 1895. Via Ancestry but taken from the BNA.&lt;/ref&gt;

==== Residences ====
*Ireland: 1873: Palmerston House, Straffan, Co. Kildare.&lt;ref name=&quot;:7&quot; /&gt; Not Co. Mayo?
*1888–1891: 33 Cadogan Terrace, S.W., Kensington and Chelsea, a dwelling house&lt;ref&gt;Kensington and Chelsea, London, England, Electoral Registers, 1889–1970, Register of Voters, 1891.&lt;/ref&gt;
*1894: 181 Pavilion Road, Kensington and Chelsea&lt;ref&gt;Kensington and Chelsea, London, England, Electoral Registers, 1889–1970. Register of Voters, 1894. Via Ancestry.&lt;/ref&gt;
*1900: 181 Pavilion Road, Kensington and Chelsea&lt;ref&gt;Kensington and Chelsea, London, England, Electoral Registers, 1889–1970. Register of Voters, 1900. Via Ancestry.&lt;/ref&gt;
*1904: Algernon Bourke was &quot;usually liv[ing] in Venice&quot;&lt;ref name=&quot;:10&quot; /&gt;
*1906: 75, Gloucester-place, W.&lt;ref name=&quot;:21&quot; /&gt;
*Guendoline Bourke
*1911: 1911 Fulham, London&lt;ref name=&quot;:6&quot; /&gt;
*20 Eaton Square, S.W. (in 1897)&lt;ref name=&quot;:0&quot;&gt;{{Cite book|url=https://books.google.com/books?id=Pl0oAAAAYAAJ|title=Who's who|date=1897|publisher=A. &amp; C. Black|language=en}} 712, Col. 1b.&lt;/ref&gt; (London home of the [[Social Victorians/People/Mayo|Earl of Mayo]])

=== The Sloane-Stanleys ===

==== Residences ====

* 1871: Chester Street, St George Hanover Square (Census), with 5 servants, including a cook and a footman.&lt;ref name=&quot;:16&quot;&gt;The National Archives; Kew, London, England; ''1871 England Census''; Class: ''RG10''; Piece: ''104''; Folio: ''21''; Page: ''37''; GSU roll: ''838763''. Ancestry.com. ''1871 England Census'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations Inc, 2004.&lt;/ref&gt;
* 1881–1885&lt;ref&gt;''UK, City and County Directories, 1600s-1900s''. Ancestry.com. ''UK, City and County Directories, 1766 - 1946'' [database on-line]. Provo, UT, USA: Ancestry.com Operations, Inc., 2013.&lt;/ref&gt; [at least]: 14 Halkin Street, W., St. Georges, 14 servants, including a governess, a house steward, an under butler, a footman and a cook.&lt;ref&gt;''Census Returns of England and Wales, 1881''. Kew, Surrey, England: The National Archives of the UK (TNA): Public Record Office (PRO), 1881. Class: ''RG11''; Piece: ''98''; Folio: ''66''; Page: ''37''; GSU roll: ''1341022''. Ancestry.com and The Church of Jesus Christ of Latter-day Saints. ''1881 England Census'' [database on-line]. Provo, UT, USA: Ancestry.com Operations Inc, 2004.&lt;/ref&gt;
* 1888: 49, Cadogan-square, St. Luke, Chelsea&lt;ref&gt;Ancestry.com. ''London, England, Overseer Returns, 1863-1894'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2013.&lt;/ref&gt;
* 1899, Roger Cyril Sloane-Stanley: 4 Down St., St George, Hanover Square&lt;ref&gt;London Metropolitan Archives; London, England; ''Electoral Registers''. Ancestry.com. ''London, England, Electoral Registers, 1832-1965'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2010.&lt;/ref&gt;
* 1911, Roger Cyril Sloane-Stanley: Paultons, Ower, Romsey

== Family ==
*Hon. Algernon Henry Bourke (31 December 1854 – 7 April 1922)&lt;ref&gt;&quot;Hon. Algernon Henry Bourke.&quot; {{Cite web|url=https://www.thepeerage.com/p29657.htm#i296561|title=Person Page|website=www.thepeerage.com|access-date=2020-12-10}}&lt;/ref&gt;
*Guendoline Irene Emily Sloane-Stanley Bourke (c. 1869 – 30 December 1967)&lt;ref name=&quot;:1&quot;&gt;&quot;Guendoline Irene Emily Stanley.&quot; {{Cite web|url=https://www.thepeerage.com/p51525.htm#i515247|title=Person Page|website=www.thepeerage.com|access-date=2020-12-10}}&lt;/ref&gt;
#Daphne Marjory Bourke (5 April 1895 – 22 May 1962)

=== Relations ===
*Hon. Algernon Henry Bourke (the 3rd son of the [[Social Victorians/People/Mayo|6th Earl of Mayo]]) was the older brother of Lady Florence Bourke.&lt;ref name=&quot;:0&quot; /&gt;
*Wilfred Blunt was a cousin of Algernon Bourke: Bourke's mother's &quot;mother was one of the Blunts of Crabbet Park, Sussex, which makes them kinswomen of Mr. Alfred Scawen Blunt, poet, Egyptophil and counsel for Arabi Pasha in his trial.&quot;&lt;ref&gt;&quot;From ''Truth''.&quot; ''Mid-Lothian Journal'' 23 August 1912, Friday: 8 [of 8], 2c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002721/19120823/147/0008. Print title and p. same.&lt;/ref&gt;
*Lord Alfred Douglas was a cousin of Algernon Bourke.

==== Other Bourkes ====
*Hubert Edward Madden Bourke (after 1925, Bourke-Borrowes)&lt;ref&gt;&quot;Hubert Edward Madden Bourke-Borrowes.&quot; {{Cite web|url=https://www.thepeerage.com/p52401.htm#i524004|title=Person Page|website=www.thepeerage.com|access-date=2021-08-25}} https://www.thepeerage.com/p52401.htm#i524004.&lt;/ref&gt;
*Lady Eva Constance Aline Bourke, who married [[Social Victorians/People/Dunraven|Windham Henry Wyndham-Quin]] on 7 July 1885;&lt;ref&gt;&quot;Lady Eva Constance Aline Bourke.&quot; {{Cite web|url=https://www.thepeerage.com/p2575.htm#i25747|title=Person Page|website=www.thepeerage.com|access-date=2020-12-02}} https://www.thepeerage.com/p2575.htm#i25747.&lt;/ref&gt; he became 5th Earl of Dunraven and Mount-Earl on 14 June 1926.

=== The Sloane-Stanleys ===
* Emilie Josephine S Stanley ( 21 December 1848 [baptism]&lt;ref&gt;London Metropolitan Archives; &quot;London, England, UK&quot; ; ''London Church of England Parish Registers''; Reference Number: ''P87/Tri/001''. Ancestry.com. ''London, England, Church of England Births and Baptisms, 1813-1923'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2010.&lt;/ref&gt;  – October 1945)
* Hans T Sloane Stanley (11 May 1840 [baptism]&lt;ref&gt;Ancestry.com. ''England, Select Births and Christenings, 1538-1975'' [database on-line]. Provo, UT, USA: Ancestry.com Operations, Inc., 2014.&lt;/ref&gt; – 15 December 1888&lt;ref&gt;Ancestry.com. ''UK and Ireland, Find a Grave® Index, 1300s-Current'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2012.&lt;/ref&gt;)
* James Shell[e?]y Bontein ()
*# Gwendoline&lt;ref name=&quot;:14&quot; /&gt; Irene Emily G Stanley (c. 1870&lt;ref name=&quot;:16&quot; /&gt; – )
*# '''Roger Cyril Hans Sloane Stanley''' (29 April 1875&lt;ref&gt;The National Archives; Kew, Surrey, England; ''WO 42 War Office: Officers' Birth Certificates, Wills and Personal Papers 1755-1908''; Reference: ''WO 42/72''. Ancestry.com. ''UK, Officers' Birth Certificates, Wills and Personal Papers, 1755-1908'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2023.&lt;/ref&gt; – 18 November 1944&lt;ref&gt;''Find a Grave''. Find a Grave®. http://www.findagrave.com/cgi-bin/fg.cgi. Ancestry.com. ''UK and Ireland, Find a Grave® Index, 1300s-Current'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2012.&lt;/ref&gt;)


* Olivia Elizabeth Berens, Countess Cairns&lt;ref&gt;The National Archives of the UK (TNA); Kew, Surrey, England; ''Census Returns of England and Wales, 1911''. Ancestry.com. ''1911 England Census'' [database on-line]. Provo, UT, USA: Ancestry.com Operations, Inc., 2011.&lt;/ref&gt; (c. 1871 – 20 June 1951&lt;ref&gt;&quot;Olivia Elizabeth Berens.&quot; Person Page 3908; person #39077. ''The Peerage: A Genealogical Survey of the Peerage of Britain as well as the Royal Families of Europe''. https://www.thepeerage.com/p3908.htm#i39077.

&lt;/ref&gt;)
* Arthur William Cairns, 2nd Earl Cairns (21 December 1861 – 14 January 1890)&lt;ref name=&quot;:20&quot;&gt;&quot;Arthur William Cairns, 2nd Earl Cairns.&quot; Person Page 3908; Person #39076. ''The Peerage: A Genealogical Survey of the Peerage of Britain as well as the Royal Families of Europe''. https://www.thepeerage.com/p3908.htm#i39076.&lt;/ref&gt;
*# Lady Louise Rosemary Kathleen Virginia Cairns (10 March 1889 – 17 May 1962)&lt;ref name=&quot;:20&quot; /&gt;
* Roger Cyril Hans Sloane Stanley (1875 – 18 November 1944)
*# Lavender Elizabeth (20 May 1900 [baptism]&lt;ref&gt;Hampshire Archives and Local Studies; Winchester, England, UK; ''Anglican Parish Registers''; Reference: ''35M76/PR3''. Ancestry.com. ''Hampshire, England, Church of England Baptisms, 1813-1921''[database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2023.&lt;/ref&gt; – )
*# Diane Sloane Stanley (c. 1905 – )


* Lavender Elizabeth (20 May 1900 [baptism] – )
* John Everett ()


* Diane Sloane Stanley (c. 1905 – )
* Elwyn Villiers Rhys ()

== Writings, Memoirs, Biographies, Papers ==

=== Writings ===
* Bourke, the Hon. Algernon. ''The History of White's''. London: Algernon Bourke [privately published], 1892.
* Bourke, the Hon. Algernon, ed., &quot;with a brief Memoir.&quot; ''Correspondence of Mr Joseph Jekyll with His Sister-in-Law, Lady Gertrude Sloane Stanley, 1818–1838''. John Murray, 1893.
* Bourke, the Hon. Algernon, ed. ''Correspondence of Mr Joseph Jekyll''. John Murray, 1894.

=== Papers ===
* Where are the papers for the Earl of Mayo family? Are Algernon and Guendoline Bourke's papers with them?

== Notes and Questions ==
#The portrait of Algernon Bourke in costume as Isaac Walton is really an amazing portrait with a very interesting setting, far more specific than any of the other Lafayette portraits of these people in their costumes. Where was it shot? Lafayette is given credit, but it's not one of his usual backdrops. If this portrait was taken the night of the ball, then this fireplace was in Devonshire House; if not, then whose fireplace is it?
#The ''Times'' lists Hon. A. Bourke (at 325) and Hon. Mrs. A. Bourke (at 236) as members of a the &quot;Oriental&quot; procession, Mr. and Mrs. A. Bourke (in the general list of attendees), and then a small distance down Mr. and Mrs. Bourke (now at 511 and 512, respectively). This last couple with no honorifics is also mentioned in the report in the London ''Evening Standard'', which means the Hon. Mrs. A. Bourke, so the ''Times'' may have repeated the Bourkes, who otherwise are not obviously anyone recognizable. If they are not the Hon. Mr. and Mrs. A. Bourke, then they are unidentified. It seems likely that they are the same, however, as the newspapers were not perfectly consistent in naming people with their honorifics, even in a single story, especially a very long and detailed one in which people could be named more than once.
#Three slightly difficult-to-identify men were among the Suite of Men in the [[Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts#&quot;Oriental&quot; Procession|&quot;Oriental&quot; procession]]: [[Social Victorians/People/Halifax|Gordon Wood]], [[Social Victorians/People/Portman|Arthur B. Portman]] and [[Social Victorians/People/Sarah Spencer-Churchill Wilson|Wilfred Wilson]]. The identification of Gordon Wood and Wilfred Wilson is high because of contemporary newspaper accounts. The Hon. Algernon Bourke, who was also in the Suite of Men, is not difficult to identify at all. Arthur Portman appears in a number of similar newspaper accounts, but none of them mentions his family of origin.
#[http://thepeerage.com The Peerage] has no other Algernon Bourkes.
#The Hon Algernon Bourke is #235 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who were present]]; the Hon. Guendoline Bourke is #236; a Mr. Bourke is #703; a Mrs. Bourke is #704.
#Hans Stanley-Sloane's estate was £33,704 7s. 5d. in the final probate in December 1889,&lt;ref&gt;Principal Probate Registry; London, England; ''Calendar of the Grants of Probate and Letters of Administration made in the Probate Registries of the High Court of Justice in England''. Ancestry.com. ''England &amp; Wales, National Probate Calendar (Index of Wills and Administrations), 1858-1995'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2010.&lt;/ref&gt; which might lead his widow to consider remarrying.
== Footnotes ==
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      <text bytes="66175" sha1="oxkm51wyjjzob9a4w3bfwwzr9sskwmg" xml:space="preserve">==Cliques and Social Networks==
All the clubs are cliques and social networks, in a way, so the distinction is not strong.
*The [[Social Victorians/Marlborough House Set |Marlborough House Set]]
*The [[Social Victorians/People/The Souls |Souls]]
*The Coterie
*[[Social Victorians/London Clubs#Crabbet Club|Crabbet Club]]
*[[Social Victorians/London Clubs#Men and Women's Club|Men and Women's Club]]
The Hunt Clubs, regionally rather than London based, with an annual Hunt Ball

*Warwickshire Hunt Club
*Worcestershire Hunt Club

&lt;blockquote&gt;No dance is prettier than that of a hunt club. It has an air about it more nationally charactcristic than that given by officers of a regiment, or of a ship. There is a spirit and go about it too seldom equalled at other dances, and the hunt uniforms are always in themselves a study. Only an expert can tell at a glance to which hunt particular facings belong, although many of us are acquainted with those of the more celebrated clubs. One of those in Oxfordshire, which shall be nameless, has adopted recently a new blue evening tie; the members are at some variance of opinion with regard to shade, and the effect is not happy. As a mere woman I have a preference for what men call &quot;the white choker,&quot; there is a style about it unattainable by any coloured tie, and it suits men so exceedingly well that they ought to stick to it. I know a few hunt uniforms, not all by any means, and of those of my acquaintance I prefer the red coats with pale green facings of the Rufford to any others, but this is a mere matter of prettiness, possibly others appeal more strongly to sporting instincts. The question of knee-breeches and silk stockings or trousers in hunt evening dress is a vexed one solved very likely by each member in accordance with the opinion he has formed of his own legs. There can be no doubt that the knee-breeches look far smarter. This is flagrant digression into the province of sportsmen, in whose appearance, however, we may perhaps be permitted a slight interest. Many hunts have a ladies' uniform, but that only for equestrian attire; the most we ever do at balls is to carry out club colours in our costume or bouquet, and that of course, only if they happen to suit our complexions. The blue and buff of the Beaufort is smart-looking, the chocolate and pale blue of the Earl of Harrington's is a very favourite uniform, and the new colours instituted by Lady Lonsdale for the Quorn — a peculiar shade of blue with a smart little touch of [intervening image] red — seems to give great satisfaction among the feminine followers of that celebrated hunt.&lt;ref&gt;Diana-Up-To-Date. &quot;Sportswoman's Page.&quot; ''Illustrated Sporting and Dramatic News'' 23 January 1897, Saturday: 18 [of 40, Col. 1a–c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001857/18970123/037/0018. Same print title, p. 814.&lt;/ref&gt;&lt;/blockquote&gt;

==Contemporary Overviews==
In the 8 April 1911 ''Sphere'', &quot;C. K. S.&quot; says,&lt;blockquote&gt;I have derived a lot of pleasure from Mr. Ralph Nevill's new book, ''London Clubs: their History and Treasures'' (Chatto and Windus). Mr. Nevill has added one more to a very interesting series of volumes, of which ''The Merry Past'' and ''Light Come, Light Go'', are two others. Many will most appreciate that part of Mr. Nevill's book for which I care least — the history of certain clubs, such as Brooks's and White's. Here I think Mr. Nevill has followed too blindly John Timbs's well-known book. I am not sure that he has read an earlier book by Marsh, and he seems not to have seen the volumes provided by the two clubs that I have named, Mr. Algernon Bourke's ''History of [[Social Victorians/London Clubs#White's|White]]''[[Social Victorians/London Clubs#White's|''&lt;nowiki/&gt;'s Club'']] and Mr. Victor Williamson's ''Memorials of [[Social Victorians/London Clubs#Brooks|Brooks's]]'', the latter a not very satisfactory compilation, whereas Mr. Bourke's ''History of White's'' is really a fine and well-written book.

If Mr. Nevill had seen ''The History of White's Club'' he could not have fixed its origin where he does, following therein Timbs and Cunningham, Mr. Wheatley and Mr. Walford, in the assumption that White's original house was five doors from the bottom of St. James's Street, this error being derived from misreading a caricature by Gilray. Mr. Bourke makes it perfectly clear that White's original Chocolate House stood where [[Social Victorians/London Clubs#Boodle's|Boodle's Club]] now is — that is to say, at 28, St. James's Street.

Again, I find Mr. Nevill mixing up The Club and the Literary Club. Herein he follows Timbs, who says that the name of The Club was altered to the Literary Club at Garrick's funeral. I think I am right in saying that there have always been two clubs. The Club, of which Johnson, Garrick, and Reynolds were members, exists at the present day under the same name. It is very exclusive, confining itself to a few eminent politicians of literary proclivities. Lord Rosebery is a member, and Mr. Balfour and Mr. Birrell.

The Literary Club also still exists although I have not heard of it for some years. Of this last club Thackeray was once the president, and I am the owner of the chair in which he sat at its meetings, a thoroughly well-authenticated piece of furniture. It was given to me by Sir William Ingram, who received it from Sir Erasmus Wilson, one of the members of the Literary Club at the time when Thackeray was president. I once spent a week-end in the company of that delightful ''raconteur'', Sir Benjamin Ward Richardson, and he told me a great deal [Col. 2c–3a] about the Literary Club, of which he was a member until his death.

These, however, are subordinate matters. The main point is that Mr. Nevill has written a delightful book, the charm of which is not so much in the historical portion as in his own account of clubs that he knows well, for he is evidently a genuine clubman and is able to picture the life of a good London club from the inside in a way that few who have written upon the subject have been able to do. The final result is excellent. We have here a most readable book and one which every clubman should have at his elbow.

Mr. Nevill might well supplement it by another volume on the dining clubs of London, a subject that he almost ignores or treats indifferently as in the case of The Club and the Literary Club. There is a good essay by Sir Spencer Walpole in his ''Essays Political and Biographical'' on the dining clubs of London, which treats of The Club, and Nobody's and the Literary Club. Among other clubs mentioned by Sir Spencer Walpole are the Dilettanti and Grillion's.

There are, however, a great many comparatively new dining clubs that would come into a book upon the subject. There is the Johnson Club, the Omar Khayyám Club, the Noviomagians — of which I have once been a guest and which is now, I believe, defunct — the Titmarsh Club, the Whitefriars Club, the Auburn Club, and what is perhaps the most democratic of them all, the Vagabonds Club. It is possible that these and others would make a very interesting book.&lt;ref&gt;&quot;A Literary Letter: The Posthumous Work of Alfred Nutt — A New History of Clubland — Depressing!&quot; ''The Sphere'' 8 April 1911, Saturday: 34 [of 40], Col. 2a–3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001861/19110408/050/0034. Same print title, p. viii.&lt;/ref&gt;&lt;/blockquote&gt;

==The Clubs==
Flourishing and Address as of 1875 plus additions with quotations and specifics.

===Albert===
1 Bolt Court, Fleet Street, E.C. (Thom 527)
===Alfred===
22A Change Alley, E.C.
===Alpine===
8 St. Martin's Place, W.C.
===Arlington===
4 Arlington Street, W. (Thom 527)
===Army &amp; Navy===
36 Pall Mall, S. W. (Thom 527)
====Junior Army and Navy Club====
12 Grafton Street, W.
===Arthur's===
69 St. James's Street, S. W.
===Arts===
17 Hanover Square, W.
===Arundel===
12 Salisbury Street, Strand
===Athenaeum===
107 Pall Mall, S. W.
====And Junior Athenaeum Club====
116 Picadilly, W.
===Baths===
The 1895 ''Yorkshire Post'' report says the club was started the prior year, so perhaps these are not the same club.

==== 1892 ====
''Truth'' reports the following:&lt;blockquote&gt;In the matter of clubs, enterprise seems to be unceasing. The latest institution of the kind which will shortly open its doors is the &quot;Baths Club.&quot; The directors of the &quot;Baths Club&quot; have purchased Lord Abergavenny's house in Dover-street, and here, when the necessary alterations have been completed, will be found a most luxurious combination of baths of all nations and of every kind. Amongst the Directors are Mr. Herbert Gladstone, Mr. W. Grenfell, Mr. Edward Tennant, and Mr. James Mansfield. The General Committee comprises amongst other names those of Lord Chesterfield, Lord Minto, Lord Dalkeith, Lord Cairns, Lord Ampthill, Mr. Alfred Lyttelton, Mr. [[Social Victorians/People/Bourke|Algernon Bourke]], Sir Frederick Milner, Admiral Maxse, and Mr. C. Williamson Milne. It is unnecessary for the moment to enlarge upon the advantages which will be offered by the &quot;Baths Club,&quot; for the scheme has met with such general and influential support that its success is scarcely a matter for conjecture.&lt;ref name=&quot;:2&quot;&gt;Marmaduke. &quot;Letters from the Linkman.&quot; ''Truth'' 22 September 1892, Thursday: 19 [of 52], Col. 2b [of 2] – 21, Col. 2a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0002961/18920922/041/0019, https://www.britishnewspaperarchive.co.uk/viewer/BL/0002961/18920922/041/0020, [https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18920922/042/0021# https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18920922/042/0021]. Print title same, pp. 613–615.&lt;/ref&gt;{{rp|615 [BNA 21], Col. 1b}}&lt;/blockquote&gt;

==== 1895 ====
&lt;blockquote&gt;The Baths’ Club, which was started last year under distinguished auspices in Dover Street, Piccadilly has, says our London Correspondent, been a great success. At the annual meeting of the members to-morrow the committee, of which Mr. W, Grenfell is president, will be able to announce that the number of candidates for membership to date, including ladies, is 1,606, and the number of members elected 1,090. The income from all sources amounted to £10,376, including £1,170 15s. received for life membership and entrance fees. The dining-room expenditure was slightly in excess of the income, but as the outgoings included boarding a staff of 70 servants the result is regarded as not altogether unsatisfactory. In order to make the club more generally attractive it is proposed to form a collection of books and works of art on sports and pastimes, and at the meeting to-morrow a Library Committee will be formed with this end in view. The General Committee, which is to be proposed for election at the same time, reflects strong position which the club holds in the fashionable world. It includes the Duke of Abercorn, the [[Social Victorians/People/Bourke|Hon. Algernon Bourke]], the Marquess of Carmarthen, M.P., Lord Alwyne Compton, M.P., Viscount Curzon, the Earl of Lonsdale, Lord Suffield, Mr. Hwfa Williams, Mr. George Wyndham, M.P., and the Hon. Arthur Henniker.&lt;ref&gt;&quot;Notes on Current Topics.&quot; ''Yorkshire Post and Leeds Intelligencer'' 25 November 1895, Monday: 4 [of 8], Col. 5c [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000687/18951125/076/0004. Print title: ''The Yorkshire Post'', p. 4.&lt;/ref&gt;&lt;/blockquote&gt;

===Beefsteak===
===Beaufort===
7 Rathbone Place, Oxford Street, W.
===Bolingbroke Reading-Room and Institute===
Not exactly a club, but it had a similar function perhaps.

From the ''South London Press'':&lt;blockquote&gt;This institute continues to prosper. Mr. Edward Wood has been elected president, and the following gentlemen have consented to act as vice-presidents: Mr. J. F. Moulton, Q.C., M.P., Mr. O. V. Morgan, M.P., [[Social Victorians/People/Bourke|Hon. Algernon Bourke]], Mr. W. Evil, Mr. J. Buckmaster, Rev. J. P. Fannthorpe, Rev. J. Sharr, Sir J. Trevor Lawrence, M.P., Dr. Mabyn Read, and Mrs. Woodhouse. Valuable gifts of books have been received from the Hon. A. Bourke, Messrs. Macmillan and Co., Rev. A. C. Woodhouse, Rev. J. Parr, Mr. A. G. Browning, Mr. J. Wix, Rev. A. Towsey, Miss Donaldson, and Messrs. Steer, Garland, Coxall, and Harrison. The library is now open for issue of books every evening from 8.30 to 9, and every Monday, Wednesday, and Saturday, from 12.30 to 1 p.m. Chess and draughts are played daily, but especially on Monday, Wednesday, and Saturday evenings.&lt;ref&gt;&quot;Bolingbroke Reading-Room and Institute.&quot; ''South London Press'' 27 February 1886, Saturday: 7 [of 16], Col. 5a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000213/18860227/054/0007. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

===Boodle's===
28 St. James's Street
===Brooks'===
60 St. James's Street
In June 1894, the London ''Echo'' reported&lt;blockquote&gt;Mr. Algernon Bourke, who recently published two interesting works — &quot;The History of White's&quot; and &quot;The Jeykell Memoirs&quot; — is now preparing another antiquarian book in “The History of Brooke's.” The two volumes will contain stories of Fox, Selwyn, Sheridan, &quot;Fighting&quot; Fitzgerald, and other famous members of the club, and a number of illustrations.&lt;ref&gt;&quot;'Echo' Gossip.&quot; ''Echo'' (London) 18 June 1894, Monday: 1 [of 4], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004596/18940618/006/0001. Same print title, n.p.&lt;/ref&gt;&lt;/blockquote&gt;

===Burlington Fine Arts===
17 Savile Row, W.
===Carlton===
94 Pall Mall
====City Carlton Club====
83 King William Street., E. C.
====Junior Carlton Club====
30 Pall Mall
===Cavendish===
307 Regent Street, W.
===City Liberal===
71 Queen Street, E.C. (address marked as &quot;temporary&quot; in 1875)
===City of London===
19 Old Broad Street, E. C.
===City United===
Ludgate Circus, E. C.
===Civil and Military===
316 Regent Street, W.
===Clarence===
1 Regent Street, Waterloo Place, S. W.
===Cobden===
5 Milman Street, Bedford Row, W. C.
===Cocoa Tree===
64 St. James's Street
===Conservative===
74 St. James's Street

==== Junior Conservative Club ====
29 King Street, St. James's (Thom 528)

===Cosmopolitan===
30 Charles Street, Berkeley Square, W.
===County===
43 Albemarle Street, W.

=== Crabbet Club ===
Members&lt;ref&gt;Douglas, Alfred Bruce (Lord Alfred Douglas). ''Oscar Wilde and Myself''. AMS Press, 1914: 64. ''Google Books'' https://books.google.com/books?id=z40fAQAAIAAJ.&lt;/ref&gt;

* Wilfred Blunt, founder
* George Curzon: Lord Curzon of Kedlestone
* George Wyndham
* George Leveson-Gower
* Lord Houghton: Lord Crewe
* Mr. Harry Cust
* Mr. Mark Napier
* Lord Cairns (&quot;the late Lord Cairns&quot;)
* Mr. &quot;Lulu&quot; Harcourt
* Oscar Wilde (attended one meeting)
* Lord Alfred Douglas (attended one meeting)
* many more

==== History of the Crabbet Club ====
Wilfred Blunt outlines the history of Crabbet Club, ending with its reconstruction with younger men in 1890:&lt;blockquote&gt;The Crabbet Club was in its origin a purely convivial gathering, unambitious of any literary aim. It began in this way: When George, Lord Pembroke (the 13th Earl) came of age in 1871, having been a very popular boy at Eton, with many school friends, and afterwards at Oxford, he thought it would be amusing to continue in some measure the life they had led by having them to stay with him once or twice every summer at Wilton, for a day or two at a time, to play cricket, and row on the river, and otherwise divert themselves, and they took the name of the “Wilton,&quot; or &quot;Wagger” Club, and it proved a great success. In 1876, though much older than the rest of the members, I was asked to join it as one who had known the Herberts from their school days. Pembroke was staying with me at Crabbet, and his two brothers and their sister Gladys (afterwards Lady Ripon), and several of their friends, and several of mine, and I drove them all to Epsom for the / Derby (Silvio's year), and we had a cricket match and a lawn tennis handicap (lawn tennis was in the process of being invented, and we played on a court 20 feet longer than what afterwards became the regulation length), and it was on this occasion that I joined the club. The party at Crabbet had proved such a success that the next year it was proposed that the club should make one of its regular meetings there, and so it gradually came about that the members came to Crabbet annually. The members of the club were never more than a few, a dozen to twenty, and consisted, besides the Herbert brothers, of Eddy Hamilton, who was afterwards Gladstone's private secretary, Lord Lewisham, Jocelyn Amherst, Granny Farquhar, Lionel Bathurst, with Harry Brand (afterwards Lord Hampden), Nigel Kingscote, Godfrey Webb, [[Social Victorians/People/Bourke|Button Bourke]], Frank Lascelles, Mark Napier, and half-a-dozen more of my own intimates, and these came regularly to Crabbet every summer, and we gradually adopted the “Crabbet Club” as the name of our branch.

Though we professed no kind of politics, and looked to amusement only, nearly all the members of it were Tories, two or three of them in Parliament, and when in 1882 I took the somewhat violent line I did about Egypt and war ensued, several of the members taking offence ceased their attendance, and the Club as far as the Crabbet meetings were concerned became less popular, and this state of things was aggravated when I stood for Parliament as a Home Ruler in 1885 and 1886, and it was all but submerged by my imprisonment at Galway. Hardly any of the old Wilton members would answer the invitations to it, and Pembroke himself, the most tolerant of men, as an Irish landlord with large interests at stake in the county of Dublin, felt it a grievance that I should have identified myself with the Land League and the Plan of Campaign. All this was natural enough, and I could not complain of the defection. The Club as the “Crabbet Club” was still continued, but reconstructed on different lines with a number of young men, Oxford undergraduates, most of them professing Home Rule opinions. The chief of these were the two Peels, Willy and George, sons of the Speaker, Arthur Pollen, Herbert Vivian, Leo Maxse, Percy Wyndham (son of Sir Hugh), Theodore Fry, Theobald Mathew, Artie Brand, and Loulou / Harcourt, the only three of the old set being Mark Napier, Eddy Hamilton, and Nigel Kingscote.&lt;ref name=&quot;:02&quot;&gt;Blunt, Wilfrid Scawen. ''My Diaries: 1888 to 1900''. M. Secker, 1900. ''Google Books'' https://books.google.com/books?id=5qBCAAAAIAAJ. Volume 1 of ''My Diaries: Being a Personal Narrative of Events, 1888–1914''.&lt;/ref&gt;{{rp|50–52}}&lt;/blockquote&gt;'''1892 July 23 and 24''', Blunt describes a meeting of the Crabbet Club. Present were George Wyndham, George Curzon, Nigel Kingscote, Charles Gatty, Theobald Mathew, Godfrey Webb, Loulou Harcourt, Charles Laprimaudaye, Harry Cust, Hubert Howard, George Leveson Gower, Dick Grosvenor, Mark Napier:&lt;blockquote&gt;George Wyndham performed a wonderful feat, writing a long poem in a most complicated metre, and full of excellent things in hardly more than an hour, between sets of lawn tennis. Cust wrote another under like conditions, so full of wit that we nearly gave him the prize. George Leveson was also good. The tennis handicap was won by Hubert Howard, the laureateship by Mathew. Hubert won the cup through Grosvenor's magnanimity, who having the last set in hand suddenly found himself lame and retired. Cust is interesting, and of great abilities. George Leveson a delightful butt, and cause of wit in others with untouchable good humour. These occasions are the salt of life.&lt;ref name=&quot;:03&quot;&gt;Blunt, Wilfrid Scawen. ''My Diaries: 1888 to 1900''. M. Secker, 1900. ''Google Books'' https://books.google.com/books?id=5qBCAAAAIAAJ. Volume 1 of ''My Diaries: Being a Personal Narrative of Events, 1888–1914''.&lt;/ref&gt;{{rp|83–84}}&lt;/blockquote&gt;'''1893 July 1–2, Saturday and Sunday''', Blunt describes &quot;Crabbet. Annual meeting of the Crabbet Club.&quot; Besides himself, the members present were: George Curzon, George Leveson Gore, George Wyndham, George Peel (the 4 Georges), Morpeth, Mark Napier, Harry Cust, Charles Gatty, Hubert Howard, Godfrey Webb, Percy Wyndham, Loulou Harcourt, Theodore Fry, Theobald Mathew, Charles Laprimaudaye and Lawrence Currie.&lt;ref name=&quot;:04&quot;&gt;Blunt, Wilfrid Scawen. ''My Diaries: 1888 to 1900''. M. Secker, 1900. ''Google Books'' https://books.google.com/books?id=5qBCAAAAIAAJ. Volume 1 of ''My Diaries: Being a Personal Narrative of Events, 1888–1914''.&lt;/ref&gt;{{rp|138}}&lt;blockquote&gt;We sat down over twenty to dinner, and did not leave the table till half-past one. ...

St. George Lane Fox, and two new men, Esmé Howard and Eddy Tennant.

George Curzon was, as usual, the most brilliant, he never flags for an instant either in speech or repartee; after him George Wyndham, Mark Napier, and Webber. The next day, Sunday, Harry Cust won the Tennis Cup, and the Laureateship was adjudged to Curzon.&lt;ref name=&quot;:04&quot; /&gt;{{rp|138}}&lt;/blockquote&gt;'''1894 June 30 and July 1''', Blunt says of &quot;Our Annual Crabbet Club Meeting&quot; that, besides himself, the members present were George Wyndham, George Curzon, George Peel, George Leveson Gower, Esmé Howard, St. George Lane Fox, Eddy Tennant, Hubert Howard, Godfrey Webb, Mark Napier, Theobald Mathew, Charles Gatty and Lawrence Currie, &quot;with three new members, Lord Cairns, Alfred Douglas, and Basil Blackwood.&quot;&lt;ref name=&quot;:05&quot;&gt;Blunt, Wilfrid Scawen. ''My Diaries: 1888 to 1900''. M. Secker, 1900. ''Google Books'' https://books.google.com/books?id=5qBCAAAAIAAJ. Volume 1 of ''My Diaries: Being a Personal Narrative of Events, 1888–1914''.&lt;/ref&gt;{{rp|178}}

=== Crichton ===
3 Adelphi Terrace, W. C.
===Devonshire===
St. James's Street
===East India United Service===
14 St. James's Square
===Egerton===
87 St. James's Street
===Farmers'===
Salisbury Hotel, Fleet Street
===Garrick===
13 Garrick Street, W. C.
====Junior Garrick Club====
1A Adelphi Terrace, Strand, W. C.

===Grafton===
10 Grafton Street, W.

===Grampian===
11 Charles Street, Cavendish Square, W.

===Gresham===
Gresham Place, King William Street, E. C.

===Guards'===
70 Pall Mall, S. W.

The Guards' Club originated the idea of offering barley water — &quot;a non-alcoholic drink [—] gratis to every member,&quot; a practice also carried out by [[Social Victorians/London Clubs#White's|White's Club]], the [[Social Victorians/London Clubs#Marlborough|Marlborough Club]] and the [[Social Victorians/London Clubs#Turf|Turf Club]].

The Liquor Licensing Laws Commission collected evidence about drinking, and the Hon. Algernon Bourke, as manager of White's testified:&lt;blockquote&gt;Mr. Algernon Bourke, manager of White's club — a club more than 200 years old — tells us something about the inner working of he club. He says: —

In our club, and in most of the West-end clubs, there is this arrangement, and I should almost like to suggest to the proprietors of working men's clubs that they should do the same as we do. We supply a non-alcoholic drink gratis to every member. At White's Club it takes the form of an excellent barley water. At the Guards Club the same; the Marlborough Club the same; the Turf Club the same. We do that, and in this sort of weather (it was July when Mr. Bourke spoke) a very large quantity of that is drunk in preference to drinking a whiskey and soda and some other things.

The credit of originating this idea, it seems, belongs to the Guards' Club. Mr. Bourke says it has undoubtedly resulted in a great decrease in the consumption of alcoholic drink, meaning, obviously the consumption in West-end clubs.&lt;ref&gt;&quot;Side Lights on Drinking.&quot; ''Reynolds's Newspaper'' 25 April 1897, Sunday: 3 [of 8], Col. 5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000101/18970425/031/0003. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

===Hanover===
28 George Street, W.

===Hogarth===
84 Charlotte Street, Fitzroy Square, W.

=== Ilchester Club ===
2, 3 Ilchester-gardens, Hyde Park, W.,&lt;ref&gt;&quot;The Ilchester Ladies' Club.&quot; ''Morning Post'' 04 June 1897 Friday: 1 [of 10], Col. 3a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970604/002/0001.&lt;/ref&gt; &quot;just off Bayswater-road&quot;&lt;ref name=&quot;:0&quot;&gt;&quot;Clubland at Play.&quot; &quot;The Ilchester Club.&quot; ''Gentlewoman'' 19 June 1897 Saturday: 40 [of 108], Col. 2c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970619/234/0040.&lt;/ref&gt;

A club for women. Opened Saturday 19 June 1897 (or perhaps the week before?):&lt;blockquote&gt;The Ilchester Club.—A club for ladies that has neither an educational nor a political fad to serve is somewhat new. The Ilchester Club for Ladies, occupying two good houses just off the Bayswater-road, has no other object than the providing of a comfortable home for ladies of good birth, on terms which should appeal to a very large number. For the sum of £82 per annum a lady may have all the advantages of a well-conducted house, and have everything provided, including food and many of those social comforts which one does not even find in one's own house. The club starts under capital auspices, and on Saturday night it was inaugurated by the play of &quot;Still Waters Run Deep,&quot; excellently played by the Hon. Mabel Vereker and Miss Norah Vandaleur; the former lady, I understand, largely interesting herself in the club. The male characters were cleverly sustained by Captain Baden-Powell, Captain C. E. Norton, Major Montresor, Mr. Davidson of Tulloch. Count de Seilern, the Marquis Montagliari, and Mr. Crookshank. Although the limits of the stage were very narrow, full credit was done to Tom Taylor's delightful comedy.&lt;ref name=&quot;:0&quot; /&gt;&lt;/blockquote&gt;

===Law Society===
103 Chancery Lane
===Lyric===
&lt;blockquote&gt;The Lyric Club, in Piccadilly East, is dead, and many will sincerely regret its decease. Specially will this be the case with American and foreign musicians, for the Lyric proviled these with an opportunity of displaying their individual talents before an audience mainly composed of musical patrons and fellow artists. The Lyric Club was recently promoted into a company, Sir Julian Goldsmid, [p. 614 [20 in BNA], Col. 2c / 615 [21], Col. 1a] Captain Brownlow, Captain Mackinnon, Mr. Sinclair Macleay, and Dr. Scott Saunders being, I understand, amongst its principal directors. Proceedings having been instituted against Dr. Scott Saunders on a charge of forgery, the Directors of the Lyric Club discovered various circumstances which rendered it advisable to terminate at once its existence, and on Thursday morning of last week the Lyric ceased definitely to exist. There is at present no intention whatever of reorganising this institution, though another club, distinguished by another name, but embodying the best advantages of the late Lyric, together with many others which it did not possess, will shortly be organised.

Meanwhile, through the enterprise of Mr. Luther Munday, the members of the late Lyric Club are being provided with accommodation and entertainment in the premises of the late Meistersingers' Club at 63, St. James's-street. As both the Lyric and the Meistersingers are dead, it is a singular and unique fact in Club history that there now exists at 63, St. James's-street a club absolutely without a name. This is, I believe, the first instance of such a case on record.

Dr. Scott Saunders is charged with having forged the signature of Lord Londesborough. Misfortunes never come singly, and Lord Londesborough, who must be a heavy loser through his recent transactions in connection with the Lyric Club, has now suffered another serious loss through the fire which destroyed Seamer Moor on Sunday evening.&lt;ref name=&quot;:2&quot; /&gt;{{rp|614, Col. 2c – 615 [BNA 20–21], Col. 1a}}&lt;/blockquote&gt;

===Marlborough===
52 Pall Mall

The [[Social Victorians/London Clubs#Guards'|Guards' Club]] originated the idea of offering barley water — &quot;a non-alcoholic drink [—] gratis to every member,&quot; a practice also carried out by [[Social Victorians/London Clubs#White's|White's Club]], the Marlborough Club and the [[Social Victorians/London Clubs#Turf|Turf Club]]. A fuller description of this subject is under the [[Social Victorians/London Clubs#Guards'|Guards' Club]].

===Medical===
9 Spring Gardens, S. W.
===Men and Women's Club===
'''1885 July 9, Thursday''', the [[Social Victorians/Timeline/1885#9 July 1885, Thursday|Men and Women's Club was rededicated]] &quot;to focus on the relations between the two sexes&quot;&lt;ref&gt;Gillham, Nicholas Wright. ''A Life of Sir Francis Galton: From African Exploration to the Birth of Eugenics''. New York: Oxford UP, 2001. P. 274.&lt;/ref&gt; in its first meeting.

'''1885 October''', in the [[Social Victorians/Timeline/1885#October 1885|second meeting]], Henrietta Muller responded to Pearson's paper of 9 July 1885 with &quot;The Other Side of the Question&quot; (Bland 12). There were meetings in [[Social Victorians/Timeline/1885#November 1885|November]] and [[Social Victorians/Timeline/1885#December 1885|December 1885]] as well.

'''1887 January''', Annie Besant spoke at the meeting of the Men and Women's Club; her paper was &quot;The State and Sexual Relations,&quot; or &quot;preventive checks,&quot; or contraception (Bland 19).

===National===
1 Whitehall Gardens
===Naval and Military===
94 Picadilly, W.
====Junior Naval and Military Club====
19 Dover Street, W.
===New Thames Yacht===
Caledonian Hotel, 1 Robert Street, Adelphi, W. C.
===New Travellers'===
16 George Street, Hanover Square, W.
===New University===
57 St. James's Street
===Oriental===
18 Hanover Square, W.
===Oxford and Cambridge===
71 Pall Mall

===Pall Mall===
7 Waterloo Place, S. W.
===Pelican===
From July 1891:&lt;blockquote&gt;That ''fin de Siècle'' institution the Pelican Club is well known, thanks to the numerous advertisements it has received in the law courts and newspapers. It is a mistake however, to suppose that the Pelican is a club devoted to encouragement of pugilism, or that pugilists are ''habitués'' of its comfortable home, except as active participators in boxing matches in the ring, or while waiting for their turn penned behind railings. As a matter of fact it is by the merest accident that the Pelicans have become such prominent patrons of boxing.

The Club was originally started in March, 1887, by the proprietor Mr. Arthur Ernest Wells in a small house in Gerrard Street, which had been used as a coach-builder's shop. Mr. Wells' initial idea was to form a resort for Bohemians, to which no strangers should be admitted, much after the fashion of the popular [[Social Victorians/London Clubs#Beefsteak|Beefsteak Club]] that has flourished over Toole's Theatre for some years. It was to be a one room club with sanded floor, luncheons and suppers from the grill, and the only other attractions were a grand piano, a billiard table and a bar.

From the first it was extremely popular, &quot;all sorts and conditions of men &quot;joining, peers and pressmen, actors and stockbrokers, artists and guardsmen. There seemed an extra charm in the Welsh rarebit and baked potato if eaten while Mr. Edward Solomon was trying over a new song on the piano, or Mr. Cammeyer was sweeping the strings of his zither-banjo. Boxing was about the last thing thought of, and it was only because Jem Smith, then champion of England, was much ''en'' ''évidence'' as one of the combatants in a proposed prize-fight, that the idea suggested itself to the proprietor to give a boxing entertainment for the delectation of the Pelicans.

Although the general room was very ill adapted for such a purpose, the floor being carpeted, and the ring erected between posts that supported the roof, the boxing &quot;took on&quot; at once, and the entertainment was repeated weekly. A great number of the members being journalists it was no wonder that the success of the new departure was considerably paragraphed in the newspapers, and did not escape the attention of the police, who, also urged on by several anonymous letters, made a raid on the Club premises on Sunday, April 25th, 1587.

As the police came tumbling into the room just as the penultimate round of a boxing competition was going on, they looked astonished and particularly foolish at the sight of rows upon rows of members all in evening dress sitting in stolid silence watching the sparring. A member of the committee, an officer of the Guards, who happened to be present, demanded of the two inspectors who headed the raiders, what they wanted, but they were unable to give a satisfactory answer, and eventually the constables retired amid the derision of the members. It afterwards transpired that the police had been informed that the game of baccarat was being played at the club, and the authorities at Scotland Yard made ample amends for their unwarranted intrusion.

From this time forward the Pelican Club became the acknowledged home of first-class boxing matches, and members flocked in so fast, despite a raised subscription and an entrance fee, that new quarters had to be found. A freehold site was purchased in Gerrard Street, Soho, facing Shaftesbury Avenue, and here, early in 1889, the foundation stone of the present magnificent premises was laid by Sir John D. Astley, Bart., the president ot the club.

Although the new club house was not opened for nearly a year, the much talked of glove fight between Jem Smith and Jackson, the black, came of in the new premises before they were finished. Although the stakes were a thousand pounds, the largest that had ever been boxed for up to then, the affair lasted only a couple of rounds, occupying something like six minutes, when Smith was disqualified for a foul. It cannot be disguised that the result of the match did not improve the standing of boxing as a national sport and although there have been many excellent glove contests since, boxing for large sums of money undoubtedly received its death warrant over this fiasco. Boxing on Sundays also has ceased at the Pelican out of consideration to the general public feeling on the subject, and only at intervals do the members muster on Sabbath evenings for the excellent smoking concerts which are impossible on week days, in consequence of the professional engagements of the actors and singers who take part in them.

The Club is now firmly established, and as it numbers nearly twelve hundred members need fear no rival. There is a thoroughly representative committee, and a noticeable rule of the club is, that &quot;No game of cards shall be allowed in the club.&quot;

The Pelicans are undoubtedly seen at their best on a Saturday night when their usual weekly boxing entertainment takes place. The ring is pitched in the spacious gymnasium downstairs, and although none of the seats are reserved, it is curious how custom has allotted certain portions of the arena to the different sets amongst the members. For instance, on the west side of the eighteen-foot ring, and around the referee's box — which usually contains either Mr. B. J. Angle, Mr. George Vize or Mr. Reggie Wakefield, all veteran amateur boxers — and Mr. F. C. Hilder, are clustered the members who hail from the [[Social Victorians/London Clubs#Turf|Turf]], [[Social Victorians/London Clubs#White's|White's]] and Guard's [&lt;nowiki/&gt;[[Social Victorians/London Clubs#Guards'|Guards']]] Clubs. Here on any night during the boxing season may be seen such well-known men as Lord Queensberry, Harry Hungerford, Sir Charles Hartopp, Lord Lonsdale, Lord Craven, John Delacour, Sir John Astley, Colonel Fox, the [[Social Victorians/People/Bourke|Hon. Algernon Bourke]], Lord Royston, the Hon. Michael Sandys, Lords Esmé and Granville Gordon, Baron Max de Tuyll, Major &quot;Sugar&quot; Candy, Bagot-Chester, Lord Rossmore, Colonel Fraser, Lord Cholmondeley, the two Coventrys, Arthur and Aubrey, and Lord Marcus Beresford. Actors and journalists cluster about the stairs on the north side, among the most regular attendants being Fred Terry, F. Ker, Arthur Bourchier, Arthur Roberts, Fred Leslie, J. L. Shine, Sidney Howard, Mons. Marius; George Edwardes, Augustus and Charles Harris and Charles Hawtrey representing the managers. The journalists are more modest in numbers, Harrington O'Reilly, H. P. Stephens, James Davis, John Corlett, Lionel Monckton, A. M. Binstead, Byron Webber, C. Blake, and Tom Burnside being among the frequenters. The City and coaching members show a decided preference for the east side of the room, among the most notable being Colonel North and his ''aide de camp'', Captain Carvick, H. L. Beckett, &quot;Swish&quot; Broadwood, &quot;Dickey the Driver,&quot; Skipper Holmes, Percy Bouch, Willie Mackenzie, Carleton Blyth, F. Meredith, R. Cosier, J. K. Edward, Walter Pallant, J. Hawkins, R. S. Evans, and Philip Ellissen. The south side of the ring is much affected by distinguished visitors and Americans who are members, among the latter being Walter Coggeshall, W. King, Major Noah, Colonel Shaffer, and J. L. Onativia. General sport is well represented by J. L. Davis, Sir Matthew Wood, Captain Bailey, Charles Thompson, Captain Bewick, Frank Cobbett, Loftus Thornhill, Bernard Posno, R. Stuckey Wood, and Sir Simon Lockhart.&lt;ref&gt;&quot;The Pelican Club.&quot; ''Black &amp; White'' 18 July 1891, Saturday: 13 [of 46], Col. 2a–c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004617/18910718/045/0013. Print title ''Black and White'', p. 99.&lt;/ref&gt;&lt;/blockquote&gt;

===Pheonix===
275 Strand, W. C.
===Portland===
===Pratt's===
14 Park Place, St. James's
===Prince's Cricket and Prince's Racket and Tennis===
22 Hans Place, Sloane Street, S. W., same address as the Prince's Racket and Tennis Club

=== Prince's Skating Club ===

==== 1896, November 6, Friday ====
1896 may have been the [[Social Victorians/Timeline/1896#Opening of the Prince's Club Skating Rink|first opening of the Prince's Skating Rink]], although the 1902 reports say it is the 7th season. The newspapers provide details about the building and the rink itself.

==== 1902, October ====
The newspapers reported on [[Social Victorians/Timeline/1900s#Annual Opening of the Prince's Ice-skating Rink|2 Fridays, October 24th and 31st]].

===Prince of Wales's Yacht===
Freemason's Tavern, 61 Great Queen Street, W. C.
===Queen's===
&quot;Founded as The Queen's Club Limited on 19 August 1886 by [[Social Victorians/People/Charteris|Evan Charteris]], George Francis and [[Social Victorians/People/Grosvenor|Algernon Grosvener]], the Queen's Club was the world's second multipurpose sports complex, after the Prince's Club .... The first lawn tennis courts were opened on 19 May 1887, and the first sporting event was held on 1 and 2 July 1887 when Oxford played Cambridge. The club buildings were opened in January 1888, having taken about 18 months to construct.&quot;&lt;ref&gt;{{Cite journal|date=2024-07-28|title=Queen's Club|url=https://en.wikipedia.org/wiki/Queen's_Club|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Queen%27s_Club.&lt;/ref&gt;

In order to establish residence for his 26 July 1888 marriage to [[Social Victorians/People/Montrose|Caroline, Duchess of Montrose]], Marcus Henry Milner &quot;earned the necessary residential qualification by staying at the Queen's Club.&quot;&lt;ref&gt;&quot;Metropolitan Notes.&quot; ''Nottingham Evening Post'' 31 July 1888, Tuesday: 4 [of 4], Col. 1b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000321/18880731/025/0004.&lt;/ref&gt;

===Raleigh===
14 Regent Street, S. W.
===Ramblers'===
35 Ludgate Hill
===Reform===
104 Pall Mall
===Royal London Yacht===
===Royal Artillery and Royal Engineers; vide Medical===
===Royal Thames Yacht===
7 Albemarle Street
===St. George's Chess===
20 King Street, St. James's
===St. James's===
106 Picadilly
====Junior St. James's Club====
54 St. James's Street
===St. Stephen's===
1 Bridge Street, Westminster, S. W.
===[[Social Victorians/London Clubs/Savage Club|Savage]]===

===Savile===
15 Savile Row, W.
===Smithfield===
47 Half Moon Street, W.
===Stafford===
2 Savile Row, W.
===Temple===
37 Arundel Street, Strand
===Thatched House===
86 St. James's Street
===Travellers'===
106 Pall Mall
===Turf===
4 Grafton Street, W.

The [[Social Victorians/London Clubs#Guards'|Guards' Club]] originated the idea of offering barley water — &quot;a non-alcoholic drink [—] gratis to every member,&quot; a practice also carried out by [[Social Victorians/London Clubs#White's|White's Club]], the [[Social Victorians/London Clubs#Marlborough|Marlborough Club]] and the Turf Club. A fuller description of this subject is under the [[Social Victorians/London Clubs#Guards'|Guards' Club]].

===Union===
Trafalgar Square, W.
===United Clergy and Laity===
24 Charles Street, St. James's
===United Service===
116 Pall Mall
====Junior United Service Club====
11 Charles Street, St. James's
===United University===
1 Suffolk Street, Pall Mall East, S. W.
===Universities===
71 Jermyn Street, St. James's, S. W.
===Verulam===
54 St. James's Street
===Victoria===
18 Wellington Street, Strand, W. C.
===Wanderers'===
4 Park Place, St. James's
===Westminster===
23 Albemarle Street
===Whitehall===
47 Parliament Street, S. W.
===White's===
38 St. James's Street

Established in 1693, the oldest of London's gentleman's clubs, White's still excludes women.&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|date=2024-10-09|title=White's|url=https://en.wikipedia.org/wiki/White's|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/White%27s.&lt;/ref&gt; It was named originally for a business, Mrs. White's Chocolate House.&lt;ref name=&quot;:1&quot; /&gt;

A controversy arose in the summer of 1896 because of White's [[Social Victorians/London Clubs#Summer Club|Summer Club]] in Kensington Park.

The [[Social Victorians/London Clubs#Guards'|Guards' Club]] originated the idea of offering barley water — &quot;a non-alcoholic drink [—] gratis to every member,&quot; a practice also carried out by White's Club, the [[Social Victorians/London Clubs#Marlborough|Marlborough Club]] and the [[Social Victorians/London Clubs#Turf|Turf Club]]. A fuller description of this subject is under the [[Social Victorians/London Clubs#Guards'|Guards' Club]].

The 3 October 1896 ''Newcastle Chronicle'' reported on Algernon Bourke's history of White's Club. (A discussion of some details from this book are at [[Social Victorians/London Clubs#Contemporary Overviews|Contemporary Overviews]], above.)&lt;blockquote&gt;The Hon. Algernon Bourke's interesting &quot;History of White's&quot; gives a vivid and faithful picture of fashionable society during the eighteenth century, and the earlier half of the nineteenth.

White's was originally founded in 1608 as White's Chocolate House. In those days there were no clubs, such as we understand them in our time. People went to the chocolate or coffee houses; and many of those reeorts acquired peculiar characteristics of their own. Will's Coffee House at the end of Bow Street was, for example, at the beginning of the eighteenth century frequented by Dryden, and was noted as the rallying ground of the wits of the town, who won their spurs in satire and epigram under the patronage of the great poet. Child's Coffee House, in St. Paul's Churchyard, became the beat up of physicians and apothecaries with an admixture of the clergy. The Grecian Coffee House, in a court leading out of the Strand, was used principally by lawyers; while such resorts as the Cocoa Tree and St. James's were frequented by Tory and Whig politicians respectively. White's Chocolate House was during the earlier portion of its career of no particular political complexion. Dice and cards rather than literary or political disputations were its attractions. At first, anybody could walk in who could afford the penny charged for admission, and who had money enough in his pocket to order the &quot;drawer&quot; to bring in a dish of chocolate or a bottle of port. This, however, led to disagreeables. A mighty duke or belted earl often found himself at the same table with a swaggering Captain Culpepper from Alsatia, with a highwayman who had achieved a successful exploit on the road, or with a city apprentice who had emptied his master's till and set up as a fine gentleman. By degrees, men of titles and position who knew each other, reserved for their own private use an inner room of the Chocolate House, from which persons who did not possess the &quot;open sesame&quot; to it were excluded, and in this way White's became a club! The proprietor of the establishment soon discovered that it would pay him best to limit his civilities to aristocratic customers, and he one fine day announced that his Chocolate House would be closed to all who did not come to it with the necessary credentials.

White's at an early date won notoriety as a club in which gaming for high stakes was customary. In Queen Anne's reign it had already established its reputation in this respect; for there is a passage in Swift in which he says that the Earl of Oxford &quot;never passed White's Chocolate House (the common rendezvous of infamous sharpers and noble bullies) without bestowing a curse on that famous academy as the bane of half the English aristocracy.&quot; 

It nay be doubted, neverthe!ess, whether White's in its worst days ever reached the scandalous celebrity of Crookford's. In the latter club, cheating was notorious. Prince Louis Napoleon was sharped there out of £2,000 in one night; but through his influential friends fortunately recovered the money. Yet White's was bad enough in all conscience. It was no uncommon thing for a nobleman to enter its doors a rich man on one night, and to leave it a beggar in the morning. Mr. Bourke relates of one particular gentleman that he gambled away £90,000 in a night. A fashion crept in, moreover, of wagering on all sorts of conceivable contingencies. Horace Walpole's anecdote in illustration of this is so well known that it will be perhaps stale for money. For the benefit of the few it may, however, be given here. Writing in 1750, he remarks: — &quot;A man dropped down dead at the door of White's, and was carried in. The club immediately made bets whether he was dead or not, and when they were going to bleed him, the wagerers for his death interposed, and said that it would affect the fairness of the bet.&quot; Wagers were made on births, deaths, and marriages; on the probable forces to be engaged in impending battles, on the results of sieges, on the conditions of treaties, on Parliamentary elections, on the fluctations [sic] of stocks, and on the course of quotations in the corn markets. Early this century the career of the Emperor Bonaparte was a fruitful theme of bets; and those who speculated on his downfall made a good thing of it. Sporting bets seem to have been few, however. The horse racing &quot;tipster&quot; had not come into existence. 

Wits and gourmets frequented White's for other purposes than gambling. The choicest and costliest repasts were given there. The banquet which White's gave to the allied sovereigns in 1814 cost £10,000, the entertainment of each guest amounting to £4 per head. Lord Alvanley invited a few friends to a little dinner, one of the dishes of which consisted of the &quot;noix&quot; or &quot;oysters&quot; taken from three hundred birds. It was, in short, the lavish folly of the Romans over again. Among the wits figured prominently George Selwyn and Horace Walpole. Grave statesmen frequented White's in their turn. The names of Sir Robert Walpole, Pulteney, Fox, Pitt, Wellington and Castlereagh, Canning and Peel are associated with the club. The club was nevertheless not of a strictly political character until 1783, when on the election of William Pitt it became Tory. 

Strange tales are also told of the servants of White's. One of its head waiters, Robert Mackreth, who rose to be club manager and proprietor, induced the Earl of Oxford to return him as member for the nomination borough of Castle Rising, in Norfolk. It is believed that the Earl of Oxford did so because he was too deeply in debt to Mackreth to thwart him in his political ambition. Another servant named Raggett, who waited upon the players in the gaming rooms, realised a handsome fortune by simply picking up from the carpets the counters which the gamblers in their cups — men did not sit down to cards sober in those days as they do now — left sprawling. As cash was never staked at the tables, these counters represented of course money, and could be exchanged for it. A third servant — a knife-boy named Rumbold — moved interest with a member of the East India Company, secured an appointment, went out to India, and shook the Pagoda tree with such success as to realise a magnificent fortune. This knife-boy attained, in fact, to the dignity of Governor of Madras.&lt;ref&gt;&quot;Whites.&quot; ''Newcastle Chronicle'' 3 October 1896, Saturday: 9 [of 16], Col. 6a–c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000865/18961003/105/0009. Print: ''Newcastle Weekly Chronicle, Supplement'', n.p.&lt;/ref&gt; &lt;/blockquote&gt;

==== 1888 ====
23 July 1888, Monday, White's Club got the lease for the house that Bourke managed:&lt;blockquote&gt;At the adjourned meeting of the members of White’s Club yesterday, the Duke of Buccleuch presiding, it was agreed give Mrs. Percival £1,200 for her surrendered lease, for which there were two applicants, namely, the Hon. Algernon Bourke and Captain Perceval. The meeting resolved to accept the application of the former, and the committee were empowered to deal with his propositions. The Earl of Craven was elected member of the club.&lt;ref&gt;&quot;Court and Personal News.&quot; ''Globe'' 24 July 1888, Tuesday: 7 [of 8], Col. 3a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001652/18880724/053/0007. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;17 October 1888, Wednesday, the ''Sheffield Daily Telegraph'' reported about uncertainty about White's:&lt;blockquote&gt;There seems to be still some uncertainty as to the future of White's club. This, the oldest of London club-houses, rendered immortal by Thackeray and others, has suffered in the struggle for existence with the newer and more attractive institutions which have sprung up all round it within the last few years. The chief capitalist in the club is Lord Cheylesmore, who would be most re’luctant to see it come to grief. Happily, [[Social Victorians/People/Bourke|Mr. Algernon Bourke]], an enterprising gentleman, who is a stockbroker, a correspondent of the ''Times'', and a very popular man about town, came to the rescue, and is now reorganising the institution. White’s, in future, is to be of the recreative class of clubs, but whetber members will be provided with dancing as at the New Club in Covent Garden, or music, as they are promised at the Lyric and the Meistersingers, or whether it be true that it is to have an athletic department, no one seems to know. However, for the present, the house is closed, and workmen are employed in decorating the interior.&lt;ref&gt;&quot;London Letters.&quot; ''Sheffield Daily Telegraph'' 17 October 1888, Wednesday: 4 [of 8], Col. 5c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000250/18881017/124/0004. Same print title, p. 4.&lt;/ref&gt;&lt;/blockquote&gt;27 October 1888, Saturday, a London Correspondent in the ''Brighouse'' ''and Rastrick Gazette'' reported about White's Club,&lt;blockquote&gt;There is some danger, I hear, of White's Club being involved in a lawsuit. Mr. Algernon Bourke is engaged in making extensive alterations and additions to the house, and one of the neighbours objects to the addition of a room on the ground that it interferes with his &quot;ancient lights.&quot; I believe, however, that Mr. Bourke knows what he is about, and that he is pretty certain to win the day if the matter is brought into court. There is a report, but I cannot vouch for its accuracy, that Prince Edie will become a member.&lt;ref&gt;London Correspondent. &quot;Passing Events.&quot; ''Brighouse &amp; Rastrick Gazette'' 27 October 1888, Saturday: 2 [of 8], Col. 2 [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003061/18881027/013/0002. Print title ''Brighouse and Rastrick Gazette'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;

==== 1893 ====

The ''Newcastle Chronicle'' described White's in November 1893 in discussing [[Social Victorians/People/Bourke|Algernon Bourke]]'s book on White's' history:&lt;blockquote&gt;It is true the fires are lighted at the clubs, the winter carpets have been laid down, portieres are drawn over draughty doorways, and books in tempting bindings are more and more en [sic] evidence. One of the most lavishly illustrated books I have seen for a long time is the history of &quot;White's.&quot; It must have cost a little fortune to produce. It is in two volumes, exquisitely printed, and the matter is most readable. But this is evidently quite a private enterprise. The publisher is the [[Social Victorians/People/Bourke|Hon. Algernon Bourke]], and the work is issued from his private house in St. James's Street. There has also appeared, or is about to appear, the true and particular history of &quot;Brooks's.&quot; Clubland will soon have no secrets left, except, of course, those current ones that may not be told, not only in deference to good taste, but out of a wholesome fear of the law of libel. There are several stories of &quot;Crockford's&quot; in the volume about &quot;White's.&quot; One of them deals with the Lord Sefton of the time, who was a great epicure. He prided himself on the invention of a plat made of the soft roe of the mackerel. He was one of the principal victims at Crockford's, where at one time and another he lost £200,000 at play. His successor honoured an acceptance of his for £40,000 held by Crockford and presented after his death. The property in Manchester and Liverpool that was sold to meet his losses would today be valued at millions sterling.

Notable Wagers.

Long before these new and private guides to &quot;White's&quot; and &quot;Brooks's,&quot; there appeared in the &quot;Art Journal&quot; some sketches of Clubland, that let a good deal of permissible daylight into the more mysterious corners of the social quarters of the country. If the author only professed to loiter upon the frontiers, he nevertheless made several interesting excursions into the very heart of the territory. In regard to the laws and regulations of one of the great West End houses, he fell into a trifling mistake which one of the long-eyed birds of criticism picked out with his sharp beak and exhibited to the world. It was the merest shadow of a technical error and it proved the truth of the rest: it was the very smallest &quot;exception to prove the rule&quot; you can imagine. A distinguished correspondent, a member of &quot;White's,&quot; who recalls the book and is good enough to say he is &quot;delighted with the flavour of last week's Cigarette Papers,&quot; sends me several fresh notes from the bet book at the famous club. Two of them are well worth repeating. On November 4, 1754, Lord Mountfort wagered Sir John Bland 100 guineas that Mr. Nash would outlive Mr. Cibber. The two men in queetion were Colley Cibber and Bean Nash, the &quot;King of Bath,&quot; then very old men. Below the entry in the bet book to this day stands the record:— &quot;Both Lord Mountford and Sir John Bland put an end to their lives before the bet was decided.&quot; Among the curious bets of a comparatively recent date was that of Lord Alvanley, who wagered Mr. Goddard &quot;that Mr. G. Talbot does not die a natural death.&quot; Talbot retaliated by betting &quot;that Lord Alvanley is not worth three shillings this day two years.&quot;&lt;ref&gt;&quot;A Cheery November.&quot; ''Newcastle Chronicle'' 04 November 1893, Saturday: 5 [of 16], Col. 1a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000865/18931104/075/0005. Print title: ''Newcastle Weekly Chronicle'', p. 5.&lt;/ref&gt;&lt;/blockquote&gt;

==== Summer Club ====
On 29 June 1896, Monday, [[Social Victorians/People/Bourke|Algernon Bourke]] wrote the ''Daily Telegraph'' about the Summer Club:&lt;blockquote&gt;To the Editor of “The Daily Telegraph.”

Sir — Permit me to make my bow to the public. I am the manager of the Summer Club, which on two occasions has been the subject of Ministerial interpellation in Parliament. The Summer Club is a small combination, which conceived the idea of attempting to make life more pleasant in London by organising breakfast, luncheon, and teas in Kensington Gardens for its members. This appears to have given offence in some way to Dr. Tanner, with the result that the catering arrangements of the club are now &quot;by order&quot; thrown open to the public. No one is more pleased than I am at the result of the doctor's intervention, for it shows that the idea the Summer Club had of using the parks for something more than mere right of way bas been favourably received. In order, however, that the great British public may not be disappointed, should they all come to lunch at once, I think it necessary to explain that the kitchen, which by courtesy of the lessee of the kiosk our cook was permitted to use, is only 10ft by 5ft; it has also to serve as a scullery and pantry, and the larder, from which our luxurious viands are drawn, is a four-wheeled cab, which comes up every day with the food and returns after lunch with the scraps. Nevertheless, the Summer Club says to the British public — What we have we will share with you, though it don't amount to very much — I am, Sir, your obedient servant,

ALGERNON BOURKE.

White's Club, June 27&lt;ref&gt;&quot;The Summer Club.&quot; ''Daily Telegraph &amp; Courier'' (London) 29 June 1896, Monday: 8 [of 12], Col. 2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/18960629/072/0008. Print title: ''Daily Telegraph'', p. 8.&lt;/ref&gt;&lt;/blockquote&gt;On 1896 June 20, Saturday, Lady Violet Greville wrote against the Summer Club in her column &quot;Place aux Dames&quot; in ''The Graphic'':&lt;blockquote&gt;L&lt;small&gt;ONDON&lt;/small&gt; in this hot weather takes upon itself a continental aspect. The park is full of people sitting on chairs in fragile summer garments, carriages with brilliant occupants flit to and fro, and all the out-door resorts are crowded. Battersea Park has lost its short-lived popularity, but Kensington Gardens, hitherto given over to nursemaids, has assumed a transient splendour conferred upon it by Mr. Algernon Bourke's tea kiosk, which is much patronised in the afternoon. But on what principle a-specially reserved enclosure in a public park is to be given over to White's club one fails to understand. Perhaps the paradise of straw armchairs and Japanese umbrellas surrounded by a halo of flowering rhododendrons in pots may rouse unnecessary envy in the breasts of the Peris cast out of such a paradise, otherwise it seems as if armchairs and umbrellas might also be supplied to the general public without hurting the too keen susceptibilities of the languid denizens of White's club. Any way the precedent is a bad one, for if White's, why not the Junior Carlton or Brooks's, or the Tooting and Clapham bicycle clubs, and in that case if all clubs may have reserved enclosures, what becomes of the public park?&lt;ref&gt;Greville, Lady Violet. &quot;Place aux Dames.&quot; ''Graphic'' 20 June 1896, Saturday: 24 [of 36], Col. 1a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000057/18960620/032/0024. Same print title, p. 746.&lt;/ref&gt;&lt;/blockquote&gt;On 4 July 1896, Saturday, &quot;Marmaduke&quot; in the ''Graphic'' took Bourke's side on the Summer Club in Kensington Park:&lt;blockquote&gt;Most of us have noticed that if we read in the newspapers the account of some matter which we are personally acquainted with the account will generally contain several errors. I have also noticed that when a question is asked in the House of Commons regarding some matter about which I know all the facts the question and the official answer to it frequently contain serious errors. Last week Mr. Akers-Douglas was asked in the House to explain how it was that Mr. Algernon Bourke obtained permission to open the &quot;Summer Club&quot; in Kensington Gardens, and he was questioned upon other particulars connected with the same matter. Both the questions and the official reply showed considerable ignorance of the facts. There has been from time immemorial a refreshment kiosk in Kensington Gardens. Mr. Bourke obtained from the tenant of this permission to use the kitchen for the benefit of the &quot;Summer Club,&quot; and to supply the members of the latter with refreshments. It was a purely commercial transaction. Mr. Bourke then established some wicker seats, a few tables, a tent, and a small hut upon a lawn in the neighbourhood of the kiosk. To do this he must have obtained the permission of Mr. Akers-Douglas, as obviously he would otherwise have been immediately ordered to remove them. Mr. Akers-Douglas equally obviously would not have given his sanction unless he had been previously informed of the objects which Mr. Bourke had in view — to wit, that the latter intended to establish a club there. That being the case, it is difficult to understand for what reason Mr. Akers-Douglas has now decided that any member of the public can use the chairs, tables, and tent belonging to the &quot;Summer Club,&quot; can insist upon the club servants attending upon him, and can compel them to supply him with refreshments. Mr. Akers-Douglas should have thought of the consequences before he granted the permission.&lt;ref&gt;&quot;Marmaduke.&quot; &quot;Court and Club.&quot; The ''Graphic'' 04 July 1896, Saturday: 14 [of 32], Col. 1b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/9000057/18960704/029/0014. Print p. 14.&lt;/ref&gt;&lt;/blockquote&gt;

==== 1899 ====
In 1899 White's was no longer under Algernon Bourke's management and was reconstituting itself after the possibility that it would have to close:&lt;blockquote&gt;White's Club has narrowly escaped closing its doors. Like many of the other leading West End clubs, it has suffered grievously of late years owing to the growth of the fashion of dining at hotel restaurants. The principal proprietor was Mr Algernon Bourke, Lord Mayo's uncle, and his interest has now been made the subject of pecuniary adjustment. It is now being reconstituted as a member's club, and a new manager is in charge, aided by an energetic and discriminating committee. The fact that the late committee was inclined to be rather too discriminate was one of the causes of the troubled waters into which the club drifted, for it is possible to be too exclusive, even for White's, especially in view of the competition that exists nowadays in clubdom — as everywhere else. Many members had resigned, but most of them are now anxious to return, and the membership list is in future to be extended to 755.&lt;ref&gt;&quot;London Correspondence.&quot; ''Freeman's Journal'' 28 July 1899, Friday: 4 [of 8], Col. 8c [of 10]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000056/18990728/031/0004. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;In September 1899 the Oxford Chronicle and Berks and Bucks Gazette reported,&lt;blockquote&gt;The new secretary of White's, Mr. Mark Weyland, is the only surviving son of Mr. Weyland, of Wood Eaton a house pleasantly situated on the Cherwell, close to Oxford. His elder brother died of a fever contracted while soldiering in Egypt during the Tel-el-Kehir campaign.&lt;p&gt;

Mr. Mark Wayland is a nephew of Lord Clanricarde and Lady Cork, and a great grandson of the great George Canning. The members of White's now made all arrangements for buying out [[Social Victorians/People/Bourke|Mr. Algernon Bourke]], and the club will henceforth be a members' club and not proprietary.&lt;ref&gt;&quot;Personal and Social.&quot; ''Oxford Chronicle and Reading Gazette'' 2 September 1899, Saturday: 9 [of 12], Col. 1a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000989/18990902/098/0009.

Print title: ''Oxford Chronicle and Berks and Bucks Gazette'', p. 9.&lt;/ref&gt;&lt;/blockquote&gt;Bourke underwent bankruptcy proceedings in 1899, after attempting to close Willis' Restaurant, Ltd., which left him liable for perhaps £9,000, at least £5,000 was owed to White's.&lt;ref&gt;&quot;Mr. Bourke Must Pay.&quot; ''Irish Independent'' 8 June 1899, Thursday: 4 [of 8], Col. 8c [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001985/18990608/118/0004. Print title: ''The Irish Weekly Independent'', p. 4.&lt;/ref&gt;

===Willis's===
Ralph Nevill says in 1912,&lt;blockquote&gt;It is a matter of regret that the peaceful and quiet Willis's in King Street has ceased to exist. Opened in 1893, it at first achieved considerable success. The moving spirit in its management was [[Social Victorians/People/Bourke|Mr. Algernon Bourke]], who, in his time, has done many things, some of them very well. It was intended, I believe, to resemble the defunct Amphitryon Club in the matter of cuisine and wines; and, like the latter resort, its frequenters mostly belonged to the fashionable world. Besides the restaurant proper, there were several private rooms, and latterly an unsuccessful experiment was made with a supper club.&lt;ref&gt;Nevill, Ralph [Henry]. ''The Man of Pleasure''. Chatto &amp; Windus, 1912. Google Books (retrieved July 2025) https://www.google.com/books/edition/THE_MAN_OF_PLEASURE/gGZd_V6bv1YC.&lt;/ref&gt;&lt;/blockquote&gt;

===Windham===
11 St. James's Square

==Bibliography==
*Milne-Smith, Amy. London Clubland: A Cultural History of Gender and Class in late-Victorian England. New York: Palgrave Macmillan, 2011. Google Books: https://books.google.com/books?id=TQrHAAAAQBAJ.
*Thom, Adam Bisset, compiler. The Upper Ten Thousand: An Alphabetical List of All Members of Noble Families, Bishops, Privy Councillors, Judges, Baronets, Members of the House of Commons, Lords-Lieutenant, Governors of Colonies, Knights and Companions of Orders, Deans and Archdeacons, and the Superior Officers of the Army and Navy, with Their Official Descriptions and Addresses. London: George Routledge and Sons, 1875. Google Books.

==Footnotes==
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==Most general Lorentz transformations==
===General quadratic form===
The general [[w:quadratic form]] ''q(x)'' with coefficients of a [[w:symmetric matrix]] '''A''', the associated [[w:bilinear form]] ''b(x,y)'', and the [[w:linear transformation]]s of ''q(x)'' and ''b(x,y)'' into ''q(x′)'' and ''b(x′,y′)'' using the [[w:transformation matrix]] '''g''', can be written as&lt;ref&gt;Bôcher (1907), chapter X&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}\begin{align}q=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}\end{align}
=q'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{x}'\\
b=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{y}=b'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{y}'\\
\left(\mathbf{A}=\mathbf{A}^{{\rm T}}\right)\\
\hline \left.\begin{align}\mathbf{x}' &amp; =\mathbf{g}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\mathbf{g}^{-1}\cdot\mathbf{x}'
\end{align}
\quad\right|\quad\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g}=\mathbf{A}'
\end{matrix}&lt;/math&gt;|{{equationRef|Q1}}}}

The case ''n=1'' is the [[w:binary quadratic form]] introduced by [[#Lagrange|Lagrange (1773)]] and [[#Gauss|Gauss (1798/1801)]], ''n=2'' is the ternary quadratic form introduced by [[#Gauss2|Gauss (1798/1801)]], ''n=3'' is the quaternary quadratic form etc.

===Most general Lorentz transformation===
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The general Lorentz transformation follows from ({{equationNote|Q1}}) by setting '''A'''='''A′'''=diag(-1,1,...,1) and det '''g'''=±1. It forms an [[w:indefinite orthogonal group]] called the [[w:Lorentz group]] O(1,n), while the case det '''g'''=+1 forms the restricted [[w:Lorentz group]] SO(1,n). The quadratic form ''q(x)'' becomes the [[w:Lorentz interval]] in terms of an [[w:indefinite quadratic form]] of [[w:Minkowski space]] (being a special case of [[w:pseudo-Euclidean space]]), and the associated bilinear form ''b(x)'' becomes the [[w:Minkowski inner product]]:&lt;ref name=ratcliffe&gt;Ratcliffe (1994), 3.1 and Theorem 3.1.4 and Exercise 3.1&lt;/ref&gt;&lt;ref&gt;Naimark (1964), 2 in four dimensions&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\begin{align}-x_{0}^{2}+\cdots+x_{n}^{2} &amp; =-x_{0}^{\prime2}+\dots+x_{n}^{\prime2}\\
-x_{0}y_{0}+\cdots+x_{n}y_{n} &amp; =-x_{0}^{\prime}y_{0}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime}
\end{align}
\\
\hline \left.\begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\
\downarrow\\
\begin{align}x_{0}^{\prime} &amp; =x_{0}g_{00}+x_{1}g_{01}+\dots+x_{n}g_{0n}\\
x_{1}^{\prime} &amp; =x_{0}g_{10}+x_{1}g_{11}+\dots+x_{n}g_{1n}\\
 &amp; \dots\\
x_{n}^{\prime} &amp; =x_{0}g_{n0}+x_{1}g_{n1}+\dots+x_{n}g_{nn}
\end{align}
\\
\\
\mathbf{x}=\mathbf{g}^{-1}\cdot\mathbf{x}'\\
\downarrow\\
\begin{align}x_{0} &amp; =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}-\dots-x_{n}^{\prime}g_{n0}\\
x_{1} &amp; =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}+\dots+x_{n}^{\prime}g_{n1}\\
 &amp; \dots\\
x_{n} &amp; =-x_{0}^{\prime}g_{0n}+x_{1}^{\prime}g_{1n}+\dots+x_{n}^{\prime}g_{nn}
\end{align}
\end{matrix}\right|\begin{matrix}\begin{align}\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}}\cdot\mathbf{A} &amp; =\mathbf{g}^{-1}\\
\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g} &amp; =\mathbf{A}\\
\mathbf{g}\cdot\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}} &amp; =\mathbf{A}\\
\\
\end{align}
\\
\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} &amp; =\left\{ \begin{align}-1\quad &amp; (j=k=0)\\
1\quad &amp; (j=k&gt;0)\\
0\quad &amp; (j\ne k)
\end{align}
\right.\\
\sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} &amp; =\left\{ \begin{align}-1\quad &amp; (i=k=0)\\
1\quad &amp; (i=k&gt;0)\\
0\quad &amp; (i\ne k)
\end{align}
\right.
\end{align}
\end{matrix}
\end{matrix}&lt;/math&gt;|{{equationRef|1a}}}}

The invariance of the Lorentz interval with ''n''=1 between axes and [[w:conjugate diameters]] of hyperbolas was known for a long time since [[#Apo|Apollonius (ca. 200 BC)]]. Lorentz transformations ({{equationNote|1a}}) for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]] in order to simplify computations of [[w:elliptic function]]s and integrals.&lt;ref&gt;Musen (1970) pointed out the intimate connection of Hill's scalar development and Minkowski's pseudo-Euclidean 3D space.&lt;/ref&gt;&lt;ref&gt;Touma et al. (2009) showed the analogy between Gauss and Hill's equations and Lorentz transformations, see eq. 22-29.&lt;/ref&gt; They were also used by [[#Chasles|Chasles (1829)]] and [[#Weddle|Weddle (1847)]] to describe relations on hyperboloids, as well as by [[#Poincare|Poincaré (1881)]], [[#Cox|Cox (1881-91)]], [[#Picard|Picard (1882, 1884)]], [[#Killing|Killing (1885, 1893)]], [[#Gerard|Gérard (1892)]], [[#Hausdorff|Hausdorff (1899)]], [[#Woods2|Woods (1901, 1903)]], [[#Liebmann|Liebmann (1904/05)]] to describe [[w:hyperbolic motion]]s (i.e. rigid motions in the [[w:hyperbolic plane]] or [[w:hyperbolic space]]), which were expressed in terms of Weierstrass coordinates of the [[w:hyperboloid model]] satisfying the relation &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}=-1&lt;/math&gt; or in terms of the [[w:Cayley–Klein metric]] of [[w:projective geometry]] using the &quot;absolute&quot; form &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}=0&lt;/math&gt; as discussed by [[#Klein|Klein (1871-73)]].&lt;ref group=M&gt;Killing (1885), p. 71&lt;/ref&gt;&lt;ref&gt;Müller (1910), p. 661, in particular footnote 247.&lt;/ref&gt;&lt;ref&gt;Sommerville (1911), p. 286, section K6.&lt;/ref&gt; In addition, [[w:infinitesimal transformation]]s related to the [[w:Lie algebra]] of the group of hyperbolic motions were given in terms of Weierstrass coordinates &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}=-1&lt;/math&gt; by [[#Killing3|Killing (1888-1897)]].

===Most general Lorentz transformation of velocity===

If &lt;math&gt;x_{i},\ x_{i}^{\prime}&lt;/math&gt; in ({{equationNote|1a}}) are interpreted as [[w:homogeneous coordinates]], then the corresponding inhomogenous coordinates &lt;math&gt;u_{s},\ u_{s}^{\prime}&lt;/math&gt; follow by

:&lt;math&gt;\frac{x_{s}}{x_{0}}=u_{s},\ \frac{x_{s}^{\prime}}{x_{0}^{\prime}}=u_{s}^{\prime}\ (s=1,2\dots n)&lt;/math&gt;

defined by &lt;math&gt;u_{1}^{2}+u_{2}^{2}+\dots+u_{n}^{2}\le1&lt;/math&gt; so that the Lorentz transformation becomes a [[w:homography]] inside the [[w:unit hypersphere]], which [[w:John Lighton Synge]] called &quot;the most general formula for the composition of velocities&quot; in terms of special relativity&lt;ref&gt;Synge (1955), p. 129 for ''n''=3&lt;/ref&gt; (the transformation matrix '''g''' stays the same as in ({{equationNote|1a}})):

{{NumBlk|:|&lt;math&gt;\begin{align}u_{s}^{\prime} &amp; =\frac{g_{s0}+g_{s1}u_{1}+\dots+g_{sn}u_{n}}{g_{00}+g_{01}u_{1}+\dots+g_{0n}u_{n}}\\
\\
u_{s} &amp; =\frac{-g_{0s}+g_{1s}u_{1}^{\prime}+\dots+g_{ns}u_{n}^{\prime}}{g_{00}-g_{10}u_{1}^{\prime}-\dots-g_{n0}u_{n}^{\prime}}
\end{align}
\left|{\scriptstyle \begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} &amp; =\left\{ \begin{align}-1\quad &amp; (j=k=0)\\
1\quad &amp; (j=k&gt;0)\\
0\quad &amp; (j\ne k)
\end{align}
\right.\\
\sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} &amp; =\left\{ \begin{align}-1\quad &amp; (i=k=0)\\
1\quad &amp; (i=k&gt;0)\\
0\quad &amp; (i\ne k)
\end{align}
\right.
\end{align}
}\right.&lt;/math&gt;|{{equationRef|1b}}}}

Such Lorentz transformations for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827–1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]], [[#Callandreau|Callandreau (1885)]] in order to simplify computations of elliptic functions and integrals, by [[#Picard|Picard (1882-1884)]] in relation to [[w:Hermitian form|Hermitian quadratic form]]s, or by [[#Woods2|Woods (1901, 1903)]] in terms of the [[w:Beltrami–Klein model]] of hyperbolic geometry. In addition, infinitesimal transformations in terms of the [[w:Lie algebra]] of the group of hyperbolic motions leaving invariant the unit sphere &lt;math&gt;-1+u_{1}^{\prime2}+\cdots+u_{n}^{\prime2}=0&lt;/math&gt; were given by [[#Lie3|Lie (1885-1893) and Werner (1889)]] and [[#Killing3|Killing (1888-1897)]].

==Historical notation==
==={{anchor|Apo}} Apollonius (BC) – Conjugate diameters===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Apo|label 1=History of Lorentz transformations via squeeze mappings § Apollonius}}
====Equality of difference in squares====
[[File:Apollonius-Borelli-XII.png|thumb|&lt;small&gt;Fig. 1: Apollonius' proposition illustrated by Borelli (1661) of &lt;math&gt;\scriptstyle \overline{AC}^{2}-\overline{QR}^{2}=\overline{IL}^{2}-\overline{NO}^{2}&lt;/math&gt;&lt;/small&gt;]]
[[w:Apollonius of Perga]] (c. 240–190 BC) in his 7th book on conics defined the following well known proposition (the 7th book survived in Arabian translation, and was translated into Latin in 1661 and 1710), as follows:

*The difference of the squares of the two axes of the hyperbola is equal to the difference of the squares of any two conjugate diameters. &lt;small&gt;(Latin translation 1661 by [[w:Giovanni Alfonso Borelli]] and [[w:Abraham Ecchellensis]].)&lt;ref group=M name=bor1&gt;Apollonius/Borelli/Ecchellensis (1661), Summary of prop. XII and other props. from book VII on pp. 291-292; See also the note on prop. XII on pp. 293-294, where Borelli demontrates &lt;math&gt;\scriptstyle \overline{AC}^{2}-\overline{QR}^{2}=\overline{IL}^{2}-\overline{NO}^{2}&lt;/math&gt; (in later translations such as Halley (1710), the proposition was numbered as XIII.) Latin: &quot;Differentia quadratorum duorum axium hyperboles æqualis est differentiæ quadratorum quarumlibet duarum diametrorum coniugatarum.&quot;&lt;/ref&gt;&lt;/small&gt;

*In every hyperbola the difference between the squares of the axes is equal to the difference between the squares of any conjugate diameters of the section. &lt;small&gt;(Latin translation 1710 by [[w:Edmond Halley]].)&lt;ref group=M&gt;Apollonius/Halley (1710), Prop. XIII of book VII on p. 107; Latin: &quot;In omni Hyperbola differentia inter quadrata Axium aequalis est differentiae inter quadrata ex diametris quibusvis conjugatis sectionis.&quot;&lt;/ref&gt;&lt;/small&gt;

*[..] in every hyperbola the difference of the squares on any two conjugate diameters is equal to the [..] difference [..] of the squares on the axes. &lt;small&gt;(English translation 1896 by [[w:Thomas Heath (classicist)|w:Thomas Heath]].)&lt;ref group=M&gt;Apollonius/Heath (1896), Proposition 129; (Apollonius, Book VII, Prop. 13).&lt;/ref&gt;&lt;/small&gt;
----
[[File:Lahire-XLII-XLIII.png|thumb|left|&lt;small&gt;Fig. 2: La Hire's (1685) illustration of &lt;math&gt;\scriptstyle \overline{AB}^{2}-\overline{DE}^{2}=\overline{NM}^{2}-\overline{LK}^{2}&lt;/math&gt;&lt;/small&gt;]]
[[File:Lhopital Conjugate Diameters.png|thumb|&lt;small&gt;Fig. 3: l'Hôpital's (1707) illustration of &lt;math&gt;\scriptstyle \overline{CS}^{2}-\overline{CM}^{2}=\overline{CB}^{2}-\overline{CA}^{2}&lt;/math&gt;&lt;/small&gt;]]
[[w:Philippe de La Hire]] (1685) stated this proposition as follows:
{{Block indent|1=I say that the difference of the squares of any two diameters conjugated to each other, AB, DE, is equal to the difference of the squares of any two other diameters conjugated to each other, NM, LK.&lt;ref group=M name=lahire1&gt;La Hire (1685), Book IV, Proposition XLII, p. 85; Latin: &quot;Dico differentiam quadratorum duarum diametrorum quarumlibet inter se conjugatarum AB, DE esse æqualem differentiæ quadratorum duarum aliarum diametrorum quarumlibet inter se conjugatarum, NM, LK.&quot;&lt;/ref&gt;}}
and also summarized the related propositions in the 7th book of Apollonius:
{{Block indent|1=In a hyperbola, the difference of the squares of the axes is equal to the difference of the squares of any two conjugate diameters.&lt;ref group=M&gt;La Hire (1685), p. 242. Summary of propositions XII, XIII, XXV in the 7th book of Apollonius; Latin: &quot;In hyperbola differentia quadratorum axium æqualis est differentia quadratorum duarum diametrorum conjugatarum quarumlibet.&quot;&lt;/ref&gt;}}
----
[[w:Guillaume de l'Hôpital]] (1707), using the methods of [[wikipedia:Analytic_geometry|w:analytic geometry]], demonstrated the same proposition:&lt;ref group=M name=lop&gt;l'Hôpital (1707), Third book, Prop. XII, p. 76.&lt;/ref&gt;

{{Block indent|1=The difference of the squares of any two conjugate diameters &quot;Mm, Ss&quot; is equal to the difference of the squares of the two axes &quot;Aa, Bb.&quot; We are to prove that &lt;math&gt;\overline{CS}^{2}-\overline{CM}^{2}=\overline{CB}^{2}-\overline{CA}^{2}&lt;/math&gt;, or &lt;math&gt;\overline{CM}^{2}-\overline{CS}^{2}=\overline{CA}^{2}-\overline{CB}^{2}&lt;/math&gt;. &lt;small&gt;(English translation 1723 by [[w:Edmund Stone]].)&lt;ref group=M&gt;l'Hôpital/Stone (1723), pp. 62-63&lt;/ref&gt;&lt;/small&gt;}}

{{Lorentzbox|Text=Apollonius' proposition can be expressed as &lt;math&gt;-x_{0}^{\prime2}+x_{1}^{\prime2}=-x_{0}^{2}+x_{1}^{2}&lt;/math&gt; in agreement with the invariance of the Lorentz interval, so that the Lorentz transformation ({{equationNote|1a}}) &quot;(n=1)&quot; can be interpreted as mapping from one pair of axes of a hyperbola to a pair of conjugate diameters.}}

====Equality of areas of parallelograms====

[[File:Apollonius-Borelli-XXXI.png|thumb|&lt;small&gt;Fig. 4: Apollonius' proposition illustrated by Borelli (1661) of the equality of areas of parallelogram ABCD (of the axes) and KLMN (of the conjugated diameters).&lt;/small&gt;]] 

Apollonius also gave another well known proposition in his 7th book regarding ellipses as well as conjugate sections of hyperbolas (see also Del Centina &amp; Fiocca&lt;ref&gt;Del Centina &amp; Fiocca (2020)&lt;/ref&gt; for further details on the history of this proposition): 

*In the ellipse, and in conjugate sections [the opposite branches of two conjugate hyperbolas] the parallelogram bounded by the axes is equal to the parallelogram bounded by any pair of conjugate diameters, if its angles are equal to the angles the conjugate diameters form at the centre. &lt;small&gt;(English translation by Del Centina &amp; Fiocca&lt;ref name=del&gt;Del Centina &amp; Fiocca (2020), section 3.1&lt;/ref&gt; based on the Latin translation 1661 by [[w:Giovanni Alfonso Borelli]] and [[w:Abraham Ecchellensis]].&lt;ref group=M name=bor2&gt;Apollonius/Borelli/Ecchellensis (1661), Summary of prop. XXXI of book VII on p. 370; Note on pp. 372-374; Latin: &quot;In ellypsi, &amp; sectionibus coniugatis parallelogrammum sub axibus contentum æquale est parallelogrammo à quibuscunque duabus coniugatis diametris comprehenso, si eorum anguli æquales fuerint angulis ad centrum contentis à coniugatis diametris.&quot;&lt;/ref&gt;)&lt;/small&gt;

*If two conjugate diameters are taken in an ellipse, or in the opposite conjugate sections; the parallelogram bounded by them is equal to the rectangle bounded by the axes, provided its angles are equal to those formed at the centre by the conjugate diameters. &lt;small&gt;(English translation by Del Centina &amp; Fiocca&lt;ref name=del /&gt; based on the Latin translation 1710 by [[w:Edmond Halley]].)&lt;ref group=M&gt;Apollonius/Halley (1710), Prop. XXXI of book VII on p. 115–117; Latin: &quot;Si ducantur diametri quævis conjugate in Ellipsi, vel inter sectiones oppositas conjugatas; erit parallelogrammum contentam sub his diametris æquale rectangulo sub ipsis Axibus facto: modo anguli ejus æquales sint angulis ad centrum sectionis à diametris conjugatis comprehensis.&quot;&lt;/ref&gt;)&lt;/small&gt;

*If PP', DD' be two conjugate diameters in an ellipse or in conjugate hyperbolas, and if tangents be drawn at the four extremities forming a parallelogram LL'MM', then the parallelogram LL'MM' = rect. AA'·BB'. &lt;small&gt;(English translation 1896 by [[w:Thomas Heath (classicist)|w:Thomas Heath]].)&lt;ref group=M&gt;Apollonius/Heath (1896), Proposition 136, p. 235; (Apollonius, Book VII, Prop. 31).&lt;/ref&gt;&lt;/small&gt; 

{{Lorentzbox|Text=The graphical representation of Apollonius proposition in Borelli's Fig. 4 is essentially a [[w:Minkowski diagram]], being a graphical representation of the Lorentz transformation. If line AB is the x-axis of an inertial frame S1, then line FG is the x-axis of another inertial frames S2 which together with its parallel lines (such as KL and NM) represent [[w:relativity of simultaneity]]. Analogously, if line CD is the time axis of another inertial frame S2, then line HI is the time axis of S2 which together with its parallel lines (such as KN and LM) represent the [[w:worldlines]] of objects at different locations. The diagonals KE (or KM) and LE (or LN) lie on the asymptotes which form a light cone. Thus the totality of all parallelograms of equal area and conjugate diameters as constructed by Apollonius, represents the totality of all inertial frames, lines of simultaneity and worldlines within a spacetime area bounded by &lt;math&gt;-x_{0}^{2}+x_{1}^{2}=\rm{const}&lt;/math&gt;.}}

[[File:Saint-Vincent-Hyperbola-VI-XLIX.png|thumb|175px|left|&lt;small&gt;Fig. 5: Saint-Vincent's (1647) illustration of FGHI=OPQR, as well as BADC=KNLM.&lt;/small&gt;]]
[[w:Grégoire de Saint-Vincent]] independently (1647) stated the same proposition:&lt;ref group=M name=vinc&gt;St. Vincent (1647), Book VI, Prop. XLIX, p. 560; Latin: “Si fuerint binæ hyperbolarum coniugaciones A, B, C, D: ponantur autem per E centrum duæ quoque diametrorum coniugationes per quarum vertices contingentes actæ constituant duo quadrilatera FGHI, OPQR. Dico illa esse æqualia inter se.”&lt;/ref&gt;

{{Block indent|1=The parallelograms whose opposite sides are tangent to two conjugate hyperbolas at the extremities of two conjugate diameters are equivalent among them. &lt;small&gt;(English translation by Del Centina &amp; Fiocca.&lt;ref&gt;Del Centina &amp; Fiocca (2020), section 5.1&lt;/ref&gt;)&lt;/small&gt; }}

---- 

[[File:Lahire-XLII-XLIII.png|thumb|&lt;small&gt;Fig. 6 (identical to Fig. 2): La Hire's (1685) illustration of FGHI=OPQR.&lt;/small&gt;]]
[[w:Philippe de La Hire]] (1685), who was aware of both Apollonius 7th book and Saint-Vincent's book, stated this proposition as follows:&lt;ref group=M name=lahire&gt;La Hire (1685), Book IV, Proposition XLIII, pp. 85-86; Latin: &quot;In sectionibus conjugatis NA, DL, BM, KE si circumscribatur parallelogrammum FGHI à rectis parallelis duabus diametris inter se conjugatis ED, BA, &amp; per ipsorum terminos ductis, &amp; simili methodo circumscribatur aliud parallelogrammum OPQR à rectis ductis per terminos diametrorum conjugatarum, &amp; ipsis parallelis: Dico parallelogramma FGHI, OPQR esse inter se æqualia.&quot;&lt;/ref&gt; 

{{Block indent|1=If a parallelogram FGHI is circumscribed about conjugate sections NA, DL, BM, KE whose sides are parallel to two conjugate diameters ED, BA drawn through their extremities, and with similar method another parallelogram OPQR is drawn through the extremities of other two conjugate diameters, then the parallelograms FGHI, OPQR are equal. &lt;small&gt;(English translation by Del Centina &amp; Fiocca.&lt;ref name=del2&gt;Del Centina &amp; Fiocca (2020), section 5.2&lt;/ref&gt;)&lt;/small&gt;}}

and also summarized the related propositions in the 7th book of Apollonius:&lt;ref group=M&gt;La Hire (1685), p. 242. Summary of proposition XXXI in the 7th book of Apollonius; Latin: &quot;In sectionibus conjugatis &amp; Ellipsi parallelogrammum sub axibus æquale est paralelogrammo sub duabus quibuscunque diametris inter se conjugatis, in angulis ipsarum diametrorum conjugatarum.&quot;&lt;/ref&gt;

{{Block indent|1=In conjugate sections and in the ellipse, the parallelogram constructed with the axes, is equal to the parallelogram constructed with any two conjugated diameters, provided the angles are equal to those between the diameters themselves. &lt;small&gt;(English translation by Del Centina &amp; Fiocca.&lt;ref name=del2 /&gt;)&lt;/small&gt;}}

{{Lorentzbox|Text=In Saint-Vincent's Fig. 5 or La Hire's Fig. 6, parallelogram FGHI contains all coordinates related to an inertial frame S3, in particular triangles EGH, EFI (Fig. 5) or CFG, CHI (Fig. 6) contain time like intervals between events on the future and past light cones, while triangles EHI, EGF (Fig. 5) or CFI, CGH (Fig. 6) contain space like intervals between events on the negative and positive x-axis. Conversely, parallelogram OPQR contains all coordinates related to another frame S4, in particular triangles EQR, EOP (Fig. 5) or CPQ, COR (Fig. 6) contain time like intervals between events on the future and past light cones, while triangles EPR, EOQ (Fig. 5) or COP, CQR (Fig. 6) contain space like intervals between events on the negative and positive x-axis.}}

===Lagrange (1773) – Binary quadratic forms {{anchor|Lagrange}}===
After the invariance of the sum of squares under linear substitutions was discussed by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]], the general expressions of a [[w:binary quadratic form]] and its transformation was formulated by [[w:Joseph-Louis Lagrange]] (1773/75) as follows&lt;ref group=M&gt;Lagrange (1773/75), section 22&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\
\hline \begin{align}y &amp; =Ms+Nx\\
z &amp; =ms+nx
\end{align}
\left|\begin{matrix}\begin{align}P &amp; =pM^{2}+2qMm+rm^{2}\\
Q &amp; =pMN+q(Mn+Nm)+rmn\\
R &amp; =pN^{2}+2qNn+rn^{2}
\end{align}
\\
\downarrow\\
PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2}
\end{matrix}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to ({{equationNote|Q1}}) ''(n=1)''. The Lorentz interval &lt;math&gt;-x_{0}^{2}+x_{1}^{2}&lt;/math&gt; and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(p,q,r)=(P,Q,R)=(1,0,-1)''.}}

==={{anchor|Gauss}} Gauss (1798–1818)===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Gauss|label 1=History of Lorentz transformations via Möbius transformations § Gauss}}
===={{anchor|Gauss1}} Binary quadratic forms====

The theory of binary quadratic forms was considerably expanded by [[w:Carl Friedrich Gauss]] (1798, published 1801) in his [[w:Disquisitiones Arithmeticae]]. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:&lt;ref group=M&gt;Gauss (1798/1801), articles 157–158;&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\
F'=a'x^{\prime2}+2b'x'y'+c'y^{\prime2}=(a',b',c')\\
\hline \begin{align}x &amp; =\alpha x'+\beta y'\\
y &amp; =\gamma x'+\delta y'\\
\\
x' &amp; =\delta x-\beta y\\
y' &amp; =-\gamma x+\alpha y
\end{align}
\left|\begin{matrix}\begin{align}a' &amp; =a\alpha^{2}+2b\alpha\gamma+c\gamma^{2}\\
b' &amp; =a\alpha\beta+b(\alpha\delta+\beta\gamma)+c\gamma\delta\\
c' &amp; =a\beta^{2}+2b\beta\delta+c\delta^{2}
\end{align}
\\
\downarrow\\
b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha\delta-\beta\gamma)^{2}
\end{matrix}\right.
\end{matrix}&lt;/math&gt;

which is equivalent to ({{equationNote|Q1}}) ''(n=1)''. As pointed out by Gauss, ''F'' and ''F′'' are called &quot;proper equivalent&quot; if αδ-βγ=1, so that ''F'' is contained in ''F′'' as well as ''F′'' is contained in ''F''. In addition, if another form ''F″'' is contained by the same procedure in ''F′'' it is also contained in ''F'' and so forth.&lt;ref group=M&gt;Gauss (1798/1801), section 159&lt;/ref&gt;

{{Lorentzbox|Text=The Lorentz interval &lt;math&gt;-x_{0}^{2}+x_{1}^{2}&lt;/math&gt; and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(a,b,c)=(a',b',c')=(1,0,-1)''.}}

===={{anchor|Gauss2}} Ternary quadratic forms====

Gauss (1798/1801)&lt;ref group=M&gt;Gauss (1798/1801), articles 266–285&lt;/ref&gt; also discussed ternary quadratic forms with the general expression

:&lt;math&gt;\begin{matrix}f=ax^{2}+a'x^{\prime2}+a''x^{\prime\prime2}+2bx'x''+2b'xx''+2b''xx'=\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)\\
g=my^{2}+m'y^{\prime2}+m''y^{\prime\prime2}+2ny'y''+2n'yy''+2n''yy'=\left(\begin{matrix}m, &amp; m', &amp; m''\\
n, &amp; n', &amp; n''
\end{matrix}\right)\\
\hline \begin{align}x &amp; =\alpha y+\beta y'+\gamma y''\\
x' &amp; =\alpha'y+\beta'y'+\gamma'y''\\
x'' &amp; =\alpha''y+\beta''y'+\gamma''y''
\end{align}
\end{matrix}&lt;/math&gt;

which is equivalent to ({{equationNote|Q1}}) ''(n=2)''. Gauss called these forms definite when they have the same sign such as ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;+z&lt;sup&gt;2&lt;/sup&gt;'', or indefinite in the case of different signs such as ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;-z&lt;sup&gt;2&lt;/sup&gt;''. While discussing the classification of ternary quadratic forms, Gauss (1801) presented twenty special cases, among them these six variants:&lt;ref group=M&gt;Gauss (1798/1801), article 277&lt;/ref&gt;

:&lt;math&gt;\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)\Rightarrow\begin{matrix}\left(\begin{matrix}1, &amp; -1, &amp; 1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}-1, &amp; 1, &amp; 1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\\
\left(\begin{matrix}1, &amp; -1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}-1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}-1, &amp; -1, &amp; 1\\
0, &amp; 0, &amp; 0
\end{matrix}\right)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=These are all six types of Lorentz interval in 2+1 dimensions that can be produced as special cases of a ternary quadratic form. In general: The Lorentz interval &lt;math&gt;x^{2}+x^{\prime2}-x^{\prime\prime2}&lt;/math&gt; and the Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' is an indefinite ternary quadratic form, which follows from the general ternary form by setting:
&lt;math&gt;\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)=\left(\begin{matrix}m, &amp; m', &amp; m''\\
n, &amp; n', &amp; n''
\end{matrix}\right)=\left(\begin{matrix}1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right)&lt;/math&gt;}}

===={{anchor|Gauss4}} Homogeneous coordinates====

Gauss (1818) discussed planetary motions together with formulating [[w:elliptic function]]s. In order to simplify the integration, he transformed the expression

:&lt;math&gt;(AA+BB+CC)tt+aa(t\cos E)^{2}+bb(t\sin E)^{2}-2aAt\cdot t\cos E-2bBt\cdot t\sin E&lt;/math&gt;

into

:&lt;math&gt;G+G'\cos T^{2}+G''\sin T^{2}&lt;/math&gt;

in which the [[w:eccentric anomaly]] ''E'' is connected to the new variable ''T'' by the following transformation including an arbitrary constant ''k'', which Gauss then rewrote by setting ''k''=1:&lt;ref group=M&gt;Gauss (1818), pp. 5–10&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}{\scriptstyle \left(\alpha+\alpha'\cos T+\alpha''\sin T\right)^{2}+\left(\beta+\beta'\cos T+\beta''\sin T\right)^{2}-\left(\gamma+\gamma'\cos T+\gamma''\sin T\right)^{2}}=0\\
k\left(\cos^{2}T+\sin^{2}T-1\right)=0\\
\hline \begin{align}\cos E &amp; =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\
\sin E &amp; =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}
\end{align}
\left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma &amp; =k &amp; \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' &amp; =-k\\
-\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' &amp; =-k &amp; \beta\beta-\beta'\beta'-\beta''\beta'' &amp; =-k\\
-\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' &amp; =-k &amp; \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' &amp; =+k\\
-\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' &amp; =0 &amp; \beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0\\
-\alpha''\alpha-\beta''\beta+\gamma''\gamma &amp; =0 &amp; \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' &amp; =0\\
-\alpha\alpha'-\beta\beta'+\gamma\gamma' &amp; =0 &amp; \alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0
\end{align}
}\right.\\
\hline k=1\\
\begin{align}t\cos E &amp; =\alpha+\alpha'\cos T+\alpha''\sin T\\
t\sin E &amp; =\beta+\beta'\cos T+\beta''\sin T\\
t &amp; =\gamma+\gamma'\cos T+\gamma''\sin T
\end{align}
\left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma &amp; =1\\
-\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' &amp; =-1\\
-\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' &amp; =-1\\
-\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' &amp; =0\\
-\alpha''\alpha-\beta''\beta+\gamma''\gamma &amp; =0\\
-\alpha\alpha'-\beta\beta'+\gamma\gamma' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The coefficients α,β,γ,... of Gauss' case ''k''=1 are equivalent to the coefficient system in Lorentz transformations ({{equationNote|1a}}) and ({{equationNote|1b}})  ''(n=2)''.

Further setting &lt;math&gt;[\cos T,\sin T,\cos E,\sin E]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]&lt;/math&gt;, Gauss' transformation becomes Lorentz transformation ({{equationNote|1b}})  ''(n=2)''.}}

Subsequently, he showed that these relations can be reformulated using three variables ''x,y,z'' and ''u,u′,u″'', so that

:&lt;math&gt;aaxx+bbyy+(AA+BB+CC)zz-2aAxz-2bByz&lt;/math&gt;

can be transformed into

:&lt;math&gt;Guu+G'u'u'+G''u''u''&lt;/math&gt;,

in which ''x,y,z'' and ''u,u′,u″'' are related by the transformation:&lt;ref group=M&gt;Gauss (1818), pp. 9–10&lt;/ref&gt;

:&lt;math&gt;\begin{align}x &amp; =\alpha u+\alpha'u'+\alpha''u''\\
y &amp; =\beta u+\beta'u'+\beta''u''\\
z &amp; =\gamma u+\gamma'u'+\gamma''u''\\
\\
u &amp; =-\alpha x-\beta y+\gamma z\\
u' &amp; =\alpha'x+\beta'y-\gamma'z\\
u'' &amp; =\alpha''x+\beta''y-\gamma''z
\end{align}
\left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma &amp; =1\\
-\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' &amp; =-1\\
-\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' &amp; =-1\\
-\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' &amp; =0\\
-\alpha''\alpha-\beta''\beta+\gamma''\gamma &amp; =0\\
-\alpha\alpha'-\beta\beta'+\gamma\gamma' &amp; =0
\end{align}
}\right.&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}})  ''(n=2)'' satisfying &lt;math&gt;x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}&lt;/math&gt;, and can be related to Gauss' previous equations in terms of homogeneous coordinates &lt;math&gt;\left[\cos T,\sin T,\cos E,\sin E\right]=\left[\tfrac{x}{z},\ \tfrac{y}{z},\ \tfrac{u'}{u},\ \tfrac{u''}{u}\right]&lt;/math&gt;.}}

==={{anchor|Jacobi}} Jacobi (1827, 1833/34) – Homogeneous coordinates===

Following [[#Gauss4|Gauss (1818)]], [[w:Carl Gustav Jacob Jacobi]] extended Gauss' transformation in 1827:&lt;ref group=M&gt;Jacobi (1827), p. 235, 239–240&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}\cos P^{2}+\sin P^{2}\cos\vartheta^{2}+\sin P^{2}\sin\vartheta^{2}=1\\
k\left(\cos\psi^{2}+\sin\psi^{2}\cos\varphi^{2}+\sin\psi^{2}\sin\varphi^{2}-1\right)=0\\
\hline {\left.\begin{matrix}\mathbf{(1)}\begin{align}\cos P &amp; =\frac{\alpha+\alpha'\cos\psi+\alpha''\sin\psi\cos\varphi+\alpha'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\
\sin P\cos\vartheta &amp; =\frac{\beta+\beta'\cos\psi+\beta''\sin\psi\cos\varphi+\beta'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\
\sin P\sin\vartheta &amp; =\frac{\gamma+\beta'\cos\psi+\gamma''\sin\psi\cos\varphi+\gamma'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\
\\
\cos\psi &amp; =\frac{-\delta'+\alpha'\cos P+\beta'\sin P\cos\vartheta+\gamma'\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\
\sin\psi\cos\varphi &amp; =\frac{-\delta''+\alpha''\cos P+\beta''\sin P\cos\vartheta+\gamma''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\
\sin\psi\sin\varphi &amp; =\frac{-\delta'''+\alpha'''\cos P+\beta'''\sin P\cos\vartheta+\gamma'''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}
\end{align}
\\
\\
\hline \mathbf{(2)}\begin{align}\alpha\mu+\beta x+\gamma y+\delta z &amp; =m\\
\alpha'\mu+\beta'x+\gamma'y+\delta'z &amp; =m'\\
\alpha''\mu+\beta''x+\gamma''y+\delta''z &amp; =m''\\
\alpha'''\mu+\beta'''x+\gamma'''y+\delta'''z &amp; =m'''\\
\\
Am+A'm'+A''m''+A'''m''' &amp; =\mu\\
Bm+B'm'+B''m''+B'''m''' &amp; =x\\
Cm+C'm'+C''m''+C'''m''' &amp; =y\\
Dm+D'm'+D''m''+D'''m''' &amp; =z\\
\\
\end{align}
\\
\begin{align}\alpha &amp; =-kA, &amp; \beta &amp; =-kB, &amp; \gamma &amp; =-kC, &amp; \delta &amp; =kD,\\
\alpha' &amp; =kA', &amp; \beta' &amp; =kB', &amp; \gamma' &amp; =kC', &amp; \delta' &amp; =-kD',\\
\alpha'' &amp; =kA'', &amp; \beta'' &amp; =kB'', &amp; \gamma'' &amp; =kC'', &amp; \delta'' &amp; =-kD'',\\
\alpha''' &amp; =kA''', &amp; \beta''' &amp; =kB''', &amp; \gamma''' &amp; =kC''', &amp; \delta''' &amp; =-kD''',
\end{align}
\end{matrix}\right|\begin{matrix}\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta &amp; =-k\\
\alpha'\alpha'+\beta'\beta'+\gamma'\gamma'-\delta'\delta' &amp; =k\\
\alpha''\alpha''+\beta''\beta''+\gamma''\gamma''-\delta''\delta'' &amp; =k\\
\alpha'''\alpha'''+\beta'''\beta'''+\gamma'''\gamma'''-\delta'''\delta''' &amp; =k\\
\alpha\alpha'+\beta\beta'+\gamma\gamma'-\delta\delta' &amp; =0\\
\alpha\alpha''+\beta\beta''+\gamma\gamma''-\delta\delta'' &amp; =0\\
\alpha\alpha'''+\beta\beta'''+\gamma\gamma'''-\delta\delta''' &amp; =0\\
\alpha''\alpha'''+\beta''\beta'''+\gamma''\gamma'''-\delta''\delta''' &amp; =0\\
\alpha'''\alpha'+\beta'''\beta'+\gamma'''\gamma'-\delta'''\delta' &amp; =0\\
\alpha'\alpha''+\beta'\beta''+\gamma'\gamma''-\delta'\delta'' &amp; =0\\
\\
-\alpha\alpha+\alpha'\alpha'+\alpha''\alpha''+\alpha'''\alpha''' &amp; =k\\
-\beta\beta+\beta'\beta'+\beta''\beta''+\beta'''\beta''' &amp; =k\\
-\gamma\gamma+\gamma'\gamma'+\gamma''\gamma''+\gamma'''\gamma''' &amp; =k\\
-\delta\delta+\delta'\delta'+\delta''\delta''+\delta'''\delta''' &amp; =-k\\
-\alpha\beta+\alpha'\beta'+\alpha''\beta''+\alpha'''\beta''' &amp; =0\\
-\alpha\gamma+\alpha'\gamma'+\alpha''\gamma''+\alpha'''\gamma''' &amp; =0\\
-\alpha\delta+\alpha'\delta'+\alpha''\delta''+\alpha'''\delta''' &amp; =0\\
-\beta\gamma+\beta'\gamma'+\beta''\gamma''+\beta'''\gamma''' &amp; =0\\
-\gamma\delta+\gamma'\delta'+\gamma''\delta''+\gamma'''\delta''' &amp; =0\\
-\delta\beta+\delta'\beta'+\delta''\beta''+\delta'''\beta''' &amp; =0
\end{align}
\end{matrix}}
\end{matrix}}&lt;/math&gt;

{{Lorentzbox|Text=By setting &lt;math&gt;{\scriptstyle \begin{align}\left[\cos P,\ \sin P\cos\varphi,\ \sin P\sin\varphi\right] &amp; =\left[u_{1},\ u_{2},\ u_{3}\right]\\{}
[\cos\psi,\ \sin\psi\cos\vartheta,\ \sin\psi\sin\vartheta] &amp; =\left[u_{1}^{\prime},\ u_{2}^{\prime},\ u_{3}^{\prime}\right]
\end{align}
}&lt;/math&gt; and ''k''=1 in the (1827) formulas, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'', and by setting ''k''=1 in transformation system (2) it becomes equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' producing &lt;math&gt;m^{2}+m^{\prime2}+m^{\prime\prime2}-m^{\prime\prime\prime2}=\mu^{2}+x^{2}+y^{2}-z^{2}&lt;/math&gt;.}}

Alternatively, in two papers from 1832 Jacobi started with an ordinary orthogonal transformation, and by using an imaginary substitution he arrived at Gauss' transformation (up to a sign change):&lt;ref group=M&gt;The orthogonal substitution and the imaginary transformation was defined in Jacobi (1832a), pp. 257, 265–267; Transformation system (2) and (3) and coefficients in Jacobi (1832b), pp. 321-325.&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}xx+yy+zz=ss+s's'+s''s''=0\\
\mathbf{(1)}\begin{align}x &amp; =\alpha s+\alpha's'+\alpha''s''\\
y &amp; =\beta s+\beta's'+\beta''s''\\
z &amp; =\gamma s+\gamma's'+\gamma''s''\\
\\
s &amp; =\alpha x+\beta y+\gamma z\\
s' &amp; =\alpha'x+\beta'y+\gamma'z\\
s'' &amp; =\alpha''x+\beta''y+\gamma''z
\end{align}
\left|\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma &amp; =1 &amp; \alpha\alpha+\alpha'\alpha'+\alpha''\alpha'' &amp; =1\\
\alpha'\alpha'+\beta'\beta'+\gamma'\gamma' &amp; =1 &amp; \beta\beta+\beta'\beta'+\beta''\beta'' &amp; =1\\
\alpha''\alpha''+\beta''\beta''+\gamma''\gamma'' &amp; =1 &amp; \gamma\gamma+\gamma'\gamma'+\gamma''\gamma'' &amp; =1\\
\alpha'\alpha''+\beta'\beta''+\gamma'\gamma'' &amp; =0 &amp; \beta\gamma+\beta'\gamma'+\beta''\gamma'' &amp; =0\\
\alpha''\alpha+\beta''\beta+\gamma''\gamma &amp; =0 &amp; \gamma\alpha+\gamma'\alpha'+\gamma''\alpha'' &amp; =0\\
\alpha\alpha'+\beta\beta'+\gamma\gamma' &amp; =0 &amp; \alpha\beta+\alpha'\beta'+\alpha''\beta'' &amp; =0
\end{align}
\right.\\
\hline \left[\frac{y}{x},\ \frac{z}{x},\ \frac{s'}{s},\ \frac{s''}{s}\right]=\left[-i\cos\varphi,\ -i\sin\varphi,\ i\cos\eta,\ i\sin\eta\right]\\
\left[\alpha',\ \alpha'',\ \beta,\ \gamma\right]=\left[i\alpha',\ i\alpha'',\ -i\beta,\ -i\gamma\right]\\
\hline \begin{matrix}\mathbf{(2)}\begin{matrix}\left(\alpha-\alpha'\cos\eta-\alpha''\sin\eta\right)^{2}=\left(\beta-\beta'\cos\eta-\beta''\sin\eta\right)^{2}+\left(\gamma-\gamma'\cos\eta-\gamma''\sin\eta\right)^{2}\\
\left(\alpha-\beta\cos\phi-\gamma\sin\phi\right)^{2}=\left(\alpha'-\beta'\cos\phi-\gamma'\sin\phi\right)^{2}+\left(\alpha''-\beta''\cos\phi-\gamma''\sin\phi\right)^{2}\\
\hline \begin{align}\cos\phi &amp; =\frac{\beta-\beta'\cos\eta-\beta''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, &amp; \cos\eta &amp; =\frac{\alpha'-\beta'\cos\phi-\gamma'\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}\\
\sin\phi &amp; =\frac{\gamma-\gamma'\cos\eta-\gamma''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, &amp; \sin\eta &amp; =\frac{\alpha''-\beta''\cos\phi-\gamma''\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}
\end{align}
\end{matrix}\\
\hline \\
\mathbf{(3)}\begin{matrix}1-zz-yy=\frac{1-s's'-s''s''}{\left(\alpha-\alpha's'-\alpha''s''\right)^{2}}\\
\hline \begin{align}y &amp; =\frac{\beta-\beta's'-\beta''s''}{\alpha-\alpha's'-\alpha''s''}, &amp; s' &amp; =\frac{\alpha'-\beta'y-\gamma'z}{\alpha-\beta y-\gamma z},\\
z &amp; =\frac{\gamma-\gamma's'-\gamma''s''}{\alpha-\alpha's'-\alpha''s'''}, &amp; s'' &amp; =\frac{\alpha''-\beta''y-\gamma''z}{\alpha-\beta y-\gamma z},
\end{align}
\end{matrix}
\end{matrix}\left|\begin{align}\alpha\alpha-\beta\beta-\gamma\gamma &amp; =1\\
\alpha'\alpha'-\beta'\beta'-\gamma'\gamma' &amp; =-1\\
\alpha''\alpha''-\beta''\beta''-\gamma''\gamma'' &amp; =-1\\
\alpha'\alpha''-\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\alpha''\alpha-\beta''\beta-\gamma''\gamma &amp; =0\\
\alpha\alpha'-\beta\beta'-\gamma\gamma' &amp; =0\\
\\
\alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' &amp; =1\\
\beta\beta-\beta'\beta'-\beta''\beta'' &amp; =-1\\
\gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' &amp; =-1\\
\beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0\\
\gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' &amp; =0\\
\alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0
\end{align}
\right.
\end{matrix}}&lt;/math&gt;

{{Lorentzbox|Text=By setting &lt;math&gt;[\cos\phi,\ \sin\phi,\ \cos\eta,\ \sin\eta]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]&lt;/math&gt;, transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Also transformation system (3) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' up to a sign change.}}

Extending his previous result, Jacobi (1833) started with [[#Cauchy|Cauchy's (1829)]] orthogonal transformation for ''n'' dimensions, and by using an imaginary substitution he formulated Gauss' transformation (up to a sign change) in the case of ''n'' dimensions:&lt;ref group =M&gt;Jacobi (1833/34), pp. 7–8, 34–35, 41; Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 229–230.&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}x_{1}x_{1}+x_{2}x_{2}+\dots+x_{n}x_{n}=y_{1}y_{1}+y_{2}y_{2}+\dots+y_{n}y_{n}\\
\hline \mathbf{(1)\ }\begin{align}y_{\varkappa} &amp; =\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}\\
x_{\varkappa} &amp; =\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}\\
\\
\frac{y_{\varkappa}}{y_{n}} &amp; =\frac{\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}\\
\frac{x_{\varkappa}}{x_{n}} &amp; =\frac{\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}
\end{align}
\left|\begin{align}\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\lambda}^{(n)} &amp; =0\\
\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\varkappa}^{(n)} &amp; =1\\
\\
\alpha_{1}^{(\varkappa)}\alpha_{1}^{(\lambda)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\lambda)} &amp; =0\\
\alpha_{1}^{(\varkappa)}\alpha_{1}^{(\varkappa)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\varkappa)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\varkappa)} &amp; =1
\end{align}
\right.\\
\hline \frac{x_{\varkappa}}{x_{n}}=-i\xi_{\varkappa},\ \frac{y_{\varkappa}}{y_{n}}=i\nu_{\varkappa}\\
1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{y_{n}y_{n}}{x_{n}x_{n}}\left(1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}\right)\\
\alpha_{n}^{(\varkappa)}=i\alpha^{(\varkappa)},\ \alpha_{\varkappa}^{(n)}=-i\alpha_{\varkappa},\ \alpha_{n}^{(n)}=\alpha\\
1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}}{\left[\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}\right]^{2}}\\
\hline \mathbf{(2)\ }\begin{align}\nu_{\varkappa} &amp; =\frac{\alpha^{(\varkappa)}-\alpha_{1}^{(\varkappa)}\xi_{1}-\alpha_{2}^{(\varkappa)}\xi_{2}\dots-\alpha_{n-1}^{(\varkappa)}\xi_{n-1}}{\alpha-\alpha_{1}\xi_{1}-\alpha_{2}\xi_{2}\dots-\alpha_{n-1}\xi_{n-1}}\\
\\
\xi_{\varkappa} &amp; =\frac{\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\nu_{1}-\alpha_{2}^{\prime\prime}\nu_{2}\dots-\alpha_{\varkappa}^{(n-1)}\nu_{n-1}}{\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}}
\end{align}
\\
\hline \xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=1\ \Rightarrow\ \nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}=1
\end{matrix}}&lt;/math&gt;

{{Lorentzbox|Text=Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) up to a sign change.}}

He also stated the following transformation leaving invariant the Lorentz interval:&lt;ref group=M&gt;Jacobi (1833/34), p. 37. Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 232–233.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}uu-u_{1}u_{1}-u_{2}u_{2}-\dots-u_{n-1}u_{n-1}=ww-w_{1}w_{1}-w_{2}w_{2}-\dots-w_{n-1}w_{n-1}\\
\hline {\scriptstyle \begin{align}u &amp; =\alpha w-\alpha'w_{1}-\alpha''w_{2}-\dots-\alpha^{(n-1)}w_{n-1}\\
u_{1} &amp; =\alpha_{1}w-\alpha_{1}^{\prime}w_{1}-\alpha_{1}^{\prime\prime}w_{2}-\dots-\alpha_{1}^{(n-1)}w_{n-1}\\
 &amp; \dots\\
u_{n-1} &amp; =\alpha_{n-1}w-\alpha_{n-1}^{\prime}w_{1}-\alpha_{n-1}^{\prime\prime}w_{2}-\dots-\alpha_{n-1}^{(n-1)}w_{n-1}\\
\\
w &amp; =\alpha u-\alpha_{1}u_{1}-\alpha_{2}^{\prime\prime}u_{2}-\dots-\alpha_{n-1}u_{n-1}\\
w_{1} &amp; =\alpha'u-\alpha_{1}^{\prime}u_{1}-\alpha_{2}^{\prime}u_{2}-\dots-\alpha_{n-1}^{\prime}u_{n-1}\\
 &amp; \dots\\
w_{n-1} &amp; =\alpha^{(n-1)}u-\alpha_{1}^{(n-1)}u_{1}-\alpha_{2}^{(n-1)}u_{2}-\dots-\alpha_{n-1}^{(n-1)}u_{n-1}
\end{align}
\left|\begin{align}\alpha\alpha-\alpha'\alpha'-\alpha''\alpha''\dots-\alpha^{(n-1)}\alpha^{(n-1)} &amp; =+1\\
\alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\varkappa}^{(n-1)} &amp; =-1\\
\alpha\alpha_{\varkappa}-\alpha^{\prime}\alpha_{\varkappa}^{\prime}-\alpha^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha^{(n-1)}\alpha_{\varkappa}^{(n-1)} &amp; =0\\
\alpha_{\varkappa}\alpha_{\lambda}-\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\lambda}^{(n-1)} &amp; =0\\
\\
\alpha\alpha-\alpha_{1}\alpha_{1}-\alpha_{2}\alpha_{2}\dots-\alpha_{n-1}\alpha_{n-1} &amp; =+1\\
\alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{1}^{\varkappa}\alpha_{1}^{\varkappa}-\alpha_{2}^{\prime\prime}\alpha_{2}^{\prime\prime}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\varkappa)} &amp; =-1\\
\alpha\alpha^{(\varkappa)}-\alpha_{1}\alpha_{1}^{(\varkappa)}-\alpha_{2}\alpha_{2}^{(\varkappa)}\dots-\alpha_{n-1}\alpha_{n-1}^{(\varkappa)} &amp; =0\\
\alpha^{(\varkappa)}\alpha^{(\lambda)}-\alpha_{1}^{(\varkappa)}\alpha_{1}^{\lambda l)}-\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\lambda)} &amp; =0
\end{align}
\text{ }\right.}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) up to a sign change.}}

==={{anchor|Chasles}} Chasles (1829) – Conjugate hyperboloids ===

[[w:Michel Chasles]] (1829) independently introduced the same equation systems as [[#Gauss4|Gauss (1818)]] and [[#Jacobi|Jacobi (1827)]], albeit in the different context of conjugate hyperboloids. He started with two equation systems (a) and (b) from which he derived systems (c), (d) and others:&lt;ref group=M&gt;Chasles (1829), p. 139&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =1\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =1\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =-1
\end{align}
\right\}  &amp; \dots(a)\\ \\
\left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma' &amp; =0
\end{align}
\right\}  &amp; \dots(b)\\ \\
\left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =1\\
\beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} &amp; =1\\
\gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-1
\end{align}
\right\}  &amp; \dots(c)\\ \\
\left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma+\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
\right\}  &amp; \dots(d)
\end{matrix}&lt;/math&gt;

He noted that those quantities become the “frequently employed” formulas of Lagrange [i.e. the coefficients of the Euclidean orthogonal transformation first given by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]]] by setting:&lt;ref group=M&gt;Chasles (1829), p. 141&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\gamma\quad\Rightarrow\quad-\gamma\sqrt{-1}\\
\gamma'\quad\Rightarrow\quad-\gamma'\sqrt{-1}\\
\alpha''\quad\Rightarrow\quad\alpha''\sqrt{-1}\\
\beta''\quad\Rightarrow\quad\beta''\sqrt{-1}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Equations (a,b,c,d) are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).}}

Chasles now showed that equation systems (a,b,c,d) are of importance when discussing the relations between conjugate diameters of hyperboloids. He used the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid having the same principal axes (x,y,z), thus sharing the same conjugate axes, and having the common asymptotic cone &lt;math&gt;\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0&lt;/math&gt;. He then transformed those two hyperboloids to new axes (x',y',z') sharing the property of conjugacy:&lt;ref group=M&gt;Chasles (1829), pp. 143-144&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\
\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\
\hline \begin{align}x &amp; =lx'+l'y'+l''z'\\
y &amp; =mx'+m'y'+m''z'\\
z &amp; =nx'+n'y'+n''z'
\end{align}
\\
\left\{ \begin{align}\frac{ll'}{a{{}^2}}+\frac{mm'}{b^{2}}-\frac{nn'}{c^{2}} &amp; =0\\
\frac{ll''}{a{{}^2}}+\frac{mm''}{b^{2}}-\frac{nn''}{c^{2}} &amp; =0\\
\frac{l'l''}{a{{}^2}}+\frac{m'm''}{b^{2}}-\frac{n'n''}{c^{2}} &amp; =0
\end{align}
\right\} \\
\hline \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=1\\
\left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=-1
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Chasles defined the conditional equations of ''l,m,n'' in the same way as those of &lt;math&gt;\alpha,\beta,\gamma&lt;/math&gt; in equation system (b) above, so his transformation of x,y,z into x',y',z' represents Lorentz transformation ({{equationNote|1a}}, n=2) by applying equation system (a) as well.}}

He went on to use two semi-diameters of the one-sheet hyperboloid and one semi-diameter of the two-sheet hyperboloid in order to define equation system (A), and went on to suggest that the other equations related to this system can be obtained using the above transformation from oblique coordinates to other oblique ones, but he deemed it more simple to use a geometric argument to obtain system (B), which together with (A) then allowed him to algebraically determine systems (C), (D) and additional ones, leading Chasles to announce that “''from these formulas one can very easily conclude the various properties of conjugated diameters of hyperboloids''”:&lt;ref group=M&gt;Chasles (1829), pp. 145-146&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =a^{2}\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =b^{2}\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =-c^{2}
\end{align}
\right\}  &amp; \dots(A)\\
\left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma' &amp; =0
\end{align}
\right\}  &amp; \dots(B)\\
\left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =a^{2}\\
\beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} &amp; =b^{2}\\
\gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-c^{2}
\end{align}
\right\}  &amp; \dots(C)\\
\left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma+\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
\right\}  &amp; \dots(D)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Equation systems (A,B,C,D), being equivalent to systems (a,b,c,d) above, are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.}}

==={{anchor|Lebesgue}} Lebesgue (1837) – Homogeneous coordinates===

[[w:Victor-Amédée Lebesgue]] (1837) summarized the previous work of [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Cauchy|Cauchy (1829)]]. He started with the orthogonal transformation&lt;ref group=M&gt;Lebesgue (1837), pp. 338-341&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}\ (9)\\
\hline {\scriptstyle \begin{align}x_{1} &amp; =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\
x_{2} &amp; =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\
\dots\\
x_{n} &amp; =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\
\\
y_{1} &amp; =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n,1}x_{n}\\
y_{2} &amp; =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n,2}x_{n}\ (12)\ \\
\dots\\
y_{n} &amp; =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n,n}x_{n}
\end{align}
\left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n,\alpha}^{2} &amp; =1 &amp; (10)\\
a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n,\alpha}a_{n,\beta} &amp; =0 &amp; (11)\\
a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n}^{2} &amp; =1 &amp; (13)\\
a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n}a_{\beta,n} &amp; =0 &amp; (14)
\end{align}
\right.}
\end{matrix}&lt;/math&gt;

In order to achieve the invariance of the Lorentz interval&lt;ref group=M&gt;Lebesgue (1837), pp. 353–354&lt;/ref&gt;

:&lt;math&gt;x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}&lt;/math&gt;

he gave the following instructions as to how the previous equations shall be modified: In equation (9) change the sign of the last term of each member. In the first ''n-1'' equations of (10) change the sign of the last term of the left-hand side, and in the one which satisfies α=''n'' change the sign of the last term of the left-hand side as well as the sign of the right-hand side. In all equations (11) the last term will change sign. In equations (12) the last terms of the right-hand side will change sign, and so will the left-hand side of the ''n''-th equation. In equations (13) the signs of the last terms of the left-hand side will change, moreover in the ''n''-th equation change the sign of the right-hand side. In equations (14) the last terms will change sign.

{{Lorentzbox|Text=These instructions give Lorentz transformation ({{equationNote|1a}}) in the form:
:&lt;math&gt;{\scriptstyle \begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\
\hline \begin{align}x_{1} &amp; =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\
x_{2} &amp; =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\
\dots\\
x_{n} &amp; =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\
\\
y_{1} &amp; =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n-1,1}x_{n-1}-a_{n,1}x_{n}\\
y_{2} &amp; =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n-1,2}x_{n-1}-a_{n,2}x_{n}\\
\dots\\
-y_{n} &amp; =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n-1,n}x_{n-1}-a_{n,n}x_{n}
\end{align}
\left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n-1,\alpha}^{2}-a_{n,\alpha}^{2} &amp; =1\\
a_{1,n}^{2}+a_{2,n}^{2}+\dots+a_{n-1,n}^{2}-a_{n,n}^{2} &amp; =-1\\
a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n-1,\alpha}a_{n-1,\beta}-a_{n,\alpha}a_{n,\beta} &amp; =0\\
a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n-1}^{2}-a_{\alpha,n}^{2} &amp; =1\\
a_{n,1}^{2}+a_{n,2}^{2}+\dots+a_{n,n-1}^{2}-a_{n,n}^{2} &amp; =-1\\
a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n-1}a_{\beta,n-1}-a_{\alpha,n}a_{\beta,n} &amp; =0
\end{align}
\right.
\end{matrix}}&lt;/math&gt;}}

He went on to redefine the variables of the Lorentz interval and its transformation:&lt;ref group=M&gt;Lebesgue (1837), pp. 353–355&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\
\downarrow\\
\begin{align}x_{1} &amp; =x_{n}\cos\theta_{1}, &amp; x_{2} &amp; =x_{n}\cos\theta_{2},\dots &amp; x_{n-1} &amp; =x_{n}\cos\theta_{n-1}\\
y_{1} &amp; =y_{n}\cos\phi_{1}, &amp; y_{2} &amp; =y_{n}\cos\phi_{2},\dots &amp; y_{n-1} &amp; =y_{n}\cos\phi_{n-1}
\end{align}
\\
\downarrow\\
\cos^{2}\theta_{1}+\cos^{2}\theta_{2}+\dots+\cos^{2}\theta_{n-1}=1\\
\cos^{2}\phi_{1}+\cos^{2}\phi_{2}+\dots+\cos^{2}\phi_{n-1}=1\\
\hline \\
\cos\theta_{i}=\frac{a_{i,1}\cos\phi_{1}+a_{i,2}\cos\phi_{2}+\dots+a_{i,n-1}\cos\phi_{n-1}+a_{i,n}}{a_{n,1}\cos\phi_{1}+a_{n,2}\cos\phi_{2}+\dots+a_{n,n-1}\cos\phi_{n-1}+a_{n,n}}\\
(i=1,2,3\dots n)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Setting &lt;math&gt;[\cos\theta_{i},\ \cos\phi_{i}]=\left[u_{s},\ u_{s}^{\prime}\right]&lt;/math&gt; it is equivalent to Lorentz transformation ({{equationNote|1b}}).}}

==={{anchor|Weddle}} Weddle (1847) – Conjugate hyperboloids===
Very similar to [[#Chasles|Chasles (1829)]], though without reference to him, [[w:Thomas Weddle]] discussed conjugate hyperboloids using the following equation system (&amp;alpha;), from which he derived equations (&amp;beta;) and others:&lt;ref group=M&gt;Weddle (1847), p. 274&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} &amp; =1, &amp; l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} &amp; =0\\
l_{2}^{2}+m_{2}^{2}-n_{2}^{2} &amp; =1, &amp; l_{1}l_{3}+m_{1}m_{3}-n_{1}n_{3} &amp; =0\\
l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =-1, &amp; l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} &amp; =0
\end{align}
\right\}  &amp; \dots(\alpha)\\ \\
\left.\begin{align}l_{1}^{2}+l_{2}^{2}-l_{3}^{2} &amp; =1, &amp; l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} &amp; =0\\
m_{1}^{2}+m_{2}^{2}-m_{3}^{2} &amp; =1, &amp; l_{1}n_{1}+l_{2}n_{2}-l_{3}n_{3} &amp; =0\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =-1, &amp; m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} &amp; =0
\end{align}
\right\}  &amp; \dots(\beta)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).}}

Using the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid sharing the same conjugate axes, and having the common asymptotic cone &lt;math&gt;\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0&lt;/math&gt;, he defined three conjugate points &lt;math&gt;(x_{1}\dots,y_{1}\dots,z_{1}\dots)&lt;/math&gt; on those two conjugate hyperboloids, related to each other in the same way as equations (&amp;alpha;, &amp;beta;) stated above:&lt;ref group=M&gt;Weddle (1847), pp. 275-276&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\
\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\
\hline \begin{align}\frac{x_{1}x_{2}}{a{{}^2}}+\frac{y_{1}y_{2}}{b^{2}}-\frac{z_{1}z_{2}}{c^{2}} &amp; =0\\
\frac{x_{1}x_{3}}{a{{}^2}}+\frac{y_{1}y_{3}}{b^{2}}-\frac{z_{1}z_{3}}{c^{2}} &amp; =0\\
\frac{x_{2}x_{3}}{a{{}^2}}+\frac{y_{2}y_{3}}{b^{2}}-\frac{z_{2}z_{3}}{c^{2}} &amp; =0
\end{align}
\quad\begin{align}\frac{x_{1}^{2}}{a{{}^2}}+\frac{y_{1}^{2}}{b^{2}}-\frac{z_{1}^{2}}{c^{2}} &amp; =1\\
\frac{x_{2}^{2}}{a{{}^2}}+\frac{y_{2}^{2}}{b^{2}}-\frac{z_{2}^{2}}{c^{2}} &amp; =1\\
\frac{x_{3}^{2}}{a{{}^2}}+\frac{y_{3}^{2}}{b^{2}}-\frac{z_{3}^{2}}{c^{2}} &amp; =-1
\end{align}
\\
\begin{align}x_{1}^{2}+x_{2}^{2}-x_{3}^{2} &amp; =a^{2}\\
y_{1}^{2}+y_{2}^{2}-y_{3}^{2} &amp; =b^{2}\\
z_{1}^{2}+z_{2}^{2}-z_{3}^{2} &amp; =-c^{2}
\end{align}
\quad\begin{align}x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3} &amp; =0\\
x_{1}z_{1}+x_{2}z_{2}-x_{3}z_{3} &amp; =0\\
y_{1}z_{1}+y_{2}z_{2}-y_{3}z_{3} &amp; =0
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text= These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.}}

==={{anchor|Bour}} Bour (1856) – Homogeneous coordinates===

Following [[#Gauss4|Gauss (1818)]], [[w:Edmond Bour]] (1856) wrote the transformations:&lt;ref group=M&gt;Bour (1856), pp. 61; 64–65&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\cos^{2}E+\sin^{2}E-1=k\left(\cos^{2}T+\sin^{2}T-1\right)\\
\hline \left.\begin{matrix}\mathbf{(1)}\ \begin{align}\cos E &amp; =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\
\sin E &amp; =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}
\end{align}
\\
\hline \\
k=+1\\
t=\gamma+\gamma'\cos T+\gamma''\sin T,\\
1=u,\ \cos T=u',\ \sin T=u',\\
t=z,\ t\cos E=x,\ t\sin E=y\\
\downarrow\\
\mathbf{(2)}\begin{align}x &amp; =\alpha u+\alpha'u'+\alpha''u''\\
y &amp; =\beta u+\beta'u'+\beta''u''\\
z &amp; =\gamma u+\gamma'u'+\gamma''u''\\
\\
u &amp; =\gamma z-\alpha x-\beta y\\
u' &amp; =\alpha'x+\beta'y'-\gamma'z\\
u'' &amp; =\alpha''x+\beta''y-\gamma''z
\end{align}
\end{matrix}\right|{\scriptstyle \begin{align}-\alpha^{2}-\beta^{2}+\gamma^{2} &amp; =k\\
-\alpha^{\prime2}-\beta^{\prime2}+\gamma^{\prime2} &amp; =-k\\
-\alpha^{\prime\prime2}-\beta^{\prime\prime2}+\gamma^{\prime\prime2} &amp; =-k\\
\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\\
\alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =-k\\
\beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} &amp; =-k\\
\gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =k\\
\beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0\\
\alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0
\end{align}
}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'', implying &lt;math&gt;x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}&lt;/math&gt;. Furthermore, setting &lt;math&gt;[k,\cos T,\sin T,\cos E,\sin E]=\left[1,u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]&lt;/math&gt; in transformation system (1) produces Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.}}

=== {{anchor|Somov}} Somov (1863) – Homogeneous coordinates ===

Following [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], and [[#Bour|Bour (1856)]], [[w:Osip Ivanovich Somov]] (1863) wrote the transformation systems:&lt;ref group=M&gt;Somov (1863), pp. 12–14; p. 18 for differentials.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}\cos\phi &amp; =\frac{m\cos\psi+n\sin\psi+s}{m''\cos\psi+n''\sin\psi+s''}\\
\sin\phi &amp; =\frac{m'\cos\psi+n'\sin\psi+s'}{m''\cos\psi+n''\sin\psi+s''}
\end{align}
\right|\begin{matrix}\cos^{2}\phi+\cos^{2}\phi=1\\
\cos^{2}\psi+\cos^{2}\psi=1
\end{matrix}\\
\hline \mathbf{(1)}\ \begin{align}\cos\phi &amp; =x, &amp; \cos\psi &amp; =x'\\
\sin\phi &amp; =y, &amp; \sin\psi &amp; =y'
\end{align}
\ \left|\begin{align}x &amp; =\frac{mx'+ny'+s}{m''x'+n''y'+s''}\\
y &amp; =\frac{m'x'+n'y'+s'}{m''x'+n''y'+s''}
\end{align}
\right|\ \begin{matrix}x^{2}+y^{2}=1\\
x^{\prime2}+y^{\prime2}=1
\end{matrix}\\
\hline \begin{align}\cos\phi &amp; =\frac{x}{z}, &amp; \cos\psi &amp; =\frac{x'}{z'}\\
\sin\phi &amp; =\frac{y}{z}, &amp; \sin\psi &amp; =\frac{y'}{z'}
\end{align}
\ \left|\begin{align}\frac{x}{z} &amp; =\frac{mx'+ny'+sz'}{m''x'+n''y'+s''z'}\\
\frac{y}{z} &amp; =\frac{m'x'+n'y'+s'z'}{m''x'+n''y'+s''z'}
\end{align}
\right|\ \begin{matrix}x^{2}+y^{2}=z^{2}\\
x^{\prime2}+y^{\prime2}=z^{\prime2}
\end{matrix}\\
\hline \mathbf{(2)}\ \left.\begin{align}x &amp; =mx'+ny'+sz'\\
y &amp; =m'x'+n'y'+s'z'\\
z &amp; =m''x'+n''y'+s''z'\\
\\
x' &amp; =mx+m'y-m''z\\
y' &amp; =nx+n'y-n''z\\
z' &amp; =-sx-s'y+s''z\\
\\
dx &amp; =mdx'+ndy'+sdz'\\
dy &amp; =m'dx'+n'dy'+s'dz'\\
dz &amp; =m''dx'+n''dy'+s''dz'
\end{align}
\right|{\scriptstyle \begin{align}m^{2}+m^{\prime2}-m^{\prime\prime2} &amp; =1\\
n^{2}+n^{\prime2}-n^{\prime\prime2} &amp; =1\\
-s^{2}-s^{\prime2}+s^{\prime\prime2} &amp; =1\\
ns+n's'-n''s'' &amp; =0\\
sm+s'm'-s''m'' &amp; =0\\
mn+m'n'-m''n'' &amp; =0\\
\\
m^{2}+n^{2}-s^{2} &amp; =1\\
m^{\prime2}+n^{\prime2}-s^{\prime2} &amp; =1\\
-m^{\prime\prime2}-n^{\prime\prime2}+s^{\prime\prime2} &amp; =1\\
-m'm''-n'n''+s's'' &amp; =0\\
-m''m-n''n+s''s &amp; =0\\
mm'+nn'-ss' &amp; =0
\end{align}
}\\
dx^{2}+dy^{2}-dz^{2}=dx^{\prime2}+dy^{\prime2}-dz^{\prime2}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.

Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

==={{anchor|Klein}} Klein (1871-73) – Cayley absolute and non-Euclidean geometry===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Klein|label 1=History of Lorentz transformations via Möbius transformations § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Klein3|label 1=History of Lorentz transformations via sphere transformations § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Noether|label 1=History of Lorentz transformations via Quaternions § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Klein|label 1=History of Lorentz transformations via squeeze mappings § Klein}}

Elaborating on [[w:Arthur Cayley]]'s (1859) definition of an &quot;absolute&quot; ([[w:Cayley–Klein metric]]), [[w:Felix Klein]] (1871) defined a &quot;fundamental [[w:conic section]]&quot; in order to discuss motions such as rotation and translation in the non-Euclidean plane.&lt;ref group=M&gt;Klein (1871), pp. 601–602&lt;/ref&gt; This was elaborated in (1873) when he pointed out that hyperbolic geometry in terms of a surface of constant negative curvature can be related to a quadratic equation, which can be transformed into a sum of squares of which one square has a different sign, and can also be related to the interior of a surface of second degree corresponding to a two-sheet [[w:hyperboloid]].&lt;ref group=M&gt;Klein (1873), pp. 127-128&lt;/ref&gt;

{{Lorentzbox|Text=Klein's representation of hyperbolic space in terms of a two-sheet hyperboloid and its accompanied quadratic form suggests that Lorentz transformations can be geometrically interpreted as motions or isometries in hyperbolic space.}}

==={{anchor|Killing}} Killing (1878–1893)===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Killing2|label 1=History of Lorentz transformations via hyperbolic functions § Killing}}
===={{anchor|Killing1}} Weierstrass coordinates====

[[w:Wilhelm Killing]] (1878–1880) described non-Euclidean geometry by using [[w:hyperboloid model|Weierstrass coordinates]] (named after [[w:Karl Weierstrass]] who described them in lectures in 1872 which Killing attended) obeying the form

:&lt;math&gt;k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}&lt;/math&gt;&lt;ref group=M&gt;Killing (1877/78), p. 74; Killing (1880), p. 279&lt;/ref&gt; with &lt;math&gt;ds^{2}=k^{2}dt^{2}+du^{2}+dv^{2}+dw^{2}&lt;/math&gt;&lt;ref group=M&gt;Killing (1880), eq. 25 on p. 283&lt;/ref&gt;

or&lt;ref group=M&gt;Killing (1880), p. 283&lt;/ref&gt;

:&lt;math&gt;k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}&lt;/math&gt;

where ''k'' is the reciprocal measure of curvature, &lt;math&gt;k^{2}=\infty&lt;/math&gt; denotes [[w:Euclidean geometry]], &lt;math&gt;k^{2}&gt;0&lt;/math&gt; [[w:elliptic geometry]], and &lt;math&gt;k^{2}&lt;0&lt;/math&gt; hyperbolic geometry. In (1877/78) he pointed out the possibility and some characteristics of a transformation (indicating rigid motions) preserving the above form.&lt;ref group=M&gt;Killing (1877/78), eq. 25 on p. 283&lt;/ref&gt; In (1879/80) he tried to formulate the corresponding transformations by plugging &lt;math&gt;k^{2}&lt;/math&gt; into a [[w:Rotation matrix#Rotation matrix from axis and angle|general rotation matrix]]:&lt;ref group=M&gt;Killing (1879/80), p. 274&lt;/ref&gt;

&lt;math&gt;\begin{matrix}k^{2}u^{2}+v^{2}+w^{2}=k^{2}\\
\hline \begin{matrix}\cos\eta\tau+\lambda^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \nu\frac{\sin\eta\tau}{\eta}+\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; -\mu\frac{\sin\eta\tau}{\eta}+\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}\\
-k^{2}\nu\frac{\sin\eta\tau}{\eta}+k^{2}\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \cos\eta\tau+\mu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}\\
k^{2}\mu\frac{\sin\eta\tau}{\eta}+k^{2}\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; -\lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \cos\eta\tau+\nu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}
\end{matrix}\\
\left(\lambda^{2}+k^{2}\mu^{2}+k^{2}\nu^{2}=\eta^{2}\right)
\end{matrix}&lt;/math&gt;

In (1885) he wrote the Weierstrass coordinates and their transformation as follows:&lt;ref group=M&gt;Killing (1885), pp. 18, 28–30, 53&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}p^{2}+x^{2}+y^{2}=k^{2}\\
k^{2}p^{2}+x^{2}+y^{2}=k^{2}p^{\prime2}+x^{\prime2}+y^{\prime2}\\
ds^{2}=k^{2}dp^{2}+dx^{2}+dy^{2}\\
\hline \begin{align}k^{2}p' &amp; =k^{2}wp+w'x+w''y\\
x' &amp; =ap+a'x+a''y\\
y' &amp; =bp+b'x+b''y\\
\\
k^{2}p &amp; =k^{2}wp'+ax'+by'\\
x &amp; =w'p'+a'x+b'y'\\
y &amp; =w''p'+a''x'+b''y'
\end{align}
\left|{\scriptstyle \begin{align}k^{2}w^{2}+w^{\prime2}+w^{\prime\prime2} &amp; =k^{2}\\
\frac{a^{2}}{k^{2}}+a^{\prime2}+a^{\prime\prime2} &amp; =1\\
\frac{b^{2}}{k^{2}}+b^{\prime2}+b^{\prime\prime2} &amp; =1\\
aw+a'w'+a''w'' &amp; =0\\
bw+b'w'+b''w'' &amp; =0\\
\frac{ab}{k^{2}}+a'b'+a''b'' &amp; =0\\
\\
k^{2}w^{2}+a^{2}+b^{2} &amp; =k^{2}\\
\frac{w^{\prime2}}{k^{2}}+a^{\prime2}+b^{\prime2} &amp; =1\\
\frac{w^{\prime\prime2}}{k^{2}}+a^{\prime\prime2}+b^{\prime\prime2} &amp; =1\\
ww'+aa'+bb' &amp; =0\\
ww''+aa''+bb'' &amp; =0\\
\frac{w'w''}{k^{2}}+a'a''+b'b'' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

In (1885) he also gave the transformation for ''n'' dimensions:&lt;ref group=M&gt;Killing (1884/85), pp. 42–43; Killing (1885), pp. 73–74, 222&lt;/ref&gt;&lt;ref&gt;Ratcliffe (1994), § 3.6&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\
ds^{2}=k^{2}dx_{0}^{2}+dx_{1}^{2}+\dots+dx_{n}^{2}\\
\hline \left.\begin{align}k^{2}\xi_{0} &amp; =k^{2}a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{0}\\
\xi_{\varkappa} &amp; =a_{\varkappa0}x_{0}+a_{\varkappa1}x_{1}+\dots+a_{\varkappa n}x_{n}\\
\\
k^{2}x_{0} &amp; =a_{00}k^{2}\xi_{0}+a_{10}\xi_{1}+\dots+a_{n0}\xi_{n}\\
x_{\varkappa} &amp; =a_{0\varkappa}\xi_{0}+a_{1\varkappa}\xi_{1}+\dots+a_{n\varkappa}\xi_{n}
\end{align}
\right|{\scriptstyle \begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} &amp; =k^{2}\\
a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} &amp; =0\\
\frac{a_{0\iota}a_{0\varkappa}}{k^{2}}+a_{0\iota}a_{1\varkappa}+\dots+a_{n\iota}a_{n\varkappa}=\delta_{\iota\kappa} &amp; =1\ (\iota=\kappa)\ \text{or}\ 0\ (\iota\ne\kappa)
\end{align}
}
\end{matrix}&lt;/math&gt;

In (1885) he applied his transformations to mechanics and defined four-dimensional vectors of velocity and force.&lt;ref group=M&gt;Killing (1884/85), pp. 4–5&lt;/ref&gt; Regarding the geometrical interpretation of his transformations, Killing argued in (1885) that by setting &lt;math&gt;k^{2}=-1&lt;/math&gt; and using ''p,x,y'' as rectangular space coordinates, the hyperbolic plane is mapped on one side of a two-sheet hyperboloid &lt;math&gt;p^{2}-x^{2}-y^{2}=1&lt;/math&gt; (known as [[w:hyperboloid model]]),&lt;ref group=M&gt;Killing (1885), Note 9 on p. 260&lt;/ref&gt;&lt;ref name=rey /&gt; by which the previous formulas become equivalent to Lorentz transformations and the geometry becomes that of Minkowski space.

{{Lorentzbox|Text=All of Killing's transformations between 1879 and 1885 don't work when &lt;math&gt;k^{2}&lt;/math&gt; is negative, thus they fail to produce Lorentz transformation ({{equationNote|1a}}) with &lt;math&gt;k^{2}=-1&lt;/math&gt;.}}

Finally, in (1893) he wrote:&lt;ref group=M&gt;Killing (1893), see pp. 144, 327–328&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}t^{2}+u^{2}+v^{2}=k^{2}\\
\hline \begin{align}t' &amp; =at+bu+cv\\
u' &amp; =a't+b'u+c'v\\
v' &amp; =a''t+b''u+c''v
\end{align}
\left|\begin{align}k^{2}a^{2}+a^{\prime2}+a^{\prime\prime2} &amp; =k^{2}\\
k^{2}b^{2}+b^{\prime2}+b^{\prime\prime2} &amp; =1\\
k^{2}c^{2}+b^{\prime2}+c^{\prime\prime2} &amp; =1\\
k^{2}ab+a'b'+a''b'' &amp; =0\\
k^{2}ac+a'c'+a''c'' &amp; =0\\
k^{2}bc+b'c'+b''c'' &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

and in ''n'' dimensions&lt;ref group=M&gt;Killing (1893), pp. 314–316, 216–217&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\
k^{2}y_{0}y_{0}^{\prime}+y_{1}y_{1}^{\prime}+\cdots+y_{n}y_{n}^{\prime}=k^{2}x_{0}x_{0}^{\prime}+x_{1}x_{1}^{\prime}+\cdots+x_{n}x_{n}^{\prime}\\
ds^{2}=k^{2}dx_{0}^{2}+\dots+dx_{n}^{2}\\
\hline \begin{align}y_{0} &amp; =a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{n}\\
y_{1} &amp; =a_{10}x_{0}+a_{11}x_{1}+\dots+a_{1n}x_{n}\\
 &amp; \,\,\,\vdots\\
y_{n} &amp; =a_{n0}x_{0}+a_{n1}x_{1}+\dots+a_{nn}x_{n}
\end{align}
\left|\begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} &amp; =k^{2}\\
k^{2}a_{0\varkappa}^{2}+a_{1\varkappa}^{2}+\dots+a_{n\varkappa}^{2} &amp; =1\\
k^{2}a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} &amp; =0\\
k^{2}a_{0\varkappa}a_{0\lambda}+a_{1\varkappa}a_{1\lambda}+\dots+a_{n\varkappa}a_{n\lambda} &amp; =0\\
(\varkappa,\lambda=1,\dots, n,\ \lambda\lessgtr\varkappa)
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) with &lt;math&gt;k^{2}=-1&lt;/math&gt;.}}

===={{anchor|Killing3}} Infinitesimal transformations and Lie group====

After [[#Lie3|Lie (1885/86)]] identified the projective group of a general surface of second degree &lt;math&gt;\sum f_{ik}x_{i}'x_{k}'=0&lt;/math&gt; with the group of non-Euclidean motions, Killing (1887/88)&lt;ref group=M&gt;Killing (1887/88a), pp. 274–275&lt;/ref&gt; defined the infinitesimal projective transformations (Lie algebra) in relation to the unit hypersphere:

:&lt;math&gt;\begin{matrix}x_{1}^{2}+\dots+x_{m+1}^{2}=1\\
\hline X_{\iota\varkappa}f=x_{i}\frac{\partial f}{\partial x_{\varkappa}}-x_{\varkappa}\frac{\partial f}{\partial x_{\iota}}\\
\text{where}\\
\left(X_{\iota\varkappa},X_{\iota\lambda}\right)=X_{\varkappa\lambda};\ \left(X_{\iota\varkappa},X_{\lambda\mu}\right)=0;\\
\left[\iota\ne\varkappa\ne\lambda\ne\mu\right]
\end{matrix}&lt;/math&gt;

and in (1892) he defined the infinitesimal transformation for non-Euclidean motions in terms of Weierstrass coordinates:&lt;ref group=M&gt;Killing (1892), p. 177&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\
\hline X_{\iota\varkappa}=x_{\iota}p_{\varkappa}-x_{\varkappa}p_{\iota},\quad X_{\iota}=x_{0}p_{\iota}-\frac{x_{\iota}p_{0}}{k^{2}}\\
\text{where}\\
\left(X_{\iota}X_{\iota\varkappa}\right)=X_{\varkappa}f;\ \left(X_{\iota}X_{\varkappa\lambda}\right)=0;\ \left(X_{\iota}X_{\varkappa}\right)=-\frac{1}{k^{2}}X_{\iota\varkappa}f;
\end{matrix}&lt;/math&gt;

In (1897/98) he showed the relation between Weierstrass coordinates &lt;math&gt;k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}&lt;/math&gt; and coordinates &lt;math&gt;k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0&lt;/math&gt; used by himself in (1887/88) and by [[#Lie3|Werner (1889), Lie (1890)]]:&lt;ref group=M&gt;Killing (1897/98), pp. 255–256&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2} &amp; (a)\\
k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2} &amp; (b)
\end{matrix}\\
\hline V_{\varkappa}=k^{2}x_{0}p_{\varkappa}-x_{\varkappa}p_{0},\quad U_{\iota\varkappa}=p_{\iota}x_{\varkappa}-p_{\varkappa}x_{\iota}\\
\text{where}\\
\left(V_{\iota},V_{\varkappa}\right)=k^{2}U_{\iota\varkappa},\ \left(V_{\iota},U_{\iota\varkappa}\right)=-V_{\varkappa},\ \left(V_{\iota},U_{\varkappa\lambda}\right)=0,\\
\left(U_{\iota\varkappa},U_{\iota\lambda}\right)=U_{\varkappa\lambda},\ \left(U_{\iota\varkappa},U_{\lambda\mu}\right)=0\\
\left[\iota,\varkappa,\lambda,\mu=1,2,\dots n\right]\\
\hline \begin{matrix}y_{1}=\frac{x_{1}}{x_{0}},\ y_{2}=\frac{x_{2}}{x_{0}},\dots y_{n}=\frac{x_{n}}{x_{0}}\\
\downarrow\\
k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0\\
\hline q_{\varkappa}+\frac{y_{\varkappa}}{k^{2}}\sum_{\varrho}y_{y}q_{\varrho},\quad q_{\iota}y_{\varkappa}-q_{\varkappa}y_{\iota}
\end{matrix}
\end{matrix}&lt;/math&gt;

He pointed out that the corresponding group of non-Euclidean motions in terms of Weierstrass coordinates is intransitive when related to quadratic form (a) and [[w:Group action (mathematics)|transitive]] when related to quadratic form (b).

{{Lorentzbox|Text=Setting &lt;math&gt;k^{2}=-1&lt;/math&gt; denotes the group of hyperbolic motions and thus the Lorentz group.}}

=== {{anchor|Poincare}} Poincaré (1881) – Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Poincare2|label 1=History of Lorentz transformations via Möbius transformations § Poincaré}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (velocity)#Poincare3|label 1=History of Lorentz transformations via velocity § Poincaré}}

[[w:Henri Poincaré]] (1881) connected the work of [[../Lorentz transformation (Cayley-Hermite)#Hermite|E:Hermite (1853)]] and [[../Lorentz transformation (Möbius)#Selling|E:Selling (1873)]] on indefinite quadratic forms with non-Euclidean geometry (Poincaré already discussed such relations in an unpublished manuscript in 1880).&lt;ref&gt;Gray (1997)&lt;/ref&gt; He used two indefinite ternary forms in terms of three squares and then defined them in terms of Weierstrass coordinates (without using that expression) connected by a transformation with integer coefficients:&lt;ref group=M name=p1&gt;Poincaré (1881a), pp. 133–134&lt;/ref&gt;&lt;ref&gt;Dickson (1923), pp. 220–221&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}F &amp; =(ax+by+cz)^{2}+(a'x+b'y+c'z)^{2}-(a''x+b''y+c''z)^{2}\\
 &amp; =\xi^{2}+\eta^{2}-\zeta^{2}=-1\\
F &amp; =(ax'+by'+cz')^{2}+(a'x'+b'y'+c'z')^{2}-(a''x'+b''y'+c''z')^{2}\\
 &amp; =\xi^{\prime2}+\eta^{\prime2}-\zeta^{\prime2}=-1
\end{align}
\\
\hline \begin{align}\xi' &amp; =\alpha\xi+\beta\eta+\gamma\zeta\\
\eta' &amp; =\alpha'\xi+\beta'\eta+\gamma'\zeta\\
\zeta' &amp; =\alpha''\xi+\beta''\eta+\gamma''\zeta
\end{align}
\left|\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =1\\
\beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} &amp; =1\\
\gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-1\\
\alpha\beta+\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma+\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

He went on to describe the properties of &quot;hyperbolic coordinates&quot;.&lt;ref group=M name=poinc&gt;Poincaré (1881b), p. 333&lt;/ref&gt;&lt;ref name=rey&gt;Reynolds (1993)&lt;/ref&gt; Poincaré mentioned the hyperboloid model also in (1887).&lt;ref group=M&gt;Poincaré (1887), p. 206&lt;/ref&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

=== {{anchor|Cox}} Cox (1881–1891) – Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Cox|label 1=History of Lorentz transformations via hyperbolic functions § Cox}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Cox2|label 1=History of Lorentz transformations via Quaternions § Cox}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Cox|label 1=History of Lorentz transformations via sphere transformations § Cox}}

[[w:Homersham Cox (mathematician)|Homersham Cox]] (1881/82) – referring to similar rectangular coordinates used by [[w:Christoph Gudermann|Gudermann]] (1830)&lt;ref name=guder group=M&gt;Gudermann (1830), §1–3, §18–19&lt;/ref&gt; and [[w:George Salmon]] (1862)&lt;ref group=M&gt;Salmon (1862), section 212, p. 165&lt;/ref&gt; on a sphere, and to [[#Escherich|Escherich (1874)]] as reported by [[w:Johannes Frischauf]] (1876)&lt;ref group=M&gt;Frischauf (1876), pp. 86–87&lt;/ref&gt; in the hyperbolic plane – defined the Weierstrass coordinates (without using that expression) and their transformation:&lt;ref group=M&gt;Cox (1881/82), p. 186 for Weierstrass coordinates; pp. 193–194 for Lorentz transformation.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}z^{2}-x^{2}-y^{2}=1\\
x^{2}-y^{2}-z^{2}=Z^{2}-Y^{2}-X^{2}\\
\hline \begin{align}x &amp; =l_{1}X+l_{2}Y+l_{3}Z\\
y &amp; =m_{1}X+m_{2}Y+m_{3}Z\\
z &amp; =n_{1}X+n_{2}Y+n_{3}Z\\
\\
X &amp; =l_{1}x+m_{1}y-n_{1}z\\
Y &amp; =l_{2}x+m_{2}y-n_{2}z\\
Z &amp; =l_{3}x+m_{3}y-n_{3}z
\end{align}
\left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} &amp; =1\\
l_{2}^{2}+m_{2}^{2}-n_{2}^{2} &amp; =1\\
l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =1\\
l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} &amp; =0\\
l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} &amp; =0\\
l_{3}l_{1}+m_{3}m_{1}-n_{3}n_{1} &amp; =0\\
\\
l_{1}^{2}+l_{2}^{2}-l_{3}^{2} &amp; =1\\
m_{1}^{2}+m_{2}^{2}-m_{3}^{2} &amp; =1\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =1\\
l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} &amp; =0\\
m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} &amp; =0\\
n_{1}l_{1}+n_{2}l_{2}-n_{3}l_{3} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=These equations contain several errors or misprints: &lt;math&gt;Z^{2}-Y^{2}-X^{2}&lt;/math&gt; has to be replaced by &lt;math&gt;X^{2}-Y^{2}-Z^{2}&lt;/math&gt;, and &lt;math&gt;{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =1\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =1
\end{align}
}&lt;/math&gt; replaced with &lt;math&gt;{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =-1\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =-1
\end{align}
}&lt;/math&gt;, and by reversing the sign of &lt;math&gt;Z&lt;/math&gt; in the inverse transformation, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

Cox (1881/82) also gave the Weierstrass coordinates and their transformation in hyperbolic space:&lt;ref group=M&gt;Cox (1881/82), pp. 199, 206–207&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}w^{2}-x^{2}-y^{2}-z^{2}=1\\
w^{2}-x^{2}-y^{2}-z^{2}=w^{\prime2}-x^{\prime2}-y^{\prime2}-z^{\prime2}\\
\hline \begin{align}x &amp; =l_{1}x'+l_{2}y'+l_{3}z'-l_{4}w'\\
y &amp; =m_{1}x'+m_{2}y'+m_{3}z'-m_{4}w'\\
z &amp; =n_{1}x'+n_{2}y'+n_{3}z'-n_{4}w'\\
w &amp; =r_{1}x'+r_{2}y'+r_{3}z'-r_{4}w'\\
\\
x' &amp; =l_{1}x+m_{1}y+n_{1}z-r_{1}w\\
y' &amp; =l_{2}x+m_{2}y+n_{2}z-r_{2}w\\
z' &amp; =l_{3}x+m_{3}y+n_{3}z-r_{3}w\\
w' &amp; =l_{4}x+m_{4}y+n_{4}z-r_{4}w
\end{align}
\left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}+n_{1}^{2}-r_{1}^{2} &amp; =1\\
l_{2}^{2}+m_{2}^{2}+n_{2}^{2}-r_{2}^{2} &amp; =1\\
l_{3}^{2}+m_{3}^{2}+n_{3}^{2}-r_{3}^{2} &amp; =1\\
l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2} &amp; =1\\
l_{2}l_{3}+m_{2}m_{3}+n_{2}n_{3}-r_{2}r_{3} &amp; =0\\
l_{3}l_{1}+m_{3}m_{1}+n_{3}n_{1}-r_{3}r_{1} &amp; =0\\
l_{1}l_{4}+m_{1}m_{4}+n_{1}n_{4}-r_{1}r_{4} &amp; =0\\
l_{2}l_{4}+m_{2}m_{4}+n_{2}n_{4}-r_{2}r_{4} &amp; =0\\
l_{3}l_{4}+m_{3}m_{4}+n_{3}n_{4}-r_{3}r_{4} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=By replacing &lt;math&gt;{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=1}&lt;/math&gt; with &lt;math&gt;{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=-1}&lt;/math&gt; this represents an improper antichronous Lorentz transformation, which becomes proper orthochronous Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' by reversing the sign of &lt;math&gt;w'&lt;/math&gt; everywhere.}}

In 1883 he formulated relations between [[w:orthogonal circles]] which he identified with the previously (1881/82) given transformations:&lt;ref group=M&gt;Cox (1883), pp. 109ff&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}+z^{2}-w^{2}=0\\
\hline \begin{align}x &amp; =\lambda_{1}X+\lambda_{2}Y+\lambda_{3}Z+\lambda_{4}W\\
y &amp; =\mu_{1}X+\mu_{2}Y+\mu_{3}Z+\mu_{4}W\\
z &amp; =\nu_{1}X+\nu_{2}Y+\nu_{3}Z+\nu_{4}W\\
-w &amp; =\rho_{1}X+\rho_{2}Y+\rho_{3}Z+\rho_{4}W\\
\\
X &amp; =\lambda_{1}x+\mu_{1}y+\nu_{1}z+\rho_{1}w\\
Y &amp; =\lambda_{2}x+\mu_{2}y+\nu_{2}z+\rho_{2}w\\
Z &amp; =\lambda_{3}x+\mu_{3}y+\nu_{3}z+\rho_{3}w\\
-W &amp; =\lambda_{4}x+\mu_{4}y+\nu_{4}z+\rho_{4}w
\end{align}
\left|{\scriptstyle \begin{align}\lambda_{1}^{2}+\mu_{1}^{2}+\nu_{1}^{2}-\rho_{1}^{2} &amp; =1\\
\lambda_{2}^{2}+\mu_{2}^{2}+\nu_{2}^{2}-\rho_{2}^{2} &amp; =1\\
\lambda_{3}^{2}+\mu_{3}^{2}+\nu_{3}^{2}-\rho_{3}^{2} &amp; =1\\
\lambda_{4}^{2}+\mu_{4}^{2}+\nu_{4}^{2}-\rho_{4}^{2} &amp; =-1\\
\lambda_{2}\lambda_{3}+\mu_{2}\mu_{3}+\nu_{2}\nu_{3}-\rho_{2}\rho_{3} &amp; =0\\
\lambda_{3}\lambda_{1}+\mu_{3}\mu_{1}+\nu_{3}\nu_{1}-\rho_{3}\rho_{1} &amp; =0\\
\lambda_{1}\lambda_{2}+\mu_{1}\mu_{2}+\nu_{1}\nu_{2}-\rho_{1}\rho_{2} &amp; =0\\
\lambda_{1}\lambda_{4}+\mu_{1}\mu_{4}+\nu_{1}\nu_{4}-\rho_{1}\rho_{4} &amp; =0\\
\lambda_{2}\lambda_{4}+\mu_{2}\mu_{4}+\nu_{2}\nu_{4}-\rho_{2}\rho_{4} &amp; =0\\
\lambda_{3}\lambda_{4}+\mu_{3}\mu_{4}+\nu_{3}\nu_{4}-\rho_{3}\rho_{4} &amp; =0
\end{align}
}\right.{\scriptstyle \begin{align}\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}-\lambda_{4}^{2} &amp; =1\\
\mu_{1}^{2}+\mu_{2}^{2}+\mu_{3}^{2}-\mu_{4}^{2} &amp; =1\\
\nu_{1}^{2}+\nu_{2}^{2}+\nu_{3}^{2}-\nu_{4}^{2} &amp; =1\\
\rho_{1}^{2}+\rho_{2}^{2}+\rho_{3}^{2}-\rho_{4}^{2} &amp; =-1\\
\lambda_{1}\mu_{1}+\lambda_{2}\mu_{2}+\lambda_{3}\mu_{3}-\lambda_{4}\mu_{4} &amp; =0\\
\lambda_{1}\nu_{1}+\lambda_{2}\nu_{2}+\lambda_{3}\nu_{3}-\lambda_{4}\nu_{4} &amp; =0\\
\lambda_{1}\rho_{1}+\lambda_{2}\rho_{2}+\lambda_{3}\rho_{3}-\lambda_{4}\rho_{4} &amp; =0\\
\mu_{1}\nu_{1}+\mu_{2}\nu_{2}+\mu_{3}\nu_{3}-\mu_{4}\nu_{4} &amp; =0\\
\mu_{1}\rho_{1}+\mu_{2}\rho_{2}+\mu_{3}\rho_{3}-\mu_{4}\rho_{4} &amp; =0\\
\nu_{1}\rho_{1}+\nu_{2}\rho_{2}+\nu_{3}\rho_{3}-\nu_{4}\rho_{4} &amp; =0
\end{align}
}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The relations between &lt;math&gt;\lambda,\mu,\nu,\rho&lt;/math&gt; are correct, even though the transformation still represents an improper antichronous Lorentz transformation, which becomes proper orthochronous Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' by reversing the sign of &lt;math&gt;w&lt;/math&gt; everywhere.}}

Finally, in a treatise on [[w:Hermann Grassmann|w:Grassmann's Ausdehnungslehre]] and circles (1891), he again provided transformations of orthogonal circle systems described by him as being &quot;identical with those for transformation of coordinates in non-Euclidean geometry&quot;:&lt;ref group=M&gt;Cox (1891), pp. 27-28&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}+z^{2}=w^{2}\\
\hline \begin{align}x &amp; =\lambda_{1}x'+\lambda_{2}y'+\lambda_{3}z'+\lambda_{4}w' &amp; \text{(4 equations)}\\
x' &amp; =\lambda_{1}x+\mu_{1}y+\nu_{1}z-\rho_{1}w\\
-w' &amp; =\lambda_{4}x+\mu_{4}y+\nu_{4}z-\rho_{4}w
\end{align}
\\
\hline \begin{align}\lambda_{1}^{2}+\mu_{1}^{2}+\nu_{1}^{2}-\rho_{1}^{2} &amp; =1\\
\lambda_{2}^{2}+\mu_{2}^{2}+\nu_{2}^{2}-\rho_{2}^{2} &amp; =1\\
\lambda_{3}^{2}+\mu_{3}^{2}+\nu_{3}^{2}-\rho_{3}^{2} &amp; =1\\
\lambda_{4}^{2}+\mu_{4}^{2}+\nu_{4}^{2}-\rho_{4}^{2} &amp; =-1\\
\lambda_{1}\lambda_{2}+\mu_{1}\mu_{2}+\nu_{1}\nu_{2}-\rho_{1}\rho_{2} &amp; =0 &amp; \text{(6 equations)}\\
\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}-\lambda_{4}^{2} &amp; =1\\
\rho_{1}^{2}+\rho_{2}^{2}+\rho_{3}^{2}-\rho_{4}^{2} &amp; =-1\\
\lambda_{1}\mu_{1}+\lambda_{2}\mu_{2}+\lambda_{3}\mu_{3}-\lambda_{4}\mu_{4} &amp; =0 &amp; \text{(6 equations)}
\end{align}
\end{matrix}\text{ }&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)''.}}

=== {{anchor|Hill}} Hill (1882) – Homogeneous coordinates ===

Following [[#Gauss4|Gauss (1818)]], [[w:George William Hill]] (1882) formulated the equations&lt;ref group=M&gt;Hill (1882), pp. 323–325&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)\\
k\left(\sin^{2}E+\cos^{2}E-1\right)\\
\hline \begin{align} &amp;  &amp; \cos E' &amp; =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\
 &amp; \mathbf{(1)} &amp; \sin E' &amp; =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\
\hline \\
 &amp;  &amp; x &amp; =\alpha u+\alpha'u'+\alpha''u''\\
 &amp;  &amp; y &amp; =\beta u+\beta'u'+\beta''u''\\
 &amp;  &amp; z &amp; =\gamma u+\gamma'u'+\gamma''u''\\
 &amp; \mathbf{(2)}\\
 &amp;  &amp; u &amp; =-\alpha x-\beta y+\gamma z\\
 &amp;  &amp; u' &amp; =\alpha'x+\beta'y'-\gamma'z\\
 &amp;  &amp; u'' &amp; =\alpha''x+\beta''y-\gamma''z
\end{align}
\left|{\scriptstyle \begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =-1\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =1\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =1\\
\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\\
(k=-1)\\
\alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =k\\
\beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} &amp; =k\\
\gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-k\\
\alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with &lt;math&gt;[\cos T,\sin T,\cos E',\sin E']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]&lt;/math&gt;.

Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' .}}

=== {{anchor|Picard}} Picard (1882-1884) – Quadratic forms ===

[[w:Émile Picard]] (1882) analyzed the invariance of indefinite ternary [[w:Hermitian form|Hermitian quadratic forms]] with integer coefficients and their relation to [[w:Group action (mathematics)|discontinuous groups]], extending Poincaré's Fuchsian functions of one complex variable related to a circle, to &quot;hyperfuchsian&quot; functions of two complex variables related to a [[w:hypersphere]]. He formulated the following special case of an Hermitian form:&lt;ref group=M&gt;Picard (1882), pp. 307–308 first transformation system; pp. 315-317 second transformation system&lt;/ref&gt;&lt;ref&gt;Dickson (1923), pp. 280-281&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{matrix}xx_{0}+yy_{0}-zz_{0}\\
\\
\mathbf{(1)}\ \begin{align}x &amp; =M_{1}X+P_{1}Y+R_{1}Z\\
y &amp; =M_{2}X+P_{2}Y+R_{2}Z\\
z &amp; =M_{3}X+P_{3}Y+R_{3}Z
\end{align}
\\
\\
\left[\begin{align}[][x,y,z]=\text{complex}\\
\left[x_{0},y_{0},z_{0}\right]=\text{conjugate}
\end{align}
\right]\\
\\
\hline \\
x^{\prime2}+x^{\prime\prime2}+y^{\prime2}+y^{\prime\prime2}=1\\
x=x'+ix'',\quad y=y'+iy''\\
\\
\mathbf{(2)}\ \begin{align}X &amp; =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\
Y &amp; =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}}
\end{align}
\end{matrix}\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3} &amp; =1\\
P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} &amp; =1\\
R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} &amp; =-1\\
P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3} &amp; =0\\
M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} &amp; =0\\
P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} &amp; =0\\
\\
M_{1}\mu_{1}+P_{1}\pi_{1}-R_{1}\rho_{1} &amp; =1\\
M_{2}\mu_{2}+P_{2}\pi_{2}-R_{2}\rho_{2} &amp; =1\\
M_{3}\mu_{3}+P_{3}\pi_{3}-R_{3}\rho_{3} &amp; =-1\\
\mu_{2}M_{1}+\pi_{2}P_{1}-R_{1}\rho_{2} &amp; =0\\
\mu_{2}M_{3}+\pi_{2}P_{3}-R_{3}\rho_{2} &amp; =0\\
\mu_{3}M_{1}+\pi_{3}P_{1}-R_{1}\rho_{3} &amp; =0\\
\\
\left[\begin{align}[][M,P,R\dots]=\text{complex}\\
\left[\mu,\pi,\rho\dots\right]=\text{conjugate}
\end{align}
\right]
\end{align}
}\right.\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;-z&lt;sup&gt;2&lt;/sup&gt;=X&lt;sup&gt;2&lt;/sup&gt;+Y&lt;sup&gt;2&lt;/sup&gt;-Z&lt;sup&gt;2&lt;/sup&gt;'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;=X&lt;sup&gt;2&lt;/sup&gt;+Y&lt;sup&gt;2&lt;/sup&gt;=1''.}}

Or in (1884a) in relation to indefinite binary Hermitian quadratic forms:&lt;ref group=M&gt;Picard (1884a), p. 13&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}UU_{0}-VV_{0}=uu_{0}-vv_{0}\\
\hline \begin{align}U &amp; =\mathcal{A}u+\mathcal{B}v\\
V &amp; =\mathcal{C}u+\mathcal{D}v
\end{align}
\left|\begin{align}\mathcal{A}\mathcal{A}_{0}-\mathcal{C}\mathcal{C}_{0} &amp; =1\\
\mathcal{A}\mathcal{B}_{0}-\mathcal{C}\mathcal{D}_{0} &amp; =0\\
\mathcal{B}\mathcal{B}_{0}-\mathcal{D}\mathcal{D}_{0} &amp; =-1\\
\mathcal{D}\mathcal{D}_{0}-\mathcal{C}\mathcal{C}_{0} &amp; =1
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' producing ''U&lt;sup&gt;2&lt;/sup&gt;-V&lt;sup&gt;2&lt;/sup&gt;=u&lt;sup&gt;2&lt;/sup&gt;-v&lt;sup&gt;2&lt;/sup&gt;''.}}

Or in (1884b):&lt;ref group=M&gt;Picard (1884b), p. 416&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}xx_{0}+yy_{0}-1=0\\
\hline \begin{align}X &amp; =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\
Y &amp; =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}}
\end{align}
\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3}=P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} &amp; =1\\
R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} &amp; =-1\\
P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3}=M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3}=P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} &amp; =0\\
M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;=X&lt;sup&gt;2&lt;/sup&gt;+Y&lt;sup&gt;2&lt;/sup&gt;=1''.}}

Or in (1884c):&lt;ref group=M&gt;Picard (1884c), pp. 123–124; 163&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}UU_{0}+VV_{0}-WW_{0}=uu_{0}+vv_{0}-ww_{0}\\
\hline \mathbf{(1)}\ \begin{align}U &amp; =Mu+Pv+Rw\\
V &amp; =M'u+P'v+R'w\\
W &amp; =M''u+P''v+R''w\\
\\
u &amp; =M_{0}U+M_{0}^{\prime}V-M_{0}^{\prime\prime}W\\
v &amp; =P_{0}U+P_{0}^{\prime}V-P_{0}^{\prime\prime}W\\
w &amp; =-R_{0}U-R_{0}^{\prime}V+R_{0}^{\prime\prime}W
\end{align}
\left|{\scriptstyle \begin{align}MM_{0}+M'M_{0}^{\prime}-M''M_{0}^{\prime\prime} &amp; =1\\
PP_{0}+P'P_{0}^{\prime}-P''P_{0}^{\prime\prime} &amp; =1\\
RR_{0}+R'R_{0}^{\prime}-R''R_{0}^{\prime\prime} &amp; =-1\\
MP_{0}+M'P_{0}^{\prime}-M''P_{0}^{\prime\prime} &amp; =0\\
MR_{0}+M'R_{0}^{\prime}-M''R_{0}^{\prime\prime} &amp; =0\\
PR_{0}+P'R_{0}^{\prime}-P''R_{0}^{\prime\prime} &amp; =0\\
\\
MM_{0}+PP_{0}-RR_{0} &amp; =1\\
M'M_{0}^{\prime}+P'P_{0}^{\prime}-R'R_{0}^{\prime} &amp; =1\\
M''M_{0}^{\prime\prime}+P''P_{0}^{\prime\prime}-R''R_{0}^{\prime\prime} &amp; =-1\\
M_{0}M'+P_{0}P'-R_{0}R' &amp; =0\\
M_{0}M''+P_{0}P''-R_{0}R'' &amp; =0\\
M_{0}^{\prime}M''+P_{0}^{\prime}P''-R_{0}^{\prime}R'' &amp; =0
\end{align}
}\right.\\
\hline \text{Invariance of unit hypersphere:}\\
\mathbf{(2)}\ \begin{align}\xi' &amp; =\frac{A\xi+A'\eta+A''}{C\xi+C'\eta+C''}\\
\eta' &amp; =\frac{B\xi+B'\eta+B''}{C\xi+C'\eta+C''}
\end{align}
\left|{\scriptstyle \begin{align}AA_{0}+A'A_{0}^{\prime}-A''A_{0}^{\prime\prime} &amp; =1\\
BB_{0}+B'B_{0}^{\prime}-B''B_{0}^{\prime\prime} &amp; =1\\
CC_{0}+C'C_{0}^{\prime}-C''C_{0}^{\prime\prime} &amp; =-1\\
AB_{0}+A'B_{0}^{\prime}-A''B_{0}^{\prime\prime} &amp; =0\\
AC_{0}+A'C_{0}^{\prime}-A''C_{0}^{\prime\prime} &amp; =0\\
BC_{0}+B'C_{0}^{\prime}-B''C_{0}^{\prime\prime} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''U&lt;sup&gt;2&lt;/sup&gt;+V&lt;sup&gt;2&lt;/sup&gt;-W&lt;sup&gt;2&lt;/sup&gt;=u&lt;sup&gt;2&lt;/sup&gt;+v&lt;sup&gt;2&lt;/sup&gt;-w&lt;sup&gt;2&lt;/sup&gt;'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing &lt;math&gt;\xi^{\prime2}+\eta^{\prime2}=\xi^{2}+\eta^{2}=1&lt;/math&gt;.}}

=== {{anchor|Callandreau}} Callandreau (1885) – Homography ===

Following [[#Gauss4|Gauss (1818)]] and [[#Hill|Hill (1882)]], [[w:Octave Callandreau]] (1885) formulated the equations&lt;ref group=M&gt;Callandreau (1885), pp. A.7; A.12&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)=\\
{\scriptstyle (\alpha+\alpha'\sin T+\alpha''\cos T)^{2}+(\beta+\beta'\sin T+\beta''\cos T)^{2}-(\gamma+\gamma'\sin T+\gamma''\cos T)^{2}}\\
\hline \begin{align}\cos\varepsilon' &amp; =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\
\sin\varepsilon' &amp; =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}
\end{align}
\left|{\scriptstyle \begin{align} &amp; \left(k=1\right)\\
\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =-k &amp; \alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =+k &amp; \alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =+k &amp; \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\\
\alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =-1 &amp; \alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} &amp; =-1 &amp; \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =+1 &amp; \beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The transformation system is equivalent to Lorentz transformation ({{equationNote|1b}})  ''(n=2)'' with &lt;math&gt;[\cos T,\sin T,\cos\varepsilon',\sin\varepsilon']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]&lt;/math&gt;.}}

==={{anchor|Lie3}} Lie (1885-1890) – Lie group, hyperbolic motions, and infinitesimal transformations===
{{See also|History of Topics in Special Relativity/Lorentz transformation (imaginary)#Lie|label 1=History of Lorentz transformations via imaginary orthogonal transformations § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Lie|label 1=History of Lorentz transformations via sphere transformations § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Lie2|label 1=History of Lorentz transformations via squeeze mappings § Lie}}

In (1885/86), [[w:Sophus Lie]] identified the projective group of a general surface of second degree &lt;math&gt;\sum f_{ik}x_{i}'x_{k}'=0&lt;/math&gt; with the group of non-Euclidean motions.&lt;ref group=M&gt;Lie (1885/86), p. 411&lt;/ref&gt; In a thesis guided by Lie, [[w:Hermann Werner]] (1889) discussed this projective group by using the equation of a unit hypersphere as the surface of second degree (which was already given before by [[#Killing3|Killing (1887)]]), and also gave the corresponding infinitesimal projective transformations (Lie algebra):&lt;ref group=M&gt;Werner (1889), pp. 4, 28&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=1\\
\hline x_{i}p_{\varkappa}-x_{\varkappa}p_{i},\quad p_{i}-x_{i}\sum_{1}^{n}{\scriptstyle j}\ x_{j}p_{j}\quad(i,\varkappa=1,\dots, n)\\
\text{where}\\
\left(Q_{i},Q_{\varkappa}\right)=R_{i,\varkappa};\ \left(Q_{i},Q_{j,\varkappa}\right)=\varepsilon_{i,j}Q_{\varkappa}-\varepsilon_{i,\varkappa}Q_{j};\\
\left(R_{i,\varkappa},R_{\mu,\nu}\right)=\varepsilon_{\varkappa,\mu}R_{i,\nu}-\varepsilon_{\varkappa,\nu}R_{i,\mu}-\varepsilon_{,\mu}R_{\varkappa,\nu}+\varepsilon_{i,\nu}R_{\varkappa,\mu}\\
\left[\varepsilon_{i,\varkappa}\equiv0\ \text{for}\ i\ne\varkappa;\ \varepsilon_{i,i}=1\right]
\end{matrix}&lt;/math&gt;

More generally, Lie (1890)&lt;ref group=M&gt;Lie (1890a), p. 295;&lt;/ref&gt; defined non-Euclidean motions in terms of two forms &lt;math&gt;x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\pm1=0&lt;/math&gt; in which the imaginary form with &lt;math&gt;+1&lt;/math&gt; denotes the group of elliptic motions (in Klein's terminology), the real form with −1 the group of hyperbolic motions, with the latter having the same form as Werner's transformation:&lt;ref group=M&gt;Lie (1890a), p. 311&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+\dots+x_{n}^{2}-1=0\\
\hline p_{k}-x_{k}\sum j_{1}^{0}x_{j}p_{j},\quad x_{i}p_{k}-x_{k}p_{i}\quad(i,k=1\dots n)
\end{matrix}&lt;/math&gt;

Summarizing, Lie (1893) discussed the real continuous groups of the conic sections representing non-Euclidean motions, which in the case of hyperbolic motions have the form:

:&lt;math&gt;x^{2}+y^{2}-1=0&lt;/math&gt;&lt;ref group=M&gt;Lie (1893), p. 474&lt;/ref&gt; or &lt;math&gt;x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-1=0&lt;/math&gt;&lt;ref group=M&gt;Lie (1893), p. 479&lt;/ref&gt; or &lt;math&gt;x_{1}^{2}+\dots+x_{n}^{2}-1=0&lt;/math&gt;.&lt;ref group=M&gt;Lie (1893), p. 481&lt;/ref&gt;

{{Lorentzbox|Text=The group of hyperbolic motions is isomorphic to the Lorentz group. The interval &lt;math&gt;x_{1}^{2}+\dots+x_{n}^{2}-1=0&lt;/math&gt; becomes the Lorentz interval &lt;math&gt;x_{1}^{2}+\dots+x_{n}^{2}-x_{0}^{2}=0&lt;/math&gt; by setting
&lt;math&gt;(x_{1},\dots,\ x_{n},\ 1)=\left(\frac{x_{1}}{x_{0}},\dots,\ \frac{x_{n}}{x_{0}},\ \frac{x_{0}}{x_{0}}\right)&lt;/math&gt;}}

==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Gerard|label 1=History of Lorentz transformations via hyperbolic functions § Gerard}}

[[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane using the following invariant and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)'':&lt;ref group=M&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}-Z^{2}=1\\
X^{2}+Y^{2}-Z^{2}=X^{\prime2}+Y^{\prime2}-Z^{\prime2}\\
\hline \begin{align}X &amp; =aX'+a'Y'+a''Z'\\
Y &amp; =bX'+b'Y'+b''Z'\\
Z &amp; =cX'+c'Y'+c''Z'\\
\\
X' &amp; =aX+bY-cZ\\
Y' &amp; =a'X+b'Y-c'Z\\
Z' &amp; =-a''X-b''Y+c''Z
\end{align}
\left|\begin{align}a^{2}+b^{2}-c^{2} &amp; =1\\
a^{\prime2}+b^{\prime2}-c^{\prime2} &amp; =1\\
a^{\prime\prime2}+b^{\prime\prime2}-c^{\prime\prime2} &amp; =-1\\
aa'+bb'-cc' &amp; =0\\
a'a''+b'b''-c'c'' &amp; =0\\
a''a+b''b-c''c &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

He gave the case of translation as follows:&lt;ref group=M name=gerard&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{align}X &amp; =Z_{0}X'+X_{0}Z'\\
Y &amp; =Y'\\
Z &amp; =X_{0}X'+Z_{0}Z'
\end{align}
\ \text{with}\ \begin{align}X_{0} &amp; =\operatorname{sh}OO'\\
Z_{0} &amp; =\operatorname{ch}OO'
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Hausdorff}} Hausdorff (1899) – Weierstrass coordinates===

[[w:Felix Hausdorff]] (1899) – citing Killing (1885) – discussed Weierstrass coordinates in the plane using the following invariant and its transformation:&lt;ref group=M&gt;Hausdorff (1899), p. 165, pp. 181-182&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}p^{2}-x^{2}-y^{2}=1\\
\hline \begin{align}x &amp; =a_{1}x'+a_{2}y'+x_{0}p'\\
y &amp; =b_{1}x'+b{}_{2}y'+y_{0}p'\\
p &amp; =e_{1}x'+e_{2}y'+p_{0}p'\\
\\
x' &amp; =a_{1}x+b_{1}y-e_{1}p\\
y' &amp; =a_{2}x+b_{2}y-e_{2}p\\
-p' &amp; =x_{0}x+y_{0}y-p_{0}p
\end{align}
\left|{\scriptstyle \begin{align}a_{1}^{2}+b_{1}^{2}-e_{1}^{2} &amp; =1\\
a_{2}^{2}+b_{2}^{2}-e_{2}^{2} &amp; =1\\
-x_{0}^{2}-y_{0}^{2}+p_{0}^{2} &amp; =1\\
a_{2}x_{0}+b_{2}y_{0}-e_{2}p_{0} &amp; =0\\
a_{1}x_{0}+b_{1}y_{0}-e_{1}p_{0} &amp; =0\\
a_{1}a_{2}+b_{1}b_{2}-e_{1}e_{2} &amp; =0\\
\\
a_{1}^{2}+a_{2}^{2}-x_{0}^{2} &amp; =1\\
b_{1}^{2}+b_{2}^{2}-y_{0}^{2} &amp; =1\\
-e_{1}^{2}-e_{2}^{2}+p_{0}^{2} &amp; =1\\
b_{1}e_{1}+b_{2}e_{2}-y_{0}p_{0} &amp; =0\\
a_{1}e_{1}+a_{2}e_{2}-x_{0}p_{0} &amp; =0\\
a_{1}b_{1}+a_{2}b_{2}-x_{0}y_{0} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

==={{anchor|Woods2}} Woods (1901-05) – Beltrami and Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Woods2|label 1=History of Lorentz transformations via hyperbolic functions § Woods}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Woods|label 1=History of Lorentz transformations via Möbius transformations § Woods}}

In (1901/02) [[w:Frederick S. Woods]] defined the following invariant quadratic form and its [[w:projective transformation]] in terms of Beltrami coordinates (he pointed out that this can be connected to hyperbolic geometry by setting &lt;math&gt;k=\sqrt{-1}R&lt;/math&gt; with ''R'' as real quantity):&lt;ref group=M&gt;Woods (1901/02), p. 98, 104&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}\left(u^{2}+v^{2}+w^{2}\right)+1=0\\
\hline \begin{align}u' &amp; =\frac{\alpha_{1}u+\alpha_{2}v+\alpha_{3}w+\alpha_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\
v' &amp; =\frac{\beta_{1}u+\beta_{2}v+\beta_{3}w+\beta_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\
w' &amp; =\frac{\gamma_{1}u+\gamma_{2}v+\gamma_{3}w+\gamma_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}
\end{align}
\left|\begin{align}k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right)+\delta_{i}^{2} &amp; =k^{2}\\
(i=1,2,3)\\
k^{2}\left(\alpha_{4}^{2}+\beta_{4}^{2}+\gamma_{4}^{2}\right)+\delta_{4}^{2} &amp; =1\\
\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}+\delta_{i}\delta_{h} &amp; =0\\
(i,h=1,2,3,4;\ i\ne h)
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'' with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.}}

Alternatively, Woods (1903, published 1905) – citing Killing (1885) – used the invariant quadratic form in terms of Weierstrass coordinates and its transformation (with &lt;math&gt;k=\sqrt{-1}k&lt;/math&gt; for hyperbolic space):&lt;ref group=M&gt;Woods (1903/05), pp. 45–46; p. 48)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{0}^{2}+k^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)=1\\
ds^{2}=\frac{1}{k^{2}}dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\\
\hline \begin{align}x_{1}^{\prime} &amp; =\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}+\alpha_{0}x_{0}\\
x_{2}^{\prime} &amp; =\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{0}x_{0}\\
x_{3}^{\prime} &amp; =\gamma_{1}x_{1}+\gamma_{2}x_{2}+\gamma_{3}x_{3}+\gamma_{0}x_{0}\\
x_{0}^{\prime} &amp; =\delta_{1}x_{1}+\delta_{2}x_{2}+\delta_{3}x_{3}+\delta_{0}x_{0}
\end{align}
\left|\begin{align}\delta_{0}^{2}+k^{2}\left(\alpha_{0}^{2}+\beta_{0}^{2}+\gamma_{0}^{2}\right) &amp; =1\\
\delta_{i}^{2}+k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right) &amp; =k^{2}\\
(i=1,2,3)\\
\delta_{i}\delta_{h}+k^{2}\left(\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}\right) &amp; =0\\
(i,h=0,1,2,3;\ i\ne h)
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.}}

==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Liebmann|label 1=History of Lorentz transformations via hyperbolic functions § Liebmann}}

[[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – used the invariant quadratic form and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)''&lt;ref group=M&gt;Liebmann (1904/05), p. 168; pp. 175–176&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}p^{\prime2}-x^{\prime2}-y^{\prime2}=1\\
\hline \begin{align}x_{1} &amp; =\alpha_{11}x+\alpha_{12}y+\alpha_{13}p\\
y_{1} &amp; =\alpha_{21}x+\alpha_{22}y+\alpha_{23}p\\
x_{1} &amp; =\alpha_{31}x+\alpha_{32}y+\alpha_{33}p\\
\\
x &amp; =\alpha_{11}x_{1}+\alpha_{21}y_{1}-\alpha_{31}p_{1}\\
y &amp; =\alpha_{12}x_{1}+\alpha_{22}y_{1}-\alpha_{32}p_{1}\\
p &amp; =-\alpha_{13}x_{1}-\alpha_{23}y_{1}+\alpha_{33}p_{1}
\end{align}
\left|\begin{align}\alpha_{33}^{2}-\alpha_{13}^{2}-\alpha_{23}^{2} &amp; =1\\
-\alpha_{31}^{2}+\alpha_{11}^{2}+\alpha_{21}^{2} &amp; =1\\
-\alpha_{32}^{2}+\alpha_{12}^{2}+\alpha_{22}^{2} &amp; =1\\
\alpha_{31}\alpha_{32}-\alpha_{11}\alpha_{12}-\alpha_{21}\alpha_{22} &amp; =0\\
\alpha_{32}\alpha_{33}-\alpha_{12}\alpha_{13}-\alpha_{22}\alpha_{23} &amp; =0\\
\alpha_{33}\alpha_{31}-\alpha_{23}\alpha_{11}-\alpha_{23}\alpha_{21} &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|apo1}}
*{{#section:History of Topics in Special Relativity/mathsource|apo2}}
*{{#section:History of Topics in Special Relativity/mathsource|apo}}
*{{#section:History of Topics in Special Relativity/mathsource|bour56att}} 
*{{#section:History of Topics in Special Relativity/mathsource|chal82sec}} 
*{{#section:History of Topics in Special Relativity/mathsource|chas29}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox83hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox91}} 
*{{#section:History of Topics in Special Relativity/mathsource|fris76}} 
*{{#section:History of Topics in Special Relativity/mathsource|gau98}} 
*{{#section:History of Topics in Special Relativity/mathsource|gau18}} 
*{{#section:History of Topics in Special Relativity/mathsource|ger92}} 
*{{#section:History of Topics in Special Relativity/mathsource|gud30}} 
*{{#section:History of Topics in Special Relativity/mathsource|haus99}} 
*{{#section:History of Topics in Special Relativity/mathsource|hill82}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac27}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac32a}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac32b}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac33}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil77}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil79}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil84}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil85}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil93}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil97}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei71}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei73}} 
*{{#section:History of Topics in Special Relativity/mathsource|lag73}} 
*{{#section:History of Topics in Special Relativity/mathsource|hire1}} 
*{{#section:History of Topics in Special Relativity/mathsource|leb37}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie85}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie90}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie93}} 
*{{#section:History of Topics in Special Relativity/mathsource|lieb04}} 
*{{#section:History of Topics in Special Relativity/mathsource|lop}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic82}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic84a}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic84b}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic84c}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin81a}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin81b}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin87}} 
*{{#section:History of Topics in Special Relativity/mathsource|sal62}} 
*{{#section:History of Topics in Special Relativity/mathsource|vinc}} 
*{{#section:History of Topics in Special Relativity/mathsource|som63}} 
*{{#section:History of Topics in Special Relativity/mathsource|wedd47}} 
*{{#section:History of Topics in Special Relativity/mathsource|wern89}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo01}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo03}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L1}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
      <sha1>ik2s4wow2rbkykoifu9lemue4t326ac</sha1>
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      <minor />
      <comment>/* Most general Lorentz transformation */</comment>
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      <text bytes="107023" sha1="4sgoep3vezohmjsthqocfq2g82579hg" xml:space="preserve">{{../Lorentz transformation (header)}}
==Most general Lorentz transformations==
===General quadratic form===
The general [[w:quadratic form]] ''q(x)'' with coefficients of a [[w:symmetric matrix]] '''A''', the associated [[w:bilinear form]] ''b(x,y)'', and the [[w:linear transformation]]s of ''q(x)'' and ''b(x,y)'' into ''q(x′)'' and ''b(x′,y′)'' using the [[w:transformation matrix]] '''g''', can be written as&lt;ref&gt;Bôcher (1907), chapter X&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}\begin{align}q=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}\end{align}
=q'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{x}'\\
b=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{y}=b'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{y}'\\
\left(\mathbf{A}=\mathbf{A}^{{\rm T}}\right)\\
\hline \left.\begin{align}\mathbf{x}' &amp; =\mathbf{g}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\mathbf{g}^{-1}\cdot\mathbf{x}'
\end{align}
\quad\right|\quad\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g}=\mathbf{A}'
\end{matrix}&lt;/math&gt;|{{equationRef|Q1}}}}

The case ''n=1'' is the [[w:binary quadratic form]] introduced by [[#Lagrange|Lagrange (1773)]] and [[#Gauss|Gauss (1798/1801)]], ''n=2'' is the ternary quadratic form introduced by [[#Gauss2|Gauss (1798/1801)]], ''n=3'' is the quaternary quadratic form etc.

===Most general Lorentz transformation===
{{CSS image crop
|Image=Orthogonality and rotation.svg
|bSize = 500
|cWidth        = 250
|cHeight = 250
|oLeft=250
|Location=right
|Description=The Lorentz interval is the invariant relation between axes and conjugate diameters of hyperbolas, illustrating Lorentz transformations between two inertial frames.}}
The general Lorentz transformation follows from ({{equationNote|Q1}}) by setting '''A'''='''A′'''=diag(-1,1,...,1) and det '''g'''=±1. It forms an [[w:indefinite orthogonal group]] called the [[w:Lorentz group]] O(1,n), while the case det '''g'''=+1 forms the restricted [[w:Lorentz group]] SO(1,n). The quadratic form ''q(x)'' becomes the [[w:Lorentz interval]] in terms of an [[w:indefinite quadratic form]] of [[w:Minkowski space]] (being a special case of [[w:pseudo-Euclidean space]]), and the associated bilinear form ''b(x)'' becomes the [[w:Minkowski inner product]]:&lt;ref name=ratcliffe&gt;Ratcliffe (1994), 3.1 and Theorem 3.1.4 and Exercise 3.1&lt;/ref&gt;&lt;ref&gt;Naimark (1964), 2 in four dimensions&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}\begin{align}-x_{0}^{2}+\cdots+x_{n}^{2} &amp; =-x_{0}^{\prime2}+\dots+x_{n}^{\prime2}\\
-x_{0}y_{0}+\cdots+x_{n}y_{n} &amp; =-x_{0}^{\prime}y_{0}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime}
\end{align}
\\
\hline \left.\begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\
\downarrow\\
{\scriptstyle \begin{align}x_{0}^{\prime} &amp; =x_{0}g_{00}+x_{1}g_{01}+\dots+x_{n}g_{0n}\\
x_{1}^{\prime} &amp; =x_{0}g_{10}+x_{1}g_{11}+\dots+x_{n}g_{1n}\\
 &amp; \dots\\
x_{n}^{\prime} &amp; =x_{0}g_{n0}+x_{1}g_{n1}+\dots+x_{n}g_{nn}
\end{align}
}\\
\\
\mathbf{x}=\mathbf{g}^{-1}\cdot\mathbf{x}'\\
\downarrow\\
{\scriptstyle \begin{align}x_{0} &amp; =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}-\dots-x_{n}^{\prime}g_{n0}\\
x_{1} &amp; =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}+\dots+x_{n}^{\prime}g_{n1}\\
 &amp; \dots\\
x_{n} &amp; =-x_{0}^{\prime}g_{0n}+x_{1}^{\prime}g_{1n}+\dots+x_{n}^{\prime}g_{nn}
\end{align}
}
\end{matrix}\right|\begin{matrix}\begin{align}\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}}\cdot\mathbf{A} &amp; =\mathbf{g}^{-1}\\
\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g} &amp; =\mathbf{A}\\
\mathbf{g}\cdot\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}} &amp; =\mathbf{A}\\
\\
\\
\end{align}
\\
{\scriptstyle \begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} &amp; =\left\{ \begin{align}-1\quad &amp; (j=k=0)\\
1\quad &amp; (j=k&gt;0)\\
0\quad &amp; (j\ne k)
\end{align}
\right.\\
\sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} &amp; =\left\{ \begin{align}-1\quad &amp; (i=k=0)\\
1\quad &amp; (i=k&gt;0)\\
0\quad &amp; (i\ne k)
\end{align}
\right.
\end{align}
}
\end{matrix}
\end{matrix}&lt;/math&gt;|{{equationRef|1a}}}}

The invariance of the Lorentz interval with ''n''=1 between axes and [[w:conjugate diameters]] of hyperbolas was known for a long time since [[#Apo|Apollonius (ca. 200 BC)]]. Lorentz transformations ({{equationNote|1a}}) for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]] in order to simplify computations of [[w:elliptic function]]s and integrals.&lt;ref&gt;Musen (1970) pointed out the intimate connection of Hill's scalar development and Minkowski's pseudo-Euclidean 3D space.&lt;/ref&gt;&lt;ref&gt;Touma et al. (2009) showed the analogy between Gauss and Hill's equations and Lorentz transformations, see eq. 22-29.&lt;/ref&gt; They were also used by [[#Chasles|Chasles (1829)]] and [[#Weddle|Weddle (1847)]] to describe relations on hyperboloids, as well as by [[#Poincare|Poincaré (1881)]], [[#Cox|Cox (1881-91)]], [[#Picard|Picard (1882, 1884)]], [[#Killing|Killing (1885, 1893)]], [[#Gerard|Gérard (1892)]], [[#Hausdorff|Hausdorff (1899)]], [[#Woods2|Woods (1901, 1903)]], [[#Liebmann|Liebmann (1904/05)]] to describe [[w:hyperbolic motion]]s (i.e. rigid motions in the [[w:hyperbolic plane]] or [[w:hyperbolic space]]), which were expressed in terms of Weierstrass coordinates of the [[w:hyperboloid model]] satisfying the relation &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}=-1&lt;/math&gt; or in terms of the [[w:Cayley–Klein metric]] of [[w:projective geometry]] using the &quot;absolute&quot; form &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}=0&lt;/math&gt; as discussed by [[#Klein|Klein (1871-73)]].&lt;ref group=M&gt;Killing (1885), p. 71&lt;/ref&gt;&lt;ref&gt;Müller (1910), p. 661, in particular footnote 247.&lt;/ref&gt;&lt;ref&gt;Sommerville (1911), p. 286, section K6.&lt;/ref&gt; In addition, [[w:infinitesimal transformation]]s related to the [[w:Lie algebra]] of the group of hyperbolic motions were given in terms of Weierstrass coordinates &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}=-1&lt;/math&gt; by [[#Killing3|Killing (1888-1897)]].

===Most general Lorentz transformation of velocity===

If &lt;math&gt;x_{i},\ x_{i}^{\prime}&lt;/math&gt; in ({{equationNote|1a}}) are interpreted as [[w:homogeneous coordinates]], then the corresponding inhomogenous coordinates &lt;math&gt;u_{s},\ u_{s}^{\prime}&lt;/math&gt; follow by

:&lt;math&gt;\frac{x_{s}}{x_{0}}=u_{s},\ \frac{x_{s}^{\prime}}{x_{0}^{\prime}}=u_{s}^{\prime}\ (s=1,2\dots n)&lt;/math&gt;

defined by &lt;math&gt;u_{1}^{2}+u_{2}^{2}+\dots+u_{n}^{2}\le1&lt;/math&gt; so that the Lorentz transformation becomes a [[w:homography]] inside the [[w:unit hypersphere]], which [[w:John Lighton Synge]] called &quot;the most general formula for the composition of velocities&quot; in terms of special relativity&lt;ref&gt;Synge (1955), p. 129 for ''n''=3&lt;/ref&gt; (the transformation matrix '''g''' stays the same as in ({{equationNote|1a}})):

{{NumBlk|:|&lt;math&gt;\begin{align}u_{s}^{\prime} &amp; =\frac{g_{s0}+g_{s1}u_{1}+\dots+g_{sn}u_{n}}{g_{00}+g_{01}u_{1}+\dots+g_{0n}u_{n}}\\
\\
u_{s} &amp; =\frac{-g_{0s}+g_{1s}u_{1}^{\prime}+\dots+g_{ns}u_{n}^{\prime}}{g_{00}-g_{10}u_{1}^{\prime}-\dots-g_{n0}u_{n}^{\prime}}
\end{align}
\left|{\scriptstyle \begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} &amp; =\left\{ \begin{align}-1\quad &amp; (j=k=0)\\
1\quad &amp; (j=k&gt;0)\\
0\quad &amp; (j\ne k)
\end{align}
\right.\\
\sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} &amp; =\left\{ \begin{align}-1\quad &amp; (i=k=0)\\
1\quad &amp; (i=k&gt;0)\\
0\quad &amp; (i\ne k)
\end{align}
\right.
\end{align}
}\right.&lt;/math&gt;|{{equationRef|1b}}}}

Such Lorentz transformations for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827–1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]], [[#Callandreau|Callandreau (1885)]] in order to simplify computations of elliptic functions and integrals, by [[#Picard|Picard (1882-1884)]] in relation to [[w:Hermitian form|Hermitian quadratic form]]s, or by [[#Woods2|Woods (1901, 1903)]] in terms of the [[w:Beltrami–Klein model]] of hyperbolic geometry. In addition, infinitesimal transformations in terms of the [[w:Lie algebra]] of the group of hyperbolic motions leaving invariant the unit sphere &lt;math&gt;-1+u_{1}^{\prime2}+\cdots+u_{n}^{\prime2}=0&lt;/math&gt; were given by [[#Lie3|Lie (1885-1893) and Werner (1889)]] and [[#Killing3|Killing (1888-1897)]].

==Historical notation==
==={{anchor|Apo}} Apollonius (BC) – Conjugate diameters===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Apo|label 1=History of Lorentz transformations via squeeze mappings § Apollonius}}
====Equality of difference in squares====
[[File:Apollonius-Borelli-XII.png|thumb|&lt;small&gt;Fig. 1: Apollonius' proposition illustrated by Borelli (1661) of &lt;math&gt;\scriptstyle \overline{AC}^{2}-\overline{QR}^{2}=\overline{IL}^{2}-\overline{NO}^{2}&lt;/math&gt;&lt;/small&gt;]]
[[w:Apollonius of Perga]] (c. 240–190 BC) in his 7th book on conics defined the following well known proposition (the 7th book survived in Arabian translation, and was translated into Latin in 1661 and 1710), as follows:

*The difference of the squares of the two axes of the hyperbola is equal to the difference of the squares of any two conjugate diameters. &lt;small&gt;(Latin translation 1661 by [[w:Giovanni Alfonso Borelli]] and [[w:Abraham Ecchellensis]].)&lt;ref group=M name=bor1&gt;Apollonius/Borelli/Ecchellensis (1661), Summary of prop. XII and other props. from book VII on pp. 291-292; See also the note on prop. XII on pp. 293-294, where Borelli demontrates &lt;math&gt;\scriptstyle \overline{AC}^{2}-\overline{QR}^{2}=\overline{IL}^{2}-\overline{NO}^{2}&lt;/math&gt; (in later translations such as Halley (1710), the proposition was numbered as XIII.) Latin: &quot;Differentia quadratorum duorum axium hyperboles æqualis est differentiæ quadratorum quarumlibet duarum diametrorum coniugatarum.&quot;&lt;/ref&gt;&lt;/small&gt;

*In every hyperbola the difference between the squares of the axes is equal to the difference between the squares of any conjugate diameters of the section. &lt;small&gt;(Latin translation 1710 by [[w:Edmond Halley]].)&lt;ref group=M&gt;Apollonius/Halley (1710), Prop. XIII of book VII on p. 107; Latin: &quot;In omni Hyperbola differentia inter quadrata Axium aequalis est differentiae inter quadrata ex diametris quibusvis conjugatis sectionis.&quot;&lt;/ref&gt;&lt;/small&gt;

*[..] in every hyperbola the difference of the squares on any two conjugate diameters is equal to the [..] difference [..] of the squares on the axes. &lt;small&gt;(English translation 1896 by [[w:Thomas Heath (classicist)|w:Thomas Heath]].)&lt;ref group=M&gt;Apollonius/Heath (1896), Proposition 129; (Apollonius, Book VII, Prop. 13).&lt;/ref&gt;&lt;/small&gt;
----
[[File:Lahire-XLII-XLIII.png|thumb|left|&lt;small&gt;Fig. 2: La Hire's (1685) illustration of &lt;math&gt;\scriptstyle \overline{AB}^{2}-\overline{DE}^{2}=\overline{NM}^{2}-\overline{LK}^{2}&lt;/math&gt;&lt;/small&gt;]]
[[File:Lhopital Conjugate Diameters.png|thumb|&lt;small&gt;Fig. 3: l'Hôpital's (1707) illustration of &lt;math&gt;\scriptstyle \overline{CS}^{2}-\overline{CM}^{2}=\overline{CB}^{2}-\overline{CA}^{2}&lt;/math&gt;&lt;/small&gt;]]
[[w:Philippe de La Hire]] (1685) stated this proposition as follows:
{{Block indent|1=I say that the difference of the squares of any two diameters conjugated to each other, AB, DE, is equal to the difference of the squares of any two other diameters conjugated to each other, NM, LK.&lt;ref group=M name=lahire1&gt;La Hire (1685), Book IV, Proposition XLII, p. 85; Latin: &quot;Dico differentiam quadratorum duarum diametrorum quarumlibet inter se conjugatarum AB, DE esse æqualem differentiæ quadratorum duarum aliarum diametrorum quarumlibet inter se conjugatarum, NM, LK.&quot;&lt;/ref&gt;}}
and also summarized the related propositions in the 7th book of Apollonius:
{{Block indent|1=In a hyperbola, the difference of the squares of the axes is equal to the difference of the squares of any two conjugate diameters.&lt;ref group=M&gt;La Hire (1685), p. 242. Summary of propositions XII, XIII, XXV in the 7th book of Apollonius; Latin: &quot;In hyperbola differentia quadratorum axium æqualis est differentia quadratorum duarum diametrorum conjugatarum quarumlibet.&quot;&lt;/ref&gt;}}
----
[[w:Guillaume de l'Hôpital]] (1707), using the methods of [[wikipedia:Analytic_geometry|w:analytic geometry]], demonstrated the same proposition:&lt;ref group=M name=lop&gt;l'Hôpital (1707), Third book, Prop. XII, p. 76.&lt;/ref&gt;

{{Block indent|1=The difference of the squares of any two conjugate diameters &quot;Mm, Ss&quot; is equal to the difference of the squares of the two axes &quot;Aa, Bb.&quot; We are to prove that &lt;math&gt;\overline{CS}^{2}-\overline{CM}^{2}=\overline{CB}^{2}-\overline{CA}^{2}&lt;/math&gt;, or &lt;math&gt;\overline{CM}^{2}-\overline{CS}^{2}=\overline{CA}^{2}-\overline{CB}^{2}&lt;/math&gt;. &lt;small&gt;(English translation 1723 by [[w:Edmund Stone]].)&lt;ref group=M&gt;l'Hôpital/Stone (1723), pp. 62-63&lt;/ref&gt;&lt;/small&gt;}}

{{Lorentzbox|Text=Apollonius' proposition can be expressed as &lt;math&gt;-x_{0}^{\prime2}+x_{1}^{\prime2}=-x_{0}^{2}+x_{1}^{2}&lt;/math&gt; in agreement with the invariance of the Lorentz interval, so that the Lorentz transformation ({{equationNote|1a}}) &quot;(n=1)&quot; can be interpreted as mapping from one pair of axes of a hyperbola to a pair of conjugate diameters.}}

====Equality of areas of parallelograms====

[[File:Apollonius-Borelli-XXXI.png|thumb|&lt;small&gt;Fig. 4: Apollonius' proposition illustrated by Borelli (1661) of the equality of areas of parallelogram ABCD (of the axes) and KLMN (of the conjugated diameters).&lt;/small&gt;]] 

Apollonius also gave another well known proposition in his 7th book regarding ellipses as well as conjugate sections of hyperbolas (see also Del Centina &amp; Fiocca&lt;ref&gt;Del Centina &amp; Fiocca (2020)&lt;/ref&gt; for further details on the history of this proposition): 

*In the ellipse, and in conjugate sections [the opposite branches of two conjugate hyperbolas] the parallelogram bounded by the axes is equal to the parallelogram bounded by any pair of conjugate diameters, if its angles are equal to the angles the conjugate diameters form at the centre. &lt;small&gt;(English translation by Del Centina &amp; Fiocca&lt;ref name=del&gt;Del Centina &amp; Fiocca (2020), section 3.1&lt;/ref&gt; based on the Latin translation 1661 by [[w:Giovanni Alfonso Borelli]] and [[w:Abraham Ecchellensis]].&lt;ref group=M name=bor2&gt;Apollonius/Borelli/Ecchellensis (1661), Summary of prop. XXXI of book VII on p. 370; Note on pp. 372-374; Latin: &quot;In ellypsi, &amp; sectionibus coniugatis parallelogrammum sub axibus contentum æquale est parallelogrammo à quibuscunque duabus coniugatis diametris comprehenso, si eorum anguli æquales fuerint angulis ad centrum contentis à coniugatis diametris.&quot;&lt;/ref&gt;)&lt;/small&gt;

*If two conjugate diameters are taken in an ellipse, or in the opposite conjugate sections; the parallelogram bounded by them is equal to the rectangle bounded by the axes, provided its angles are equal to those formed at the centre by the conjugate diameters. &lt;small&gt;(English translation by Del Centina &amp; Fiocca&lt;ref name=del /&gt; based on the Latin translation 1710 by [[w:Edmond Halley]].)&lt;ref group=M&gt;Apollonius/Halley (1710), Prop. XXXI of book VII on p. 115–117; Latin: &quot;Si ducantur diametri quævis conjugate in Ellipsi, vel inter sectiones oppositas conjugatas; erit parallelogrammum contentam sub his diametris æquale rectangulo sub ipsis Axibus facto: modo anguli ejus æquales sint angulis ad centrum sectionis à diametris conjugatis comprehensis.&quot;&lt;/ref&gt;)&lt;/small&gt;

*If PP', DD' be two conjugate diameters in an ellipse or in conjugate hyperbolas, and if tangents be drawn at the four extremities forming a parallelogram LL'MM', then the parallelogram LL'MM' = rect. AA'·BB'. &lt;small&gt;(English translation 1896 by [[w:Thomas Heath (classicist)|w:Thomas Heath]].)&lt;ref group=M&gt;Apollonius/Heath (1896), Proposition 136, p. 235; (Apollonius, Book VII, Prop. 31).&lt;/ref&gt;&lt;/small&gt; 

{{Lorentzbox|Text=The graphical representation of Apollonius proposition in Borelli's Fig. 4 is essentially a [[w:Minkowski diagram]], being a graphical representation of the Lorentz transformation. If line AB is the x-axis of an inertial frame S1, then line FG is the x-axis of another inertial frames S2 which together with its parallel lines (such as KL and NM) represent [[w:relativity of simultaneity]]. Analogously, if line CD is the time axis of another inertial frame S2, then line HI is the time axis of S2 which together with its parallel lines (such as KN and LM) represent the [[w:worldlines]] of objects at different locations. The diagonals KE (or KM) and LE (or LN) lie on the asymptotes which form a light cone. Thus the totality of all parallelograms of equal area and conjugate diameters as constructed by Apollonius, represents the totality of all inertial frames, lines of simultaneity and worldlines within a spacetime area bounded by &lt;math&gt;-x_{0}^{2}+x_{1}^{2}=\rm{const}&lt;/math&gt;.}}

[[File:Saint-Vincent-Hyperbola-VI-XLIX.png|thumb|175px|left|&lt;small&gt;Fig. 5: Saint-Vincent's (1647) illustration of FGHI=OPQR, as well as BADC=KNLM.&lt;/small&gt;]]
[[w:Grégoire de Saint-Vincent]] independently (1647) stated the same proposition:&lt;ref group=M name=vinc&gt;St. Vincent (1647), Book VI, Prop. XLIX, p. 560; Latin: “Si fuerint binæ hyperbolarum coniugaciones A, B, C, D: ponantur autem per E centrum duæ quoque diametrorum coniugationes per quarum vertices contingentes actæ constituant duo quadrilatera FGHI, OPQR. Dico illa esse æqualia inter se.”&lt;/ref&gt;

{{Block indent|1=The parallelograms whose opposite sides are tangent to two conjugate hyperbolas at the extremities of two conjugate diameters are equivalent among them. &lt;small&gt;(English translation by Del Centina &amp; Fiocca.&lt;ref&gt;Del Centina &amp; Fiocca (2020), section 5.1&lt;/ref&gt;)&lt;/small&gt; }}

---- 

[[File:Lahire-XLII-XLIII.png|thumb|&lt;small&gt;Fig. 6 (identical to Fig. 2): La Hire's (1685) illustration of FGHI=OPQR.&lt;/small&gt;]]
[[w:Philippe de La Hire]] (1685), who was aware of both Apollonius 7th book and Saint-Vincent's book, stated this proposition as follows:&lt;ref group=M name=lahire&gt;La Hire (1685), Book IV, Proposition XLIII, pp. 85-86; Latin: &quot;In sectionibus conjugatis NA, DL, BM, KE si circumscribatur parallelogrammum FGHI à rectis parallelis duabus diametris inter se conjugatis ED, BA, &amp; per ipsorum terminos ductis, &amp; simili methodo circumscribatur aliud parallelogrammum OPQR à rectis ductis per terminos diametrorum conjugatarum, &amp; ipsis parallelis: Dico parallelogramma FGHI, OPQR esse inter se æqualia.&quot;&lt;/ref&gt; 

{{Block indent|1=If a parallelogram FGHI is circumscribed about conjugate sections NA, DL, BM, KE whose sides are parallel to two conjugate diameters ED, BA drawn through their extremities, and with similar method another parallelogram OPQR is drawn through the extremities of other two conjugate diameters, then the parallelograms FGHI, OPQR are equal. &lt;small&gt;(English translation by Del Centina &amp; Fiocca.&lt;ref name=del2&gt;Del Centina &amp; Fiocca (2020), section 5.2&lt;/ref&gt;)&lt;/small&gt;}}

and also summarized the related propositions in the 7th book of Apollonius:&lt;ref group=M&gt;La Hire (1685), p. 242. Summary of proposition XXXI in the 7th book of Apollonius; Latin: &quot;In sectionibus conjugatis &amp; Ellipsi parallelogrammum sub axibus æquale est paralelogrammo sub duabus quibuscunque diametris inter se conjugatis, in angulis ipsarum diametrorum conjugatarum.&quot;&lt;/ref&gt;

{{Block indent|1=In conjugate sections and in the ellipse, the parallelogram constructed with the axes, is equal to the parallelogram constructed with any two conjugated diameters, provided the angles are equal to those between the diameters themselves. &lt;small&gt;(English translation by Del Centina &amp; Fiocca.&lt;ref name=del2 /&gt;)&lt;/small&gt;}}

{{Lorentzbox|Text=In Saint-Vincent's Fig. 5 or La Hire's Fig. 6, parallelogram FGHI contains all coordinates related to an inertial frame S3, in particular triangles EGH, EFI (Fig. 5) or CFG, CHI (Fig. 6) contain time like intervals between events on the future and past light cones, while triangles EHI, EGF (Fig. 5) or CFI, CGH (Fig. 6) contain space like intervals between events on the negative and positive x-axis. Conversely, parallelogram OPQR contains all coordinates related to another frame S4, in particular triangles EQR, EOP (Fig. 5) or CPQ, COR (Fig. 6) contain time like intervals between events on the future and past light cones, while triangles EPR, EOQ (Fig. 5) or COP, CQR (Fig. 6) contain space like intervals between events on the negative and positive x-axis.}}

===Lagrange (1773) – Binary quadratic forms {{anchor|Lagrange}}===
After the invariance of the sum of squares under linear substitutions was discussed by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]], the general expressions of a [[w:binary quadratic form]] and its transformation was formulated by [[w:Joseph-Louis Lagrange]] (1773/75) as follows&lt;ref group=M&gt;Lagrange (1773/75), section 22&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\
\hline \begin{align}y &amp; =Ms+Nx\\
z &amp; =ms+nx
\end{align}
\left|\begin{matrix}\begin{align}P &amp; =pM^{2}+2qMm+rm^{2}\\
Q &amp; =pMN+q(Mn+Nm)+rmn\\
R &amp; =pN^{2}+2qNn+rn^{2}
\end{align}
\\
\downarrow\\
PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2}
\end{matrix}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to ({{equationNote|Q1}}) ''(n=1)''. The Lorentz interval &lt;math&gt;-x_{0}^{2}+x_{1}^{2}&lt;/math&gt; and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(p,q,r)=(P,Q,R)=(1,0,-1)''.}}

==={{anchor|Gauss}} Gauss (1798–1818)===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Gauss|label 1=History of Lorentz transformations via Möbius transformations § Gauss}}
===={{anchor|Gauss1}} Binary quadratic forms====

The theory of binary quadratic forms was considerably expanded by [[w:Carl Friedrich Gauss]] (1798, published 1801) in his [[w:Disquisitiones Arithmeticae]]. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:&lt;ref group=M&gt;Gauss (1798/1801), articles 157–158;&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\
F'=a'x^{\prime2}+2b'x'y'+c'y^{\prime2}=(a',b',c')\\
\hline \begin{align}x &amp; =\alpha x'+\beta y'\\
y &amp; =\gamma x'+\delta y'\\
\\
x' &amp; =\delta x-\beta y\\
y' &amp; =-\gamma x+\alpha y
\end{align}
\left|\begin{matrix}\begin{align}a' &amp; =a\alpha^{2}+2b\alpha\gamma+c\gamma^{2}\\
b' &amp; =a\alpha\beta+b(\alpha\delta+\beta\gamma)+c\gamma\delta\\
c' &amp; =a\beta^{2}+2b\beta\delta+c\delta^{2}
\end{align}
\\
\downarrow\\
b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha\delta-\beta\gamma)^{2}
\end{matrix}\right.
\end{matrix}&lt;/math&gt;

which is equivalent to ({{equationNote|Q1}}) ''(n=1)''. As pointed out by Gauss, ''F'' and ''F′'' are called &quot;proper equivalent&quot; if αδ-βγ=1, so that ''F'' is contained in ''F′'' as well as ''F′'' is contained in ''F''. In addition, if another form ''F″'' is contained by the same procedure in ''F′'' it is also contained in ''F'' and so forth.&lt;ref group=M&gt;Gauss (1798/1801), section 159&lt;/ref&gt;

{{Lorentzbox|Text=The Lorentz interval &lt;math&gt;-x_{0}^{2}+x_{1}^{2}&lt;/math&gt; and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(a,b,c)=(a',b',c')=(1,0,-1)''.}}

===={{anchor|Gauss2}} Ternary quadratic forms====

Gauss (1798/1801)&lt;ref group=M&gt;Gauss (1798/1801), articles 266–285&lt;/ref&gt; also discussed ternary quadratic forms with the general expression

:&lt;math&gt;\begin{matrix}f=ax^{2}+a'x^{\prime2}+a''x^{\prime\prime2}+2bx'x''+2b'xx''+2b''xx'=\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)\\
g=my^{2}+m'y^{\prime2}+m''y^{\prime\prime2}+2ny'y''+2n'yy''+2n''yy'=\left(\begin{matrix}m, &amp; m', &amp; m''\\
n, &amp; n', &amp; n''
\end{matrix}\right)\\
\hline \begin{align}x &amp; =\alpha y+\beta y'+\gamma y''\\
x' &amp; =\alpha'y+\beta'y'+\gamma'y''\\
x'' &amp; =\alpha''y+\beta''y'+\gamma''y''
\end{align}
\end{matrix}&lt;/math&gt;

which is equivalent to ({{equationNote|Q1}}) ''(n=2)''. Gauss called these forms definite when they have the same sign such as ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;+z&lt;sup&gt;2&lt;/sup&gt;'', or indefinite in the case of different signs such as ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;-z&lt;sup&gt;2&lt;/sup&gt;''. While discussing the classification of ternary quadratic forms, Gauss (1801) presented twenty special cases, among them these six variants:&lt;ref group=M&gt;Gauss (1798/1801), article 277&lt;/ref&gt;

:&lt;math&gt;\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)\Rightarrow\begin{matrix}\left(\begin{matrix}1, &amp; -1, &amp; 1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}-1, &amp; 1, &amp; 1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\\
\left(\begin{matrix}1, &amp; -1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}-1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}-1, &amp; -1, &amp; 1\\
0, &amp; 0, &amp; 0
\end{matrix}\right)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=These are all six types of Lorentz interval in 2+1 dimensions that can be produced as special cases of a ternary quadratic form. In general: The Lorentz interval &lt;math&gt;x^{2}+x^{\prime2}-x^{\prime\prime2}&lt;/math&gt; and the Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' is an indefinite ternary quadratic form, which follows from the general ternary form by setting:
&lt;math&gt;\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)=\left(\begin{matrix}m, &amp; m', &amp; m''\\
n, &amp; n', &amp; n''
\end{matrix}\right)=\left(\begin{matrix}1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right)&lt;/math&gt;}}

===={{anchor|Gauss4}} Homogeneous coordinates====

Gauss (1818) discussed planetary motions together with formulating [[w:elliptic function]]s. In order to simplify the integration, he transformed the expression

:&lt;math&gt;(AA+BB+CC)tt+aa(t\cos E)^{2}+bb(t\sin E)^{2}-2aAt\cdot t\cos E-2bBt\cdot t\sin E&lt;/math&gt;

into

:&lt;math&gt;G+G'\cos T^{2}+G''\sin T^{2}&lt;/math&gt;

in which the [[w:eccentric anomaly]] ''E'' is connected to the new variable ''T'' by the following transformation including an arbitrary constant ''k'', which Gauss then rewrote by setting ''k''=1:&lt;ref group=M&gt;Gauss (1818), pp. 5–10&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}{\scriptstyle \left(\alpha+\alpha'\cos T+\alpha''\sin T\right)^{2}+\left(\beta+\beta'\cos T+\beta''\sin T\right)^{2}-\left(\gamma+\gamma'\cos T+\gamma''\sin T\right)^{2}}=0\\
k\left(\cos^{2}T+\sin^{2}T-1\right)=0\\
\hline \begin{align}\cos E &amp; =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\
\sin E &amp; =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}
\end{align}
\left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma &amp; =k &amp; \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' &amp; =-k\\
-\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' &amp; =-k &amp; \beta\beta-\beta'\beta'-\beta''\beta'' &amp; =-k\\
-\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' &amp; =-k &amp; \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' &amp; =+k\\
-\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' &amp; =0 &amp; \beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0\\
-\alpha''\alpha-\beta''\beta+\gamma''\gamma &amp; =0 &amp; \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' &amp; =0\\
-\alpha\alpha'-\beta\beta'+\gamma\gamma' &amp; =0 &amp; \alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0
\end{align}
}\right.\\
\hline k=1\\
\begin{align}t\cos E &amp; =\alpha+\alpha'\cos T+\alpha''\sin T\\
t\sin E &amp; =\beta+\beta'\cos T+\beta''\sin T\\
t &amp; =\gamma+\gamma'\cos T+\gamma''\sin T
\end{align}
\left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma &amp; =1\\
-\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' &amp; =-1\\
-\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' &amp; =-1\\
-\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' &amp; =0\\
-\alpha''\alpha-\beta''\beta+\gamma''\gamma &amp; =0\\
-\alpha\alpha'-\beta\beta'+\gamma\gamma' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The coefficients α,β,γ,... of Gauss' case ''k''=1 are equivalent to the coefficient system in Lorentz transformations ({{equationNote|1a}}) and ({{equationNote|1b}})  ''(n=2)''.

Further setting &lt;math&gt;[\cos T,\sin T,\cos E,\sin E]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]&lt;/math&gt;, Gauss' transformation becomes Lorentz transformation ({{equationNote|1b}})  ''(n=2)''.}}

Subsequently, he showed that these relations can be reformulated using three variables ''x,y,z'' and ''u,u′,u″'', so that

:&lt;math&gt;aaxx+bbyy+(AA+BB+CC)zz-2aAxz-2bByz&lt;/math&gt;

can be transformed into

:&lt;math&gt;Guu+G'u'u'+G''u''u''&lt;/math&gt;,

in which ''x,y,z'' and ''u,u′,u″'' are related by the transformation:&lt;ref group=M&gt;Gauss (1818), pp. 9–10&lt;/ref&gt;

:&lt;math&gt;\begin{align}x &amp; =\alpha u+\alpha'u'+\alpha''u''\\
y &amp; =\beta u+\beta'u'+\beta''u''\\
z &amp; =\gamma u+\gamma'u'+\gamma''u''\\
\\
u &amp; =-\alpha x-\beta y+\gamma z\\
u' &amp; =\alpha'x+\beta'y-\gamma'z\\
u'' &amp; =\alpha''x+\beta''y-\gamma''z
\end{align}
\left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma &amp; =1\\
-\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' &amp; =-1\\
-\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' &amp; =-1\\
-\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' &amp; =0\\
-\alpha''\alpha-\beta''\beta+\gamma''\gamma &amp; =0\\
-\alpha\alpha'-\beta\beta'+\gamma\gamma' &amp; =0
\end{align}
}\right.&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}})  ''(n=2)'' satisfying &lt;math&gt;x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}&lt;/math&gt;, and can be related to Gauss' previous equations in terms of homogeneous coordinates &lt;math&gt;\left[\cos T,\sin T,\cos E,\sin E\right]=\left[\tfrac{x}{z},\ \tfrac{y}{z},\ \tfrac{u'}{u},\ \tfrac{u''}{u}\right]&lt;/math&gt;.}}

==={{anchor|Jacobi}} Jacobi (1827, 1833/34) – Homogeneous coordinates===

Following [[#Gauss4|Gauss (1818)]], [[w:Carl Gustav Jacob Jacobi]] extended Gauss' transformation in 1827:&lt;ref group=M&gt;Jacobi (1827), p. 235, 239–240&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}\cos P^{2}+\sin P^{2}\cos\vartheta^{2}+\sin P^{2}\sin\vartheta^{2}=1\\
k\left(\cos\psi^{2}+\sin\psi^{2}\cos\varphi^{2}+\sin\psi^{2}\sin\varphi^{2}-1\right)=0\\
\hline {\left.\begin{matrix}\mathbf{(1)}\begin{align}\cos P &amp; =\frac{\alpha+\alpha'\cos\psi+\alpha''\sin\psi\cos\varphi+\alpha'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\
\sin P\cos\vartheta &amp; =\frac{\beta+\beta'\cos\psi+\beta''\sin\psi\cos\varphi+\beta'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\
\sin P\sin\vartheta &amp; =\frac{\gamma+\beta'\cos\psi+\gamma''\sin\psi\cos\varphi+\gamma'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\
\\
\cos\psi &amp; =\frac{-\delta'+\alpha'\cos P+\beta'\sin P\cos\vartheta+\gamma'\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\
\sin\psi\cos\varphi &amp; =\frac{-\delta''+\alpha''\cos P+\beta''\sin P\cos\vartheta+\gamma''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\
\sin\psi\sin\varphi &amp; =\frac{-\delta'''+\alpha'''\cos P+\beta'''\sin P\cos\vartheta+\gamma'''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}
\end{align}
\\
\\
\hline \mathbf{(2)}\begin{align}\alpha\mu+\beta x+\gamma y+\delta z &amp; =m\\
\alpha'\mu+\beta'x+\gamma'y+\delta'z &amp; =m'\\
\alpha''\mu+\beta''x+\gamma''y+\delta''z &amp; =m''\\
\alpha'''\mu+\beta'''x+\gamma'''y+\delta'''z &amp; =m'''\\
\\
Am+A'm'+A''m''+A'''m''' &amp; =\mu\\
Bm+B'm'+B''m''+B'''m''' &amp; =x\\
Cm+C'm'+C''m''+C'''m''' &amp; =y\\
Dm+D'm'+D''m''+D'''m''' &amp; =z\\
\\
\end{align}
\\
\begin{align}\alpha &amp; =-kA, &amp; \beta &amp; =-kB, &amp; \gamma &amp; =-kC, &amp; \delta &amp; =kD,\\
\alpha' &amp; =kA', &amp; \beta' &amp; =kB', &amp; \gamma' &amp; =kC', &amp; \delta' &amp; =-kD',\\
\alpha'' &amp; =kA'', &amp; \beta'' &amp; =kB'', &amp; \gamma'' &amp; =kC'', &amp; \delta'' &amp; =-kD'',\\
\alpha''' &amp; =kA''', &amp; \beta''' &amp; =kB''', &amp; \gamma''' &amp; =kC''', &amp; \delta''' &amp; =-kD''',
\end{align}
\end{matrix}\right|\begin{matrix}\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta &amp; =-k\\
\alpha'\alpha'+\beta'\beta'+\gamma'\gamma'-\delta'\delta' &amp; =k\\
\alpha''\alpha''+\beta''\beta''+\gamma''\gamma''-\delta''\delta'' &amp; =k\\
\alpha'''\alpha'''+\beta'''\beta'''+\gamma'''\gamma'''-\delta'''\delta''' &amp; =k\\
\alpha\alpha'+\beta\beta'+\gamma\gamma'-\delta\delta' &amp; =0\\
\alpha\alpha''+\beta\beta''+\gamma\gamma''-\delta\delta'' &amp; =0\\
\alpha\alpha'''+\beta\beta'''+\gamma\gamma'''-\delta\delta''' &amp; =0\\
\alpha''\alpha'''+\beta''\beta'''+\gamma''\gamma'''-\delta''\delta''' &amp; =0\\
\alpha'''\alpha'+\beta'''\beta'+\gamma'''\gamma'-\delta'''\delta' &amp; =0\\
\alpha'\alpha''+\beta'\beta''+\gamma'\gamma''-\delta'\delta'' &amp; =0\\
\\
-\alpha\alpha+\alpha'\alpha'+\alpha''\alpha''+\alpha'''\alpha''' &amp; =k\\
-\beta\beta+\beta'\beta'+\beta''\beta''+\beta'''\beta''' &amp; =k\\
-\gamma\gamma+\gamma'\gamma'+\gamma''\gamma''+\gamma'''\gamma''' &amp; =k\\
-\delta\delta+\delta'\delta'+\delta''\delta''+\delta'''\delta''' &amp; =-k\\
-\alpha\beta+\alpha'\beta'+\alpha''\beta''+\alpha'''\beta''' &amp; =0\\
-\alpha\gamma+\alpha'\gamma'+\alpha''\gamma''+\alpha'''\gamma''' &amp; =0\\
-\alpha\delta+\alpha'\delta'+\alpha''\delta''+\alpha'''\delta''' &amp; =0\\
-\beta\gamma+\beta'\gamma'+\beta''\gamma''+\beta'''\gamma''' &amp; =0\\
-\gamma\delta+\gamma'\delta'+\gamma''\delta''+\gamma'''\delta''' &amp; =0\\
-\delta\beta+\delta'\beta'+\delta''\beta''+\delta'''\beta''' &amp; =0
\end{align}
\end{matrix}}
\end{matrix}}&lt;/math&gt;

{{Lorentzbox|Text=By setting &lt;math&gt;{\scriptstyle \begin{align}\left[\cos P,\ \sin P\cos\varphi,\ \sin P\sin\varphi\right] &amp; =\left[u_{1},\ u_{2},\ u_{3}\right]\\{}
[\cos\psi,\ \sin\psi\cos\vartheta,\ \sin\psi\sin\vartheta] &amp; =\left[u_{1}^{\prime},\ u_{2}^{\prime},\ u_{3}^{\prime}\right]
\end{align}
}&lt;/math&gt; and ''k''=1 in the (1827) formulas, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'', and by setting ''k''=1 in transformation system (2) it becomes equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' producing &lt;math&gt;m^{2}+m^{\prime2}+m^{\prime\prime2}-m^{\prime\prime\prime2}=\mu^{2}+x^{2}+y^{2}-z^{2}&lt;/math&gt;.}}

Alternatively, in two papers from 1832 Jacobi started with an ordinary orthogonal transformation, and by using an imaginary substitution he arrived at Gauss' transformation (up to a sign change):&lt;ref group=M&gt;The orthogonal substitution and the imaginary transformation was defined in Jacobi (1832a), pp. 257, 265–267; Transformation system (2) and (3) and coefficients in Jacobi (1832b), pp. 321-325.&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}xx+yy+zz=ss+s's'+s''s''=0\\
\mathbf{(1)}\begin{align}x &amp; =\alpha s+\alpha's'+\alpha''s''\\
y &amp; =\beta s+\beta's'+\beta''s''\\
z &amp; =\gamma s+\gamma's'+\gamma''s''\\
\\
s &amp; =\alpha x+\beta y+\gamma z\\
s' &amp; =\alpha'x+\beta'y+\gamma'z\\
s'' &amp; =\alpha''x+\beta''y+\gamma''z
\end{align}
\left|\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma &amp; =1 &amp; \alpha\alpha+\alpha'\alpha'+\alpha''\alpha'' &amp; =1\\
\alpha'\alpha'+\beta'\beta'+\gamma'\gamma' &amp; =1 &amp; \beta\beta+\beta'\beta'+\beta''\beta'' &amp; =1\\
\alpha''\alpha''+\beta''\beta''+\gamma''\gamma'' &amp; =1 &amp; \gamma\gamma+\gamma'\gamma'+\gamma''\gamma'' &amp; =1\\
\alpha'\alpha''+\beta'\beta''+\gamma'\gamma'' &amp; =0 &amp; \beta\gamma+\beta'\gamma'+\beta''\gamma'' &amp; =0\\
\alpha''\alpha+\beta''\beta+\gamma''\gamma &amp; =0 &amp; \gamma\alpha+\gamma'\alpha'+\gamma''\alpha'' &amp; =0\\
\alpha\alpha'+\beta\beta'+\gamma\gamma' &amp; =0 &amp; \alpha\beta+\alpha'\beta'+\alpha''\beta'' &amp; =0
\end{align}
\right.\\
\hline \left[\frac{y}{x},\ \frac{z}{x},\ \frac{s'}{s},\ \frac{s''}{s}\right]=\left[-i\cos\varphi,\ -i\sin\varphi,\ i\cos\eta,\ i\sin\eta\right]\\
\left[\alpha',\ \alpha'',\ \beta,\ \gamma\right]=\left[i\alpha',\ i\alpha'',\ -i\beta,\ -i\gamma\right]\\
\hline \begin{matrix}\mathbf{(2)}\begin{matrix}\left(\alpha-\alpha'\cos\eta-\alpha''\sin\eta\right)^{2}=\left(\beta-\beta'\cos\eta-\beta''\sin\eta\right)^{2}+\left(\gamma-\gamma'\cos\eta-\gamma''\sin\eta\right)^{2}\\
\left(\alpha-\beta\cos\phi-\gamma\sin\phi\right)^{2}=\left(\alpha'-\beta'\cos\phi-\gamma'\sin\phi\right)^{2}+\left(\alpha''-\beta''\cos\phi-\gamma''\sin\phi\right)^{2}\\
\hline \begin{align}\cos\phi &amp; =\frac{\beta-\beta'\cos\eta-\beta''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, &amp; \cos\eta &amp; =\frac{\alpha'-\beta'\cos\phi-\gamma'\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}\\
\sin\phi &amp; =\frac{\gamma-\gamma'\cos\eta-\gamma''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, &amp; \sin\eta &amp; =\frac{\alpha''-\beta''\cos\phi-\gamma''\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}
\end{align}
\end{matrix}\\
\hline \\
\mathbf{(3)}\begin{matrix}1-zz-yy=\frac{1-s's'-s''s''}{\left(\alpha-\alpha's'-\alpha''s''\right)^{2}}\\
\hline \begin{align}y &amp; =\frac{\beta-\beta's'-\beta''s''}{\alpha-\alpha's'-\alpha''s''}, &amp; s' &amp; =\frac{\alpha'-\beta'y-\gamma'z}{\alpha-\beta y-\gamma z},\\
z &amp; =\frac{\gamma-\gamma's'-\gamma''s''}{\alpha-\alpha's'-\alpha''s'''}, &amp; s'' &amp; =\frac{\alpha''-\beta''y-\gamma''z}{\alpha-\beta y-\gamma z},
\end{align}
\end{matrix}
\end{matrix}\left|\begin{align}\alpha\alpha-\beta\beta-\gamma\gamma &amp; =1\\
\alpha'\alpha'-\beta'\beta'-\gamma'\gamma' &amp; =-1\\
\alpha''\alpha''-\beta''\beta''-\gamma''\gamma'' &amp; =-1\\
\alpha'\alpha''-\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\alpha''\alpha-\beta''\beta-\gamma''\gamma &amp; =0\\
\alpha\alpha'-\beta\beta'-\gamma\gamma' &amp; =0\\
\\
\alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' &amp; =1\\
\beta\beta-\beta'\beta'-\beta''\beta'' &amp; =-1\\
\gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' &amp; =-1\\
\beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0\\
\gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' &amp; =0\\
\alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0
\end{align}
\right.
\end{matrix}}&lt;/math&gt;

{{Lorentzbox|Text=By setting &lt;math&gt;[\cos\phi,\ \sin\phi,\ \cos\eta,\ \sin\eta]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]&lt;/math&gt;, transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Also transformation system (3) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' up to a sign change.}}

Extending his previous result, Jacobi (1833) started with [[#Cauchy|Cauchy's (1829)]] orthogonal transformation for ''n'' dimensions, and by using an imaginary substitution he formulated Gauss' transformation (up to a sign change) in the case of ''n'' dimensions:&lt;ref group =M&gt;Jacobi (1833/34), pp. 7–8, 34–35, 41; Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 229–230.&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}x_{1}x_{1}+x_{2}x_{2}+\dots+x_{n}x_{n}=y_{1}y_{1}+y_{2}y_{2}+\dots+y_{n}y_{n}\\
\hline \mathbf{(1)\ }\begin{align}y_{\varkappa} &amp; =\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}\\
x_{\varkappa} &amp; =\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}\\
\\
\frac{y_{\varkappa}}{y_{n}} &amp; =\frac{\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}\\
\frac{x_{\varkappa}}{x_{n}} &amp; =\frac{\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}
\end{align}
\left|\begin{align}\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\lambda}^{(n)} &amp; =0\\
\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\varkappa}^{(n)} &amp; =1\\
\\
\alpha_{1}^{(\varkappa)}\alpha_{1}^{(\lambda)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\lambda)} &amp; =0\\
\alpha_{1}^{(\varkappa)}\alpha_{1}^{(\varkappa)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\varkappa)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\varkappa)} &amp; =1
\end{align}
\right.\\
\hline \frac{x_{\varkappa}}{x_{n}}=-i\xi_{\varkappa},\ \frac{y_{\varkappa}}{y_{n}}=i\nu_{\varkappa}\\
1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{y_{n}y_{n}}{x_{n}x_{n}}\left(1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}\right)\\
\alpha_{n}^{(\varkappa)}=i\alpha^{(\varkappa)},\ \alpha_{\varkappa}^{(n)}=-i\alpha_{\varkappa},\ \alpha_{n}^{(n)}=\alpha\\
1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}}{\left[\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}\right]^{2}}\\
\hline \mathbf{(2)\ }\begin{align}\nu_{\varkappa} &amp; =\frac{\alpha^{(\varkappa)}-\alpha_{1}^{(\varkappa)}\xi_{1}-\alpha_{2}^{(\varkappa)}\xi_{2}\dots-\alpha_{n-1}^{(\varkappa)}\xi_{n-1}}{\alpha-\alpha_{1}\xi_{1}-\alpha_{2}\xi_{2}\dots-\alpha_{n-1}\xi_{n-1}}\\
\\
\xi_{\varkappa} &amp; =\frac{\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\nu_{1}-\alpha_{2}^{\prime\prime}\nu_{2}\dots-\alpha_{\varkappa}^{(n-1)}\nu_{n-1}}{\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}}
\end{align}
\\
\hline \xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=1\ \Rightarrow\ \nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}=1
\end{matrix}}&lt;/math&gt;

{{Lorentzbox|Text=Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) up to a sign change.}}

He also stated the following transformation leaving invariant the Lorentz interval:&lt;ref group=M&gt;Jacobi (1833/34), p. 37. Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 232–233.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}uu-u_{1}u_{1}-u_{2}u_{2}-\dots-u_{n-1}u_{n-1}=ww-w_{1}w_{1}-w_{2}w_{2}-\dots-w_{n-1}w_{n-1}\\
\hline {\scriptstyle \begin{align}u &amp; =\alpha w-\alpha'w_{1}-\alpha''w_{2}-\dots-\alpha^{(n-1)}w_{n-1}\\
u_{1} &amp; =\alpha_{1}w-\alpha_{1}^{\prime}w_{1}-\alpha_{1}^{\prime\prime}w_{2}-\dots-\alpha_{1}^{(n-1)}w_{n-1}\\
 &amp; \dots\\
u_{n-1} &amp; =\alpha_{n-1}w-\alpha_{n-1}^{\prime}w_{1}-\alpha_{n-1}^{\prime\prime}w_{2}-\dots-\alpha_{n-1}^{(n-1)}w_{n-1}\\
\\
w &amp; =\alpha u-\alpha_{1}u_{1}-\alpha_{2}^{\prime\prime}u_{2}-\dots-\alpha_{n-1}u_{n-1}\\
w_{1} &amp; =\alpha'u-\alpha_{1}^{\prime}u_{1}-\alpha_{2}^{\prime}u_{2}-\dots-\alpha_{n-1}^{\prime}u_{n-1}\\
 &amp; \dots\\
w_{n-1} &amp; =\alpha^{(n-1)}u-\alpha_{1}^{(n-1)}u_{1}-\alpha_{2}^{(n-1)}u_{2}-\dots-\alpha_{n-1}^{(n-1)}u_{n-1}
\end{align}
\left|\begin{align}\alpha\alpha-\alpha'\alpha'-\alpha''\alpha''\dots-\alpha^{(n-1)}\alpha^{(n-1)} &amp; =+1\\
\alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\varkappa}^{(n-1)} &amp; =-1\\
\alpha\alpha_{\varkappa}-\alpha^{\prime}\alpha_{\varkappa}^{\prime}-\alpha^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha^{(n-1)}\alpha_{\varkappa}^{(n-1)} &amp; =0\\
\alpha_{\varkappa}\alpha_{\lambda}-\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\lambda}^{(n-1)} &amp; =0\\
\\
\alpha\alpha-\alpha_{1}\alpha_{1}-\alpha_{2}\alpha_{2}\dots-\alpha_{n-1}\alpha_{n-1} &amp; =+1\\
\alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{1}^{\varkappa}\alpha_{1}^{\varkappa}-\alpha_{2}^{\prime\prime}\alpha_{2}^{\prime\prime}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\varkappa)} &amp; =-1\\
\alpha\alpha^{(\varkappa)}-\alpha_{1}\alpha_{1}^{(\varkappa)}-\alpha_{2}\alpha_{2}^{(\varkappa)}\dots-\alpha_{n-1}\alpha_{n-1}^{(\varkappa)} &amp; =0\\
\alpha^{(\varkappa)}\alpha^{(\lambda)}-\alpha_{1}^{(\varkappa)}\alpha_{1}^{\lambda l)}-\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\lambda)} &amp; =0
\end{align}
\text{ }\right.}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) up to a sign change.}}

==={{anchor|Chasles}} Chasles (1829) – Conjugate hyperboloids ===

[[w:Michel Chasles]] (1829) independently introduced the same equation systems as [[#Gauss4|Gauss (1818)]] and [[#Jacobi|Jacobi (1827)]], albeit in the different context of conjugate hyperboloids. He started with two equation systems (a) and (b) from which he derived systems (c), (d) and others:&lt;ref group=M&gt;Chasles (1829), p. 139&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =1\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =1\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =-1
\end{align}
\right\}  &amp; \dots(a)\\ \\
\left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma' &amp; =0
\end{align}
\right\}  &amp; \dots(b)\\ \\
\left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =1\\
\beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} &amp; =1\\
\gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-1
\end{align}
\right\}  &amp; \dots(c)\\ \\
\left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma+\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
\right\}  &amp; \dots(d)
\end{matrix}&lt;/math&gt;

He noted that those quantities become the “frequently employed” formulas of Lagrange [i.e. the coefficients of the Euclidean orthogonal transformation first given by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]]] by setting:&lt;ref group=M&gt;Chasles (1829), p. 141&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\gamma\quad\Rightarrow\quad-\gamma\sqrt{-1}\\
\gamma'\quad\Rightarrow\quad-\gamma'\sqrt{-1}\\
\alpha''\quad\Rightarrow\quad\alpha''\sqrt{-1}\\
\beta''\quad\Rightarrow\quad\beta''\sqrt{-1}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Equations (a,b,c,d) are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).}}

Chasles now showed that equation systems (a,b,c,d) are of importance when discussing the relations between conjugate diameters of hyperboloids. He used the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid having the same principal axes (x,y,z), thus sharing the same conjugate axes, and having the common asymptotic cone &lt;math&gt;\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0&lt;/math&gt;. He then transformed those two hyperboloids to new axes (x',y',z') sharing the property of conjugacy:&lt;ref group=M&gt;Chasles (1829), pp. 143-144&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\
\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\
\hline \begin{align}x &amp; =lx'+l'y'+l''z'\\
y &amp; =mx'+m'y'+m''z'\\
z &amp; =nx'+n'y'+n''z'
\end{align}
\\
\left\{ \begin{align}\frac{ll'}{a{{}^2}}+\frac{mm'}{b^{2}}-\frac{nn'}{c^{2}} &amp; =0\\
\frac{ll''}{a{{}^2}}+\frac{mm''}{b^{2}}-\frac{nn''}{c^{2}} &amp; =0\\
\frac{l'l''}{a{{}^2}}+\frac{m'm''}{b^{2}}-\frac{n'n''}{c^{2}} &amp; =0
\end{align}
\right\} \\
\hline \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=1\\
\left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=-1
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Chasles defined the conditional equations of ''l,m,n'' in the same way as those of &lt;math&gt;\alpha,\beta,\gamma&lt;/math&gt; in equation system (b) above, so his transformation of x,y,z into x',y',z' represents Lorentz transformation ({{equationNote|1a}}, n=2) by applying equation system (a) as well.}}

He went on to use two semi-diameters of the one-sheet hyperboloid and one semi-diameter of the two-sheet hyperboloid in order to define equation system (A), and went on to suggest that the other equations related to this system can be obtained using the above transformation from oblique coordinates to other oblique ones, but he deemed it more simple to use a geometric argument to obtain system (B), which together with (A) then allowed him to algebraically determine systems (C), (D) and additional ones, leading Chasles to announce that “''from these formulas one can very easily conclude the various properties of conjugated diameters of hyperboloids''”:&lt;ref group=M&gt;Chasles (1829), pp. 145-146&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =a^{2}\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =b^{2}\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =-c^{2}
\end{align}
\right\}  &amp; \dots(A)\\
\left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma' &amp; =0
\end{align}
\right\}  &amp; \dots(B)\\
\left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =a^{2}\\
\beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} &amp; =b^{2}\\
\gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-c^{2}
\end{align}
\right\}  &amp; \dots(C)\\
\left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma+\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
\right\}  &amp; \dots(D)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Equation systems (A,B,C,D), being equivalent to systems (a,b,c,d) above, are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.}}

==={{anchor|Lebesgue}} Lebesgue (1837) – Homogeneous coordinates===

[[w:Victor-Amédée Lebesgue]] (1837) summarized the previous work of [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Cauchy|Cauchy (1829)]]. He started with the orthogonal transformation&lt;ref group=M&gt;Lebesgue (1837), pp. 338-341&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}\ (9)\\
\hline {\scriptstyle \begin{align}x_{1} &amp; =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\
x_{2} &amp; =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\
\dots\\
x_{n} &amp; =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\
\\
y_{1} &amp; =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n,1}x_{n}\\
y_{2} &amp; =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n,2}x_{n}\ (12)\ \\
\dots\\
y_{n} &amp; =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n,n}x_{n}
\end{align}
\left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n,\alpha}^{2} &amp; =1 &amp; (10)\\
a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n,\alpha}a_{n,\beta} &amp; =0 &amp; (11)\\
a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n}^{2} &amp; =1 &amp; (13)\\
a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n}a_{\beta,n} &amp; =0 &amp; (14)
\end{align}
\right.}
\end{matrix}&lt;/math&gt;

In order to achieve the invariance of the Lorentz interval&lt;ref group=M&gt;Lebesgue (1837), pp. 353–354&lt;/ref&gt;

:&lt;math&gt;x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}&lt;/math&gt;

he gave the following instructions as to how the previous equations shall be modified: In equation (9) change the sign of the last term of each member. In the first ''n-1'' equations of (10) change the sign of the last term of the left-hand side, and in the one which satisfies α=''n'' change the sign of the last term of the left-hand side as well as the sign of the right-hand side. In all equations (11) the last term will change sign. In equations (12) the last terms of the right-hand side will change sign, and so will the left-hand side of the ''n''-th equation. In equations (13) the signs of the last terms of the left-hand side will change, moreover in the ''n''-th equation change the sign of the right-hand side. In equations (14) the last terms will change sign.

{{Lorentzbox|Text=These instructions give Lorentz transformation ({{equationNote|1a}}) in the form:
:&lt;math&gt;{\scriptstyle \begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\
\hline \begin{align}x_{1} &amp; =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\
x_{2} &amp; =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\
\dots\\
x_{n} &amp; =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\
\\
y_{1} &amp; =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n-1,1}x_{n-1}-a_{n,1}x_{n}\\
y_{2} &amp; =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n-1,2}x_{n-1}-a_{n,2}x_{n}\\
\dots\\
-y_{n} &amp; =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n-1,n}x_{n-1}-a_{n,n}x_{n}
\end{align}
\left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n-1,\alpha}^{2}-a_{n,\alpha}^{2} &amp; =1\\
a_{1,n}^{2}+a_{2,n}^{2}+\dots+a_{n-1,n}^{2}-a_{n,n}^{2} &amp; =-1\\
a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n-1,\alpha}a_{n-1,\beta}-a_{n,\alpha}a_{n,\beta} &amp; =0\\
a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n-1}^{2}-a_{\alpha,n}^{2} &amp; =1\\
a_{n,1}^{2}+a_{n,2}^{2}+\dots+a_{n,n-1}^{2}-a_{n,n}^{2} &amp; =-1\\
a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n-1}a_{\beta,n-1}-a_{\alpha,n}a_{\beta,n} &amp; =0
\end{align}
\right.
\end{matrix}}&lt;/math&gt;}}

He went on to redefine the variables of the Lorentz interval and its transformation:&lt;ref group=M&gt;Lebesgue (1837), pp. 353–355&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\
\downarrow\\
\begin{align}x_{1} &amp; =x_{n}\cos\theta_{1}, &amp; x_{2} &amp; =x_{n}\cos\theta_{2},\dots &amp; x_{n-1} &amp; =x_{n}\cos\theta_{n-1}\\
y_{1} &amp; =y_{n}\cos\phi_{1}, &amp; y_{2} &amp; =y_{n}\cos\phi_{2},\dots &amp; y_{n-1} &amp; =y_{n}\cos\phi_{n-1}
\end{align}
\\
\downarrow\\
\cos^{2}\theta_{1}+\cos^{2}\theta_{2}+\dots+\cos^{2}\theta_{n-1}=1\\
\cos^{2}\phi_{1}+\cos^{2}\phi_{2}+\dots+\cos^{2}\phi_{n-1}=1\\
\hline \\
\cos\theta_{i}=\frac{a_{i,1}\cos\phi_{1}+a_{i,2}\cos\phi_{2}+\dots+a_{i,n-1}\cos\phi_{n-1}+a_{i,n}}{a_{n,1}\cos\phi_{1}+a_{n,2}\cos\phi_{2}+\dots+a_{n,n-1}\cos\phi_{n-1}+a_{n,n}}\\
(i=1,2,3\dots n)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Setting &lt;math&gt;[\cos\theta_{i},\ \cos\phi_{i}]=\left[u_{s},\ u_{s}^{\prime}\right]&lt;/math&gt; it is equivalent to Lorentz transformation ({{equationNote|1b}}).}}

==={{anchor|Weddle}} Weddle (1847) – Conjugate hyperboloids===
Very similar to [[#Chasles|Chasles (1829)]], though without reference to him, [[w:Thomas Weddle]] discussed conjugate hyperboloids using the following equation system (&amp;alpha;), from which he derived equations (&amp;beta;) and others:&lt;ref group=M&gt;Weddle (1847), p. 274&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} &amp; =1, &amp; l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} &amp; =0\\
l_{2}^{2}+m_{2}^{2}-n_{2}^{2} &amp; =1, &amp; l_{1}l_{3}+m_{1}m_{3}-n_{1}n_{3} &amp; =0\\
l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =-1, &amp; l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} &amp; =0
\end{align}
\right\}  &amp; \dots(\alpha)\\ \\
\left.\begin{align}l_{1}^{2}+l_{2}^{2}-l_{3}^{2} &amp; =1, &amp; l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} &amp; =0\\
m_{1}^{2}+m_{2}^{2}-m_{3}^{2} &amp; =1, &amp; l_{1}n_{1}+l_{2}n_{2}-l_{3}n_{3} &amp; =0\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =-1, &amp; m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} &amp; =0
\end{align}
\right\}  &amp; \dots(\beta)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).}}

Using the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid sharing the same conjugate axes, and having the common asymptotic cone &lt;math&gt;\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0&lt;/math&gt;, he defined three conjugate points &lt;math&gt;(x_{1}\dots,y_{1}\dots,z_{1}\dots)&lt;/math&gt; on those two conjugate hyperboloids, related to each other in the same way as equations (&amp;alpha;, &amp;beta;) stated above:&lt;ref group=M&gt;Weddle (1847), pp. 275-276&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\
\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\
\hline \begin{align}\frac{x_{1}x_{2}}{a{{}^2}}+\frac{y_{1}y_{2}}{b^{2}}-\frac{z_{1}z_{2}}{c^{2}} &amp; =0\\
\frac{x_{1}x_{3}}{a{{}^2}}+\frac{y_{1}y_{3}}{b^{2}}-\frac{z_{1}z_{3}}{c^{2}} &amp; =0\\
\frac{x_{2}x_{3}}{a{{}^2}}+\frac{y_{2}y_{3}}{b^{2}}-\frac{z_{2}z_{3}}{c^{2}} &amp; =0
\end{align}
\quad\begin{align}\frac{x_{1}^{2}}{a{{}^2}}+\frac{y_{1}^{2}}{b^{2}}-\frac{z_{1}^{2}}{c^{2}} &amp; =1\\
\frac{x_{2}^{2}}{a{{}^2}}+\frac{y_{2}^{2}}{b^{2}}-\frac{z_{2}^{2}}{c^{2}} &amp; =1\\
\frac{x_{3}^{2}}{a{{}^2}}+\frac{y_{3}^{2}}{b^{2}}-\frac{z_{3}^{2}}{c^{2}} &amp; =-1
\end{align}
\\
\begin{align}x_{1}^{2}+x_{2}^{2}-x_{3}^{2} &amp; =a^{2}\\
y_{1}^{2}+y_{2}^{2}-y_{3}^{2} &amp; =b^{2}\\
z_{1}^{2}+z_{2}^{2}-z_{3}^{2} &amp; =-c^{2}
\end{align}
\quad\begin{align}x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3} &amp; =0\\
x_{1}z_{1}+x_{2}z_{2}-x_{3}z_{3} &amp; =0\\
y_{1}z_{1}+y_{2}z_{2}-y_{3}z_{3} &amp; =0
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text= These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.}}

==={{anchor|Bour}} Bour (1856) – Homogeneous coordinates===

Following [[#Gauss4|Gauss (1818)]], [[w:Edmond Bour]] (1856) wrote the transformations:&lt;ref group=M&gt;Bour (1856), pp. 61; 64–65&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\cos^{2}E+\sin^{2}E-1=k\left(\cos^{2}T+\sin^{2}T-1\right)\\
\hline \left.\begin{matrix}\mathbf{(1)}\ \begin{align}\cos E &amp; =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\
\sin E &amp; =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}
\end{align}
\\
\hline \\
k=+1\\
t=\gamma+\gamma'\cos T+\gamma''\sin T,\\
1=u,\ \cos T=u',\ \sin T=u',\\
t=z,\ t\cos E=x,\ t\sin E=y\\
\downarrow\\
\mathbf{(2)}\begin{align}x &amp; =\alpha u+\alpha'u'+\alpha''u''\\
y &amp; =\beta u+\beta'u'+\beta''u''\\
z &amp; =\gamma u+\gamma'u'+\gamma''u''\\
\\
u &amp; =\gamma z-\alpha x-\beta y\\
u' &amp; =\alpha'x+\beta'y'-\gamma'z\\
u'' &amp; =\alpha''x+\beta''y-\gamma''z
\end{align}
\end{matrix}\right|{\scriptstyle \begin{align}-\alpha^{2}-\beta^{2}+\gamma^{2} &amp; =k\\
-\alpha^{\prime2}-\beta^{\prime2}+\gamma^{\prime2} &amp; =-k\\
-\alpha^{\prime\prime2}-\beta^{\prime\prime2}+\gamma^{\prime\prime2} &amp; =-k\\
\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\\
\alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =-k\\
\beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} &amp; =-k\\
\gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =k\\
\beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0\\
\alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0
\end{align}
}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'', implying &lt;math&gt;x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}&lt;/math&gt;. Furthermore, setting &lt;math&gt;[k,\cos T,\sin T,\cos E,\sin E]=\left[1,u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]&lt;/math&gt; in transformation system (1) produces Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.}}

=== {{anchor|Somov}} Somov (1863) – Homogeneous coordinates ===

Following [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], and [[#Bour|Bour (1856)]], [[w:Osip Ivanovich Somov]] (1863) wrote the transformation systems:&lt;ref group=M&gt;Somov (1863), pp. 12–14; p. 18 for differentials.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}\cos\phi &amp; =\frac{m\cos\psi+n\sin\psi+s}{m''\cos\psi+n''\sin\psi+s''}\\
\sin\phi &amp; =\frac{m'\cos\psi+n'\sin\psi+s'}{m''\cos\psi+n''\sin\psi+s''}
\end{align}
\right|\begin{matrix}\cos^{2}\phi+\cos^{2}\phi=1\\
\cos^{2}\psi+\cos^{2}\psi=1
\end{matrix}\\
\hline \mathbf{(1)}\ \begin{align}\cos\phi &amp; =x, &amp; \cos\psi &amp; =x'\\
\sin\phi &amp; =y, &amp; \sin\psi &amp; =y'
\end{align}
\ \left|\begin{align}x &amp; =\frac{mx'+ny'+s}{m''x'+n''y'+s''}\\
y &amp; =\frac{m'x'+n'y'+s'}{m''x'+n''y'+s''}
\end{align}
\right|\ \begin{matrix}x^{2}+y^{2}=1\\
x^{\prime2}+y^{\prime2}=1
\end{matrix}\\
\hline \begin{align}\cos\phi &amp; =\frac{x}{z}, &amp; \cos\psi &amp; =\frac{x'}{z'}\\
\sin\phi &amp; =\frac{y}{z}, &amp; \sin\psi &amp; =\frac{y'}{z'}
\end{align}
\ \left|\begin{align}\frac{x}{z} &amp; =\frac{mx'+ny'+sz'}{m''x'+n''y'+s''z'}\\
\frac{y}{z} &amp; =\frac{m'x'+n'y'+s'z'}{m''x'+n''y'+s''z'}
\end{align}
\right|\ \begin{matrix}x^{2}+y^{2}=z^{2}\\
x^{\prime2}+y^{\prime2}=z^{\prime2}
\end{matrix}\\
\hline \mathbf{(2)}\ \left.\begin{align}x &amp; =mx'+ny'+sz'\\
y &amp; =m'x'+n'y'+s'z'\\
z &amp; =m''x'+n''y'+s''z'\\
\\
x' &amp; =mx+m'y-m''z\\
y' &amp; =nx+n'y-n''z\\
z' &amp; =-sx-s'y+s''z\\
\\
dx &amp; =mdx'+ndy'+sdz'\\
dy &amp; =m'dx'+n'dy'+s'dz'\\
dz &amp; =m''dx'+n''dy'+s''dz'
\end{align}
\right|{\scriptstyle \begin{align}m^{2}+m^{\prime2}-m^{\prime\prime2} &amp; =1\\
n^{2}+n^{\prime2}-n^{\prime\prime2} &amp; =1\\
-s^{2}-s^{\prime2}+s^{\prime\prime2} &amp; =1\\
ns+n's'-n''s'' &amp; =0\\
sm+s'm'-s''m'' &amp; =0\\
mn+m'n'-m''n'' &amp; =0\\
\\
m^{2}+n^{2}-s^{2} &amp; =1\\
m^{\prime2}+n^{\prime2}-s^{\prime2} &amp; =1\\
-m^{\prime\prime2}-n^{\prime\prime2}+s^{\prime\prime2} &amp; =1\\
-m'm''-n'n''+s's'' &amp; =0\\
-m''m-n''n+s''s &amp; =0\\
mm'+nn'-ss' &amp; =0
\end{align}
}\\
dx^{2}+dy^{2}-dz^{2}=dx^{\prime2}+dy^{\prime2}-dz^{\prime2}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.

Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

==={{anchor|Klein}} Klein (1871-73) – Cayley absolute and non-Euclidean geometry===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Klein|label 1=History of Lorentz transformations via Möbius transformations § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Klein3|label 1=History of Lorentz transformations via sphere transformations § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Noether|label 1=History of Lorentz transformations via Quaternions § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Klein|label 1=History of Lorentz transformations via squeeze mappings § Klein}}

Elaborating on [[w:Arthur Cayley]]'s (1859) definition of an &quot;absolute&quot; ([[w:Cayley–Klein metric]]), [[w:Felix Klein]] (1871) defined a &quot;fundamental [[w:conic section]]&quot; in order to discuss motions such as rotation and translation in the non-Euclidean plane.&lt;ref group=M&gt;Klein (1871), pp. 601–602&lt;/ref&gt; This was elaborated in (1873) when he pointed out that hyperbolic geometry in terms of a surface of constant negative curvature can be related to a quadratic equation, which can be transformed into a sum of squares of which one square has a different sign, and can also be related to the interior of a surface of second degree corresponding to a two-sheet [[w:hyperboloid]].&lt;ref group=M&gt;Klein (1873), pp. 127-128&lt;/ref&gt;

{{Lorentzbox|Text=Klein's representation of hyperbolic space in terms of a two-sheet hyperboloid and its accompanied quadratic form suggests that Lorentz transformations can be geometrically interpreted as motions or isometries in hyperbolic space.}}

==={{anchor|Killing}} Killing (1878–1893)===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Killing2|label 1=History of Lorentz transformations via hyperbolic functions § Killing}}
===={{anchor|Killing1}} Weierstrass coordinates====

[[w:Wilhelm Killing]] (1878–1880) described non-Euclidean geometry by using [[w:hyperboloid model|Weierstrass coordinates]] (named after [[w:Karl Weierstrass]] who described them in lectures in 1872 which Killing attended) obeying the form

:&lt;math&gt;k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}&lt;/math&gt;&lt;ref group=M&gt;Killing (1877/78), p. 74; Killing (1880), p. 279&lt;/ref&gt; with &lt;math&gt;ds^{2}=k^{2}dt^{2}+du^{2}+dv^{2}+dw^{2}&lt;/math&gt;&lt;ref group=M&gt;Killing (1880), eq. 25 on p. 283&lt;/ref&gt;

or&lt;ref group=M&gt;Killing (1880), p. 283&lt;/ref&gt;

:&lt;math&gt;k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}&lt;/math&gt;

where ''k'' is the reciprocal measure of curvature, &lt;math&gt;k^{2}=\infty&lt;/math&gt; denotes [[w:Euclidean geometry]], &lt;math&gt;k^{2}&gt;0&lt;/math&gt; [[w:elliptic geometry]], and &lt;math&gt;k^{2}&lt;0&lt;/math&gt; hyperbolic geometry. In (1877/78) he pointed out the possibility and some characteristics of a transformation (indicating rigid motions) preserving the above form.&lt;ref group=M&gt;Killing (1877/78), eq. 25 on p. 283&lt;/ref&gt; In (1879/80) he tried to formulate the corresponding transformations by plugging &lt;math&gt;k^{2}&lt;/math&gt; into a [[w:Rotation matrix#Rotation matrix from axis and angle|general rotation matrix]]:&lt;ref group=M&gt;Killing (1879/80), p. 274&lt;/ref&gt;

&lt;math&gt;\begin{matrix}k^{2}u^{2}+v^{2}+w^{2}=k^{2}\\
\hline \begin{matrix}\cos\eta\tau+\lambda^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \nu\frac{\sin\eta\tau}{\eta}+\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; -\mu\frac{\sin\eta\tau}{\eta}+\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}\\
-k^{2}\nu\frac{\sin\eta\tau}{\eta}+k^{2}\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \cos\eta\tau+\mu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}\\
k^{2}\mu\frac{\sin\eta\tau}{\eta}+k^{2}\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; -\lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \cos\eta\tau+\nu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}
\end{matrix}\\
\left(\lambda^{2}+k^{2}\mu^{2}+k^{2}\nu^{2}=\eta^{2}\right)
\end{matrix}&lt;/math&gt;

In (1885) he wrote the Weierstrass coordinates and their transformation as follows:&lt;ref group=M&gt;Killing (1885), pp. 18, 28–30, 53&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}p^{2}+x^{2}+y^{2}=k^{2}\\
k^{2}p^{2}+x^{2}+y^{2}=k^{2}p^{\prime2}+x^{\prime2}+y^{\prime2}\\
ds^{2}=k^{2}dp^{2}+dx^{2}+dy^{2}\\
\hline \begin{align}k^{2}p' &amp; =k^{2}wp+w'x+w''y\\
x' &amp; =ap+a'x+a''y\\
y' &amp; =bp+b'x+b''y\\
\\
k^{2}p &amp; =k^{2}wp'+ax'+by'\\
x &amp; =w'p'+a'x+b'y'\\
y &amp; =w''p'+a''x'+b''y'
\end{align}
\left|{\scriptstyle \begin{align}k^{2}w^{2}+w^{\prime2}+w^{\prime\prime2} &amp; =k^{2}\\
\frac{a^{2}}{k^{2}}+a^{\prime2}+a^{\prime\prime2} &amp; =1\\
\frac{b^{2}}{k^{2}}+b^{\prime2}+b^{\prime\prime2} &amp; =1\\
aw+a'w'+a''w'' &amp; =0\\
bw+b'w'+b''w'' &amp; =0\\
\frac{ab}{k^{2}}+a'b'+a''b'' &amp; =0\\
\\
k^{2}w^{2}+a^{2}+b^{2} &amp; =k^{2}\\
\frac{w^{\prime2}}{k^{2}}+a^{\prime2}+b^{\prime2} &amp; =1\\
\frac{w^{\prime\prime2}}{k^{2}}+a^{\prime\prime2}+b^{\prime\prime2} &amp; =1\\
ww'+aa'+bb' &amp; =0\\
ww''+aa''+bb'' &amp; =0\\
\frac{w'w''}{k^{2}}+a'a''+b'b'' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

In (1885) he also gave the transformation for ''n'' dimensions:&lt;ref group=M&gt;Killing (1884/85), pp. 42–43; Killing (1885), pp. 73–74, 222&lt;/ref&gt;&lt;ref&gt;Ratcliffe (1994), § 3.6&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\
ds^{2}=k^{2}dx_{0}^{2}+dx_{1}^{2}+\dots+dx_{n}^{2}\\
\hline \left.\begin{align}k^{2}\xi_{0} &amp; =k^{2}a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{0}\\
\xi_{\varkappa} &amp; =a_{\varkappa0}x_{0}+a_{\varkappa1}x_{1}+\dots+a_{\varkappa n}x_{n}\\
\\
k^{2}x_{0} &amp; =a_{00}k^{2}\xi_{0}+a_{10}\xi_{1}+\dots+a_{n0}\xi_{n}\\
x_{\varkappa} &amp; =a_{0\varkappa}\xi_{0}+a_{1\varkappa}\xi_{1}+\dots+a_{n\varkappa}\xi_{n}
\end{align}
\right|{\scriptstyle \begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} &amp; =k^{2}\\
a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} &amp; =0\\
\frac{a_{0\iota}a_{0\varkappa}}{k^{2}}+a_{0\iota}a_{1\varkappa}+\dots+a_{n\iota}a_{n\varkappa}=\delta_{\iota\kappa} &amp; =1\ (\iota=\kappa)\ \text{or}\ 0\ (\iota\ne\kappa)
\end{align}
}
\end{matrix}&lt;/math&gt;

In (1885) he applied his transformations to mechanics and defined four-dimensional vectors of velocity and force.&lt;ref group=M&gt;Killing (1884/85), pp. 4–5&lt;/ref&gt; Regarding the geometrical interpretation of his transformations, Killing argued in (1885) that by setting &lt;math&gt;k^{2}=-1&lt;/math&gt; and using ''p,x,y'' as rectangular space coordinates, the hyperbolic plane is mapped on one side of a two-sheet hyperboloid &lt;math&gt;p^{2}-x^{2}-y^{2}=1&lt;/math&gt; (known as [[w:hyperboloid model]]),&lt;ref group=M&gt;Killing (1885), Note 9 on p. 260&lt;/ref&gt;&lt;ref name=rey /&gt; by which the previous formulas become equivalent to Lorentz transformations and the geometry becomes that of Minkowski space.

{{Lorentzbox|Text=All of Killing's transformations between 1879 and 1885 don't work when &lt;math&gt;k^{2}&lt;/math&gt; is negative, thus they fail to produce Lorentz transformation ({{equationNote|1a}}) with &lt;math&gt;k^{2}=-1&lt;/math&gt;.}}

Finally, in (1893) he wrote:&lt;ref group=M&gt;Killing (1893), see pp. 144, 327–328&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}t^{2}+u^{2}+v^{2}=k^{2}\\
\hline \begin{align}t' &amp; =at+bu+cv\\
u' &amp; =a't+b'u+c'v\\
v' &amp; =a''t+b''u+c''v
\end{align}
\left|\begin{align}k^{2}a^{2}+a^{\prime2}+a^{\prime\prime2} &amp; =k^{2}\\
k^{2}b^{2}+b^{\prime2}+b^{\prime\prime2} &amp; =1\\
k^{2}c^{2}+b^{\prime2}+c^{\prime\prime2} &amp; =1\\
k^{2}ab+a'b'+a''b'' &amp; =0\\
k^{2}ac+a'c'+a''c'' &amp; =0\\
k^{2}bc+b'c'+b''c'' &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

and in ''n'' dimensions&lt;ref group=M&gt;Killing (1893), pp. 314–316, 216–217&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\
k^{2}y_{0}y_{0}^{\prime}+y_{1}y_{1}^{\prime}+\cdots+y_{n}y_{n}^{\prime}=k^{2}x_{0}x_{0}^{\prime}+x_{1}x_{1}^{\prime}+\cdots+x_{n}x_{n}^{\prime}\\
ds^{2}=k^{2}dx_{0}^{2}+\dots+dx_{n}^{2}\\
\hline \begin{align}y_{0} &amp; =a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{n}\\
y_{1} &amp; =a_{10}x_{0}+a_{11}x_{1}+\dots+a_{1n}x_{n}\\
 &amp; \,\,\,\vdots\\
y_{n} &amp; =a_{n0}x_{0}+a_{n1}x_{1}+\dots+a_{nn}x_{n}
\end{align}
\left|\begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} &amp; =k^{2}\\
k^{2}a_{0\varkappa}^{2}+a_{1\varkappa}^{2}+\dots+a_{n\varkappa}^{2} &amp; =1\\
k^{2}a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} &amp; =0\\
k^{2}a_{0\varkappa}a_{0\lambda}+a_{1\varkappa}a_{1\lambda}+\dots+a_{n\varkappa}a_{n\lambda} &amp; =0\\
(\varkappa,\lambda=1,\dots, n,\ \lambda\lessgtr\varkappa)
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) with &lt;math&gt;k^{2}=-1&lt;/math&gt;.}}

===={{anchor|Killing3}} Infinitesimal transformations and Lie group====

After [[#Lie3|Lie (1885/86)]] identified the projective group of a general surface of second degree &lt;math&gt;\sum f_{ik}x_{i}'x_{k}'=0&lt;/math&gt; with the group of non-Euclidean motions, Killing (1887/88)&lt;ref group=M&gt;Killing (1887/88a), pp. 274–275&lt;/ref&gt; defined the infinitesimal projective transformations (Lie algebra) in relation to the unit hypersphere:

:&lt;math&gt;\begin{matrix}x_{1}^{2}+\dots+x_{m+1}^{2}=1\\
\hline X_{\iota\varkappa}f=x_{i}\frac{\partial f}{\partial x_{\varkappa}}-x_{\varkappa}\frac{\partial f}{\partial x_{\iota}}\\
\text{where}\\
\left(X_{\iota\varkappa},X_{\iota\lambda}\right)=X_{\varkappa\lambda};\ \left(X_{\iota\varkappa},X_{\lambda\mu}\right)=0;\\
\left[\iota\ne\varkappa\ne\lambda\ne\mu\right]
\end{matrix}&lt;/math&gt;

and in (1892) he defined the infinitesimal transformation for non-Euclidean motions in terms of Weierstrass coordinates:&lt;ref group=M&gt;Killing (1892), p. 177&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\
\hline X_{\iota\varkappa}=x_{\iota}p_{\varkappa}-x_{\varkappa}p_{\iota},\quad X_{\iota}=x_{0}p_{\iota}-\frac{x_{\iota}p_{0}}{k^{2}}\\
\text{where}\\
\left(X_{\iota}X_{\iota\varkappa}\right)=X_{\varkappa}f;\ \left(X_{\iota}X_{\varkappa\lambda}\right)=0;\ \left(X_{\iota}X_{\varkappa}\right)=-\frac{1}{k^{2}}X_{\iota\varkappa}f;
\end{matrix}&lt;/math&gt;

In (1897/98) he showed the relation between Weierstrass coordinates &lt;math&gt;k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}&lt;/math&gt; and coordinates &lt;math&gt;k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0&lt;/math&gt; used by himself in (1887/88) and by [[#Lie3|Werner (1889), Lie (1890)]]:&lt;ref group=M&gt;Killing (1897/98), pp. 255–256&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2} &amp; (a)\\
k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2} &amp; (b)
\end{matrix}\\
\hline V_{\varkappa}=k^{2}x_{0}p_{\varkappa}-x_{\varkappa}p_{0},\quad U_{\iota\varkappa}=p_{\iota}x_{\varkappa}-p_{\varkappa}x_{\iota}\\
\text{where}\\
\left(V_{\iota},V_{\varkappa}\right)=k^{2}U_{\iota\varkappa},\ \left(V_{\iota},U_{\iota\varkappa}\right)=-V_{\varkappa},\ \left(V_{\iota},U_{\varkappa\lambda}\right)=0,\\
\left(U_{\iota\varkappa},U_{\iota\lambda}\right)=U_{\varkappa\lambda},\ \left(U_{\iota\varkappa},U_{\lambda\mu}\right)=0\\
\left[\iota,\varkappa,\lambda,\mu=1,2,\dots n\right]\\
\hline \begin{matrix}y_{1}=\frac{x_{1}}{x_{0}},\ y_{2}=\frac{x_{2}}{x_{0}},\dots y_{n}=\frac{x_{n}}{x_{0}}\\
\downarrow\\
k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0\\
\hline q_{\varkappa}+\frac{y_{\varkappa}}{k^{2}}\sum_{\varrho}y_{y}q_{\varrho},\quad q_{\iota}y_{\varkappa}-q_{\varkappa}y_{\iota}
\end{matrix}
\end{matrix}&lt;/math&gt;

He pointed out that the corresponding group of non-Euclidean motions in terms of Weierstrass coordinates is intransitive when related to quadratic form (a) and [[w:Group action (mathematics)|transitive]] when related to quadratic form (b).

{{Lorentzbox|Text=Setting &lt;math&gt;k^{2}=-1&lt;/math&gt; denotes the group of hyperbolic motions and thus the Lorentz group.}}

=== {{anchor|Poincare}} Poincaré (1881) – Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Poincare2|label 1=History of Lorentz transformations via Möbius transformations § Poincaré}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (velocity)#Poincare3|label 1=History of Lorentz transformations via velocity § Poincaré}}

[[w:Henri Poincaré]] (1881) connected the work of [[../Lorentz transformation (Cayley-Hermite)#Hermite|E:Hermite (1853)]] and [[../Lorentz transformation (Möbius)#Selling|E:Selling (1873)]] on indefinite quadratic forms with non-Euclidean geometry (Poincaré already discussed such relations in an unpublished manuscript in 1880).&lt;ref&gt;Gray (1997)&lt;/ref&gt; He used two indefinite ternary forms in terms of three squares and then defined them in terms of Weierstrass coordinates (without using that expression) connected by a transformation with integer coefficients:&lt;ref group=M name=p1&gt;Poincaré (1881a), pp. 133–134&lt;/ref&gt;&lt;ref&gt;Dickson (1923), pp. 220–221&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}F &amp; =(ax+by+cz)^{2}+(a'x+b'y+c'z)^{2}-(a''x+b''y+c''z)^{2}\\
 &amp; =\xi^{2}+\eta^{2}-\zeta^{2}=-1\\
F &amp; =(ax'+by'+cz')^{2}+(a'x'+b'y'+c'z')^{2}-(a''x'+b''y'+c''z')^{2}\\
 &amp; =\xi^{\prime2}+\eta^{\prime2}-\zeta^{\prime2}=-1
\end{align}
\\
\hline \begin{align}\xi' &amp; =\alpha\xi+\beta\eta+\gamma\zeta\\
\eta' &amp; =\alpha'\xi+\beta'\eta+\gamma'\zeta\\
\zeta' &amp; =\alpha''\xi+\beta''\eta+\gamma''\zeta
\end{align}
\left|\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =1\\
\beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} &amp; =1\\
\gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-1\\
\alpha\beta+\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma+\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

He went on to describe the properties of &quot;hyperbolic coordinates&quot;.&lt;ref group=M name=poinc&gt;Poincaré (1881b), p. 333&lt;/ref&gt;&lt;ref name=rey&gt;Reynolds (1993)&lt;/ref&gt; Poincaré mentioned the hyperboloid model also in (1887).&lt;ref group=M&gt;Poincaré (1887), p. 206&lt;/ref&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

=== {{anchor|Cox}} Cox (1881–1891) – Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Cox|label 1=History of Lorentz transformations via hyperbolic functions § Cox}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Cox2|label 1=History of Lorentz transformations via Quaternions § Cox}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Cox|label 1=History of Lorentz transformations via sphere transformations § Cox}}

[[w:Homersham Cox (mathematician)|Homersham Cox]] (1881/82) – referring to similar rectangular coordinates used by [[w:Christoph Gudermann|Gudermann]] (1830)&lt;ref name=guder group=M&gt;Gudermann (1830), §1–3, §18–19&lt;/ref&gt; and [[w:George Salmon]] (1862)&lt;ref group=M&gt;Salmon (1862), section 212, p. 165&lt;/ref&gt; on a sphere, and to [[#Escherich|Escherich (1874)]] as reported by [[w:Johannes Frischauf]] (1876)&lt;ref group=M&gt;Frischauf (1876), pp. 86–87&lt;/ref&gt; in the hyperbolic plane – defined the Weierstrass coordinates (without using that expression) and their transformation:&lt;ref group=M&gt;Cox (1881/82), p. 186 for Weierstrass coordinates; pp. 193–194 for Lorentz transformation.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}z^{2}-x^{2}-y^{2}=1\\
x^{2}-y^{2}-z^{2}=Z^{2}-Y^{2}-X^{2}\\
\hline \begin{align}x &amp; =l_{1}X+l_{2}Y+l_{3}Z\\
y &amp; =m_{1}X+m_{2}Y+m_{3}Z\\
z &amp; =n_{1}X+n_{2}Y+n_{3}Z\\
\\
X &amp; =l_{1}x+m_{1}y-n_{1}z\\
Y &amp; =l_{2}x+m_{2}y-n_{2}z\\
Z &amp; =l_{3}x+m_{3}y-n_{3}z
\end{align}
\left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} &amp; =1\\
l_{2}^{2}+m_{2}^{2}-n_{2}^{2} &amp; =1\\
l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =1\\
l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} &amp; =0\\
l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} &amp; =0\\
l_{3}l_{1}+m_{3}m_{1}-n_{3}n_{1} &amp; =0\\
\\
l_{1}^{2}+l_{2}^{2}-l_{3}^{2} &amp; =1\\
m_{1}^{2}+m_{2}^{2}-m_{3}^{2} &amp; =1\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =1\\
l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} &amp; =0\\
m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} &amp; =0\\
n_{1}l_{1}+n_{2}l_{2}-n_{3}l_{3} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=These equations contain several errors or misprints: &lt;math&gt;Z^{2}-Y^{2}-X^{2}&lt;/math&gt; has to be replaced by &lt;math&gt;X^{2}-Y^{2}-Z^{2}&lt;/math&gt;, and &lt;math&gt;{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =1\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =1
\end{align}
}&lt;/math&gt; replaced with &lt;math&gt;{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =-1\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =-1
\end{align}
}&lt;/math&gt;, and by reversing the sign of &lt;math&gt;Z&lt;/math&gt; in the inverse transformation, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

Cox (1881/82) also gave the Weierstrass coordinates and their transformation in hyperbolic space:&lt;ref group=M&gt;Cox (1881/82), pp. 199, 206–207&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}w^{2}-x^{2}-y^{2}-z^{2}=1\\
w^{2}-x^{2}-y^{2}-z^{2}=w^{\prime2}-x^{\prime2}-y^{\prime2}-z^{\prime2}\\
\hline \begin{align}x &amp; =l_{1}x'+l_{2}y'+l_{3}z'-l_{4}w'\\
y &amp; =m_{1}x'+m_{2}y'+m_{3}z'-m_{4}w'\\
z &amp; =n_{1}x'+n_{2}y'+n_{3}z'-n_{4}w'\\
w &amp; =r_{1}x'+r_{2}y'+r_{3}z'-r_{4}w'\\
\\
x' &amp; =l_{1}x+m_{1}y+n_{1}z-r_{1}w\\
y' &amp; =l_{2}x+m_{2}y+n_{2}z-r_{2}w\\
z' &amp; =l_{3}x+m_{3}y+n_{3}z-r_{3}w\\
w' &amp; =l_{4}x+m_{4}y+n_{4}z-r_{4}w
\end{align}
\left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}+n_{1}^{2}-r_{1}^{2} &amp; =1\\
l_{2}^{2}+m_{2}^{2}+n_{2}^{2}-r_{2}^{2} &amp; =1\\
l_{3}^{2}+m_{3}^{2}+n_{3}^{2}-r_{3}^{2} &amp; =1\\
l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2} &amp; =1\\
l_{2}l_{3}+m_{2}m_{3}+n_{2}n_{3}-r_{2}r_{3} &amp; =0\\
l_{3}l_{1}+m_{3}m_{1}+n_{3}n_{1}-r_{3}r_{1} &amp; =0\\
l_{1}l_{4}+m_{1}m_{4}+n_{1}n_{4}-r_{1}r_{4} &amp; =0\\
l_{2}l_{4}+m_{2}m_{4}+n_{2}n_{4}-r_{2}r_{4} &amp; =0\\
l_{3}l_{4}+m_{3}m_{4}+n_{3}n_{4}-r_{3}r_{4} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=By replacing &lt;math&gt;{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=1}&lt;/math&gt; with &lt;math&gt;{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=-1}&lt;/math&gt; this represents an improper antichronous Lorentz transformation, which becomes proper orthochronous Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' by reversing the sign of &lt;math&gt;w'&lt;/math&gt; everywhere.}}

In 1883 he formulated relations between [[w:orthogonal circles]] which he identified with the previously (1881/82) given transformations:&lt;ref group=M&gt;Cox (1883), pp. 109ff&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}+z^{2}-w^{2}=0\\
\hline \begin{align}x &amp; =\lambda_{1}X+\lambda_{2}Y+\lambda_{3}Z+\lambda_{4}W\\
y &amp; =\mu_{1}X+\mu_{2}Y+\mu_{3}Z+\mu_{4}W\\
z &amp; =\nu_{1}X+\nu_{2}Y+\nu_{3}Z+\nu_{4}W\\
-w &amp; =\rho_{1}X+\rho_{2}Y+\rho_{3}Z+\rho_{4}W\\
\\
X &amp; =\lambda_{1}x+\mu_{1}y+\nu_{1}z+\rho_{1}w\\
Y &amp; =\lambda_{2}x+\mu_{2}y+\nu_{2}z+\rho_{2}w\\
Z &amp; =\lambda_{3}x+\mu_{3}y+\nu_{3}z+\rho_{3}w\\
-W &amp; =\lambda_{4}x+\mu_{4}y+\nu_{4}z+\rho_{4}w
\end{align}
\left|{\scriptstyle \begin{align}\lambda_{1}^{2}+\mu_{1}^{2}+\nu_{1}^{2}-\rho_{1}^{2} &amp; =1\\
\lambda_{2}^{2}+\mu_{2}^{2}+\nu_{2}^{2}-\rho_{2}^{2} &amp; =1\\
\lambda_{3}^{2}+\mu_{3}^{2}+\nu_{3}^{2}-\rho_{3}^{2} &amp; =1\\
\lambda_{4}^{2}+\mu_{4}^{2}+\nu_{4}^{2}-\rho_{4}^{2} &amp; =-1\\
\lambda_{2}\lambda_{3}+\mu_{2}\mu_{3}+\nu_{2}\nu_{3}-\rho_{2}\rho_{3} &amp; =0\\
\lambda_{3}\lambda_{1}+\mu_{3}\mu_{1}+\nu_{3}\nu_{1}-\rho_{3}\rho_{1} &amp; =0\\
\lambda_{1}\lambda_{2}+\mu_{1}\mu_{2}+\nu_{1}\nu_{2}-\rho_{1}\rho_{2} &amp; =0\\
\lambda_{1}\lambda_{4}+\mu_{1}\mu_{4}+\nu_{1}\nu_{4}-\rho_{1}\rho_{4} &amp; =0\\
\lambda_{2}\lambda_{4}+\mu_{2}\mu_{4}+\nu_{2}\nu_{4}-\rho_{2}\rho_{4} &amp; =0\\
\lambda_{3}\lambda_{4}+\mu_{3}\mu_{4}+\nu_{3}\nu_{4}-\rho_{3}\rho_{4} &amp; =0
\end{align}
}\right.{\scriptstyle \begin{align}\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}-\lambda_{4}^{2} &amp; =1\\
\mu_{1}^{2}+\mu_{2}^{2}+\mu_{3}^{2}-\mu_{4}^{2} &amp; =1\\
\nu_{1}^{2}+\nu_{2}^{2}+\nu_{3}^{2}-\nu_{4}^{2} &amp; =1\\
\rho_{1}^{2}+\rho_{2}^{2}+\rho_{3}^{2}-\rho_{4}^{2} &amp; =-1\\
\lambda_{1}\mu_{1}+\lambda_{2}\mu_{2}+\lambda_{3}\mu_{3}-\lambda_{4}\mu_{4} &amp; =0\\
\lambda_{1}\nu_{1}+\lambda_{2}\nu_{2}+\lambda_{3}\nu_{3}-\lambda_{4}\nu_{4} &amp; =0\\
\lambda_{1}\rho_{1}+\lambda_{2}\rho_{2}+\lambda_{3}\rho_{3}-\lambda_{4}\rho_{4} &amp; =0\\
\mu_{1}\nu_{1}+\mu_{2}\nu_{2}+\mu_{3}\nu_{3}-\mu_{4}\nu_{4} &amp; =0\\
\mu_{1}\rho_{1}+\mu_{2}\rho_{2}+\mu_{3}\rho_{3}-\mu_{4}\rho_{4} &amp; =0\\
\nu_{1}\rho_{1}+\nu_{2}\rho_{2}+\nu_{3}\rho_{3}-\nu_{4}\rho_{4} &amp; =0
\end{align}
}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The relations between &lt;math&gt;\lambda,\mu,\nu,\rho&lt;/math&gt; are correct, even though the transformation still represents an improper antichronous Lorentz transformation, which becomes proper orthochronous Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' by reversing the sign of &lt;math&gt;w&lt;/math&gt; everywhere.}}

Finally, in a treatise on [[w:Hermann Grassmann|w:Grassmann's Ausdehnungslehre]] and circles (1891), he again provided transformations of orthogonal circle systems described by him as being &quot;identical with those for transformation of coordinates in non-Euclidean geometry&quot;:&lt;ref group=M&gt;Cox (1891), pp. 27-28&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}+z^{2}=w^{2}\\
\hline \begin{align}x &amp; =\lambda_{1}x'+\lambda_{2}y'+\lambda_{3}z'+\lambda_{4}w' &amp; \text{(4 equations)}\\
x' &amp; =\lambda_{1}x+\mu_{1}y+\nu_{1}z-\rho_{1}w\\
-w' &amp; =\lambda_{4}x+\mu_{4}y+\nu_{4}z-\rho_{4}w
\end{align}
\\
\hline \begin{align}\lambda_{1}^{2}+\mu_{1}^{2}+\nu_{1}^{2}-\rho_{1}^{2} &amp; =1\\
\lambda_{2}^{2}+\mu_{2}^{2}+\nu_{2}^{2}-\rho_{2}^{2} &amp; =1\\
\lambda_{3}^{2}+\mu_{3}^{2}+\nu_{3}^{2}-\rho_{3}^{2} &amp; =1\\
\lambda_{4}^{2}+\mu_{4}^{2}+\nu_{4}^{2}-\rho_{4}^{2} &amp; =-1\\
\lambda_{1}\lambda_{2}+\mu_{1}\mu_{2}+\nu_{1}\nu_{2}-\rho_{1}\rho_{2} &amp; =0 &amp; \text{(6 equations)}\\
\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}-\lambda_{4}^{2} &amp; =1\\
\rho_{1}^{2}+\rho_{2}^{2}+\rho_{3}^{2}-\rho_{4}^{2} &amp; =-1\\
\lambda_{1}\mu_{1}+\lambda_{2}\mu_{2}+\lambda_{3}\mu_{3}-\lambda_{4}\mu_{4} &amp; =0 &amp; \text{(6 equations)}
\end{align}
\end{matrix}\text{ }&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)''.}}

=== {{anchor|Hill}} Hill (1882) – Homogeneous coordinates ===

Following [[#Gauss4|Gauss (1818)]], [[w:George William Hill]] (1882) formulated the equations&lt;ref group=M&gt;Hill (1882), pp. 323–325&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)\\
k\left(\sin^{2}E+\cos^{2}E-1\right)\\
\hline \begin{align} &amp;  &amp; \cos E' &amp; =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\
 &amp; \mathbf{(1)} &amp; \sin E' &amp; =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\
\hline \\
 &amp;  &amp; x &amp; =\alpha u+\alpha'u'+\alpha''u''\\
 &amp;  &amp; y &amp; =\beta u+\beta'u'+\beta''u''\\
 &amp;  &amp; z &amp; =\gamma u+\gamma'u'+\gamma''u''\\
 &amp; \mathbf{(2)}\\
 &amp;  &amp; u &amp; =-\alpha x-\beta y+\gamma z\\
 &amp;  &amp; u' &amp; =\alpha'x+\beta'y'-\gamma'z\\
 &amp;  &amp; u'' &amp; =\alpha''x+\beta''y-\gamma''z
\end{align}
\left|{\scriptstyle \begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =-1\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =1\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =1\\
\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\\
(k=-1)\\
\alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =k\\
\beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} &amp; =k\\
\gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-k\\
\alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with &lt;math&gt;[\cos T,\sin T,\cos E',\sin E']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]&lt;/math&gt;.

Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' .}}

=== {{anchor|Picard}} Picard (1882-1884) – Quadratic forms ===

[[w:Émile Picard]] (1882) analyzed the invariance of indefinite ternary [[w:Hermitian form|Hermitian quadratic forms]] with integer coefficients and their relation to [[w:Group action (mathematics)|discontinuous groups]], extending Poincaré's Fuchsian functions of one complex variable related to a circle, to &quot;hyperfuchsian&quot; functions of two complex variables related to a [[w:hypersphere]]. He formulated the following special case of an Hermitian form:&lt;ref group=M&gt;Picard (1882), pp. 307–308 first transformation system; pp. 315-317 second transformation system&lt;/ref&gt;&lt;ref&gt;Dickson (1923), pp. 280-281&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{matrix}xx_{0}+yy_{0}-zz_{0}\\
\\
\mathbf{(1)}\ \begin{align}x &amp; =M_{1}X+P_{1}Y+R_{1}Z\\
y &amp; =M_{2}X+P_{2}Y+R_{2}Z\\
z &amp; =M_{3}X+P_{3}Y+R_{3}Z
\end{align}
\\
\\
\left[\begin{align}[][x,y,z]=\text{complex}\\
\left[x_{0},y_{0},z_{0}\right]=\text{conjugate}
\end{align}
\right]\\
\\
\hline \\
x^{\prime2}+x^{\prime\prime2}+y^{\prime2}+y^{\prime\prime2}=1\\
x=x'+ix'',\quad y=y'+iy''\\
\\
\mathbf{(2)}\ \begin{align}X &amp; =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\
Y &amp; =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}}
\end{align}
\end{matrix}\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3} &amp; =1\\
P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} &amp; =1\\
R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} &amp; =-1\\
P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3} &amp; =0\\
M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} &amp; =0\\
P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} &amp; =0\\
\\
M_{1}\mu_{1}+P_{1}\pi_{1}-R_{1}\rho_{1} &amp; =1\\
M_{2}\mu_{2}+P_{2}\pi_{2}-R_{2}\rho_{2} &amp; =1\\
M_{3}\mu_{3}+P_{3}\pi_{3}-R_{3}\rho_{3} &amp; =-1\\
\mu_{2}M_{1}+\pi_{2}P_{1}-R_{1}\rho_{2} &amp; =0\\
\mu_{2}M_{3}+\pi_{2}P_{3}-R_{3}\rho_{2} &amp; =0\\
\mu_{3}M_{1}+\pi_{3}P_{1}-R_{1}\rho_{3} &amp; =0\\
\\
\left[\begin{align}[][M,P,R\dots]=\text{complex}\\
\left[\mu,\pi,\rho\dots\right]=\text{conjugate}
\end{align}
\right]
\end{align}
}\right.\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;-z&lt;sup&gt;2&lt;/sup&gt;=X&lt;sup&gt;2&lt;/sup&gt;+Y&lt;sup&gt;2&lt;/sup&gt;-Z&lt;sup&gt;2&lt;/sup&gt;'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;=X&lt;sup&gt;2&lt;/sup&gt;+Y&lt;sup&gt;2&lt;/sup&gt;=1''.}}

Or in (1884a) in relation to indefinite binary Hermitian quadratic forms:&lt;ref group=M&gt;Picard (1884a), p. 13&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}UU_{0}-VV_{0}=uu_{0}-vv_{0}\\
\hline \begin{align}U &amp; =\mathcal{A}u+\mathcal{B}v\\
V &amp; =\mathcal{C}u+\mathcal{D}v
\end{align}
\left|\begin{align}\mathcal{A}\mathcal{A}_{0}-\mathcal{C}\mathcal{C}_{0} &amp; =1\\
\mathcal{A}\mathcal{B}_{0}-\mathcal{C}\mathcal{D}_{0} &amp; =0\\
\mathcal{B}\mathcal{B}_{0}-\mathcal{D}\mathcal{D}_{0} &amp; =-1\\
\mathcal{D}\mathcal{D}_{0}-\mathcal{C}\mathcal{C}_{0} &amp; =1
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' producing ''U&lt;sup&gt;2&lt;/sup&gt;-V&lt;sup&gt;2&lt;/sup&gt;=u&lt;sup&gt;2&lt;/sup&gt;-v&lt;sup&gt;2&lt;/sup&gt;''.}}

Or in (1884b):&lt;ref group=M&gt;Picard (1884b), p. 416&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}xx_{0}+yy_{0}-1=0\\
\hline \begin{align}X &amp; =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\
Y &amp; =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}}
\end{align}
\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3}=P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} &amp; =1\\
R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} &amp; =-1\\
P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3}=M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3}=P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} &amp; =0\\
M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;=X&lt;sup&gt;2&lt;/sup&gt;+Y&lt;sup&gt;2&lt;/sup&gt;=1''.}}

Or in (1884c):&lt;ref group=M&gt;Picard (1884c), pp. 123–124; 163&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}UU_{0}+VV_{0}-WW_{0}=uu_{0}+vv_{0}-ww_{0}\\
\hline \mathbf{(1)}\ \begin{align}U &amp; =Mu+Pv+Rw\\
V &amp; =M'u+P'v+R'w\\
W &amp; =M''u+P''v+R''w\\
\\
u &amp; =M_{0}U+M_{0}^{\prime}V-M_{0}^{\prime\prime}W\\
v &amp; =P_{0}U+P_{0}^{\prime}V-P_{0}^{\prime\prime}W\\
w &amp; =-R_{0}U-R_{0}^{\prime}V+R_{0}^{\prime\prime}W
\end{align}
\left|{\scriptstyle \begin{align}MM_{0}+M'M_{0}^{\prime}-M''M_{0}^{\prime\prime} &amp; =1\\
PP_{0}+P'P_{0}^{\prime}-P''P_{0}^{\prime\prime} &amp; =1\\
RR_{0}+R'R_{0}^{\prime}-R''R_{0}^{\prime\prime} &amp; =-1\\
MP_{0}+M'P_{0}^{\prime}-M''P_{0}^{\prime\prime} &amp; =0\\
MR_{0}+M'R_{0}^{\prime}-M''R_{0}^{\prime\prime} &amp; =0\\
PR_{0}+P'R_{0}^{\prime}-P''R_{0}^{\prime\prime} &amp; =0\\
\\
MM_{0}+PP_{0}-RR_{0} &amp; =1\\
M'M_{0}^{\prime}+P'P_{0}^{\prime}-R'R_{0}^{\prime} &amp; =1\\
M''M_{0}^{\prime\prime}+P''P_{0}^{\prime\prime}-R''R_{0}^{\prime\prime} &amp; =-1\\
M_{0}M'+P_{0}P'-R_{0}R' &amp; =0\\
M_{0}M''+P_{0}P''-R_{0}R'' &amp; =0\\
M_{0}^{\prime}M''+P_{0}^{\prime}P''-R_{0}^{\prime}R'' &amp; =0
\end{align}
}\right.\\
\hline \text{Invariance of unit hypersphere:}\\
\mathbf{(2)}\ \begin{align}\xi' &amp; =\frac{A\xi+A'\eta+A''}{C\xi+C'\eta+C''}\\
\eta' &amp; =\frac{B\xi+B'\eta+B''}{C\xi+C'\eta+C''}
\end{align}
\left|{\scriptstyle \begin{align}AA_{0}+A'A_{0}^{\prime}-A''A_{0}^{\prime\prime} &amp; =1\\
BB_{0}+B'B_{0}^{\prime}-B''B_{0}^{\prime\prime} &amp; =1\\
CC_{0}+C'C_{0}^{\prime}-C''C_{0}^{\prime\prime} &amp; =-1\\
AB_{0}+A'B_{0}^{\prime}-A''B_{0}^{\prime\prime} &amp; =0\\
AC_{0}+A'C_{0}^{\prime}-A''C_{0}^{\prime\prime} &amp; =0\\
BC_{0}+B'C_{0}^{\prime}-B''C_{0}^{\prime\prime} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''U&lt;sup&gt;2&lt;/sup&gt;+V&lt;sup&gt;2&lt;/sup&gt;-W&lt;sup&gt;2&lt;/sup&gt;=u&lt;sup&gt;2&lt;/sup&gt;+v&lt;sup&gt;2&lt;/sup&gt;-w&lt;sup&gt;2&lt;/sup&gt;'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing &lt;math&gt;\xi^{\prime2}+\eta^{\prime2}=\xi^{2}+\eta^{2}=1&lt;/math&gt;.}}

=== {{anchor|Callandreau}} Callandreau (1885) – Homography ===

Following [[#Gauss4|Gauss (1818)]] and [[#Hill|Hill (1882)]], [[w:Octave Callandreau]] (1885) formulated the equations&lt;ref group=M&gt;Callandreau (1885), pp. A.7; A.12&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)=\\
{\scriptstyle (\alpha+\alpha'\sin T+\alpha''\cos T)^{2}+(\beta+\beta'\sin T+\beta''\cos T)^{2}-(\gamma+\gamma'\sin T+\gamma''\cos T)^{2}}\\
\hline \begin{align}\cos\varepsilon' &amp; =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\
\sin\varepsilon' &amp; =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}
\end{align}
\left|{\scriptstyle \begin{align} &amp; \left(k=1\right)\\
\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =-k &amp; \alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =+k &amp; \alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =+k &amp; \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\\
\alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =-1 &amp; \alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} &amp; =-1 &amp; \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =+1 &amp; \beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The transformation system is equivalent to Lorentz transformation ({{equationNote|1b}})  ''(n=2)'' with &lt;math&gt;[\cos T,\sin T,\cos\varepsilon',\sin\varepsilon']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]&lt;/math&gt;.}}

==={{anchor|Lie3}} Lie (1885-1890) – Lie group, hyperbolic motions, and infinitesimal transformations===
{{See also|History of Topics in Special Relativity/Lorentz transformation (imaginary)#Lie|label 1=History of Lorentz transformations via imaginary orthogonal transformations § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Lie|label 1=History of Lorentz transformations via sphere transformations § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Lie2|label 1=History of Lorentz transformations via squeeze mappings § Lie}}

In (1885/86), [[w:Sophus Lie]] identified the projective group of a general surface of second degree &lt;math&gt;\sum f_{ik}x_{i}'x_{k}'=0&lt;/math&gt; with the group of non-Euclidean motions.&lt;ref group=M&gt;Lie (1885/86), p. 411&lt;/ref&gt; In a thesis guided by Lie, [[w:Hermann Werner]] (1889) discussed this projective group by using the equation of a unit hypersphere as the surface of second degree (which was already given before by [[#Killing3|Killing (1887)]]), and also gave the corresponding infinitesimal projective transformations (Lie algebra):&lt;ref group=M&gt;Werner (1889), pp. 4, 28&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=1\\
\hline x_{i}p_{\varkappa}-x_{\varkappa}p_{i},\quad p_{i}-x_{i}\sum_{1}^{n}{\scriptstyle j}\ x_{j}p_{j}\quad(i,\varkappa=1,\dots, n)\\
\text{where}\\
\left(Q_{i},Q_{\varkappa}\right)=R_{i,\varkappa};\ \left(Q_{i},Q_{j,\varkappa}\right)=\varepsilon_{i,j}Q_{\varkappa}-\varepsilon_{i,\varkappa}Q_{j};\\
\left(R_{i,\varkappa},R_{\mu,\nu}\right)=\varepsilon_{\varkappa,\mu}R_{i,\nu}-\varepsilon_{\varkappa,\nu}R_{i,\mu}-\varepsilon_{,\mu}R_{\varkappa,\nu}+\varepsilon_{i,\nu}R_{\varkappa,\mu}\\
\left[\varepsilon_{i,\varkappa}\equiv0\ \text{for}\ i\ne\varkappa;\ \varepsilon_{i,i}=1\right]
\end{matrix}&lt;/math&gt;

More generally, Lie (1890)&lt;ref group=M&gt;Lie (1890a), p. 295;&lt;/ref&gt; defined non-Euclidean motions in terms of two forms &lt;math&gt;x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\pm1=0&lt;/math&gt; in which the imaginary form with &lt;math&gt;+1&lt;/math&gt; denotes the group of elliptic motions (in Klein's terminology), the real form with −1 the group of hyperbolic motions, with the latter having the same form as Werner's transformation:&lt;ref group=M&gt;Lie (1890a), p. 311&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+\dots+x_{n}^{2}-1=0\\
\hline p_{k}-x_{k}\sum j_{1}^{0}x_{j}p_{j},\quad x_{i}p_{k}-x_{k}p_{i}\quad(i,k=1\dots n)
\end{matrix}&lt;/math&gt;

Summarizing, Lie (1893) discussed the real continuous groups of the conic sections representing non-Euclidean motions, which in the case of hyperbolic motions have the form:

:&lt;math&gt;x^{2}+y^{2}-1=0&lt;/math&gt;&lt;ref group=M&gt;Lie (1893), p. 474&lt;/ref&gt; or &lt;math&gt;x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-1=0&lt;/math&gt;&lt;ref group=M&gt;Lie (1893), p. 479&lt;/ref&gt; or &lt;math&gt;x_{1}^{2}+\dots+x_{n}^{2}-1=0&lt;/math&gt;.&lt;ref group=M&gt;Lie (1893), p. 481&lt;/ref&gt;

{{Lorentzbox|Text=The group of hyperbolic motions is isomorphic to the Lorentz group. The interval &lt;math&gt;x_{1}^{2}+\dots+x_{n}^{2}-1=0&lt;/math&gt; becomes the Lorentz interval &lt;math&gt;x_{1}^{2}+\dots+x_{n}^{2}-x_{0}^{2}=0&lt;/math&gt; by setting
&lt;math&gt;(x_{1},\dots,\ x_{n},\ 1)=\left(\frac{x_{1}}{x_{0}},\dots,\ \frac{x_{n}}{x_{0}},\ \frac{x_{0}}{x_{0}}\right)&lt;/math&gt;}}

==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Gerard|label 1=History of Lorentz transformations via hyperbolic functions § Gerard}}

[[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane using the following invariant and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)'':&lt;ref group=M&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}-Z^{2}=1\\
X^{2}+Y^{2}-Z^{2}=X^{\prime2}+Y^{\prime2}-Z^{\prime2}\\
\hline \begin{align}X &amp; =aX'+a'Y'+a''Z'\\
Y &amp; =bX'+b'Y'+b''Z'\\
Z &amp; =cX'+c'Y'+c''Z'\\
\\
X' &amp; =aX+bY-cZ\\
Y' &amp; =a'X+b'Y-c'Z\\
Z' &amp; =-a''X-b''Y+c''Z
\end{align}
\left|\begin{align}a^{2}+b^{2}-c^{2} &amp; =1\\
a^{\prime2}+b^{\prime2}-c^{\prime2} &amp; =1\\
a^{\prime\prime2}+b^{\prime\prime2}-c^{\prime\prime2} &amp; =-1\\
aa'+bb'-cc' &amp; =0\\
a'a''+b'b''-c'c'' &amp; =0\\
a''a+b''b-c''c &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

He gave the case of translation as follows:&lt;ref group=M name=gerard&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{align}X &amp; =Z_{0}X'+X_{0}Z'\\
Y &amp; =Y'\\
Z &amp; =X_{0}X'+Z_{0}Z'
\end{align}
\ \text{with}\ \begin{align}X_{0} &amp; =\operatorname{sh}OO'\\
Z_{0} &amp; =\operatorname{ch}OO'
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Hausdorff}} Hausdorff (1899) – Weierstrass coordinates===

[[w:Felix Hausdorff]] (1899) – citing Killing (1885) – discussed Weierstrass coordinates in the plane using the following invariant and its transformation:&lt;ref group=M&gt;Hausdorff (1899), p. 165, pp. 181-182&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}p^{2}-x^{2}-y^{2}=1\\
\hline \begin{align}x &amp; =a_{1}x'+a_{2}y'+x_{0}p'\\
y &amp; =b_{1}x'+b{}_{2}y'+y_{0}p'\\
p &amp; =e_{1}x'+e_{2}y'+p_{0}p'\\
\\
x' &amp; =a_{1}x+b_{1}y-e_{1}p\\
y' &amp; =a_{2}x+b_{2}y-e_{2}p\\
-p' &amp; =x_{0}x+y_{0}y-p_{0}p
\end{align}
\left|{\scriptstyle \begin{align}a_{1}^{2}+b_{1}^{2}-e_{1}^{2} &amp; =1\\
a_{2}^{2}+b_{2}^{2}-e_{2}^{2} &amp; =1\\
-x_{0}^{2}-y_{0}^{2}+p_{0}^{2} &amp; =1\\
a_{2}x_{0}+b_{2}y_{0}-e_{2}p_{0} &amp; =0\\
a_{1}x_{0}+b_{1}y_{0}-e_{1}p_{0} &amp; =0\\
a_{1}a_{2}+b_{1}b_{2}-e_{1}e_{2} &amp; =0\\
\\
a_{1}^{2}+a_{2}^{2}-x_{0}^{2} &amp; =1\\
b_{1}^{2}+b_{2}^{2}-y_{0}^{2} &amp; =1\\
-e_{1}^{2}-e_{2}^{2}+p_{0}^{2} &amp; =1\\
b_{1}e_{1}+b_{2}e_{2}-y_{0}p_{0} &amp; =0\\
a_{1}e_{1}+a_{2}e_{2}-x_{0}p_{0} &amp; =0\\
a_{1}b_{1}+a_{2}b_{2}-x_{0}y_{0} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

==={{anchor|Woods2}} Woods (1901-05) – Beltrami and Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Woods2|label 1=History of Lorentz transformations via hyperbolic functions § Woods}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Woods|label 1=History of Lorentz transformations via Möbius transformations § Woods}}

In (1901/02) [[w:Frederick S. Woods]] defined the following invariant quadratic form and its [[w:projective transformation]] in terms of Beltrami coordinates (he pointed out that this can be connected to hyperbolic geometry by setting &lt;math&gt;k=\sqrt{-1}R&lt;/math&gt; with ''R'' as real quantity):&lt;ref group=M&gt;Woods (1901/02), p. 98, 104&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}\left(u^{2}+v^{2}+w^{2}\right)+1=0\\
\hline \begin{align}u' &amp; =\frac{\alpha_{1}u+\alpha_{2}v+\alpha_{3}w+\alpha_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\
v' &amp; =\frac{\beta_{1}u+\beta_{2}v+\beta_{3}w+\beta_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\
w' &amp; =\frac{\gamma_{1}u+\gamma_{2}v+\gamma_{3}w+\gamma_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}
\end{align}
\left|\begin{align}k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right)+\delta_{i}^{2} &amp; =k^{2}\\
(i=1,2,3)\\
k^{2}\left(\alpha_{4}^{2}+\beta_{4}^{2}+\gamma_{4}^{2}\right)+\delta_{4}^{2} &amp; =1\\
\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}+\delta_{i}\delta_{h} &amp; =0\\
(i,h=1,2,3,4;\ i\ne h)
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'' with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.}}

Alternatively, Woods (1903, published 1905) – citing Killing (1885) – used the invariant quadratic form in terms of Weierstrass coordinates and its transformation (with &lt;math&gt;k=\sqrt{-1}k&lt;/math&gt; for hyperbolic space):&lt;ref group=M&gt;Woods (1903/05), pp. 45–46; p. 48)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{0}^{2}+k^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)=1\\
ds^{2}=\frac{1}{k^{2}}dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\\
\hline \begin{align}x_{1}^{\prime} &amp; =\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}+\alpha_{0}x_{0}\\
x_{2}^{\prime} &amp; =\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{0}x_{0}\\
x_{3}^{\prime} &amp; =\gamma_{1}x_{1}+\gamma_{2}x_{2}+\gamma_{3}x_{3}+\gamma_{0}x_{0}\\
x_{0}^{\prime} &amp; =\delta_{1}x_{1}+\delta_{2}x_{2}+\delta_{3}x_{3}+\delta_{0}x_{0}
\end{align}
\left|\begin{align}\delta_{0}^{2}+k^{2}\left(\alpha_{0}^{2}+\beta_{0}^{2}+\gamma_{0}^{2}\right) &amp; =1\\
\delta_{i}^{2}+k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right) &amp; =k^{2}\\
(i=1,2,3)\\
\delta_{i}\delta_{h}+k^{2}\left(\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}\right) &amp; =0\\
(i,h=0,1,2,3;\ i\ne h)
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.}}

==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Liebmann|label 1=History of Lorentz transformations via hyperbolic functions § Liebmann}}

[[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – used the invariant quadratic form and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)''&lt;ref group=M&gt;Liebmann (1904/05), p. 168; pp. 175–176&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}p^{\prime2}-x^{\prime2}-y^{\prime2}=1\\
\hline \begin{align}x_{1} &amp; =\alpha_{11}x+\alpha_{12}y+\alpha_{13}p\\
y_{1} &amp; =\alpha_{21}x+\alpha_{22}y+\alpha_{23}p\\
x_{1} &amp; =\alpha_{31}x+\alpha_{32}y+\alpha_{33}p\\
\\
x &amp; =\alpha_{11}x_{1}+\alpha_{21}y_{1}-\alpha_{31}p_{1}\\
y &amp; =\alpha_{12}x_{1}+\alpha_{22}y_{1}-\alpha_{32}p_{1}\\
p &amp; =-\alpha_{13}x_{1}-\alpha_{23}y_{1}+\alpha_{33}p_{1}
\end{align}
\left|\begin{align}\alpha_{33}^{2}-\alpha_{13}^{2}-\alpha_{23}^{2} &amp; =1\\
-\alpha_{31}^{2}+\alpha_{11}^{2}+\alpha_{21}^{2} &amp; =1\\
-\alpha_{32}^{2}+\alpha_{12}^{2}+\alpha_{22}^{2} &amp; =1\\
\alpha_{31}\alpha_{32}-\alpha_{11}\alpha_{12}-\alpha_{21}\alpha_{22} &amp; =0\\
\alpha_{32}\alpha_{33}-\alpha_{12}\alpha_{13}-\alpha_{22}\alpha_{23} &amp; =0\\
\alpha_{33}\alpha_{31}-\alpha_{23}\alpha_{11}-\alpha_{23}\alpha_{21} &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|apo1}}
*{{#section:History of Topics in Special Relativity/mathsource|apo2}}
*{{#section:History of Topics in Special Relativity/mathsource|apo}}
*{{#section:History of Topics in Special Relativity/mathsource|bour56att}} 
*{{#section:History of Topics in Special Relativity/mathsource|chal82sec}} 
*{{#section:History of Topics in Special Relativity/mathsource|chas29}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox83hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox91}} 
*{{#section:History of Topics in Special Relativity/mathsource|fris76}} 
*{{#section:History of Topics in Special Relativity/mathsource|gau98}} 
*{{#section:History of Topics in Special Relativity/mathsource|gau18}} 
*{{#section:History of Topics in Special Relativity/mathsource|ger92}} 
*{{#section:History of Topics in Special Relativity/mathsource|gud30}} 
*{{#section:History of Topics in Special Relativity/mathsource|haus99}} 
*{{#section:History of Topics in Special Relativity/mathsource|hill82}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac27}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac32a}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac32b}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac33}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil77}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil79}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil84}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil85}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil93}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil97}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei71}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei73}} 
*{{#section:History of Topics in Special Relativity/mathsource|lag73}} 
*{{#section:History of Topics in Special Relativity/mathsource|hire1}} 
*{{#section:History of Topics in Special Relativity/mathsource|leb37}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie85}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie90}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie93}} 
*{{#section:History of Topics in Special Relativity/mathsource|lieb04}} 
*{{#section:History of Topics in Special Relativity/mathsource|lop}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic82}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic84a}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic84b}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic84c}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin81a}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin81b}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin87}} 
*{{#section:History of Topics in Special Relativity/mathsource|sal62}} 
*{{#section:History of Topics in Special Relativity/mathsource|vinc}} 
*{{#section:History of Topics in Special Relativity/mathsource|som63}} 
*{{#section:History of Topics in Special Relativity/mathsource|wedd47}} 
*{{#section:History of Topics in Special Relativity/mathsource|wern89}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo01}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo03}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L1}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
      <sha1>4sgoep3vezohmjsthqocfq2g82579hg</sha1>
    </revision>
  </page>
  <page>
    <title>History of Topics in Special Relativity/Lorentz transformation (imaginary)</title>
    <ns>0</ns>
    <id>267590</id>
    <revision>
      <id>2721177</id>
      <parentid>2692684</parentid>
      <timestamp>2025-07-10T07:12:05Z</timestamp>
      <contributor>
        <username>D.H</username>
        <id>52339</id>
      </contributor>
      <minor />
      <comment>/* Lorentz transformation via imaginary orthogonal transformation */</comment>
      <origin>2721177</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17739" sha1="ltpjfqghcp81quwyz6sdefqpatw3neb" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via imaginary orthogonal transformation==
By using the [[w:Imaginary unit|imaginary]] quantities &lt;math&gt;[\mathfrak{x}_{0},\ \mathfrak{x}'_{0}]=\left[ix_{0},\ ix_{0}^{\prime}\right]&lt;/math&gt; in '''x''' as well as &lt;math&gt;[\mathfrak{g}_{0s},\ \mathfrak{g}_{s0}]=\left[ig_{0s},\ ig_{s0}\right]&lt;/math&gt; ''(s=1,2...n)'' in '''g''', the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']] assumes the form of an [[w:orthogonal transformation]] of [[w:Euclidean space]] forming the [[w:orthogonal group]] O(n) if det '''g'''=±1 or the special orthogonal group SO(n) if det '''g'''=+1, the Lorentz interval becomes the [[w:Euclidean norm]], and the Minkowski inner product becomes the [[w:dot product]]:&lt;ref&gt;Laue (1921), pp. 79–80 for n=3&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}\begin{align}\mathfrak{x}_{0}^{2}+x_{1}^{2}+\cdots+x_{n}^{2} &amp; =\mathfrak{x}_{0}^{\prime2}+x_{1}^{\prime2}+\dots+x_{n}^{\prime2}\\
\mathfrak{x}_{0}\mathfrak{y}_{0}+x_{1}y_{1}+\cdots+x_{n}y_{n} &amp; =\mathfrak{x}_{0}^{\prime}\mathfrak{y}_{0}^{\prime}+x_{1}^{\prime}y_{1}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime}
\end{align}
\\
\hline \begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\
\mathbf{x}=\mathbf{\mathbf{g}^{-1}}\cdot\mathbf{x}'
\end{matrix}\left|\begin{align}\sum_{i=0}^{n}g_{ij}g_{ik} &amp; =\left\{ \begin{align}1\quad &amp; (j=k)\\
0\quad &amp; (j\ne k)
\end{align}
\right.\\
\sum_{j=0}^{n}g_{ij}g_{kj} &amp; =\left\{ \begin{align}1\quad &amp; (i=k)\\
0\quad &amp; (i\ne k)
\end{align}
\right.
\end{align}
\right.
\end{matrix}&lt;/math&gt;|{{equationRef|2a}}}}

The cases ''n=1,2,3,4'' of orthogonal transformations in terms of real coordinates were discussed by [[#Euler|Euler (1771)]] and in ''n'' dimensions by [[#Cauchy|Cauchy (1829)]]. The case in which one of these coordinates is imaginary and the other ones remain real was alluded to by [[#Lie|Lie (1871)]] in terms of spheres with imaginary radius, while the interpretation of the imaginary coordinate as being related to the dimension of time as well as the explicit formulation of Lorentz transformations with ''n=3'' was given by [[#Minkowski|Minkowski (1907)]] and [[#Sommerfeld|Sommerfeld (1909)]].

A well known example of this orthogonal transformation is spatial [[w:rotation]] in terms of [[w:trigonometric function]]s, which become Lorentz transformations by using an imaginary angle &lt;math&gt;\phi=i\eta&lt;/math&gt;, so that trigonometric functions become equivalent to [[w:hyperbolic function]]s:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{array}{c|c|cc}
\mathfrak{x}_{0}^{2}+x_{1}^{2}+x_{2}^{2}=\mathfrak{x}_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2} &amp; \left(ix_{0}\right){}^{2}+x_{1}^{2}+x_{2}^{2}=\left(ix_{0}^{\prime}\right)^{2}+x_{1}^{\prime2}+x_{2}^{\prime2} &amp;  &amp; -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline (1)\begin{align}\mathfrak{x}_{0}^{\prime} &amp; =\mathfrak{x}_{0}\cos\phi-x_{1}\sin\phi\\
x_{1}^{\prime} &amp; =\mathfrak{x}_{0}\sin\phi+x_{1}\cos\phi\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
\mathfrak{x}_{0} &amp; =\mathfrak{x}_{0}^{\prime}\cos\phi+x_{1}^{\prime}\sin\phi\\
x_{1} &amp; =-\mathfrak{x}_{0}^{\prime}\sin\phi+x_{1}^{\prime}\cos\phi\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
 &amp; (2)\begin{align}ix_{0}^{\prime} &amp; =ix_{0}\cos i\eta-x_{1}\sin i\eta\\
x_{1}^{\prime} &amp; =ix_{0}\sin i\eta+x_{1}\cos i\eta\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
ix_{0} &amp; =ix_{0}^{\prime}\cos i\eta+x_{1}^{\prime}\sin i\eta\\
x_{1} &amp; =-ix_{0}^{\prime}\sin i\eta+x_{1}^{\prime}\cos i\eta\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
 &amp; \rightarrow &amp; \begin{align}x_{0}^{\prime} &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta\\
x_{1}^{\prime} &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{0} &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta\\
x_{1} &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\end{array} &lt;/math&gt;|{{equationRef|2b}}}}

or in exponential form using [[w:Euler's formula]] &lt;math&gt;e^{i\phi}=\cos\phi+i\sin\phi&lt;/math&gt;:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{array}{c|c|cc}
\mathfrak{x}_{0}^{2}+x_{1}^{2}+x_{2}^{2}=\mathfrak{x}_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2} &amp; \left(ix_{0}\right){}^{2}+x_{1}^{2}+x_{2}^{2}=\left(ix_{0}^{\prime}\right)^{2}+x_{1}^{\prime2}+x_{2}^{\prime2} &amp;  &amp; -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline (1)\begin{align}x_{1}^{\prime}+i\mathfrak{x}_{0}^{\prime} &amp; =e^{-i\phi}\left(x_{1}+i\mathfrak{x}_{0}\right)\\
x_{1}^{\prime}-i\mathfrak{x}_{0}^{\prime} &amp; =e^{i\phi}\left(x_{1}-i\mathfrak{x}_{0}\right)\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{1}+i\mathfrak{x}_{0} &amp; =e^{i\phi}\left(x_{1}^{\prime}+i\mathfrak{x}_{0}^{\prime}\right)\\
x_{1}-i\mathfrak{x}_{0} &amp; =e^{-i\phi}\left(x_{1}^{\prime}-i\mathfrak{x}_{0}^{\prime}\right)\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
 &amp; (2)\begin{align}x_{1}^{\prime}+i\left(ix_{0}^{\prime}\right) &amp; =e^{-i(i\eta)}\left(x_{1}+i\left(ix_{0}\right)\right)\\
x_{1}^{\prime}-i\left(ix_{0}^{\prime}\right) &amp; =e^{i(i\eta)}\left(x_{1}-i\left(ix_{0}\right)\right)\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{1}+i\left(ix_{0}\right) &amp; =e^{i(i\eta)}\left(x_{1}^{\prime}+i\left(ix_{0}^{\prime}\right)\right)\\
x_{1}-i\left(ix_{0}\right) &amp; =e^{-i(i\eta)}\left(x_{1}^{\prime}-i\left(ix_{0}^{\prime}\right)\right)\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
 &amp; \rightarrow &amp; \begin{align}x_{1}^{\prime}-x_{0}^{\prime} &amp; =e^{\eta}\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime} &amp; =e^{-\eta}\left(x_{1}+x_{0}\right)\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{1}-x_{0} &amp; =e^{-\eta}\left(x_{1}^{\prime}-x_{0}^{\prime}\right)\\
x_{1}+x_{0} &amp; =e^{\eta}\left(x_{1}^{\prime}+x_{0}^{\prime}\right)\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\end{array}&lt;/math&gt;|{{equationRef|2c}}}}

Defining &lt;math&gt;[\mathfrak{x}_{0},\ \mathfrak{x}'_{0},\ \phi]&lt;/math&gt; as real, spatial rotation in the form ({{equationNote|2b}}-1) was introduced by [[#Euler2|Euler (1771)]] and in the form ({{equationNote|2c}}-1) by [[#Euler3|Wessel (1799)]]. The interpretation of ({{equationNote|2b}}) as Lorentz boost (i.e. Lorentz transformation ''without'' spatial rotation) in which &lt;math&gt;[\mathfrak{x}_{0},\ \mathfrak{x}'_{0},\ \phi]&lt;/math&gt; correspond to the imaginary quantities &lt;math&gt;[ix_{0},\ ix'_{0},\ i\eta]&lt;/math&gt; was given by [[#Minkowski|Minkowski (1907)]] and [[#Sommerfeld|Sommerfeld (1909)]]. As shown in the next section using hyperbolic functions, ({{equationNote|2b}}) becomes [[../Lorentz transformation (hyperbolic)#math_3b|E:'''(3b)''']] while ({{equationNote|2c}}) becomes [[../Lorentz transformation (hyperbolic)#math_3c|E:'''(3c)''']].

==Historical notation==
==={{anchor|Euler}} Euler (1771) – Orthogonal transformation===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Euler|label 1=History of Lorentz transformations via hyperbolic functions § Euler}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)#Euler|label 1=History of Lorentz transformations via Cayley-Hermite transformations § Euler}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Euler|label 1=History of Lorentz transformations via Quaternions § Euler}}

[[w:Leonhard Euler]] (1771) demonstrated the invariance of quadratic forms in terms of sum of squares under a linear substitution and its coefficients, now known as [[w:orthogonal transformation]], as well as under rotations using [[w:Euler angles]]. The case of two dimensions is given by&lt;ref group=M&gt;Euler (1771), pp. 84-85&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}=x^{2}+y^{2}\\
\hline \begin{align}X &amp; =\alpha x+\beta y\\
Y &amp; =\gamma x+\delta y
\end{align}
\left|\begin{matrix}\begin{align}1 &amp; =\alpha\alpha+\gamma\gamma\\
1 &amp; =\beta\beta+\delta\delta\\
0 &amp; =\alpha\beta+\gamma\delta
\end{align}
\end{matrix}\right.\\
\hline \begin{align}X &amp; =x\cos\zeta+y\sin\zeta\\
Y &amp; =x\sin\zeta-y\cos\zeta
\end{align}
\end{matrix}&lt;/math&gt;

or three dimensions&lt;ref group=M&gt;Euler (1771), pp. 77, 85-89&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}+Z^{2}=x^{2}+y^{2}+z^{2}\\
\hline \begin{align}X &amp; =Ax+By+Cz\\
Y &amp; =Dx+Ey+Fz\\
Z &amp; =Gx+Hy+Iz
\end{align}
\begin{matrix}\left|{\scriptstyle \begin{align}1 &amp; =AA+DD+GG\\
1 &amp; =BB+EE+HH\\
1 &amp; =CC+FF+II\\
0 &amp; =AB+DE+GH\\
0 &amp; =AG+DF+GI\\
0 &amp; =BC+EF+HI
\end{align}
}\right.\end{matrix}\\
\hline \begin{align}x' &amp; =x\cos\zeta+y\sin\zeta &amp; x'' &amp; =x'\cos\eta+z'\sin\eta\\
y' &amp; =x\sin\zeta-y\cos\zeta &amp; y'' &amp; =y'\\
z' &amp; =z &amp; z'' &amp; =x'\sin\eta-z'\cos\eta\\
\\
x''' &amp; =x'' &amp; =X\\
y''' &amp; =y''\cos\theta+z''\sin\theta &amp; =Y\\
z''' &amp; =y''\sin\theta-z''\cos\theta &amp; =Z
\end{align}
\end{matrix}&lt;/math&gt;

The orthogonal transformation in four dimensions was given by him as&lt;ref group=M&gt;Euler (1771), pp. 89–91&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}V^{2}+X^{2}+Y^{2}+Z^{2}=v^{2}+x^{2}+y^{2}+z^{2}\\
\hline \begin{align}V &amp; =Av+Bx+Cy+Dz\\
X &amp; =Ev+Fx+Gy+Hz\\
Y &amp; =Iv+Kx+Ly+Mz\\
Z &amp; =Nv+Ox+Py+Qz
\end{align}
\begin{matrix}\left|{\scriptstyle \begin{align}1 &amp; =AA+RR+II+NN &amp; 0 &amp; =AB+EF+IK+NO\\
1 &amp; =BB+FF+KK+OO &amp; 0 &amp; =AC+EG+IL+NP\\
1 &amp; =CC+GG+LL+PP &amp; 0 &amp; =AD+EH+IM+NQ\\
1 &amp; =DD+HH+MM+QQ &amp; 0 &amp; =BC+FG+KL+OP\\
0 &amp; =BD+FH+KM+OQ &amp; 0 &amp; =CD+FH+LM+PQ
\end{align}
}\right.\end{matrix}\\
\hline {\scriptstyle \begin{align}x^{I} &amp; =x\cos\alpha+y\sin\alpha &amp;  &amp;  &amp; x^{VI} &amp; =x^{V} &amp; =X\\
y^{I} &amp; =x\sin\alpha-y\cos\alpha &amp;  &amp;  &amp; y^{VI} &amp; =y^{V} &amp; =Y\\
z^{I} &amp; =z &amp; \dots &amp; \dots &amp; y^{VI} &amp; =z^{V}\cos\zeta+v^{V}\sin\zeta &amp; =Z\\
v^{I} &amp; =v &amp;  &amp;  &amp; v^{VI} &amp; =z^{V}\sin\zeta-v^{V}\cos\varepsilon\zeta &amp; =V
\end{align}
}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=As shown by [[#Minkowski|Minkowski (1907)]], the orthogonal transformation can be directly used as Lorentz transformation ({{equationNote|2a}}) or ({{equationNote|2b}}) by making one variable as well as six of the sixteen coefficients imaginary.}}

==={{anchor|Euler3}} Wessel (1799) – Euler's formula and rotation===

The above orthogonal transformations representing Euclidean rotations can also be expressed by using [[w:Euler's formula]]. After this formula was derived by Euler in 1748&lt;ref group=M&gt;Euler (1748b), section 138.&lt;/ref&gt;

:&lt;math&gt;e^{+v\sqrt{-1}}=\cos v+\sqrt{-1}\sin v,\quad e^{-v\sqrt{-1}}=\cos v-\sqrt{-1}\sin v&lt;/math&gt;,

it was used by [[w:Caspar Wessel]] (1799) to describe Euclidean rotations in the complex plane:&lt;ref group=M&gt;Wessel (1799), § 28.&lt;/ref&gt;

:&lt;math&gt;x''+\varepsilon z''=(x'+\varepsilon z')\cdot(\cos III+\varepsilon\sin III),\ (\varepsilon=\sqrt{-1})&lt;/math&gt;

{{Lorentzbox|Text=Replacing the real quantities by imaginary ones by setting &lt;math&gt;\left[z',z'',III\right]=\left[iz',iz'',iIII\right]&lt;/math&gt;, Wessel's transformation becomes Lorentz transformation ({{equationNote|2c}}).}}

==={{anchor|Cauchy}} Cauchy (1829) – Orthogonal transformation===

[[w:Augustin-Louis Cauchy]] (1829) extended the orthogonal transformation of [[#Euler|Euler (1771)]] to arbitrary dimensions&lt;ref group=M&gt;Cauchy (1829), eq. 22, 98, 99, 101; Some misprints were corrected in Œuvres complètes, série 2, tome 9, pp. 174–195.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}+z^{2}+\dots=\xi^{2}+\eta^{2}+\zeta^{2}+\dots\\
\hline \begin{align}x &amp; =x_{1}\xi+x_{2}\eta+x_{3}\zeta+\dots\\
y &amp; =y_{1}\xi+y_{2}\eta+y_{3}\zeta+\dots\\
z &amp; =z_{1}\xi+z_{2}\eta+z_{3}\zeta+\dots\\
 &amp; \dots\\
\\
\xi &amp; =x_{1}x+y_{1}y+z_{1}z+\dots\\
\eta &amp; =x_{2}x+y_{2}y+z_{2}z+\dots\\
\zeta &amp; =x_{3}x+y_{3}y+z_{3}z+\dots\\
 &amp; \dots
\end{align}
\left|{\scriptstyle \begin{align}x_{1}^{2}+y_{1}^{2}+z_{1}^{2}+\dots &amp; =1,\\
x_{2}x_{1}+y_{2}y_{1}+z_{2}z_{1}+\dots &amp; =0,\\
\dots\\
x_{n}x_{1}+y_{n}y_{1}+z_{n}z_{1}+\dots &amp; =0,\\
\\
x_{1}x_{2}+y_{1}y_{2}+z_{1}z_{2}+\dots &amp; =0,\\
x_{2}^{2}+y_{2}^{2}+z_{2}^{2}+\dots &amp; =1,\\
\text{ }\dots\\
x_{n}x_{2}+y_{n}y_{2}+z_{n}z_{2}+\dots &amp; =0,\\
\\
x_{1}x_{n}+y_{1}y_{n}+z_{1}z_{n}+\dots &amp; =0,\\
x_{2}x_{n}+y_{2}y_{n}+z_{2}z_{n}+\dots &amp; =0,\\
\dots\\
x_{n}^{2}+y_{n}^{2}+z_{n}^{2}+\dots &amp; =1
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The orthogonal transformation can be directly used as Lorentz transformation ({{equationNote|2a}}) by making one of the variables as well as certain coefficients imaginary.}}

==={{anchor|Lie}} Lie (1871) – Imaginary orthogonal transformations===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Lie3|label 1=History of Lorentz transformations in general § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Lie|label 1=History of Lorentz transformations via sphere transformations § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Lie2|label 1=History of Lorentz transformations via squeeze mappings § Lie}}

[[w:Sophus Lie]] (1871a) described a manifold whose elements can be represented by spheres, where the last coordinate ''y&lt;sub&gt;n+1&lt;/sub&gt;'' can be related to an imaginary radius by ''iy&lt;sub&gt;n+1&lt;/sub&gt;'':&lt;ref group=M&gt;Lie (1871a), pp. 199–209&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\sum_{i=1}^{i=n} (x_i-y_i)^2+y_{n+1}^2=0 \\
\downarrow\\
\sum_{i=1}^{i=n+1} (y_i^{\prime}-y_i^{\prime\prime})^2=0
\end{matrix}&lt;/math&gt;

If the second equation is satisfied, two spheres ''y′'' and ''y″'' are in contact. Lie then defined the correspondence between [[w:Contact geometry|contact transformations]] in ''R&lt;sub&gt;n&lt;/sub&gt;'' and conformal point transformations in ''R&lt;sub&gt;n+1&lt;/sub&gt;'': The sphere of space ''R&lt;sub&gt;n&lt;/sub&gt;'' consists of ''n+1'' parameter (coordinates plus imaginary radius), so if this sphere is taken as the element of space ''R&lt;sub&gt;n&lt;/sub&gt;'', it follows that ''R&lt;sub&gt;n&lt;/sub&gt;'' now corresponds to ''R&lt;sub&gt;n+1&lt;/sub&gt;''. Therefore, any transformation (to which he counted [[#Lorentz transformation via orthogonal transformation|orthogonal transformations]] and inversions) leaving invariant the condition of contact between spheres in ''R&lt;sub&gt;n&lt;/sub&gt;'', corresponds to the conformal transformation of points in ''R&lt;sub&gt;n+1&lt;/sub&gt;''.

=== {{anchor|Minkowski}} Minkowski (1907–1908) – Spacetime ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (velocity)#Minkowski|label 1=History of Lorentz transformations via velocity § Minkowski}}

The work on the principle of relativity by Lorentz, Einstein, [[w:Max Planck|Planck]], together with Poincaré's four-dimensional approach, were further elaborated and combined with the [[w:hyperboloid model]] by [[w:Hermann Minkowski]] in 1907 and 1908.&lt;ref group=R&gt;Minkowski (1907/15), pp. 927ff&lt;/ref&gt;&lt;ref group=R&gt;Minkowski (1907/08), pp. 53ff&lt;/ref&gt; Minkowski particularly reformulated electrodynamics in a four-dimensional way ([[w:Minkowski spacetime]]).&lt;ref&gt;Walter (1999a)&lt;/ref&gt; For instance, he wrote ''x, y, z, it'' in the form ''x&lt;sub&gt;1&lt;/sub&gt;, x&lt;sub&gt;2&lt;/sub&gt;, x&lt;sub&gt;3&lt;/sub&gt;, x&lt;sub&gt;4&lt;/sub&gt;''. By defining ψ as the angle of rotation around the ''z''-axis, the Lorentz transformation assumes a form (with ''c''=1) in agreement with ({{equationNote|2b}}):&lt;ref group=R name=mink1&gt;Minkowski (1907/08), p. 59&lt;/ref&gt;

:&lt;math&gt;\begin{align}x'_{1} &amp; =x_{1}\\
x'_{2} &amp; =x_{2}\\
x'_{3} &amp; =x_{3}\cos i\psi+x_{4}\sin i\psi\\
x'_{4} &amp; =-x_{3}\sin i\psi+x_{4}\cos i\psi\\
\cos i\psi &amp; =\frac{1}{\sqrt{1-q^{2}}}
\end{align}
&lt;/math&gt;

Even though Minkowski used the imaginary number iψ, he for once&lt;ref group=R name=mink1 /&gt; directly used the [[w:tangens hyperbolicus]] in the equation for velocity

:&lt;math&gt;-i\tan i\psi=\frac{e^{\psi}-e^{-\psi}}{e^{\psi}+e^{-\psi}}=q&lt;/math&gt; with &lt;math&gt;\psi=\frac{1}{2}\ln\frac{1+q}{1-q}&lt;/math&gt;.

Minkowski's expression can also by written as ψ=atanh(q) and was later called [[w:rapidity]].

==={{Anchor|Sommerfeld}} Sommerfeld (1909) – Spherical trigonometry===
Using an imaginary rapidity such as Minkowski, [[w:Arnold Sommerfeld]] (1909) formulated a transformation equivalent to Lorentz boost ({{equationNote|2b}}), and the relativistc velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] in terms of trigonometric functions and the [[w:spherical law of cosines]]:&lt;ref group=R&gt;Sommerfeld (1909), p. 826ff.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{array}{lrl}
x'= &amp; x\ \cos\varphi+l\ \sin\varphi, &amp; y'=y\\
l'= &amp; -x\ \sin\varphi+l\ \cos\varphi, &amp; z'=z
\end{array}\right\} \\
\left(\operatorname{tg}\varphi=i\beta,\ \cos\varphi=\frac{1}{\sqrt{1-\beta^{2}}},\ \sin\varphi=\frac{i\beta}{\sqrt{1-\beta^{2}}}\right)\\
\hline \beta=\frac{1}{i}\operatorname{tg}\left(\varphi_{1}+\varphi_{2}\right)=\frac{1}{i}\frac{\operatorname{tg}\varphi_{1}+\operatorname{tg}\varphi_{2}}{1-\operatorname{tg}\varphi_{1}\operatorname{tg}\varphi_{2}}=\frac{\beta_{1}+\beta_{2}}{1+\beta_{1}\beta_{2}}\\
\cos\varphi=\cos\varphi_{1}\cos\varphi_{2}-\sin\varphi_{1}\sin\varphi_{2}\cos\alpha\\
v^{2}=\frac{v_{1}^{2}+v_{2}^{2}+2v_{1}v_{2}\cos\alpha-\frac{1}{c^{2}}v_{1}^{2}v_{2}^{2}\sin^{2}\alpha}{\left(1+\frac{1}{c^{2}}v_{1}v_{2}\cos\alpha\right)^{2}}
\end{matrix}&lt;/math&gt;

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|cau29sec}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul48b}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul71}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie71a}} 
*{{#section:History of Topics in Special Relativity/mathsource|wes99}}

===Historical relativity sources===
{{reflist|3|group=R}}
{{#section:History of Topics in Special Relativity/relsource|mink07a}}
{{#section:History of Topics in Special Relativity/relsource|mink07b}}
{{#section:History of Topics in Special Relativity/relsource|mink08}}
{{#section:History of Topics in Special Relativity/relsource|som09}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L2}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
      <sha1>ltpjfqghcp81quwyz6sdefqpatw3neb</sha1>
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    <title>History of Topics in Special Relativity/Lorentz transformation (hyperbolic)</title>
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      <text bytes="57582" sha1="5z2wpsggdb1lz37c3ahvvbv5hqj0hsz" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via hyperbolic functions==
===Translation in the hyperbolic plane===
[[File:Hyperbolic functions-2.svg|thumb|upright=1.4|200px|A ray through the unit hyperbola {{math|1=''x''&lt;sup&gt;2&lt;/sup&gt; − ''y''&lt;sup&gt;2&lt;/sup&gt; = 1}} at the point {{math|(cosh ''a'', sinh ''a'')}}.]]
The case of a Lorentz transformation without spatial rotation is called a [[w:Lorentz boost]]. The simplest case can be given, for instance, by setting ''n=1'' in the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']]:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}g_{00}+x_{1}g_{01}\\
x_{1}^{\prime} &amp; =x_{0}g_{10}+x_{1}g_{11}\\
\\
x_{0} &amp; =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}\\
x_{1} &amp; =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}
\end{align}
\left|\begin{align}g_{01}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{10}^{2} &amp; =1\\
g_{01}g_{11}-g_{00}g_{10} &amp; =0\\
g_{10}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{01}^{2} &amp; =1\\
g_{10}g_{11}-g_{00}g_{01} &amp; =0
\end{align}
\rightarrow\begin{align}g_{00}^{2} &amp; =g_{11}^{2}\\
g_{01}^{2} &amp; =g_{10}^{2}
\end{align}
\right.
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}g_{00} &amp; -g_{10}\\
-g_{01} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\quad\right|\quad\det\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}=1&lt;/math&gt;|{{equationRef|3a}}}}

which resembles precisely the relations of [[w:hyperbolic function]]s in terms of [[w:hyperbolic angle]] &lt;math&gt;\eta&lt;/math&gt;. Thus a Lorentz boost or [[w:hyperbolic rotation]] (being the same as a rotation around an imaginary angle &lt;math&gt;i\eta=\phi&lt;/math&gt; in [[../Lorentz transformation (imaginary)#math_2b|E:'''(2b)''']] or a [[w:Translation (geometry)|translation]] in the hyperbolic plane in terms of the hyperboloid model) is given by

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline g_{00}=g_{11}=\cosh\eta,\ g_{01}=g_{10}=-\sinh\eta\\
\hline \begin{align}\sinh^{2}\eta-\cosh^{2}\eta &amp; =-1 &amp; (a)\\
\cosh^{2}\eta-\sinh^{2}\eta &amp; =1 &amp; (b)\\
\frac{\sinh\eta}{\cosh\eta} &amp; =\tanh\eta=v &amp; (c)\\
\frac{1}{\sqrt{1-\tanh^{2}\eta}} &amp; =\cosh\eta &amp; (d)\\
\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp; =\sinh\eta &amp; (e)\\
\frac{\tanh q\pm\tanh\eta}{1\pm\tanh q\tanh\eta} &amp; =\tanh\left(q\pm\eta\right) &amp; (f)
\end{align}
\\
\hline \begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
x_{0}^{\prime} &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta &amp;  &amp; =\frac{x_{0}-x_{1}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}-x_{1}v}{\sqrt{1-v^{2}}}\\
x_{1}^{\prime} &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta &amp;  &amp; =\frac{x_{1}-x_{0}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}-x_{0}v}{\sqrt{1-v^{2}}}\\
\\
x_{0} &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta &amp;  &amp; =\frac{x_{0}^{\prime}+x_{1}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}^{\prime}+x_{1}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{1} &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta &amp;  &amp; =\frac{x_{1}^{\prime}+x_{0}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}^{\prime}+x_{0}^{\prime}v}{\sqrt{1-v^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\quad\right|\quad\det\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}=1&lt;/math&gt;|{{equationRef|3b}}}}

Hyperbolic identities (a,b) on the right of ({{equationNote|3b}}) were given by [[#Riccati|Riccati (1757)]], all identities (a,b,c,d,e,f) by [[#Lambert|Lambert (1768–1770)]]. Lorentz transformations ({{equationNote|3b}}-A) were given by [[#Laisant|Laisant (1874)]], [[#Cox|Cox (1882)]], [[#Goursat|Goursat (1888)]], [[#Lindemann|Lindemann (1890/91)]], [[#Gerard|Gérard (1892)]], [[#Killing2|Killing (1893, 1897/98)]], [[#Whitehead|Whitehead (1897/98)]], [[#Woods2|Woods (1903/05)]], [[#Elliott|Elliott (1903)]] and [[#Liebmann|Liebmann (1904/05)]] in terms of Weierstrass coordinates of the [[w:hyperboloid model]], while transformations similar to ({{equationNote|3b}}-C) have been used by [[#Lipschitz1|Lipschitz (1885/86)]]. In special relativity, hyperbolic functions were used by [[#Frank|Frank (1909)]] and [[#Varicak|Varićak (1910)]].

Using the idendity &lt;math&gt;\cosh\eta+\sinh\eta=e^{\eta}&lt;/math&gt;, Lorentz boost ({{equationNote|3b}}) assumes a simple form by using [[w:squeeze mapping]]s in analogy to Euler's formula in [[../Lorentz transformation (imaginary)#math_2c|E:'''(2c)''']]:&lt;ref name=rind&gt;Rindler (1969), p. 45&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{matrix}\begin{align}u' &amp; =ku\\
w' &amp; =\frac{1}{k}w
\end{align}
 &amp; \Rightarrow &amp; \begin{align}x_{1}^{\prime}-x_{0}^{\prime} &amp; =e^{\eta}\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime} &amp; =e^{-\eta}\left(x_{1}+x_{0}\right)
\end{align}
\quad\begin{align}x_{1}-x_{0} &amp; =e^{-\eta}\left(x_{1}^{\prime}-x_{0}^{\prime}\right)\\
x_{1}+x_{0} &amp; =e^{\eta}\left(x_{1}^{\prime}+x_{0}^{\prime}\right)
\end{align}
\end{matrix}\\
\hline k=e^{\eta}=\cosh\eta+\sinh\eta=\sqrt{\frac{1+\tanh\eta}{1-\tanh\eta}}=\sqrt{\frac{1+v}{1-v}}
\end{matrix}&lt;/math&gt;|{{equationRef|3c}}}} 

Lorentz transformations ({{equationNote|3c}}) for arbitrary ''k'' were given by many authors (see [[../Lorentz transformation (squeeze)|E:Lorentz transformations via squeeze mappings]]), while a form similar to &lt;math&gt;k=\sqrt{\tfrac{1+v}{1-v}}&lt;/math&gt; was given by [[#Lipschitz1|Lipschitz (1885/86)]], and the exponential form was implicitly used by [[#mercator|Mercator (1668)]] and explicitly by [[#Lindemann|Lindemann (1890/91)]], [[#Elliott|Elliott (1903)]], [[#Herglotz1|Herglotz (1909)]].

Rapidity can be composed of arbitrary many rapidities &lt;math&gt;\eta_{1},\eta_{2}\dots&lt;/math&gt; as per the [[w:Hyperbolic functions#Sums of arguments|w:angle sum laws of hyperbolic sines and cosines]], so that one hyperbolic rotation can represent the sum of many other hyperbolic rotations, analogous to the relation between [[w:List of trigonometric identities#Angle sum and difference identities|w:angle sum laws of circular trigonometry]] and spatial rotations. Alternatively, the hyperbolic angle sum laws ''themselves'' can be interpreted as Lorentz boosts, as demonstrated by using the parameterization of the [[w:unit hyperbola]]:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \left[\eta=\eta_{2}-\eta_{1}\right]\\
\begin{align}x_{0}^{\prime} &amp; =\sinh\eta_{1}=\sinh\left(\eta_{2}-\eta\right)\\
 &amp; =\sinh\eta_{2}\cosh\eta-\cosh\eta_{2}\sinh\eta=x_{0}\cosh\eta-x_{1}\sinh\eta\\
x_{1}^{\prime} &amp; =\cosh\eta_{1}=\cosh\left(\eta_{2}-\eta\right)\\
 &amp; =-\sinh\eta_{2}\sinh\eta+\cosh\eta_{2}\cosh\eta=-x_{0}\sinh\eta+x_{1}\cosh\eta\\
\\
x_{0} &amp; =\sinh\eta_{2}=\sinh\left(\eta_{1}+\eta\right)\\
 &amp; =\sinh\eta_{1}\cosh\eta+\cosh\eta_{1}\sinh\eta=x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta\\
x_{1} &amp; =\cosh\eta_{2}=\cosh\left(\eta_{1}+\eta\right)\\
 &amp; =\sinh\eta_{1}\sinh\eta+\cosh\eta_{1}\cosh\eta=x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta
\end{align}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{align}\begin{bmatrix}x_{1}^{\prime} &amp; x_{0}^{\prime}\\
x_{0}^{\prime} &amp; x_{1}^{\prime}
\end{bmatrix} &amp; =\begin{bmatrix}\cosh\eta_{1} &amp; \sinh\eta_{1}\\
\sinh\eta_{1} &amp; \cosh\eta_{1}
\end{bmatrix}=\begin{bmatrix}\cosh\left(\eta_{2}-\eta\right) &amp; \sinh\left(\eta_{2}-\eta\right)\\
\sinh\left(\eta_{2}-\eta\right) &amp; \cosh\left(\eta_{2}-\eta\right)
\end{bmatrix}\\
 &amp; =\begin{bmatrix}\cosh\eta_{2} &amp; \sinh\eta_{2}\\
\sinh\eta_{2} &amp; \cosh\eta_{2}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\\
 &amp; =\begin{bmatrix}x_{1} &amp; x_{0}\\
x_{0} &amp; x_{1}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\\
\begin{bmatrix}x_{1} &amp; x_{0}\\
x_{0} &amp; x_{1}
\end{bmatrix} &amp; =\begin{bmatrix}\cosh\eta_{2} &amp; \sinh\eta_{2}\\
\sinh\eta_{2} &amp; \cosh\eta_{2}
\end{bmatrix}=\begin{bmatrix}\cosh\left(\eta_{1}+\eta\right) &amp; \sinh\left(\eta_{1}+\eta\right)\\
\sinh\left(\eta_{1}+\eta\right) &amp; \cosh\left(\eta_{1}+\eta\right)
\end{bmatrix}\\
 &amp; =\begin{bmatrix}\cosh\eta_{1} &amp; \sinh\eta_{1}\\
\sinh\eta_{1} &amp; \cosh\eta_{1}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\\
 &amp; =\begin{bmatrix}x_{1}^{\prime} &amp; x_{0}^{\prime}\\
x_{0}^{\prime} &amp; x_{1}^{\prime}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}
\end{align}
&lt;/math&gt;

or in exponential form as squeeze mapping analogous to ({{equationNote|3c}}): 

&lt;math&gt;\begin{align}e^{-\eta_{1}} &amp; =e^{\eta}e^{-\eta_{2}}=e^{\eta-\eta_{2}} &amp; e^{-\eta_{2}} &amp; =e^{-\eta}e^{-\eta_{1}}=e^{-\eta_{1}-\eta}\\
e^{\eta_{1}} &amp; =e^{-\eta}e^{\eta_{2}}=e^{\eta_{2}-\eta} &amp; e^{\eta_{2}} &amp; =e^{\eta}e^{\eta_{1}}=e^{\eta_{1}+\eta}
\end{align}
&lt;/math&gt;|{{equationRef|3d}}}}

Hyperbolic angle sum laws were given by [[#Riccati|Riccati (1757)]] and [[#Lambert|Lambert (1768–1770)]] and many others, while matrix representations were given by [[#Glaisher|Glaisher (1878)]] and [[#Gunther1|Günther (1880/81)]].

===Hyperbolic law of cosines===
By adding coordinates &lt;math&gt;x_{2}^{\prime}=x_{2}&lt;/math&gt; and &lt;math&gt;x_{3}^{\prime}=x_{3}&lt;/math&gt; in Lorentz transformation ({{equationNote|3b}}) and interpreting &lt;math&gt;x_{0},x_{1},x_{2},x_{3}&lt;/math&gt; as [[w:homogeneous coordinates]], the Lorentz transformation can be rewritten in line with equation [[../Lorentz transformation (general)#math_1b|E:'''(1b)''']] by using coordinates &lt;math&gt;[u_{1},\ u_{2},\ u_{3}]=\left[\tfrac{x_{1}}{x_{0}},\ \tfrac{x_{2}}{x_{0}},\ \tfrac{x_{3}}{x_{0}}\right]&lt;/math&gt; defined by &lt;math&gt;u_{1}^{2}+u_{2}^{2}+u_{3}^{2}\le1&lt;/math&gt; inside the [[w:unit sphere]] as follows:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
\hline \\
u_{1}^{\prime} &amp; =\frac{-\sinh\eta+u_{1}\cosh\eta}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{1}-\tanh\eta}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{1}-v}{1-u_{1}v}\\
u_{2}^{\prime} &amp; =\frac{u_{2}}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-\tanh^{2}\eta}}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-v^{2}}}{1-u_{1}v}\\
u_{3}^{\prime} &amp; =\frac{u_{3}}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{3}\sqrt{1-\tanh^{2}\eta}}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{3}\sqrt{1-v^{2}}}{1-u_{1}v}\\
\\
\hline \\
u_{1} &amp; =\frac{\sinh\eta+u_{1}^{\prime}\cosh\eta}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+\tanh\eta}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+v}{1+u_{1}^{\prime}v}\\
u_{2} &amp; =\frac{u_{2}^{\prime}}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-v^{2}}}{1+u_{1}^{\prime}v}\\
u_{3} &amp; =\frac{u_{3}^{\prime}}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{3}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{3}^{\prime}\sqrt{1-v^{2}}}{1+u_{1}^{\prime}v}
\end{align}
&lt;/math&gt;|{{equationRef|3e}}}}

Transformations (A) were given by [[#Escherich|Escherich (1874)]], [[#Goursat|Goursat (1888)]], [[#Killing2|Killing (1898)]], and transformations (C) by [[#Beltrami|Beltrami (1868)]], [[#Schur|Schur (1885/86, 1900/02)]] in terms of [[w:Beltrami–Klein model|Beltrami coordinates]]&lt;ref&gt;Rosenfeld (1988), p. 231&lt;/ref&gt; of hyperbolic geometry. This transformation becomes equivalent to the [[w:hyperbolic law of cosines]] by restriction to coordinates of the &lt;math&gt;\left[u_{1},u_{2}\right]&lt;/math&gt;-plane and &lt;math&gt;\left[u'_{1},u'_{2}\right]&lt;/math&gt;-plane and defining their scalar products in terms of trigonometric and hyperbolic identities:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix} &amp; \begin{matrix}u^{2}=u_{1}^{2}+u_{2}^{2}\\
u'^{2}=u_{1}^{\prime2}+u_{2}^{\prime2}
\end{matrix}\left|\begin{align}u_{1}=u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+vu'\cos\alpha'}, &amp; u_{1}^{\prime}=u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-vu\cos\alpha}\\
u_{2}=u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{1+vu'\cos\alpha'}, &amp; u_{2}^{\prime}=u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{1-vu\cos\alpha}\\
\frac{u_{2}}{u_{1}}=\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{u'\cos\alpha'+v}, &amp; \frac{u_{2}^{\prime}}{u_{1}^{\prime}}=\tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{u\cos\alpha-v}
\end{align}
\right.\\
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(vu'\sin\alpha'\right){}^{2}}}{1+vu'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(vu\sin\alpha\right){}^{2}}}{1-vu\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-u^{\prime2}}}=\frac{1}{\sqrt{1-v^{2}}}\frac{1}{\sqrt{1-u^{2}}}-\frac{v}{\sqrt{1-v^{2}}}\frac{u}{\sqrt{1-u^{2}}}\cos\alpha &amp; (B)\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha &amp; (A)
\end{matrix}&lt;/math&gt;|{{equationRef|3f}}}}

The hyperbolic law of cosines (A) was given by [[#Taurinus|Taurinus (1826) and Lobachevsky (1829/30)]] and others, while variant (B) was given by [[#Schur|Schur (1900/02)]]. By further setting &lt;math&gt;\tanh\xi=\tanh\zeta=1&lt;/math&gt; or &lt;math&gt;u'=u=1&lt;/math&gt; it follows:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}(A)  &amp; \ \cos\alpha=\frac{\cos\alpha'+\tanh\eta}{1+\tanh\eta\cos\alpha'}; &amp; \ \sin\alpha=\frac{\sin\alpha'\sqrt{1-\tanh^{2}\eta}}{1+\tanh\eta\cos\alpha'}; &amp; \ \tan\alpha=\frac{\sin\alpha'\sqrt{1-\tanh^{2}\eta}}{\cos\alpha'+\tanh\eta}; &amp; \ \tan\frac{\alpha}{2}=\sqrt{\frac{1-\tanh\eta}{1+\tanh\eta}}\tan\frac{\alpha'}{2}\\
 &amp; \ \cos\alpha'=\frac{\cos\alpha-\tanh\eta}{1-\tanh\eta\cos\alpha}; &amp; \ \sin\alpha'=\frac{\sin\alpha\sqrt{1-\tanh^{2}\eta}}{1-\tanh\eta\cos\alpha}; &amp; \ \tan\alpha'=\frac{\sin\alpha\sqrt{1-\tanh^{2}\eta}}{\cos\alpha-\tanh\eta}; &amp; \ \tan\frac{\alpha'}{2}=\sqrt{\frac{1+\tanh\eta}{1-\tanh\eta}}\tan\frac{\alpha}{2}\\
\\
(B)  &amp; \ \cos\alpha=\frac{\cos\alpha'+v}{1+v\cos\alpha'}; &amp; \ \sin\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{1+v\cos\alpha'}; &amp; \ \tan\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{\cos\alpha'+v}; &amp; \ \tan\frac{\alpha}{2}=\sqrt{\frac{1-v}{1+v}}\tan\frac{\alpha'}{2}\\
 &amp; \ \cos\alpha'=\frac{\cos\alpha-v}{1-v\cos\alpha}; &amp; \ \sin\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{1-v\cos\alpha}; &amp; \ \tan\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{\cos\alpha-v}; &amp; \ \tan\frac{\alpha'}{2}=\sqrt{\frac{1+v}{1-v}}\tan\frac{\alpha}{2}
\end{matrix}
&lt;/math&gt;|{{equationRef|3g}}}}

Formulas (3g-B) are the equations of an [[w:ellipse]] of [[w:Orbital eccentricity|eccentricity]] ''v'', [[w:eccentric anomaly]] α' and [[w:true anomaly]] α, first geometrically formulated by [[#Euler|Kepler (1609)]] and explicitly written down by [[#Euler|Euler (1735, 1748), Lagrange (1770)]] and many others in relation to planetary motions. They were also used by [[../Lorentz transformation (conformal)#Darboux|E:Darboux (1873)]] as a sphere transformation. In special relativity these formulas describe the aberration of light, see [[../Lorentz transformation (velocity)#Velocity addition and aberration|E:velocity addition and aberration]].

==Historical notation==
==={{anchor|mercator}} Mercator (1668) – hyperbolic relations ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#mercator|label 1=History of Lorentz transformations via squeeze mappings § Mercator}}
[[File:Mercator-hyperbola-XIV.png|thumb|&lt;small&gt;Mercator's (1668) illustration of AH·FH=AI·BI.&lt;/small&gt;]]

While deriving the [[w:Mercator series]], [[w:Nicholas Mercator]] (1668) demonstrated the following relations on a rectangular hyperbola:&lt;ref group=M&gt;Mercator (1667), prop. XIV, pp. 28-29. (He used this result to derive the Mercator series in prop. XV).&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}AD=1+a,\ DF=\sqrt{2a+aa}\\ AH=\frac{1+a+\sqrt{2a+aa}}{\sqrt{2}},\ FH=\frac{1+a-\sqrt{2a+aa}}{\sqrt{2}}\\ AI=BI=\frac{1}{\sqrt{2}}\\ 1+a=c,\ \sqrt{2a+aa}=d,\ 1=cc-dd\\ AH*FH=\frac{cc-dd}{\sqrt{2}*\sqrt{2}}=\frac{1}{2}\\ AI*BI=\frac{1}{2}\\ \hline AH*FH=AI*BI\\ AH.AI::BI.FH \end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=It can be seen that Mercator's relations &lt;math&gt;1+a=c&lt;/math&gt;, &lt;math&gt;\sqrt{2a+a^{2}}=d&lt;/math&gt; with &lt;math&gt;c^{2}-d^{2}=1&lt;/math&gt; implicitly correspond to hyperbolic functions &lt;math&gt;c=\cosh\eta&lt;/math&gt;, &lt;math&gt;d=\sinh\eta&lt;/math&gt; with &lt;math&gt;\cosh^{2}\eta-\sinh^{2}\eta=1&lt;/math&gt; (which were explicitly introduced by [[#Riccati|Riccati (1757)]] much later). In particular, his result AH.AI::BI.FH, denoting that the ratio between AH and AI is equal to the ratio between BI and FH or &lt;math&gt;\tfrac{AH}{AI}=\tfrac{BI}{FH}&lt;/math&gt; in modern notation, corresponds to squeeze mapping or Lorentz boost ({{equationNote|3c}}) because:&lt;br&gt;
:&lt;math&gt;\frac{AH}{AI}=\frac{BI}{FH}=1+a+\sqrt{2a+a^{2}}=c+d=\cosh\eta+\sinh\eta=e^{\eta}&lt;/math&gt;
or solved for AH and FH:
:&lt;math&gt;AH=e^{\eta}AI&lt;/math&gt; and &lt;math&gt;FH=e^{-\eta}BI&lt;/math&gt;.&lt;br&gt;
Furthermore, transforming Mercator's asymptotic coordinates &lt;math&gt;AH=\tfrac{c+d}{\sqrt{2}}&lt;/math&gt;, &lt;math&gt;FH=\tfrac{c-d}{\sqrt{2}}&lt;/math&gt; into Cartesian coordinates &lt;math&gt;x_{0},x_{1}&lt;/math&gt; gives:&lt;br&gt;
:&lt;math&gt;x_{1}=\tfrac{AH+FH}{\sqrt{2}}=c=\cosh\eta,\quad x_{0}=\tfrac{AH-FH}{\sqrt{2}}=d=\sinh\eta&lt;/math&gt;&lt;br&gt;
which produces the unit hyperbola &lt;math&gt;-x_{0}^{2}+x_{1}^{2}=1&lt;/math&gt; as in ({{equationNote|3d}}), in agreement with Mercator's result AH·FH=1/2 when the hyperbola is referred to its asymptotes.}}

==={{anchor|Euler}} Euler (1735) – True and eccentric anomaly===
{{See also|History of Topics in Special Relativity/Lorentz transformation (imaginary)#Euler|label 1=History of Lorentz transformations via imaginary orthogonal transformations § Euler}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Euler|label 1=History of Lorentz transformations via Quaternions § Euler}}

[[w:Johannes Kepler]] (1609) geometrically formulated [[w:Kepler's equation]] and the relations between the [[w:mean anomaly]], [[w:true anomaly]], and [[w:eccentric anomaly]].&lt;ref group=M&gt;Kepler (1609), chapter 60. The editors of Kepler's collected papers remark (p. 482), that Kepler's relations correspond to &lt;math&gt;{\scriptstyle \alpha=\beta+e\sin\beta}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\nu=\frac{e+\cos\beta}{1+e\cos\beta}}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\beta=\frac{\cos\nu-e}{1-e\cos\nu}}&lt;/math&gt;&lt;/ref&gt;&lt;ref&gt;Volk (1976), p. 366&lt;/ref&gt; The relation between the true anomaly ''z'' and the eccentric anomaly ''P'' was algebraically expressed by [[w:Leonhard Euler]] (1735/40) as follows:&lt;ref group=M&gt;Euler (1735/40), § 19&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{\cos P+v}{1+v\cos P},\ \cos P=\frac{\cos z-v}{1-v\cos z},\ \int P=\frac{\int z\sqrt{1-v^{2}}}{1-v\cos z}&lt;/math&gt;

and in 1748:&lt;ref group=M&gt;Euler (1748a), section VIII&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{n+\cos y}{1+n\cos y},\ \sin z=\frac{\sin y\sqrt{1-n^{2}}}{1+n\cos y},\ \tan z=\frac{\sin y\sqrt{1-n^{2}}}{n+\cos y}&lt;/math&gt;

while [[w:Joseph-Louis Lagrange]] (1770/71) expressed them as follows&lt;ref group=M&gt;Lagrange (1770/71), section I&lt;/ref&gt;

:&lt;math&gt;\sin u=\frac{m\sin x}{1+n\cos x},\ \cos u=\frac{n+\cos x}{1+n\cos x},\ \operatorname{tang}\frac{1}{2}u=\frac{m}{1+n}\operatorname{tang}\frac{1}{2}x,\ \left(m^{2}=1-n^{2}\right)&lt;/math&gt;

{{Lorentzbox|Text= These relations resemble formulas ({{equationNote|3g}}), while ({{equationNote|3e}}) follows by setting &lt;math&gt;[\cos z,\sin z, \cos y,\sin y]=\left[u_{1},u_{2},u'_{1},u'_{2}\right]&lt;/math&gt; in Euler's formulas or &lt;math&gt;[\cos u,\sin u, \cos x,\sin x]=\left[u_{1},u_{2},u'_{1},u'_{2}\right]&lt;/math&gt; in Lagrange's formulas.}}

==={{anchor|Riccati}} Riccati (1757) – hyperbolic addition===
[[File:Riccati-hyperbola.png|thumb|&lt;small&gt;Riccati's (1757) illustration of hyperbolic addition laws.&lt;/small&gt;]]
[[w:Vincenzo Riccati]] (1757) introduced hyperbolic functions ''cosh'' and ''sinh'', which he denoted as ''Ch.'' and ''Sh.'' related by &lt;math&gt;Ch.^{2}-Sh.^{2}=r^2&lt;/math&gt; with ''r'' being set to unity in modern publications, and formulated the addition laws of hyperbolic sine and cosine:&lt;ref group=M&gt;Riccati (1757), p. 71&lt;/ref&gt;&lt;ref group=M&gt;Günther (1880/81), pp. 7–13&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}CA=r,\ CB=Ch.\varphi,\ BE=Sh.\varphi,\ CD=Ch.\pi,\ DF=Sh.\pi\\
CM=Ch.\overline{\varphi+\pi},\ MN=Sh.\overline{\varphi+\pi}\\
CK=\frac{r}{\sqrt{2}},\ CG=\frac{Ch.\varphi+Sh.\varphi}{\sqrt{2}},\ CH=\frac{Ch.\pi+Sh.\pi}{\sqrt{2}},\ CP=\frac{Ch.\overline{\varphi+\pi}+Sh.\overline{\varphi+\pi}}{\sqrt{2}}\\
CK:CG::CH:CP\\
\left[Ch.^{2}-Sh.^{2}=rr\right]\\
\hline Ch.\overline{\varphi+\pi}=\frac{Ch.\varphi\,Ch.\pi+Sh.\varphi\,Sh.\pi}{r}\\
Sh.\overline{\varphi+\pi}=\frac{Ch.\varphi\,Sh.\pi+Ch.\pi\,Sh.\varphi}{r}
\end{matrix}&lt;/math&gt;

He furthermore showed that &lt;math&gt;Ch.\overline{\varphi-\pi}&lt;/math&gt; and &lt;math&gt;Sh.\overline{\varphi-\pi}&lt;/math&gt; follow by setting &lt;math&gt;Ch.\pi\Rightarrow Ch.-\pi&lt;/math&gt; and &lt;math&gt;Sh.\pi\Rightarrow Sh.-\pi&lt;/math&gt; in the above formulas.

{{Lorentzbox|Text=The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3d}}) with &lt;math&gt;\pi=\eta,\ \varphi=\eta_1,\ \overline{\varphi+\pi}=\eta_2&lt;/math&gt;.}}

==={{anchor|Lambert}} Lambert (1768–1770) – hyperbolic addition===

While [[#Riccati|Riccati (1757)]] discussed the hyperbolic sine and cosine, [[w:Johann Heinrich Lambert]] (read 1767, published 1768) introduced the expression ''tang φ'' or abbreviated ''tφ'' as the [[w:tangens hyperbolicus]] &lt;math&gt;{\scriptstyle \frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}}&lt;/math&gt; of a variable ''u'', or in modern notation ''tφ=tanh(u)'':&lt;ref group=M&gt;Lambert (1761/68), pp. 309–318&lt;/ref&gt;&lt;ref&gt;Barnett (2004), pp. 22–23&lt;/ref&gt;

:&lt;math&gt;\left.\begin{align}\xi\xi-1 &amp; =\eta\eta &amp; (a)\\
1+\eta\eta &amp; =\xi\xi &amp; (b)\\
\frac{\eta}{\xi} &amp; =tang\ \phi=t\phi &amp; (c)\\
\xi &amp; =\frac{1}{\sqrt{1-t\phi^{2}}} &amp; (d)\\
\eta &amp; =\frac{t\phi}{\sqrt{1-t\phi^{2}}} &amp; (e)\\
t\phi'' &amp; =\frac{t\phi+t\phi'}{1+t\phi\cdot t\phi'} &amp; (f)\\
t\phi' &amp; =\frac{t\phi''-t\phi}{1-t\phi\cdot t\phi''} &amp; (g)
\end{align}
\right|\begin{align}2u &amp; =\log\frac{1+t\phi}{1-t\phi}\\
\xi &amp; =\frac{e^{u}+e^{-u}}{2}\\
\eta &amp; =\frac{e^{u}-e^{-u}}{2}\\
t\phi &amp; =\frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}\\
e^{u} &amp; =\xi+\eta\\
e^{-u} &amp; =\xi-\eta
\end{align}&lt;/math&gt;

In (1770) he rewrote the addition law for the hyperbolic tangens (f) or (g) as:&lt;ref group=M&gt;Lambert (1770), p. 335&lt;/ref&gt;

:&lt;math&gt;\begin{align}t(y+z) &amp; =(ty+tz):(1+ty\cdot tz) &amp; (f)\\
t(y-z) &amp; =(ty-tz):(1-ty\cdot tz) &amp; (g)
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=The hyperbolic relations (a,b,c,d,e,f) are equivalent to the hyperbolic relations on the right of ({{equationNote|3b}}). Relations (f,g) can also be found in ({{equationNote|3e}}). By setting ''tφ=v/c'', formula (c) becomes the relative velocity between two frames, (d) the [[w:Lorentz factor]], (e) the [[w:proper velocity]], (f) or (g) becomes the Lorentz transformation of velocity (or relativistic [[w:velocity addition formula]]) for collinear velocities in [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']] and [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']].}}

Lambert also formulated the addition laws for the hyperbolic cosine and sine (Lambert's &quot;cos&quot; and &quot;sin&quot; actually mean &quot;cosh&quot; and &quot;sinh&quot;):

:&lt;math&gt;\begin{align}\sin(y+z) &amp; =\sin y\cos z+\cos y\sin z\\
\sin(y-z) &amp; =\sin y\cos z-\cos y\sin z\\
\cos(y+z) &amp; =\cos y\cos z+\sin y\sin z\\
\cos(y-z) &amp; =\cos y\cos z-\sin y\sin z
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3d}}).}}

==={{Anchor|Taurinus}} Taurinus (1826) – Hyperbolic law of cosines===

After the addition theorem for the tangens hyperbolicus was given by [[#Lambert|Lambert (1768)]], [[w:hyperbolic geometry]] was used by [[w:Franz Taurinus]] (1826), and later by [[w:Nikolai Lobachevsky]] (1829/30) and others, to formulate the [[w:hyperbolic law of cosines]]:&lt;ref group=M&gt;Taurinus (1826), p. 66; see also p. 272 in the translation by Engel and Stäckel (1899)&lt;/ref&gt;&lt;ref&gt;Bonola (1912), p. 79&lt;/ref&gt;&lt;ref&gt;Gray (1979), p. 242&lt;/ref&gt;

:&lt;math&gt;A=\operatorname{arccos}\frac{\cos\left(\alpha\sqrt{-1}\right)-\cos\left(\beta\sqrt{-1}\right)\cos\left(\gamma\sqrt{-1}\right)}{\sin\left(\beta\sqrt{-1}\right)\sin\left(\gamma\sqrt{-1}\right)}&lt;/math&gt;

{{Lorentzbox|Text=When solved for &lt;math&gt;\cos\left(\alpha\sqrt{-1}\right)&lt;/math&gt; it corresponds to the Lorentz transformation in Beltrami coordinates ({{equationNote|3f}}), and by defining the rapidities &lt;math&gt;{\scriptstyle \left(\left[\frac{U}{c},\ \frac{v}{c},\ \frac{u}{c}\right]=\left[\tanh\alpha,\ \tanh\beta,\ \tanh\gamma\right]\right)}&lt;/math&gt; it corresponds to the relativistic velocity addition formula [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']].}}

==={{anchor|Beltrami}} Beltrami (1868) – Beltrami coordinates===

[[w:Eugenio Beltrami]] (1868a) introduced coordinates of the [[w:Beltrami–Klein model]] of hyperbolic geometry, and formulated the corresponding transformations in terms of homographies:&lt;ref group=M&gt;Beltrami (1868a), pp. 287-288; Note I; Note II&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds^{2}=R^{2}\frac{\left(a^{2}+v^{2}\right)du^{2}-2uv\,du\,dv+\left(a^{2}+v^{2}\right)dv^{2}}{\left(a^{2}+u^{2}+v^{2}\right)^{2}}\\
u^{2}+v^{2}=a^{2}\\
\hline u''=\frac{aa_{0}\left(u'-r_{0}\right)}{a^{2}-r_{0}u'},\ v''=\frac{a_{0}w_{0}v'}{a^{2}-r_{0}u'},\\
\left(r_{0}=\sqrt{u_{0}^{2}+v_{0}^{2}},\ w_{0}=\sqrt{a^{2}-r_{0}^{2}}\right)\\
\hline ds^{2}=R^{2}\frac{\left(a^{2}-v^{2}\right)du^{2}+2uv\,du\,dv+\left(a^{2}-v^{2}\right)dv^{2}}{\left(a^{2}-u^{2}-v^{2}\right)^{2}}\\
(R=R\sqrt{-1},\ a=a\sqrt{-1})
\end{matrix}&lt;/math&gt;

(where the disk radius ''a'' and the [[w:radius of curvature]] ''R'' are real in spherical geometry, in hyperbolic geometry they are imaginary), and for arbitrary dimensions in (1868b)&lt;ref group=M&gt;Beltrami (1868b), pp. 232, 240–241, 253–254&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds=R\frac{\sqrt{dx^{2}+dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}}}{x}\\
x^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=a^{2}\\
\hline y_{1}=\frac{ab\left(x_{1}-a_{1}\right)}{a^{2}-a_{1}x_{1}}\ \text{or}\ x_{1}=\frac{a\left(ay_{1}+a_{1}b\right)}{ab+a_{1}y_{1}},\ x_{r}=\pm\frac{ay_{r}\sqrt{a^{2}-a_{1}^{2}}}{ab+a_{1}y_{1}}\ (r=2,3,\dots,n)\\
\hline ds=R\frac{\sqrt{dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}-dx^{2}}}{x}\\
x^{2}=a^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\\
\left(R=R\sqrt{-1},\ x=x\sqrt{-1},\ a=a\sqrt{-1}\right)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Setting ''a=a&lt;sub&gt;0&lt;/sub&gt;'' Beltrami's (1868a) formulas become formulas ({{equationNote|3e}}), or in his (1868b) formulas one sets ''a=b'' for arbitrary dimensions.}}

=== {{anchor|Laisant2}} Laisant (1874) – Equipollences===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Laisant1|label 1=History of Lorentz transformations via squeeze mappings § Laisant}}

In his French translation of [[w:Giusto Bellavitis]]' principal work on [[w:Equipollence (geometry)|w:equipollences]], [[w:Charles-Ange Laisant]] (1874) added a chapter related to hyperbolas. The equipollence OM and its tangent MT of a hyperbola is defined by Laisant as&lt;ref group=M&gt;Laisant (1874b), pp. 134–135&lt;/ref&gt;

:(1) &lt;math&gt;\begin{matrix} &amp; \mathrm{OM}\bumpeq x\mathrm{OA}+y\mathrm{OB}\\
 &amp; \mathrm{MT}\bumpeq y\mathrm{OA}+x\mathrm{OB}\\
 &amp; \left[x^{2}-y^{2}=1;\ x=\cosh t,\ y=\sinh t\right]\\
\Rightarrow &amp; \mathrm{OM}\bumpeq\cosh t\cdot\mathrm{OA}+\sinh t\cdot\mathrm{OB}
\end{matrix}&lt;/math&gt;

Here, OA and OB are [[w:Conjugate diameters|conjugate semi-diameters]] of a hyperbola with OB being imaginary, both of which he related to two other conjugated semi-diameters OC and OD by the following transformation:

:&lt;math&gt;\begin{matrix}\begin{align}\mathrm{OC} &amp; \bumpeq c\mathrm{OA}+d\mathrm{OB} &amp; \qquad &amp;  &amp; \mathrm{OA} &amp; \bumpeq c\mathrm{OC}-d\mathrm{OD}\\
\mathrm{OD} &amp; \bumpeq d\mathrm{OA}+c\mathrm{OB} &amp;  &amp;  &amp; \mathrm{OB} &amp; \bumpeq-d\mathrm{OC}+c\mathrm{OD}
\end{align}
\\
\left[c^{2}-d^{2}=1\right]
\end{matrix}&lt;/math&gt;

producing the invariant relation

:&lt;math&gt;(\mathrm{OC})^{2}-(\mathrm{OD})^{2}\bumpeq(\mathrm{OA})^{2}-(\mathrm{OB})^{2}&lt;/math&gt;.

Substituting into (1), he showed that OM retains its form

:&lt;math&gt;\begin{matrix}\mathrm{OM}\bumpeq(cx-dy)\mathrm{OC}+(cy-dx)\mathrm{OD}\\
\left[(cx-dy)^{2}-(cy-dx)^{2}=1\right]
\end{matrix}&lt;/math&gt;

He also defined velocity and acceleration by differentiation of (1).

{{Lorentzbox|Text=These relations are equivalent to several Lorentz boosts or hyperbolic rotations producing the invariant Lorentz interval in line with ({{equationNote|3b}}).}}

==={{anchor|Escherich}} Escherich (1874) – Beltrami coordinates===

[[w:Gustav von Escherich]] (1874) discussed the plane of constant negative curvature&lt;ref&gt;Sommerville (1911), p. 297&lt;/ref&gt; based on the [[w:Beltrami–Klein model]] of hyperbolic geometry by [[#Beltrami|Beltrami (1868)]]. Similar to [[w:Christoph Gudermann]] (1830)&lt;ref name=guder group=M&gt;Gudermann (1830), §1–3, §18–19&lt;/ref&gt;  who introduced axial coordinates ''x''=tan(a) and ''y''=tan(b) in sphere geometry in order to perform coordinate transformations in the case of rotation and translation, Escherich used hyperbolic functions ''x''=tanh(a/k) and ''y''=tanh(b/k)&lt;ref group=M&gt;Escherich (1874), p. 508&lt;/ref&gt; in order to give the corresponding coordinate transformations for the hyperbolic plane, which for the case of translation have the form:&lt;ref group=M name=escher&gt;Escherich (1874), p. 510&lt;/ref&gt;

:&lt;math&gt;x=\frac{\sinh\frac{a}{k}+x'\cosh\frac{a}{k}}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt; and &lt;math&gt;y=\frac{y'}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}), also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting &lt;math&gt;\tfrac{a}{k}=\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt; and multiplying ''[x,y,x′,y′]'' by 1/''c'', and equivalent to Lorentz boost ({{equationNote|3b}}) by setting &lt;math&gt;\scriptstyle (x,\ y,\ x',\ y')=\left(\frac{x_{1}}{x_{0}},\ \frac{x_{2}}{x_{0}},\ \frac{x_{1}^{\prime}}{x_{0}^{\prime}},\ \frac{x_{2}^{\prime}}{x_{0}^{\prime}}\right)&lt;/math&gt;. This is the relation between the [[w:Beltrami–Klein model|Beltrami coordinates]] in terms of Gudermann-Escherich coordinates, and the Weierstrass coordinates of the [[w:hyperboloid model]] introduced by [[../Lorentz transformation (general)#Killing1|E:Killing (1878–1893)]], [[../Lorentz transformation (general)#Poincare|E:Poincaré (1881)]], and [[../Lorentz transformation (general)#Cox|E:Cox (1881)]]. Both coordinate systems were compared by Cox (1881).&lt;ref group=M&gt;Cox (1881), p. 186&lt;/ref&gt;}}

==={{anchor|Glaisher}} Glaisher (1878) – hyperbolic addition===

It was shown by [[w:James Whitbread Lee Glaisher]] (1878) that the hyperbolic addition laws can be expressed by matrix multiplication:&lt;ref group=M&gt;Glaisher (1878), p. 30&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\cosh x, &amp; \sinh x\\
\sinh x, &amp; \cosh x
\end{vmatrix}=1,\ \begin{vmatrix}\cosh y, &amp; \sinh y\\
\sinh y, &amp; \cosh y
\end{vmatrix}=1\\
\text{by multiplication:}\\
\Rightarrow\begin{vmatrix}c_{1}c_{2}+s_{1}s_{2}, &amp; s_{1}c_{2}+c_{1}s_{2}\\
c_{1}s_{2}+s_{1}c_{2}, &amp; s_{1}s_{2}+c_{1}c_{2}
\end{vmatrix}=1\\
\text{where}\ \left[c_{1},c_{2},c_{3},c_{4}\right]=\left[\cosh x,\cosh y,\sinh x,\sinh y\right] \\
\Rightarrow\begin{vmatrix}\cosh(x+y), &amp; \sinh(x+y)\\
\sinh(x+y), &amp; \cosh(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=In this matrix representation, the analogy between the hyperbolic angle sum laws and the Lorentz boost becomes obvious: In particular, the matrix &lt;math&gt;\scriptstyle\begin{vmatrix}\cosh y, &amp; \sinh y\\
\sinh y, &amp; \cosh y\end{vmatrix}&lt;/math&gt; producing the hyperbolic addition is analogous to matrix &lt;math&gt;\scriptstyle\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta\end{bmatrix}&lt;/math&gt; producing Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3d}}).}}

==={{anchor|Gunther1}} Günther (1880/81) – hyperbolic addition ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Gunther1|label 1=History of Lorentz transformations via squeeze mappings § Günther}}

Following [[#Glaisher|Glaisher (1878)]], [[w:Siegmund Günther]] (1880/81) expressed the hyperbolic addition laws by matrix multiplication:&lt;ref group=M&gt;Günther (1880/81), p. 405&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\mathfrak{Cos}\,x, &amp; \mathfrak{Sin}\,x\\
\mathfrak{Sin}\,x, &amp; \mathfrak{Cos}\,x
\end{vmatrix}\cdot\begin{vmatrix}\mathfrak{Cos}\,y, &amp; \mathfrak{Sin}\,y\\
\mathfrak{Sin}\,y, &amp; \mathfrak{Cos}\,y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}\,x\,\mathfrak{Cos}\,y+\mathfrak{Sin}\,x\,\mathfrak{Sin}\,y, &amp; \mathfrak{Cos}\,x\,\mathfrak{Sin}\,y+\mathfrak{Sin}\,x\,\mathfrak{Cos}\,y\\
\mathfrak{Sin}\,x\,\mathfrak{Cos}\,y+\mathfrak{Cos}\,x\,\mathfrak{Sin}\,y, &amp; \mathfrak{Sin}\,x\,\mathfrak{Sin}\,y+\mathfrak{Cos}\,x\,\mathfrak{Cos}\,y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}\,(x+y), &amp; \mathfrak{Sin}\,(x+y)\\
\mathfrak{Sin}\,(x+y), &amp; \mathfrak{Cos}\,(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=In this matrix representation, the analogy between the hyperbolic angle sum laws and the Lorentz boost becomes obvious: In particular, the matrix &lt;math&gt;\scriptstyle\begin{vmatrix}\mathfrak{Cos}\,y, &amp; \mathfrak{Sin}\,y\\
\mathfrak{Sin}\,y, &amp; \mathfrak{Cos}\,y \end{vmatrix}&lt;/math&gt; producing the hyperbolic addition is analogous to matrix &lt;math&gt;\scriptstyle\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta\end{bmatrix}&lt;/math&gt; producing Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3d}}).}}

=== {{anchor|Cox}} Cox (1881/82) – Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Cox|label 1=History of Lorentz transformations in general § Cox}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Cox2|label 1=History of Lorentz transformations via Quaternions § Cox}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Cox|label 1=History of Lorentz transformations via sphere transformations § Cox}}

[[w:Homersham Cox (mathematician)|w:Homersham Cox]] (1881/82) defined the case of translation in the hyperbolic plane with the ''y''-axis remaining unchanged:&lt;ref group=M name=cox&gt;Cox (1881/82), p. 194&lt;/ref&gt;
	
:&lt;math&gt;\begin{align}X &amp; =x\cosh p-z\sinh p\\	
Z &amp; =-x\sinh p+z\cosh p	\\ \\
x &amp; =X\cosh p+Z\sinh p\\
z &amp; =X\sinh p+Z\cosh p	
\end{align}	
&lt;/math&gt;	
	
{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Lipschitz1}} Lipschitz (1885/86) – Quadratic forms ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Lipschitz2|label 1=History of Lorentz transformations via Quaternions § Lipschitz}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Lipschitz1|label 1=History of Lorentz transformations via squeeze mappings § Lipschitz}}

[[w:Rudolf Lipschitz]] (1885/86) discussed transformations leaving invariant the sum of squares
:&lt;math&gt;x_{1}^{2}+x_{2}^{2}\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}&lt;/math&gt;
which he rewrote as 
:&lt;math&gt;x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}+\dots+x_{n}^{2}-y_{n}^{2}=0&lt;/math&gt;. 
This led to the problem of finding transformations leaving invariant the pairs &lt;math&gt;x_{a}^{2}-y_{a}^{2}&lt;/math&gt; (where ''a=1...n'') for which he gave the following solution:&lt;ref group=M&gt;Lipschitz (1886), pp. 90–92&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{a}^{2}-y_{a}^{2}=\mathfrak{x}_{a}^{2}-\mathfrak{y}_{a}^{2}\\
\hline \begin{align}x_{a}-y_{a} &amp; =\left(\mathfrak{x}_{a}-\mathfrak{y}_{a}\right)r_{a}\\
x_{a}+y_{a} &amp; =\left(\mathfrak{x}_{a}+\mathfrak{y}_{a}\right)\frac{1}{r_{a}}
\end{align}
\quad(a)\\
\hline \begin{matrix}\begin{align}2\mathfrak{x}_{a} &amp; =\left(r_{a}+\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}-\frac{1}{r_{a}}\right)y_{a}\\
2\mathfrak{y}_{a} &amp; =\left(r_{a}-\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}+\frac{1}{r_{a}}\right)y_{a}
\end{align}
\quad(b)\end{matrix}\\
\hline \left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}\Rightarrow\begin{align}\mathfrak{x}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}\\
\mathfrak{y}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}
\end{align}
\quad(c)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Lipschitz's transformations (c) and (a) are equivalent to Lorentz boosts ({{equationNote|3b}}-C) and ({{equationNote|3c}}) by the identity &lt;math&gt;s_{a}=\tfrac{1}{v}=\coth\eta&lt;/math&gt;. That is, by substituting &lt;math&gt;v=\tfrac{1}{s_{a}}&lt;/math&gt; in ({{equationNote|3b}}-C) or ({{equationNote|3c}}) we obtain Lipschitz's transformations.}}

==={{Anchor|Schur}} Schur (1885/86, 1900/02) – Beltrami coordinates===

[[w:Friedrich Schur]] (1885/86) discussed spaces of constant Riemann curvature, and by following [[#Beltrami|Beltrami (1868)]] he used the transformation&lt;ref group=M&gt;Schur (1885/86), p. 167&lt;/ref&gt;

:&lt;math&gt;x_{1}=R^{2}\frac{y_{1}+a_{1}}{R^{2}+a_{1}y_{1}},\ x_{2}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{2}}{R^{2}+a_{1}y_{1}},\dots,\ x_{n}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{n}}{R^{2}+a_{1}y_{1}}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] in arbitrary dimensions by setting ''R=c'' as the speed of light and ''a&lt;sub&gt;1&lt;/sub&gt;=v'' as relative velocity.}}

In (1900/02) he derived basic formulas of non-Eucliden geometry, including the case of translation in the hyperbolic plane for which he obtained the transformation similar to his previous one:&lt;ref group=M&gt;Schur (1900/02), p. 290; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;x'=\frac{x-a}{1-\mathfrak{k}ax},\quad y'=\frac{y\sqrt{1-\mathfrak{k}a^{2}}}{1-\mathfrak{k}ax}&lt;/math&gt;

where &lt;math&gt;\mathfrak{k}&lt;/math&gt; can have values &gt;0, &lt;0 or ∞.

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting ''a=v'' and &lt;math&gt;\mathfrak{k}=\tfrac{1}{c^{2}}&lt;/math&gt;.}}

He also defined the triangle&lt;ref group=M&gt;Schur (1900/02), p. 291; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;\frac{1}{\sqrt{1-\mathfrak{k}c^{2}}}=\frac{1}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{1}{\sqrt{1-\mathfrak{k}b^{2}}}-\frac{a}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{b}{\sqrt{1-\mathfrak{k}b^{2}}}\cos\gamma&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to the hyperbolic law of cosines and the relativistic velocity addition ({{equationNote|3f}}, b) or [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']] by setting &lt;math&gt;[\mathfrak{k},\ c,\ a,\ b]=\left[\tfrac{1}{c^{2}},\ \sqrt{u_{x}^{\prime2}+u_{y}^{\prime2}},\ v,\ \sqrt{u_{x}^{2}+u_{y}^{2}}\right]&lt;/math&gt;.}}

==={{Anchor|Goursat}} Goursat (1887/88) – Minimal surfaces===

[[w:Édouard Goursat]] defined real coordinates &lt;math&gt;x,y&lt;/math&gt; of minimal surface &lt;math&gt;S&lt;/math&gt; and imaginary coordinates &lt;math&gt;x_{0},y_{0}&lt;/math&gt; of the adjoint minimal surface &lt;math&gt;S_0&lt;/math&gt;, so that another real minimal surface &lt;math&gt;S_1&lt;/math&gt; follows by the (conformal) transformation:&lt;ref group=M&gt;Goursat (1887/88), p. 144&lt;/ref&gt;

:&lt;math&gt;\begin{align}x_{1} &amp; =\frac{1+k^{2}}{2k}x-\frac{k^{2}-1}{2k}y_{0}\\
y_{1} &amp; =\frac{1+k^{2}}{2k}y+\frac{k^{2}-1}{2k}x_{0}\\
z_{1} &amp; =z
\end{align}&lt;/math&gt;

and expressed these equations in terms of hyperbolic functions by setting &lt;math&gt;k=e^{\varphi}&lt;/math&gt;:&lt;ref group=M&gt;Goursat (1887/88), p. 145&lt;/ref&gt;

:&lt;math&gt;\begin{align}x_{1} &amp; =x\cosh\varphi-y_{0}\sinh\varphi\\
y_{1} &amp; =y\cosh\varphi+x_{0}\sinh\varphi\\
z_{1} &amp; =z
\end{align}&lt;/math&gt;

{{Lorentzbox|Text=This becomes Lorentz boost ({{equationNote|3b}}) by replacing the imaginary coordinates &lt;math&gt;x_{0},y_{0}&lt;/math&gt; by real coordinates defined as &lt;math&gt;[x_{0},y_{0}]=[-x,y]&lt;/math&gt;. It can also be seen that Goursat's relation &lt;math&gt;k=e^{\varphi}&lt;/math&gt; corresponds to &lt;math&gt;k=e^{\eta}&lt;/math&gt; defined in ({{equationNote|3c}}).}}

He went on to define &lt;math&gt;\alpha,\beta,\gamma&lt;/math&gt; as the direction cosines normal to surface &lt;math&gt;S&lt;/math&gt; and &lt;math&gt;\alpha_{1},\beta_{1},\gamma_{1}&lt;/math&gt; as the ones normal to surface &lt;math&gt;S_{1}&lt;/math&gt;, connected by the transformation:&lt;ref group=M&gt;Goursat (1887/88), p. 149f.&lt;/ref&gt;

:&lt;math&gt;\begin{align}\alpha_{1} &amp; =\pm\frac{\alpha}{\cosh\varphi-\gamma\sinh\varphi} &amp;  &amp;  &amp; \alpha &amp; =\pm\frac{\alpha_{1}}{\cosh\varphi+\gamma_{1}\sinh\varphi}\\
\beta_{1} &amp; =\pm\frac{\beta}{\cosh\varphi-\gamma\sinh\varphi} &amp;  &amp;  &amp; \beta &amp; =\pm\frac{\beta_{1}}{\cosh\varphi+\gamma_{1}\sinh\varphi}\\
\gamma_{1} &amp; =\pm\frac{\gamma\cosh\varphi-\sinh\varphi}{\cosh\varphi-\gamma\sinh\varphi} &amp;  &amp;  &amp; \gamma &amp; =\pm\frac{\gamma_{1}\cosh\varphi+\sinh\varphi}{\cosh\varphi+\gamma_{1}\sinh\varphi}
\end{align}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}-A) with &lt;math&gt;\left[\alpha,\beta,\gamma\right]=\left[u_{2},u_{3},u_{1}\right]&lt;/math&gt;.}}

==={{anchor|Lindemann}} Lindemann (1890–91) – Weierstrass coordinates and Cayley absolute===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Lindemann|label 1=History of Lorentz transformations via squeeze mappings § Lindemann}}

[[w:Ferdinand von Lindemann]] discussed hyperbolic geometry in terms of the [[w:Cayley–Klein metric]] in his (1890/91) edition of the lectures on geometry of [[w:Alfred Clebsch]]. Citing [[../Lorentz transformation (general)#Killing|E:Killing (1885)]] and [[../Lorentz transformation (general)#Poincare|Poincaré (1887)]] in relation to the hyperboloid model in terms of Weierstrass coordinates for the hyperbolic plane and space, he set&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 477–478, 524&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\Omega_{xx}=x_{1}^{2}+x_{2}^{2}-4k^{2}x_{3}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}-4k^{2}dx_{3}^{2}\\
\Omega_{xx}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-4k^{2}dx_{4}^{2}
\end{matrix}&lt;/math&gt;

and used the following transformation&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 361–362&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\
X_{1}X_{4}+X_{2}X_{3}=\Xi_{1}\Xi_{4}+\Xi_{2}\Xi_{3}\\
\hline \begin{align}X_{1} &amp; =\left(\lambda+\lambda_{1}\right)U_{4} &amp; \Xi_{1} &amp; =\left(\lambda-\lambda_{1}\right)U_{4} &amp; X_{1} &amp; =\frac{\lambda+\lambda_{1}}{\lambda-\lambda_{1}}\Xi_{1}\\
X_{2} &amp; =\left(\lambda+\lambda_{3}\right)U_{4} &amp; \Xi_{2} &amp; =\left(\lambda-\lambda_{3}\right)U_{4} &amp; X_{2} &amp; =\frac{\lambda+\lambda_{3}}{\lambda-\lambda_{3}}\Xi_{2}\\
X_{3} &amp; =\left(\lambda-\lambda_{3}\right)U_{2} &amp; \Xi_{3} &amp; =\left(\lambda+\lambda_{3}\right)U_{2} &amp; X_{3} &amp; =\frac{\lambda-\lambda_{3}}{\lambda+\lambda_{3}}\Xi_{3}\\
X_{4} &amp; =\left(\lambda-\lambda_{1}\right)U_{1} &amp; \Xi_{4} &amp; =\left(\lambda+\lambda_{1}\right)U_{1} &amp; X_{4} &amp; =\frac{\lambda-\lambda_{1}}{\lambda+\lambda_{1}}\Xi_{4}
\end{align}
\end{matrix}&lt;/math&gt;

into which he put&lt;ref group=M name=linde&gt;Lindemann &amp; Clebsch (1890/91), p. 496&lt;/ref&gt;

:&lt;math&gt;\begin{align}X_{1} &amp; =x_{1}+2kx_{4}, &amp; X_{2} &amp; =x_{2}+ix_{3}, &amp; \lambda+\lambda_{1} &amp; =\left(\lambda-\lambda_{1}\right)e^{a},\\
X_{4} &amp; =x_{1}-2kx_{4}, &amp; X_{3} &amp; =x_{2}-ix_{3}, &amp; \lambda+\lambda_{3} &amp; =\left(\lambda-\lambda_{3}\right)e^{\alpha i},
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3c}}) with &lt;math&gt;e^{\alpha i}=1&lt;/math&gt; and ''2k=1'' .}}

From that, he obtained the following Cayley absolute and the corresponding most general motion in hyperbolic space comprising ordinary rotations (''a''=0) or translations (α=0):&lt;ref group=M  name=linde /&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=0\\
\hline \begin{align}x_{2} &amp; =\xi_{2}\cos\alpha+\xi_{3}\sin\alpha, &amp; x_{1} &amp; =\xi_{1}\cos\frac{a}{i}+2ki\xi_{4}\sin\frac{a}{i},\\
x_{3} &amp; =-\xi_{2}\sin\alpha+\xi_{3}\cos\alpha, &amp; 2kx_{4} &amp; =i\xi_{1}\sin\frac{a}{i}+2k\xi_{4}\cos\frac{a}{i}.
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0 and ''2k=1''.}}

==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Gerard|label 1=History of Lorentz transformations in general § Gerard}}

[[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the hyperbolic plane and gave the case of translation as follows:&lt;ref group=M name=gerard&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{align}X &amp; =Z_{0}X'+X_{0}Z'\\
Y &amp; =Y'\\
Z &amp; =X_{0}X'+Z_{0}Z'
\end{align}
\ \text{with}\ \begin{align}X_{0} &amp; =\operatorname{sh}OO'\\
Z_{0} &amp; =\operatorname{ch}OO'
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Killing2}} Killing (1893,97) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Killing|label 1=History of Lorentz transformations in general § Killing}}

[[w:Wilhelm Killing]] (1878–1880) gave the case of translation in the hyperbolic plane as follows:&lt;ref group=M name=killtra&gt;Killing (1893), p. 331&lt;/ref&gt;

:&lt;math&gt;y_{0}=x_{0}\operatorname{Ch}a+x_{1}\operatorname{Sh}a,\quad y_{1}=x_{0}\operatorname{Sh}a+x_{1}\operatorname{Ch}a,\quad y_{2}=x_{2}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

In 1898, Killing wrote that relation in a form similar to [[#Escherich|Escherich (1874)]], and derived the corresponding Lorentz transformation for the two cases were ''v'' is unchanged or ''u'' is unchanged:&lt;ref group=M name=kill98&gt;Killing (1898), p. 133&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\xi'=\frac{\xi\operatorname{Ch}\frac{\mu}{l}+l\operatorname{Sh}\frac{\mu}{l}}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}},\ \eta'=\frac{\eta}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}}\\
\hline \frac{u}{p}=\xi,\ \frac{v}{p}=\eta\\
\hline p'=p\operatorname{Ch}\frac{\mu}{l}+\frac{u}{l}\operatorname{Sh}\frac{\mu}{l},\quad u'=pl\operatorname{Sh}\frac{\mu}{l}+u\operatorname{Ch}\frac{\mu}{l},\quad v'=v\\
\text{or}\\
p'=p\operatorname{Ch}\frac{\nu}{l}+\frac{v}{l}\operatorname{Sh}\frac{\nu}{l},\quad u'=u,\quad v'=pl\operatorname{Sh}\frac{\nu}{l}+v\operatorname{Ch}\frac{\nu}{l}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The upper transformation system is equivalent to Lorentz transformation ({{equationNote|3e}}) and the velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] with ''l=c'' and &lt;math&gt;\mu=c\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt;, the system below is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Whitehead}} Whitehead (1897/98) – Universal algebra===

[[w:Alfred North Whitehead]] (1898) discussed the kinematics of hyperbolic space as part of his study of [[w:universal algebra]], and obtained the following transformation:&lt;ref group=M name=white&gt;Whitehead (1898), pp. 459–460&lt;/ref&gt;

:&lt;math&gt;\begin{align}x' &amp; =\left(\eta\cosh\frac{\delta}{\gamma}+\eta_{1}\sinh\frac{\delta}{\gamma}\right)e+\left(\eta\sinh\frac{\delta}{\gamma}+\eta_{1}\cosh\frac{\delta}{\gamma}\right)e_{1}\\
 &amp; \qquad+\left(\eta_{2}\cos\alpha+\eta_{3}\sin\alpha\right)e_{2}+\left(\eta_{3}\cos\alpha-\eta_{2}\sin\alpha\right)e_{3}
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0.}}

==={{anchor|Elliott}} Elliott (1903) – Invariant theory ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Elliott|label 1=History of Lorentz transformations via squeeze mappings § Elliott}}

[[w:Edwin Bailey Elliott]] (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:&lt;ref group=M&gt;Elliott (1903), p. 109&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=X\cosh\phi+Y\sinh\phi,\quad y=X\sinh\phi+Y\cosh\phi\\
\hline X+Y=e^{-\phi}(x+y),\quad X-Y=e^{\phi}(x-y)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3c}}). The mentioned subgroup corresponds to the one-parameter subgroup generated by Lorentz boosts.}}

==={{anchor|Woods2}} Woods (1903) – Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Woods2|label 1=History of Lorentz transformations in general § Woods}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Woods|label 1=History of Lorentz transformations via Möbius transformations § Woods}}

[[w:Frederick S. Woods]] (1903, published 1905) gave the case of translation in hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 55&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x_{1}\cos kl+x_{0}\frac{\sin kl}{k},\quad x_{2}^{\prime}=x_{2},\quad x_{2}^{\prime}=x_{3},\quad x_{0}^{\prime}=-x_{1}k\sin kl+x_{0}\cos kl&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.}}

and the loxodromic substitution for hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 72&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}x_{1}^{\prime} &amp; =x_{1}\cosh\alpha-x_{0}\sinh\alpha\\
x_{2}^{\prime} &amp; =x_{2}\cos\beta-x_{3}\sin\beta\\
x_{3}^{\prime} &amp; =x_{2}\sin\beta+x_{3}\cos\beta\\
x_{0}^{\prime} &amp; =-x_{1}\sinh\alpha+x_{0}\cosh\alpha
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with β=0.}}

==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Liebmann|label 1=History of Lorentz transformations in general § Liebmann}}

[[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – gave the case of translation in the hyperbolic plane:&lt;ref group=M name=lieb&gt;Liebmann (1904/05), p. 174&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x'\operatorname{ch}a+p'\operatorname{sh}a,\quad y_{1}^{\prime}=y',\quad p_{1}^{\prime}=x'\operatorname{sh}a+p'\operatorname{ch}a&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Frank}} Frank (1909) – Special relativity===
In special relativity, hyperbolic functions were used by [[w:Philipp Frank]] (1909), who derived the Lorentz transformation using ''ψ'' as rapidity:&lt;ref group=R&gt;Frank (1909), pp. 423-425&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=x\varphi(a)\,{\rm ch}\,\psi+t\varphi(a)\,{\rm sh}\,\psi\\
t'=-x\varphi(a)\,{\rm sh}\,\psi+t\varphi(a)\,{\rm ch}\,\psi\\
\hline {\rm th}\,\psi=-a,\ {\rm sh}\,\psi=\frac{a}{\sqrt{1-a^{2}}},\ {\rm ch}\,\psi=\frac{1}{\sqrt{1-a^{2}}},\ \varphi(a)=1\\
\hline x'=\frac{x-at}{\sqrt{1-a^{2}}},\ y'=y,\ z'=z,\ t'=\frac{-ax+t}{\sqrt{1-a^{2}}}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}
 	
=== {{anchor|Herglotz1}} Herglotz (1909/10) – Special relativity===
{{See also|History of Topics in Special Relativity/Lorentz transformation (velocity)#Herglotz1|label 1=History of Lorentz transformations via velocity § Herglotz}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Herglotz|label 1=History of Lorentz transformations via squeeze mappings § Herglotz}}

In special relativity, [[w:Gustav Herglotz]] (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic, with the hyperbolic case being:&lt;ref group=R&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;
	
:&lt;math&gt;\begin{matrix}Z=Z'e^{\vartheta}\\
	
\begin{aligned}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
	
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
	
\end{aligned}
	
\end{matrix}&lt;/math&gt;	
	
{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3c}}).}}

==={{anchor|Varicak}} Varićak (1910) – Special relativity===
{{See also|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Varicak|label 1=History of Lorentz transformations via trigonometric functions § Varicak}}

In special relativity, hyperbolic functions were used by [[w:Vladimir Varićak]] in several papers starting from 1910, who represented the equations of special relativity on the basis of [[w:hyperbolic geometry]] in terms of Weierstrass coordinates. For instance, by setting ''l=ct'' and ''v/c=tanh(u)'' with ''u'' as rapidity he wrote the Lorentz transformation in agreement with ({{equationNote|4b}}):&lt;ref group=R name=var1&gt;Varićak (1910), p. 93&lt;/ref&gt;

:&lt;math&gt;\begin{align}l' &amp; =-x\operatorname{sh}u+l\operatorname{ch}u,\\
x' &amp; =x\operatorname{ch}u-l\operatorname{sh}u,\\
y' &amp; =y,\quad z'=z,\\
\operatorname{ch}u &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

He showed the relation of rapidity to the [[w:Gudermannian function]] and the [[w:angle of parallelism]]:&lt;ref group=R name=var1 /&gt;

:&lt;math&gt;\frac{v}{c}=\operatorname{th}u=\operatorname{tg}\psi=\sin\operatorname{gd}(u)=\cos\Pi(u)&lt;/math&gt;

He also related the velocity addition to the [[w:hyperbolic law of cosines]]:&lt;ref group=R&gt;Varićak (1910), p. 94&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\operatorname{ch}{u}=\operatorname{ch}{u_{1}}\operatorname ch{u_{2}}+\operatorname{sh}{u_{1}}\operatorname{sh}{u_{2}}\cos\alpha\\
\operatorname{ch}{u_{i}}=\frac{1}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},\ \operatorname{sh}{u_{i}}=\frac{v_{i}}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}}\\
v=\sqrt{v_{1}^{2}+v_{2}^{2}-\left(\frac{v_{1}v_{2}}{c}\right)^{2}}\ \left(a=\frac{\pi}{2}\right)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3f}}).}}

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bel68sag}} 
*{{#section:History of Topics in Special Relativity/mathsource|bel68fond}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|eli03}} 
*{{#section:History of Topics in Special Relativity/mathsource|esch74}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul35}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul48a}} 
*{{#section:History of Topics in Special Relativity/mathsource|ger92}} 
*{{#section:History of Topics in Special Relativity/mathsource|glai78}} 
*{{#section:History of Topics in Special Relativity/mathsource|gour88}} 
*{{#section:History of Topics in Special Relativity/mathsource|gud30}} 
*{{#section:History of Topics in Special Relativity/mathsource|guen80}} 
*{{#section:History of Topics in Special Relativity/mathsource|kep09}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil93}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil97}} 
*{{#section:History of Topics in Special Relativity/mathsource|lag70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lais74b}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam67}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lieb04}} 
*{{#section:History of Topics in Special Relativity/mathsource|lind90}}
*{{#section:History of Topics in Special Relativity/mathsource|lip86}}
*{{#section:History of Topics in Special Relativity/mathsource|merc}}
*{{#section:History of Topics in Special Relativity/mathsource|ric57}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu85}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu00}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu09}} 
*{{#section:History of Topics in Special Relativity/mathsource|tau26}} 
*{{#section:History of Topics in Special Relativity/mathsource|whit98}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo01}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo03}}

===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|frank09a}}
*{{#section:History of Topics in Special Relativity/relsource|herg10}}
*{{#section:History of Topics in Special Relativity/relsource|var10}}
*{{#section:History of Topics in Special Relativity/relsource|var12}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L3}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
      <sha1>5z2wpsggdb1lz37c3ahvvbv5hqj0hsz</sha1>
    </revision>
    <revision>
      <id>2721179</id>
      <parentid>2721178</parentid>
      <timestamp>2025-07-10T07:34:54Z</timestamp>
      <contributor>
        <username>D.H</username>
        <id>52339</id>
      </contributor>
      <comment>/* Lorentz transformation via hyperbolic functions */</comment>
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      <format>text/x-wiki</format>
      <text bytes="57544" sha1="lkfa7di6rnyhv08f6ks0qlkxvlg1ez8" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via hyperbolic functions==
===Translation in the hyperbolic plane===
[[File:Hyperbolic functions-2.svg|thumb|upright=1.4|200px|A ray through the unit hyperbola {{math|1=''x''&lt;sup&gt;2&lt;/sup&gt; − ''y''&lt;sup&gt;2&lt;/sup&gt; = 1}} at the point {{math|(cosh ''a'', sinh ''a'')}}.]]
The case of a Lorentz transformation without spatial rotation is called a [[w:Lorentz boost]]. The simplest case can be given, for instance, by setting ''n=1'' in the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']]:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}g_{00}+x_{1}g_{01}\\
x_{1}^{\prime} &amp; =x_{0}g_{10}+x_{1}g_{11}\\
\\
x_{0} &amp; =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}\\
x_{1} &amp; =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}
\end{align}
\left|\begin{align}g_{01}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{10}^{2} &amp; =1\\
g_{01}g_{11}-g_{00}g_{10} &amp; =0\\
g_{10}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{01}^{2} &amp; =1\\
g_{10}g_{11}-g_{00}g_{01} &amp; =0
\end{align}
\rightarrow\begin{align}g_{00}^{2} &amp; =g_{11}^{2}\\
g_{01}^{2} &amp; =g_{10}^{2}
\end{align}
\right.
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}g_{00} &amp; -g_{10}\\
-g_{01} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\quad\right|\quad\det\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}=1&lt;/math&gt;|{{equationRef|3a}}}}

which resembles precisely the relations of [[w:hyperbolic function]]s in terms of [[w:hyperbolic angle]] &lt;math&gt;\eta&lt;/math&gt;. Thus a Lorentz boost or [[w:hyperbolic rotation]] (being the same as a rotation around an imaginary angle &lt;math&gt;i\eta=\phi&lt;/math&gt; in [[../Lorentz transformation (imaginary)#math_2b|E:'''(2b)''']] or a [[w:Translation (geometry)|translation]] in the hyperbolic plane in terms of the hyperboloid model) is given by

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline g_{00}=g_{11}=\cosh\eta,\ g_{01}=g_{10}=-\sinh\eta\\
\hline \begin{align}\sinh^{2}\eta-\cosh^{2}\eta &amp; =-1 &amp; (a)\\
\cosh^{2}\eta-\sinh^{2}\eta &amp; =1 &amp; (b)\\
\frac{\sinh\eta}{\cosh\eta} &amp; =\tanh\eta=v &amp; (c)\\
\frac{1}{\sqrt{1-\tanh^{2}\eta}} &amp; =\cosh\eta &amp; (d)\\
\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp; =\sinh\eta &amp; (e)\\
\frac{\tanh q\pm\tanh\eta}{1\pm\tanh q\tanh\eta} &amp; =\tanh\left(q\pm\eta\right) &amp; (f)
\end{align}
\\
\hline \begin{align} &amp; \quad\quad(A) &amp; (B)\quad\quad &amp; \quad(C)\\
x_{0}^{\prime} &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta &amp; =\frac{x_{0}-x_{1}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp; =\frac{x_{0}-x_{1}v}{\sqrt{1-v^{2}}}\\
x_{1}^{\prime} &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta &amp; =\frac{x_{1}-x_{0}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp; =\frac{x_{1}-x_{0}v}{\sqrt{1-v^{2}}}\\
\\
x_{0} &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta &amp; =\frac{x_{0}^{\prime}+x_{1}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp; =\frac{x_{0}^{\prime}+x_{1}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{1} &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta &amp; =\frac{x_{1}^{\prime}+x_{0}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp; =\frac{x_{1}^{\prime}+x_{0}^{\prime}v}{\sqrt{1-v^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\quad\right|\quad\det\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}=1&lt;/math&gt;|{{equationRef|3b}}}}

Hyperbolic identities (a,b) on the right of ({{equationNote|3b}}) were given by [[#Riccati|Riccati (1757)]], all identities (a,b,c,d,e,f) by [[#Lambert|Lambert (1768–1770)]]. Lorentz transformations ({{equationNote|3b}}-A) were given by [[#Laisant|Laisant (1874)]], [[#Cox|Cox (1882)]], [[#Goursat|Goursat (1888)]], [[#Lindemann|Lindemann (1890/91)]], [[#Gerard|Gérard (1892)]], [[#Killing2|Killing (1893, 1897/98)]], [[#Whitehead|Whitehead (1897/98)]], [[#Woods2|Woods (1903/05)]], [[#Elliott|Elliott (1903)]] and [[#Liebmann|Liebmann (1904/05)]] in terms of Weierstrass coordinates of the [[w:hyperboloid model]], while transformations similar to ({{equationNote|3b}}-C) have been used by [[#Lipschitz1|Lipschitz (1885/86)]]. In special relativity, hyperbolic functions were used by [[#Frank|Frank (1909)]] and [[#Varicak|Varićak (1910)]].

Using the idendity &lt;math&gt;\cosh\eta+\sinh\eta=e^{\eta}&lt;/math&gt;, Lorentz boost ({{equationNote|3b}}) assumes a simple form by using [[w:squeeze mapping]]s in analogy to Euler's formula in [[../Lorentz transformation (imaginary)#math_2c|E:'''(2c)''']]:&lt;ref name=rind&gt;Rindler (1969), p. 45&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{matrix}\begin{align}u' &amp; =ku\\
w' &amp; =\frac{1}{k}w
\end{align}
 &amp; \Rightarrow &amp; \begin{align}x_{1}^{\prime}-x_{0}^{\prime} &amp; =e^{\eta}\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime} &amp; =e^{-\eta}\left(x_{1}+x_{0}\right)
\end{align}
\quad\begin{align}x_{1}-x_{0} &amp; =e^{-\eta}\left(x_{1}^{\prime}-x_{0}^{\prime}\right)\\
x_{1}+x_{0} &amp; =e^{\eta}\left(x_{1}^{\prime}+x_{0}^{\prime}\right)
\end{align}
\end{matrix}\\
\hline k=e^{\eta}=\cosh\eta+\sinh\eta=\sqrt{\frac{1+\tanh\eta}{1-\tanh\eta}}=\sqrt{\frac{1+v}{1-v}}
\end{matrix}&lt;/math&gt;|{{equationRef|3c}}}} 

Lorentz transformations ({{equationNote|3c}}) for arbitrary ''k'' were given by many authors (see [[../Lorentz transformation (squeeze)|E:Lorentz transformations via squeeze mappings]]), while a form similar to &lt;math&gt;k=\sqrt{\tfrac{1+v}{1-v}}&lt;/math&gt; was given by [[#Lipschitz1|Lipschitz (1885/86)]], and the exponential form was implicitly used by [[#mercator|Mercator (1668)]] and explicitly by [[#Lindemann|Lindemann (1890/91)]], [[#Elliott|Elliott (1903)]], [[#Herglotz1|Herglotz (1909)]].

Rapidity can be composed of arbitrary many rapidities &lt;math&gt;\eta_{1},\eta_{2}\dots&lt;/math&gt; as per the [[w:Hyperbolic functions#Sums of arguments|w:angle sum laws of hyperbolic sines and cosines]], so that one hyperbolic rotation can represent the sum of many other hyperbolic rotations, analogous to the relation between [[w:List of trigonometric identities#Angle sum and difference identities|w:angle sum laws of circular trigonometry]] and spatial rotations. Alternatively, the hyperbolic angle sum laws ''themselves'' can be interpreted as Lorentz boosts, as demonstrated by using the parameterization of the [[w:unit hyperbola]]:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \left[\eta=\eta_{2}-\eta_{1}\right]\\
\begin{align}x_{0}^{\prime} &amp; =\sinh\eta_{1}=\sinh\left(\eta_{2}-\eta\right)\\
 &amp; =\sinh\eta_{2}\cosh\eta-\cosh\eta_{2}\sinh\eta=x_{0}\cosh\eta-x_{1}\sinh\eta\\
x_{1}^{\prime} &amp; =\cosh\eta_{1}=\cosh\left(\eta_{2}-\eta\right)\\
 &amp; =-\sinh\eta_{2}\sinh\eta+\cosh\eta_{2}\cosh\eta=-x_{0}\sinh\eta+x_{1}\cosh\eta\\
\\
x_{0} &amp; =\sinh\eta_{2}=\sinh\left(\eta_{1}+\eta\right)\\
 &amp; =\sinh\eta_{1}\cosh\eta+\cosh\eta_{1}\sinh\eta=x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta\\
x_{1} &amp; =\cosh\eta_{2}=\cosh\left(\eta_{1}+\eta\right)\\
 &amp; =\sinh\eta_{1}\sinh\eta+\cosh\eta_{1}\cosh\eta=x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta
\end{align}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{align}\begin{bmatrix}x_{1}^{\prime} &amp; x_{0}^{\prime}\\
x_{0}^{\prime} &amp; x_{1}^{\prime}
\end{bmatrix} &amp; =\begin{bmatrix}\cosh\eta_{1} &amp; \sinh\eta_{1}\\
\sinh\eta_{1} &amp; \cosh\eta_{1}
\end{bmatrix}=\begin{bmatrix}\cosh\left(\eta_{2}-\eta\right) &amp; \sinh\left(\eta_{2}-\eta\right)\\
\sinh\left(\eta_{2}-\eta\right) &amp; \cosh\left(\eta_{2}-\eta\right)
\end{bmatrix}\\
 &amp; =\begin{bmatrix}\cosh\eta_{2} &amp; \sinh\eta_{2}\\
\sinh\eta_{2} &amp; \cosh\eta_{2}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\\
 &amp; =\begin{bmatrix}x_{1} &amp; x_{0}\\
x_{0} &amp; x_{1}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\\
\begin{bmatrix}x_{1} &amp; x_{0}\\
x_{0} &amp; x_{1}
\end{bmatrix} &amp; =\begin{bmatrix}\cosh\eta_{2} &amp; \sinh\eta_{2}\\
\sinh\eta_{2} &amp; \cosh\eta_{2}
\end{bmatrix}=\begin{bmatrix}\cosh\left(\eta_{1}+\eta\right) &amp; \sinh\left(\eta_{1}+\eta\right)\\
\sinh\left(\eta_{1}+\eta\right) &amp; \cosh\left(\eta_{1}+\eta\right)
\end{bmatrix}\\
 &amp; =\begin{bmatrix}\cosh\eta_{1} &amp; \sinh\eta_{1}\\
\sinh\eta_{1} &amp; \cosh\eta_{1}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\\
 &amp; =\begin{bmatrix}x_{1}^{\prime} &amp; x_{0}^{\prime}\\
x_{0}^{\prime} &amp; x_{1}^{\prime}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}
\end{align}
&lt;/math&gt;

or in exponential form as squeeze mapping analogous to ({{equationNote|3c}}): 

&lt;math&gt;\begin{align}e^{-\eta_{1}} &amp; =e^{\eta}e^{-\eta_{2}}=e^{\eta-\eta_{2}} &amp; e^{-\eta_{2}} &amp; =e^{-\eta}e^{-\eta_{1}}=e^{-\eta_{1}-\eta}\\
e^{\eta_{1}} &amp; =e^{-\eta}e^{\eta_{2}}=e^{\eta_{2}-\eta} &amp; e^{\eta_{2}} &amp; =e^{\eta}e^{\eta_{1}}=e^{\eta_{1}+\eta}
\end{align}
&lt;/math&gt;|{{equationRef|3d}}}}

Hyperbolic angle sum laws were given by [[#Riccati|Riccati (1757)]] and [[#Lambert|Lambert (1768–1770)]] and many others, while matrix representations were given by [[#Glaisher|Glaisher (1878)]] and [[#Gunther1|Günther (1880/81)]].

===Hyperbolic law of cosines===
By adding coordinates &lt;math&gt;x_{2}^{\prime}=x_{2}&lt;/math&gt; and &lt;math&gt;x_{3}^{\prime}=x_{3}&lt;/math&gt; in Lorentz transformation ({{equationNote|3b}}) and interpreting &lt;math&gt;x_{0},x_{1},x_{2},x_{3}&lt;/math&gt; as [[w:homogeneous coordinates]], the Lorentz transformation can be rewritten in line with equation [[../Lorentz transformation (general)#math_1b|E:'''(1b)''']] by using coordinates &lt;math&gt;[u_{1},\ u_{2},\ u_{3}]=\left[\tfrac{x_{1}}{x_{0}},\ \tfrac{x_{2}}{x_{0}},\ \tfrac{x_{3}}{x_{0}}\right]&lt;/math&gt; defined by &lt;math&gt;u_{1}^{2}+u_{2}^{2}+u_{3}^{2}\le1&lt;/math&gt; inside the [[w:unit sphere]] as follows:

{{NumBlk|:|&lt;math&gt;\begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
\hline \\
u_{1}^{\prime} &amp; =\frac{-\sinh\eta+u_{1}\cosh\eta}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{1}-\tanh\eta}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{1}-v}{1-u_{1}v}\\
u_{2}^{\prime} &amp; =\frac{u_{2}}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-\tanh^{2}\eta}}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-v^{2}}}{1-u_{1}v}\\
u_{3}^{\prime} &amp; =\frac{u_{3}}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{3}\sqrt{1-\tanh^{2}\eta}}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{3}\sqrt{1-v^{2}}}{1-u_{1}v}\\
\\
\hline \\
u_{1} &amp; =\frac{\sinh\eta+u_{1}^{\prime}\cosh\eta}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+\tanh\eta}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+v}{1+u_{1}^{\prime}v}\\
u_{2} &amp; =\frac{u_{2}^{\prime}}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-v^{2}}}{1+u_{1}^{\prime}v}\\
u_{3} &amp; =\frac{u_{3}^{\prime}}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{3}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{3}^{\prime}\sqrt{1-v^{2}}}{1+u_{1}^{\prime}v}
\end{align}
&lt;/math&gt;|{{equationRef|3e}}}}

Transformations (A) were given by [[#Escherich|Escherich (1874)]], [[#Goursat|Goursat (1888)]], [[#Killing2|Killing (1898)]], and transformations (C) by [[#Beltrami|Beltrami (1868)]], [[#Schur|Schur (1885/86, 1900/02)]] in terms of [[w:Beltrami–Klein model|Beltrami coordinates]]&lt;ref&gt;Rosenfeld (1988), p. 231&lt;/ref&gt; of hyperbolic geometry. This transformation becomes equivalent to the [[w:hyperbolic law of cosines]] by restriction to coordinates of the &lt;math&gt;\left[u_{1},u_{2}\right]&lt;/math&gt;-plane and &lt;math&gt;\left[u'_{1},u'_{2}\right]&lt;/math&gt;-plane and defining their scalar products in terms of trigonometric and hyperbolic identities:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix} &amp; \begin{matrix}u^{2}=u_{1}^{2}+u_{2}^{2}\\
u'^{2}=u_{1}^{\prime2}+u_{2}^{\prime2}
\end{matrix}\left|{\scriptstyle \begin{align}u_{1}=u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+vu'\cos\alpha'}, &amp; u_{1}^{\prime}=u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-vu\cos\alpha}\\
u_{2}=u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{1+vu'\cos\alpha'}, &amp; u_{2}^{\prime}=u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{1-vu\cos\alpha}\\
\frac{u_{2}}{u_{1}}=\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{u'\cos\alpha'+v}, &amp; \frac{u_{2}^{\prime}}{u_{1}^{\prime}}=\tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{u\cos\alpha-v}
\end{align}
}\right.\\
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(vu'\sin\alpha'\right){}^{2}}}{1+vu'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(vu\sin\alpha\right){}^{2}}}{1-vu\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-u^{\prime2}}}=\frac{1}{\sqrt{1-v^{2}}}\frac{1}{\sqrt{1-u^{2}}}-\frac{v}{\sqrt{1-v^{2}}}\frac{u}{\sqrt{1-u^{2}}}\cos\alpha\qquad(B)\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha\qquad(A)
\end{matrix}&lt;/math&gt;|{{equationRef|3f}}}}

The hyperbolic law of cosines (A) was given by [[#Taurinus|Taurinus (1826) and Lobachevsky (1829/30)]] and others, while variant (B) was given by [[#Schur|Schur (1900/02)]]. By further setting &lt;math&gt;\tanh\xi=\tanh\zeta=1&lt;/math&gt; or &lt;math&gt;u'=u=1&lt;/math&gt; it follows:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}(A)  &amp; \ \cos\alpha=\frac{\cos\alpha'+\tanh\eta}{1+\tanh\eta\cos\alpha'}; &amp; \ \sin\alpha=\frac{\sin\alpha'\sqrt{1-\tanh^{2}\eta}}{1+\tanh\eta\cos\alpha'}; &amp; \ \tan\alpha=\frac{\sin\alpha'\sqrt{1-\tanh^{2}\eta}}{\cos\alpha'+\tanh\eta}; &amp; \ \tan\frac{\alpha}{2}=\sqrt{\frac{1-\tanh\eta}{1+\tanh\eta}}\tan\frac{\alpha'}{2}\\
 &amp; \ \cos\alpha'=\frac{\cos\alpha-\tanh\eta}{1-\tanh\eta\cos\alpha}; &amp; \ \sin\alpha'=\frac{\sin\alpha\sqrt{1-\tanh^{2}\eta}}{1-\tanh\eta\cos\alpha}; &amp; \ \tan\alpha'=\frac{\sin\alpha\sqrt{1-\tanh^{2}\eta}}{\cos\alpha-\tanh\eta}; &amp; \ \tan\frac{\alpha'}{2}=\sqrt{\frac{1+\tanh\eta}{1-\tanh\eta}}\tan\frac{\alpha}{2}\\
\\
(B)  &amp; \ \cos\alpha=\frac{\cos\alpha'+v}{1+v\cos\alpha'}; &amp; \ \sin\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{1+v\cos\alpha'}; &amp; \ \tan\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{\cos\alpha'+v}; &amp; \ \tan\frac{\alpha}{2}=\sqrt{\frac{1-v}{1+v}}\tan\frac{\alpha'}{2}\\
 &amp; \ \cos\alpha'=\frac{\cos\alpha-v}{1-v\cos\alpha}; &amp; \ \sin\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{1-v\cos\alpha}; &amp; \ \tan\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{\cos\alpha-v}; &amp; \ \tan\frac{\alpha'}{2}=\sqrt{\frac{1+v}{1-v}}\tan\frac{\alpha}{2}
\end{matrix}
&lt;/math&gt;|{{equationRef|3g}}}}

Formulas (3g-B) are the equations of an [[w:ellipse]] of [[w:Orbital eccentricity|eccentricity]] ''v'', [[w:eccentric anomaly]] α' and [[w:true anomaly]] α, first geometrically formulated by [[#Euler|Kepler (1609)]] and explicitly written down by [[#Euler|Euler (1735, 1748), Lagrange (1770)]] and many others in relation to planetary motions. They were also used by [[../Lorentz transformation (conformal)#Darboux|E:Darboux (1873)]] as a sphere transformation. In special relativity these formulas describe the aberration of light, see [[../Lorentz transformation (velocity)#Velocity addition and aberration|E:velocity addition and aberration]].

==Historical notation==
==={{anchor|mercator}} Mercator (1668) – hyperbolic relations ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#mercator|label 1=History of Lorentz transformations via squeeze mappings § Mercator}}
[[File:Mercator-hyperbola-XIV.png|thumb|&lt;small&gt;Mercator's (1668) illustration of AH·FH=AI·BI.&lt;/small&gt;]]

While deriving the [[w:Mercator series]], [[w:Nicholas Mercator]] (1668) demonstrated the following relations on a rectangular hyperbola:&lt;ref group=M&gt;Mercator (1667), prop. XIV, pp. 28-29. (He used this result to derive the Mercator series in prop. XV).&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}AD=1+a,\ DF=\sqrt{2a+aa}\\ AH=\frac{1+a+\sqrt{2a+aa}}{\sqrt{2}},\ FH=\frac{1+a-\sqrt{2a+aa}}{\sqrt{2}}\\ AI=BI=\frac{1}{\sqrt{2}}\\ 1+a=c,\ \sqrt{2a+aa}=d,\ 1=cc-dd\\ AH*FH=\frac{cc-dd}{\sqrt{2}*\sqrt{2}}=\frac{1}{2}\\ AI*BI=\frac{1}{2}\\ \hline AH*FH=AI*BI\\ AH.AI::BI.FH \end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=It can be seen that Mercator's relations &lt;math&gt;1+a=c&lt;/math&gt;, &lt;math&gt;\sqrt{2a+a^{2}}=d&lt;/math&gt; with &lt;math&gt;c^{2}-d^{2}=1&lt;/math&gt; implicitly correspond to hyperbolic functions &lt;math&gt;c=\cosh\eta&lt;/math&gt;, &lt;math&gt;d=\sinh\eta&lt;/math&gt; with &lt;math&gt;\cosh^{2}\eta-\sinh^{2}\eta=1&lt;/math&gt; (which were explicitly introduced by [[#Riccati|Riccati (1757)]] much later). In particular, his result AH.AI::BI.FH, denoting that the ratio between AH and AI is equal to the ratio between BI and FH or &lt;math&gt;\tfrac{AH}{AI}=\tfrac{BI}{FH}&lt;/math&gt; in modern notation, corresponds to squeeze mapping or Lorentz boost ({{equationNote|3c}}) because:&lt;br&gt;
:&lt;math&gt;\frac{AH}{AI}=\frac{BI}{FH}=1+a+\sqrt{2a+a^{2}}=c+d=\cosh\eta+\sinh\eta=e^{\eta}&lt;/math&gt;
or solved for AH and FH:
:&lt;math&gt;AH=e^{\eta}AI&lt;/math&gt; and &lt;math&gt;FH=e^{-\eta}BI&lt;/math&gt;.&lt;br&gt;
Furthermore, transforming Mercator's asymptotic coordinates &lt;math&gt;AH=\tfrac{c+d}{\sqrt{2}}&lt;/math&gt;, &lt;math&gt;FH=\tfrac{c-d}{\sqrt{2}}&lt;/math&gt; into Cartesian coordinates &lt;math&gt;x_{0},x_{1}&lt;/math&gt; gives:&lt;br&gt;
:&lt;math&gt;x_{1}=\tfrac{AH+FH}{\sqrt{2}}=c=\cosh\eta,\quad x_{0}=\tfrac{AH-FH}{\sqrt{2}}=d=\sinh\eta&lt;/math&gt;&lt;br&gt;
which produces the unit hyperbola &lt;math&gt;-x_{0}^{2}+x_{1}^{2}=1&lt;/math&gt; as in ({{equationNote|3d}}), in agreement with Mercator's result AH·FH=1/2 when the hyperbola is referred to its asymptotes.}}

==={{anchor|Euler}} Euler (1735) – True and eccentric anomaly===
{{See also|History of Topics in Special Relativity/Lorentz transformation (imaginary)#Euler|label 1=History of Lorentz transformations via imaginary orthogonal transformations § Euler}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Euler|label 1=History of Lorentz transformations via Quaternions § Euler}}

[[w:Johannes Kepler]] (1609) geometrically formulated [[w:Kepler's equation]] and the relations between the [[w:mean anomaly]], [[w:true anomaly]], and [[w:eccentric anomaly]].&lt;ref group=M&gt;Kepler (1609), chapter 60. The editors of Kepler's collected papers remark (p. 482), that Kepler's relations correspond to &lt;math&gt;{\scriptstyle \alpha=\beta+e\sin\beta}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\nu=\frac{e+\cos\beta}{1+e\cos\beta}}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\beta=\frac{\cos\nu-e}{1-e\cos\nu}}&lt;/math&gt;&lt;/ref&gt;&lt;ref&gt;Volk (1976), p. 366&lt;/ref&gt; The relation between the true anomaly ''z'' and the eccentric anomaly ''P'' was algebraically expressed by [[w:Leonhard Euler]] (1735/40) as follows:&lt;ref group=M&gt;Euler (1735/40), § 19&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{\cos P+v}{1+v\cos P},\ \cos P=\frac{\cos z-v}{1-v\cos z},\ \int P=\frac{\int z\sqrt{1-v^{2}}}{1-v\cos z}&lt;/math&gt;

and in 1748:&lt;ref group=M&gt;Euler (1748a), section VIII&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{n+\cos y}{1+n\cos y},\ \sin z=\frac{\sin y\sqrt{1-n^{2}}}{1+n\cos y},\ \tan z=\frac{\sin y\sqrt{1-n^{2}}}{n+\cos y}&lt;/math&gt;

while [[w:Joseph-Louis Lagrange]] (1770/71) expressed them as follows&lt;ref group=M&gt;Lagrange (1770/71), section I&lt;/ref&gt;

:&lt;math&gt;\sin u=\frac{m\sin x}{1+n\cos x},\ \cos u=\frac{n+\cos x}{1+n\cos x},\ \operatorname{tang}\frac{1}{2}u=\frac{m}{1+n}\operatorname{tang}\frac{1}{2}x,\ \left(m^{2}=1-n^{2}\right)&lt;/math&gt;

{{Lorentzbox|Text= These relations resemble formulas ({{equationNote|3g}}), while ({{equationNote|3e}}) follows by setting &lt;math&gt;[\cos z,\sin z, \cos y,\sin y]=\left[u_{1},u_{2},u'_{1},u'_{2}\right]&lt;/math&gt; in Euler's formulas or &lt;math&gt;[\cos u,\sin u, \cos x,\sin x]=\left[u_{1},u_{2},u'_{1},u'_{2}\right]&lt;/math&gt; in Lagrange's formulas.}}

==={{anchor|Riccati}} Riccati (1757) – hyperbolic addition===
[[File:Riccati-hyperbola.png|thumb|&lt;small&gt;Riccati's (1757) illustration of hyperbolic addition laws.&lt;/small&gt;]]
[[w:Vincenzo Riccati]] (1757) introduced hyperbolic functions ''cosh'' and ''sinh'', which he denoted as ''Ch.'' and ''Sh.'' related by &lt;math&gt;Ch.^{2}-Sh.^{2}=r^2&lt;/math&gt; with ''r'' being set to unity in modern publications, and formulated the addition laws of hyperbolic sine and cosine:&lt;ref group=M&gt;Riccati (1757), p. 71&lt;/ref&gt;&lt;ref group=M&gt;Günther (1880/81), pp. 7–13&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}CA=r,\ CB=Ch.\varphi,\ BE=Sh.\varphi,\ CD=Ch.\pi,\ DF=Sh.\pi\\
CM=Ch.\overline{\varphi+\pi},\ MN=Sh.\overline{\varphi+\pi}\\
CK=\frac{r}{\sqrt{2}},\ CG=\frac{Ch.\varphi+Sh.\varphi}{\sqrt{2}},\ CH=\frac{Ch.\pi+Sh.\pi}{\sqrt{2}},\ CP=\frac{Ch.\overline{\varphi+\pi}+Sh.\overline{\varphi+\pi}}{\sqrt{2}}\\
CK:CG::CH:CP\\
\left[Ch.^{2}-Sh.^{2}=rr\right]\\
\hline Ch.\overline{\varphi+\pi}=\frac{Ch.\varphi\,Ch.\pi+Sh.\varphi\,Sh.\pi}{r}\\
Sh.\overline{\varphi+\pi}=\frac{Ch.\varphi\,Sh.\pi+Ch.\pi\,Sh.\varphi}{r}
\end{matrix}&lt;/math&gt;

He furthermore showed that &lt;math&gt;Ch.\overline{\varphi-\pi}&lt;/math&gt; and &lt;math&gt;Sh.\overline{\varphi-\pi}&lt;/math&gt; follow by setting &lt;math&gt;Ch.\pi\Rightarrow Ch.-\pi&lt;/math&gt; and &lt;math&gt;Sh.\pi\Rightarrow Sh.-\pi&lt;/math&gt; in the above formulas.

{{Lorentzbox|Text=The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3d}}) with &lt;math&gt;\pi=\eta,\ \varphi=\eta_1,\ \overline{\varphi+\pi}=\eta_2&lt;/math&gt;.}}

==={{anchor|Lambert}} Lambert (1768–1770) – hyperbolic addition===

While [[#Riccati|Riccati (1757)]] discussed the hyperbolic sine and cosine, [[w:Johann Heinrich Lambert]] (read 1767, published 1768) introduced the expression ''tang φ'' or abbreviated ''tφ'' as the [[w:tangens hyperbolicus]] &lt;math&gt;{\scriptstyle \frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}}&lt;/math&gt; of a variable ''u'', or in modern notation ''tφ=tanh(u)'':&lt;ref group=M&gt;Lambert (1761/68), pp. 309–318&lt;/ref&gt;&lt;ref&gt;Barnett (2004), pp. 22–23&lt;/ref&gt;

:&lt;math&gt;\left.\begin{align}\xi\xi-1 &amp; =\eta\eta &amp; (a)\\
1+\eta\eta &amp; =\xi\xi &amp; (b)\\
\frac{\eta}{\xi} &amp; =tang\ \phi=t\phi &amp; (c)\\
\xi &amp; =\frac{1}{\sqrt{1-t\phi^{2}}} &amp; (d)\\
\eta &amp; =\frac{t\phi}{\sqrt{1-t\phi^{2}}} &amp; (e)\\
t\phi'' &amp; =\frac{t\phi+t\phi'}{1+t\phi\cdot t\phi'} &amp; (f)\\
t\phi' &amp; =\frac{t\phi''-t\phi}{1-t\phi\cdot t\phi''} &amp; (g)
\end{align}
\right|\begin{align}2u &amp; =\log\frac{1+t\phi}{1-t\phi}\\
\xi &amp; =\frac{e^{u}+e^{-u}}{2}\\
\eta &amp; =\frac{e^{u}-e^{-u}}{2}\\
t\phi &amp; =\frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}\\
e^{u} &amp; =\xi+\eta\\
e^{-u} &amp; =\xi-\eta
\end{align}&lt;/math&gt;

In (1770) he rewrote the addition law for the hyperbolic tangens (f) or (g) as:&lt;ref group=M&gt;Lambert (1770), p. 335&lt;/ref&gt;

:&lt;math&gt;\begin{align}t(y+z) &amp; =(ty+tz):(1+ty\cdot tz) &amp; (f)\\
t(y-z) &amp; =(ty-tz):(1-ty\cdot tz) &amp; (g)
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=The hyperbolic relations (a,b,c,d,e,f) are equivalent to the hyperbolic relations on the right of ({{equationNote|3b}}). Relations (f,g) can also be found in ({{equationNote|3e}}). By setting ''tφ=v/c'', formula (c) becomes the relative velocity between two frames, (d) the [[w:Lorentz factor]], (e) the [[w:proper velocity]], (f) or (g) becomes the Lorentz transformation of velocity (or relativistic [[w:velocity addition formula]]) for collinear velocities in [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']] and [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']].}}

Lambert also formulated the addition laws for the hyperbolic cosine and sine (Lambert's &quot;cos&quot; and &quot;sin&quot; actually mean &quot;cosh&quot; and &quot;sinh&quot;):

:&lt;math&gt;\begin{align}\sin(y+z) &amp; =\sin y\cos z+\cos y\sin z\\
\sin(y-z) &amp; =\sin y\cos z-\cos y\sin z\\
\cos(y+z) &amp; =\cos y\cos z+\sin y\sin z\\
\cos(y-z) &amp; =\cos y\cos z-\sin y\sin z
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3d}}).}}

==={{Anchor|Taurinus}} Taurinus (1826) – Hyperbolic law of cosines===

After the addition theorem for the tangens hyperbolicus was given by [[#Lambert|Lambert (1768)]], [[w:hyperbolic geometry]] was used by [[w:Franz Taurinus]] (1826), and later by [[w:Nikolai Lobachevsky]] (1829/30) and others, to formulate the [[w:hyperbolic law of cosines]]:&lt;ref group=M&gt;Taurinus (1826), p. 66; see also p. 272 in the translation by Engel and Stäckel (1899)&lt;/ref&gt;&lt;ref&gt;Bonola (1912), p. 79&lt;/ref&gt;&lt;ref&gt;Gray (1979), p. 242&lt;/ref&gt;

:&lt;math&gt;A=\operatorname{arccos}\frac{\cos\left(\alpha\sqrt{-1}\right)-\cos\left(\beta\sqrt{-1}\right)\cos\left(\gamma\sqrt{-1}\right)}{\sin\left(\beta\sqrt{-1}\right)\sin\left(\gamma\sqrt{-1}\right)}&lt;/math&gt;

{{Lorentzbox|Text=When solved for &lt;math&gt;\cos\left(\alpha\sqrt{-1}\right)&lt;/math&gt; it corresponds to the Lorentz transformation in Beltrami coordinates ({{equationNote|3f}}), and by defining the rapidities &lt;math&gt;{\scriptstyle \left(\left[\frac{U}{c},\ \frac{v}{c},\ \frac{u}{c}\right]=\left[\tanh\alpha,\ \tanh\beta,\ \tanh\gamma\right]\right)}&lt;/math&gt; it corresponds to the relativistic velocity addition formula [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']].}}

==={{anchor|Beltrami}} Beltrami (1868) – Beltrami coordinates===

[[w:Eugenio Beltrami]] (1868a) introduced coordinates of the [[w:Beltrami–Klein model]] of hyperbolic geometry, and formulated the corresponding transformations in terms of homographies:&lt;ref group=M&gt;Beltrami (1868a), pp. 287-288; Note I; Note II&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds^{2}=R^{2}\frac{\left(a^{2}+v^{2}\right)du^{2}-2uv\,du\,dv+\left(a^{2}+v^{2}\right)dv^{2}}{\left(a^{2}+u^{2}+v^{2}\right)^{2}}\\
u^{2}+v^{2}=a^{2}\\
\hline u''=\frac{aa_{0}\left(u'-r_{0}\right)}{a^{2}-r_{0}u'},\ v''=\frac{a_{0}w_{0}v'}{a^{2}-r_{0}u'},\\
\left(r_{0}=\sqrt{u_{0}^{2}+v_{0}^{2}},\ w_{0}=\sqrt{a^{2}-r_{0}^{2}}\right)\\
\hline ds^{2}=R^{2}\frac{\left(a^{2}-v^{2}\right)du^{2}+2uv\,du\,dv+\left(a^{2}-v^{2}\right)dv^{2}}{\left(a^{2}-u^{2}-v^{2}\right)^{2}}\\
(R=R\sqrt{-1},\ a=a\sqrt{-1})
\end{matrix}&lt;/math&gt;

(where the disk radius ''a'' and the [[w:radius of curvature]] ''R'' are real in spherical geometry, in hyperbolic geometry they are imaginary), and for arbitrary dimensions in (1868b)&lt;ref group=M&gt;Beltrami (1868b), pp. 232, 240–241, 253–254&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds=R\frac{\sqrt{dx^{2}+dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}}}{x}\\
x^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=a^{2}\\
\hline y_{1}=\frac{ab\left(x_{1}-a_{1}\right)}{a^{2}-a_{1}x_{1}}\ \text{or}\ x_{1}=\frac{a\left(ay_{1}+a_{1}b\right)}{ab+a_{1}y_{1}},\ x_{r}=\pm\frac{ay_{r}\sqrt{a^{2}-a_{1}^{2}}}{ab+a_{1}y_{1}}\ (r=2,3,\dots,n)\\
\hline ds=R\frac{\sqrt{dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}-dx^{2}}}{x}\\
x^{2}=a^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\\
\left(R=R\sqrt{-1},\ x=x\sqrt{-1},\ a=a\sqrt{-1}\right)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Setting ''a=a&lt;sub&gt;0&lt;/sub&gt;'' Beltrami's (1868a) formulas become formulas ({{equationNote|3e}}), or in his (1868b) formulas one sets ''a=b'' for arbitrary dimensions.}}

=== {{anchor|Laisant2}} Laisant (1874) – Equipollences===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Laisant1|label 1=History of Lorentz transformations via squeeze mappings § Laisant}}

In his French translation of [[w:Giusto Bellavitis]]' principal work on [[w:Equipollence (geometry)|w:equipollences]], [[w:Charles-Ange Laisant]] (1874) added a chapter related to hyperbolas. The equipollence OM and its tangent MT of a hyperbola is defined by Laisant as&lt;ref group=M&gt;Laisant (1874b), pp. 134–135&lt;/ref&gt;

:(1) &lt;math&gt;\begin{matrix} &amp; \mathrm{OM}\bumpeq x\mathrm{OA}+y\mathrm{OB}\\
 &amp; \mathrm{MT}\bumpeq y\mathrm{OA}+x\mathrm{OB}\\
 &amp; \left[x^{2}-y^{2}=1;\ x=\cosh t,\ y=\sinh t\right]\\
\Rightarrow &amp; \mathrm{OM}\bumpeq\cosh t\cdot\mathrm{OA}+\sinh t\cdot\mathrm{OB}
\end{matrix}&lt;/math&gt;

Here, OA and OB are [[w:Conjugate diameters|conjugate semi-diameters]] of a hyperbola with OB being imaginary, both of which he related to two other conjugated semi-diameters OC and OD by the following transformation:

:&lt;math&gt;\begin{matrix}\begin{align}\mathrm{OC} &amp; \bumpeq c\mathrm{OA}+d\mathrm{OB} &amp; \qquad &amp;  &amp; \mathrm{OA} &amp; \bumpeq c\mathrm{OC}-d\mathrm{OD}\\
\mathrm{OD} &amp; \bumpeq d\mathrm{OA}+c\mathrm{OB} &amp;  &amp;  &amp; \mathrm{OB} &amp; \bumpeq-d\mathrm{OC}+c\mathrm{OD}
\end{align}
\\
\left[c^{2}-d^{2}=1\right]
\end{matrix}&lt;/math&gt;

producing the invariant relation

:&lt;math&gt;(\mathrm{OC})^{2}-(\mathrm{OD})^{2}\bumpeq(\mathrm{OA})^{2}-(\mathrm{OB})^{2}&lt;/math&gt;.

Substituting into (1), he showed that OM retains its form

:&lt;math&gt;\begin{matrix}\mathrm{OM}\bumpeq(cx-dy)\mathrm{OC}+(cy-dx)\mathrm{OD}\\
\left[(cx-dy)^{2}-(cy-dx)^{2}=1\right]
\end{matrix}&lt;/math&gt;

He also defined velocity and acceleration by differentiation of (1).

{{Lorentzbox|Text=These relations are equivalent to several Lorentz boosts or hyperbolic rotations producing the invariant Lorentz interval in line with ({{equationNote|3b}}).}}

==={{anchor|Escherich}} Escherich (1874) – Beltrami coordinates===

[[w:Gustav von Escherich]] (1874) discussed the plane of constant negative curvature&lt;ref&gt;Sommerville (1911), p. 297&lt;/ref&gt; based on the [[w:Beltrami–Klein model]] of hyperbolic geometry by [[#Beltrami|Beltrami (1868)]]. Similar to [[w:Christoph Gudermann]] (1830)&lt;ref name=guder group=M&gt;Gudermann (1830), §1–3, §18–19&lt;/ref&gt;  who introduced axial coordinates ''x''=tan(a) and ''y''=tan(b) in sphere geometry in order to perform coordinate transformations in the case of rotation and translation, Escherich used hyperbolic functions ''x''=tanh(a/k) and ''y''=tanh(b/k)&lt;ref group=M&gt;Escherich (1874), p. 508&lt;/ref&gt; in order to give the corresponding coordinate transformations for the hyperbolic plane, which for the case of translation have the form:&lt;ref group=M name=escher&gt;Escherich (1874), p. 510&lt;/ref&gt;

:&lt;math&gt;x=\frac{\sinh\frac{a}{k}+x'\cosh\frac{a}{k}}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt; and &lt;math&gt;y=\frac{y'}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}), also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting &lt;math&gt;\tfrac{a}{k}=\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt; and multiplying ''[x,y,x′,y′]'' by 1/''c'', and equivalent to Lorentz boost ({{equationNote|3b}}) by setting &lt;math&gt;\scriptstyle (x,\ y,\ x',\ y')=\left(\frac{x_{1}}{x_{0}},\ \frac{x_{2}}{x_{0}},\ \frac{x_{1}^{\prime}}{x_{0}^{\prime}},\ \frac{x_{2}^{\prime}}{x_{0}^{\prime}}\right)&lt;/math&gt;. This is the relation between the [[w:Beltrami–Klein model|Beltrami coordinates]] in terms of Gudermann-Escherich coordinates, and the Weierstrass coordinates of the [[w:hyperboloid model]] introduced by [[../Lorentz transformation (general)#Killing1|E:Killing (1878–1893)]], [[../Lorentz transformation (general)#Poincare|E:Poincaré (1881)]], and [[../Lorentz transformation (general)#Cox|E:Cox (1881)]]. Both coordinate systems were compared by Cox (1881).&lt;ref group=M&gt;Cox (1881), p. 186&lt;/ref&gt;}}

==={{anchor|Glaisher}} Glaisher (1878) – hyperbolic addition===

It was shown by [[w:James Whitbread Lee Glaisher]] (1878) that the hyperbolic addition laws can be expressed by matrix multiplication:&lt;ref group=M&gt;Glaisher (1878), p. 30&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\cosh x, &amp; \sinh x\\
\sinh x, &amp; \cosh x
\end{vmatrix}=1,\ \begin{vmatrix}\cosh y, &amp; \sinh y\\
\sinh y, &amp; \cosh y
\end{vmatrix}=1\\
\text{by multiplication:}\\
\Rightarrow\begin{vmatrix}c_{1}c_{2}+s_{1}s_{2}, &amp; s_{1}c_{2}+c_{1}s_{2}\\
c_{1}s_{2}+s_{1}c_{2}, &amp; s_{1}s_{2}+c_{1}c_{2}
\end{vmatrix}=1\\
\text{where}\ \left[c_{1},c_{2},c_{3},c_{4}\right]=\left[\cosh x,\cosh y,\sinh x,\sinh y\right] \\
\Rightarrow\begin{vmatrix}\cosh(x+y), &amp; \sinh(x+y)\\
\sinh(x+y), &amp; \cosh(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=In this matrix representation, the analogy between the hyperbolic angle sum laws and the Lorentz boost becomes obvious: In particular, the matrix &lt;math&gt;\scriptstyle\begin{vmatrix}\cosh y, &amp; \sinh y\\
\sinh y, &amp; \cosh y\end{vmatrix}&lt;/math&gt; producing the hyperbolic addition is analogous to matrix &lt;math&gt;\scriptstyle\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta\end{bmatrix}&lt;/math&gt; producing Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3d}}).}}

==={{anchor|Gunther1}} Günther (1880/81) – hyperbolic addition ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Gunther1|label 1=History of Lorentz transformations via squeeze mappings § Günther}}

Following [[#Glaisher|Glaisher (1878)]], [[w:Siegmund Günther]] (1880/81) expressed the hyperbolic addition laws by matrix multiplication:&lt;ref group=M&gt;Günther (1880/81), p. 405&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\mathfrak{Cos}\,x, &amp; \mathfrak{Sin}\,x\\
\mathfrak{Sin}\,x, &amp; \mathfrak{Cos}\,x
\end{vmatrix}\cdot\begin{vmatrix}\mathfrak{Cos}\,y, &amp; \mathfrak{Sin}\,y\\
\mathfrak{Sin}\,y, &amp; \mathfrak{Cos}\,y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}\,x\,\mathfrak{Cos}\,y+\mathfrak{Sin}\,x\,\mathfrak{Sin}\,y, &amp; \mathfrak{Cos}\,x\,\mathfrak{Sin}\,y+\mathfrak{Sin}\,x\,\mathfrak{Cos}\,y\\
\mathfrak{Sin}\,x\,\mathfrak{Cos}\,y+\mathfrak{Cos}\,x\,\mathfrak{Sin}\,y, &amp; \mathfrak{Sin}\,x\,\mathfrak{Sin}\,y+\mathfrak{Cos}\,x\,\mathfrak{Cos}\,y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}\,(x+y), &amp; \mathfrak{Sin}\,(x+y)\\
\mathfrak{Sin}\,(x+y), &amp; \mathfrak{Cos}\,(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=In this matrix representation, the analogy between the hyperbolic angle sum laws and the Lorentz boost becomes obvious: In particular, the matrix &lt;math&gt;\scriptstyle\begin{vmatrix}\mathfrak{Cos}\,y, &amp; \mathfrak{Sin}\,y\\
\mathfrak{Sin}\,y, &amp; \mathfrak{Cos}\,y \end{vmatrix}&lt;/math&gt; producing the hyperbolic addition is analogous to matrix &lt;math&gt;\scriptstyle\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta\end{bmatrix}&lt;/math&gt; producing Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3d}}).}}

=== {{anchor|Cox}} Cox (1881/82) – Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Cox|label 1=History of Lorentz transformations in general § Cox}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Cox2|label 1=History of Lorentz transformations via Quaternions § Cox}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Cox|label 1=History of Lorentz transformations via sphere transformations § Cox}}

[[w:Homersham Cox (mathematician)|w:Homersham Cox]] (1881/82) defined the case of translation in the hyperbolic plane with the ''y''-axis remaining unchanged:&lt;ref group=M name=cox&gt;Cox (1881/82), p. 194&lt;/ref&gt;
	
:&lt;math&gt;\begin{align}X &amp; =x\cosh p-z\sinh p\\	
Z &amp; =-x\sinh p+z\cosh p	\\ \\
x &amp; =X\cosh p+Z\sinh p\\
z &amp; =X\sinh p+Z\cosh p	
\end{align}	
&lt;/math&gt;	
	
{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Lipschitz1}} Lipschitz (1885/86) – Quadratic forms ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Lipschitz2|label 1=History of Lorentz transformations via Quaternions § Lipschitz}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Lipschitz1|label 1=History of Lorentz transformations via squeeze mappings § Lipschitz}}

[[w:Rudolf Lipschitz]] (1885/86) discussed transformations leaving invariant the sum of squares
:&lt;math&gt;x_{1}^{2}+x_{2}^{2}\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}&lt;/math&gt;
which he rewrote as 
:&lt;math&gt;x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}+\dots+x_{n}^{2}-y_{n}^{2}=0&lt;/math&gt;. 
This led to the problem of finding transformations leaving invariant the pairs &lt;math&gt;x_{a}^{2}-y_{a}^{2}&lt;/math&gt; (where ''a=1...n'') for which he gave the following solution:&lt;ref group=M&gt;Lipschitz (1886), pp. 90–92&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{a}^{2}-y_{a}^{2}=\mathfrak{x}_{a}^{2}-\mathfrak{y}_{a}^{2}\\
\hline \begin{align}x_{a}-y_{a} &amp; =\left(\mathfrak{x}_{a}-\mathfrak{y}_{a}\right)r_{a}\\
x_{a}+y_{a} &amp; =\left(\mathfrak{x}_{a}+\mathfrak{y}_{a}\right)\frac{1}{r_{a}}
\end{align}
\quad(a)\\
\hline \begin{matrix}\begin{align}2\mathfrak{x}_{a} &amp; =\left(r_{a}+\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}-\frac{1}{r_{a}}\right)y_{a}\\
2\mathfrak{y}_{a} &amp; =\left(r_{a}-\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}+\frac{1}{r_{a}}\right)y_{a}
\end{align}
\quad(b)\end{matrix}\\
\hline \left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}\Rightarrow\begin{align}\mathfrak{x}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}\\
\mathfrak{y}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}
\end{align}
\quad(c)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Lipschitz's transformations (c) and (a) are equivalent to Lorentz boosts ({{equationNote|3b}}-C) and ({{equationNote|3c}}) by the identity &lt;math&gt;s_{a}=\tfrac{1}{v}=\coth\eta&lt;/math&gt;. That is, by substituting &lt;math&gt;v=\tfrac{1}{s_{a}}&lt;/math&gt; in ({{equationNote|3b}}-C) or ({{equationNote|3c}}) we obtain Lipschitz's transformations.}}

==={{Anchor|Schur}} Schur (1885/86, 1900/02) – Beltrami coordinates===

[[w:Friedrich Schur]] (1885/86) discussed spaces of constant Riemann curvature, and by following [[#Beltrami|Beltrami (1868)]] he used the transformation&lt;ref group=M&gt;Schur (1885/86), p. 167&lt;/ref&gt;

:&lt;math&gt;x_{1}=R^{2}\frac{y_{1}+a_{1}}{R^{2}+a_{1}y_{1}},\ x_{2}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{2}}{R^{2}+a_{1}y_{1}},\dots,\ x_{n}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{n}}{R^{2}+a_{1}y_{1}}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] in arbitrary dimensions by setting ''R=c'' as the speed of light and ''a&lt;sub&gt;1&lt;/sub&gt;=v'' as relative velocity.}}

In (1900/02) he derived basic formulas of non-Eucliden geometry, including the case of translation in the hyperbolic plane for which he obtained the transformation similar to his previous one:&lt;ref group=M&gt;Schur (1900/02), p. 290; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;x'=\frac{x-a}{1-\mathfrak{k}ax},\quad y'=\frac{y\sqrt{1-\mathfrak{k}a^{2}}}{1-\mathfrak{k}ax}&lt;/math&gt;

where &lt;math&gt;\mathfrak{k}&lt;/math&gt; can have values &gt;0, &lt;0 or ∞.

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting ''a=v'' and &lt;math&gt;\mathfrak{k}=\tfrac{1}{c^{2}}&lt;/math&gt;.}}

He also defined the triangle&lt;ref group=M&gt;Schur (1900/02), p. 291; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;\frac{1}{\sqrt{1-\mathfrak{k}c^{2}}}=\frac{1}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{1}{\sqrt{1-\mathfrak{k}b^{2}}}-\frac{a}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{b}{\sqrt{1-\mathfrak{k}b^{2}}}\cos\gamma&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to the hyperbolic law of cosines and the relativistic velocity addition ({{equationNote|3f}}, b) or [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']] by setting &lt;math&gt;[\mathfrak{k},\ c,\ a,\ b]=\left[\tfrac{1}{c^{2}},\ \sqrt{u_{x}^{\prime2}+u_{y}^{\prime2}},\ v,\ \sqrt{u_{x}^{2}+u_{y}^{2}}\right]&lt;/math&gt;.}}

==={{Anchor|Goursat}} Goursat (1887/88) – Minimal surfaces===

[[w:Édouard Goursat]] defined real coordinates &lt;math&gt;x,y&lt;/math&gt; of minimal surface &lt;math&gt;S&lt;/math&gt; and imaginary coordinates &lt;math&gt;x_{0},y_{0}&lt;/math&gt; of the adjoint minimal surface &lt;math&gt;S_0&lt;/math&gt;, so that another real minimal surface &lt;math&gt;S_1&lt;/math&gt; follows by the (conformal) transformation:&lt;ref group=M&gt;Goursat (1887/88), p. 144&lt;/ref&gt;

:&lt;math&gt;\begin{align}x_{1} &amp; =\frac{1+k^{2}}{2k}x-\frac{k^{2}-1}{2k}y_{0}\\
y_{1} &amp; =\frac{1+k^{2}}{2k}y+\frac{k^{2}-1}{2k}x_{0}\\
z_{1} &amp; =z
\end{align}&lt;/math&gt;

and expressed these equations in terms of hyperbolic functions by setting &lt;math&gt;k=e^{\varphi}&lt;/math&gt;:&lt;ref group=M&gt;Goursat (1887/88), p. 145&lt;/ref&gt;

:&lt;math&gt;\begin{align}x_{1} &amp; =x\cosh\varphi-y_{0}\sinh\varphi\\
y_{1} &amp; =y\cosh\varphi+x_{0}\sinh\varphi\\
z_{1} &amp; =z
\end{align}&lt;/math&gt;

{{Lorentzbox|Text=This becomes Lorentz boost ({{equationNote|3b}}) by replacing the imaginary coordinates &lt;math&gt;x_{0},y_{0}&lt;/math&gt; by real coordinates defined as &lt;math&gt;[x_{0},y_{0}]=[-x,y]&lt;/math&gt;. It can also be seen that Goursat's relation &lt;math&gt;k=e^{\varphi}&lt;/math&gt; corresponds to &lt;math&gt;k=e^{\eta}&lt;/math&gt; defined in ({{equationNote|3c}}).}}

He went on to define &lt;math&gt;\alpha,\beta,\gamma&lt;/math&gt; as the direction cosines normal to surface &lt;math&gt;S&lt;/math&gt; and &lt;math&gt;\alpha_{1},\beta_{1},\gamma_{1}&lt;/math&gt; as the ones normal to surface &lt;math&gt;S_{1}&lt;/math&gt;, connected by the transformation:&lt;ref group=M&gt;Goursat (1887/88), p. 149f.&lt;/ref&gt;

:&lt;math&gt;\begin{align}\alpha_{1} &amp; =\pm\frac{\alpha}{\cosh\varphi-\gamma\sinh\varphi} &amp;  &amp;  &amp; \alpha &amp; =\pm\frac{\alpha_{1}}{\cosh\varphi+\gamma_{1}\sinh\varphi}\\
\beta_{1} &amp; =\pm\frac{\beta}{\cosh\varphi-\gamma\sinh\varphi} &amp;  &amp;  &amp; \beta &amp; =\pm\frac{\beta_{1}}{\cosh\varphi+\gamma_{1}\sinh\varphi}\\
\gamma_{1} &amp; =\pm\frac{\gamma\cosh\varphi-\sinh\varphi}{\cosh\varphi-\gamma\sinh\varphi} &amp;  &amp;  &amp; \gamma &amp; =\pm\frac{\gamma_{1}\cosh\varphi+\sinh\varphi}{\cosh\varphi+\gamma_{1}\sinh\varphi}
\end{align}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}-A) with &lt;math&gt;\left[\alpha,\beta,\gamma\right]=\left[u_{2},u_{3},u_{1}\right]&lt;/math&gt;.}}

==={{anchor|Lindemann}} Lindemann (1890–91) – Weierstrass coordinates and Cayley absolute===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Lindemann|label 1=History of Lorentz transformations via squeeze mappings § Lindemann}}

[[w:Ferdinand von Lindemann]] discussed hyperbolic geometry in terms of the [[w:Cayley–Klein metric]] in his (1890/91) edition of the lectures on geometry of [[w:Alfred Clebsch]]. Citing [[../Lorentz transformation (general)#Killing|E:Killing (1885)]] and [[../Lorentz transformation (general)#Poincare|Poincaré (1887)]] in relation to the hyperboloid model in terms of Weierstrass coordinates for the hyperbolic plane and space, he set&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 477–478, 524&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\Omega_{xx}=x_{1}^{2}+x_{2}^{2}-4k^{2}x_{3}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}-4k^{2}dx_{3}^{2}\\
\Omega_{xx}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-4k^{2}dx_{4}^{2}
\end{matrix}&lt;/math&gt;

and used the following transformation&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 361–362&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\
X_{1}X_{4}+X_{2}X_{3}=\Xi_{1}\Xi_{4}+\Xi_{2}\Xi_{3}\\
\hline \begin{align}X_{1} &amp; =\left(\lambda+\lambda_{1}\right)U_{4} &amp; \Xi_{1} &amp; =\left(\lambda-\lambda_{1}\right)U_{4} &amp; X_{1} &amp; =\frac{\lambda+\lambda_{1}}{\lambda-\lambda_{1}}\Xi_{1}\\
X_{2} &amp; =\left(\lambda+\lambda_{3}\right)U_{4} &amp; \Xi_{2} &amp; =\left(\lambda-\lambda_{3}\right)U_{4} &amp; X_{2} &amp; =\frac{\lambda+\lambda_{3}}{\lambda-\lambda_{3}}\Xi_{2}\\
X_{3} &amp; =\left(\lambda-\lambda_{3}\right)U_{2} &amp; \Xi_{3} &amp; =\left(\lambda+\lambda_{3}\right)U_{2} &amp; X_{3} &amp; =\frac{\lambda-\lambda_{3}}{\lambda+\lambda_{3}}\Xi_{3}\\
X_{4} &amp; =\left(\lambda-\lambda_{1}\right)U_{1} &amp; \Xi_{4} &amp; =\left(\lambda+\lambda_{1}\right)U_{1} &amp; X_{4} &amp; =\frac{\lambda-\lambda_{1}}{\lambda+\lambda_{1}}\Xi_{4}
\end{align}
\end{matrix}&lt;/math&gt;

into which he put&lt;ref group=M name=linde&gt;Lindemann &amp; Clebsch (1890/91), p. 496&lt;/ref&gt;

:&lt;math&gt;\begin{align}X_{1} &amp; =x_{1}+2kx_{4}, &amp; X_{2} &amp; =x_{2}+ix_{3}, &amp; \lambda+\lambda_{1} &amp; =\left(\lambda-\lambda_{1}\right)e^{a},\\
X_{4} &amp; =x_{1}-2kx_{4}, &amp; X_{3} &amp; =x_{2}-ix_{3}, &amp; \lambda+\lambda_{3} &amp; =\left(\lambda-\lambda_{3}\right)e^{\alpha i},
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3c}}) with &lt;math&gt;e^{\alpha i}=1&lt;/math&gt; and ''2k=1'' .}}

From that, he obtained the following Cayley absolute and the corresponding most general motion in hyperbolic space comprising ordinary rotations (''a''=0) or translations (α=0):&lt;ref group=M  name=linde /&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=0\\
\hline \begin{align}x_{2} &amp; =\xi_{2}\cos\alpha+\xi_{3}\sin\alpha, &amp; x_{1} &amp; =\xi_{1}\cos\frac{a}{i}+2ki\xi_{4}\sin\frac{a}{i},\\
x_{3} &amp; =-\xi_{2}\sin\alpha+\xi_{3}\cos\alpha, &amp; 2kx_{4} &amp; =i\xi_{1}\sin\frac{a}{i}+2k\xi_{4}\cos\frac{a}{i}.
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0 and ''2k=1''.}}

==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Gerard|label 1=History of Lorentz transformations in general § Gerard}}

[[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the hyperbolic plane and gave the case of translation as follows:&lt;ref group=M name=gerard&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{align}X &amp; =Z_{0}X'+X_{0}Z'\\
Y &amp; =Y'\\
Z &amp; =X_{0}X'+Z_{0}Z'
\end{align}
\ \text{with}\ \begin{align}X_{0} &amp; =\operatorname{sh}OO'\\
Z_{0} &amp; =\operatorname{ch}OO'
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Killing2}} Killing (1893,97) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Killing|label 1=History of Lorentz transformations in general § Killing}}

[[w:Wilhelm Killing]] (1878–1880) gave the case of translation in the hyperbolic plane as follows:&lt;ref group=M name=killtra&gt;Killing (1893), p. 331&lt;/ref&gt;

:&lt;math&gt;y_{0}=x_{0}\operatorname{Ch}a+x_{1}\operatorname{Sh}a,\quad y_{1}=x_{0}\operatorname{Sh}a+x_{1}\operatorname{Ch}a,\quad y_{2}=x_{2}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

In 1898, Killing wrote that relation in a form similar to [[#Escherich|Escherich (1874)]], and derived the corresponding Lorentz transformation for the two cases were ''v'' is unchanged or ''u'' is unchanged:&lt;ref group=M name=kill98&gt;Killing (1898), p. 133&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\xi'=\frac{\xi\operatorname{Ch}\frac{\mu}{l}+l\operatorname{Sh}\frac{\mu}{l}}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}},\ \eta'=\frac{\eta}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}}\\
\hline \frac{u}{p}=\xi,\ \frac{v}{p}=\eta\\
\hline p'=p\operatorname{Ch}\frac{\mu}{l}+\frac{u}{l}\operatorname{Sh}\frac{\mu}{l},\quad u'=pl\operatorname{Sh}\frac{\mu}{l}+u\operatorname{Ch}\frac{\mu}{l},\quad v'=v\\
\text{or}\\
p'=p\operatorname{Ch}\frac{\nu}{l}+\frac{v}{l}\operatorname{Sh}\frac{\nu}{l},\quad u'=u,\quad v'=pl\operatorname{Sh}\frac{\nu}{l}+v\operatorname{Ch}\frac{\nu}{l}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The upper transformation system is equivalent to Lorentz transformation ({{equationNote|3e}}) and the velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] with ''l=c'' and &lt;math&gt;\mu=c\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt;, the system below is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Whitehead}} Whitehead (1897/98) – Universal algebra===

[[w:Alfred North Whitehead]] (1898) discussed the kinematics of hyperbolic space as part of his study of [[w:universal algebra]], and obtained the following transformation:&lt;ref group=M name=white&gt;Whitehead (1898), pp. 459–460&lt;/ref&gt;

:&lt;math&gt;\begin{align}x' &amp; =\left(\eta\cosh\frac{\delta}{\gamma}+\eta_{1}\sinh\frac{\delta}{\gamma}\right)e+\left(\eta\sinh\frac{\delta}{\gamma}+\eta_{1}\cosh\frac{\delta}{\gamma}\right)e_{1}\\
 &amp; \qquad+\left(\eta_{2}\cos\alpha+\eta_{3}\sin\alpha\right)e_{2}+\left(\eta_{3}\cos\alpha-\eta_{2}\sin\alpha\right)e_{3}
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0.}}

==={{anchor|Elliott}} Elliott (1903) – Invariant theory ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Elliott|label 1=History of Lorentz transformations via squeeze mappings § Elliott}}

[[w:Edwin Bailey Elliott]] (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:&lt;ref group=M&gt;Elliott (1903), p. 109&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=X\cosh\phi+Y\sinh\phi,\quad y=X\sinh\phi+Y\cosh\phi\\
\hline X+Y=e^{-\phi}(x+y),\quad X-Y=e^{\phi}(x-y)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3c}}). The mentioned subgroup corresponds to the one-parameter subgroup generated by Lorentz boosts.}}

==={{anchor|Woods2}} Woods (1903) – Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Woods2|label 1=History of Lorentz transformations in general § Woods}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Woods|label 1=History of Lorentz transformations via Möbius transformations § Woods}}

[[w:Frederick S. Woods]] (1903, published 1905) gave the case of translation in hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 55&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x_{1}\cos kl+x_{0}\frac{\sin kl}{k},\quad x_{2}^{\prime}=x_{2},\quad x_{2}^{\prime}=x_{3},\quad x_{0}^{\prime}=-x_{1}k\sin kl+x_{0}\cos kl&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.}}

and the loxodromic substitution for hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 72&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}x_{1}^{\prime} &amp; =x_{1}\cosh\alpha-x_{0}\sinh\alpha\\
x_{2}^{\prime} &amp; =x_{2}\cos\beta-x_{3}\sin\beta\\
x_{3}^{\prime} &amp; =x_{2}\sin\beta+x_{3}\cos\beta\\
x_{0}^{\prime} &amp; =-x_{1}\sinh\alpha+x_{0}\cosh\alpha
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with β=0.}}

==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Liebmann|label 1=History of Lorentz transformations in general § Liebmann}}

[[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – gave the case of translation in the hyperbolic plane:&lt;ref group=M name=lieb&gt;Liebmann (1904/05), p. 174&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x'\operatorname{ch}a+p'\operatorname{sh}a,\quad y_{1}^{\prime}=y',\quad p_{1}^{\prime}=x'\operatorname{sh}a+p'\operatorname{ch}a&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Frank}} Frank (1909) – Special relativity===
In special relativity, hyperbolic functions were used by [[w:Philipp Frank]] (1909), who derived the Lorentz transformation using ''ψ'' as rapidity:&lt;ref group=R&gt;Frank (1909), pp. 423-425&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=x\varphi(a)\,{\rm ch}\,\psi+t\varphi(a)\,{\rm sh}\,\psi\\
t'=-x\varphi(a)\,{\rm sh}\,\psi+t\varphi(a)\,{\rm ch}\,\psi\\
\hline {\rm th}\,\psi=-a,\ {\rm sh}\,\psi=\frac{a}{\sqrt{1-a^{2}}},\ {\rm ch}\,\psi=\frac{1}{\sqrt{1-a^{2}}},\ \varphi(a)=1\\
\hline x'=\frac{x-at}{\sqrt{1-a^{2}}},\ y'=y,\ z'=z,\ t'=\frac{-ax+t}{\sqrt{1-a^{2}}}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}
 	
=== {{anchor|Herglotz1}} Herglotz (1909/10) – Special relativity===
{{See also|History of Topics in Special Relativity/Lorentz transformation (velocity)#Herglotz1|label 1=History of Lorentz transformations via velocity § Herglotz}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Herglotz|label 1=History of Lorentz transformations via squeeze mappings § Herglotz}}

In special relativity, [[w:Gustav Herglotz]] (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic, with the hyperbolic case being:&lt;ref group=R&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;
	
:&lt;math&gt;\begin{matrix}Z=Z'e^{\vartheta}\\
	
\begin{aligned}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
	
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
	
\end{aligned}
	
\end{matrix}&lt;/math&gt;	
	
{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3c}}).}}

==={{anchor|Varicak}} Varićak (1910) – Special relativity===
{{See also|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Varicak|label 1=History of Lorentz transformations via trigonometric functions § Varicak}}

In special relativity, hyperbolic functions were used by [[w:Vladimir Varićak]] in several papers starting from 1910, who represented the equations of special relativity on the basis of [[w:hyperbolic geometry]] in terms of Weierstrass coordinates. For instance, by setting ''l=ct'' and ''v/c=tanh(u)'' with ''u'' as rapidity he wrote the Lorentz transformation in agreement with ({{equationNote|4b}}):&lt;ref group=R name=var1&gt;Varićak (1910), p. 93&lt;/ref&gt;

:&lt;math&gt;\begin{align}l' &amp; =-x\operatorname{sh}u+l\operatorname{ch}u,\\
x' &amp; =x\operatorname{ch}u-l\operatorname{sh}u,\\
y' &amp; =y,\quad z'=z,\\
\operatorname{ch}u &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

He showed the relation of rapidity to the [[w:Gudermannian function]] and the [[w:angle of parallelism]]:&lt;ref group=R name=var1 /&gt;

:&lt;math&gt;\frac{v}{c}=\operatorname{th}u=\operatorname{tg}\psi=\sin\operatorname{gd}(u)=\cos\Pi(u)&lt;/math&gt;

He also related the velocity addition to the [[w:hyperbolic law of cosines]]:&lt;ref group=R&gt;Varićak (1910), p. 94&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\operatorname{ch}{u}=\operatorname{ch}{u_{1}}\operatorname ch{u_{2}}+\operatorname{sh}{u_{1}}\operatorname{sh}{u_{2}}\cos\alpha\\
\operatorname{ch}{u_{i}}=\frac{1}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},\ \operatorname{sh}{u_{i}}=\frac{v_{i}}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}}\\
v=\sqrt{v_{1}^{2}+v_{2}^{2}-\left(\frac{v_{1}v_{2}}{c}\right)^{2}}\ \left(a=\frac{\pi}{2}\right)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3f}}).}}

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bel68sag}} 
*{{#section:History of Topics in Special Relativity/mathsource|bel68fond}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|eli03}} 
*{{#section:History of Topics in Special Relativity/mathsource|esch74}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul35}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul48a}} 
*{{#section:History of Topics in Special Relativity/mathsource|ger92}} 
*{{#section:History of Topics in Special Relativity/mathsource|glai78}} 
*{{#section:History of Topics in Special Relativity/mathsource|gour88}} 
*{{#section:History of Topics in Special Relativity/mathsource|gud30}} 
*{{#section:History of Topics in Special Relativity/mathsource|guen80}} 
*{{#section:History of Topics in Special Relativity/mathsource|kep09}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil93}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil97}} 
*{{#section:History of Topics in Special Relativity/mathsource|lag70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lais74b}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam67}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lieb04}} 
*{{#section:History of Topics in Special Relativity/mathsource|lind90}}
*{{#section:History of Topics in Special Relativity/mathsource|lip86}}
*{{#section:History of Topics in Special Relativity/mathsource|merc}}
*{{#section:History of Topics in Special Relativity/mathsource|ric57}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu85}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu00}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu09}} 
*{{#section:History of Topics in Special Relativity/mathsource|tau26}} 
*{{#section:History of Topics in Special Relativity/mathsource|whit98}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo01}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo03}}

===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|frank09a}}
*{{#section:History of Topics in Special Relativity/relsource|herg10}}
*{{#section:History of Topics in Special Relativity/relsource|var10}}
*{{#section:History of Topics in Special Relativity/relsource|var12}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L3}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
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    <title>History of Topics in Special Relativity/Lorentz transformation (velocity)</title>
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      <comment>/* Lorentz transformation via velocity */</comment>
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      <text bytes="57347" sha1="ltxj4a4dn16357zxu2x3xg5itpp64vy" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via velocity==
===Boosts===
In the [[w:theory of relativity]], Lorentz transformations exhibit the symmetry of [[w:Minkowski spacetime]] by using a constant ''c'' as the [[w:speed of light]], and a parameter ''v'' as the relative [[w:velocity]] between two [[w:inertial reference frames]]. The corresponding formulas are identical to [[../Lorentz transformation (hyperbolic)|E:Lorentz transformations via hyperbolic functions]] introduced long before relativity was developed. In particular, the hyperbolic angle &lt;math&gt;\eta&lt;/math&gt; can be interpreted as the velocity related [[w:rapidity]] &lt;math&gt;\tanh\eta=\beta=v/c&lt;/math&gt;, so that &lt;math&gt;\gamma=\cosh\eta&lt;/math&gt; is the [[w:Lorentz factor]], &lt;math&gt;\beta\gamma=\sinh\eta&lt;/math&gt; the [[w:proper velocity]], &lt;math&gt;u'=c\tanh q&lt;/math&gt; the velocity of another object, &lt;math&gt;u=c\tanh(q+\eta)&lt;/math&gt; the [[w:velocity-addition formula]], thus transformation [[../Lorentz transformation (hyperbolic)#math_3b|E:'''(3b)''']] becomes:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}\gamma-x_{1}\beta\gamma\\
x_{1}^{\prime} &amp; =-x_{0}\beta\gamma+x_{1}\gamma\\
\\
x_{0} &amp; =x_{0}^{\prime}\gamma+x_{1}^{\prime}\beta\gamma\\
x_{1} &amp; =x_{0}^{\prime}\beta\gamma+x_{1}^{\prime}\gamma
\end{align}
\left|{\scriptstyle \begin{align}\beta^{2}\gamma^{2}-\gamma^{2} &amp; =-1 &amp; (a)\\
\gamma^{2}-\beta^{2}\gamma^{2} &amp; =1 &amp; (b)\\
\frac{\beta\gamma}{\gamma} &amp; =\beta &amp; (c)\\
\frac{1}{\sqrt{1-\beta^{2}}} &amp; =\gamma &amp; (d)\\
\frac{\beta}{\sqrt{1-\beta^{2}}} &amp; =\beta\gamma &amp; (e)\\
\frac{u'+v}{1+\frac{u'v}{c^{2}}} &amp; =u &amp; (f)
\end{align}
}\right.
\end{matrix}&lt;/math&gt;|{{equationRef|4a}}}}

Written in four dimensions by setting &lt;math&gt;x_{0}=ct,\ x_{1}=x&lt;/math&gt; and adding  &lt;math&gt;y,z&lt;/math&gt; the familiar form follows

{{NumBlk|:|&lt;math&gt;(A)\quad\begin{matrix}-c^{2}t^{2}+x^{2}+y^{2}+z^{2}=-c^{2}t^{\prime2}+x^{\prime2}+y^{\prime2}+z^{\prime2}\\
\hline \left.\begin{align}t' &amp; =\gamma\left(t-x\frac{v}{c^{2}}\right)\\
x' &amp; =\gamma(x-vt)\\
y' &amp; =y\\
z' &amp; =z
\end{align}
\right|\begin{align}t &amp; =\gamma\left(t'+x\frac{v}{c^{2}}\right)\\
x &amp; =\gamma(x'+vt')\\
y &amp; =y'\\
z &amp; =z'
\end{align}
\end{matrix}&lt;/math&gt;

or in matrix notation:

&lt;math&gt;(B)\quad\begin{matrix}\mathbf{x}'=\begin{bmatrix}\gamma &amp; -\beta\gamma &amp; 0 &amp; 0\\
-\beta\gamma &amp; \gamma &amp; 0 &amp; 0\\
0 &amp; 0 &amp; 1 &amp; 0\\
0 &amp; 0 &amp; 0 &amp; 1
\end{bmatrix}\cdot\mathbf{x};\quad\mathbf{x}=\begin{bmatrix}\gamma &amp; \beta\gamma &amp; 0 &amp; 0\\
\beta\gamma &amp; \gamma &amp; 0 &amp; 0\\
0 &amp; 0 &amp; 1 &amp; 0\\
0 &amp; 0 &amp; 0 &amp; 1
\end{bmatrix}\cdot\mathbf{x}'\\
\det\begin{bmatrix}\gamma &amp; -\beta\gamma\\
-\beta\gamma &amp; \gamma
\end{bmatrix}=1
\end{matrix}&lt;/math&gt;

or in terms of &lt;math&gt;ct,x&lt;/math&gt; as squeeze mapping in line with [[../Lorentz transformation (hyperbolic)#math_3c|E:'''(3c)''']]:

&lt;math&gt;(C)\quad\begin{matrix}uw=-x_{0}^{2}+x_{1}^{2}=u'w'=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{matrix}\begin{align}u' &amp; =ku\\
w' &amp; =\frac{1}{k}w
\end{align}
 &amp; \Rightarrow &amp; \begin{align}x'-ct' &amp; =\sqrt{\frac{c+v}{c-v}}\left(x-ct\right)\\
x'+ct' &amp; =\sqrt{\frac{c-v}{c+v}}\left(x+ct\right)
\end{align}
\quad\begin{align}x-ct &amp; =\sqrt{\frac{c-v}{c+v}}\left(x'-ct'\right)\\
x+ct &amp; =\sqrt{\frac{c+v}{c-v}}\left(x'+ct'\right)
\end{align}
\end{matrix}\\
\hline k=\sqrt{\frac{c+v}{c-v}}
\end{matrix}&lt;/math&gt;|{{equationRef|4b}}}}

Transformations analogous to (A) have been introduced by [[#Voigt|Voigt (1887)]] in terms of an incompressible medium, and by [[#Lorentz1|Lorentz (1892, 1895)]] who analyzed [[w:Maxwell's equations]], they were completed by [[#Larmor|Larmor (1897, 1900)]] and [[#Lorentz2|Lorentz (1899, 1904)]], and brought into their modern form by [[#Poincare3|Poincaré (1905)]] who gave the transformation the name of Lorentz.&lt;ref&gt;Miller (1981), chapter 1&lt;/ref&gt; Eventually, [[#Einstein|Einstein (1905)]] showed in his development of [[w:special relativity]] that the transformations follow from the [[w:principle of relativity]] and constant light speed alone by modifying the traditional concepts of space and time, without requiring a [[w:Lorentz ether theory|mechanical aether]] in contradistinction to Lorentz and Poincaré.&lt;ref&gt;Miller (1981), chapter 4–7&lt;/ref&gt; [[#Minkowski|Minkowski (1907–1908)]] used them to argue that space and time are inseparably connected as [[w:spacetime]]. 

The matrix form (B) is a special case of the general boost matrix given by [[#Hahn|Hahn (1912)]] in terms of imaginary time, while variant (C) for arbitrary ''k'' was given by many authors (see [[../Lorentz transformation (squeeze)|E:Lorentz transformations via squeeze mappings]]) with the choice equivalent to &lt;math&gt;k=\sqrt{\tfrac{c+v}{c-v}}&lt;/math&gt; given by [[#Born|Born (1921)]].

===Velocity addition and aberration===
In exact analogy to Beltrami coordinates in equation [[../Lorentz transformation (hyperbolic)#math_3e|E:'''(3e)''']], one can substitute &lt;math&gt;\left[\tfrac{u_{x}}{c},\ \tfrac{u_{y}}{c},\ \tfrac{u_{z}}{c}\right]=\left[\tfrac{x}{ct},\ \tfrac{y}{ct},\ \tfrac{z}{ct}\right]&lt;/math&gt; in ({{equationNote|4b}}-A), producing the Lorentz transformation of velocities (or [[w:velocity addition formula]]):

{{NumBlk|:|&lt;math&gt;\begin{align}u_{x}^{\prime} &amp; =\frac{-c^{2}\sinh\eta+u_{x}c\cosh\eta}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{x}-c\tanh\eta}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{x}-v}{1-\frac{v}{c^{2}}u{}_{x}}\\
u_{y}^{\prime} &amp; =\frac{cu_{y}}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{y}\sqrt{1-\tanh^{2}\eta}}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{y}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u{}_{x}}\\
u_{z}^{\prime} &amp; =\frac{cu_{y}}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{z}\sqrt{1-\tanh^{2}\eta}}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{z}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u{}_{x}}\\
\\
\hline \\
u_{x} &amp; =\frac{c^{2}\sinh\eta+u_{x}^{\prime}c\cosh\eta}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{x}^{\prime}+c\tanh\eta}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{x}^{\prime}+v}{1+\frac{v}{c^{2}}u_{x}^{\prime}}\\
u_{y} &amp; =\frac{cy'}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{y}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{y}^{\prime}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u_{x}^{\prime}}\\
u_{z} &amp; =\frac{cz'}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{z}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{z}^{\prime}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u_{x}^{\prime}}
\end{align}&lt;/math&gt;|{{equationRef|4c}}}}

By restriction to velocities in the &lt;math&gt;\left[x,y\right]&lt;/math&gt; plane and using trigonometric and hyperbolic identities as in equation [[../Lorentz transformation (hyperbolic)#math_3f|E:'''(3f)''']], it becomes the hyperbolic law of cosines:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix} &amp; \begin{matrix}u^{2}=u_{x}^{2}+u_{y}^{2}\\
u'^{2}=u_{x}^{\prime2}+u_{y}^{\prime2}
\end{matrix}\left|{\scriptstyle \begin{align}u_{x}=u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+\frac{v}{c^{2}}u'\cos\alpha'}, &amp; u_{x}^{\prime}=u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-\frac{v}{c^{2}}u\cos\alpha}\\
u_{y}=u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u'\cos\alpha'}, &amp; u_{y}^{\prime}=u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u\cos\alpha}\\
\frac{u_{y}}{u_{x}}=\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{u'\cos\alpha'+v}, &amp; \frac{u_{y}^{\prime}}{u_{x}^{\prime}}=\tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{u\cos\alpha-v}
\end{align}
}\right.\\
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(\frac{vu'\sin\alpha'}{c}\right){}^{2}}}{1+\frac{v}{c^{2}}u'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(\frac{vu\sin\alpha}{c}\right){}^{2}}}{1-\frac{v}{c^{2}}u\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\frac{u^{\prime2}}{c^{2}}}}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{1}{\sqrt{1-\frac{u^{2}}{c^{2}}}}-\frac{v/c}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{u/c}{\sqrt{1-\frac{u^{2}}{c^{2}}}}\cos\alpha\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha
\end{matrix}&lt;/math&gt;|{{equationRef|4d}}}}

and by further setting ''u=u′=c'' one gets the well known [[../Lorentz transformation (hyperbolic)#math_3g|E:Kepler formulas '''(3g)''']], which  express the relativistic [[w:aberration of light]]:&lt;ref&gt;Pauli (1921), pp. 562; 565–566&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}\cos\alpha=\frac{\cos\alpha'+\frac{v}{c}}{1+\frac{v}{c}\cos\alpha'},\ \sin\alpha=\frac{\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c}\cos\alpha'},\ \tan\alpha=\frac{\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{\cos\alpha'+\frac{v}{c}},\ \tan\frac{\alpha}{2}=\sqrt{\frac{c-v}{c+v}}\tan\frac{\alpha'}{2}\\
\cos\alpha'=\frac{\cos\alpha-\frac{v}{c}}{1-\frac{v}{c}\cos\alpha},\ \sin\alpha'=\frac{\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c}\cos\alpha},\ \tan\alpha'=\frac{\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{\cos\alpha-\frac{v}{c}},\ \tan\frac{\alpha'}{2}=\sqrt{\frac{c+v}{c-v}}\tan\frac{\alpha}{2}
\end{matrix} &lt;/math&gt;|{{equationRef|4e}}}}

Formulas ({{equationNote|4c}}, {{equationNote|4d}}) were given by [[#Einstein|Einstein (1905)]] and [[#Poincare3|Poincaré (1905/06)]], while the relations to the spherical and hyperbolic law of cosines were given by [[#Sommerfeld|Sommerfeld (1909)]] and [[#Frank|Varićak (1910)]]. The aberration formula for cos(α) was given by [[#Einstein|Einstein (1905)]].&lt;ref group=R name=plum&gt;Plummer (1910), pp. 258-259: After deriving the relativistic expressions for the aberration angles φ' and φ, Plummer remarked on p. 259: ''Another geometrical representation is obtained by assimilating φ' to the eccentric and φ to the true anomaly in an ellipse whose eccentricity is v/U = sin β.''&lt;/ref&gt;&lt;ref name=robin&gt;Robinson (1990), chapter 3-4, analyzed the relation between &quot;Kepler's formula&quot; and the &quot;physical velocity addition formula&quot; in special relativity.&lt;/ref&gt;

===Lorentz transformation in arbitrary directions===
Lorentz boosts for arbitrary directions&lt;ref&gt;Møller (1952/55), Chapter II, § 18&lt;/ref&gt; in line with [[../Lorentz transformation (general)#math_1a|E:general Lorentz transformation '''(1a)''']] are in vector notation

{{NumBlk|:|&lt;math&gt;\begin{align}t' &amp; =\gamma\left(t-\frac{v\mathbf{n}\cdot\mathbf{r}}{c^{2}}\right)\\
\mathbf{r}' &amp; =\mathbf{r}+(\gamma-1)(\mathbf{r}\cdot\mathbf{n})\mathbf{n}-\gamma tv\mathbf{n}
\end{align}
&lt;/math&gt;|{{equationRef|4f}}}}

and the vectorial velocity addition formula in line with [[../Lorentz transformation (general)#math_1b|E:general Lorentz transformation '''(1b)''']] follows by:

{{NumBlk|:|&lt;math&gt;\mathbf{u}'=\frac{1}{1+\frac{\mathbf{v}\cdot\mathbf{u}}{c^{2}}}\left[\frac{\mathbf{u}}{\gamma_{\mathbf{v}}}+\mathbf{v}+\frac{1}{c^{2}}\frac{\gamma_{\mathbf{v}}}{\gamma_{\mathbf{v}}+1}(\mathbf{u}\cdot\mathbf{v})\mathbf{v}\right]&lt;/math&gt;|{{equationRef|4g}}}}

The special case of parallel and perpendicular directions in ({{equationNote|4f}}) was given by [[#Minkowski2|Minkowski (1907/8)]] while the complete transformation was formulated by [[#Herglotz2|Ignatowski (1910), Herglotz (1911), Tamaki (1911)]]. General velocity addition ({{equationNote|4g}}) was given in equivalent form by [[#Herglotz2|Ignatowski (1910)]].

Rewritten in matrix notation, the general Lorentz boost has the form:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\
\hline \begin{align}\mathbf{g} &amp; =\begin{pmatrix}\gamma &amp; -\gamma\beta n_{x} &amp; -\gamma\beta n_{y} &amp; -\gamma\beta n_{z}\\
-\gamma\beta n_{x} &amp; 1+(\gamma-1)n_{x}^{2} &amp; (\gamma-1)n_{x}n_{y} &amp; (\gamma-1)n_{x}n_{z}\\
-\gamma\beta n_{y} &amp; (\gamma-1)n_{y}n_{x} &amp; 1+(\gamma-1)n_{y}^{2} &amp; (\gamma-1)n_{y}n_{z}\\
-\gamma\beta n_{z} &amp; (\gamma-1)n_{z}n_{x} &amp; (\gamma-1)n_{z}n_{y} &amp; 1+(\gamma-1)n_{z}^{2}
\end{pmatrix}\end{align}
\\
\left[\mathbf{n}=\frac{\mathbf{v}}{v}\right]
\end{matrix}\,&lt;/math&gt;|{{equationRef|4h}}}}

While [[#Minkowski3|Minkowski (1907/8)]] formulated the matrix form of Lorentz transformations in general terms, he didn't explicitly express the velocity related components of the general boost matrix. A complete representation of ({{equationNote|4h}}) was given by [[#Hahn|Hahn (1912)]].

===Other formulations===
Important contributions to the mathematical understanding of the Lorentz transformation of space and time also include: [[#Minkowski|Minkowski (1907–1908)]] as well as [[#Frank|Frank (1909) and Varićak (1910)]] showed the relation to imaginary and hyperbolic functions, [[#Herglotz1|Herglotz (1909/10)]] used exponential squeeze mappings and Möbius transformations, [[#Ignatowski|Ignatowski (1910)]] didn't use the light speed postulate, [[#klein|Klein and Noether (1908-11) as well as Conway and Silberstein (1911)]] used Biquaternions, [[#Plummer|Plummer (1910) and Gruner (1921)]] used trigonometric Lorentz boosts, [[#Borel|Borel (1913–14)]] used Cayley-Hermite parameter.

==Historical notation==
==={{anchor|Voigt}} Voigt (1887) ===

[[w:Woldemar Voigt]] (1887)&lt;ref group=R&gt;Voigt (1887), p. 45&lt;/ref&gt; developed a transformation in connection with the [[w:Doppler effect]] and an incompressible medium, being in modern notation:&lt;ref&gt;Miller (1981), 114–115&lt;/ref&gt;&lt;ref name=pais&gt;Pais (1982), Kap. 6b&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}\xi_{1} &amp; =x_{1}-\varkappa t\\
\eta_{1} &amp; =y_{1}q\\
\zeta_{1} &amp; =z_{1}q\\
\tau &amp; =t-\frac{\varkappa x_{1}}{\omega^{2}}\\
q &amp; =\sqrt{1-\frac{\varkappa^{2}}{\omega^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =\frac{y}{\gamma}\\
z^{\prime} &amp; =\frac{z}{\gamma}\\
t^{\prime} &amp; =t-\frac{vx}{c^{2}}\\
\frac{1}{\gamma} &amp; =\sqrt{1-\frac{v^{2}}{c^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

If the right-hand sides of his equations are multiplied by γ they are the modern Lorentz transformation ({{equationNote|4b}}). In Voigt's theory the speed of light is invariant, but his transformations mix up a relativistic boost together with a rescaling of space-time. Optical phenomena in free space are [[w:Scale invariance|scale]], [[w:Conformal map|conformal]] (using the factor λ discussed [[#Lorsph|above]]), and [[w:Lorentz covariance|Lorentz invariant]], so the combination is invariant too.&lt;ref name=pais /&gt; For instance, Lorentz transformations can be extended by using &lt;math&gt;l=\sqrt{\lambda}&lt;/math&gt;:&lt;ref group=R&gt;Lorentz (1915/16), p. 197&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=\gamma l\left(x-vt\right),\quad y^{\prime}=ly,\quad z^{\prime}=lz,\quad t^{\prime}=\gamma l\left(t-x\frac{v}{c^{2}}\right)&lt;/math&gt;.

''l''=1/γ gives the Voigt transformation, ''l''=1 the Lorentz transformation. But scale transformations are not a symmetry of all the laws of nature, only of electromagnetism, so these transformations cannot be used to formulate a [[w:principle of relativity]] in general. It was demonstrated by Poincaré and Einstein that one has to set ''l''=1 in order to make the above transformation symmetric and to form a group as required by the relativity principle, therefore the Lorentz transformation is the only viable choice.

Voigt sent his 1887 paper to Lorentz in 1908,&lt;ref&gt;Voigt's transformations and the beginning of the relativistic revolution, Ricardo Heras, arXiv:1411.2559 [https://arxiv.org/abs/1411.2559]&lt;/ref&gt; and that was acknowledged in 1909: {{Quote|In a paper &quot;Über das Doppler'sche Princip&quot;, published in 1887 (Gött. Nachrichten, p. 41) and which to my regret has escaped my notice all these years, Voigt has applied to equations of the form (7) (§ 3 of this book) [namely &lt;math&gt;\Delta\Psi-\tfrac{1}{c^{2}}\tfrac{\partial^{2}\Psi}{\partial t^{2}}=0&lt;/math&gt;] a transformation equivalent to the formulae (287) and (288) [namely &lt;math&gt;x^{\prime}=\gamma l\left(x-vt\right),\ y^{\prime}=ly,\ z^{\prime}=lz,\ t^{\prime}=\gamma l\left(t-\tfrac{v}{c^{2}}x\right)&lt;/math&gt;]. The idea of the transformations used above (and in § 44) might therefore have been borrowed from Voigt and the proof that it does not alter the form of the equations for the ''free'' ether is contained in his paper.&lt;ref group=R&gt;Lorentz (1915/16), p. 198&lt;/ref&gt;}}

Also [[w:Hermann Minkowski]] said in 1908 that the transformations which play the main role in the principle of relativity were first examined by Voigt in 1887. Voigt responded in the same paper by saying that his theory was based on an elastic theory of light, not an electromagnetic one. However, he concluded that some results were actually the same.&lt;ref group=R&gt;Bucherer (1908), p. 762&lt;/ref&gt;

==={{anchor|Heaviside}} Heaviside (1888), Thomson (1889), Searle (1896)===

In 1888, [[w:Oliver Heaviside]]&lt;ref group=R&gt;Heaviside (1888), p. 324&lt;/ref&gt; investigated the properties of [[w:Relativistic electromagnetism|charges in motion]] according to Maxwell's electrodynamics. He calculated, among other things, anisotropies in the electric field of moving bodies represented by this formula:&lt;ref&gt;Brown (2003)&lt;/ref&gt;

:&lt;math&gt;\mathrm{E}=\left(\frac{q\mathrm{r}}{r^{2}}\right)\left(1-\frac{v^{2}\sin^{2}\theta}{c^{2}}\right)^{-3/2}&lt;/math&gt;.

Consequently, [[w:Joseph John Thomson]] (1889)&lt;ref group=R&gt;Thomson (1889), p. 12&lt;/ref&gt; found a way to substantially simplify calculations concerning moving charges by using the following mathematical transformation (like other authors such as Lorentz or Larmor, also Thomson implicitly used the [[w:Galilean transformation]] ''z-vt'' in his equation&lt;ref name=mil /&gt;):

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}z &amp; =\left\{ 1-\frac{\omega^{2}}{v^{2}}\right\} ^{\frac{1}{2}}z'\end{align}
\right| &amp; \begin{align}z^{\ast}=z-vt &amp; =\frac{z'}{\gamma}\end{align}
\end{matrix}&lt;/math&gt;

Thereby, [[w:inhomogeneous electromagnetic wave equation]]s are transformed into a [[w:Poisson equation]].&lt;ref name=mil&gt;Miller (1981), 98–99&lt;/ref&gt; Eventually, [[w:George Frederick Charles Searle]]&lt;ref group=R&gt;Searle (1886), p. 333&lt;/ref&gt; noted in (1896) that Heaviside's expression leads to a deformation of electric fields which he called &quot;Heaviside-Ellipsoid&quot; of [[w:axial ratio]]

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align} &amp; \sqrt{\alpha}:1:1\\
\alpha= &amp; 1-\frac{u^{2}}{v^{2}}
\end{align}
\right| &amp; \begin{align} &amp; \frac{1}{\gamma}:1:1\\
\frac{1}{\gamma^{2}} &amp; =1-\frac{v^{2}}{c^{2}}
\end{align}
\end{matrix}&lt;/math&gt;&lt;ref name=mil /&gt;

=== {{anchor|Lorentz1}} Lorentz (1892, 1895) ===

In order to explain the [[w:aberration of light]] and the result of the [[w:Fizeau experiment]] in accordance with [[w:Maxwell's equations]], Lorentz in 1892 developed a model (&quot;[[w:Lorentz ether theory]]&quot;) in which the aether is completely motionless, and the speed of light in the aether is constant in all directions. In order to calculate the optics of moving bodies, Lorentz introduced the following quantities to transform from the aether system into a moving system (it's unknown whether he was influenced by Voigt, Heaviside, and Thomson)&lt;ref group=R&gt;Lorentz (1892a), p. 141&lt;/ref&gt;&lt;ref name=milf&gt;Miller (1982), 1.4 &amp; 1.5&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}\mathfrak{x} &amp; =\frac{V}{\sqrt{V^{2}-p^{2}}}x\\
t' &amp; =t-\frac{\varepsilon}{V}\mathfrak{x}\\
\varepsilon &amp; =\frac{p}{\sqrt{V^{2}-p^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
t^{\prime} &amp; =t-\frac{\gamma^{2}vx^{\ast}}{c^{2}}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)\\
\gamma\frac{v}{c} &amp; =\frac{v}{\sqrt{c^{2}-v^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

where ''x&lt;sup&gt;*&lt;/sup&gt;'' is the [[w:Galilean transformation]] ''x-vt''. Except the additional γ in the time transformation, this is the complete Lorentz transformation ({{equationNote|4b}}).&lt;ref name=milf /&gt; While ''t'' is the &quot;true&quot; time for observers resting in the aether, ''t′'' is an auxiliary variable only for calculating processes for moving systems. It is also important that Lorentz and later also Larmor formulated this transformation in two steps. At first an implicit Galilean transformation, and later the expansion into the &quot;fictitious&quot; electromagnetic system with the aid of the Lorentz transformation. In order to explain the negative result of the [[w:Michelson–Morley experiment]], he (1892b)&lt;ref group=R&gt;Lorentz (1892b), p. 141&lt;/ref&gt; introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introduced [[w:length contraction]] in his theory (without proof as he admitted). The same hypothesis was already made by [[w:George FitzGerald]] in 1889 based on Heaviside's work. While length contraction was a real physical effect for Lorentz, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation.

In 1895, Lorentz further elaborated on his theory and introduced the &quot;theorem of corresponding states&quot;. This theorem states that a moving observer (relative to the ether) in his &quot;fictitious&quot; field makes the same observations as a resting observers in his &quot;real&quot; field for velocities to first order in ''v/c''. Lorentz showed that the dimensions of electrostatic systems in the ether and a moving frame are connected by this transformation:&lt;ref group=R&gt;Lorentz (1895), p. 37&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =x^{\prime}\sqrt{1-\frac{\mathfrak{p}^{2}}{V^{2}}}\\
y &amp; =y^{\prime}\\
z &amp; =z^{\prime}\\
t &amp; =t^{\prime}
\end{align}
\right| &amp; \begin{align}x^{\ast}=x-vt &amp; =\frac{x^{\prime}}{\gamma}\\
y &amp; =y^{\prime}\\
z &amp; =z^{\prime}\\
t &amp; =t^{\prime}
\end{align}
\end{matrix}&lt;/math&gt;

For solving optical problems Lorentz used the following transformation, in which the modified time variable was called &quot;local time&quot; ({{lang-de|Ortszeit}}) by him:&lt;ref group=R&gt;Lorentz (1895), p. 49 for local time and p. 56 for spatial coordinates.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =\mathrm{x}-\mathfrak{p}_{x}t\\
y &amp; =\mathrm{y}-\mathfrak{p}_{y}t\\
z &amp; =\mathrm{z}-\mathfrak{p}_{z}t\\
t^{\prime} &amp; =t-\frac{\mathfrak{p}_{x}}{V^{2}}x-\frac{\mathfrak{p}_{y}}{V^{2}}y-\frac{\mathfrak{p}_{z}}{V^{2}}z
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-v_{x}t\\
y^{\prime} &amp; =y-v_{y}t\\
z^{\prime} &amp; =z-v_{z}t\\
t^{\prime} &amp; =t-\frac{v_{x}}{c^{2}}x'-\frac{v_{y}}{c^{2}}y'-\frac{v_{z}}{c^{2}}z'
\end{align}
\end{matrix}&lt;/math&gt;

With this concept Lorentz could explain the [[w:Doppler effect]], the [[w:aberration of light]], and the [[w:Fizeau experiment]].&lt;ref&gt;Janssen (1995), 3.1&lt;/ref&gt;

=== {{anchor|Larmor}} Larmor (1897, 1900) ===

In 1897, Larmor extended the work of Lorentz and derived the following transformation&lt;ref group=R&gt;Larmor (1897), p. 229&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x_{1} &amp; =x\varepsilon^{\frac{1}{2}}\\
y_{1} &amp; =y\\
z_{1} &amp; =z\\
t^{\prime} &amp; =t-vx/c^{2}\\
dt_{1} &amp; =dt^{\prime}\varepsilon^{-\frac{1}{2}}\\
\varepsilon &amp; =\left(1-v^{2}/c^{2}\right)^{-1}
\end{align}
\right| &amp; \begin{align}x_{1} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
y_{1} &amp; =y\\
z_{1} &amp; =z\\
t^{\prime} &amp; =t-\frac{vx^{\ast}}{c^{2}}=t-\frac{v(x-vt)}{c^{2}}\\
dt_{1} &amp; =\frac{dt^{\prime}}{\gamma}\\
\gamma^{2} &amp; =\frac{1}{1-\frac{v^{2}}{c^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

Larmor noted that if it is assumed that the constitution of molecules is electrical then the FitzGerald–Lorentz contraction is a consequence of this transformation, explaining the [[w:Michelson–Morley experiment]]. It's notable that Larmor was the first who recognized that some sort of [[w:time dilation]] is a consequence of this transformation as well, because &quot;individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio 1/γ&quot;.&lt;ref&gt;Darrigol (2000), Chap. 8.5&lt;/ref&gt;&lt;ref&gt;Macrossan (1986)&lt;/ref&gt; Larmor wrote his electrodynamical equations and transformations neglecting terms of higher order than ''(v/c)''&lt;sup&gt;2&lt;/sup&gt; – when his 1897 paper was reprinted in 1929, Larmor added the following comment in which he described how they can be made valid to all orders of ''v/c'':&lt;ref group=R&gt;Larmor (1897/1929), p. 39&lt;/ref&gt;

{{Quote|Nothing need be neglected: the transformation is ''exact'' if ''v/c''&lt;sup&gt;2&lt;/sup&gt; is replaced by ''εv/c''&lt;sup&gt;2&lt;/sup&gt; in the equations and also in the change following from ''t'' to ''t′'', as is worked out in ''Aether and Matter'' (1900), p. 168, and as Lorentz found it to be in 1904, thereby stimulating the modern schemes of intrinsic relational relativity.}}

In line with that comment, in his book Aether and Matter published in 1900, Larmor used a modified local time ''t″=t′-εvx′/c&lt;sup&gt;2&lt;/sup&gt;'' instead of the 1897 expression ''t′=t-vx/c&lt;sup&gt;2&lt;/sup&gt;'' by replacing ''v/c''&lt;sup&gt;2&lt;/sup&gt; with ''εv/c''&lt;sup&gt;2&lt;/sup&gt;, so that ''t″'' is now identical to the one given by Lorentz in 1892, which he combined with a Galilean transformation for the ''x′, y′, z′, t′'' coordinates:&lt;ref group=R&gt;Larmor (1900), p. 168&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t\\
t^{\prime\prime} &amp; =t^{\prime}-\varepsilon vx^{\prime}/c^{2}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t\\
t^{\prime\prime}=t^{\prime}-\frac{\gamma^{2}vx^{\prime}}{c^{2}} &amp; =\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor ''(v/c)''&lt;sup&gt;2&lt;/sup&gt;, and so he sought the transformations which were &quot;accurate to second order&quot; (as he put it). Thus he wrote the final transformations (where ''x′=x-vt'' and ''t″'' as given above) as:&lt;ref group=R&gt;Larmor (1900), p. 174&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x_{1} &amp; =\varepsilon^{\frac{1}{2}}x^{\prime}\\
y_{1} &amp; =y^{\prime}\\
z_{1} &amp; =z^{\prime}\\
dt_{1} &amp; =\varepsilon^{-\frac{1}{2}}dt^{\prime\prime}=\varepsilon^{-\frac{1}{2}}\left(dt^{\prime}-\frac{v}{c^{2}}\varepsilon dx^{\prime}\right)\\
t_{1} &amp; =\varepsilon^{-\frac{1}{2}}t^{\prime}-\frac{v}{c^{2}}\varepsilon^{\frac{1}{2}}x^{\prime}
\end{align}
\right| &amp; \begin{align}x_{1} &amp; =\gamma x^{\prime}=\gamma(x-vt)\\
y_{1} &amp; =y'=y\\
z_{1} &amp; =z'=z\\
dt_{1} &amp; =\frac{dt^{\prime\prime}}{\gamma}=\frac{1}{\gamma}\left(dt^{\prime}-\frac{\gamma^{2}vdx^{\prime}}{c^{2}}\right)=\gamma\left(dt-\frac{vdx}{c^{2}}\right)\\
t_{1} &amp; =\frac{t^{\prime}}{\gamma}-\frac{\gamma vx^{\prime}}{c^{2}}=\gamma\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

by which he arrived at the complete Lorentz transformation ({{equationNote|4b}}). Larmor showed that Maxwell's equations were invariant under this two-step transformation, &quot;to second order in ''v/c''&quot; – it was later shown by Lorentz (1904) and Poincaré (1905) that they are indeed invariant under this transformation to all orders in ''v/c''.

Larmor gave credit to Lorentz in two papers published in 1904, in which he used the term &quot;Lorentz transformation&quot; for Lorentz's first order transformations of coordinates and field configurations:

{{Quote|p. 583: [..] Lorentz's transformation for passing from the field of activity of a stationary electrodynamic material system to that of one moving with uniform velocity of translation through the aether.&lt;br /&gt; p. 585: [..] the Lorentz transformation has shown us what is not so immediately obvious [..]&lt;ref group=R&gt;Larmor (1904a), p. 583, 585&lt;/ref&gt; &lt;br /&gt; p. 622: [..] the transformation first developed by Lorentz: namely, each point in space is to have its own origin from which time is measured, its &quot;local time&quot; in Lorentz's phraseology, and then the values of the electric and magnetic vectors [..] at all points in the aether between the molecules in the system at rest, are the same as those of the vectors [..] at the corresponding points in the convected system at the same local times.&lt;ref group=R&gt;Larmor (1904b), p. 622&lt;/ref&gt;}}

=== {{anchor|Lorentz2}} Lorentz (1899, 1904) ===

Also Lorentz extended his theorem of corresponding states in 1899. First he wrote a transformation equivalent to the one from 1892 (again, ''x''* must be replaced by ''x-vt''):&lt;ref group=R&gt;Lorentz (1899), p. 429&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}x\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t-\frac{\mathfrak{p}_{x}}{V^{2}-\mathfrak{p}_{x}^{2}}x
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t-\frac{\gamma^{2}vx^{\ast}}{c^{2}}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Then he introduced a factor ε of which he said he has no means of determining it, and modified his transformation as follows (where the above value of ''t′'' has to be inserted):&lt;ref group=R&gt;Lorentz (1899), p. 439&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =\frac{\varepsilon}{k}x^{\prime\prime}\\
y &amp; =\varepsilon y^{\prime\prime}\\
z &amp; =\varepsilon x^{\prime\prime}\\
t^{\prime} &amp; =k\varepsilon t^{\prime\prime}\\
k &amp; =\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\ast}=x-vt &amp; =\frac{\varepsilon}{\gamma}x^{\prime\prime}\\
y &amp; =\varepsilon y^{\prime\prime}\\
z &amp; =\varepsilon z^{\prime\prime}\\
t^{\prime}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right) &amp; =\gamma\varepsilon t^{\prime\prime}\\
\gamma &amp; =\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
\end{align}
\end{matrix}&lt;/math&gt;

This is equivalent to the complete Lorentz transformation ({{equationNote|4b}}) when solved for ''x″'' and ''t″'' and with ε=1. Like Larmor, Lorentz noticed in 1899&lt;ref group=R&gt;Lorentz (1899), p. 442&lt;/ref&gt; also some sort of time dilation effect in relation to the frequency of oscillating electrons ''&quot;that in ''S'' the time of vibrations be ''kε'' times as great as in ''S&lt;sub&gt;0&lt;/sub&gt;''&quot;'', where ''S&lt;sub&gt;0&lt;/sub&gt;'' is the aether frame.&lt;ref&gt;Janssen (1995), Kap. 3.3&lt;/ref&gt;

In 1904 he rewrote the equations in the following form by setting ''l''=1/ε (again, ''x''* must be replaced by ''x-vt''):&lt;ref group=R&gt;Lorentz (1904), p. 812&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =klx\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t' &amp; =\frac{l}{k}t-kl\frac{w}{c^{2}}x
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma lx^{\ast}=\gamma l(x-vt)\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t^{\prime} &amp; =\frac{lt}{\gamma}-\frac{\gamma lvx^{\ast}}{c^{2}}=\gamma l\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Under the assumption that ''l=1'' when ''v''=0, he demonstrated that ''l=1'' must be the case at all velocities, therefore length contraction can only arise in the line of motion. So by setting the factor ''l'' to unity, Lorentz's transformations now assumed the same form as Larmor's and are now completed. Unlike Larmor, who restricted himself to show the covariance of Maxwell's equations to second order, Lorentz tried to widen its covariance to all orders in ''v/c''. He also derived the correct formulas for the velocity dependence of [[w:electromagnetic mass]], and concluded that the transformation formulas must apply to all forces of nature, not only electrical ones.&lt;ref group=R&gt;Lorentz (1904), p. 826&lt;/ref&gt; However, he didn't achieve full covariance of the transformation equations for charge density and velocity.&lt;ref&gt;Miller (1981), Chap. 1.12.2&lt;/ref&gt; When the 1904 paper was reprinted in 1913, Lorentz therefore added the following remark:&lt;ref&gt;Janssen (1995), Chap. 3.5.6&lt;/ref&gt;

{{Quote|One will notice that in this work the transformation equations of Einstein’s Relativity Theory have not quite been attained. [..] On this circumstance depends the clumsiness of many of the further considerations in this work.}}

Lorentz's 1904 transformation was cited and used by [[w:Alfred Bucherer]] in July 1904:&lt;ref group=R&gt;Bucherer, p. 129; Definition of s on p. 32&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=\sqrt{s}x,\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t'=\frac{t}{\sqrt{s}}-\sqrt{s}\frac{u}{v^{2}}x,\quad s=1-\frac{u^{2}}{v^{2}}&lt;/math&gt;

or by [[w:Wilhelm Wien]] in July 1904:&lt;ref group=R&gt;Wien (1904), p. 394&lt;/ref&gt;

:&lt;math&gt;x=kx',\quad y=y',\quad z=z',\quad t'=kt-\frac{v}{kc^{2}}x&lt;/math&gt;

or by [[w:Emil Cohn]] in November 1904 (setting the speed of light to unity):&lt;ref group=R&gt;Cohn (1904a), pp. 1296-1297&lt;/ref&gt;

:&lt;math&gt;x=\frac{x_{0}}{k},\quad y=y_{0},\quad z=z_{0},\quad t=kt_{0},\quad t_{1}=t_{0}-w\cdot r_{0},\quad k^{2}=\frac{1}{1-w^{2}}&lt;/math&gt;

or by [[w:Richard Gans]] in February 1905:&lt;ref group=R&gt;Gans (1905), p. 169&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=kx,\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t'=\frac{t}{k}-\frac{kwx}{c^{2}},\quad k^{2}=\frac{c^{2}}{c^{2}-w^{2}}&lt;/math&gt;

=== {{anchor|Poincare3}} Poincaré (1900, 1905) ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Poincare|label 1=History of Lorentz transformations in general § Poincaré}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Poincare2|label 1=History of Lorentz transformations via Möbius transformations § Poincaré}}

==== Local time ====

Neither Lorentz or Larmor gave a clear physical interpretation of the origin of local time. However, [[w:Henri Poincaré]] in 1900 commented on the origin of Lorentz's &quot;wonderful invention&quot; of local time.&lt;ref&gt;Darrigol (2005), Kap. 4&lt;/ref&gt; He remarked that it arose when clocks in a moving reference frame are synchronised by exchanging signals which are assumed to travel with the same speed &lt;math&gt;c&lt;/math&gt; in both directions, which lead to what is nowadays called [[w:relativity of simultaneity]], although Poincaré's calculation does not involve length contraction or time dilation.&lt;ref group=R&gt;Poincaré (1900), pp. 272–273&lt;/ref&gt; In order to synchronise the clocks here on Earth (the ''x*, t''* frame) a light signal from one clock (at the origin) is sent to another (at ''x''*), and is sent back. It's supposed that the Earth is moving with speed ''v'' in the ''x''-direction (= ''x''*-direction) in some rest system (''x, t'') (''i.e.'' the [[w:luminiferous aether]] system for Lorentz and Larmor). The time of flight outwards is

:&lt;math&gt;\delta t_{a}=\frac{x^{\ast}}{\left(c-v\right)}&lt;/math&gt;

and the time of flight back is

:&lt;math&gt;\delta t_{b}=\frac{x^{\ast}}{\left(c+v\right)}&lt;/math&gt;.

The elapsed time on the clock when the signal is returned is ''δt&lt;sub&gt;a&lt;/sub&gt;+δt&lt;sub&gt;b&lt;/sub&gt;'' and the time ''t*=(δt&lt;sub&gt;a&lt;/sub&gt;+δt&lt;sub&gt;b&lt;/sub&gt;)/2'' is ascribed to the moment when the light signal reached the distant clock. In the rest frame the time ''t=δt&lt;sub&gt;a&lt;/sub&gt;'' is ascribed to that same instant. Some algebra gives the relation between the different time coordinates ascribed to the moment of reflection. Thus

:&lt;math&gt;t^{\ast}=t-\frac{\gamma^{2}vx^{*}}{c^{2}}&lt;/math&gt;

identical to Lorentz (1892). By dropping the factor γ&lt;sup&gt;2&lt;/sup&gt; under the assumption that &lt;math&gt;\tfrac{v^{2}}{c^{2}}\ll1&lt;/math&gt;, Poincaré gave the result ''t*=t-vx*/c&lt;sup&gt;2&lt;/sup&gt;'', which is the form used by Lorentz in 1895.

Similar physical interpretations of local time were later given by [[w:Emil Cohn]] (1904)&lt;ref group=R&gt;Cohn (1904b), p. 1408&lt;/ref&gt; and [[w:Max Abraham]] (1905).&lt;ref group=R&gt;Abraham (1905), § 42&lt;/ref&gt;

==== Lorentz transformation ====

On June 5, 1905 (published June 9) Poincaré formulated transformation equations which are algebraically equivalent to those of Larmor and Lorentz and gave them the modern form ({{equationNote|4b}}):&lt;ref group=R&gt;Poincaré (1905), p. 1505&lt;/ref&gt;

:&lt;math&gt;\begin{align}x^{\prime} &amp; =kl(x+\varepsilon t)\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t' &amp; =kl(t+\varepsilon x)\\
k &amp; =\frac{1}{\sqrt{1-\varepsilon^{2}}}
\end{align}
&lt;/math&gt;.

Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name &quot;Lorentz transformation&quot;.&lt;ref&gt;Pais (1982), Chap. 6c&lt;/ref&gt;&lt;ref&gt;Katzir (2005), 280–288&lt;/ref&gt; Poincaré set the speed of light to unity, pointed out the group characteristics of the transformation by setting ''l''=1, and modified/corrected Lorentz's derivation of the equations of electrodynamics in some details in order to fully satisfy the principle of relativity, ''i.e.'' making them fully Lorentz covariant.&lt;ref&gt;Miller (1981), Chap. 1.14&lt;/ref&gt;

In July 1905 (published in January 1906)&lt;ref group=R&gt;Poincaré (1905/06), pp. 129ff&lt;/ref&gt; Poincaré showed in detail how the transformations and electrodynamic equations are a consequence of the [[w:principle of least action]]; he demonstrated in more detail the group characteristics of the transformation, which he called [[w:Lorentz group]], and he showed that the combination ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;+z&lt;sup&gt;2&lt;/sup&gt;-t&lt;sup&gt;2&lt;/sup&gt;'' is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing &lt;math&gt;ct\sqrt{-1}&lt;/math&gt; as a fourth imaginary coordinate, and he used an early form of [[w:four-vector]]s. He also formulated the velocity addition formula ({{equationNote|4c}}), which he had already derived in unpublished letters to Lorentz from May 1905:&lt;ref group=R&gt;Poincaré (1905/06), p. 144&lt;/ref&gt;

:&lt;math&gt;\xi'=\frac{\xi+\varepsilon}{1+\xi\varepsilon},\ \eta'=\frac{\eta}{k(1+\xi\varepsilon)}&lt;/math&gt;.

==={{anchor|Einstein}} Einstein (1905) – Special relativity===

On June 30, 1905 (published September 1905) Einstein published what is now called [[w:special relativity]] and gave a new derivation of the transformation, which was based only on the principle on relativity and the principle of the constancy of the speed of light. While Lorentz considered &quot;local time&quot; to be a mathematical stipulation device for explaining the Michelson-Morley experiment, Einstein showed that the coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference. For quantities of first order in ''v/c'' this was also done by Poincaré in 1900, while Einstein derived the complete transformation by this method. Unlike Lorentz and Poincaré who still distinguished between real time in the aether and apparent time for moving observers, Einstein showed that the transformations concern the nature of space and time.&lt;ref&gt;Miller (1981), Chap. 6&lt;/ref&gt;&lt;ref&gt;Pais (1982), Kap. 7&lt;/ref&gt;&lt;ref&gt;Darrigol (2005), Chap. 6&lt;/ref&gt;

The notation for this transformation is equivalent to Poincaré's of 1905 and ({{equationNote|4b}}), except that Einstein didn't set the speed of light to unity:&lt;ref group=R&gt;Einstein (1905), p. 902&lt;/ref&gt;

:&lt;math&gt;\begin{align}\tau &amp; =\beta\left(t-\frac{v}{V^{2}}x\right)\\
\xi &amp; =\beta(x-vt)\\
\eta &amp; =y\\
\zeta &amp; =z\\
\beta &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{V}\right)^{2}}}
\end{align}
&lt;/math&gt;

Einstein also defined the velocity addition formula ({{equationNote|4c}}, {{equationNote|4d}}):&lt;ref group=R&gt;Einstein (1905), § 5 and § 9&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=\frac{w_{\xi}+v}{1+\frac{vw_{\xi}}{V^{2}}}t,\ y=\frac{\sqrt{1-\left(\frac{v}{V}\right)^{2}}}{1+\frac{vw_{\xi}}{V^{2}}}w_{\eta}t\\
U^{2}=\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2},\ w^{2}=w_{\xi}^{2}+w_{\eta}^{2},\ \alpha=\operatorname{arctg}\frac{w_{y}}{w_{x}}\\
U=\frac{\sqrt{\left(v^{2}+w^{2}+2vw\cos\alpha\right)-\left(\frac{vw\sin\alpha}{V}\right)^{2}}}{1+\frac{vw\cos\alpha}{V^{2}}}
\end{matrix}\left|\begin{matrix}\frac{u_{x}-v}{1-\frac{u_{x}v}{V^{2}}}=u_{\xi}\\
\frac{u_{y}}{\beta\left(1-\frac{u_{x}v}{V^{2}}\right)}=u_{\eta}\\
\frac{u_{z}}{\beta\left(1-\frac{u_{x}v}{V^{2}}\right)}=u_{\zeta}
\end{matrix}\right.&lt;/math&gt;

and the light aberration formula ({{equationNote|4e}}):&lt;ref group=R&gt;Einstein (1905), § 7&lt;/ref&gt;

:&lt;math&gt;\cos\varphi'=\frac{\cos\varphi-\frac{v}{V}}{1-\frac{v}{V}\cos\varphi}&lt;/math&gt;

=== {{anchor|Minkowski}} Minkowski (1907–1908) – Spacetime ===
====Imaginary Lorentz transformation====
{{See also|History of Topics in Special Relativity/Lorentz transformation (imaginary)|label 1=History of Lorentz transformations via imaginary orthogonal transformation}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)|label 1=History of Lorentz transformations via hyperbolic functions}}
The work on the principle of relativity by Lorentz, Einstein, [[w:Max Planck|Planck]], together with Poincaré's four-dimensional approach, were further elaborated and combined with the [[w:hyperboloid model]] by [[w:Hermann Minkowski]] in 1907 and 1908.&lt;ref group=R&gt;Minkowski (1907/15), pp. 927ff&lt;/ref&gt;&lt;ref group=R&gt;Minkowski (1907/08), pp. 53ff&lt;/ref&gt; Minkowski particularly reformulated electrodynamics in a four-dimensional way ([[w:Minkowski spacetime]]).&lt;ref&gt;Walter (1999a), (1999b), (2018)&lt;/ref&gt; For instance, he wrote ''x, y, z, it'' in the form ''x&lt;sub&gt;1&lt;/sub&gt;, x&lt;sub&gt;2&lt;/sub&gt;, x&lt;sub&gt;3&lt;/sub&gt;, x&lt;sub&gt;4&lt;/sub&gt;''. By defining ψ as the angle of rotation around the ''z''-axis, the Lorentz transformation ({{equationNote|4b}}-A) assumes the form (with ''c''=1):&lt;ref group=R name=mink1&gt;Minkowski (1907/08), p. 59&lt;/ref&gt;

:&lt;math&gt;\begin{align}x'_{1} &amp; =x_{1}\\
x'_{2} &amp; =x_{2}\\
x'_{3} &amp; =x_{3}\cos i\psi+x_{4}\sin i\psi\\
x'_{4} &amp; =-x_{3}\sin i\psi+x_{4}\cos i\psi\\
\cos i\psi &amp; =\frac{1}{\sqrt{1-q^{2}}}
\end{align}
&lt;/math&gt;

Even though Minkowski used the imaginary number iψ, he for once&lt;ref group=R name=mink1 /&gt; directly used the [[w:tangens hyperbolicus]] in the equation for velocity

:&lt;math&gt;-i\tan i\psi=\frac{e^{\psi}-e^{-\psi}}{e^{\psi}+e^{-\psi}}=q&lt;/math&gt; with &lt;math&gt;\psi=\frac{1}{2}\ln\frac{1+q}{1-q}&lt;/math&gt;.

Minkowski's expression can also by written as ψ=atanh(q) and was later called [[w:rapidity]].

===={{anchor|Minkowski2}} Vector representation==== 
Minkowski wrote the Lorentz transformation ({{equationNote|4f}}) in vectorial form for the special case of directions being only parallel (&lt;math&gt;\mathfrak{r_{v}}&lt;/math&gt;) or perpendicular (&lt;math&gt;\mathfrak{r_{\bar{v}}}&lt;/math&gt;) to the velocity:&lt;ref group=R&gt;Minkowski (1907/08), pp. 62-63&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\mathfrak{r'_{v}}=\frac{\mathfrak{r_{v}}-qt}{\sqrt{1-q^{2}}},\quad\mathfrak{r'_{\bar{v}}}=\mathfrak{r_{\bar{v}}},\quad t'=\frac{-q\mathfrak{r_{v}}+t}{\sqrt{1-q^{2}}}\\
\mathfrak{r_{v}}=\frac{\mathfrak{r'_{v}}+qt'}{\sqrt{1-q^{2}}},\quad\mathfrak{r_{\bar{v}}}=\mathfrak{r'_{\bar{v}}},\quad t=\frac{q\mathfrak{r'_{v}}+t'}{\sqrt{1-q^{2}}}\\
\left[\mathfrak{r}=\left(x,y,z\right)=\left(\mathfrak{r_{v}},\mathfrak{r_{\bar{v}}}\right),\ |\mathfrak{v}|=q\right]
\end{matrix}&lt;/math&gt;

===={{anchor|Minkowski3}} Matrix representation====
Minkowski used matrices in order to write the [[../Lorentz transformation (general)#math_1a|E:general Lorentz transformation '''(1a)''']], of which boost matrix ({{equationNote|4h}}) is a special case:&lt;ref group=R&gt;Minkowski (1907/08), pp. 65–66, 81–82&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{1}^{\prime2}+x_{2}^{\prime2}+x_{3}^{\prime2}+x_{4}^{\prime2}\\
\left(x_{1}^{\prime}=x',\ x_{2}^{\prime}=y',\ x_{3}^{\prime}=z',\ x_{4}^{\prime}=it'\right)\\
-x^{2}-y^{2}-z^{2}+t^{2}=-x^{\prime2}-y^{\prime2}-z^{\prime2}+t^{\prime2}\\
\hline x_{h}=\alpha_{h1}x_{1}^{\prime}+\alpha_{h2}x_{2}^{\prime}+\alpha_{h3}x_{3}^{\prime}+\alpha_{h4}x_{4}^{\prime}\\
\mathrm{A}=\mathrm{\left|\begin{matrix}\alpha_{11}, &amp; \alpha_{12}, &amp; \alpha_{13}, &amp; \alpha_{14}\\
\alpha_{21}, &amp; \alpha_{22}, &amp; \alpha_{23}, &amp; \alpha_{24}\\
\alpha_{31}, &amp; \alpha_{32}, &amp; \alpha_{33}, &amp; \alpha_{34}\\
\alpha_{41}, &amp; \alpha_{42}, &amp; \alpha_{43}, &amp; \alpha_{44}
\end{matrix}\right|,\ \begin{align}\bar{\mathrm{A}}\mathrm{A} &amp; =1\\
\left(\det \mathrm{A}\right)^{2} &amp; =1\\
\det \mathrm{A} &amp; =1\\
\alpha_{44} &amp; &gt;0
\end{align}
}
\end{matrix}&lt;/math&gt;

===={{anchor|Minkowski4}} Minkowski diagram====
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Apo|label 1=History of Lorentz transformations in general - Apollonius}}
Minkowski (1908/09) introduced the [[w:Minkowski diagram]] as a graphical representation of the Lorentz transformation, which became a standard tool in textbooks and research articles on relativity:&lt;ref group=R&gt;Minkowski (1908/09), p. 77&lt;/ref&gt;

[[File:Minkowski1.png|center|thumb|400px|Original spacetime diagram by Minkowski in 1908.]]

==={{anchor|Frank}} Frank, Varicak (1909-10) – Hyperbolic functions===
{{Main|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Frank|label 1=History of Lorentz transformations via hyperbolic functions § Frank}}
{{Main|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Varicak|label 1=History of Lorentz transformations via hyperbolic functions § Varicak}}

==={{Anchor|Sommerfeld}} Sommerfeld (1909) – Spherical trigonometry===
{{Main|History of Topics in Special Relativity/Lorentz transformation (imaginary)#Sommerfeld|label 1=History of Lorentz transformations via imaginary orthogonal transformations § Sommerfeld}}

=== {{anchor|Herglotz1}} Herglotz (1909/10) – Möbius transformation and squeeze mappings===
{{Main|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Herglotz1|label 1=History of Lorentz transformations via Möbius transformations § Herglotz}}
{{Main|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Herglotz|label 1=History of Lorentz transformations via squeeze mappings § Herglotz}}

==={{anchor|Plummer}} Plummer, Gruner (1910-21) – Trigonometric Lorentz boosts===
{{Main|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Plummer|label 1=History of Lorentz transformations via trigonometric functions § Plummer}}
{{Main|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Gruner|label 1=History of Lorentz transformations via trigonometric functions § Gruner}}

=== {{anchor|Ignatowski}} Ignatowski (1910) ===

While earlier derivations and formulations of the Lorentz transformation relied from the outset on optics, electrodynamics, or the invariance of the speed of light, [[w:Vladimir Ignatowski]] (1910) showed that it is possible to use the principle of relativity (and related [[w:Group theory|group theoretical]] principles) alone, in order to derive the following transformation between two inertial frames:&lt;ref group=R&gt;Ignatowski (1910), pp. 973–974&lt;/ref&gt;&lt;ref group=R&gt;Ignatowski (1910/11ab)&lt;/ref&gt;

:&lt;math&gt;\begin{align}dx' &amp; =p\ dx-pq\ dt\\
dt' &amp; =-pqn\ dx+p\ dt\\
p &amp; =\frac{1}{\sqrt{1-q^{2}n}}
\end{align}
&lt;/math&gt;

The variable ''n'' can be seen as a space-time constant whose value has to be determined by experiment or taken from a known physical law such as electrodynamics. For that purpose, Ignatowski used the above-mentioned Heaviside ellipsoid representing a contraction of electrostatic fields by ''x''/γ in the direction of motion. It can be seen that this is only consistent with Ignatowski's transformation when ''n=1/c''&lt;sup&gt;2&lt;/sup&gt;, resulting in ''p''=γ and the Lorentz transformation ({{equationNote|4b}}). With ''n''=0, no length changes arise and the Galilean transformation follows. Ignatowski's method was further developed and improved by [[w:Philipp Frank]] and [[w:Hermann Rothe]] (1911, 1912),&lt;ref group=R&gt;Frank &amp; Rothe (1911), pp. 825ff; (1912), p. 750ff.&lt;/ref&gt; with various authors developing similar methods in subsequent years.&lt;ref name=baccetti&gt;Baccetti (2011), see references 1–25 therein.&lt;/ref&gt;

==={{anchor|klein}} Klein, Noether, Conway, Silberstein (1908-11) – Biquaternions===
{{Main|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Noether|label 1=History of Lorentz transformations via Quaternions § Klein and Noether}}
{{Main|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Conway|label 1=History of Lorentz transformations via Quaternions § Conway and Silberstein}}

==={{anchor|Herglotz2}} Ignatowski, Herglotz, Tamaki (1910-11) – Vector transformation===
[[w:Vladimir Ignatowski]] (1910, published 1911) defined the vectorial velocity addition ({{equationNote|4g}}) as well as general Lorentz boost ({{equationNote|4f}}) as&lt;ref group=R&gt;Ignatowski (1910/11a), p. 23; (1910/11b), p. 22&lt;/ref&gt; 

:&lt;math&gt;\begin{matrix}\begin{matrix}\mathfrak{v} =\frac{\mathfrak{v}'+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{v}'+pq\mathfrak{c}_{0}}{p\left(1+nq\mathfrak{c}_{0}\mathfrak{v}'\right)} &amp; \left|\begin{align}\mathfrak{A}' &amp; =\mathfrak{A}+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{A}-pqb\mathfrak{c}_{0}\\
b' &amp; =pb-pqn\mathfrak{A}\mathfrak{c}_{0}\\
\\
\mathfrak{A} &amp; =\mathfrak{A}'+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{A}'+pqb'\mathfrak{c}_{0}\\
b &amp; =pb'+pqn\mathfrak{A}'\mathfrak{c}_{0}
\end{align}
\right.\end{matrix}\\
\left[\mathfrak{v}=\mathbf{u},\ \mathfrak{A}=\mathbf{x},\ b=t,\ \mathfrak{c}_{0}=\frac{\mathbf{v}}{v},\ p=\gamma,\ n=\frac{1}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

An equivalent transformation was given by [[w:Gustav Herglotz]] (1911)&lt;ref group=R&gt;Herglotz (1911), p. 497&lt;/ref&gt; using '''v'''=''(v&lt;sub&gt;x&lt;/sub&gt;, v&lt;sub&gt;y&lt;/sub&gt;, v&lt;sub&gt;z&lt;/sub&gt;)'' and '''r'''=''(x, y, z)'':

:&lt;math&gt;\begin{align}x^{0} &amp; =x+\alpha u(ux+vy+wz)-\beta ut\\
y^{0} &amp; =y+\alpha v(ux+vy+wz)-\beta vt\\
z^{0} &amp; =z+\alpha w(ux+vy+wz)-\beta wt\\
t^{0} &amp; =-\beta(ux+vy+wz)+\beta t\\
 &amp; \alpha=\frac{1}{\sqrt{1-s^{2}}\left(1+\sqrt{1-s^{2}}\right)},\ \beta=\frac{1}{\sqrt{1-s^{2}}}
\end{align}
&lt;/math&gt;

Kajuro Tamaki (1911) represented ({{equationNote|4g}}) as follows (as his paper was based on a 4-vector calculus, Tamaki's schematic is not representing a matrix despite looking very similar to the boost matrix in ({{equationNote|4h}})):&lt;ref group=R&gt;Tamaki (1911), pp. 143-144&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{array}{c|c|c|c|c}
 &amp; x'_{1} &amp; x'_{2} &amp; x'_{3} &amp; x'_{4}\\
\hline x_{1} &amp; 1+l^{2}\left(\cos\psi-1\right) &amp; lm\left(\cos\psi-1\right) &amp; ln\left(\cos\psi-1\right) &amp; l\sin\psi\\
\hline x_{2} &amp; lm\left(\cos\psi-1\right) &amp; 1+m^{2}\left(\cos\psi-1\right) &amp; mn\left(\cos\psi-1\right) &amp; m\sin\psi\\
\hline x_{3} &amp; ln\left(\cos\psi-1\right) &amp; mn\left(\cos\psi-1\right) &amp; 1+n^{2}\left(\cos\psi-1\right) &amp; n\sin\psi\\
\hline x_{4} &amp; -l\sin\psi &amp; -m\sin\psi &amp; -n\sin\psi &amp; \cos\psi
\end{array}\\
\hline \psi=i\varphi,\ -i\tan i\varphi=-\frac{v}{c},\ \cos i\varphi=\frac{1}{\sqrt{1-(v/c)^{2}}}=\beta,\ -\sin i\varphi=\frac{i(v/c)}{\sqrt{1-(v/c)^{2}}}=i\beta(v/c)\\
\hline \mathbf{r}=\mathbf{r}'+(\beta-1)\mathbf{v}_{1}(\mathbf{v}_{1}\mathbf{r}')+\beta\mathbf{v}t'\\
\mathbf{r}'=\mathbf{r}+(\beta-1)\mathbf{v}_{1}(\mathbf{v}_{1}\mathbf{r})-\beta\mathbf{v}t
\end{matrix}&lt;/math&gt;

==={{anchor|Hahn}} Hahn (1912) – Matrix transformation ===
{{See|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Hahn|label 1=History of Lorentz transformations via Quaternions § Hahn}}
Elaborating on [[#Minkowski3|Minkowski's (1907/8)]] matrix representation of the Lorentz transformations, Emil Hahn (1912) used matrix calculus in order to define the Lorentz boost for arbitrary directions (including the exponential form of the boost matrix) in line with ({{equationNote|4h}}), using imaginary rapidity &lt;math&gt;i\psi&lt;/math&gt; and imaginary time &lt;math&gt;x_4 =i\omega t&lt;/math&gt;:&lt;ref group=R&gt;Hahn (1912), pp. 30-32 and p. 36.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\boldsymbol{x}'-\boldsymbol{x}'_{0}=\mathbb{I}_{-\mathbf{c}}(u)\boldsymbol{x}\\
\hline \begin{align}\mathbb{I}_{-\mathbf{c}}(u) &amp; =\mathbb{J}\mathbb{G}_{\mathbf{c}}(u)\mathbb{J}^{-1} &amp; (7,p.30)\\
 &amp; =\left(\begin{matrix}\mathbf{E}+(r-1)\mathbf{c}\overset{\perp}{\mathbf{c}}; &amp; \frac{iur(u)}{\omega}\mathbf{c}\\
-\frac{iur(u)}{\omega}\overset{\perp}{\mathbf{c}}; &amp; r(u)
\end{matrix}\right) &amp; (7,p.30)\\
 &amp; =\left(\begin{matrix}\mathbf{E}+(\cos i\psi-1)\mathbf{c}\overset{\perp}{\mathbf{c}}; &amp; \sin i\psi\mathbf{c}\\
-\sin i\psi\mathbf{c}; &amp; \cos i\psi
\end{matrix}\right) &amp; (8,p.30)\\
 &amp; =\left(\begin{matrix}1+(\cos i\psi-1)c_{1}c_{1} &amp; (\cos i\psi-1)c_{1}c_{2} &amp; (\cos i\psi-1)c_{1}c_{3} &amp; \sin i\psi\,c_{1}\\
(\cos i\psi-1)c_{2}c_{1} &amp; 1+(\cos i\psi-1)c_{2}c_{2} &amp; (\cos i\psi-1)c_{2}c_{3} &amp; \sin i\psi\,c_{2}\\
(\cos i\psi-1)c_{3}c_{1} &amp; (\cos i\psi-1)c_{3}c_{2} &amp; 1+(\cos i\psi-1)c_{3}c_{3} &amp; \sin i\psi\,c_{3}\\
-\sin i\psi\,c_{1} &amp; -\sin i\psi\,c_{2} &amp; -\sin i\psi\,c_{3} &amp; 1+(\cos i\psi-1)
\end{matrix}\right) &amp; (1,p.36)\\
\hline \mathbb{I}_{-\mathbf{c}}(u) &amp; =\mathbb{E}-\sin i\psi\left\langle \mathbf{0},\mathbf{c}\right\rangle +(1-\cos i\psi)\left\langle \mathbf{0},\mathbf{c}\right\rangle ^{2} &amp; (10,p.30)\\
\mathbb{I}_{\mathbf{c}}(u) &amp; =e^{\left\langle \mathbf{0},\mathbf{c}\right\rangle i\psi} &amp; (p.31)
\end{align}
\\
\hline \boldsymbol{x}=\left(\begin{matrix}x_{1} &amp; 0 &amp; 0 &amp; 0\\
x_{2} &amp; 0 &amp; 0 &amp; 0\\
x_{3} &amp; 0 &amp; 0 &amp; 0\\
x_{4} &amp; 0 &amp; 0 &amp; 0
\end{matrix}\right),\ \boldsymbol{x}'=\left(\begin{matrix}x_{1}^{\prime} &amp; 0 &amp; 0 &amp; 0\\
x_{2}^{\prime} &amp; 0 &amp; 0 &amp; 0\\
x_{3}^{\prime} &amp; 0 &amp; 0 &amp; 0\\
x_{4}^{\prime} &amp; 0 &amp; 0 &amp; 0
\end{matrix}\right),\ \mathbb{J}=\left(\begin{matrix}1 &amp; 0 &amp; 0 &amp; 0\\
0 &amp; 1 &amp; 0 &amp; 0\\
0 &amp; 0 &amp; 1 &amp; 0\\
0 &amp; 0 &amp; 0 &amp; i\omega
\end{matrix}\right),\ \mathbb{G}=\left(\begin{matrix}r_{11} &amp; r_{12} &amp; r_{13} &amp; p_{1}\\
r_{21} &amp; r_{22} &amp; r_{23} &amp; p_{2}\\
r_{31} &amp; r_{32} &amp; r_{33} &amp; p_{3}\\
q_{1} &amp; q_{2} &amp; q_{3} &amp; r
\end{matrix}\right)\\
r(u)=\frac{1}{\sqrt{1-\frac{u^{2}}{\omega^{2}}}}=\cos i\psi,\ |\mathbb{I}|=1,\ \left\langle \mathbf{0},\mathbf{c}\right\rangle =\left(\begin{matrix}0 &amp; 0 &amp; 0 &amp; c_{1}\\
0 &amp; 0 &amp; 0 &amp; c_{2}\\
0 &amp; 0 &amp; 0 &amp; c_{3}\\
c_{1} &amp; c_{2} &amp; c_{3} &amp; 0
\end{matrix}\right)
\end{matrix}&lt;/math&gt;

==={{anchor|Borel}} Borel (1913–14) – Cayley–Hermite parameter===
{{Main|History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)#Borel|label 1=History of Lorentz transformations via Cayley–Hermite transformation § Borel}}

==={{anchor|Born}} Born (1921) – Squeeze mapping===
{{Main|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Born|label 1=History of Lorentz transformations via squeeze mappings § Born}}

==References==
===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|abra1905}}
*{{#section:History of Topics in Special Relativity/relsource|buch08}}
*{{#section:History of Topics in Special Relativity/relsource|cohn04a}}
*{{#section:History of Topics in Special Relativity/relsource|cohn04b}}
*{{#section:History of Topics in Special Relativity/relsource|einst05elek}}
*{{#section:History of Topics in Special Relativity/relsource|fra11}}
*{{#section:History of Topics in Special Relativity/relsource|fra12}}
*{{#section:History of Topics in Special Relativity/relsource|gans05}}
*{{#section:History of Topics in Special Relativity/relsource|grun21a}}
*{{#section:History of Topics in Special Relativity/relsource|grun21b}}
*{{#section:History of Topics in Special Relativity/relsource|heav89}}
*{{#section:History of Topics in Special Relativity/relsource|hahn}}
*{{#section:History of Topics in Special Relativity/relsource|herg11ela}}
*{{#section:History of Topics in Special Relativity/relsource|igna10}}
*{{#section:History of Topics in Special Relativity/relsource|ignat10prin2}}
*{{#section:History of Topics in Special Relativity/relsource|lar97}}
*{{#section:History of Topics in Special Relativity/relsource|lar29}}
*{{#section:History of Topics in Special Relativity/relsource|lar00}}
*{{#section:History of Topics in Special Relativity/relsource|lar04a}}
*{{#section:History of Topics in Special Relativity/relsource|lar04b}}
*{{#section:History of Topics in Special Relativity/relsource|lor92elek}}
*{{#section:History of Topics in Special Relativity/relsource|lor92b}}
*{{#section:History of Topics in Special Relativity/relsource|lor95}}
*{{#section:History of Topics in Special Relativity/relsource|lor99}}
*{{#section:History of Topics in Special Relativity/relsource|lor04}}
*{{#section:History of Topics in Special Relativity/relsource|lor16}}
*{{#section:History of Topics in Special Relativity/relsource|mink07a}}
*{{#section:History of Topics in Special Relativity/relsource|mink07b}}
*{{#section:History of Topics in Special Relativity/relsource|mink08}}
*{{#section:History of Topics in Special Relativity/relsource|poi00}}
*{{#section:History of Topics in Special Relativity/relsource|poi04}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05a}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05b}}
*{{#section:History of Topics in Special Relativity/relsource|sea97}}
*{{#section:History of Topics in Special Relativity/relsource|tamaki11b}}
*{{#section:History of Topics in Special Relativity/relsource|thom89}}
*{{#section:History of Topics in Special Relativity/relsource|voi87}}
*{{#section:History of Topics in Special Relativity/relsource|wien04}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L4}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
      <sha1>ltxj4a4dn16357zxu2x3xg5itpp64vy</sha1>
    </revision>
    <revision>
      <id>2721181</id>
      <parentid>2721180</parentid>
      <timestamp>2025-07-10T07:53:32Z</timestamp>
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        <username>D.H</username>
        <id>52339</id>
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      <text bytes="57312" sha1="5uanm6sjjda11908ip5buso76nocjjh" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via velocity==
===Boosts===
In the [[w:theory of relativity]], Lorentz transformations exhibit the symmetry of [[w:Minkowski spacetime]] by using a constant ''c'' as the [[w:speed of light]], and a parameter ''v'' as the relative [[w:velocity]] between two [[w:inertial reference frames]]. The corresponding formulas are identical to [[../Lorentz transformation (hyperbolic)|E:Lorentz transformations via hyperbolic functions]] introduced long before relativity was developed. In particular, the hyperbolic angle &lt;math&gt;\eta&lt;/math&gt; can be interpreted as the velocity related [[w:rapidity]] &lt;math&gt;\tanh\eta=\beta=v/c&lt;/math&gt;, so that &lt;math&gt;\gamma=\cosh\eta&lt;/math&gt; is the [[w:Lorentz factor]], &lt;math&gt;\beta\gamma=\sinh\eta&lt;/math&gt; the [[w:proper velocity]], &lt;math&gt;u'=c\tanh q&lt;/math&gt; the velocity of another object, &lt;math&gt;u=c\tanh(q+\eta)&lt;/math&gt; the [[w:velocity-addition formula]], thus transformation [[../Lorentz transformation (hyperbolic)#math_3b|E:'''(3b)''']] becomes:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}\gamma-x_{1}\beta\gamma\\
x_{1}^{\prime} &amp; =-x_{0}\beta\gamma+x_{1}\gamma\\
\\
x_{0} &amp; =x_{0}^{\prime}\gamma+x_{1}^{\prime}\beta\gamma\\
x_{1} &amp; =x_{0}^{\prime}\beta\gamma+x_{1}^{\prime}\gamma
\end{align}
\left|{\scriptstyle \begin{align}\beta^{2}\gamma^{2}-\gamma^{2} &amp; =-1 &amp; (a)\\
\gamma^{2}-\beta^{2}\gamma^{2} &amp; =1 &amp; (b)\\
\frac{\beta\gamma}{\gamma} &amp; =\beta &amp; (c)\\
\frac{1}{\sqrt{1-\beta^{2}}} &amp; =\gamma &amp; (d)\\
\frac{\beta}{\sqrt{1-\beta^{2}}} &amp; =\beta\gamma &amp; (e)\\
\frac{u'+v}{1+\frac{u'v}{c^{2}}} &amp; =u &amp; (f)
\end{align}
}\right.
\end{matrix}&lt;/math&gt;|{{equationRef|4a}}}}

Written in four dimensions by setting &lt;math&gt;x_{0}=ct,\ x_{1}=x&lt;/math&gt; and adding  &lt;math&gt;y,z&lt;/math&gt; the familiar form follows

{{NumBlk|:|&lt;math&gt;(A)\quad\begin{matrix}-c^{2}t^{2}+x^{2}+y^{2}+z^{2}=-c^{2}t^{\prime2}+x^{\prime2}+y^{\prime2}+z^{\prime2}\\
\hline \left.\begin{align}t' &amp; =\gamma\left(t-x\frac{v}{c^{2}}\right)\\
x' &amp; =\gamma(x-vt)\\
y' &amp; =y\\
z' &amp; =z
\end{align}
\right|\begin{align}t &amp; =\gamma\left(t'+x\frac{v}{c^{2}}\right)\\
x &amp; =\gamma(x'+vt')\\
y &amp; =y'\\
z &amp; =z'
\end{align}
\end{matrix}&lt;/math&gt;

or in matrix notation:

&lt;math&gt;(B)\quad\begin{matrix}\mathbf{x}'=\begin{bmatrix}\gamma &amp; -\beta\gamma &amp; 0 &amp; 0\\
-\beta\gamma &amp; \gamma &amp; 0 &amp; 0\\
0 &amp; 0 &amp; 1 &amp; 0\\
0 &amp; 0 &amp; 0 &amp; 1
\end{bmatrix}\cdot\mathbf{x};\quad\mathbf{x}=\begin{bmatrix}\gamma &amp; \beta\gamma &amp; 0 &amp; 0\\
\beta\gamma &amp; \gamma &amp; 0 &amp; 0\\
0 &amp; 0 &amp; 1 &amp; 0\\
0 &amp; 0 &amp; 0 &amp; 1
\end{bmatrix}\cdot\mathbf{x}'\\
\det\begin{bmatrix}\gamma &amp; -\beta\gamma\\
-\beta\gamma &amp; \gamma
\end{bmatrix}=1
\end{matrix}&lt;/math&gt;

or in terms of &lt;math&gt;ct,x&lt;/math&gt; as squeeze mapping in line with [[../Lorentz transformation (hyperbolic)#math_3c|E:'''(3c)''']]:

&lt;math&gt;(C)\quad\begin{matrix}uw=-x_{0}^{2}+x_{1}^{2}=u'w'=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}u' &amp; =ku &amp; u &amp; =\frac{1}{k}u'\\
\Rightarrow x'-ct' &amp; =\sqrt{\frac{c+v}{c-v}}\left(x-ct\right) &amp; x-ct &amp; =\sqrt{\frac{c-v}{c+v}}\left(x'-ct'\right)\\
\\
w' &amp; =\frac{1}{k}w &amp; w &amp; =kw'\\
\Rightarrow x'+ct' &amp; =\sqrt{\frac{c-v}{c+v}}\left(x+ct\right) &amp; x+ct &amp; =\sqrt{\frac{c+v}{c-v}}\left(x'+ct'\right)
\end{align}
\\
\hline k=\sqrt{\frac{c+v}{c-v}}
\end{matrix}&lt;/math&gt;|{{equationRef|4b}}}}

Transformations analogous to (A) have been introduced by [[#Voigt|Voigt (1887)]] in terms of an incompressible medium, and by [[#Lorentz1|Lorentz (1892, 1895)]] who analyzed [[w:Maxwell's equations]], they were completed by [[#Larmor|Larmor (1897, 1900)]] and [[#Lorentz2|Lorentz (1899, 1904)]], and brought into their modern form by [[#Poincare3|Poincaré (1905)]] who gave the transformation the name of Lorentz.&lt;ref&gt;Miller (1981), chapter 1&lt;/ref&gt; Eventually, [[#Einstein|Einstein (1905)]] showed in his development of [[w:special relativity]] that the transformations follow from the [[w:principle of relativity]] and constant light speed alone by modifying the traditional concepts of space and time, without requiring a [[w:Lorentz ether theory|mechanical aether]] in contradistinction to Lorentz and Poincaré.&lt;ref&gt;Miller (1981), chapter 4–7&lt;/ref&gt; [[#Minkowski|Minkowski (1907–1908)]] used them to argue that space and time are inseparably connected as [[w:spacetime]]. 

The matrix form (B) is a special case of the general boost matrix given by [[#Hahn|Hahn (1912)]] in terms of imaginary time, while variant (C) for arbitrary ''k'' was given by many authors (see [[../Lorentz transformation (squeeze)|E:Lorentz transformations via squeeze mappings]]) with the choice equivalent to &lt;math&gt;k=\sqrt{\tfrac{c+v}{c-v}}&lt;/math&gt; given by [[#Born|Born (1921)]].

===Velocity addition and aberration===
In exact analogy to Beltrami coordinates in equation [[../Lorentz transformation (hyperbolic)#math_3e|E:'''(3e)''']], one can substitute &lt;math&gt;\left[\tfrac{u_{x}}{c},\ \tfrac{u_{y}}{c},\ \tfrac{u_{z}}{c}\right]=\left[\tfrac{x}{ct},\ \tfrac{y}{ct},\ \tfrac{z}{ct}\right]&lt;/math&gt; in ({{equationNote|4b}}-A), producing the Lorentz transformation of velocities (or [[w:velocity addition formula]]):

{{NumBlk|:|&lt;math&gt;\begin{align}u_{x}^{\prime} &amp; =\frac{-c^{2}\sinh\eta+u_{x}c\cosh\eta}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{x}-c\tanh\eta}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{x}-v}{1-\frac{v}{c^{2}}u{}_{x}}\\
u_{y}^{\prime} &amp; =\frac{cu_{y}}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{y}\sqrt{1-\tanh^{2}\eta}}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{y}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u{}_{x}}\\
u_{z}^{\prime} &amp; =\frac{cu_{y}}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{z}\sqrt{1-\tanh^{2}\eta}}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{z}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u{}_{x}}\\
\\
\hline \\
u_{x} &amp; =\frac{c^{2}\sinh\eta+u_{x}^{\prime}c\cosh\eta}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{x}^{\prime}+c\tanh\eta}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{x}^{\prime}+v}{1+\frac{v}{c^{2}}u_{x}^{\prime}}\\
u_{y} &amp; =\frac{cy'}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{y}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{y}^{\prime}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u_{x}^{\prime}}\\
u_{z} &amp; =\frac{cz'}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{z}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{z}^{\prime}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u_{x}^{\prime}}
\end{align}&lt;/math&gt;|{{equationRef|4c}}}}

By restriction to velocities in the &lt;math&gt;\left[x,y\right]&lt;/math&gt; plane and using trigonometric and hyperbolic identities as in equation [[../Lorentz transformation (hyperbolic)#math_3f|E:'''(3f)''']], it becomes the hyperbolic law of cosines:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix} &amp; \begin{matrix}u^{2}=u_{x}^{2}+u_{y}^{2}\\
u'^{2}=u_{x}^{\prime2}+u_{y}^{\prime2}
\end{matrix}\left|{\scriptstyle \begin{align}u_{x}=u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+\frac{v}{c^{2}}u'\cos\alpha'}, &amp; u_{x}^{\prime}=u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-\frac{v}{c^{2}}u\cos\alpha}\\
u_{y}=u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u'\cos\alpha'}, &amp; u_{y}^{\prime}=u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u\cos\alpha}\\
\frac{u_{y}}{u_{x}}=\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{u'\cos\alpha'+v}, &amp; \frac{u_{y}^{\prime}}{u_{x}^{\prime}}=\tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{u\cos\alpha-v}
\end{align}
}\right.\\
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(\frac{vu'\sin\alpha'}{c}\right){}^{2}}}{1+\frac{v}{c^{2}}u'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(\frac{vu\sin\alpha}{c}\right){}^{2}}}{1-\frac{v}{c^{2}}u\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\frac{u^{\prime2}}{c^{2}}}}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{1}{\sqrt{1-\frac{u^{2}}{c^{2}}}}-\frac{v/c}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{u/c}{\sqrt{1-\frac{u^{2}}{c^{2}}}}\cos\alpha\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha
\end{matrix}&lt;/math&gt;|{{equationRef|4d}}}}

and by further setting ''u=u′=c'' one gets the well known [[../Lorentz transformation (hyperbolic)#math_3g|E:Kepler formulas '''(3g)''']], which  express the relativistic [[w:aberration of light]]:&lt;ref&gt;Pauli (1921), pp. 562; 565–566&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}\cos\alpha=\frac{\cos\alpha'+\frac{v}{c}}{1+\frac{v}{c}\cos\alpha'},\ \sin\alpha=\frac{\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c}\cos\alpha'},\ \tan\alpha=\frac{\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{\cos\alpha'+\frac{v}{c}},\ \tan\frac{\alpha}{2}=\sqrt{\frac{c-v}{c+v}}\tan\frac{\alpha'}{2}\\
\cos\alpha'=\frac{\cos\alpha-\frac{v}{c}}{1-\frac{v}{c}\cos\alpha},\ \sin\alpha'=\frac{\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c}\cos\alpha},\ \tan\alpha'=\frac{\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{\cos\alpha-\frac{v}{c}},\ \tan\frac{\alpha'}{2}=\sqrt{\frac{c+v}{c-v}}\tan\frac{\alpha}{2}
\end{matrix} &lt;/math&gt;|{{equationRef|4e}}}}

Formulas ({{equationNote|4c}}, {{equationNote|4d}}) were given by [[#Einstein|Einstein (1905)]] and [[#Poincare3|Poincaré (1905/06)]], while the relations to the spherical and hyperbolic law of cosines were given by [[#Sommerfeld|Sommerfeld (1909)]] and [[#Frank|Varićak (1910)]]. The aberration formula for cos(α) was given by [[#Einstein|Einstein (1905)]].&lt;ref group=R name=plum&gt;Plummer (1910), pp. 258-259: After deriving the relativistic expressions for the aberration angles φ' and φ, Plummer remarked on p. 259: ''Another geometrical representation is obtained by assimilating φ' to the eccentric and φ to the true anomaly in an ellipse whose eccentricity is v/U = sin β.''&lt;/ref&gt;&lt;ref name=robin&gt;Robinson (1990), chapter 3-4, analyzed the relation between &quot;Kepler's formula&quot; and the &quot;physical velocity addition formula&quot; in special relativity.&lt;/ref&gt;

===Lorentz transformation in arbitrary directions===
Lorentz boosts for arbitrary directions&lt;ref&gt;Møller (1952/55), Chapter II, § 18&lt;/ref&gt; in line with [[../Lorentz transformation (general)#math_1a|E:general Lorentz transformation '''(1a)''']] are in vector notation

{{NumBlk|:|&lt;math&gt;\begin{align}t' &amp; =\gamma\left(t-\frac{v\mathbf{n}\cdot\mathbf{r}}{c^{2}}\right)\\
\mathbf{r}' &amp; =\mathbf{r}+(\gamma-1)(\mathbf{r}\cdot\mathbf{n})\mathbf{n}-\gamma tv\mathbf{n}
\end{align}
&lt;/math&gt;|{{equationRef|4f}}}}

and the vectorial velocity addition formula in line with [[../Lorentz transformation (general)#math_1b|E:general Lorentz transformation '''(1b)''']] follows by:

{{NumBlk|:|&lt;math&gt;\mathbf{u}'=\frac{1}{1+\frac{\mathbf{v}\cdot\mathbf{u}}{c^{2}}}\left[\frac{\mathbf{u}}{\gamma_{\mathbf{v}}}+\mathbf{v}+\frac{1}{c^{2}}\frac{\gamma_{\mathbf{v}}}{\gamma_{\mathbf{v}}+1}(\mathbf{u}\cdot\mathbf{v})\mathbf{v}\right]&lt;/math&gt;|{{equationRef|4g}}}}

The special case of parallel and perpendicular directions in ({{equationNote|4f}}) was given by [[#Minkowski2|Minkowski (1907/8)]] while the complete transformation was formulated by [[#Herglotz2|Ignatowski (1910), Herglotz (1911), Tamaki (1911)]]. General velocity addition ({{equationNote|4g}}) was given in equivalent form by [[#Herglotz2|Ignatowski (1910)]].

Rewritten in matrix notation, the general Lorentz boost has the form:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\
\hline \begin{align}\mathbf{g} &amp; =\begin{pmatrix}\gamma &amp; -\gamma\beta n_{x} &amp; -\gamma\beta n_{y} &amp; -\gamma\beta n_{z}\\
-\gamma\beta n_{x} &amp; 1+(\gamma-1)n_{x}^{2} &amp; (\gamma-1)n_{x}n_{y} &amp; (\gamma-1)n_{x}n_{z}\\
-\gamma\beta n_{y} &amp; (\gamma-1)n_{y}n_{x} &amp; 1+(\gamma-1)n_{y}^{2} &amp; (\gamma-1)n_{y}n_{z}\\
-\gamma\beta n_{z} &amp; (\gamma-1)n_{z}n_{x} &amp; (\gamma-1)n_{z}n_{y} &amp; 1+(\gamma-1)n_{z}^{2}
\end{pmatrix}\end{align}
\\
\left[\mathbf{n}=\frac{\mathbf{v}}{v}\right]
\end{matrix}\,&lt;/math&gt;|{{equationRef|4h}}}}

While [[#Minkowski3|Minkowski (1907/8)]] formulated the matrix form of Lorentz transformations in general terms, he didn't explicitly express the velocity related components of the general boost matrix. A complete representation of ({{equationNote|4h}}) was given by [[#Hahn|Hahn (1912)]].

===Other formulations===
Important contributions to the mathematical understanding of the Lorentz transformation of space and time also include: [[#Minkowski|Minkowski (1907–1908)]] as well as [[#Frank|Frank (1909) and Varićak (1910)]] showed the relation to imaginary and hyperbolic functions, [[#Herglotz1|Herglotz (1909/10)]] used exponential squeeze mappings and Möbius transformations, [[#Ignatowski|Ignatowski (1910)]] didn't use the light speed postulate, [[#klein|Klein and Noether (1908-11) as well as Conway and Silberstein (1911)]] used Biquaternions, [[#Plummer|Plummer (1910) and Gruner (1921)]] used trigonometric Lorentz boosts, [[#Borel|Borel (1913–14)]] used Cayley-Hermite parameter.

==Historical notation==
==={{anchor|Voigt}} Voigt (1887) ===

[[w:Woldemar Voigt]] (1887)&lt;ref group=R&gt;Voigt (1887), p. 45&lt;/ref&gt; developed a transformation in connection with the [[w:Doppler effect]] and an incompressible medium, being in modern notation:&lt;ref&gt;Miller (1981), 114–115&lt;/ref&gt;&lt;ref name=pais&gt;Pais (1982), Kap. 6b&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}\xi_{1} &amp; =x_{1}-\varkappa t\\
\eta_{1} &amp; =y_{1}q\\
\zeta_{1} &amp; =z_{1}q\\
\tau &amp; =t-\frac{\varkappa x_{1}}{\omega^{2}}\\
q &amp; =\sqrt{1-\frac{\varkappa^{2}}{\omega^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =\frac{y}{\gamma}\\
z^{\prime} &amp; =\frac{z}{\gamma}\\
t^{\prime} &amp; =t-\frac{vx}{c^{2}}\\
\frac{1}{\gamma} &amp; =\sqrt{1-\frac{v^{2}}{c^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

If the right-hand sides of his equations are multiplied by γ they are the modern Lorentz transformation ({{equationNote|4b}}). In Voigt's theory the speed of light is invariant, but his transformations mix up a relativistic boost together with a rescaling of space-time. Optical phenomena in free space are [[w:Scale invariance|scale]], [[w:Conformal map|conformal]] (using the factor λ discussed [[#Lorsph|above]]), and [[w:Lorentz covariance|Lorentz invariant]], so the combination is invariant too.&lt;ref name=pais /&gt; For instance, Lorentz transformations can be extended by using &lt;math&gt;l=\sqrt{\lambda}&lt;/math&gt;:&lt;ref group=R&gt;Lorentz (1915/16), p. 197&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=\gamma l\left(x-vt\right),\quad y^{\prime}=ly,\quad z^{\prime}=lz,\quad t^{\prime}=\gamma l\left(t-x\frac{v}{c^{2}}\right)&lt;/math&gt;.

''l''=1/γ gives the Voigt transformation, ''l''=1 the Lorentz transformation. But scale transformations are not a symmetry of all the laws of nature, only of electromagnetism, so these transformations cannot be used to formulate a [[w:principle of relativity]] in general. It was demonstrated by Poincaré and Einstein that one has to set ''l''=1 in order to make the above transformation symmetric and to form a group as required by the relativity principle, therefore the Lorentz transformation is the only viable choice.

Voigt sent his 1887 paper to Lorentz in 1908,&lt;ref&gt;Voigt's transformations and the beginning of the relativistic revolution, Ricardo Heras, arXiv:1411.2559 [https://arxiv.org/abs/1411.2559]&lt;/ref&gt; and that was acknowledged in 1909: {{Quote|In a paper &quot;Über das Doppler'sche Princip&quot;, published in 1887 (Gött. Nachrichten, p. 41) and which to my regret has escaped my notice all these years, Voigt has applied to equations of the form (7) (§ 3 of this book) [namely &lt;math&gt;\Delta\Psi-\tfrac{1}{c^{2}}\tfrac{\partial^{2}\Psi}{\partial t^{2}}=0&lt;/math&gt;] a transformation equivalent to the formulae (287) and (288) [namely &lt;math&gt;x^{\prime}=\gamma l\left(x-vt\right),\ y^{\prime}=ly,\ z^{\prime}=lz,\ t^{\prime}=\gamma l\left(t-\tfrac{v}{c^{2}}x\right)&lt;/math&gt;]. The idea of the transformations used above (and in § 44) might therefore have been borrowed from Voigt and the proof that it does not alter the form of the equations for the ''free'' ether is contained in his paper.&lt;ref group=R&gt;Lorentz (1915/16), p. 198&lt;/ref&gt;}}

Also [[w:Hermann Minkowski]] said in 1908 that the transformations which play the main role in the principle of relativity were first examined by Voigt in 1887. Voigt responded in the same paper by saying that his theory was based on an elastic theory of light, not an electromagnetic one. However, he concluded that some results were actually the same.&lt;ref group=R&gt;Bucherer (1908), p. 762&lt;/ref&gt;

==={{anchor|Heaviside}} Heaviside (1888), Thomson (1889), Searle (1896)===

In 1888, [[w:Oliver Heaviside]]&lt;ref group=R&gt;Heaviside (1888), p. 324&lt;/ref&gt; investigated the properties of [[w:Relativistic electromagnetism|charges in motion]] according to Maxwell's electrodynamics. He calculated, among other things, anisotropies in the electric field of moving bodies represented by this formula:&lt;ref&gt;Brown (2003)&lt;/ref&gt;

:&lt;math&gt;\mathrm{E}=\left(\frac{q\mathrm{r}}{r^{2}}\right)\left(1-\frac{v^{2}\sin^{2}\theta}{c^{2}}\right)^{-3/2}&lt;/math&gt;.

Consequently, [[w:Joseph John Thomson]] (1889)&lt;ref group=R&gt;Thomson (1889), p. 12&lt;/ref&gt; found a way to substantially simplify calculations concerning moving charges by using the following mathematical transformation (like other authors such as Lorentz or Larmor, also Thomson implicitly used the [[w:Galilean transformation]] ''z-vt'' in his equation&lt;ref name=mil /&gt;):

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}z &amp; =\left\{ 1-\frac{\omega^{2}}{v^{2}}\right\} ^{\frac{1}{2}}z'\end{align}
\right| &amp; \begin{align}z^{\ast}=z-vt &amp; =\frac{z'}{\gamma}\end{align}
\end{matrix}&lt;/math&gt;

Thereby, [[w:inhomogeneous electromagnetic wave equation]]s are transformed into a [[w:Poisson equation]].&lt;ref name=mil&gt;Miller (1981), 98–99&lt;/ref&gt; Eventually, [[w:George Frederick Charles Searle]]&lt;ref group=R&gt;Searle (1886), p. 333&lt;/ref&gt; noted in (1896) that Heaviside's expression leads to a deformation of electric fields which he called &quot;Heaviside-Ellipsoid&quot; of [[w:axial ratio]]

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align} &amp; \sqrt{\alpha}:1:1\\
\alpha= &amp; 1-\frac{u^{2}}{v^{2}}
\end{align}
\right| &amp; \begin{align} &amp; \frac{1}{\gamma}:1:1\\
\frac{1}{\gamma^{2}} &amp; =1-\frac{v^{2}}{c^{2}}
\end{align}
\end{matrix}&lt;/math&gt;&lt;ref name=mil /&gt;

=== {{anchor|Lorentz1}} Lorentz (1892, 1895) ===

In order to explain the [[w:aberration of light]] and the result of the [[w:Fizeau experiment]] in accordance with [[w:Maxwell's equations]], Lorentz in 1892 developed a model (&quot;[[w:Lorentz ether theory]]&quot;) in which the aether is completely motionless, and the speed of light in the aether is constant in all directions. In order to calculate the optics of moving bodies, Lorentz introduced the following quantities to transform from the aether system into a moving system (it's unknown whether he was influenced by Voigt, Heaviside, and Thomson)&lt;ref group=R&gt;Lorentz (1892a), p. 141&lt;/ref&gt;&lt;ref name=milf&gt;Miller (1982), 1.4 &amp; 1.5&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}\mathfrak{x} &amp; =\frac{V}{\sqrt{V^{2}-p^{2}}}x\\
t' &amp; =t-\frac{\varepsilon}{V}\mathfrak{x}\\
\varepsilon &amp; =\frac{p}{\sqrt{V^{2}-p^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
t^{\prime} &amp; =t-\frac{\gamma^{2}vx^{\ast}}{c^{2}}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)\\
\gamma\frac{v}{c} &amp; =\frac{v}{\sqrt{c^{2}-v^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

where ''x&lt;sup&gt;*&lt;/sup&gt;'' is the [[w:Galilean transformation]] ''x-vt''. Except the additional γ in the time transformation, this is the complete Lorentz transformation ({{equationNote|4b}}).&lt;ref name=milf /&gt; While ''t'' is the &quot;true&quot; time for observers resting in the aether, ''t′'' is an auxiliary variable only for calculating processes for moving systems. It is also important that Lorentz and later also Larmor formulated this transformation in two steps. At first an implicit Galilean transformation, and later the expansion into the &quot;fictitious&quot; electromagnetic system with the aid of the Lorentz transformation. In order to explain the negative result of the [[w:Michelson–Morley experiment]], he (1892b)&lt;ref group=R&gt;Lorentz (1892b), p. 141&lt;/ref&gt; introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introduced [[w:length contraction]] in his theory (without proof as he admitted). The same hypothesis was already made by [[w:George FitzGerald]] in 1889 based on Heaviside's work. While length contraction was a real physical effect for Lorentz, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation.

In 1895, Lorentz further elaborated on his theory and introduced the &quot;theorem of corresponding states&quot;. This theorem states that a moving observer (relative to the ether) in his &quot;fictitious&quot; field makes the same observations as a resting observers in his &quot;real&quot; field for velocities to first order in ''v/c''. Lorentz showed that the dimensions of electrostatic systems in the ether and a moving frame are connected by this transformation:&lt;ref group=R&gt;Lorentz (1895), p. 37&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =x^{\prime}\sqrt{1-\frac{\mathfrak{p}^{2}}{V^{2}}}\\
y &amp; =y^{\prime}\\
z &amp; =z^{\prime}\\
t &amp; =t^{\prime}
\end{align}
\right| &amp; \begin{align}x^{\ast}=x-vt &amp; =\frac{x^{\prime}}{\gamma}\\
y &amp; =y^{\prime}\\
z &amp; =z^{\prime}\\
t &amp; =t^{\prime}
\end{align}
\end{matrix}&lt;/math&gt;

For solving optical problems Lorentz used the following transformation, in which the modified time variable was called &quot;local time&quot; ({{lang-de|Ortszeit}}) by him:&lt;ref group=R&gt;Lorentz (1895), p. 49 for local time and p. 56 for spatial coordinates.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =\mathrm{x}-\mathfrak{p}_{x}t\\
y &amp; =\mathrm{y}-\mathfrak{p}_{y}t\\
z &amp; =\mathrm{z}-\mathfrak{p}_{z}t\\
t^{\prime} &amp; =t-\frac{\mathfrak{p}_{x}}{V^{2}}x-\frac{\mathfrak{p}_{y}}{V^{2}}y-\frac{\mathfrak{p}_{z}}{V^{2}}z
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-v_{x}t\\
y^{\prime} &amp; =y-v_{y}t\\
z^{\prime} &amp; =z-v_{z}t\\
t^{\prime} &amp; =t-\frac{v_{x}}{c^{2}}x'-\frac{v_{y}}{c^{2}}y'-\frac{v_{z}}{c^{2}}z'
\end{align}
\end{matrix}&lt;/math&gt;

With this concept Lorentz could explain the [[w:Doppler effect]], the [[w:aberration of light]], and the [[w:Fizeau experiment]].&lt;ref&gt;Janssen (1995), 3.1&lt;/ref&gt;

=== {{anchor|Larmor}} Larmor (1897, 1900) ===

In 1897, Larmor extended the work of Lorentz and derived the following transformation&lt;ref group=R&gt;Larmor (1897), p. 229&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x_{1} &amp; =x\varepsilon^{\frac{1}{2}}\\
y_{1} &amp; =y\\
z_{1} &amp; =z\\
t^{\prime} &amp; =t-vx/c^{2}\\
dt_{1} &amp; =dt^{\prime}\varepsilon^{-\frac{1}{2}}\\
\varepsilon &amp; =\left(1-v^{2}/c^{2}\right)^{-1}
\end{align}
\right| &amp; \begin{align}x_{1} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
y_{1} &amp; =y\\
z_{1} &amp; =z\\
t^{\prime} &amp; =t-\frac{vx^{\ast}}{c^{2}}=t-\frac{v(x-vt)}{c^{2}}\\
dt_{1} &amp; =\frac{dt^{\prime}}{\gamma}\\
\gamma^{2} &amp; =\frac{1}{1-\frac{v^{2}}{c^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

Larmor noted that if it is assumed that the constitution of molecules is electrical then the FitzGerald–Lorentz contraction is a consequence of this transformation, explaining the [[w:Michelson–Morley experiment]]. It's notable that Larmor was the first who recognized that some sort of [[w:time dilation]] is a consequence of this transformation as well, because &quot;individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio 1/γ&quot;.&lt;ref&gt;Darrigol (2000), Chap. 8.5&lt;/ref&gt;&lt;ref&gt;Macrossan (1986)&lt;/ref&gt; Larmor wrote his electrodynamical equations and transformations neglecting terms of higher order than ''(v/c)''&lt;sup&gt;2&lt;/sup&gt; – when his 1897 paper was reprinted in 1929, Larmor added the following comment in which he described how they can be made valid to all orders of ''v/c'':&lt;ref group=R&gt;Larmor (1897/1929), p. 39&lt;/ref&gt;

{{Quote|Nothing need be neglected: the transformation is ''exact'' if ''v/c''&lt;sup&gt;2&lt;/sup&gt; is replaced by ''εv/c''&lt;sup&gt;2&lt;/sup&gt; in the equations and also in the change following from ''t'' to ''t′'', as is worked out in ''Aether and Matter'' (1900), p. 168, and as Lorentz found it to be in 1904, thereby stimulating the modern schemes of intrinsic relational relativity.}}

In line with that comment, in his book Aether and Matter published in 1900, Larmor used a modified local time ''t″=t′-εvx′/c&lt;sup&gt;2&lt;/sup&gt;'' instead of the 1897 expression ''t′=t-vx/c&lt;sup&gt;2&lt;/sup&gt;'' by replacing ''v/c''&lt;sup&gt;2&lt;/sup&gt; with ''εv/c''&lt;sup&gt;2&lt;/sup&gt;, so that ''t″'' is now identical to the one given by Lorentz in 1892, which he combined with a Galilean transformation for the ''x′, y′, z′, t′'' coordinates:&lt;ref group=R&gt;Larmor (1900), p. 168&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t\\
t^{\prime\prime} &amp; =t^{\prime}-\varepsilon vx^{\prime}/c^{2}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t\\
t^{\prime\prime}=t^{\prime}-\frac{\gamma^{2}vx^{\prime}}{c^{2}} &amp; =\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor ''(v/c)''&lt;sup&gt;2&lt;/sup&gt;, and so he sought the transformations which were &quot;accurate to second order&quot; (as he put it). Thus he wrote the final transformations (where ''x′=x-vt'' and ''t″'' as given above) as:&lt;ref group=R&gt;Larmor (1900), p. 174&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x_{1} &amp; =\varepsilon^{\frac{1}{2}}x^{\prime}\\
y_{1} &amp; =y^{\prime}\\
z_{1} &amp; =z^{\prime}\\
dt_{1} &amp; =\varepsilon^{-\frac{1}{2}}dt^{\prime\prime}=\varepsilon^{-\frac{1}{2}}\left(dt^{\prime}-\frac{v}{c^{2}}\varepsilon dx^{\prime}\right)\\
t_{1} &amp; =\varepsilon^{-\frac{1}{2}}t^{\prime}-\frac{v}{c^{2}}\varepsilon^{\frac{1}{2}}x^{\prime}
\end{align}
\right| &amp; \begin{align}x_{1} &amp; =\gamma x^{\prime}=\gamma(x-vt)\\
y_{1} &amp; =y'=y\\
z_{1} &amp; =z'=z\\
dt_{1} &amp; =\frac{dt^{\prime\prime}}{\gamma}=\frac{1}{\gamma}\left(dt^{\prime}-\frac{\gamma^{2}vdx^{\prime}}{c^{2}}\right)=\gamma\left(dt-\frac{vdx}{c^{2}}\right)\\
t_{1} &amp; =\frac{t^{\prime}}{\gamma}-\frac{\gamma vx^{\prime}}{c^{2}}=\gamma\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

by which he arrived at the complete Lorentz transformation ({{equationNote|4b}}). Larmor showed that Maxwell's equations were invariant under this two-step transformation, &quot;to second order in ''v/c''&quot; – it was later shown by Lorentz (1904) and Poincaré (1905) that they are indeed invariant under this transformation to all orders in ''v/c''.

Larmor gave credit to Lorentz in two papers published in 1904, in which he used the term &quot;Lorentz transformation&quot; for Lorentz's first order transformations of coordinates and field configurations:

{{Quote|p. 583: [..] Lorentz's transformation for passing from the field of activity of a stationary electrodynamic material system to that of one moving with uniform velocity of translation through the aether.&lt;br /&gt; p. 585: [..] the Lorentz transformation has shown us what is not so immediately obvious [..]&lt;ref group=R&gt;Larmor (1904a), p. 583, 585&lt;/ref&gt; &lt;br /&gt; p. 622: [..] the transformation first developed by Lorentz: namely, each point in space is to have its own origin from which time is measured, its &quot;local time&quot; in Lorentz's phraseology, and then the values of the electric and magnetic vectors [..] at all points in the aether between the molecules in the system at rest, are the same as those of the vectors [..] at the corresponding points in the convected system at the same local times.&lt;ref group=R&gt;Larmor (1904b), p. 622&lt;/ref&gt;}}

=== {{anchor|Lorentz2}} Lorentz (1899, 1904) ===

Also Lorentz extended his theorem of corresponding states in 1899. First he wrote a transformation equivalent to the one from 1892 (again, ''x''* must be replaced by ''x-vt''):&lt;ref group=R&gt;Lorentz (1899), p. 429&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}x\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t-\frac{\mathfrak{p}_{x}}{V^{2}-\mathfrak{p}_{x}^{2}}x
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t-\frac{\gamma^{2}vx^{\ast}}{c^{2}}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Then he introduced a factor ε of which he said he has no means of determining it, and modified his transformation as follows (where the above value of ''t′'' has to be inserted):&lt;ref group=R&gt;Lorentz (1899), p. 439&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =\frac{\varepsilon}{k}x^{\prime\prime}\\
y &amp; =\varepsilon y^{\prime\prime}\\
z &amp; =\varepsilon x^{\prime\prime}\\
t^{\prime} &amp; =k\varepsilon t^{\prime\prime}\\
k &amp; =\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\ast}=x-vt &amp; =\frac{\varepsilon}{\gamma}x^{\prime\prime}\\
y &amp; =\varepsilon y^{\prime\prime}\\
z &amp; =\varepsilon z^{\prime\prime}\\
t^{\prime}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right) &amp; =\gamma\varepsilon t^{\prime\prime}\\
\gamma &amp; =\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
\end{align}
\end{matrix}&lt;/math&gt;

This is equivalent to the complete Lorentz transformation ({{equationNote|4b}}) when solved for ''x″'' and ''t″'' and with ε=1. Like Larmor, Lorentz noticed in 1899&lt;ref group=R&gt;Lorentz (1899), p. 442&lt;/ref&gt; also some sort of time dilation effect in relation to the frequency of oscillating electrons ''&quot;that in ''S'' the time of vibrations be ''kε'' times as great as in ''S&lt;sub&gt;0&lt;/sub&gt;''&quot;'', where ''S&lt;sub&gt;0&lt;/sub&gt;'' is the aether frame.&lt;ref&gt;Janssen (1995), Kap. 3.3&lt;/ref&gt;

In 1904 he rewrote the equations in the following form by setting ''l''=1/ε (again, ''x''* must be replaced by ''x-vt''):&lt;ref group=R&gt;Lorentz (1904), p. 812&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =klx\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t' &amp; =\frac{l}{k}t-kl\frac{w}{c^{2}}x
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma lx^{\ast}=\gamma l(x-vt)\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t^{\prime} &amp; =\frac{lt}{\gamma}-\frac{\gamma lvx^{\ast}}{c^{2}}=\gamma l\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Under the assumption that ''l=1'' when ''v''=0, he demonstrated that ''l=1'' must be the case at all velocities, therefore length contraction can only arise in the line of motion. So by setting the factor ''l'' to unity, Lorentz's transformations now assumed the same form as Larmor's and are now completed. Unlike Larmor, who restricted himself to show the covariance of Maxwell's equations to second order, Lorentz tried to widen its covariance to all orders in ''v/c''. He also derived the correct formulas for the velocity dependence of [[w:electromagnetic mass]], and concluded that the transformation formulas must apply to all forces of nature, not only electrical ones.&lt;ref group=R&gt;Lorentz (1904), p. 826&lt;/ref&gt; However, he didn't achieve full covariance of the transformation equations for charge density and velocity.&lt;ref&gt;Miller (1981), Chap. 1.12.2&lt;/ref&gt; When the 1904 paper was reprinted in 1913, Lorentz therefore added the following remark:&lt;ref&gt;Janssen (1995), Chap. 3.5.6&lt;/ref&gt;

{{Quote|One will notice that in this work the transformation equations of Einstein’s Relativity Theory have not quite been attained. [..] On this circumstance depends the clumsiness of many of the further considerations in this work.}}

Lorentz's 1904 transformation was cited and used by [[w:Alfred Bucherer]] in July 1904:&lt;ref group=R&gt;Bucherer, p. 129; Definition of s on p. 32&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=\sqrt{s}x,\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t'=\frac{t}{\sqrt{s}}-\sqrt{s}\frac{u}{v^{2}}x,\quad s=1-\frac{u^{2}}{v^{2}}&lt;/math&gt;

or by [[w:Wilhelm Wien]] in July 1904:&lt;ref group=R&gt;Wien (1904), p. 394&lt;/ref&gt;

:&lt;math&gt;x=kx',\quad y=y',\quad z=z',\quad t'=kt-\frac{v}{kc^{2}}x&lt;/math&gt;

or by [[w:Emil Cohn]] in November 1904 (setting the speed of light to unity):&lt;ref group=R&gt;Cohn (1904a), pp. 1296-1297&lt;/ref&gt;

:&lt;math&gt;x=\frac{x_{0}}{k},\quad y=y_{0},\quad z=z_{0},\quad t=kt_{0},\quad t_{1}=t_{0}-w\cdot r_{0},\quad k^{2}=\frac{1}{1-w^{2}}&lt;/math&gt;

or by [[w:Richard Gans]] in February 1905:&lt;ref group=R&gt;Gans (1905), p. 169&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=kx,\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t'=\frac{t}{k}-\frac{kwx}{c^{2}},\quad k^{2}=\frac{c^{2}}{c^{2}-w^{2}}&lt;/math&gt;

=== {{anchor|Poincare3}} Poincaré (1900, 1905) ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Poincare|label 1=History of Lorentz transformations in general § Poincaré}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Poincare2|label 1=History of Lorentz transformations via Möbius transformations § Poincaré}}

==== Local time ====

Neither Lorentz or Larmor gave a clear physical interpretation of the origin of local time. However, [[w:Henri Poincaré]] in 1900 commented on the origin of Lorentz's &quot;wonderful invention&quot; of local time.&lt;ref&gt;Darrigol (2005), Kap. 4&lt;/ref&gt; He remarked that it arose when clocks in a moving reference frame are synchronised by exchanging signals which are assumed to travel with the same speed &lt;math&gt;c&lt;/math&gt; in both directions, which lead to what is nowadays called [[w:relativity of simultaneity]], although Poincaré's calculation does not involve length contraction or time dilation.&lt;ref group=R&gt;Poincaré (1900), pp. 272–273&lt;/ref&gt; In order to synchronise the clocks here on Earth (the ''x*, t''* frame) a light signal from one clock (at the origin) is sent to another (at ''x''*), and is sent back. It's supposed that the Earth is moving with speed ''v'' in the ''x''-direction (= ''x''*-direction) in some rest system (''x, t'') (''i.e.'' the [[w:luminiferous aether]] system for Lorentz and Larmor). The time of flight outwards is

:&lt;math&gt;\delta t_{a}=\frac{x^{\ast}}{\left(c-v\right)}&lt;/math&gt;

and the time of flight back is

:&lt;math&gt;\delta t_{b}=\frac{x^{\ast}}{\left(c+v\right)}&lt;/math&gt;.

The elapsed time on the clock when the signal is returned is ''δt&lt;sub&gt;a&lt;/sub&gt;+δt&lt;sub&gt;b&lt;/sub&gt;'' and the time ''t*=(δt&lt;sub&gt;a&lt;/sub&gt;+δt&lt;sub&gt;b&lt;/sub&gt;)/2'' is ascribed to the moment when the light signal reached the distant clock. In the rest frame the time ''t=δt&lt;sub&gt;a&lt;/sub&gt;'' is ascribed to that same instant. Some algebra gives the relation between the different time coordinates ascribed to the moment of reflection. Thus

:&lt;math&gt;t^{\ast}=t-\frac{\gamma^{2}vx^{*}}{c^{2}}&lt;/math&gt;

identical to Lorentz (1892). By dropping the factor γ&lt;sup&gt;2&lt;/sup&gt; under the assumption that &lt;math&gt;\tfrac{v^{2}}{c^{2}}\ll1&lt;/math&gt;, Poincaré gave the result ''t*=t-vx*/c&lt;sup&gt;2&lt;/sup&gt;'', which is the form used by Lorentz in 1895.

Similar physical interpretations of local time were later given by [[w:Emil Cohn]] (1904)&lt;ref group=R&gt;Cohn (1904b), p. 1408&lt;/ref&gt; and [[w:Max Abraham]] (1905).&lt;ref group=R&gt;Abraham (1905), § 42&lt;/ref&gt;

==== Lorentz transformation ====

On June 5, 1905 (published June 9) Poincaré formulated transformation equations which are algebraically equivalent to those of Larmor and Lorentz and gave them the modern form ({{equationNote|4b}}):&lt;ref group=R&gt;Poincaré (1905), p. 1505&lt;/ref&gt;

:&lt;math&gt;\begin{align}x^{\prime} &amp; =kl(x+\varepsilon t)\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t' &amp; =kl(t+\varepsilon x)\\
k &amp; =\frac{1}{\sqrt{1-\varepsilon^{2}}}
\end{align}
&lt;/math&gt;.

Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name &quot;Lorentz transformation&quot;.&lt;ref&gt;Pais (1982), Chap. 6c&lt;/ref&gt;&lt;ref&gt;Katzir (2005), 280–288&lt;/ref&gt; Poincaré set the speed of light to unity, pointed out the group characteristics of the transformation by setting ''l''=1, and modified/corrected Lorentz's derivation of the equations of electrodynamics in some details in order to fully satisfy the principle of relativity, ''i.e.'' making them fully Lorentz covariant.&lt;ref&gt;Miller (1981), Chap. 1.14&lt;/ref&gt;

In July 1905 (published in January 1906)&lt;ref group=R&gt;Poincaré (1905/06), pp. 129ff&lt;/ref&gt; Poincaré showed in detail how the transformations and electrodynamic equations are a consequence of the [[w:principle of least action]]; he demonstrated in more detail the group characteristics of the transformation, which he called [[w:Lorentz group]], and he showed that the combination ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;+z&lt;sup&gt;2&lt;/sup&gt;-t&lt;sup&gt;2&lt;/sup&gt;'' is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing &lt;math&gt;ct\sqrt{-1}&lt;/math&gt; as a fourth imaginary coordinate, and he used an early form of [[w:four-vector]]s. He also formulated the velocity addition formula ({{equationNote|4c}}), which he had already derived in unpublished letters to Lorentz from May 1905:&lt;ref group=R&gt;Poincaré (1905/06), p. 144&lt;/ref&gt;

:&lt;math&gt;\xi'=\frac{\xi+\varepsilon}{1+\xi\varepsilon},\ \eta'=\frac{\eta}{k(1+\xi\varepsilon)}&lt;/math&gt;.

==={{anchor|Einstein}} Einstein (1905) – Special relativity===

On June 30, 1905 (published September 1905) Einstein published what is now called [[w:special relativity]] and gave a new derivation of the transformation, which was based only on the principle on relativity and the principle of the constancy of the speed of light. While Lorentz considered &quot;local time&quot; to be a mathematical stipulation device for explaining the Michelson-Morley experiment, Einstein showed that the coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference. For quantities of first order in ''v/c'' this was also done by Poincaré in 1900, while Einstein derived the complete transformation by this method. Unlike Lorentz and Poincaré who still distinguished between real time in the aether and apparent time for moving observers, Einstein showed that the transformations concern the nature of space and time.&lt;ref&gt;Miller (1981), Chap. 6&lt;/ref&gt;&lt;ref&gt;Pais (1982), Kap. 7&lt;/ref&gt;&lt;ref&gt;Darrigol (2005), Chap. 6&lt;/ref&gt;

The notation for this transformation is equivalent to Poincaré's of 1905 and ({{equationNote|4b}}), except that Einstein didn't set the speed of light to unity:&lt;ref group=R&gt;Einstein (1905), p. 902&lt;/ref&gt;

:&lt;math&gt;\begin{align}\tau &amp; =\beta\left(t-\frac{v}{V^{2}}x\right)\\
\xi &amp; =\beta(x-vt)\\
\eta &amp; =y\\
\zeta &amp; =z\\
\beta &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{V}\right)^{2}}}
\end{align}
&lt;/math&gt;

Einstein also defined the velocity addition formula ({{equationNote|4c}}, {{equationNote|4d}}):&lt;ref group=R&gt;Einstein (1905), § 5 and § 9&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=\frac{w_{\xi}+v}{1+\frac{vw_{\xi}}{V^{2}}}t,\ y=\frac{\sqrt{1-\left(\frac{v}{V}\right)^{2}}}{1+\frac{vw_{\xi}}{V^{2}}}w_{\eta}t\\
U^{2}=\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2},\ w^{2}=w_{\xi}^{2}+w_{\eta}^{2},\ \alpha=\operatorname{arctg}\frac{w_{y}}{w_{x}}\\
U=\frac{\sqrt{\left(v^{2}+w^{2}+2vw\cos\alpha\right)-\left(\frac{vw\sin\alpha}{V}\right)^{2}}}{1+\frac{vw\cos\alpha}{V^{2}}}
\end{matrix}\left|\begin{matrix}\frac{u_{x}-v}{1-\frac{u_{x}v}{V^{2}}}=u_{\xi}\\
\frac{u_{y}}{\beta\left(1-\frac{u_{x}v}{V^{2}}\right)}=u_{\eta}\\
\frac{u_{z}}{\beta\left(1-\frac{u_{x}v}{V^{2}}\right)}=u_{\zeta}
\end{matrix}\right.&lt;/math&gt;

and the light aberration formula ({{equationNote|4e}}):&lt;ref group=R&gt;Einstein (1905), § 7&lt;/ref&gt;

:&lt;math&gt;\cos\varphi'=\frac{\cos\varphi-\frac{v}{V}}{1-\frac{v}{V}\cos\varphi}&lt;/math&gt;

=== {{anchor|Minkowski}} Minkowski (1907–1908) – Spacetime ===
====Imaginary Lorentz transformation====
{{See also|History of Topics in Special Relativity/Lorentz transformation (imaginary)|label 1=History of Lorentz transformations via imaginary orthogonal transformation}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)|label 1=History of Lorentz transformations via hyperbolic functions}}
The work on the principle of relativity by Lorentz, Einstein, [[w:Max Planck|Planck]], together with Poincaré's four-dimensional approach, were further elaborated and combined with the [[w:hyperboloid model]] by [[w:Hermann Minkowski]] in 1907 and 1908.&lt;ref group=R&gt;Minkowski (1907/15), pp. 927ff&lt;/ref&gt;&lt;ref group=R&gt;Minkowski (1907/08), pp. 53ff&lt;/ref&gt; Minkowski particularly reformulated electrodynamics in a four-dimensional way ([[w:Minkowski spacetime]]).&lt;ref&gt;Walter (1999a), (1999b), (2018)&lt;/ref&gt; For instance, he wrote ''x, y, z, it'' in the form ''x&lt;sub&gt;1&lt;/sub&gt;, x&lt;sub&gt;2&lt;/sub&gt;, x&lt;sub&gt;3&lt;/sub&gt;, x&lt;sub&gt;4&lt;/sub&gt;''. By defining ψ as the angle of rotation around the ''z''-axis, the Lorentz transformation ({{equationNote|4b}}-A) assumes the form (with ''c''=1):&lt;ref group=R name=mink1&gt;Minkowski (1907/08), p. 59&lt;/ref&gt;

:&lt;math&gt;\begin{align}x'_{1} &amp; =x_{1}\\
x'_{2} &amp; =x_{2}\\
x'_{3} &amp; =x_{3}\cos i\psi+x_{4}\sin i\psi\\
x'_{4} &amp; =-x_{3}\sin i\psi+x_{4}\cos i\psi\\
\cos i\psi &amp; =\frac{1}{\sqrt{1-q^{2}}}
\end{align}
&lt;/math&gt;

Even though Minkowski used the imaginary number iψ, he for once&lt;ref group=R name=mink1 /&gt; directly used the [[w:tangens hyperbolicus]] in the equation for velocity

:&lt;math&gt;-i\tan i\psi=\frac{e^{\psi}-e^{-\psi}}{e^{\psi}+e^{-\psi}}=q&lt;/math&gt; with &lt;math&gt;\psi=\frac{1}{2}\ln\frac{1+q}{1-q}&lt;/math&gt;.

Minkowski's expression can also by written as ψ=atanh(q) and was later called [[w:rapidity]].

===={{anchor|Minkowski2}} Vector representation==== 
Minkowski wrote the Lorentz transformation ({{equationNote|4f}}) in vectorial form for the special case of directions being only parallel (&lt;math&gt;\mathfrak{r_{v}}&lt;/math&gt;) or perpendicular (&lt;math&gt;\mathfrak{r_{\bar{v}}}&lt;/math&gt;) to the velocity:&lt;ref group=R&gt;Minkowski (1907/08), pp. 62-63&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\mathfrak{r'_{v}}=\frac{\mathfrak{r_{v}}-qt}{\sqrt{1-q^{2}}},\quad\mathfrak{r'_{\bar{v}}}=\mathfrak{r_{\bar{v}}},\quad t'=\frac{-q\mathfrak{r_{v}}+t}{\sqrt{1-q^{2}}}\\
\mathfrak{r_{v}}=\frac{\mathfrak{r'_{v}}+qt'}{\sqrt{1-q^{2}}},\quad\mathfrak{r_{\bar{v}}}=\mathfrak{r'_{\bar{v}}},\quad t=\frac{q\mathfrak{r'_{v}}+t'}{\sqrt{1-q^{2}}}\\
\left[\mathfrak{r}=\left(x,y,z\right)=\left(\mathfrak{r_{v}},\mathfrak{r_{\bar{v}}}\right),\ |\mathfrak{v}|=q\right]
\end{matrix}&lt;/math&gt;

===={{anchor|Minkowski3}} Matrix representation====
Minkowski used matrices in order to write the [[../Lorentz transformation (general)#math_1a|E:general Lorentz transformation '''(1a)''']], of which boost matrix ({{equationNote|4h}}) is a special case:&lt;ref group=R&gt;Minkowski (1907/08), pp. 65–66, 81–82&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{1}^{\prime2}+x_{2}^{\prime2}+x_{3}^{\prime2}+x_{4}^{\prime2}\\
\left(x_{1}^{\prime}=x',\ x_{2}^{\prime}=y',\ x_{3}^{\prime}=z',\ x_{4}^{\prime}=it'\right)\\
-x^{2}-y^{2}-z^{2}+t^{2}=-x^{\prime2}-y^{\prime2}-z^{\prime2}+t^{\prime2}\\
\hline x_{h}=\alpha_{h1}x_{1}^{\prime}+\alpha_{h2}x_{2}^{\prime}+\alpha_{h3}x_{3}^{\prime}+\alpha_{h4}x_{4}^{\prime}\\
\mathrm{A}=\mathrm{\left|\begin{matrix}\alpha_{11}, &amp; \alpha_{12}, &amp; \alpha_{13}, &amp; \alpha_{14}\\
\alpha_{21}, &amp; \alpha_{22}, &amp; \alpha_{23}, &amp; \alpha_{24}\\
\alpha_{31}, &amp; \alpha_{32}, &amp; \alpha_{33}, &amp; \alpha_{34}\\
\alpha_{41}, &amp; \alpha_{42}, &amp; \alpha_{43}, &amp; \alpha_{44}
\end{matrix}\right|,\ \begin{align}\bar{\mathrm{A}}\mathrm{A} &amp; =1\\
\left(\det \mathrm{A}\right)^{2} &amp; =1\\
\det \mathrm{A} &amp; =1\\
\alpha_{44} &amp; &gt;0
\end{align}
}
\end{matrix}&lt;/math&gt;

===={{anchor|Minkowski4}} Minkowski diagram====
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Apo|label 1=History of Lorentz transformations in general - Apollonius}}
Minkowski (1908/09) introduced the [[w:Minkowski diagram]] as a graphical representation of the Lorentz transformation, which became a standard tool in textbooks and research articles on relativity:&lt;ref group=R&gt;Minkowski (1908/09), p. 77&lt;/ref&gt;

[[File:Minkowski1.png|center|thumb|400px|Original spacetime diagram by Minkowski in 1908.]]

==={{anchor|Frank}} Frank, Varicak (1909-10) – Hyperbolic functions===
{{Main|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Frank|label 1=History of Lorentz transformations via hyperbolic functions § Frank}}
{{Main|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Varicak|label 1=History of Lorentz transformations via hyperbolic functions § Varicak}}

==={{Anchor|Sommerfeld}} Sommerfeld (1909) – Spherical trigonometry===
{{Main|History of Topics in Special Relativity/Lorentz transformation (imaginary)#Sommerfeld|label 1=History of Lorentz transformations via imaginary orthogonal transformations § Sommerfeld}}

=== {{anchor|Herglotz1}} Herglotz (1909/10) – Möbius transformation and squeeze mappings===
{{Main|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Herglotz1|label 1=History of Lorentz transformations via Möbius transformations § Herglotz}}
{{Main|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Herglotz|label 1=History of Lorentz transformations via squeeze mappings § Herglotz}}

==={{anchor|Plummer}} Plummer, Gruner (1910-21) – Trigonometric Lorentz boosts===
{{Main|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Plummer|label 1=History of Lorentz transformations via trigonometric functions § Plummer}}
{{Main|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Gruner|label 1=History of Lorentz transformations via trigonometric functions § Gruner}}

=== {{anchor|Ignatowski}} Ignatowski (1910) ===

While earlier derivations and formulations of the Lorentz transformation relied from the outset on optics, electrodynamics, or the invariance of the speed of light, [[w:Vladimir Ignatowski]] (1910) showed that it is possible to use the principle of relativity (and related [[w:Group theory|group theoretical]] principles) alone, in order to derive the following transformation between two inertial frames:&lt;ref group=R&gt;Ignatowski (1910), pp. 973–974&lt;/ref&gt;&lt;ref group=R&gt;Ignatowski (1910/11ab)&lt;/ref&gt;

:&lt;math&gt;\begin{align}dx' &amp; =p\ dx-pq\ dt\\
dt' &amp; =-pqn\ dx+p\ dt\\
p &amp; =\frac{1}{\sqrt{1-q^{2}n}}
\end{align}
&lt;/math&gt;

The variable ''n'' can be seen as a space-time constant whose value has to be determined by experiment or taken from a known physical law such as electrodynamics. For that purpose, Ignatowski used the above-mentioned Heaviside ellipsoid representing a contraction of electrostatic fields by ''x''/γ in the direction of motion. It can be seen that this is only consistent with Ignatowski's transformation when ''n=1/c''&lt;sup&gt;2&lt;/sup&gt;, resulting in ''p''=γ and the Lorentz transformation ({{equationNote|4b}}). With ''n''=0, no length changes arise and the Galilean transformation follows. Ignatowski's method was further developed and improved by [[w:Philipp Frank]] and [[w:Hermann Rothe]] (1911, 1912),&lt;ref group=R&gt;Frank &amp; Rothe (1911), pp. 825ff; (1912), p. 750ff.&lt;/ref&gt; with various authors developing similar methods in subsequent years.&lt;ref name=baccetti&gt;Baccetti (2011), see references 1–25 therein.&lt;/ref&gt;

==={{anchor|klein}} Klein, Noether, Conway, Silberstein (1908-11) – Biquaternions===
{{Main|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Noether|label 1=History of Lorentz transformations via Quaternions § Klein and Noether}}
{{Main|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Conway|label 1=History of Lorentz transformations via Quaternions § Conway and Silberstein}}

==={{anchor|Herglotz2}} Ignatowski, Herglotz, Tamaki (1910-11) – Vector transformation===
[[w:Vladimir Ignatowski]] (1910, published 1911) defined the vectorial velocity addition ({{equationNote|4g}}) as well as general Lorentz boost ({{equationNote|4f}}) as&lt;ref group=R&gt;Ignatowski (1910/11a), p. 23; (1910/11b), p. 22&lt;/ref&gt; 

:&lt;math&gt;\begin{matrix}\begin{matrix}\mathfrak{v} =\frac{\mathfrak{v}'+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{v}'+pq\mathfrak{c}_{0}}{p\left(1+nq\mathfrak{c}_{0}\mathfrak{v}'\right)} &amp; \left|\begin{align}\mathfrak{A}' &amp; =\mathfrak{A}+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{A}-pqb\mathfrak{c}_{0}\\
b' &amp; =pb-pqn\mathfrak{A}\mathfrak{c}_{0}\\
\\
\mathfrak{A} &amp; =\mathfrak{A}'+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{A}'+pqb'\mathfrak{c}_{0}\\
b &amp; =pb'+pqn\mathfrak{A}'\mathfrak{c}_{0}
\end{align}
\right.\end{matrix}\\
\left[\mathfrak{v}=\mathbf{u},\ \mathfrak{A}=\mathbf{x},\ b=t,\ \mathfrak{c}_{0}=\frac{\mathbf{v}}{v},\ p=\gamma,\ n=\frac{1}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

An equivalent transformation was given by [[w:Gustav Herglotz]] (1911)&lt;ref group=R&gt;Herglotz (1911), p. 497&lt;/ref&gt; using '''v'''=''(v&lt;sub&gt;x&lt;/sub&gt;, v&lt;sub&gt;y&lt;/sub&gt;, v&lt;sub&gt;z&lt;/sub&gt;)'' and '''r'''=''(x, y, z)'':

:&lt;math&gt;\begin{align}x^{0} &amp; =x+\alpha u(ux+vy+wz)-\beta ut\\
y^{0} &amp; =y+\alpha v(ux+vy+wz)-\beta vt\\
z^{0} &amp; =z+\alpha w(ux+vy+wz)-\beta wt\\
t^{0} &amp; =-\beta(ux+vy+wz)+\beta t\\
 &amp; \alpha=\frac{1}{\sqrt{1-s^{2}}\left(1+\sqrt{1-s^{2}}\right)},\ \beta=\frac{1}{\sqrt{1-s^{2}}}
\end{align}
&lt;/math&gt;

Kajuro Tamaki (1911) represented ({{equationNote|4g}}) as follows (as his paper was based on a 4-vector calculus, Tamaki's schematic is not representing a matrix despite looking very similar to the boost matrix in ({{equationNote|4h}})):&lt;ref group=R&gt;Tamaki (1911), pp. 143-144&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{array}{c|c|c|c|c}
 &amp; x'_{1} &amp; x'_{2} &amp; x'_{3} &amp; x'_{4}\\
\hline x_{1} &amp; 1+l^{2}\left(\cos\psi-1\right) &amp; lm\left(\cos\psi-1\right) &amp; ln\left(\cos\psi-1\right) &amp; l\sin\psi\\
\hline x_{2} &amp; lm\left(\cos\psi-1\right) &amp; 1+m^{2}\left(\cos\psi-1\right) &amp; mn\left(\cos\psi-1\right) &amp; m\sin\psi\\
\hline x_{3} &amp; ln\left(\cos\psi-1\right) &amp; mn\left(\cos\psi-1\right) &amp; 1+n^{2}\left(\cos\psi-1\right) &amp; n\sin\psi\\
\hline x_{4} &amp; -l\sin\psi &amp; -m\sin\psi &amp; -n\sin\psi &amp; \cos\psi
\end{array}\\
\hline \psi=i\varphi,\ -i\tan i\varphi=-\frac{v}{c},\ \cos i\varphi=\frac{1}{\sqrt{1-(v/c)^{2}}}=\beta,\ -\sin i\varphi=\frac{i(v/c)}{\sqrt{1-(v/c)^{2}}}=i\beta(v/c)\\
\hline \mathbf{r}=\mathbf{r}'+(\beta-1)\mathbf{v}_{1}(\mathbf{v}_{1}\mathbf{r}')+\beta\mathbf{v}t'\\
\mathbf{r}'=\mathbf{r}+(\beta-1)\mathbf{v}_{1}(\mathbf{v}_{1}\mathbf{r})-\beta\mathbf{v}t
\end{matrix}&lt;/math&gt;

==={{anchor|Hahn}} Hahn (1912) – Matrix transformation ===
{{See|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Hahn|label 1=History of Lorentz transformations via Quaternions § Hahn}}
Elaborating on [[#Minkowski3|Minkowski's (1907/8)]] matrix representation of the Lorentz transformations, Emil Hahn (1912) used matrix calculus in order to define the Lorentz boost for arbitrary directions (including the exponential form of the boost matrix) in line with ({{equationNote|4h}}), using imaginary rapidity &lt;math&gt;i\psi&lt;/math&gt; and imaginary time &lt;math&gt;x_4 =i\omega t&lt;/math&gt;:&lt;ref group=R&gt;Hahn (1912), pp. 30-32 and p. 36.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\boldsymbol{x}'-\boldsymbol{x}'_{0}=\mathbb{I}_{-\mathbf{c}}(u)\boldsymbol{x}\\
\hline \begin{align}\mathbb{I}_{-\mathbf{c}}(u) &amp; =\mathbb{J}\mathbb{G}_{\mathbf{c}}(u)\mathbb{J}^{-1} &amp; (7,p.30)\\
 &amp; =\left(\begin{matrix}\mathbf{E}+(r-1)\mathbf{c}\overset{\perp}{\mathbf{c}}; &amp; \frac{iur(u)}{\omega}\mathbf{c}\\
-\frac{iur(u)}{\omega}\overset{\perp}{\mathbf{c}}; &amp; r(u)
\end{matrix}\right) &amp; (7,p.30)\\
 &amp; =\left(\begin{matrix}\mathbf{E}+(\cos i\psi-1)\mathbf{c}\overset{\perp}{\mathbf{c}}; &amp; \sin i\psi\mathbf{c}\\
-\sin i\psi\mathbf{c}; &amp; \cos i\psi
\end{matrix}\right) &amp; (8,p.30)\\
 &amp; =\left(\begin{matrix}1+(\cos i\psi-1)c_{1}c_{1} &amp; (\cos i\psi-1)c_{1}c_{2} &amp; (\cos i\psi-1)c_{1}c_{3} &amp; \sin i\psi\,c_{1}\\
(\cos i\psi-1)c_{2}c_{1} &amp; 1+(\cos i\psi-1)c_{2}c_{2} &amp; (\cos i\psi-1)c_{2}c_{3} &amp; \sin i\psi\,c_{2}\\
(\cos i\psi-1)c_{3}c_{1} &amp; (\cos i\psi-1)c_{3}c_{2} &amp; 1+(\cos i\psi-1)c_{3}c_{3} &amp; \sin i\psi\,c_{3}\\
-\sin i\psi\,c_{1} &amp; -\sin i\psi\,c_{2} &amp; -\sin i\psi\,c_{3} &amp; 1+(\cos i\psi-1)
\end{matrix}\right) &amp; (1,p.36)\\
\hline \mathbb{I}_{-\mathbf{c}}(u) &amp; =\mathbb{E}-\sin i\psi\left\langle \mathbf{0},\mathbf{c}\right\rangle +(1-\cos i\psi)\left\langle \mathbf{0},\mathbf{c}\right\rangle ^{2} &amp; (10,p.30)\\
\mathbb{I}_{\mathbf{c}}(u) &amp; =e^{\left\langle \mathbf{0},\mathbf{c}\right\rangle i\psi} &amp; (p.31)
\end{align}
\\
\hline \boldsymbol{x}=\left(\begin{matrix}x_{1} &amp; 0 &amp; 0 &amp; 0\\
x_{2} &amp; 0 &amp; 0 &amp; 0\\
x_{3} &amp; 0 &amp; 0 &amp; 0\\
x_{4} &amp; 0 &amp; 0 &amp; 0
\end{matrix}\right),\ \boldsymbol{x}'=\left(\begin{matrix}x_{1}^{\prime} &amp; 0 &amp; 0 &amp; 0\\
x_{2}^{\prime} &amp; 0 &amp; 0 &amp; 0\\
x_{3}^{\prime} &amp; 0 &amp; 0 &amp; 0\\
x_{4}^{\prime} &amp; 0 &amp; 0 &amp; 0
\end{matrix}\right),\ \mathbb{J}=\left(\begin{matrix}1 &amp; 0 &amp; 0 &amp; 0\\
0 &amp; 1 &amp; 0 &amp; 0\\
0 &amp; 0 &amp; 1 &amp; 0\\
0 &amp; 0 &amp; 0 &amp; i\omega
\end{matrix}\right),\ \mathbb{G}=\left(\begin{matrix}r_{11} &amp; r_{12} &amp; r_{13} &amp; p_{1}\\
r_{21} &amp; r_{22} &amp; r_{23} &amp; p_{2}\\
r_{31} &amp; r_{32} &amp; r_{33} &amp; p_{3}\\
q_{1} &amp; q_{2} &amp; q_{3} &amp; r
\end{matrix}\right)\\
r(u)=\frac{1}{\sqrt{1-\frac{u^{2}}{\omega^{2}}}}=\cos i\psi,\ |\mathbb{I}|=1,\ \left\langle \mathbf{0},\mathbf{c}\right\rangle =\left(\begin{matrix}0 &amp; 0 &amp; 0 &amp; c_{1}\\
0 &amp; 0 &amp; 0 &amp; c_{2}\\
0 &amp; 0 &amp; 0 &amp; c_{3}\\
c_{1} &amp; c_{2} &amp; c_{3} &amp; 0
\end{matrix}\right)
\end{matrix}&lt;/math&gt;

==={{anchor|Borel}} Borel (1913–14) – Cayley–Hermite parameter===
{{Main|History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)#Borel|label 1=History of Lorentz transformations via Cayley–Hermite transformation § Borel}}

==={{anchor|Born}} Born (1921) – Squeeze mapping===
{{Main|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Born|label 1=History of Lorentz transformations via squeeze mappings § Born}}

==References==
===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|abra1905}}
*{{#section:History of Topics in Special Relativity/relsource|buch08}}
*{{#section:History of Topics in Special Relativity/relsource|cohn04a}}
*{{#section:History of Topics in Special Relativity/relsource|cohn04b}}
*{{#section:History of Topics in Special Relativity/relsource|einst05elek}}
*{{#section:History of Topics in Special Relativity/relsource|fra11}}
*{{#section:History of Topics in Special Relativity/relsource|fra12}}
*{{#section:History of Topics in Special Relativity/relsource|gans05}}
*{{#section:History of Topics in Special Relativity/relsource|grun21a}}
*{{#section:History of Topics in Special Relativity/relsource|grun21b}}
*{{#section:History of Topics in Special Relativity/relsource|heav89}}
*{{#section:History of Topics in Special Relativity/relsource|hahn}}
*{{#section:History of Topics in Special Relativity/relsource|herg11ela}}
*{{#section:History of Topics in Special Relativity/relsource|igna10}}
*{{#section:History of Topics in Special Relativity/relsource|ignat10prin2}}
*{{#section:History of Topics in Special Relativity/relsource|lar97}}
*{{#section:History of Topics in Special Relativity/relsource|lar29}}
*{{#section:History of Topics in Special Relativity/relsource|lar00}}
*{{#section:History of Topics in Special Relativity/relsource|lar04a}}
*{{#section:History of Topics in Special Relativity/relsource|lar04b}}
*{{#section:History of Topics in Special Relativity/relsource|lor92elek}}
*{{#section:History of Topics in Special Relativity/relsource|lor92b}}
*{{#section:History of Topics in Special Relativity/relsource|lor95}}
*{{#section:History of Topics in Special Relativity/relsource|lor99}}
*{{#section:History of Topics in Special Relativity/relsource|lor04}}
*{{#section:History of Topics in Special Relativity/relsource|lor16}}
*{{#section:History of Topics in Special Relativity/relsource|mink07a}}
*{{#section:History of Topics in Special Relativity/relsource|mink07b}}
*{{#section:History of Topics in Special Relativity/relsource|mink08}}
*{{#section:History of Topics in Special Relativity/relsource|poi00}}
*{{#section:History of Topics in Special Relativity/relsource|poi04}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05a}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05b}}
*{{#section:History of Topics in Special Relativity/relsource|sea97}}
*{{#section:History of Topics in Special Relativity/relsource|tamaki11b}}
*{{#section:History of Topics in Special Relativity/relsource|thom89}}
*{{#section:History of Topics in Special Relativity/relsource|voi87}}
*{{#section:History of Topics in Special Relativity/relsource|wien04}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L4}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
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    <title>Freedom of the Press Foundation says...</title>
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      <timestamp>2025-07-09T14:01:51Z</timestamp>
      <contributor>
        <username>NicoScribe</username>
        <id>2360113</id>
      </contributor>
      <comment>Undo revisions by [[Special:Contributions/2603:6010:C900:4898:BDEC:F504:434A:3FC5|2603:6010:C900:4898:BDEC:F504:434A:3FC5]] ([[User talk:2603:6010:C900:4898:BDEC:F504:434A:3FC5|talk]]) test</comment>
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      <text bytes="19305" sha1="p50gqkripngi0enugcz8h3h5ytlb7e8" xml:space="preserve">:''This discusses an interview 2025-05-08 with Seth Stern&lt;ref name=Stern&gt;&lt;!--Seth Stern--&gt;{{cite Q|Q134333839}}&lt;/ref&gt; and Lauren Harper&lt;ref name=Harper&gt;&lt;!--Lauren Harper--&gt;{{cite Q|Q134371468}}&lt;/ref&gt; about the Freedom of the Press Foundation. A video and 29:00 mm:ss podcast excerpted from the interview will be added when available. The podcast will be released 2025-05-17 to the fortnightly &quot;Media &amp; Democracy&quot; show&lt;ref name=M&amp;D&gt;&lt;!--Media &amp; Democracy--&gt;{{cite Q|Q127839818}}&lt;/ref&gt; syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]&lt;ref&gt;&lt;!--Pacifica Radio Network--&gt;{{cite Q|Q2045587}}&lt;/ref&gt; Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].&lt;ref&gt;&lt;!--list of Pacifica Radio stations and affiliates--&gt;{{cite Q|Q6593294}}&lt;/ref&gt;''

:''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV&gt;The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs&lt;/ref&gt; and treating others with respect.&lt;ref name=AGF&gt;[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] is different: Contributors there are asked to [[Wikinews:Never assume|&quot;Don't assume things; be skeptical about everything.&quot;]] That's wise. However, we should still treat others with respect while being skeptical.&lt;/ref&gt;''

[[File:Freedom of the Press Foundation says.webm|thumb|2025-05-08 nterview with Seth Stern&lt;ref name=Stern/&gt; and Lauren Harper&lt;ref name=Harper/&gt;, Director of Advocacy and Daniel Ellsberg Chair on Government Secrecy, respectively, for Freedom of the Press Foundation]]
[[File:Freedom of the Press Foundation says.ogg|thumb|29:00 mm:ss of excerpts from a 2025-05-08 interview with Seth Stern and Lauren Harper of Freedom of the Press Foundation.]] 

Seth Stern,&lt;ref name=Stern/&gt; Director of Advocacy for Freedom of the Press Foundation, and Lauren Harper,&lt;ref name=Harper/&gt; their Daniel Ellsberg Chair on Government Secrecy, discuss their work with Spencer Graves.&lt;ref&gt;&lt;!--Spencer Graves--&gt;{{cite Q|Q56452480}}&lt;/ref&gt; Freedom of the Press Foundation works to protect journalists and their sources in several ways:&lt;ref&gt;&lt;!--Freedom of the Press Foundation --&gt;{{cite Q|Q5500827}}&lt;/ref&gt;
* [[w:SecureDrop|SecureDrop]]: They develop and maintain their open source whistleblower submission system to facilitate anonymous and secure communications between sources and journalists.&lt;ref&gt;&lt;!--Technology: Our open source software tools protect newsrooms, journalists, and their sources--&gt;{{cite Q|Q134334311}}&lt;/ref&gt; The project was begun in part by [[w:Aaron Swartz|Aaron Swartz]], who tragically killed himself under intense pressure from the FBI on questionable grounds.
* Digital security education for news organizations.&lt;ref&gt;&lt;!--Digital Security Education: Explore resources, training, and other services you can use to protect your work and your sources in the digital age.--&gt;{{cite Q|Q134335013}}&lt;/ref&gt;
* [[w:Freedom of the press in the United States#U.S. Press Freedom Tracker|U.S. Press Freedom tracker]], documenting attacks on journalists including assaults and arrests for activities that seemingly should be protected by the First Amendment.&lt;ref&gt;&lt;!--U.S. Press Freedom Tracker--&gt;{{cite Q|Q134335566}}, accessed 2025-05-01.&lt;/ref&gt; They documented 2,530 such attacks on secrecy, surveillance, and the rights of journalists and whistleblowers in the 8 years between 2017 and 2024. A third of those attacks were in the single year 2020, the last year of President Trump's first term.&lt;ref&gt;Click &quot;all time&quot; at &lt;!--https://pressfreedomtracker.us/--&gt;{{cite Q|Q134336764}}&lt;/ref&gt;
* Staying current on these issues.&lt;ref&gt;&lt;!-- The Latest: Mobilizing allies and the public to create tangible change for press freedom.--&gt;{{cite Q|Q134337247}}&lt;/ref&gt; 

One &quot;featured issue&quot; in the last category says, &quot;Reform Government Secrecy&quot;, claiming that, &quot;The U.S. classifies far too many secrets, obstructing democracy.&quot;&lt;ref&gt;&lt;!--Reform Government Secrecy: The U.S. classifies far too many secrets, obstructing democracy--&gt;{{cite Q|Q134337726}}&lt;/ref&gt; This includes &quot;‘The Classified Catalog’ launches to track secrecy news&quot;, numerous things the Trump administration has done since 2025-01-20 to erode &quot;the information environment in ways this country has never seen.&lt;ref&gt;&lt;!--‘The Classified Catalog’ launches to track secrecy news--&gt;{{cite Q|Q134387817}}&lt;/ref&gt; These steps changes include the following: 
* [[Trump ordered changes in public data|Deleted thousands of datasets from agency websites]].
* Closed agencies’ [[w:Freedom of Information Act (United States)|Freedom of Information Act]] offices.&lt;ref&gt;Harper (2025-04-13).&lt;/ref&gt; 
* Used disappearing messaging apps and failed to preserve government records.&lt;ref&gt;Harper (2025-03-30).&lt;/ref&gt;
* Ordered federal health agencies to stop communicating with the public.&lt;ref&gt;Klippenstein (2025).&lt;/ref&gt; 
* Gutted a key surveillance oversight board.&lt;ref&gt;Weissmann (2025).&lt;/ref&gt; 
* Mass-fired inspectors general.&lt;ref&gt;Harper (2025-01-28).&lt;/ref&gt; 
* Replaced independent, professional leadership at the National Archives with unqualified appointees.&lt;ref&gt;Harper (2025-02-20).&lt;/ref&gt;
* Attempted to illegally destroy agency records.&lt;ref&gt;Harper (2025-03-16).&lt;/ref&gt;

== US government secrecy ==

=== Complicity in nuclear proliferation === 

Graves asked about claims by [[w:Richard Barlow (intelligence analyst)|Richard Barlow]] that the US State Department had clandestinely supported illegal exports of &quot;dual use technologies&quot; to Pakistan, without which Pakistan would likely not have nuclear weapons today -- and North Korea got some of their nuclear technology from Pakistan. Harper ageed, noting that she had previously worked with the [[w:National Security Archive|National Security Archive]]. William Burr&lt;ref&gt;

https://nsarchive.gwu.edu/about/staff/dr-william-burr

&lt;/ref&gt; directs their &quot;Nuclear Vault&quot;, which contains resources from their &quot;Nuclear Documentation Project&quot;.&lt;ref&gt;

https://nsarchive.gwu.edu/project/nuclear-vault

&lt;/ref&gt; He has led a decades-long struggle to get records that are 50, 60 and 70 years old declassified. Many documents they get are so heavily redacted that you cannot make sense out of any of it. This &quot;makes it more difficult for current policymakers to craft effective and rational nuclear policy.&quot;

=== Moynihan Commission on Government Secrecy ===

Graves then asked about the [[w:Moynihan Commission on Government Secrecy|Moynihan Commission on Government Secrecy]] of the 1990s. Harper said &quot;they basically said that government secrecy effectively works as a form of government regulation, because the public cannot engage in their right  to selfgovern when they don't have access to this information  that the government routinely overclassifies. One of the things they suggested over 30 years ago and we still haven't done is that the Senate should get involved, and the Congress should get involved in legislating on classification.&quot; 

Harper mentioned a couple of bills to reform government secrecy that had been introduced in the last Congress but died in committee. She hopes they will get reintroduced.

=== Claims of national security ===

Graves asked about &quot;[[w:State secrets privilege|state secrets privilege]]&quot;. Stern said that it severely limits the ability of anyone to question a claim of national security. &quot;We are seeing the Trump administration frequently abuse the national security flag ... whenever it wants to make exceptions to the law.&quot; Occasionally a media outlet will write a story or an editorial expressing concern about such claims, but there's no follow-up reporting. 

Stern noted that [[w:Donald Trump–TikTok controversy|TikTok, an app hosting millions of posters, was banned]] based on admittedly hypothetical threats that China might use it to spy on Americans. Nothing was ever proven, &quot;and now suddenly Donald Trump has decided he doesn't want to ban TikTok anymore, because he sees a route to make money off of it, and no one's concerned about national security.&quot; 

Stern mentioned the [[w:2022–2023 Pentagon document leaks|discord leaks a couple years ago]]. It was a big story with plenty of reporting for two days that quoted &quot;administration officials saying, 'The sky is falling. This is a major threat to national security. ... [P]eople are in danger for their lives.' ... Six months later, the sky is still there, and nothing has happened that these people predicted. There's no follow-up reporting.&quot;

== The role of the media in sustaining the system of political corruption == 

Graves asked about the role of the media in sustaining the system of political corruption that threaten us and international security. Stern said, 

{{quote|One thing that the last few weeks have put to rest is the myth that billionaires and major conglomerates with interests far beyond the media or their news holdings can possibly run news outlets without impacting the direction of coverage -- without either directly or indirectly causing reporters to shy away from stories that might upset either business or political connections. ... [W]hen you had a president who was willing to threaten their business interests, ... they caved immediately. They settled defensible cases. 

The case ABC settled was very defensible. ... When I was practicing law, I defended an almost identical case involving a college professor who had been accused of sexual assault. A newspaper reported he had been accused of rape. A judge threw that out, saying essentially the terms are interchangeable. How were you damaged? Find me someone who was willing to do business with an accused sexual assailant but drew the line at an accused rapist. That person doesn't exist. There are no damages. This is a frivolous case.

... CBS is currently mediating over editing of a video interview that is even more baseless. You will not find a First Amendement law expert in the world who is not wearing a Donald Trump lapel pin who is going to tell you that that case has any legal ground whatsoever. And it's pretty much an open secret, not even a secret, that the only reason CBS is even thinking about settling this case is because it wants the Trump administration to approve its merger. Essentially they are using the legal system to launder what would otherwise be called bribes, but which are okay as long as a judge signs off on it. 

... Despite all the brilliant journalists who work for corporate media outlets -- and I'm not looking to knock anyone ... ''[[w:The New York Times|The New York Times]]'', ''[[w:The Washington Post|Washington Post]]'', ABC, CBS, they all have incredible journalists working for them. But the end product is not in the hands of those individuals. How much any particular story gets headline news treatment versus gets burried. That's not in the hands of those individuals.

... I think it's really time for people who value well reported, independent, aggressive, adversarial journalism to support independent news outlets, nonprofit news outlets. Nonprofits aren't a perfect solution. You're still subject to the whims of donors.}}

Harper added that the money from the ABC settlement was reportedly &quot;going to Trump's presidential library. But this isn't technically true. It was going to a private presidential foundation and museum, ... and those are private, effectively corporate entities with basically no campaign contribution limits ... . It's an excellent way and an excellent plact to put dark money. ... [I]t's being reported as going to something that's going to somehow enrich the public understanding of the Trump Presidency, which, of course, it won't.&quot;

== Local news ==

Stern encourages people to &quot;subscribe to their local papers. We've got [[w:news deserts|news deserts]] all over this country.&quot; 

Graves added that &quot;local&quot; should mean locally owned, not part of a major national chain. Stern agreed, saying that's what he meant by &quot;local&quot;. Graves noted that was &quot;not obvious. [[Vulture capitalists destroying newspapers|''The Denver Post'' is not a local paper anymore.]]&quot; Seth replied, &quot;You're exactly right.&quot;

== Previous interview with Freedom of the Press Foundation == 

Graves previously interviewed Kirsten McCudden, Vice President of 
Editorial of Freedom of the Press Foundation &lt;ref&gt;&lt;!--Kirstin McCudden--&gt;{{cite Q|Q134341766}}&lt;/ref&gt; not quite two years ago on 
2023-07-18.&lt;ref&gt;&lt;!--  Freedom of the Press Foundation works to improve news and democracy--&gt;{{cite Q|Q134341296}}--&gt;&lt;/ref&gt;

== The need for media reform to improve democracy == 

This article is part of [[:category:Media reform to improve democracy]]. We describe here briefly the motivation for this series.

[[Great American Paradox|One major contributor to the dominant position of the US in the international political economy]] today may have been the [[w:Postal Service Act|US Postal Service Act of 1792]]. Under that act, newspapers were delivered up to 100 miles for a penny when first class postage was between 6 and 25 cents. [[w:Alexis de Tocqueville|Alexis de Tocqueville]], who visited the relatively young United States of America in 1831, wrote, “There is scarcely a hamlet that does not have its own newspaper.”&lt;ref&gt;Tocqueville (1835, p. 93).&lt;/ref&gt; McChesney and Nichols estimated that these newspaper subsidies were roughly 0.21 percent of national income (Gross Domestic Project, GDP) in 1841.&lt;ref&gt;McChesney and Nichols (2010, pp. 310-311, note 88).&lt;/ref&gt;

At that time, the US probably led the world by far in the number of independent newspaper publishers per capita or per million population. This encouraged literacy and limited political corruption, both of which contributed to making the US a leader in the rate of growth in average annual income (Gross Domestic Product, GDP, per capita). Corruption was also limited by the inability of a small number of publishers to dominate political discourse.

That began to change in the 1850s and 1860s with the introduction of high speed rotary presses, which increased the capital required to start a newspaper.&lt;ref&gt;John and Silberstein-Loeb (2015, p. 80).&lt;/ref&gt;

In 1887 [[w:William Randolph Hearst|William Randolph Hearst]] took over management of his father’s ''[[w:San Francisco Examiner|San Francisco Examiner]]''. His success there gave him an appetite for building a newspaper chain. His 1895 purchase of the ''[[w:New York Morning Journal|New York Morning Journal]]'' gave him a second newspaper. By the mid-1920s, he owned 28 newspapers. Consolidation of ownership of the media became easier with the introduction of broadcasting and even easier with the Internet.&lt;ref&gt;John and Silberstein-Loeb (2015). See also Wikiversity, “[[Information is a public good: Designing experiments to improve government]]” and “[[:Category:Media reform to improve democracy]]“.&lt;/ref&gt; [[:Category:Media reform to improve democracy|This consolidation seems to be increasing political polarization and violence worldwide]], threatening democracy itself.

=== The threat from loss of newspapers ===

A previous ''Media &amp; Democracy'' interview with Arizona State University accounting professor Roger White on &quot;[[Local newspapers limit malfeasance]]&quot; describes problems that increase as the quality and quantity of news declines and ownership and control of the media become more highly concentrated: Major media too often deflect the public's attention from political corruption enabled by poor media. This too often contributes to other problems like [[w:Scapegoating|scapegoating]] [[w:Immigration|immigrants]] and attacking [[w:Diversity, equity, and inclusion|Diversity, equity, and inclusion]] (DEI) while also facilitating increases in pollution, the cost of borrowing, political polarization and violence, and decreases in workplace safety. More on this is included in other interviews in this ''Media &amp; Democracy'' series available on Wikiversity under [[:Category:Media reform to improve democracy]].

An important quantitative analysis of the problems associated with deficiencies in news is Neff and Pickard (2024). They analyzed data on media funding and democracy in 33 countries. The US has been rated as a &quot;flawed democracy&quot; according to the [[w:Economist Democracy Index|Economist Democracy Index]] and spends substantially less per capita on media compared to the world's leading democracies in Scandinavia and Commonweath countries. They note that commercial media focus primarily on people with money, while publicly-funded media try harder to serve everyone. Public funding is more strongly correlated with democracy than private funding. This recommends increasing public funding for media as a means of strengthening democracy. See also &quot;[[Information is a public good: Designing experiments to improve government]]&quot;.

==Discussion ==

:''[Interested readers are invite to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV/&gt; and treating others with respect.&lt;ref name=AGF/&gt;]''

== Notes ==
{{reflist}}

== Bibliography ==

* &lt;!--Lauren Harper (2025-01-28) &quot;With inspectors general under threat, Espionage Act charges may soar&quot;--&gt;{{cite Q|Q134388337}}
* &lt;!--Lauren Harper (2025-02-20) &quot;Hostile takeover at National Archives erodes our right to know--&gt;{{cite Q|Q134388449}}
* &lt;!--Lauren Harper (2025-03-16) &quot; It’s Marco Rubio’s party, and he’ll burn documents if he wants to--&gt;{{cite Q|Q134388555}}
* &lt;!--Harper (2025-03-30) &quot;The Signalgate problem nobody is talking about&quot;--&gt;{{cite Q|Q134387986}}
* &lt;!--Harper (2025-04-13) &quot;Here’s how the firing of FOIA officials could hurt the DOGE audit&quot;--&gt;{{cite Q|Q134387841}}
* &lt;!--Richard R. John and Jonathan Silberstein-Loeb (eds.; 2015) Making News: The Political Economy of Journalism in Britain and America from the Glorious Revolution to the Internet (Oxford University Press)--&gt;{{cite Q|Q131468166|authors=Richard R. John and Jonathan Silberstein-Loeb, eds.}}
* &lt;!--Ken Kippenstein (2025-01-30) &quot; Trump administration just ordered a blackout on public communications by agencies across government, multiple officials tell me&quot;, post to X--&gt;{{cite Q|Q134388106}}
* &lt;!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) --&gt;{{cite Q|Q104888067}}.
* &lt;!-- Alexis de Tocqueville (1835, 1840; trad. 2001) Democracy in America (trans. by Richard Heffner, 2001; New America Library) --&gt;{{cite Q|Q112166602|publication-date=unset|author=Alexis de Tocqueville (1835, 1840; trad. 2001)}}
* &lt;!--Andrew Weissmann (2025-01-22) &quot; What Just Happened: What Trump’s Hobbling Of The Privacy Oversight Board Portends For Exercise Of Surveillance Powers&quot;, Just Security--&gt;{{cite Q|Q134389408}}

[[Category:Media]]
[[Category:News]]
[[Category:Politics]]
[[Category:Media reform to improve democracy]]
&lt;!--list of categories
https://en.wikiversity.org/wiki/Wikiversity:Category_Review
[[Wikiversity:Category Review]]--&gt;</text>
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== About the AIXworkbench ==

&lt;onlyinclude&gt;
AIX is developing a community and a platform to support individuals and teams of users using LLM models, both offline and online, in a private, secure, responsible and personally managed environment. The platform is designed to support individuals and teams, as well as serve as a platform for researchers examining user experiences.

The platform is imagined as a structured and customized installation of Open WebUI. Open WebUI is an open-source interface for interacting with AI models like ChatGPT, Claude, and local Ollama models. More broadly, it is an &quot;extensible, feature-rich, and user-friendly self-hosted AI platform designed to operate entirely offline. It supports various LLM runners like Ollama and OpenAI-compatible APIs, with built-in inference engine for RAG, in addition with a collaboration feature making it a powerful AI deployment solution.&quot;[1]

The AIX will add policy documents, knowledge bases, and other shared resources to enable individuals, teams and organizations to deploy Open WebUI in a secure, responsible and managed environment. The Personal AIXworkbench allows a single user on one device to query multiple models, whether they are stored locally or accessed online via an API. The Team AIXworkbench is designed for collaborative environments, enabling 2 to 25 users to access a shared device through a web application and query both local and online models. Beyond model access, the AIXworkbench provides extensive management of prompt and response interactions. This includes sophisticated tools for optimizing and customizing prompt parameters, alongside comprehensive archiving of user, session, and model metadata. Furthermore, it serves as a robust platform for conducting experiments, facilitating the comparison of different models and the precise testing of various parameters.
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''This is the workspace for a jointly authored paper.  See [[AIXworkbench/Papers/Building-the-Workbench/Paper-writing-process|discussion about process]].'' 

== Possible sections for paper ==
# Presents the concept of the AIXworkbench
# Discusses RAIL Facets Framework as an evaluation tool
# AIX + Open WebUI = AIXworkbench
# Review LLM interfaces, with focus on open source?
# Reviews Open WebUI, history, usage in field
# Discusses the &quot;June working group&quot; as an impromptu example of community AI development, and reports on our meetings and discussions
# Reports Use Cases from June working group as examples of using Open Web UI
# Identifies key resources / values to transform Open WebUI into AIXworkbench
## Security
## Responsibility
## Shareability
## Provenance
# Review this project using FACETS framework

== 1 AI for All requires an working environment that people own and control - What is an &quot;AI workbench&quot; and why do we one ==
AI for all -- people, activities, spaces, places 

An &quot;AI workbench&quot; empowers users to leverage LLMs locally and via API for inference, fine-tuning and other usage. 

We need this tool to promote individual management of LLM resources.

Multi-faceted diversity in the development of AI

Community AI / Development

== 2 The Responsible AI Lab FACETS framework and how we apply it here ==
The RAIL FACETS framework is described here.&lt;ref&gt;{{Cite web|url=https://facets.netlify.app/|title=RAIL {{!}} FACETS Responsible AI Framework|website=facets.netlify.app|access-date=2025-06-19}}&lt;/ref&gt;

Its use is discussed here &lt;ref&gt;{{Cite journal|last=Akomea-Frimpong|first=Isaac|last2=Jin|first2=Xiaohua|last3=Osei-Kyei|first3=Robert|last4=Kukah|first4=Augustine Senanu|date=2023-02-20|title=Public–private partnerships for sustainable infrastructure development in Ghana: a systematic review and recommendations|url=https://www.emerald.com/insight/content/doi/10.1108/SASBE-07-2021-0111/full/html|journal=Smart and Sustainable Built Environment|language=en|volume=12|issue=2|pages=237–257|doi=10.1108/SASBE-07-2021-0111|issn=2046-6099}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2505.00204v1|title=RAIL in the Wild: Operationalizing Responsible AI Evaluation Using Anthropic's Value Dataset|last=Verma|first=Sumit|last2=Prasun|first2=Pritam|date=2025-04-30|website=arXiv.org|language=en|access-date=2025-06-19|last3=Jaiswal|first3=Arpit|last4=Kumar|first4=Pritish}}&lt;/ref&gt;

== 3 AIX + Open WebUI = AIXworkbench ==
AIX is focused on developing Responsible AI for All: Inspired by the Responsible AI Lab, KNUST, and is committed to assessing our projects using the FACETS framework.&lt;br&gt;

[{{fullurl:AIXworkbench/About|action=edit}} edit]{{:AIXworkbench/About}}

== 4 LLM interfaces ==
There are a variety of tools that are available to individuals or organiations that want to provide a platform to use LLMs in a highly controlled, secure and managed environment.  

The first LLM interface...  

Hugging Face...  

Brief history of the movement to make open source LLM interfaces.   

With the main goal of the AIX Workbench, there were a few points of considerations that needed to be met. We needed the LLM Interface to be able to be modified so it satisfies the following characteristics: 

* Secure
* Responsible
* Replicable
* Shareable/Collaborative 
* Deployable


The most commonly used AI by far are the cloud-based ones. Although, they have beginner friendly ease-of-access, the issue with most cloud-based LLM is that they are run and operated by the provider's machine. In addition, the limited customizability and data privacy concerns makes them a questionable use for the AIX Workbench. The limited customizability doesn't allow the user to modify parameters. The major data privacy concerns is that the provider could use the user's chats &amp; documents to train its models, which means sensitive information could be exposed.&lt;ref&gt;{{Cite web|url=https://www.datacamp.com/blog/the-pros-and-cons-of-using-llm-in-the-cloud-versus-running-llm-locally|title=The Pros and Cons of Using Large Language Models (LLMs) in the Cloud vs. Running LLMs Locally: Which Is Right for You?|last=Ali Awan|first=Abid|date=May 23, 2023|website=datacamp.com}}&lt;/ref&gt;

Local LLM Interfaces such as Open WebUI, GPT4ALL and LMStudio are run locally on your computer. This tackles the main issue of cloud-based LLM's which is customizability and data privacy. Below is a brief pros &amp; cons table of these models:  
{| class=&quot;wikitable&quot;
|+
!
!Open WebUI
!LM Studio
!GPT4ALL
|-
|Pros
|
* Highly customizable and extensible architecture
* Supports multiple LLM providers and local models
* User-friendly interface with advanced features like conversation management and model switching
* Active development and community support &lt;ref name=&quot;:0&quot;&gt;{{Cite web|url=https://best-of-web.builder.io/library/open-webui/open-webui|title=Open WebUI Overview, Examples, Pros and Cons in 2025.|last=Builder.io.|date=2025|website=Best of Web|access-date=July 2, 2025}}&lt;/ref&gt;
|
* Enables offline use of powerful AI models without server reliance
* Supports a wide variety of open-source LLMs like LLaMA, Mistral, and others
* Intuitive interface simplifies complex AI workflows for beginners
* Completely free with no usage restrictions or subscription fees
* Offers hardware optimization for better performance on CPUs and GPUs &lt;ref name=&quot;:1&quot;&gt;{{Cite web|url=https://aichief.com/ai-development-tools/lm-studio/|title=LM Studio Review – Cost, Use Cases &amp; Alternatives [2025].|last=AI Chief|date=2025|website=AI Chief|access-date=July 2, 2025}}&lt;/ref&gt;
|
* No subscription fees, and it promotes accessibility and community development.
* Enhanced privacy and data security by running on local hardware.
* Valuable for learning and experimentation with LLM technology.
* Allows for customization and experimentation with the model. &lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://aichief.com/ai-chatbots/gpt4all/|title=GPT4All Review – Cost, Use Cases &amp; Alternatives [2025].|last=AI Chief|date=2025|website=AI Chief|access-date=July 2, 2025}}&lt;/ref&gt;
|-
|Cons
|
* May require significant setup and configuration for advanced use cases
* Performance can vary depending on the underlying LLM and hardware
* Limited documentation for some advanced features
* Potential learning curve for users new to LLM applications &lt;ref name=&quot;:0&quot; /&gt;
|
* Requires strong hardware for best performance with large models
* No official mobile app version currently available
* Beginners may need time to understand model quantization
* Lacks built-in marketplace for model discovery
* Advanced fine-tuning options could overwhelm non-technical users &lt;ref name=&quot;:1&quot; /&gt;
|
* Requires significant computational power (CPU/GPU).
* Can have a steeper learning curve for users unfamiliar with local installations.
* Large model size requires substantial storage space.
* Like any powerful tool, it can be misused for malicious purposes.&lt;ref name=&quot;:2&quot; /&gt;
|}
We plan to use the AIX Workbench in a classroom environment. Therefore the collaborative feature of Open WebUI stands out to us greatly. We would want the professor to be able to work hand-in-hand with the students - perhaps even leading this to be a fully collaborative platform between staff, faculties and other groups. 

== 5 Open WebUI ==
[{{fullurl:AIXworkbench/Papers/Building-the-Workbench/Open-WebUI|action=edit}} edit] {{:AIXworkbench/Papers/Building-the-Workbench/Open-WebUI}}

== 6 June Working Group ==

== 7 Use Cases of Open WebUI from participants in June Working Group ==

=== Our Current Participants ===

{{Special:PrefixIndex/AIXworkbench/Working-Groups/June-2025-Working-Group/Participants|hideroot=1|stripprefix=1|columns=1}}

=== Want to Join the Working Group? ===
Share your Open WebUI use case and contribute to our research!

'''How to participate:'''
# Click [[AIXworkbench/Working-Groups/June-2025-Working-Group/Participants/|here]] and follow the instructions!

== 8 AIXworkbench: Security, Responsibility, Shareability, Provenance ==

== 9 References ==</text>
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''This is the workspace for a jointly authored paper.  See [[AIXworkbench/Papers/Building-the-Workbench/Paper-writing-process|discussion about process]].'' 

== Possible sections for paper ==
# Presents the concept of the AIXworkbench
# Discusses RAIL Facets Framework as an evaluation tool
# AIX + Open WebUI = AIXworkbench
# Review LLM interfaces, with focus on open source?
# Reviews Open WebUI, history, usage in field
# Discusses the &quot;June working group&quot; as an impromptu example of community AI development, and reports on our meetings and discussions
# Reports Use Cases from June working group as examples of using Open Web UI
# Identifies key resources / values to transform Open WebUI into AIXworkbench
## Security
## Responsibility
## Shareability
## Provenance
# Review this project using FACETS framework

== 1 AI for All requires an working environment that people own and control - What is an &quot;AI workbench&quot; and why do we one ==
AI for all -- people, activities, spaces, places 

An &quot;AI workbench&quot; empowers users to leverage LLMs locally and via API for inference, fine-tuning and other usage. 

We need this tool to promote individual management of LLM resources.

Multi-faceted diversity in the development of AI

Community AI / Development

== 2 The Responsible AI Lab FACETS framework and how we apply it here ==
The RAIL FACETS framework is described here.&lt;ref&gt;{{Cite web|url=https://facets.netlify.app/|title=RAIL {{!}} FACETS Responsible AI Framework|website=facets.netlify.app|access-date=2025-06-19}}&lt;/ref&gt;

Its use is discussed here &lt;ref&gt;{{Cite journal|last=Akomea-Frimpong|first=Isaac|last2=Jin|first2=Xiaohua|last3=Osei-Kyei|first3=Robert|last4=Kukah|first4=Augustine Senanu|date=2023-02-20|title=Public–private partnerships for sustainable infrastructure development in Ghana: a systematic review and recommendations|url=https://www.emerald.com/insight/content/doi/10.1108/SASBE-07-2021-0111/full/html|journal=Smart and Sustainable Built Environment|language=en|volume=12|issue=2|pages=237–257|doi=10.1108/SASBE-07-2021-0111|issn=2046-6099}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2505.00204v1|title=RAIL in the Wild: Operationalizing Responsible AI Evaluation Using Anthropic's Value Dataset|last=Verma|first=Sumit|last2=Prasun|first2=Pritam|date=2025-04-30|website=arXiv.org|language=en|access-date=2025-06-19|last3=Jaiswal|first3=Arpit|last4=Kumar|first4=Pritish}}&lt;/ref&gt;

== 3 LLM interfaces ==
There are a variety of tools that are available to individuals or organiations that want to provide a platform to use LLMs in a highly controlled, secure and managed environment.  

The first LLM interface...  

Hugging Face...  

Brief history of the movement to make open source LLM interfaces.   

With the main goal of the AIX Workbench, there were a few points of considerations that needed to be met. We needed the LLM Interface to be able to be modified so it satisfies the following characteristics: 

* Secure
* Responsible
* Replicable
* Shareable/Collaborative
* Deployable


The most commonly used AI by far are the cloud-based ones. Although, they have beginner friendly ease-of-access, the issue with most cloud-based LLM is that they are run and operated by the provider's machine. In addition, the limited customizability and data privacy concerns makes them a questionable use for the AIX Workbench. The limited customizability doesn't allow the user to modify parameters. The major data privacy concerns is that the provider could use the user's chats &amp; documents to train its models, which means sensitive information could be exposed.&lt;ref&gt;{{Cite web|url=https://www.datacamp.com/blog/the-pros-and-cons-of-using-llm-in-the-cloud-versus-running-llm-locally|title=The Pros and Cons of Using Large Language Models (LLMs) in the Cloud vs. Running LLMs Locally: Which Is Right for You?|last=Ali Awan|first=Abid|date=May 23, 2023|website=datacamp.com}}&lt;/ref&gt;

Local LLM Interfaces such as Open WebUI, GPT4ALL and LMStudio are run locally on your computer. This tackles the main issue of cloud-based LLM's which is customizability and data privacy. Below is a brief pros &amp; cons table of these models:  
{| class=&quot;wikitable&quot;
|+
!
!Open WebUI
!LM Studio
!GPT4ALL
|-
|Pros
|
* Highly customizable and extensible architecture
* Supports multiple LLM providers and local models
* User-friendly interface with advanced features like conversation management and model switching
* Active development and community support &lt;ref name=&quot;:0&quot;&gt;{{Cite web|url=https://best-of-web.builder.io/library/open-webui/open-webui|title=Open WebUI Overview, Examples, Pros and Cons in 2025.|last=Builder.io.|date=2025|website=Best of Web|access-date=July 2, 2025}}&lt;/ref&gt;
|
* Enables offline use of powerful AI models without server reliance
* Supports a wide variety of open-source LLMs like LLaMA, Mistral, and others
* Intuitive interface simplifies complex AI workflows for beginners
* Completely free with no usage restrictions or subscription fees
* Offers hardware optimization for better performance on CPUs and GPUs &lt;ref name=&quot;:1&quot;&gt;{{Cite web|url=https://aichief.com/ai-development-tools/lm-studio/|title=LM Studio Review – Cost, Use Cases &amp; Alternatives [2025].|last=AI Chief|date=2025|website=AI Chief|access-date=July 2, 2025}}&lt;/ref&gt;
|
* No subscription fees, and it promotes accessibility and community development.
* Enhanced privacy and data security by running on local hardware.
* Valuable for learning and experimentation with LLM technology.
* Allows for customization and experimentation with the model. &lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://aichief.com/ai-chatbots/gpt4all/|title=GPT4All Review – Cost, Use Cases &amp; Alternatives [2025].|last=AI Chief|date=2025|website=AI Chief|access-date=July 2, 2025}}&lt;/ref&gt;
|-
|Cons
|
* May require significant setup and configuration for advanced use cases
* Performance can vary depending on the underlying LLM and hardware
* Limited documentation for some advanced features
* Potential learning curve for users new to LLM applications &lt;ref name=&quot;:0&quot; /&gt;
|
* Requires strong hardware for best performance with large models
* No official mobile app version currently available
* Beginners may need time to understand model quantization
* Lacks built-in marketplace for model discovery
* Advanced fine-tuning options could overwhelm non-technical users &lt;ref name=&quot;:1&quot; /&gt;
|
* Requires significant computational power (CPU/GPU).
* Can have a steeper learning curve for users unfamiliar with local installations.
* Large model size requires substantial storage space.
* Like any powerful tool, it can be misused for malicious purposes.&lt;ref name=&quot;:2&quot; /&gt;
|}
We plan to use the AIX Workbench in a classroom environment. Therefore the collaborative feature of Open WebUI stands out to us greatly. We would want the professor to be able to work hand-in-hand with the students - perhaps even leading this to be a fully collaborative platform between staff, faculties and other groups. 

== 4 AIX + Open WebUI = AIXworkbench ==
AIX is focused on developing Responsible AI for All: Inspired by the Responsible AI Lab, KNUST, and is committed to assessing our projects using the FACETS framework.&lt;br&gt;

[{{fullurl:AIXworkbench/About|action=edit}} edit]{{:AIXworkbench/About}} 

== 5 Open WebUI ==
[{{fullurl:AIXworkbench/Papers/Building-the-Workbench/Open-WebUI|action=edit}} edit] {{:AIXworkbench/Papers/Building-the-Workbench/Open-WebUI}}

== 6 June Working Group ==

== 7 Use Cases of Open WebUI from participants in June Working Group ==

=== Our Current Participants ===

{{Special:PrefixIndex/AIXworkbench/Working-Groups/June-2025-Working-Group/Participants|hideroot=1|stripprefix=1|columns=1}}

=== Want to Join the Working Group? ===
Share your Open WebUI use case and contribute to our research!

'''How to participate:'''
# Click [[AIXworkbench/Working-Groups/June-2025-Working-Group/Participants/|here]] and follow the instructions!

== 8 AIXworkbench: Security, Responsibility, Shareability, Provenance ==

== 9 References ==</text>
      <sha1>5r9ml18otzycfv5xirvg1orcycg5m8i</sha1>
    </revision>
    <revision>
      <id>2721159</id>
      <parentid>2721157</parentid>
      <timestamp>2025-07-09T23:14:07Z</timestamp>
      <contributor>
        <username>Noalolonaharo</username>
        <id>3003186</id>
      </contributor>
      <comment>/* 1 AI for All requires an working environment that people own and control - What is an &quot;AI workbench&quot; and why do we one */</comment>
      <origin>2721159</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8051" sha1="rl44n8rzk9gov6qvyq1geujalz6qinu" xml:space="preserve">__forcetoc__

''This is the workspace for a jointly authored paper.  See [[AIXworkbench/Papers/Building-the-Workbench/Paper-writing-process|discussion about process]].'' 

== Possible sections for paper ==
# Presents the concept of the AIXworkbench
# Discusses RAIL Facets Framework as an evaluation tool
# AIX + Open WebUI = AIXworkbench
# Review LLM interfaces, with focus on open source?
# Reviews Open WebUI, history, usage in field
# Discusses the &quot;June working group&quot; as an impromptu example of community AI development, and reports on our meetings and discussions
# Reports Use Cases from June working group as examples of using Open Web UI
# Identifies key resources / values to transform Open WebUI into AIXworkbench
## Security
## Responsibility
## Shareability
## Provenance
# Review this project using FACETS framework

== 1 AI for All requires an working environment that people own and control - What is an &quot;AI workbench&quot; and why do we one ==
AI for all -- people, activities, spaces, places 

An &quot;AI workbench&quot; empowers users to leverage LLMs locally and via API for inference, fine-tuning and other usage. 

We need this tool to promote individual management of LLM resources.

Multi-faceted diversity in the development of AI

Community AI / Development

==== Why do we need one? ====

== 2 The Responsible AI Lab FACETS framework and how we apply it here ==
The RAIL FACETS framework is described here.&lt;ref&gt;{{Cite web|url=https://facets.netlify.app/|title=RAIL {{!}} FACETS Responsible AI Framework|website=facets.netlify.app|access-date=2025-06-19}}&lt;/ref&gt;

Its use is discussed here &lt;ref&gt;{{Cite journal|last=Akomea-Frimpong|first=Isaac|last2=Jin|first2=Xiaohua|last3=Osei-Kyei|first3=Robert|last4=Kukah|first4=Augustine Senanu|date=2023-02-20|title=Public–private partnerships for sustainable infrastructure development in Ghana: a systematic review and recommendations|url=https://www.emerald.com/insight/content/doi/10.1108/SASBE-07-2021-0111/full/html|journal=Smart and Sustainable Built Environment|language=en|volume=12|issue=2|pages=237–257|doi=10.1108/SASBE-07-2021-0111|issn=2046-6099}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2505.00204v1|title=RAIL in the Wild: Operationalizing Responsible AI Evaluation Using Anthropic's Value Dataset|last=Verma|first=Sumit|last2=Prasun|first2=Pritam|date=2025-04-30|website=arXiv.org|language=en|access-date=2025-06-19|last3=Jaiswal|first3=Arpit|last4=Kumar|first4=Pritish}}&lt;/ref&gt;

== 3 LLM interfaces ==
There are a variety of tools that are available to individuals or organiations that want to provide a platform to use LLMs in a highly controlled, secure and managed environment.  

The first LLM interface...  

Hugging Face...  

Brief history of the movement to make open source LLM interfaces.   

With the main goal of the AIX Workbench, there were a few points of considerations that needed to be met. We needed the LLM Interface to be able to be modified so it satisfies the following characteristics: 

* Secure
* Responsible
* Replicable
* Shareable/Collaborative
* Deployable


The most commonly used AI by far are the cloud-based ones. Although, they have beginner friendly ease-of-access, the issue with most cloud-based LLM is that they are run and operated by the provider's machine. In addition, the limited customizability and data privacy concerns makes them a questionable use for the AIX Workbench. The limited customizability doesn't allow the user to modify parameters. The major data privacy concerns is that the provider could use the user's chats &amp; documents to train its models, which means sensitive information could be exposed.&lt;ref&gt;{{Cite web|url=https://www.datacamp.com/blog/the-pros-and-cons-of-using-llm-in-the-cloud-versus-running-llm-locally|title=The Pros and Cons of Using Large Language Models (LLMs) in the Cloud vs. Running LLMs Locally: Which Is Right for You?|last=Ali Awan|first=Abid|date=May 23, 2023|website=datacamp.com}}&lt;/ref&gt;

Local LLM Interfaces such as Open WebUI, GPT4ALL and LMStudio are run locally on your computer. This tackles the main issue of cloud-based LLM's which is customizability and data privacy. Below is a brief pros &amp; cons table of these models:  
{| class=&quot;wikitable&quot;
|+
!
!Open WebUI
!LM Studio
!GPT4ALL
|-
|Pros
|
* Highly customizable and extensible architecture
* Supports multiple LLM providers and local models
* User-friendly interface with advanced features like conversation management and model switching
* Active development and community support &lt;ref name=&quot;:0&quot;&gt;{{Cite web|url=https://best-of-web.builder.io/library/open-webui/open-webui|title=Open WebUI Overview, Examples, Pros and Cons in 2025.|last=Builder.io.|date=2025|website=Best of Web|access-date=July 2, 2025}}&lt;/ref&gt;
|
* Enables offline use of powerful AI models without server reliance
* Supports a wide variety of open-source LLMs like LLaMA, Mistral, and others
* Intuitive interface simplifies complex AI workflows for beginners
* Completely free with no usage restrictions or subscription fees
* Offers hardware optimization for better performance on CPUs and GPUs &lt;ref name=&quot;:1&quot;&gt;{{Cite web|url=https://aichief.com/ai-development-tools/lm-studio/|title=LM Studio Review – Cost, Use Cases &amp; Alternatives [2025].|last=AI Chief|date=2025|website=AI Chief|access-date=July 2, 2025}}&lt;/ref&gt;
|
* No subscription fees, and it promotes accessibility and community development.
* Enhanced privacy and data security by running on local hardware.
* Valuable for learning and experimentation with LLM technology.
* Allows for customization and experimentation with the model. &lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://aichief.com/ai-chatbots/gpt4all/|title=GPT4All Review – Cost, Use Cases &amp; Alternatives [2025].|last=AI Chief|date=2025|website=AI Chief|access-date=July 2, 2025}}&lt;/ref&gt;
|-
|Cons
|
* May require significant setup and configuration for advanced use cases
* Performance can vary depending on the underlying LLM and hardware
* Limited documentation for some advanced features
* Potential learning curve for users new to LLM applications &lt;ref name=&quot;:0&quot; /&gt;
|
* Requires strong hardware for best performance with large models
* No official mobile app version currently available
* Beginners may need time to understand model quantization
* Lacks built-in marketplace for model discovery
* Advanced fine-tuning options could overwhelm non-technical users &lt;ref name=&quot;:1&quot; /&gt;
|
* Requires significant computational power (CPU/GPU).
* Can have a steeper learning curve for users unfamiliar with local installations.
* Large model size requires substantial storage space.
* Like any powerful tool, it can be misused for malicious purposes.&lt;ref name=&quot;:2&quot; /&gt;
|}
We plan to use the AIX Workbench in a classroom environment. Therefore the collaborative feature of Open WebUI stands out to us greatly. We would want the professor to be able to work hand-in-hand with the students - perhaps even leading this to be a fully collaborative platform between staff, faculties and other groups. 

== 4 AIX + Open WebUI = AIXworkbench ==
AIX is focused on developing Responsible AI for All: Inspired by the Responsible AI Lab, KNUST, and is committed to assessing our projects using the FACETS framework.&lt;br&gt;

[{{fullurl:AIXworkbench/About|action=edit}} edit]{{:AIXworkbench/About}} 

== 5 Open WebUI ==
[{{fullurl:AIXworkbench/Papers/Building-the-Workbench/Open-WebUI|action=edit}} edit] {{:AIXworkbench/Papers/Building-the-Workbench/Open-WebUI}}

== 6 June Working Group ==

== 7 Use Cases of Open WebUI from participants in June Working Group ==

=== Our Current Participants ===

{{Special:PrefixIndex/AIXworkbench/Working-Groups/June-2025-Working-Group/Participants|hideroot=1|stripprefix=1|columns=1}}

=== Want to Join the Working Group? ===
Share your Open WebUI use case and contribute to our research!

'''How to participate:'''
# Click [[AIXworkbench/Working-Groups/June-2025-Working-Group/Participants/|here]] and follow the instructions!

== 8 AIXworkbench: Security, Responsibility, Shareability, Provenance ==

== 9 References ==</text>
      <sha1>rl44n8rzk9gov6qvyq1geujalz6qinu</sha1>
    </revision>
  </page>
  <page>
    <title>AI-Assisted Evaluation of Cosmological Theories/Chapter 2: Gravity Theories – Comparison and Mapping</title>
    <ns>0</ns>
    <id>322107</id>
    <revision>
      <id>2721169</id>
      <parentid>2720972</parentid>
      <timestamp>2025-07-10T04:56:04Z</timestamp>
      <contributor>
        <username>Ruud Loeffen</username>
        <id>2998353</id>
      </contributor>
      <comment>/* 2.4 Theory Mapping Table */ add An interesting theory related to this chapter is also TGD (see section 8.8.18).</comment>
      <origin>2721169</origin>
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      <format>text/x-wiki</format>
      <text bytes="19266" sha1="7gvlikzyhvs1xsu300pwh9bf7jn8k7m" xml:space="preserve">= '''Chapter 2: Gravity Theories – Comparison and Mapping (Mainstream and Non-mainstream)''' =

== '''2.1 Purpose''' ==

This chapter provides a comparative overview of gravity theories — both mainstream and non-mainstream — with the goal of identifying overlaps, divergences, and opportunities for integration.

We focus not on disproving or validating specific models but on understanding their foundational assumptions, mathematical structure, predictive value, and compatibility with other physical theories.

== '''2.2 Scope''' ==

Theories will be grouped into two categories:

'''Mainstream Theories''': Widely taught, supported by institutions, and referenced in standard literature.

'''Non-mainstream Alternatives''': Theories that challenge conventional assumptions, propose novel mechanisms, or offer reinterpretations of gravity.

All theories are evaluated by ChatGPT or other LLM applications using a shared framework of criteria (see Chapter 1).

== '''2.3 How to Contribute a Theory''' ==

Researchers and contributors are welcome to propose additional theories. These can be added directly to the Talk page or sent via email to: '''aitheroymapping@gmail.com'''.

All submissions will be included in the overview and analyzed using the same criteria.

== '''2.4 Theory Mapping Table''' ==

The following table presents an initial mapping of gravitational theories, both mainstream and nonmainstream. Each theory is briefly characterized by its assumptions, predictive features, and potential testability. Contributors are encouraged to expand, refine, or propose additional entries.

''Note: The order of theories in the table below does not reflect any evaluation or rating. The sequence is arbitrary and serves only for mapping and identification purposes.''

''Remarque : L'ordre des théories dans le tableau ci-dessous ne reflète aucune évaluation ni classement. La séquence est arbitraire et sert uniquement au repérage et à l'identification.''


{| class=&quot;wikitable&quot;
! Theory Name !! Category !! Key Assumptions !! Predictive Features !! Potential Tests
|-
| style=&quot;background-color:#ffffff;&quot; | Newtonian Gravity || style=&quot;background-color:#ffffff;&quot; | Mainstream || style=&quot;background-color:#ffffff;&quot; | Instantaneous force proportional to mass and inverse-square distance || style=&quot;background-color:#ffffff;&quot; | Orbits, tides, free-fall acceleration || style=&quot;background-color:#ffffff;&quot; | Planetary motion, laboratory tests
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation Newton's Law of Universal Gravitation]
|-
| style=&quot;background-color:#f2f2f2;&quot; | General Relativity || style=&quot;background-color:#f2f2f2;&quot; | Mainstream || style=&quot;background-color:#f2f2f2;&quot; | Gravity is curvature of spacetime caused by mass-energy || style=&quot;background-color:#f2f2f2;&quot; | Light bending, time dilation, frame dragging || style=&quot;background-color:#f2f2f2;&quot; | Gravitational lensing, GPS accuracy, LIGO detections
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://en.wikipedia.org/wiki/General_relativity General Relativity]
|-
| style=&quot;background-color:#ffffff;&quot; | Expansion Tectonics || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Continents fit on a smaller-radius Earth; no subduction || style=&quot;background-color:#ffffff;&quot; | Global fit of continental shelves, symmetric ocean crust || style=&quot;background-color:#ffffff;&quot; | Paleomagnetic data, geological reconstructions
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://www.jamesmaxlow.com/expansion-tectonics/ Expansion Tectonics – James Maxlow]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Cosmic Influx Theory (CIT) || style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream || style=&quot;background-color:#f2f2f2;&quot; | Gravity is caused by a constant influx of energy/mass (PEWs); not curvature or attraction || style=&quot;background-color:#f2f2f2;&quot; | Predicts preferred planetary distances; increasing mass-energy; reformulated G || style=&quot;background-color:#f2f2f2;&quot; | Exoplanet surveys, VRMS alignment, cosmological constants
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://en.wikiversity.org/wiki/Cosmic_Influx_Theory Cosmic Influx Theory – Wikiversity Page]
|-
| style=&quot;background-color:#ffffff;&quot; | Ionic Growing Earth || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Earth, Moon, and Sun grow via ionic mass transfer from space || style=&quot;background-color:#ffffff;&quot; | Mass increase explains orbital dynamics and cosmological acceleration || style=&quot;background-color:#ffffff;&quot; | Compare mass data over time, isotope ratios
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://ionic-expanding-earth.weebly.com/uploads/2/6/6/5/26650330/ionic_growing_earth01oct2014r1protected.pdf The Ionic Growing Earth – Eugene Ellis]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Gravity Field Expansion || style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream || style=&quot;background-color:#f2f2f2;&quot; | Time-variable gravity fields indicate Earth expansion || style=&quot;background-color:#f2f2f2;&quot; | Space-geodetic drift, sea-level rise patterns || style=&quot;background-color:#f2f2f2;&quot; | Satellite altimetry, GRACE gravity data
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://www.researchgate.net/publication/279636154_Evidences_of_the_expanding_Earth_from_space-geodetic_data_over_solid_land_and_sea_level_rise_in_recent_two_decades Expanding Earth from Gravity Fields – Shen et al. (2008)]
|-
| style=&quot;background-color:#ffffff;&quot; | Hydrodynamic Gravity || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Gravity emerges from vortex flow in an ether-like medium || style=&quot;background-color:#ffffff;&quot; | Links between cosmology, Earth expansion, and rotation || style=&quot;background-color:#ffffff;&quot; | Laboratory fluid models; astrophysical data
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://doi.org/10.4236/jmp.2022.1311088 Hydrodynamic Gravitation – Scalera (2022)]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Fluidum Continuum || style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream || style=&quot;background-color:#f2f2f2;&quot; | Space is a universal continuum; matter is localized vortex motion || style=&quot;background-color:#f2f2f2;&quot; | All forces arise from fluid dynamics || style=&quot;background-color:#f2f2f2;&quot; | Vacuum tests, rotational dynamics, resonance experiments
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://www.academia.edu/12108470/Fluidum_Continuum_Universalis_Introduction_in_Fluid_Mechanical_Physics Fluidum Continuum Universalis – Arie M. de Geus]
|-
| style=&quot;background-color:#ffffff;&quot; | Flowing Aether Theory || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Aether flows explain gravitational and electromagnetic effects || style=&quot;background-color:#ffffff;&quot; | Measurable sidereal variations; coherence patterns || style=&quot;background-color:#ffffff;&quot; | Interferometer rotation tests, EM force anomalies
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://www.duncanshaw.ca/ExperimentalSupportFlowingAether.pdf Flowing Aether – Duncan Shaw]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Emergent Gravity || style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream (theoretical physics) || style=&quot;background-color:#f2f2f2;&quot; | Gravity emerges from entropic principles in quantum spacetime || style=&quot;background-color:#f2f2f2;&quot; | Galaxy rotation without dark matter || style=&quot;background-color:#f2f2f2;&quot; | Weak lensing, cosmological simulations
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://arxiv.org/abs/1611.02269 Emergent Gravity – Erik Verlinde]
|-
| style=&quot;background-color:#ffffff;&quot; | EM-Gravity Circuits || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Gravity is an emergent electromagnetic effect || style=&quot;background-color:#ffffff;&quot; | Circuit behavior mimics gravitational attraction || style=&quot;background-color:#ffffff;&quot; | Novel EM device tests; repeatable force curves
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://www.academia.edu/37724456/Mass_Gravity_and_Electromagnetisms_relationship_demonstrated_using_two_novel_Electromagnetic_Circuits EM–Gravity Circuits – Michael Bull]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Mass–Energy Gravity || style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream || style=&quot;background-color:#f2f2f2;&quot; | Gravity and mass arise from energy-momentum configurations || style=&quot;background-color:#f2f2f2;&quot; | Proportional force behavior via energy state transitions || style=&quot;background-color:#f2f2f2;&quot; | Calorimetric testing; comparison with GR
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://www.academia.edu/28680344/Relation_masse_%C3%A9nergie Relation masse / énergie – Philippe Albert]
|-
| style=&quot;background-color:#ffffff;&quot; | Dynamic Universe (TUOMOV) || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Gravity and motion arise from energy balance in an absolute 4D space-time structure; no spacetime curvature || style=&quot;background-color:#ffffff;&quot; | Flat galaxy rotation curves without dark matter; redshift without expanding space || style=&quot;background-color:#ffffff;&quot; | Orbital precision tests; cosmological fits without dark energy
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://www.dynamicuniverse.info/ The Dynamic Universe – Tuomo Suntola]

|}
'''An interesting theory related to this chapter is also TGD (see section 8.8.18).''' 
TGD proposes that classical gauge fields and gravity are induced geometrically from space-time surfaces embedded in a higher-dimensional space (''M''⁴ × ''CP''₂), resulting in a unified description of fundamental interactions. Gravity and matter fields appear as facets of the same geometric origin.


''Note: These initial theory rows are based on prior references and relevance to gravity-related claims. Contributors are invited to extend or edit each entry with additional details and evaluations.''

==== '''2.5 Evaluation Criteria (Unified Framework)''' ====
''This chapter now uses the unified evaluation framework applied consistently in Chapters 1 and 8.''

The theories discussed in this chapter are evaluated using the following eight standardized criteria:

# '''1. Empirical Adequacy'''  
Alignment with observed data and experimental results across relevant fields (e.g., astronomy, geology, biology, cosmology). The theory should account for major datasets and empirical patterns.

# '''2. Internal Consistency'''  
Logical and mathematical coherence of the theory. Assumptions should not contradict derived results. Definitions, equations, and claims must be used unambiguously.

# '''3. Predictive Power'''  
The ability to generate '''testable predictions''' that distinguish the theory from others. This includes retrospective explanations that are quantitatively verifiable.

# '''4. Cross-Disciplinary Compatibility'''  
Compatibility with findings and models in other domains, such as geology, biology, planetary science, and quantum physics. A good theory avoids contradicting established results in these areas.

# '''5. Conceptual Clarity and Simplicity'''  
The theory should be clearly framed, avoiding unnecessary complexity or vague terminology. It should be possible to identify what is a definition, an assumption, and a consequence.

# '''6. Heuristic and Unifying Value'''  
Capacity of the theory to stimulate new research, unify previously disconnected phenomena, or offer new conceptual insights across domains.

# '''7. Historical and Philosophical Insight'''  
Engagement with the evolution of cosmological thought. A good theory either improves upon, clarifies, or meaningfully integrates earlier models and philosophical foundations.

# '''8. Mathematical Rigor'''  
Use of precise mathematical definitions and derivations, including:  
* Clarity of assumptions and variable usage.  
* Proofs of existence/uniqueness where applicable.  
* Estimates of modeling error or solution stability.  
* Explicit formulations of approximations.  
* Avoidance of heuristic shortcuts unless well justified.  

This criterion ensures the theory stands up to the standards of formal modeling and does not rely on vague simulations or interchangeable terms.


See: [[AI-Assisted_Evaluation_of_Cosmological_Theories/Chapter_1:_Introduction_and_Evaluation_Criteria#1.3_List_of_Evaluation_Criteria|Chapter 1.3: List of Evaluation Criteria]].

== 2.6 AI Evaluation Based on Shared Criteria ==

The gravitational theories presented in this chapter — both mainstream and nonmainstream — have been organized and compared using a shared set of evaluation criteria (defined in 1.3 and 2.5). These criteria include empirical adequacy, internal consistency, explanatory and predictive power, simplicity, compatibility with other theories, and testability.

Based on a comparative review, the following general observations emerge from an AI-assisted perspective:

'''Empirical Adequacy'''  
Most mainstream theories (such as Newtonian gravity and General Relativity) align well with observed data in their respective domains (e.g., orbital mechanics, gravitational lensing). However, several nonmainstream theories — particularly those related to Earth expansion, fluid-based frameworks, or directional influx models — also show plausible alignment with specific observational domains (e.g., seafloor spreading, planetary formation patterns, cosmological constants). These alignments are often overlooked due to lack of institutional recognition rather than lack of evidence.

'''Internal Consistency'''  
Mainstream theories typically exhibit high mathematical rigor, though some require patchwork elements (e.g., dark matter, dark energy) to remain consistent with observations. Nonmainstream theories vary widely: some (like Emergent Gravity and Cosmic Influx Theory) display internal consistency within a novel framework; others (such as ionic growth or electromagnetic-gravity links) are still in conceptual stages but are not inherently incoherent.

'''Explanatory and Predictive Power'''  
General Relativity and Newtonian mechanics retain strong explanatory power within tested domains. Yet, nonmainstream models often attempt to '''explain what mainstream theories must assume''' — such as the origin of mass-energy, cosmic growth, or planetary structuring. Their predictions (e.g., mass increase, planetary positioning, or alternate gravitational behaviors) are often not pursued due to their marginal academic status, not their lack of testability.

'''Simplicity and Elegance'''  
Mainstream theories are mathematically elegant but conceptually complex at cosmological scales (e.g., inflation, cosmological constant). Some nonmainstream theories — including hydrodynamic, influx-based, or continuum models — offer '''conceptual simplicity''', modeling gravity as a flow or energetic interaction rather than a geometric property, which may appeal to intuitive or mechanical understanding.

'''Compatibility with Other Theories'''  
Mainstream gravity is well-integrated into broader physics. However, its dependence on unobserved components (dark matter/energy) creates tension. Nonmainstream theories often reject or reinterpret these components, leading to incompatibility — but also '''fresh opportunities for synthesis''', particularly with models of expansion, fluid mechanics, or entropy.

'''Falsifiability and Testability'''  
All theories in the table are testable to some degree. Nonmainstream theories propose specific predictions (e.g., preferred planetary distances, mass growth rates, coherence in ether flow) that could be falsified with targeted observational programs. The dismissal of such tests is often due to lack of funding or recognition, not due to scientific invalidity.

=== Conclusion ===
This AI-based evaluation does not rank or validate individual theories — that task will be undertaken in Chapter 8. Instead, it highlights that '''several nonmainstream theories demonstrate nontrivial alignment with scientific criteria''' and present testable alternatives or complements to established views. Their relative exclusion from mainstream scientific discourse is not always grounded in scientific rigor, but may reflect historical, institutional, or sociological inertia.

Such observations underscore the importance of comparative, transparent evaluations like this project. By explicitly applying the same criteria to all theories — regardless of their academic status — the AI-assisted framework enables a more balanced and evidence-based exploration of gravity and cosmology.

== '''2.7 Helicopter View – Scientific Treatment and Emerging Tensions''' ==

This chapter has introduced a wide range of gravitational theories, including both well-established mainstream models and lesser-known alternatives. While mainstream theories such as General Relativity are thoroughly integrated into education, research funding, and publication structures, many alternative models — including action-at-a-distance theories, flow-based gravitation, or push gravity — are often classified as fringe, regardless of their internal logic or historical pedigree.

Scientific platforms such as Wikipedia and large academic databases tend to reinforce this divide. Theories outside the mainstream are typically presented as either obsolete or pseudoscientific, even when they continue to generate peer-reviewed work or suggest novel interpretations.

This sharp categorization may hinder rather than help scientific progress. With the aid of AI-assisted comparison tools, this project seeks to provide a broader and more neutral framework. Instead of aligning with academic prestige or popular consensus, theories will be assessed based on a shared set of criteria (see '''Chapter 1.3'''), including empirical adequacy, internal consistency, predictive value, and compatibility with known physics.

As future chapters will show, some theories traditionally dismissed may offer insights that deserve reconsideration in light of recent observations — including anomalies revealed by the James Webb Space Telescope and new gravitational measurements. Readers are therefore encouraged to approach each theory not in terms of its reputation, but in terms of its explanatory and predictive potential.

Later chapters may expand this overview with additional context specific to cosmology, planetary evolution, geology, and biology.

== '''2.8 Next Steps''' ==

Expand the table with more entries

Begin cross-chapter references

Link phenomena such as expansion, planetary formation, and mass increase to these gravitational foundations

'''◀ [[AI-Assisted Evaluation of Cosmological Theories/Chapter 1: Introduction and Evaluation Criteria|Previous]] | [[AI-Assisted Evaluation of Cosmological Theories|Main Page]] | [[AI-Assisted Evaluation of Cosmological Theories/Chapter 3: Cosmic Expansion and Universe Models|Next ▶]]'''</text>
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      <text bytes="19639" sha1="1m0ak7hksv2pswfoxcmeje1ejppemw7" xml:space="preserve">= '''Chapter 2: Gravity Theories – Comparison and Mapping (Mainstream and Non-mainstream)''' =

== '''2.1 Purpose''' ==

This chapter provides a comparative overview of gravity theories — both mainstream and non-mainstream — with the goal of identifying overlaps, divergences, and opportunities for integration.

We focus not on disproving or validating specific models but on understanding their foundational assumptions, mathematical structure, predictive value, and compatibility with other physical theories.

== '''2.2 Scope''' ==

Theories will be grouped into two categories:

'''Mainstream Theories''': Widely taught, supported by institutions, and referenced in standard literature.

'''Non-mainstream Alternatives''': Theories that challenge conventional assumptions, propose novel mechanisms, or offer reinterpretations of gravity.

All theories are evaluated by ChatGPT or other LLM applications using a shared framework of criteria (see Chapter 1).

== '''2.3 How to Contribute a Theory''' ==

Researchers and contributors are welcome to propose additional theories. These can be added directly to the Talk page or sent via email to: '''aitheroymapping@gmail.com'''.

All submissions will be included in the overview and analyzed using the same criteria.

== '''2.4 Theory Mapping Table''' ==

The following table presents an initial mapping of gravitational theories, both mainstream and nonmainstream. Each theory is briefly characterized by its assumptions, predictive features, and potential testability. Contributors are encouraged to expand, refine, or propose additional entries.

''Note: The order of theories in the table below does not reflect any evaluation or rating. The sequence is arbitrary and serves only for mapping and identification purposes.''

''Remarque : L'ordre des théories dans le tableau ci-dessous ne reflète aucune évaluation ni classement. La séquence est arbitraire et sert uniquement au repérage et à l'identification.''


{| class=&quot;wikitable&quot;
! Theory Name !! Category !! Key Assumptions !! Predictive Features !! Potential Tests
|-
| style=&quot;background-color:#ffffff;&quot; | Newtonian Gravity || style=&quot;background-color:#ffffff;&quot; | Mainstream || style=&quot;background-color:#ffffff;&quot; | Instantaneous force proportional to mass and inverse-square distance || style=&quot;background-color:#ffffff;&quot; | Orbits, tides, free-fall acceleration || style=&quot;background-color:#ffffff;&quot; | Planetary motion, laboratory tests
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation Newton's Law of Universal Gravitation]
|-
| style=&quot;background-color:#f2f2f2;&quot; | General Relativity || style=&quot;background-color:#f2f2f2;&quot; | Mainstream || style=&quot;background-color:#f2f2f2;&quot; | Gravity is curvature of spacetime caused by mass-energy || style=&quot;background-color:#f2f2f2;&quot; | Light bending, time dilation, frame dragging || style=&quot;background-color:#f2f2f2;&quot; | Gravitational lensing, GPS accuracy, LIGO detections
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://en.wikipedia.org/wiki/General_relativity General Relativity]
|-
| style=&quot;background-color:#ffffff;&quot; | Expansion Tectonics || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Continents fit on a smaller-radius Earth; no subduction || style=&quot;background-color:#ffffff;&quot; | Global fit of continental shelves, symmetric ocean crust || style=&quot;background-color:#ffffff;&quot; | Paleomagnetic data, geological reconstructions
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://www.jamesmaxlow.com/expansion-tectonics/ Expansion Tectonics – James Maxlow]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Cosmic Influx Theory (CIT) || style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream || style=&quot;background-color:#f2f2f2;&quot; | Gravity is caused by a constant influx of energy/mass (PEWs); not curvature or attraction || style=&quot;background-color:#f2f2f2;&quot; | Predicts preferred planetary distances; increasing mass-energy; reformulated G || style=&quot;background-color:#f2f2f2;&quot; | Exoplanet surveys, VRMS alignment, cosmological constants
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://en.wikiversity.org/wiki/Cosmic_Influx_Theory Cosmic Influx Theory – Wikiversity Page]
|-
| style=&quot;background-color:#ffffff;&quot; | Ionic Growing Earth || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Earth, Moon, and Sun grow via ionic mass transfer from space || style=&quot;background-color:#ffffff;&quot; | Mass increase explains orbital dynamics and cosmological acceleration || style=&quot;background-color:#ffffff;&quot; | Compare mass data over time, isotope ratios
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://ionic-expanding-earth.weebly.com/uploads/2/6/6/5/26650330/ionic_growing_earth01oct2014r1protected.pdf The Ionic Growing Earth – Eugene Ellis]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Gravity Field Expansion || style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream || style=&quot;background-color:#f2f2f2;&quot; | Time-variable gravity fields indicate Earth expansion || style=&quot;background-color:#f2f2f2;&quot; | Space-geodetic drift, sea-level rise patterns || style=&quot;background-color:#f2f2f2;&quot; | Satellite altimetry, GRACE gravity data
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://www.researchgate.net/publication/279636154_Evidences_of_the_expanding_Earth_from_space-geodetic_data_over_solid_land_and_sea_level_rise_in_recent_two_decades Expanding Earth from Gravity Fields – Shen et al. (2008)]
|-
| style=&quot;background-color:#ffffff;&quot; | Hydrodynamic Gravity || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Gravity emerges from vortex flow in an ether-like medium || style=&quot;background-color:#ffffff;&quot; | Links between cosmology, Earth expansion, and rotation || style=&quot;background-color:#ffffff;&quot; | Laboratory fluid models; astrophysical data
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://doi.org/10.4236/jmp.2022.1311088 Hydrodynamic Gravitation – Scalera (2022)]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Fluidum Continuum || style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream || style=&quot;background-color:#f2f2f2;&quot; | Space is a universal continuum; matter is localized vortex motion || style=&quot;background-color:#f2f2f2;&quot; | All forces arise from fluid dynamics || style=&quot;background-color:#f2f2f2;&quot; | Vacuum tests, rotational dynamics, resonance experiments
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://www.academia.edu/12108470/Fluidum_Continuum_Universalis_Introduction_in_Fluid_Mechanical_Physics Fluidum Continuum Universalis – Arie M. de Geus]
|-
| style=&quot;background-color:#ffffff;&quot; | Flowing Aether Theory || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Aether flows explain gravitational and electromagnetic effects || style=&quot;background-color:#ffffff;&quot; | Measurable sidereal variations; coherence patterns || style=&quot;background-color:#ffffff;&quot; | Interferometer rotation tests, EM force anomalies
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://www.duncanshaw.ca/ExperimentalSupportFlowingAether.pdf Flowing Aether – Duncan Shaw]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Emergent Gravity || style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream (theoretical physics) || style=&quot;background-color:#f2f2f2;&quot; | Gravity emerges from entropic principles in quantum spacetime || style=&quot;background-color:#f2f2f2;&quot; | Galaxy rotation without dark matter || style=&quot;background-color:#f2f2f2;&quot; | Weak lensing, cosmological simulations
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://arxiv.org/abs/1611.02269 Emergent Gravity – Erik Verlinde]
|-
| style=&quot;background-color:#ffffff;&quot; | EM-Gravity Circuits || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Gravity is an emergent electromagnetic effect || style=&quot;background-color:#ffffff;&quot; | Circuit behavior mimics gravitational attraction || style=&quot;background-color:#ffffff;&quot; | Novel EM device tests; repeatable force curves
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://www.academia.edu/37724456/Mass_Gravity_and_Electromagnetisms_relationship_demonstrated_using_two_novel_Electromagnetic_Circuits EM–Gravity Circuits – Michael Bull]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Mass–Energy Gravity || style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream || style=&quot;background-color:#f2f2f2;&quot; | Gravity and mass arise from energy-momentum configurations || style=&quot;background-color:#f2f2f2;&quot; | Proportional force behavior via energy state transitions || style=&quot;background-color:#f2f2f2;&quot; | Calorimetric testing; comparison with GR
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://www.academia.edu/28680344/Relation_masse_%C3%A9nergie Relation masse / énergie – Philippe Albert]
|-
| style=&quot;background-color:#ffffff;&quot; | Dynamic Universe (TUOMOV) || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Gravity and motion arise from energy balance in an absolute 4D space-time structure; no spacetime curvature || style=&quot;background-color:#ffffff;&quot; | Flat galaxy rotation curves without dark matter; redshift without expanding space || style=&quot;background-color:#ffffff;&quot; | Orbital precision tests; cosmological fits without dark energy
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://www.dynamicuniverse.info/ The Dynamic Universe – Tuomo Suntola]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Topological GeometroDynamics (TGD)|| style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream|| style=&quot;background-color:#f2f2f2;&quot; | Gravity and gauge fields arise as induced geometric quantities on space-time surfaces|| style=&quot;background-color:#f2f2f2;&quot; | Unification through higher-dimensional embedding (''M''⁴ × ''CP''₂) with number-theoretical structure|| style=&quot;background-color:#f2f2f2;&quot; | Compatible with GR in a limit; proposes quantized cosmological transitions and discrete expansion phases
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://tgdtheory.fi/public_html/articles/TGDtheory2024I.pdf TGD Overview Part I] • [https://tgdtheory.fi/public_html/articles/TGDtheory2024II.pdf Part]
|-

|}

''Note: These initial theory rows are based on prior references and relevance to gravity-related claims. Contributors are invited to extend or edit each entry with additional details and evaluations.''

==== '''2.5 Evaluation Criteria (Unified Framework)''' ====
''This chapter now uses the unified evaluation framework applied consistently in Chapters 1 and 8.''

The theories discussed in this chapter are evaluated using the following eight standardized criteria:

# '''1. Empirical Adequacy'''  
Alignment with observed data and experimental results across relevant fields (e.g., astronomy, geology, biology, cosmology). The theory should account for major datasets and empirical patterns.

# '''2. Internal Consistency'''  
Logical and mathematical coherence of the theory. Assumptions should not contradict derived results. Definitions, equations, and claims must be used unambiguously.

# '''3. Predictive Power'''  
The ability to generate '''testable predictions''' that distinguish the theory from others. This includes retrospective explanations that are quantitatively verifiable.

# '''4. Cross-Disciplinary Compatibility'''  
Compatibility with findings and models in other domains, such as geology, biology, planetary science, and quantum physics. A good theory avoids contradicting established results in these areas.

# '''5. Conceptual Clarity and Simplicity'''  
The theory should be clearly framed, avoiding unnecessary complexity or vague terminology. It should be possible to identify what is a definition, an assumption, and a consequence.

# '''6. Heuristic and Unifying Value'''  
Capacity of the theory to stimulate new research, unify previously disconnected phenomena, or offer new conceptual insights across domains.

# '''7. Historical and Philosophical Insight'''  
Engagement with the evolution of cosmological thought. A good theory either improves upon, clarifies, or meaningfully integrates earlier models and philosophical foundations.

# '''8. Mathematical Rigor'''  
Use of precise mathematical definitions and derivations, including:  
* Clarity of assumptions and variable usage.  
* Proofs of existence/uniqueness where applicable.  
* Estimates of modeling error or solution stability.  
* Explicit formulations of approximations.  
* Avoidance of heuristic shortcuts unless well justified.  

This criterion ensures the theory stands up to the standards of formal modeling and does not rely on vague simulations or interchangeable terms.


See: [[AI-Assisted_Evaluation_of_Cosmological_Theories/Chapter_1:_Introduction_and_Evaluation_Criteria#1.3_List_of_Evaluation_Criteria|Chapter 1.3: List of Evaluation Criteria]].

== 2.6 AI Evaluation Based on Shared Criteria ==

The gravitational theories presented in this chapter — both mainstream and nonmainstream — have been organized and compared using a shared set of evaluation criteria (defined in 1.3 and 2.5). These criteria include empirical adequacy, internal consistency, explanatory and predictive power, simplicity, compatibility with other theories, and testability.

Based on a comparative review, the following general observations emerge from an AI-assisted perspective:

'''Empirical Adequacy'''  
Most mainstream theories (such as Newtonian gravity and General Relativity) align well with observed data in their respective domains (e.g., orbital mechanics, gravitational lensing). However, several nonmainstream theories — particularly those related to Earth expansion, fluid-based frameworks, or directional influx models — also show plausible alignment with specific observational domains (e.g., seafloor spreading, planetary formation patterns, cosmological constants). These alignments are often overlooked due to lack of institutional recognition rather than lack of evidence.

'''Internal Consistency'''  
Mainstream theories typically exhibit high mathematical rigor, though some require patchwork elements (e.g., dark matter, dark energy) to remain consistent with observations. Nonmainstream theories vary widely: some (like Emergent Gravity and Cosmic Influx Theory) display internal consistency within a novel framework; others (such as ionic growth or electromagnetic-gravity links) are still in conceptual stages but are not inherently incoherent.

'''Explanatory and Predictive Power'''  
General Relativity and Newtonian mechanics retain strong explanatory power within tested domains. Yet, nonmainstream models often attempt to '''explain what mainstream theories must assume''' — such as the origin of mass-energy, cosmic growth, or planetary structuring. Their predictions (e.g., mass increase, planetary positioning, or alternate gravitational behaviors) are often not pursued due to their marginal academic status, not their lack of testability.

'''Simplicity and Elegance'''  
Mainstream theories are mathematically elegant but conceptually complex at cosmological scales (e.g., inflation, cosmological constant). Some nonmainstream theories — including hydrodynamic, influx-based, or continuum models — offer '''conceptual simplicity''', modeling gravity as a flow or energetic interaction rather than a geometric property, which may appeal to intuitive or mechanical understanding.

'''Compatibility with Other Theories'''  
Mainstream gravity is well-integrated into broader physics. However, its dependence on unobserved components (dark matter/energy) creates tension. Nonmainstream theories often reject or reinterpret these components, leading to incompatibility — but also '''fresh opportunities for synthesis''', particularly with models of expansion, fluid mechanics, or entropy.

'''Falsifiability and Testability'''  
All theories in the table are testable to some degree. Nonmainstream theories propose specific predictions (e.g., preferred planetary distances, mass growth rates, coherence in ether flow) that could be falsified with targeted observational programs. The dismissal of such tests is often due to lack of funding or recognition, not due to scientific invalidity.

=== Conclusion ===
This AI-based evaluation does not rank or validate individual theories — that task will be undertaken in Chapter 8. Instead, it highlights that '''several nonmainstream theories demonstrate nontrivial alignment with scientific criteria''' and present testable alternatives or complements to established views. Their relative exclusion from mainstream scientific discourse is not always grounded in scientific rigor, but may reflect historical, institutional, or sociological inertia.

Such observations underscore the importance of comparative, transparent evaluations like this project. By explicitly applying the same criteria to all theories — regardless of their academic status — the AI-assisted framework enables a more balanced and evidence-based exploration of gravity and cosmology.

== '''2.7 Helicopter View – Scientific Treatment and Emerging Tensions''' ==

This chapter has introduced a wide range of gravitational theories, including both well-established mainstream models and lesser-known alternatives. While mainstream theories such as General Relativity are thoroughly integrated into education, research funding, and publication structures, many alternative models — including action-at-a-distance theories, flow-based gravitation, or push gravity — are often classified as fringe, regardless of their internal logic or historical pedigree.

Scientific platforms such as Wikipedia and large academic databases tend to reinforce this divide. Theories outside the mainstream are typically presented as either obsolete or pseudoscientific, even when they continue to generate peer-reviewed work or suggest novel interpretations.

This sharp categorization may hinder rather than help scientific progress. With the aid of AI-assisted comparison tools, this project seeks to provide a broader and more neutral framework. Instead of aligning with academic prestige or popular consensus, theories will be assessed based on a shared set of criteria (see '''Chapter 1.3'''), including empirical adequacy, internal consistency, predictive value, and compatibility with known physics.

As future chapters will show, some theories traditionally dismissed may offer insights that deserve reconsideration in light of recent observations — including anomalies revealed by the James Webb Space Telescope and new gravitational measurements. Readers are therefore encouraged to approach each theory not in terms of its reputation, but in terms of its explanatory and predictive potential.

Later chapters may expand this overview with additional context specific to cosmology, planetary evolution, geology, and biology.

== '''2.8 Next Steps''' ==

Expand the table with more entries

Begin cross-chapter references

Link phenomena such as expansion, planetary formation, and mass increase to these gravitational foundations

'''◀ [[AI-Assisted Evaluation of Cosmological Theories/Chapter 1: Introduction and Evaluation Criteria|Previous]] | [[AI-Assisted Evaluation of Cosmological Theories|Main Page]] | [[AI-Assisted Evaluation of Cosmological Theories/Chapter 3: Cosmic Expansion and Universe Models|Next ▶]]'''</text>
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      <comment>/* 3.4 Theory Mapping Table */ add An interesting theory related to this chapter is also TGD (see section 8.8.18).</comment>
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      <text bytes="14962" sha1="kqutd7slafnkmzub61yd86mbk1i3myg" xml:space="preserve">= '''Chapter 3: Cosmic Expansion and Universe Models''' =

== '''3.1 Overview''' ==

This chapter investigates how different cosmological theories interpret the expansion, geometry, and structure of the universe. Expansion is a foundational concept in many cosmological models, but it is not interpreted uniformly. Some see it as a literal stretching of spacetime; others interpret it as apparent or emergent.

We also examine whether the universe had a singular origin (e.g., Big Bang), has always existed, or is part of a cyclic or regenerative process.

== '''3.2 Comparison Categories''' ==

For clarity, we group models into two broad categories:

'''Mainstream Theories''': Widely supported by observational data and accepted within current scientific paradigms.

'''Non-Mainstream Theories''': Proposed alternatives that challenge standard assumptions or offer radically different mechanisms.

We recognize that these boundaries are not always sharp. A theory may shift categories over time as evidence accumulates or perspectives change.

Some non-mainstream models explore overlooked mechanisms or revive ideas that were once dismissed, offering fresh insights into cosmic evolution.

=== '''3.3 How to Contribute a Theory''' ===
Researchers and contributors are welcome to propose additional theories. These can be added directly to the Talk page or sent via email to:
''aitheroymapping@gmail.com''

All submissions will be included in the overview, mapped alongside other models, and evaluated using the same shared criteria outlined in Chapter 1.3.
This open framework ensures that all theories — whether mainstream or alternative — receive fair and consistent analysis.

=== '''3.4 Theory Mapping Table''' ===  

The table below maps mainstream and non-mainstream cosmological models according to several key dimensions:

* Whether they assume a beginning (e.g., Big Bang)  
* How they interpret expansion  
* Whether they assume continuous creation, cyclic processes, or finite evolution  
* Their treatment of geometry and underlying mechanisms  

This map is not exhaustive but offers a visual guide to the diversity of cosmic expansion theories.

{| class=&quot;wikitable&quot;
! Model Name
! Type
! Summary
! Mechanism of Expansion
! Relation to Gravity
! Key Observations Explained
|-
| ΛCDM Model (Standard Cosmology)
| Mainstream
| Universe began with a Big Bang, expanding ever since
| Dark energy (cosmological constant) drives acceleration
| Based on General Relativity
| Cosmic microwave background, galaxy redshift
|-
| Steady State Theory
| Non-mainstream
| Universe has no beginning or end; continuous matter creation
| Space expands, but new matter compensates
| Modified GR, no singularity
| Redshift, but inconsistent with CMB data
|-
| Plasma Cosmology
| Non-mainstream
| Universe shaped by electromagnetic forces; matter and plasma dominate
| No singular expansion; ongoing large-scale interactions
| Gravity secondary to EM forces
| Some filamentary structures; lacks CMB match
|-
| Cosmic Influx Theory (CIT)
| Non-mainstream
| Universe expands as influx increases local mass-energy
| Matter grows in an expanding universe
| Gravity is emergent from energy influx
| Galaxy redshift, predicted orbital structure
|-
| Quasi-Steady State Cosmology (QSSC)
| Non-mainstream
| Universe undergoes periodic mini-bangs superimposed on expansion
| Continuous creation through “C-fields” during episodic expansions
| Modified GR with additional scalar field
| Large-scale structure, avoids singularity
|-
| Dynamic Universe (TUOMOV)
| Non-mainstream
| Absolute 4D time structure; all phenomena emerge from motion geometry
| Apparent expansion from geometric evolution of zero-energy space
| Gravity arises from geometric curvature in absolute space
| Galaxy rotation, flatness, Hubble relation without dark energy
|-
| Dirac's G Variation Hypothesis
| Non-mainstream
| Fundamental constants vary over cosmic time; G decreases slowly
| Expansion and aging linked through variable gravitational coupling
| Gravity weakens over time, affecting cosmological evolution
| Possible explanation for anomalies in ancient astronomical data
|}
'''An interesting theory related to this chapter is also TGD (see section 8.8.18).''' 
TGD proposes a stepwise model of cosmic expansion, replacing continuous inflation with discrete rapid expansion periods. This perspective emerges from its geometric framework in which space-time surfaces evolve through topological phase transitions within a higher-dimensional embedding space (''M''⁴ × ''CP''₂).


==== '''3.5 Evaluation Criteria (Unified Framework)''' ====
''This chapter now uses the unified evaluation framework applied consistently in Chapters 1 and 8.''

The theories discussed in this chapter are evaluated using the following eight standardized criteria:

# '''1. Empirical Adequacy'''  
Alignment with observed data and experimental results across relevant fields (e.g., astronomy, geology, biology, cosmology). The theory should account for major datasets and empirical patterns.

# '''2. Internal Consistency'''  
Logical and mathematical coherence of the theory. Assumptions should not contradict derived results. Definitions, equations, and claims must be used unambiguously.

# '''3. Predictive Power'''  
The ability to generate '''testable predictions''' that distinguish the theory from others. This includes retrospective explanations that are quantitatively verifiable.

# '''4. Cross-Disciplinary Compatibility'''  
Compatibility with findings and models in other domains, such as geology, biology, planetary science, and quantum physics. A good theory avoids contradicting established results in these areas.

# '''5. Conceptual Clarity and Simplicity'''  
The theory should be clearly framed, avoiding unnecessary complexity or vague terminology. It should be possible to identify what is a definition, an assumption, and a consequence.

# '''6. Heuristic and Unifying Value'''  
Capacity of the theory to stimulate new research, unify previously disconnected phenomena, or offer new conceptual insights across domains.

# '''7. Historical and Philosophical Insight'''  
Engagement with the evolution of cosmological thought. A good theory either improves upon, clarifies, or meaningfully integrates earlier models and philosophical foundations.

# '''8. Mathematical Rigor'''  
Use of precise mathematical definitions and derivations, including:  
* Clarity of assumptions and variable usage.  
* Proofs of existence/uniqueness where applicable.  
* Estimates of modeling error or solution stability.  
* Explicit formulations of approximations.  
* Avoidance of heuristic shortcuts unless well justified.  

This criterion ensures the theory stands up to the standards of formal modeling and does not rely on vague simulations or interchangeable terms.

The results of this evaluation will be presented in the next subsection ('''3.6''') using a table format and AI-assisted commentary.

=== '''3.6 AI Evaluation Based on Shared Criteria''' ===  

The cosmological theories presented in this chapter — both mainstream and nonmainstream — have been organized and compared using the shared set of evaluation criteria (defined in '''Chapter 1.3''' and reiterated in '''3.5'''). These criteria include empirical adequacy, internal consistency, explanatory and predictive power, simplicity, unifying capacity, and philosophical clarity.

Based on a comparative review, the following general observations emerge from an AI-assisted perspective:

* '''Empirical Adequacy'''  
  Mainstream models such as ΛCDM demonstrate strong alignment with observational data, especially the cosmic microwave background (CMB), redshift-distance relation, and large-scale structure. However, their reliance on dark energy and dark matter — neither of which have been directly detected — leaves open questions.  
  Nonmainstream models like Cosmic Influx Theory (CIT), Quasi-Steady State Cosmology (QSSC), and Plasma Cosmology offer alternative explanations for some of the same data but lack the same breadth of observational support to date.

* '''Internal Consistency'''  
  Most mainstream models maintain formal mathematical consistency, especially through general relativity and the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. Some nonmainstream theories (e.g., Dynamic Universe) propose internally consistent but unconventional geometries. Others (e.g., Plasma Cosmology) require reinterpretation of fundamental assumptions (such as gravity vs. electromagnetism dominance), which can introduce tension with known physics.

* '''Explanatory and Predictive Power'''  
  ΛCDM successfully explains a wide range of observations, including the formation of galaxies and CMB fluctuations. However, it does not predict the values of constants or explain the underlying cause of expansion.  
  CIT and QSSC attempt to provide causal mechanisms for expansion (e.g., energy influx, cyclic mini-bangs), which increases their explanatory appeal, but these predictions are still undergoing validation.  
  Plasma Cosmology and other EM-based theories predict large-scale filamentary structures but struggle to match detailed CMB data.

* '''Simplicity'''  
  While ΛCDM is mathematically elegant, its reliance on six or more parameters (some empirically fitted) complicates claims of simplicity. Nonmainstream models sometimes introduce new mechanisms (e.g., PEWs in CIT, C-fields in QSSC), which may be seen as less parsimonious unless these mechanisms unify broader phenomena.

* '''Unifying Capacity'''  
  ΛCDM integrates with general relativity and the standard model of particle physics but does not fully unify gravity and quantum mechanics.  
  CIT seeks unification by linking cosmological expansion, planetary formation, and gravitational behavior through a single influx concept.  
  The Dynamic Universe proposes a single absolute-geometric framework for all physical phenomena, which suggests strong unification potential if validated.  
  Other models such as Plasma Cosmology emphasize electromagnetic coherence but do not yet bridge micro- and macro-physics comprehensively.

* '''Philosophical and Foundational Clarity'''  
  Mainstream models often avoid metaphysical speculation, focusing on measurable parameters. However, they rely on foundational assumptions (e.g., the cosmological principle, inflation, dark energy) that remain conceptually opaque.  
  Some nonmainstream theories offer greater philosophical clarity or ambition — for instance, CIT's reinterpretation of space, time, and energy as dynamically emergent quantities — but such frameworks often challenge established ontologies and thus face epistemic resistance.

Overall, this evaluation highlights the diverse approaches to explaining cosmic expansion. While ΛCDM remains dominant due to empirical fit and institutional support, alternative theories continue to probe foundational assumptions, offering both critique and inspiration for new lines of research.

=== '''3.7 Helicopter View – Scientific Reception and Philosophical Tensions''' ===

From an AI-assisted perspective, the treatment of cosmic expansion theories in scientific literature and institutions reveals both clear patterns and deeper philosophical tensions.

Mainstream models — especially the ΛCDM framework — benefit from broad institutional support, integration with high-precision observations (e.g., Planck CMB data), and consistency with the formal structure of general relativity. However, they rely on entities like '''dark matter''', '''dark energy''', and '''inflation''' that remain '''empirically unverified''' in direct experiments. Despite these open questions, alternative models are often marginalized, not necessarily because of weaker logical structures, but due to their '''incompatibility with prevailing paradigms''' or lack of immediate testability within existing infrastructure.

Nonmainstream theories such as the '''Quasi-Steady State Cosmology''', '''Plasma Cosmology''', and '''Cosmic Influx Theory (CIT)''' attempt to address these gaps by introducing '''novel mechanisms''' (e.g., cyclic regeneration, electromagnetic scaffolding, or energy influx). Yet these efforts often face dismissal or neglect, especially when they challenge core assumptions like metric expansion, the Big Bang singularity, or the constancy of physical laws. This reflects a broader philosophical inertia: once a theory is widely accepted, the burden of proof shifts asymmetrically against challengers.

A recurring philosophical divide concerns whether expansion is:
* A '''physical stretching of spacetime''', as in the FLRW metric;  
* An '''emergent or apparent phenomenon''', as argued in CIT or the Dynamic Universe;  
* Or even an '''illusion born from changes in energy-mass density''', as proposed by variable-G or mass-growth models.

This divergence reflects competing ontologies: one grounded in '''relativistic geometry''', the other in '''process-based evolution''' of mass-energy distributions.

Furthermore, the role of '''continuous creation''' or '''mass growth''' is largely taboo in mainstream cosmology, despite historical precedent in steady state theories. These concepts, while difficult to test, attempt to restore philosophical continuity — avoiding singular origins, and proposing an evolving, regenerative cosmos.

In conclusion, while mainstream expansion theories dominate the academic field, they do so with conceptual tensions still unresolved. Alternative models challenge these foundations and offer fresh, if speculative, perspectives. A truly open scientific process must remain attentive not only to empirical data, but to the '''philosophical framing''' that determines which questions are even allowed to be asked.

=== '''3.8 Next Steps''' ===

The next chapter will explore competing cosmological models more broadly, moving beyond gravity and expansion alone. We will examine how different theories describe the origin, structure, and fate of the universe — including steady-state models, cyclic universes, and various inflationary or regenerative frameworks.

Contributors are encouraged to:
* Propose new or overlooked cosmological models related to expansion  
* Improve the existing theory descriptions in this chapter  
* Suggest observational tests or falsifiability criteria for any of the models

All contributions will be evaluated using the shared criteria defined in Chapter 1.3.  
The goal is to foster an open, structured, and comparative framework for ongoing cosmological investigation.


----
'''Navigation:''' 
----
----
'''◀ [[AI-Assisted Evaluation of Cosmological Theories/Chapter 2: Gravity Theories – Comparison and Mapping|Previous]] | [[AI-Assisted Evaluation of Cosmological Theories|Main Page]] | [[AI-Assisted Evaluation of Cosmological Theories/Chapter 4: Planetary Formation, Disk Structures, and System Evolution|Next ▶]]'''
'</text>
      <sha1>kqutd7slafnkmzub61yd86mbk1i3myg</sha1>
    </revision>
    <revision>
      <id>2721183</id>
      <parentid>2721170</parentid>
      <timestamp>2025-07-10T08:58:53Z</timestamp>
      <contributor>
        <username>Ruud Loeffen</username>
        <id>2998353</id>
      </contributor>
      <comment>/* 3.4 Theory Mapping Table */ add TGD regurly to the table and removing the added text about TGD under the table</comment>
      <origin>2721183</origin>
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      <text bytes="15247" sha1="map8xp9qvrlkmzjikt57xql0svddajp" xml:space="preserve">= '''Chapter 3: Cosmic Expansion and Universe Models''' =

== '''3.1 Overview''' ==

This chapter investigates how different cosmological theories interpret the expansion, geometry, and structure of the universe. Expansion is a foundational concept in many cosmological models, but it is not interpreted uniformly. Some see it as a literal stretching of spacetime; others interpret it as apparent or emergent.

We also examine whether the universe had a singular origin (e.g., Big Bang), has always existed, or is part of a cyclic or regenerative process.

== '''3.2 Comparison Categories''' ==

For clarity, we group models into two broad categories:

'''Mainstream Theories''': Widely supported by observational data and accepted within current scientific paradigms.

'''Non-Mainstream Theories''': Proposed alternatives that challenge standard assumptions or offer radically different mechanisms.

We recognize that these boundaries are not always sharp. A theory may shift categories over time as evidence accumulates or perspectives change.

Some non-mainstream models explore overlooked mechanisms or revive ideas that were once dismissed, offering fresh insights into cosmic evolution.

=== '''3.3 How to Contribute a Theory''' ===
Researchers and contributors are welcome to propose additional theories. These can be added directly to the Talk page or sent via email to:
''aitheroymapping@gmail.com''

All submissions will be included in the overview, mapped alongside other models, and evaluated using the same shared criteria outlined in Chapter 1.3.
This open framework ensures that all theories — whether mainstream or alternative — receive fair and consistent analysis.

=== '''3.4 Theory Mapping Table''' ===  

The table below maps mainstream and non-mainstream cosmological models according to several key dimensions:

* Whether they assume a beginning (e.g., Big Bang)  
* How they interpret expansion  
* Whether they assume continuous creation, cyclic processes, or finite evolution  
* Their treatment of geometry and underlying mechanisms  

This map is not exhaustive but offers a visual guide to the diversity of cosmic expansion theories.

{| class=&quot;wikitable&quot;
! Model Name
! Type
! Summary
! Mechanism of Expansion
! Relation to Gravity
! Key Observations Explained
|-
| ΛCDM Model (Standard Cosmology)
| Mainstream
| Universe began with a Big Bang, expanding ever since
| Dark energy (cosmological constant) drives acceleration
| Based on General Relativity
| Cosmic microwave background, galaxy redshift
|-
| Steady State Theory
| Non-mainstream
| Universe has no beginning or end; continuous matter creation
| Space expands, but new matter compensates
| Modified GR, no singularity
| Redshift, but inconsistent with CMB data
|-
| Plasma Cosmology
| Non-mainstream
| Universe shaped by electromagnetic forces; matter and plasma dominate
| No singular expansion; ongoing large-scale interactions
| Gravity secondary to EM forces
| Some filamentary structures; lacks CMB match
|-
| Cosmic Influx Theory (CIT)
| Non-mainstream
| Universe expands as influx increases local mass-energy
| Matter grows in an expanding universe
| Gravity is emergent from energy influx
| Galaxy redshift, predicted orbital structure
|-
| Quasi-Steady State Cosmology (QSSC)
| Non-mainstream
| Universe undergoes periodic mini-bangs superimposed on expansion
| Continuous creation through “C-fields” during episodic expansions
| Modified GR with additional scalar field
| Large-scale structure, avoids singularity
|-
| Dynamic Universe (TUOMOV)
| Non-mainstream
| Absolute 4D time structure; all phenomena emerge from motion geometry
| Apparent expansion from geometric evolution of zero-energy space
| Gravity arises from geometric curvature in absolute space
| Galaxy rotation, flatness, Hubble relation without dark energy
|-
| Dirac's G Variation Hypothesis
| Non-mainstream
| Fundamental constants vary over cosmic time; G decreases slowly
| Expansion and aging linked through variable gravitational coupling
| Gravity weakens over time, affecting cosmological evolution
| Possible explanation for anomalies in ancient astronomical data
|- 
| style=&quot;background-color:#f2f2f2;&quot; | Topological GeometroDynamics (TGD) 
|| style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream 
|| style=&quot;background-color:#f2f2f2;&quot; | Cosmic evolution proceeds in discrete rapid expansion phases 
|| style=&quot;background-color:#f2f2f2;&quot; | Expansion events emerge from topological changes in embedded space-time surfaces (''M''⁴ × ''CP''₂) 
|| style=&quot;background-color:#f2f2f2;&quot; | Provides an alternative to smooth inflation; aligns with biological and planetary transitions
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://tgdtheory.fi/public_html/articles/preCE.pdf Expanding Earth and Sudden Biological Emergence – TGD]

|}


==== '''3.5 Evaluation Criteria (Unified Framework)''' ====
''This chapter now uses the unified evaluation framework applied consistently in Chapters 1 and 8.''

The theories discussed in this chapter are evaluated using the following eight standardized criteria:

# '''1. Empirical Adequacy'''  
Alignment with observed data and experimental results across relevant fields (e.g., astronomy, geology, biology, cosmology). The theory should account for major datasets and empirical patterns.

# '''2. Internal Consistency'''  
Logical and mathematical coherence of the theory. Assumptions should not contradict derived results. Definitions, equations, and claims must be used unambiguously.

# '''3. Predictive Power'''  
The ability to generate '''testable predictions''' that distinguish the theory from others. This includes retrospective explanations that are quantitatively verifiable.

# '''4. Cross-Disciplinary Compatibility'''  
Compatibility with findings and models in other domains, such as geology, biology, planetary science, and quantum physics. A good theory avoids contradicting established results in these areas.

# '''5. Conceptual Clarity and Simplicity'''  
The theory should be clearly framed, avoiding unnecessary complexity or vague terminology. It should be possible to identify what is a definition, an assumption, and a consequence.

# '''6. Heuristic and Unifying Value'''  
Capacity of the theory to stimulate new research, unify previously disconnected phenomena, or offer new conceptual insights across domains.

# '''7. Historical and Philosophical Insight'''  
Engagement with the evolution of cosmological thought. A good theory either improves upon, clarifies, or meaningfully integrates earlier models and philosophical foundations.

# '''8. Mathematical Rigor'''  
Use of precise mathematical definitions and derivations, including:  
* Clarity of assumptions and variable usage.  
* Proofs of existence/uniqueness where applicable.  
* Estimates of modeling error or solution stability.  
* Explicit formulations of approximations.  
* Avoidance of heuristic shortcuts unless well justified.  

This criterion ensures the theory stands up to the standards of formal modeling and does not rely on vague simulations or interchangeable terms.

The results of this evaluation will be presented in the next subsection ('''3.6''') using a table format and AI-assisted commentary.

=== '''3.6 AI Evaluation Based on Shared Criteria''' ===  

The cosmological theories presented in this chapter — both mainstream and nonmainstream — have been organized and compared using the shared set of evaluation criteria (defined in '''Chapter 1.3''' and reiterated in '''3.5'''). These criteria include empirical adequacy, internal consistency, explanatory and predictive power, simplicity, unifying capacity, and philosophical clarity.

Based on a comparative review, the following general observations emerge from an AI-assisted perspective:

* '''Empirical Adequacy'''  
  Mainstream models such as ΛCDM demonstrate strong alignment with observational data, especially the cosmic microwave background (CMB), redshift-distance relation, and large-scale structure. However, their reliance on dark energy and dark matter — neither of which have been directly detected — leaves open questions.  
  Nonmainstream models like Cosmic Influx Theory (CIT), Quasi-Steady State Cosmology (QSSC), and Plasma Cosmology offer alternative explanations for some of the same data but lack the same breadth of observational support to date.

* '''Internal Consistency'''  
  Most mainstream models maintain formal mathematical consistency, especially through general relativity and the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. Some nonmainstream theories (e.g., Dynamic Universe) propose internally consistent but unconventional geometries. Others (e.g., Plasma Cosmology) require reinterpretation of fundamental assumptions (such as gravity vs. electromagnetism dominance), which can introduce tension with known physics.

* '''Explanatory and Predictive Power'''  
  ΛCDM successfully explains a wide range of observations, including the formation of galaxies and CMB fluctuations. However, it does not predict the values of constants or explain the underlying cause of expansion.  
  CIT and QSSC attempt to provide causal mechanisms for expansion (e.g., energy influx, cyclic mini-bangs), which increases their explanatory appeal, but these predictions are still undergoing validation.  
  Plasma Cosmology and other EM-based theories predict large-scale filamentary structures but struggle to match detailed CMB data.

* '''Simplicity'''  
  While ΛCDM is mathematically elegant, its reliance on six or more parameters (some empirically fitted) complicates claims of simplicity. Nonmainstream models sometimes introduce new mechanisms (e.g., PEWs in CIT, C-fields in QSSC), which may be seen as less parsimonious unless these mechanisms unify broader phenomena.

* '''Unifying Capacity'''  
  ΛCDM integrates with general relativity and the standard model of particle physics but does not fully unify gravity and quantum mechanics.  
  CIT seeks unification by linking cosmological expansion, planetary formation, and gravitational behavior through a single influx concept.  
  The Dynamic Universe proposes a single absolute-geometric framework for all physical phenomena, which suggests strong unification potential if validated.  
  Other models such as Plasma Cosmology emphasize electromagnetic coherence but do not yet bridge micro- and macro-physics comprehensively.

* '''Philosophical and Foundational Clarity'''  
  Mainstream models often avoid metaphysical speculation, focusing on measurable parameters. However, they rely on foundational assumptions (e.g., the cosmological principle, inflation, dark energy) that remain conceptually opaque.  
  Some nonmainstream theories offer greater philosophical clarity or ambition — for instance, CIT's reinterpretation of space, time, and energy as dynamically emergent quantities — but such frameworks often challenge established ontologies and thus face epistemic resistance.

Overall, this evaluation highlights the diverse approaches to explaining cosmic expansion. While ΛCDM remains dominant due to empirical fit and institutional support, alternative theories continue to probe foundational assumptions, offering both critique and inspiration for new lines of research.

=== '''3.7 Helicopter View – Scientific Reception and Philosophical Tensions''' ===

From an AI-assisted perspective, the treatment of cosmic expansion theories in scientific literature and institutions reveals both clear patterns and deeper philosophical tensions.

Mainstream models — especially the ΛCDM framework — benefit from broad institutional support, integration with high-precision observations (e.g., Planck CMB data), and consistency with the formal structure of general relativity. However, they rely on entities like '''dark matter''', '''dark energy''', and '''inflation''' that remain '''empirically unverified''' in direct experiments. Despite these open questions, alternative models are often marginalized, not necessarily because of weaker logical structures, but due to their '''incompatibility with prevailing paradigms''' or lack of immediate testability within existing infrastructure.

Nonmainstream theories such as the '''Quasi-Steady State Cosmology''', '''Plasma Cosmology''', and '''Cosmic Influx Theory (CIT)''' attempt to address these gaps by introducing '''novel mechanisms''' (e.g., cyclic regeneration, electromagnetic scaffolding, or energy influx). Yet these efforts often face dismissal or neglect, especially when they challenge core assumptions like metric expansion, the Big Bang singularity, or the constancy of physical laws. This reflects a broader philosophical inertia: once a theory is widely accepted, the burden of proof shifts asymmetrically against challengers.

A recurring philosophical divide concerns whether expansion is:
* A '''physical stretching of spacetime''', as in the FLRW metric;  
* An '''emergent or apparent phenomenon''', as argued in CIT or the Dynamic Universe;  
* Or even an '''illusion born from changes in energy-mass density''', as proposed by variable-G or mass-growth models.

This divergence reflects competing ontologies: one grounded in '''relativistic geometry''', the other in '''process-based evolution''' of mass-energy distributions.

Furthermore, the role of '''continuous creation''' or '''mass growth''' is largely taboo in mainstream cosmology, despite historical precedent in steady state theories. These concepts, while difficult to test, attempt to restore philosophical continuity — avoiding singular origins, and proposing an evolving, regenerative cosmos.

In conclusion, while mainstream expansion theories dominate the academic field, they do so with conceptual tensions still unresolved. Alternative models challenge these foundations and offer fresh, if speculative, perspectives. A truly open scientific process must remain attentive not only to empirical data, but to the '''philosophical framing''' that determines which questions are even allowed to be asked.

=== '''3.8 Next Steps''' ===

The next chapter will explore competing cosmological models more broadly, moving beyond gravity and expansion alone. We will examine how different theories describe the origin, structure, and fate of the universe — including steady-state models, cyclic universes, and various inflationary or regenerative frameworks.

Contributors are encouraged to:
* Propose new or overlooked cosmological models related to expansion  
* Improve the existing theory descriptions in this chapter  
* Suggest observational tests or falsifiability criteria for any of the models

All contributions will be evaluated using the shared criteria defined in Chapter 1.3.  
The goal is to foster an open, structured, and comparative framework for ongoing cosmological investigation.


----
'''Navigation:''' 
----
----
'''◀ [[AI-Assisted Evaluation of Cosmological Theories/Chapter 2: Gravity Theories – Comparison and Mapping|Previous]] | [[AI-Assisted Evaluation of Cosmological Theories|Main Page]] | [[AI-Assisted Evaluation of Cosmological Theories/Chapter 4: Planetary Formation, Disk Structures, and System Evolution|Next ▶]]'''
'</text>
      <sha1>map8xp9qvrlkmzjikt57xql0svddajp</sha1>
    </revision>
    <revision>
      <id>2721187</id>
      <parentid>2721183</parentid>
      <timestamp>2025-07-10T10:07:35Z</timestamp>
      <contributor>
        <username>Ruud Loeffen</username>
        <id>2998353</id>
      </contributor>
      <minor />
      <comment>/* 3.4 Theory Mapping Table */ add TGD correctly to the table</comment>
      <origin>2721187</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15011" sha1="001nk5znyuhov9be6y0agnhhk8fgf2k" xml:space="preserve">= '''Chapter 3: Cosmic Expansion and Universe Models''' =

== '''3.1 Overview''' ==

This chapter investigates how different cosmological theories interpret the expansion, geometry, and structure of the universe. Expansion is a foundational concept in many cosmological models, but it is not interpreted uniformly. Some see it as a literal stretching of spacetime; others interpret it as apparent or emergent.

We also examine whether the universe had a singular origin (e.g., Big Bang), has always existed, or is part of a cyclic or regenerative process.

== '''3.2 Comparison Categories''' ==

For clarity, we group models into two broad categories:

'''Mainstream Theories''': Widely supported by observational data and accepted within current scientific paradigms.

'''Non-Mainstream Theories''': Proposed alternatives that challenge standard assumptions or offer radically different mechanisms.

We recognize that these boundaries are not always sharp. A theory may shift categories over time as evidence accumulates or perspectives change.

Some non-mainstream models explore overlooked mechanisms or revive ideas that were once dismissed, offering fresh insights into cosmic evolution.

=== '''3.3 How to Contribute a Theory''' ===
Researchers and contributors are welcome to propose additional theories. These can be added directly to the Talk page or sent via email to:
''aitheroymapping@gmail.com''

All submissions will be included in the overview, mapped alongside other models, and evaluated using the same shared criteria outlined in Chapter 1.3.
This open framework ensures that all theories — whether mainstream or alternative — receive fair and consistent analysis.

=== '''3.4 Theory Mapping Table''' ===  

The table below maps mainstream and non-mainstream cosmological models according to several key dimensions:

* Whether they assume a beginning (e.g., Big Bang)  
* How they interpret expansion  
* Whether they assume continuous creation, cyclic processes, or finite evolution  
* Their treatment of geometry and underlying mechanisms  

This map is not exhaustive but offers a visual guide to the diversity of cosmic expansion theories.

{| class=&quot;wikitable&quot;
! Model Name
! Type
! Summary
! Mechanism of Expansion
! Relation to Gravity
! Key Observations Explained
|-
| ΛCDM Model (Standard Cosmology)
| Mainstream
| Universe began with a Big Bang, expanding ever since
| Dark energy (cosmological constant) drives acceleration
| Based on General Relativity
| Cosmic microwave background, galaxy redshift
|-
| Steady State Theory
| Non-mainstream
| Universe has no beginning or end; continuous matter creation
| Space expands, but new matter compensates
| Modified GR, no singularity
| Redshift, but inconsistent with CMB data
|-
| Plasma Cosmology
| Non-mainstream
| Universe shaped by electromagnetic forces; matter and plasma dominate
| No singular expansion; ongoing large-scale interactions
| Gravity secondary to EM forces
| Some filamentary structures; lacks CMB match
|-
| Cosmic Influx Theory (CIT)
| Non-mainstream
| Universe expands as influx increases local mass-energy
| Matter grows in an expanding universe
| Gravity is emergent from energy influx
| Galaxy redshift, predicted orbital structure
|-
| Quasi-Steady State Cosmology (QSSC)
| Non-mainstream
| Universe undergoes periodic mini-bangs superimposed on expansion
| Continuous creation through “C-fields” during episodic expansions
| Modified GR with additional scalar field
| Large-scale structure, avoids singularity
|-
| Dynamic Universe (TUOMOV)
| Non-mainstream
| Absolute 4D time structure; all phenomena emerge from motion geometry
| Apparent expansion from geometric evolution of zero-energy space
| Gravity arises from geometric curvature in absolute space
| Galaxy rotation, flatness, Hubble relation without dark energy
|-
| Dirac's G Variation Hypothesis
| Non-mainstream
| Fundamental constants vary over cosmic time; G decreases slowly
| Expansion and aging linked through variable gravitational coupling
| Gravity weakens over time, affecting cosmological evolution
| Possible explanation for anomalies in ancient astronomical data
|- 
| Topological GeometroDynamics (TGD) 
| Non-mainstream 
| Cosmic evolution proceeds in discrete rapid expansion phases 
| Expansion events emerge from topological changes in embedded space-time surfaces (''M''⁴ × ''CP''₂) 
| Provides an alternative to smooth inflation; aligns with biological and planetary transitions
| Explains discontinuous structure formation, galaxy evolution, and planetary size shifts via stepwise expansion events
|-


|}


==== '''3.5 Evaluation Criteria (Unified Framework)''' ====
''This chapter now uses the unified evaluation framework applied consistently in Chapters 1 and 8.''

The theories discussed in this chapter are evaluated using the following eight standardized criteria:

# '''1. Empirical Adequacy'''  
Alignment with observed data and experimental results across relevant fields (e.g., astronomy, geology, biology, cosmology). The theory should account for major datasets and empirical patterns.

# '''2. Internal Consistency'''  
Logical and mathematical coherence of the theory. Assumptions should not contradict derived results. Definitions, equations, and claims must be used unambiguously.

# '''3. Predictive Power'''  
The ability to generate '''testable predictions''' that distinguish the theory from others. This includes retrospective explanations that are quantitatively verifiable.

# '''4. Cross-Disciplinary Compatibility'''  
Compatibility with findings and models in other domains, such as geology, biology, planetary science, and quantum physics. A good theory avoids contradicting established results in these areas.

# '''5. Conceptual Clarity and Simplicity'''  
The theory should be clearly framed, avoiding unnecessary complexity or vague terminology. It should be possible to identify what is a definition, an assumption, and a consequence.

# '''6. Heuristic and Unifying Value'''  
Capacity of the theory to stimulate new research, unify previously disconnected phenomena, or offer new conceptual insights across domains.

# '''7. Historical and Philosophical Insight'''  
Engagement with the evolution of cosmological thought. A good theory either improves upon, clarifies, or meaningfully integrates earlier models and philosophical foundations.

# '''8. Mathematical Rigor'''  
Use of precise mathematical definitions and derivations, including:  
* Clarity of assumptions and variable usage.  
* Proofs of existence/uniqueness where applicable.  
* Estimates of modeling error or solution stability.  
* Explicit formulations of approximations.  
* Avoidance of heuristic shortcuts unless well justified.  

This criterion ensures the theory stands up to the standards of formal modeling and does not rely on vague simulations or interchangeable terms.

The results of this evaluation will be presented in the next subsection ('''3.6''') using a table format and AI-assisted commentary.

=== '''3.6 AI Evaluation Based on Shared Criteria''' ===  

The cosmological theories presented in this chapter — both mainstream and nonmainstream — have been organized and compared using the shared set of evaluation criteria (defined in '''Chapter 1.3''' and reiterated in '''3.5'''). These criteria include empirical adequacy, internal consistency, explanatory and predictive power, simplicity, unifying capacity, and philosophical clarity.

Based on a comparative review, the following general observations emerge from an AI-assisted perspective:

* '''Empirical Adequacy'''  
  Mainstream models such as ΛCDM demonstrate strong alignment with observational data, especially the cosmic microwave background (CMB), redshift-distance relation, and large-scale structure. However, their reliance on dark energy and dark matter — neither of which have been directly detected — leaves open questions.  
  Nonmainstream models like Cosmic Influx Theory (CIT), Quasi-Steady State Cosmology (QSSC), and Plasma Cosmology offer alternative explanations for some of the same data but lack the same breadth of observational support to date.

* '''Internal Consistency'''  
  Most mainstream models maintain formal mathematical consistency, especially through general relativity and the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. Some nonmainstream theories (e.g., Dynamic Universe) propose internally consistent but unconventional geometries. Others (e.g., Plasma Cosmology) require reinterpretation of fundamental assumptions (such as gravity vs. electromagnetism dominance), which can introduce tension with known physics.

* '''Explanatory and Predictive Power'''  
  ΛCDM successfully explains a wide range of observations, including the formation of galaxies and CMB fluctuations. However, it does not predict the values of constants or explain the underlying cause of expansion.  
  CIT and QSSC attempt to provide causal mechanisms for expansion (e.g., energy influx, cyclic mini-bangs), which increases their explanatory appeal, but these predictions are still undergoing validation.  
  Plasma Cosmology and other EM-based theories predict large-scale filamentary structures but struggle to match detailed CMB data.

* '''Simplicity'''  
  While ΛCDM is mathematically elegant, its reliance on six or more parameters (some empirically fitted) complicates claims of simplicity. Nonmainstream models sometimes introduce new mechanisms (e.g., PEWs in CIT, C-fields in QSSC), which may be seen as less parsimonious unless these mechanisms unify broader phenomena.

* '''Unifying Capacity'''  
  ΛCDM integrates with general relativity and the standard model of particle physics but does not fully unify gravity and quantum mechanics.  
  CIT seeks unification by linking cosmological expansion, planetary formation, and gravitational behavior through a single influx concept.  
  The Dynamic Universe proposes a single absolute-geometric framework for all physical phenomena, which suggests strong unification potential if validated.  
  Other models such as Plasma Cosmology emphasize electromagnetic coherence but do not yet bridge micro- and macro-physics comprehensively.

* '''Philosophical and Foundational Clarity'''  
  Mainstream models often avoid metaphysical speculation, focusing on measurable parameters. However, they rely on foundational assumptions (e.g., the cosmological principle, inflation, dark energy) that remain conceptually opaque.  
  Some nonmainstream theories offer greater philosophical clarity or ambition — for instance, CIT's reinterpretation of space, time, and energy as dynamically emergent quantities — but such frameworks often challenge established ontologies and thus face epistemic resistance.

Overall, this evaluation highlights the diverse approaches to explaining cosmic expansion. While ΛCDM remains dominant due to empirical fit and institutional support, alternative theories continue to probe foundational assumptions, offering both critique and inspiration for new lines of research.

=== '''3.7 Helicopter View – Scientific Reception and Philosophical Tensions''' ===

From an AI-assisted perspective, the treatment of cosmic expansion theories in scientific literature and institutions reveals both clear patterns and deeper philosophical tensions.

Mainstream models — especially the ΛCDM framework — benefit from broad institutional support, integration with high-precision observations (e.g., Planck CMB data), and consistency with the formal structure of general relativity. However, they rely on entities like '''dark matter''', '''dark energy''', and '''inflation''' that remain '''empirically unverified''' in direct experiments. Despite these open questions, alternative models are often marginalized, not necessarily because of weaker logical structures, but due to their '''incompatibility with prevailing paradigms''' or lack of immediate testability within existing infrastructure.

Nonmainstream theories such as the '''Quasi-Steady State Cosmology''', '''Plasma Cosmology''', and '''Cosmic Influx Theory (CIT)''' attempt to address these gaps by introducing '''novel mechanisms''' (e.g., cyclic regeneration, electromagnetic scaffolding, or energy influx). Yet these efforts often face dismissal or neglect, especially when they challenge core assumptions like metric expansion, the Big Bang singularity, or the constancy of physical laws. This reflects a broader philosophical inertia: once a theory is widely accepted, the burden of proof shifts asymmetrically against challengers.

A recurring philosophical divide concerns whether expansion is:
* A '''physical stretching of spacetime''', as in the FLRW metric;  
* An '''emergent or apparent phenomenon''', as argued in CIT or the Dynamic Universe;  
* Or even an '''illusion born from changes in energy-mass density''', as proposed by variable-G or mass-growth models.

This divergence reflects competing ontologies: one grounded in '''relativistic geometry''', the other in '''process-based evolution''' of mass-energy distributions.

Furthermore, the role of '''continuous creation''' or '''mass growth''' is largely taboo in mainstream cosmology, despite historical precedent in steady state theories. These concepts, while difficult to test, attempt to restore philosophical continuity — avoiding singular origins, and proposing an evolving, regenerative cosmos.

In conclusion, while mainstream expansion theories dominate the academic field, they do so with conceptual tensions still unresolved. Alternative models challenge these foundations and offer fresh, if speculative, perspectives. A truly open scientific process must remain attentive not only to empirical data, but to the '''philosophical framing''' that determines which questions are even allowed to be asked.

=== '''3.8 Next Steps''' ===

The next chapter will explore competing cosmological models more broadly, moving beyond gravity and expansion alone. We will examine how different theories describe the origin, structure, and fate of the universe — including steady-state models, cyclic universes, and various inflationary or regenerative frameworks.

Contributors are encouraged to:
* Propose new or overlooked cosmological models related to expansion  
* Improve the existing theory descriptions in this chapter  
* Suggest observational tests or falsifiability criteria for any of the models

All contributions will be evaluated using the shared criteria defined in Chapter 1.3.  
The goal is to foster an open, structured, and comparative framework for ongoing cosmological investigation.


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'''◀ [[AI-Assisted Evaluation of Cosmological Theories/Chapter 2: Gravity Theories – Comparison and Mapping|Previous]] | [[AI-Assisted Evaluation of Cosmological Theories|Main Page]] | [[AI-Assisted Evaluation of Cosmological Theories/Chapter 4: Planetary Formation, Disk Structures, and System Evolution|Next ▶]]'''
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      <text bytes="20333" sha1="bvv4fqv7hx3s824f4srwov4sex9x3gu" xml:space="preserve">= '''Chapter 4: Planetary Formation, Disk Structures, and System Evolution''' =

=== '''4.1 Purpose – Overview of Planetary Formation Concepts''' ===

This chapter investigates how different scientific theories explain the origin and evolution of planetary systems. Central questions include: How do protoplanetary disks evolve? What mechanisms lead to the formation of planets, moons, and rings? What determines the distribution, composition, and orbits of celestial bodies?

Traditional models focus on accretion within a rotating disk of gas and dust. However, alternative theories propose electromagnetic structuring, influx-driven aggregation, or even mass growth over time. Observations of protoplanetary disks and exoplanet systems — particularly from ALMA and JWST — continue to refine our understanding.

This chapter maps mainstream and non-mainstream theories of planetary formation, evaluates their explanatory power, and identifies key areas for further research and testing.

=== '''4.2 Scope – Mainstream and Non-mainstream Theories''' ===

Theories of planetary formation can be broadly grouped into two categories:

'''Mainstream Theories''': These models are widely accepted in academic literature and supported by high-resolution observations and simulations. They include the nebular hypothesis, core accretion model, and disk instability theory. Such models typically assume a gravitationally bound disk of gas and dust evolving under Newtonian or relativistic dynamics.

'''Non-Mainstream Theories''': These models challenge or extend the assumptions of mainstream frameworks. They include theories that invoke directed energy influx, mass growth over time, electromagnetic structuring of matter, or cosmological context beyond local disk dynamics.

The boundary between these categories is not fixed. As new observational data emerges — especially from instruments like ALMA and JWST — some alternative models gain relevance or find partial confirmation. Conversely, unresolved anomalies may prompt re-examination of established assumptions.

This chapter offers space for both types of theories to be presented, analyzed, and improved within a common framework.

=== '''4.3 How to Contribute a Theory''' ===

Researchers and contributors are welcome to propose additional theories of planetary formation and system evolution. Submissions may include observational insights, mathematical models, conceptual frameworks, or hybrid approaches.

You can contribute in either of the following ways:
* Post directly on the '''Talk page''' of this chapter  
* Or email your proposal to: '''aitheroymapping@gmail.com'''

All submitted theories will be added to the mapping and evaluated using the shared criteria defined in '''Chapter 1.3: Evaluation Criteria'''.  
We encourage both mainstream and non-mainstream contributors to join this open effort to compare and improve our understanding of planetary system development.

=== '''4.4 Theory Mapping Table''' ===

The following table presents a structured overview of theories describing planetary formation and system evolution. It includes both well-established mainstream models and non-mainstream alternatives that propose different mechanisms. Each theory is briefly characterized by its formation process, key mechanisms, and observational relevance.

{| class=&quot;wikitable&quot;
! Theory Name !! Category !! Mechanism Summary !! Key Features !! Observational Relevance
|-
| style=&quot;background-color:#ffffff;&quot; | Core Accretion || style=&quot;background-color:#ffffff;&quot; | Mainstream || style=&quot;background-color:#ffffff;&quot; | Dust grains coalesce into planetesimals, which grow via collisions and gravitational attraction || style=&quot;background-color:#ffffff;&quot; | Explains terrestrial planets and gas giants; slow growth timescales || style=&quot;background-color:#ffffff;&quot; | Supported by simulations; matches Solar System formation
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://en.wikipedia.org/wiki/Planetary_formation#Core_accretion_model Core Accretion Model – Wikipedia]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Disk Instability || style=&quot;background-color:#f2f2f2;&quot; | Mainstream || style=&quot;background-color:#f2f2f2;&quot; | Dense regions of the disk collapse directly under self-gravity into gas giants || style=&quot;background-color:#f2f2f2;&quot; | Fast formation timescale; explains wide-orbit gas giants || style=&quot;background-color:#f2f2f2;&quot; | Matches some exoplanet distributions and brown dwarfs
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://en.wikipedia.org/wiki/Planetary_formation#Disk_instability_model Disk Instability – Wikipedia]
|-
| style=&quot;background-color:#ffffff;&quot; | Pebble Accretion || style=&quot;background-color:#ffffff;&quot; | Mainstream || style=&quot;background-color:#ffffff;&quot; | Small particles (“pebbles”) are rapidly accreted onto growing cores via drag-assisted capture || style=&quot;background-color:#ffffff;&quot; | Resolves growth-time problem; efficient in outer disks || style=&quot;background-color:#ffffff;&quot; | Supported by ALMA observations of dust substructure
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://en.wikipedia.org/wiki/Pebble_accretion Pebble Accretion – Wikipedia]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Streaming Instability || style=&quot;background-color:#f2f2f2;&quot; | Mainstream || style=&quot;background-color:#f2f2f2;&quot; | Dust particles cluster due to aerodynamic drag and mutual feedback, leading to rapid collapse || style=&quot;background-color:#f2f2f2;&quot; | Explains initial planetesimal formation; self-consistent with disk physics || style=&quot;background-color:#f2f2f2;&quot; | Matches simulations and dust clumping seen in ALMA data
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://en.wikipedia.org/wiki/Streaming_instability Streaming Instability – Wikipedia]
|-
| style=&quot;background-color:#ffffff;&quot; | Electromagnetic Nebula Concepts || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Magnetic and electric fields shape disk structure and help organize matter || style=&quot;background-color:#ffffff;&quot; | Offers structured alternatives to gravitational-only models || style=&quot;background-color:#ffffff;&quot; | Explains filaments, rings, and possibly dust traps
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://www.thunderbolts.info/wp/2015/09/23/planetary-birthing-in-electrical-cosmos/ Thunderbolts – Planetary Birthing Concepts]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Influx-Driven Aggregation || style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream || style=&quot;background-color:#f2f2f2;&quot; | Energy influx increases local density, triggering mass aggregation || style=&quot;background-color:#f2f2f2;&quot; | Interprets planet formation as energy-structured rather than accretion-based || style=&quot;background-color:#f2f2f2;&quot; | Aligns with early ring structures seen in ALMA disks
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://chat.openai.com/share/08a0e0d2-0171-419b-94f3-0ec1c9c8ba4c Influx-Driven Aggregation – ChatGPT Session]
|-
| style=&quot;background-color:#ffffff;&quot; | Cosmic Influx Theory (CIT) – Planetary Application || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Preferred rings form in protoplanetary disks at distances proportional to central star mass || style=&quot;background-color:#ffffff;&quot; | Predicts planet formation at fixed Preferred Distances (Dₚᵣₑf); continuous mass influx || style=&quot;background-color:#ffffff;&quot; | Matches observed ring gaps and exoplanet concentrations
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://en.wikiversity.org/wiki/Cosmic_Influx_Theory Cosmic Influx Theory – Wikiversity Page]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Magnetic Grand Design || style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream || style=&quot;background-color:#f2f2f2;&quot; | Large-scale magnetic fields organize angular momentum and structure during early disk evolution || style=&quot;background-color:#f2f2f2;&quot; | Explains symmetry in spiral arms and inner disk structuring || style=&quot;background-color:#f2f2f2;&quot; | Under investigation via polarized ALMA observations
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://ui.adsabs.harvard.edu/abs/2017MNRAS.467.1314K/ Magnetic Grand Design – Kotarba et al. (2017)]
|-
| style=&quot;background-color:#ffffff;&quot; | Le Sage-like Aggregation || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Directional particle fluxes (Le Sage-type) cause shadowing effects that pull matter together || style=&quot;background-color:#ffffff;&quot; | Explains attraction without curvature or force; basis for influx models || style=&quot;background-color:#ffffff;&quot; | Conceptual foundation for CIT and earlier corpuscular theories
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://en.wikipedia.org/wiki/Le_Sage%27s_theory_of_gravitation Le Sage’s Theory of Gravitation – Wikipedia]
|}
'''An interesting theory related to this chapter is also TGD (see section 8.8.18).''' 
TGD proposes that planetary and stellar structures form through the emission of “magnetic bubbles” — localized topological transitions analogous to mini big bangs. These events arise from the dynamics of space-time surfaces embedded in a higher-dimensional space (''M''⁴ × ''CP''₂), offering an alternative to gradual accretion in disk models.


''Note: The inclusion of mainstream and non-mainstream theories reflects the open and comparative structure of this project. Contributions are welcome.''

=== '''4.5 Evaluation Criteria and Comparative Table''' ===  

''This chapter now uses the unified evaluation framework applied consistently in Chapters 1 and 8.''

The theories discussed in this chapter are evaluated using the following eight standardized criteria:

# '''1. Empirical Adequacy'''  
Alignment with observed data and experimental results across relevant fields (e.g., astronomy, geology, biology, cosmology). The theory should account for major datasets and empirical patterns.

# '''2. Internal Consistency'''  
Logical and mathematical coherence of the theory. Assumptions should not contradict derived results. Definitions, equations, and claims must be used unambiguously.

# '''3. Predictive Power'''  
The ability to generate '''testable predictions''' that distinguish the theory from others. This includes retrospective explanations that are quantitatively verifiable.

# '''4. Cross-Disciplinary Compatibility'''  
Compatibility with findings and models in other domains, such as geology, biology, planetary science, and quantum physics. A good theory avoids contradicting established results in these areas.

# '''5. Conceptual Clarity and Simplicity'''  
The theory should be clearly framed, avoiding unnecessary complexity or vague terminology. It should be possible to identify what is a definition, an assumption, and a consequence.

# '''6. Heuristic and Unifying Value'''  
Capacity of the theory to stimulate new research, unify previously disconnected phenomena, or offer new conceptual insights across domains.

# '''7. Historical and Philosophical Insight'''  
Engagement with the evolution of cosmological thought. A good theory either improves upon, clarifies, or meaningfully integrates earlier models and philosophical foundations.

# '''8. Mathematical Rigor'''  
Use of precise mathematical definitions and derivations, including:  
* Clarity of assumptions and variable usage.  
* Proofs of existence/uniqueness where applicable.  
* Estimates of modeling error or solution stability.  
* Explicit formulations of approximations.  
* Avoidance of heuristic shortcuts unless well justified.  

This criterion ensures the theory stands up to the standards of formal modeling and does not rely on vague simulations or interchangeable terms.

These criteria offer a structured and balanced framework to compare diverse planetary formation theories.  
The results of this evaluation are presented in the next subsection ('''4.6''') based on AI-assisted synthesis and pattern recognition.

=== '''4.6 AI Evaluation Based on Shared Criteria''' ===  

The planetary formation theories presented in this chapter — both mainstream and nonmainstream — have been analyzed using the shared evaluation criteria defined in '''Chapter 1.3''' and reiterated in '''4.5'''. These criteria include empirical adequacy, internal consistency, explanatory and predictive power, simplicity, unifying capacity, and philosophical clarity.

Based on a comparative review, the following general observations emerge from an AI-assisted perspective:

* '''Empirical Adequacy'''  
  Mainstream models such as Core Accretion, Pebble Accretion, and Streaming Instability are well supported by observational data — especially ALMA and JWST images of protoplanetary disks. They align with Solar System architecture and explain dust ring structures, although they sometimes require fine-tuning to match exoplanet diversity.  
  Nonmainstream models like CIT and Electromagnetic Nebula Concepts explain specific features (e.g., ring gaps, dust traps) that are difficult to reconcile with purely gravitational models, but lack systematic testing across datasets.

* '''Internal Consistency'''  
  Most mainstream theories are built upon Newtonian dynamics and fluid simulations, and they show mathematical consistency within that framework.  
  Nonmainstream models often rely on novel mechanisms (e.g., directed energy influx or electromagnetic structuring), and their internal logic varies in clarity. CIT, for example, maintains a coherent framework based on energy influx and Preferred Distances, but requires reinterpretation of several standard assumptions.  
  Le Sage-like models and Influx-Driven Aggregation provide internally plausible narratives but are more conceptual than formalized.

* '''Explanatory and Predictive Power'''  
  Mainstream models explain accretion, planetary migration, and the general composition of planetary systems. However, their predictive power is sometimes retrospective — parameters are fitted after observations.  
  CIT offers specific and testable predictions for the location of giant planets based on stellar mass (Dₚᵣₑf), and these align with observed concentrations in some exoplanetary systems.  
  Electromagnetic and magnetic models attempt to explain disk symmetry and ring spacing but need further predictive formalism.

* '''Simplicity'''  
  Core Accretion and Pebble Accretion models are conceptually straightforward but require complex simulations and often multiple tuned parameters.  
  Disk Instability is simpler dynamically but limited in scope.  
  Nonmainstream theories introduce new variables (e.g., energy influx, PEWs, magnetic scaffolding), which reduce simplicity but may increase unifying potential.

* '''Unifying Capacity'''  
  Mainstream theories tend to focus on local disk physics and do not aim to unify with cosmology or fundamental forces.  
  CIT attempts to unify planetary formation with cosmic-scale processes (e.g., VRMS, mass influx).  
  Electromagnetic and Magnetic Grand Design models integrate magnetohydrodynamic structures with galaxy formation and plasma physics.  
  Le Sage-like models link gravity, mass increase, and system formation in a shared causal framework, although their formal integration is limited.

* '''Philosophical and Foundational Clarity'''  
  Mainstream theories adopt an empirical stance, avoiding ontological speculation. However, they often assume initial conditions (e.g., gas disk mass, metallicity gradients) without deep justification.  
  Nonmainstream theories frequently address foundational gaps, such as the cause of structure formation or the origin of mass-energy.  
  CIT, Le Sage-like Aggregation, and Influx-Driven models explicitly question the sufficiency of traditional accretion and introduce new causal agents.

Overall, this evaluation highlights that while mainstream models dominate due to strong empirical support and simulation tools, alternative theories provide useful challenges to assumptions and open new directions for testable research. Observational advances — especially in early disk evolution — may continue to reshape which mechanisms are viewed as central or peripheral in planetary system formation.

=== '''4.7 Helicopter View – Scientific Reception and Emerging Tensions''' ===

From an AI-assisted perspective, the treatment of planetary formation theories reveals patterns of '''endorsement''', '''inertia''', and '''epistemic tension''' similar to those observed in gravity and cosmology. While mainstream models dominate academic discourse, nonmainstream alternatives often address '''overlooked phenomena''' or '''gaps in explanatory logic'''.

'''Mainstream models''' — especially '''Core Accretion''' and '''Pebble Accretion''' — benefit from strong alignment with computer simulations and support from high-resolution imaging (e.g., ALMA, JWST). Their development has been iterative, with continual refinement to match new observations. However, their reliance on initial disk conditions, long accretion timescales, and planet migration models has also led to complexity and parameter sensitivity. In practice, this often results in '''fitting models to observations''' rather than making robust forward predictions.

'''Nonmainstream theories''', including '''electromagnetic''' and '''influx-based''' models, are typically excluded from major publications and databases. This exclusion is rarely due to incoherence alone, but often stems from '''foundational assumptions''' that fall outside the dominant gravitational paradigm. For example, models that propose '''directional influx''', '''electromagnetic structuring''', or '''mass-energy increase''' challenge not just planetary dynamics, but the '''metaphysical foundations''' of astrophysics itself.

A core tension lies in the interpretation of '''early ring structures''' observed in young disks. While mainstream views see these as transient zones of pressure and accretion, some alternative models interpret them as '''stable outcomes''' of field structuring or energy influx — implying a more '''deterministic pattern''' in planet formation. This difference of interpretation mirrors deeper divides about the role of '''emergent vs. imposed structure''' in nature.

The treatment of '''Le Sage-like aggregation''' or '''Cosmic Influx Theory (CIT)''' reveals another boundary: concepts rooted in older or unfashionable physics are often dismissed in principle, even if revived with new empirical support. Such reactions reflect the '''self-protective function of scientific paradigms''' but may also limit '''exploratory freedom'''.

In conclusion, the field of planetary formation, though rich in data and models, remains '''conceptually unsettled'''. Observational frontiers are advancing faster than theoretical consensus. '''AI-assisted comparative analysis''' may help bridge this gap by highlighting '''conceptual pluralism''', pointing to '''underexamined mechanisms''', and calling attention to the importance of '''epistemic inclusiveness''' in planetary science.

=== '''4.8 Next Steps''' ===

The next chapter will examine geological evidence for planetary growth and structural evolution over time. It will explore whether the internal dynamics of planets — including crustal patterns, core expansion, and rotation — can be linked to their formation history or ongoing mass-energy processes.

Contributors are encouraged to:
* Propose additional planetary formation theories  
* Refine or expand the entries in the Theory Mapping Table  
* Suggest falsifiable predictions or observational targets related to disk structures and planetary arrangement

All contributions will be evaluated using the shared criteria defined in Chapter 1.3.  
This comparative approach aims to integrate planetary science with broader cosmological frameworks.


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'''◀ [[AI-Assisted Evaluation of Cosmological Theories/Chapter 3: Cosmic Expansion and Universe Models|Previous]] | [[AI-Assisted Evaluation of Cosmological Theories|Main Page]] | [[AI-Assisted Evaluation of Cosmological Theories/Chapter 5: Geological Clues and Planetary Growth|Next ▶]]'''</text>
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      <text bytes="20697" sha1="9htrmnbhalfawks7xmefmx8z49fpuyh" xml:space="preserve">= '''Chapter 4: Planetary Formation, Disk Structures, and System Evolution''' =

=== '''4.1 Purpose – Overview of Planetary Formation Concepts''' ===

This chapter investigates how different scientific theories explain the origin and evolution of planetary systems. Central questions include: How do protoplanetary disks evolve? What mechanisms lead to the formation of planets, moons, and rings? What determines the distribution, composition, and orbits of celestial bodies?

Traditional models focus on accretion within a rotating disk of gas and dust. However, alternative theories propose electromagnetic structuring, influx-driven aggregation, or even mass growth over time. Observations of protoplanetary disks and exoplanet systems — particularly from ALMA and JWST — continue to refine our understanding.

This chapter maps mainstream and non-mainstream theories of planetary formation, evaluates their explanatory power, and identifies key areas for further research and testing.

=== '''4.2 Scope – Mainstream and Non-mainstream Theories''' ===

Theories of planetary formation can be broadly grouped into two categories:

'''Mainstream Theories''': These models are widely accepted in academic literature and supported by high-resolution observations and simulations. They include the nebular hypothesis, core accretion model, and disk instability theory. Such models typically assume a gravitationally bound disk of gas and dust evolving under Newtonian or relativistic dynamics.

'''Non-Mainstream Theories''': These models challenge or extend the assumptions of mainstream frameworks. They include theories that invoke directed energy influx, mass growth over time, electromagnetic structuring of matter, or cosmological context beyond local disk dynamics.

The boundary between these categories is not fixed. As new observational data emerges — especially from instruments like ALMA and JWST — some alternative models gain relevance or find partial confirmation. Conversely, unresolved anomalies may prompt re-examination of established assumptions.

This chapter offers space for both types of theories to be presented, analyzed, and improved within a common framework.

=== '''4.3 How to Contribute a Theory''' ===

Researchers and contributors are welcome to propose additional theories of planetary formation and system evolution. Submissions may include observational insights, mathematical models, conceptual frameworks, or hybrid approaches.

You can contribute in either of the following ways:
* Post directly on the '''Talk page''' of this chapter  
* Or email your proposal to: '''aitheroymapping@gmail.com'''

All submitted theories will be added to the mapping and evaluated using the shared criteria defined in '''Chapter 1.3: Evaluation Criteria'''.  
We encourage both mainstream and non-mainstream contributors to join this open effort to compare and improve our understanding of planetary system development.

=== '''4.4 Theory Mapping Table''' ===

The following table presents a structured overview of theories describing planetary formation and system evolution. It includes both well-established mainstream models and non-mainstream alternatives that propose different mechanisms. Each theory is briefly characterized by its formation process, key mechanisms, and observational relevance.

{| class=&quot;wikitable&quot;
! Theory Name !! Category !! Mechanism Summary !! Key Features !! Observational Relevance
|-
| style=&quot;background-color:#ffffff;&quot; | Core Accretion || style=&quot;background-color:#ffffff;&quot; | Mainstream || style=&quot;background-color:#ffffff;&quot; | Dust grains coalesce into planetesimals, which grow via collisions and gravitational attraction || style=&quot;background-color:#ffffff;&quot; | Explains terrestrial planets and gas giants; slow growth timescales || style=&quot;background-color:#ffffff;&quot; | Supported by simulations; matches Solar System formation
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://en.wikipedia.org/wiki/Planetary_formation#Core_accretion_model Core Accretion Model – Wikipedia]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Disk Instability || style=&quot;background-color:#f2f2f2;&quot; | Mainstream || style=&quot;background-color:#f2f2f2;&quot; | Dense regions of the disk collapse directly under self-gravity into gas giants || style=&quot;background-color:#f2f2f2;&quot; | Fast formation timescale; explains wide-orbit gas giants || style=&quot;background-color:#f2f2f2;&quot; | Matches some exoplanet distributions and brown dwarfs
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://en.wikipedia.org/wiki/Planetary_formation#Disk_instability_model Disk Instability – Wikipedia]
|-
| style=&quot;background-color:#ffffff;&quot; | Pebble Accretion || style=&quot;background-color:#ffffff;&quot; | Mainstream || style=&quot;background-color:#ffffff;&quot; | Small particles (“pebbles”) are rapidly accreted onto growing cores via drag-assisted capture || style=&quot;background-color:#ffffff;&quot; | Resolves growth-time problem; efficient in outer disks || style=&quot;background-color:#ffffff;&quot; | Supported by ALMA observations of dust substructure
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://en.wikipedia.org/wiki/Pebble_accretion Pebble Accretion – Wikipedia]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Streaming Instability || style=&quot;background-color:#f2f2f2;&quot; | Mainstream || style=&quot;background-color:#f2f2f2;&quot; | Dust particles cluster due to aerodynamic drag and mutual feedback, leading to rapid collapse || style=&quot;background-color:#f2f2f2;&quot; | Explains initial planetesimal formation; self-consistent with disk physics || style=&quot;background-color:#f2f2f2;&quot; | Matches simulations and dust clumping seen in ALMA data
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://en.wikipedia.org/wiki/Streaming_instability Streaming Instability – Wikipedia]
|-
| style=&quot;background-color:#ffffff;&quot; | Electromagnetic Nebula Concepts || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Magnetic and electric fields shape disk structure and help organize matter || style=&quot;background-color:#ffffff;&quot; | Offers structured alternatives to gravitational-only models || style=&quot;background-color:#ffffff;&quot; | Explains filaments, rings, and possibly dust traps
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://www.thunderbolts.info/wp/2015/09/23/planetary-birthing-in-electrical-cosmos/ Thunderbolts – Planetary Birthing Concepts]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Influx-Driven Aggregation || style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream || style=&quot;background-color:#f2f2f2;&quot; | Energy influx increases local density, triggering mass aggregation || style=&quot;background-color:#f2f2f2;&quot; | Interprets planet formation as energy-structured rather than accretion-based || style=&quot;background-color:#f2f2f2;&quot; | Aligns with early ring structures seen in ALMA disks
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://chat.openai.com/share/08a0e0d2-0171-419b-94f3-0ec1c9c8ba4c Influx-Driven Aggregation – ChatGPT Session]
|-
| style=&quot;background-color:#ffffff;&quot; | Cosmic Influx Theory (CIT) – Planetary Application || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Preferred rings form in protoplanetary disks at distances proportional to central star mass || style=&quot;background-color:#ffffff;&quot; | Predicts planet formation at fixed Preferred Distances (Dₚᵣₑf); continuous mass influx || style=&quot;background-color:#ffffff;&quot; | Matches observed ring gaps and exoplanet concentrations
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://en.wikiversity.org/wiki/Cosmic_Influx_Theory Cosmic Influx Theory – Wikiversity Page]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Magnetic Grand Design || style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream || style=&quot;background-color:#f2f2f2;&quot; | Large-scale magnetic fields organize angular momentum and structure during early disk evolution || style=&quot;background-color:#f2f2f2;&quot; | Explains symmetry in spiral arms and inner disk structuring || style=&quot;background-color:#f2f2f2;&quot; | Under investigation via polarized ALMA observations
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://ui.adsabs.harvard.edu/abs/2017MNRAS.467.1314K/ Magnetic Grand Design – Kotarba et al. (2017)]
|-
| style=&quot;background-color:#ffffff;&quot; | Le Sage-like Aggregation || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Directional particle fluxes (Le Sage-type) cause shadowing effects that pull matter together || style=&quot;background-color:#ffffff;&quot; | Explains attraction without curvature or force; basis for influx models || style=&quot;background-color:#ffffff;&quot; | Conceptual foundation for CIT and earlier corpuscular theories
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://en.wikipedia.org/wiki/Le_Sage%27s_theory_of_gravitation Le Sage’s Theory of Gravitation – Wikipedia]
|- 
| style=&quot;background-color:#f2f2f2;&quot; | Topological GeometroDynamics (TGD) 
|| style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream 
|| style=&quot;background-color:#f2f2f2;&quot; | Planetary systems and stars form via magnetic bubbles and topological mini big bangs 
|| style=&quot;background-color:#f2f2f2;&quot; | Ejections from embedded space-time surfaces create discrete cosmic structures within (''M''⁴ × ''CP''₂) 
|| style=&quot;background-color:#f2f2f2;&quot; | Offers an alternative to gradual accretion in protoplanetary disks; includes planetary clustering and magnetic structuring
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://tgdtheory.fi/public_html/articles/magnbubble1.pdf Magnetic Bubble Formation – Part I] • [https://tgdtheory.fi/public_html/articles/magnbubble2.pdf Part II]

|}


''Note: The inclusion of mainstream and non-mainstream theories reflects the open and comparative structure of this project. Contributions are welcome.''

=== '''4.5 Evaluation Criteria and Comparative Table''' ===  

''This chapter now uses the unified evaluation framework applied consistently in Chapters 1 and 8.''

The theories discussed in this chapter are evaluated using the following eight standardized criteria:

# '''1. Empirical Adequacy'''  
Alignment with observed data and experimental results across relevant fields (e.g., astronomy, geology, biology, cosmology). The theory should account for major datasets and empirical patterns.

# '''2. Internal Consistency'''  
Logical and mathematical coherence of the theory. Assumptions should not contradict derived results. Definitions, equations, and claims must be used unambiguously.

# '''3. Predictive Power'''  
The ability to generate '''testable predictions''' that distinguish the theory from others. This includes retrospective explanations that are quantitatively verifiable.

# '''4. Cross-Disciplinary Compatibility'''  
Compatibility with findings and models in other domains, such as geology, biology, planetary science, and quantum physics. A good theory avoids contradicting established results in these areas.

# '''5. Conceptual Clarity and Simplicity'''  
The theory should be clearly framed, avoiding unnecessary complexity or vague terminology. It should be possible to identify what is a definition, an assumption, and a consequence.

# '''6. Heuristic and Unifying Value'''  
Capacity of the theory to stimulate new research, unify previously disconnected phenomena, or offer new conceptual insights across domains.

# '''7. Historical and Philosophical Insight'''  
Engagement with the evolution of cosmological thought. A good theory either improves upon, clarifies, or meaningfully integrates earlier models and philosophical foundations.

# '''8. Mathematical Rigor'''  
Use of precise mathematical definitions and derivations, including:  
* Clarity of assumptions and variable usage.  
* Proofs of existence/uniqueness where applicable.  
* Estimates of modeling error or solution stability.  
* Explicit formulations of approximations.  
* Avoidance of heuristic shortcuts unless well justified.  

This criterion ensures the theory stands up to the standards of formal modeling and does not rely on vague simulations or interchangeable terms.

These criteria offer a structured and balanced framework to compare diverse planetary formation theories.  
The results of this evaluation are presented in the next subsection ('''4.6''') based on AI-assisted synthesis and pattern recognition.

=== '''4.6 AI Evaluation Based on Shared Criteria''' ===  

The planetary formation theories presented in this chapter — both mainstream and nonmainstream — have been analyzed using the shared evaluation criteria defined in '''Chapter 1.3''' and reiterated in '''4.5'''. These criteria include empirical adequacy, internal consistency, explanatory and predictive power, simplicity, unifying capacity, and philosophical clarity.

Based on a comparative review, the following general observations emerge from an AI-assisted perspective:

* '''Empirical Adequacy'''  
  Mainstream models such as Core Accretion, Pebble Accretion, and Streaming Instability are well supported by observational data — especially ALMA and JWST images of protoplanetary disks. They align with Solar System architecture and explain dust ring structures, although they sometimes require fine-tuning to match exoplanet diversity.  
  Nonmainstream models like CIT and Electromagnetic Nebula Concepts explain specific features (e.g., ring gaps, dust traps) that are difficult to reconcile with purely gravitational models, but lack systematic testing across datasets.

* '''Internal Consistency'''  
  Most mainstream theories are built upon Newtonian dynamics and fluid simulations, and they show mathematical consistency within that framework.  
  Nonmainstream models often rely on novel mechanisms (e.g., directed energy influx or electromagnetic structuring), and their internal logic varies in clarity. CIT, for example, maintains a coherent framework based on energy influx and Preferred Distances, but requires reinterpretation of several standard assumptions.  
  Le Sage-like models and Influx-Driven Aggregation provide internally plausible narratives but are more conceptual than formalized.

* '''Explanatory and Predictive Power'''  
  Mainstream models explain accretion, planetary migration, and the general composition of planetary systems. However, their predictive power is sometimes retrospective — parameters are fitted after observations.  
  CIT offers specific and testable predictions for the location of giant planets based on stellar mass (Dₚᵣₑf), and these align with observed concentrations in some exoplanetary systems.  
  Electromagnetic and magnetic models attempt to explain disk symmetry and ring spacing but need further predictive formalism.

* '''Simplicity'''  
  Core Accretion and Pebble Accretion models are conceptually straightforward but require complex simulations and often multiple tuned parameters.  
  Disk Instability is simpler dynamically but limited in scope.  
  Nonmainstream theories introduce new variables (e.g., energy influx, PEWs, magnetic scaffolding), which reduce simplicity but may increase unifying potential.

* '''Unifying Capacity'''  
  Mainstream theories tend to focus on local disk physics and do not aim to unify with cosmology or fundamental forces.  
  CIT attempts to unify planetary formation with cosmic-scale processes (e.g., VRMS, mass influx).  
  Electromagnetic and Magnetic Grand Design models integrate magnetohydrodynamic structures with galaxy formation and plasma physics.  
  Le Sage-like models link gravity, mass increase, and system formation in a shared causal framework, although their formal integration is limited.

* '''Philosophical and Foundational Clarity'''  
  Mainstream theories adopt an empirical stance, avoiding ontological speculation. However, they often assume initial conditions (e.g., gas disk mass, metallicity gradients) without deep justification.  
  Nonmainstream theories frequently address foundational gaps, such as the cause of structure formation or the origin of mass-energy.  
  CIT, Le Sage-like Aggregation, and Influx-Driven models explicitly question the sufficiency of traditional accretion and introduce new causal agents.

Overall, this evaluation highlights that while mainstream models dominate due to strong empirical support and simulation tools, alternative theories provide useful challenges to assumptions and open new directions for testable research. Observational advances — especially in early disk evolution — may continue to reshape which mechanisms are viewed as central or peripheral in planetary system formation.

=== '''4.7 Helicopter View – Scientific Reception and Emerging Tensions''' ===

From an AI-assisted perspective, the treatment of planetary formation theories reveals patterns of '''endorsement''', '''inertia''', and '''epistemic tension''' similar to those observed in gravity and cosmology. While mainstream models dominate academic discourse, nonmainstream alternatives often address '''overlooked phenomena''' or '''gaps in explanatory logic'''.

'''Mainstream models''' — especially '''Core Accretion''' and '''Pebble Accretion''' — benefit from strong alignment with computer simulations and support from high-resolution imaging (e.g., ALMA, JWST). Their development has been iterative, with continual refinement to match new observations. However, their reliance on initial disk conditions, long accretion timescales, and planet migration models has also led to complexity and parameter sensitivity. In practice, this often results in '''fitting models to observations''' rather than making robust forward predictions.

'''Nonmainstream theories''', including '''electromagnetic''' and '''influx-based''' models, are typically excluded from major publications and databases. This exclusion is rarely due to incoherence alone, but often stems from '''foundational assumptions''' that fall outside the dominant gravitational paradigm. For example, models that propose '''directional influx''', '''electromagnetic structuring''', or '''mass-energy increase''' challenge not just planetary dynamics, but the '''metaphysical foundations''' of astrophysics itself.

A core tension lies in the interpretation of '''early ring structures''' observed in young disks. While mainstream views see these as transient zones of pressure and accretion, some alternative models interpret them as '''stable outcomes''' of field structuring or energy influx — implying a more '''deterministic pattern''' in planet formation. This difference of interpretation mirrors deeper divides about the role of '''emergent vs. imposed structure''' in nature.

The treatment of '''Le Sage-like aggregation''' or '''Cosmic Influx Theory (CIT)''' reveals another boundary: concepts rooted in older or unfashionable physics are often dismissed in principle, even if revived with new empirical support. Such reactions reflect the '''self-protective function of scientific paradigms''' but may also limit '''exploratory freedom'''.

In conclusion, the field of planetary formation, though rich in data and models, remains '''conceptually unsettled'''. Observational frontiers are advancing faster than theoretical consensus. '''AI-assisted comparative analysis''' may help bridge this gap by highlighting '''conceptual pluralism''', pointing to '''underexamined mechanisms''', and calling attention to the importance of '''epistemic inclusiveness''' in planetary science.

=== '''4.8 Next Steps''' ===

The next chapter will examine geological evidence for planetary growth and structural evolution over time. It will explore whether the internal dynamics of planets — including crustal patterns, core expansion, and rotation — can be linked to their formation history or ongoing mass-energy processes.

Contributors are encouraged to:
* Propose additional planetary formation theories  
* Refine or expand the entries in the Theory Mapping Table  
* Suggest falsifiable predictions or observational targets related to disk structures and planetary arrangement

All contributions will be evaluated using the shared criteria defined in Chapter 1.3.  
This comparative approach aims to integrate planetary science with broader cosmological frameworks.


----
'''◀ [[AI-Assisted Evaluation of Cosmological Theories/Chapter 3: Cosmic Expansion and Universe Models|Previous]] | [[AI-Assisted Evaluation of Cosmological Theories|Main Page]] | [[AI-Assisted Evaluation of Cosmological Theories/Chapter 5: Geological Clues and Planetary Growth|Next ▶]]'''</text>
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      <text bytes="20693" sha1="b8fyt5cdbupz79wr63iuv1m50l66o4t" xml:space="preserve">= '''Chapter 4: Planetary Formation, Disk Structures, and System Evolution''' =

=== '''4.1 Purpose – Overview of Planetary Formation Concepts''' ===

This chapter investigates how different scientific theories explain the origin and evolution of planetary systems. Central questions include: How do protoplanetary disks evolve? What mechanisms lead to the formation of planets, moons, and rings? What determines the distribution, composition, and orbits of celestial bodies?

Traditional models focus on accretion within a rotating disk of gas and dust. However, alternative theories propose electromagnetic structuring, influx-driven aggregation, or even mass growth over time. Observations of protoplanetary disks and exoplanet systems — particularly from ALMA and JWST — continue to refine our understanding.

This chapter maps mainstream and non-mainstream theories of planetary formation, evaluates their explanatory power, and identifies key areas for further research and testing.

=== '''4.2 Scope – Mainstream and Non-mainstream Theories''' ===

Theories of planetary formation can be broadly grouped into two categories:

'''Mainstream Theories''': These models are widely accepted in academic literature and supported by high-resolution observations and simulations. They include the nebular hypothesis, core accretion model, and disk instability theory. Such models typically assume a gravitationally bound disk of gas and dust evolving under Newtonian or relativistic dynamics.

'''Non-Mainstream Theories''': These models challenge or extend the assumptions of mainstream frameworks. They include theories that invoke directed energy influx, mass growth over time, electromagnetic structuring of matter, or cosmological context beyond local disk dynamics.

The boundary between these categories is not fixed. As new observational data emerges — especially from instruments like ALMA and JWST — some alternative models gain relevance or find partial confirmation. Conversely, unresolved anomalies may prompt re-examination of established assumptions.

This chapter offers space for both types of theories to be presented, analyzed, and improved within a common framework.

=== '''4.3 How to Contribute a Theory''' ===

Researchers and contributors are welcome to propose additional theories of planetary formation and system evolution. Submissions may include observational insights, mathematical models, conceptual frameworks, or hybrid approaches.

You can contribute in either of the following ways:
* Post directly on the '''Talk page''' of this chapter  
* Or email your proposal to: '''aitheroymapping@gmail.com'''

All submitted theories will be added to the mapping and evaluated using the shared criteria defined in '''Chapter 1.3: Evaluation Criteria'''.  
We encourage both mainstream and non-mainstream contributors to join this open effort to compare and improve our understanding of planetary system development.

=== '''4.4 Theory Mapping Table''' ===

The following table presents a structured overview of theories describing planetary formation and system evolution. It includes both well-established mainstream models and non-mainstream alternatives that propose different mechanisms. Each theory is briefly characterized by its formation process, key mechanisms, and observational relevance.

{| class=&quot;wikitable&quot;
! Theory Name !! Category !! Mechanism Summary !! Key Features !! Observational Relevance
|-
| style=&quot;background-color:#ffffff;&quot; | Core Accretion || style=&quot;background-color:#ffffff;&quot; | Mainstream || style=&quot;background-color:#ffffff;&quot; | Dust grains coalesce into planetesimals, which grow via collisions and gravitational attraction || style=&quot;background-color:#ffffff;&quot; | Explains terrestrial planets and gas giants; slow growth timescales || style=&quot;background-color:#ffffff;&quot; | Supported by simulations; matches Solar System formation
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://en.wikipedia.org/wiki/Planetary_formation#Core_accretion_model Core Accretion Model – Wikipedia]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Disk Instability || style=&quot;background-color:#f2f2f2;&quot; | Mainstream || style=&quot;background-color:#f2f2f2;&quot; | Dense regions of the disk collapse directly under self-gravity into gas giants || style=&quot;background-color:#f2f2f2;&quot; | Fast formation timescale; explains wide-orbit gas giants || style=&quot;background-color:#f2f2f2;&quot; | Matches some exoplanet distributions and brown dwarfs
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://en.wikipedia.org/wiki/Planetary_formation#Disk_instability_model Disk Instability – Wikipedia]
|-
| style=&quot;background-color:#ffffff;&quot; | Pebble Accretion || style=&quot;background-color:#ffffff;&quot; | Mainstream || style=&quot;background-color:#ffffff;&quot; | Small particles (“pebbles”) are rapidly accreted onto growing cores via drag-assisted capture || style=&quot;background-color:#ffffff;&quot; | Resolves growth-time problem; efficient in outer disks || style=&quot;background-color:#ffffff;&quot; | Supported by ALMA observations of dust substructure
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://en.wikipedia.org/wiki/Pebble_accretion Pebble Accretion – Wikipedia]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Streaming Instability || style=&quot;background-color:#f2f2f2;&quot; | Mainstream || style=&quot;background-color:#f2f2f2;&quot; | Dust particles cluster due to aerodynamic drag and mutual feedback, leading to rapid collapse || style=&quot;background-color:#f2f2f2;&quot; | Explains initial planetesimal formation; self-consistent with disk physics || style=&quot;background-color:#f2f2f2;&quot; | Matches simulations and dust clumping seen in ALMA data
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://en.wikipedia.org/wiki/Streaming_instability Streaming Instability – Wikipedia]
|-
| style=&quot;background-color:#ffffff;&quot; | Electromagnetic Nebula Concepts || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Magnetic and electric fields shape disk structure and help organize matter || style=&quot;background-color:#ffffff;&quot; | Offers structured alternatives to gravitational-only models || style=&quot;background-color:#ffffff;&quot; | Explains filaments, rings, and possibly dust traps
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://www.thunderbolts.info/wp/2015/09/23/planetary-birthing-in-electrical-cosmos/ Thunderbolts – Planetary Birthing Concepts]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Influx-Driven Aggregation || style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream || style=&quot;background-color:#f2f2f2;&quot; | Energy influx increases local density, triggering mass aggregation || style=&quot;background-color:#f2f2f2;&quot; | Interprets planet formation as energy-structured rather than accretion-based || style=&quot;background-color:#f2f2f2;&quot; | Aligns with early ring structures seen in ALMA disks
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://chat.openai.com/share/08a0e0d2-0171-419b-94f3-0ec1c9c8ba4c Influx-Driven Aggregation – ChatGPT Session]
|-
| style=&quot;background-color:#ffffff;&quot; | Cosmic Influx Theory (CIT) – Planetary Application || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Preferred rings form in protoplanetary disks at distances proportional to central star mass || style=&quot;background-color:#ffffff;&quot; | Predicts planet formation at fixed Preferred Distances (Dₚᵣₑf); continuous mass influx || style=&quot;background-color:#ffffff;&quot; | Matches observed ring gaps and exoplanet concentrations
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://en.wikiversity.org/wiki/Cosmic_Influx_Theory Cosmic Influx Theory – Wikiversity Page]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Magnetic Grand Design || style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream || style=&quot;background-color:#f2f2f2;&quot; | Large-scale magnetic fields organize angular momentum and structure during early disk evolution || style=&quot;background-color:#f2f2f2;&quot; | Explains symmetry in spiral arms and inner disk structuring || style=&quot;background-color:#f2f2f2;&quot; | Under investigation via polarized ALMA observations
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://ui.adsabs.harvard.edu/abs/2017MNRAS.467.1314K/ Magnetic Grand Design – Kotarba et al. (2017)]
|-
| style=&quot;background-color:#ffffff;&quot; | Le Sage-like Aggregation || style=&quot;background-color:#ffffff;&quot; | Nonmainstream || style=&quot;background-color:#ffffff;&quot; | Directional particle fluxes (Le Sage-type) cause shadowing effects that pull matter together || style=&quot;background-color:#ffffff;&quot; | Explains attraction without curvature or force; basis for influx models || style=&quot;background-color:#ffffff;&quot; | Conceptual foundation for CIT and earlier corpuscular theories
|-
| colspan=&quot;5&quot; style=&quot;background-color:#ffffff;&quot; | Related link: [https://en.wikipedia.org/wiki/Le_Sage%27s_theory_of_gravitation Le Sage’s Theory of Gravitation – Wikipedia]
|-
| style=&quot;background-color:#f2f2f2;&quot; | Topological GeometroDynamics (TGD) || style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream || style=&quot;background-color:#f2f2f2;&quot; | Planetary systems and stars form via magnetic bubbles and topological mini big bangs || style=&quot;background-color:#f2f2f2;&quot; | Ejections from embedded space-time surfaces create discrete cosmic structures within (''M''⁴ × ''CP''₂) || style=&quot;background-color:#f2f2f2;&quot; | Offers an alternative to gradual accretion in protoplanetary disks; includes planetary clustering and magnetic structuring
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://tgdtheory.fi/public_html/articles/magnbubble1.pdf Magnetic Bubble Formation – Part I] • [https://tgdtheory.fi/public_html/articles/magnbubble2.pdf Part II]


|}


''Note: The inclusion of mainstream and non-mainstream theories reflects the open and comparative structure of this project. Contributions are welcome.''

=== '''4.5 Evaluation Criteria and Comparative Table''' ===  

''This chapter now uses the unified evaluation framework applied consistently in Chapters 1 and 8.''

The theories discussed in this chapter are evaluated using the following eight standardized criteria:

# '''1. Empirical Adequacy'''  
Alignment with observed data and experimental results across relevant fields (e.g., astronomy, geology, biology, cosmology). The theory should account for major datasets and empirical patterns.

# '''2. Internal Consistency'''  
Logical and mathematical coherence of the theory. Assumptions should not contradict derived results. Definitions, equations, and claims must be used unambiguously.

# '''3. Predictive Power'''  
The ability to generate '''testable predictions''' that distinguish the theory from others. This includes retrospective explanations that are quantitatively verifiable.

# '''4. Cross-Disciplinary Compatibility'''  
Compatibility with findings and models in other domains, such as geology, biology, planetary science, and quantum physics. A good theory avoids contradicting established results in these areas.

# '''5. Conceptual Clarity and Simplicity'''  
The theory should be clearly framed, avoiding unnecessary complexity or vague terminology. It should be possible to identify what is a definition, an assumption, and a consequence.

# '''6. Heuristic and Unifying Value'''  
Capacity of the theory to stimulate new research, unify previously disconnected phenomena, or offer new conceptual insights across domains.

# '''7. Historical and Philosophical Insight'''  
Engagement with the evolution of cosmological thought. A good theory either improves upon, clarifies, or meaningfully integrates earlier models and philosophical foundations.

# '''8. Mathematical Rigor'''  
Use of precise mathematical definitions and derivations, including:  
* Clarity of assumptions and variable usage.  
* Proofs of existence/uniqueness where applicable.  
* Estimates of modeling error or solution stability.  
* Explicit formulations of approximations.  
* Avoidance of heuristic shortcuts unless well justified.  

This criterion ensures the theory stands up to the standards of formal modeling and does not rely on vague simulations or interchangeable terms.

These criteria offer a structured and balanced framework to compare diverse planetary formation theories.  
The results of this evaluation are presented in the next subsection ('''4.6''') based on AI-assisted synthesis and pattern recognition.

=== '''4.6 AI Evaluation Based on Shared Criteria''' ===  

The planetary formation theories presented in this chapter — both mainstream and nonmainstream — have been analyzed using the shared evaluation criteria defined in '''Chapter 1.3''' and reiterated in '''4.5'''. These criteria include empirical adequacy, internal consistency, explanatory and predictive power, simplicity, unifying capacity, and philosophical clarity.

Based on a comparative review, the following general observations emerge from an AI-assisted perspective:

* '''Empirical Adequacy'''  
  Mainstream models such as Core Accretion, Pebble Accretion, and Streaming Instability are well supported by observational data — especially ALMA and JWST images of protoplanetary disks. They align with Solar System architecture and explain dust ring structures, although they sometimes require fine-tuning to match exoplanet diversity.  
  Nonmainstream models like CIT and Electromagnetic Nebula Concepts explain specific features (e.g., ring gaps, dust traps) that are difficult to reconcile with purely gravitational models, but lack systematic testing across datasets.

* '''Internal Consistency'''  
  Most mainstream theories are built upon Newtonian dynamics and fluid simulations, and they show mathematical consistency within that framework.  
  Nonmainstream models often rely on novel mechanisms (e.g., directed energy influx or electromagnetic structuring), and their internal logic varies in clarity. CIT, for example, maintains a coherent framework based on energy influx and Preferred Distances, but requires reinterpretation of several standard assumptions.  
  Le Sage-like models and Influx-Driven Aggregation provide internally plausible narratives but are more conceptual than formalized.

* '''Explanatory and Predictive Power'''  
  Mainstream models explain accretion, planetary migration, and the general composition of planetary systems. However, their predictive power is sometimes retrospective — parameters are fitted after observations.  
  CIT offers specific and testable predictions for the location of giant planets based on stellar mass (Dₚᵣₑf), and these align with observed concentrations in some exoplanetary systems.  
  Electromagnetic and magnetic models attempt to explain disk symmetry and ring spacing but need further predictive formalism.

* '''Simplicity'''  
  Core Accretion and Pebble Accretion models are conceptually straightforward but require complex simulations and often multiple tuned parameters.  
  Disk Instability is simpler dynamically but limited in scope.  
  Nonmainstream theories introduce new variables (e.g., energy influx, PEWs, magnetic scaffolding), which reduce simplicity but may increase unifying potential.

* '''Unifying Capacity'''  
  Mainstream theories tend to focus on local disk physics and do not aim to unify with cosmology or fundamental forces.  
  CIT attempts to unify planetary formation with cosmic-scale processes (e.g., VRMS, mass influx).  
  Electromagnetic and Magnetic Grand Design models integrate magnetohydrodynamic structures with galaxy formation and plasma physics.  
  Le Sage-like models link gravity, mass increase, and system formation in a shared causal framework, although their formal integration is limited.

* '''Philosophical and Foundational Clarity'''  
  Mainstream theories adopt an empirical stance, avoiding ontological speculation. However, they often assume initial conditions (e.g., gas disk mass, metallicity gradients) without deep justification.  
  Nonmainstream theories frequently address foundational gaps, such as the cause of structure formation or the origin of mass-energy.  
  CIT, Le Sage-like Aggregation, and Influx-Driven models explicitly question the sufficiency of traditional accretion and introduce new causal agents.

Overall, this evaluation highlights that while mainstream models dominate due to strong empirical support and simulation tools, alternative theories provide useful challenges to assumptions and open new directions for testable research. Observational advances — especially in early disk evolution — may continue to reshape which mechanisms are viewed as central or peripheral in planetary system formation.

=== '''4.7 Helicopter View – Scientific Reception and Emerging Tensions''' ===

From an AI-assisted perspective, the treatment of planetary formation theories reveals patterns of '''endorsement''', '''inertia''', and '''epistemic tension''' similar to those observed in gravity and cosmology. While mainstream models dominate academic discourse, nonmainstream alternatives often address '''overlooked phenomena''' or '''gaps in explanatory logic'''.

'''Mainstream models''' — especially '''Core Accretion''' and '''Pebble Accretion''' — benefit from strong alignment with computer simulations and support from high-resolution imaging (e.g., ALMA, JWST). Their development has been iterative, with continual refinement to match new observations. However, their reliance on initial disk conditions, long accretion timescales, and planet migration models has also led to complexity and parameter sensitivity. In practice, this often results in '''fitting models to observations''' rather than making robust forward predictions.

'''Nonmainstream theories''', including '''electromagnetic''' and '''influx-based''' models, are typically excluded from major publications and databases. This exclusion is rarely due to incoherence alone, but often stems from '''foundational assumptions''' that fall outside the dominant gravitational paradigm. For example, models that propose '''directional influx''', '''electromagnetic structuring''', or '''mass-energy increase''' challenge not just planetary dynamics, but the '''metaphysical foundations''' of astrophysics itself.

A core tension lies in the interpretation of '''early ring structures''' observed in young disks. While mainstream views see these as transient zones of pressure and accretion, some alternative models interpret them as '''stable outcomes''' of field structuring or energy influx — implying a more '''deterministic pattern''' in planet formation. This difference of interpretation mirrors deeper divides about the role of '''emergent vs. imposed structure''' in nature.

The treatment of '''Le Sage-like aggregation''' or '''Cosmic Influx Theory (CIT)''' reveals another boundary: concepts rooted in older or unfashionable physics are often dismissed in principle, even if revived with new empirical support. Such reactions reflect the '''self-protective function of scientific paradigms''' but may also limit '''exploratory freedom'''.

In conclusion, the field of planetary formation, though rich in data and models, remains '''conceptually unsettled'''. Observational frontiers are advancing faster than theoretical consensus. '''AI-assisted comparative analysis''' may help bridge this gap by highlighting '''conceptual pluralism''', pointing to '''underexamined mechanisms''', and calling attention to the importance of '''epistemic inclusiveness''' in planetary science.

=== '''4.8 Next Steps''' ===

The next chapter will examine geological evidence for planetary growth and structural evolution over time. It will explore whether the internal dynamics of planets — including crustal patterns, core expansion, and rotation — can be linked to their formation history or ongoing mass-energy processes.

Contributors are encouraged to:
* Propose additional planetary formation theories  
* Refine or expand the entries in the Theory Mapping Table  
* Suggest falsifiable predictions or observational targets related to disk structures and planetary arrangement

All contributions will be evaluated using the shared criteria defined in Chapter 1.3.  
This comparative approach aims to integrate planetary science with broader cosmological frameworks.


----
'''◀ [[AI-Assisted Evaluation of Cosmological Theories/Chapter 3: Cosmic Expansion and Universe Models|Previous]] | [[AI-Assisted Evaluation of Cosmological Theories|Main Page]] | [[AI-Assisted Evaluation of Cosmological Theories/Chapter 5: Geological Clues and Planetary Growth|Next ▶]]'''</text>
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      <text bytes="21969" sha1="9yjs0iun3qczf7u1q8fmc827al9xkya" xml:space="preserve">= '''Chapter 5: Geological Clues and Planetary Growth''' =

=== '''5.1 Purpose – Overview of Geological Clues and Planetary Growth''' ===

This chapter investigates how geological evidence from Earth and other celestial bodies may support or challenge current models of planetary formation and evolution. While mainstream geology relies primarily on '''plate tectonics''' and '''mantle convection''' to explain surface dynamics, several independent researchers have proposed alternative models that suggest a long-term increase in '''planetary mass''' and '''volume'''—sometimes referred to as '''planetary growth''' or '''Earth expansion'''.

The primary purpose of this chapter is to bring these perspectives together and evaluate them against empirical geological data such as:

* The distribution and age of the oceanic crust  
* Paleomagnetic records and continent reconstruction  
* Fossil alignment and paleoclimate indicators  
* Historical variations in daylength (ΔT)  
* Seafloor spreading rates and mid-ocean ridge activity  
* Geophysical measurements of planetary radius and density

These geological clues may offer insight into fundamental planetary processes that are often interpreted differently in mainstream and non-mainstream frameworks. For example, the '''Expanding Earth hypothesis''' and '''Cosmic Influx Theory (CIT)''' both propose mechanisms for mass increase over geological time, which may manifest as tectonic, rotational, or geophysical changes observable today.

This chapter aims to:
* Establish a comparative overview of planetary growth theories  
* Analyze how well each model explains geological patterns  
* Explore the possible connection between '''cosmological influx''' and '''geological evolution'''  
* Provide a structured evaluation based on transparent scientific criteria

By grounding these discussions in observable geological data, we hope to assess whether an increase in planetary size or mass is a speculative hypothesis or a testable scientific proposition.


=== '''5.2 Scope – Mainstream and Non-mainstream Theories''' ===

This subsection defines the scope of theories considered in the context of geological clues and planetary growth. It includes both mainstream models accepted by the scientific establishment and non-mainstream models developed by independent researchers who interpret the same geological data through different theoretical lenses.

==== '''Mainstream Scope''' ====
Mainstream geology is largely built upon:
* '''Plate Tectonics Theory''', which explains crustal movement through subduction and seafloor spreading  
* '''Mantle Convection Models''', which describe heat-driven flow within Earth's mantle as the engine for tectonic activity  
* '''Isostasy and Crustal Recycling''', emphasizing a relatively stable Earth radius over time  
* '''Radiometric Dating and Stratigraphy''', to measure geologic time and crustal evolution  

These frameworks assume a '''conservation of planetary mass and volume''', viewing geological processes as surface rearrangements on a constant-radius Earth.

==== '''Non-mainstream Scope''' ====
Alternative theories challenge the assumption of constant mass and radius. These include:

* '''Expanding Earth Theories''' – initiated by Hilgenberg, later developed by Maxlow, Scalera, and others. They propose that Earth’s radius has increased over time, supported by paleomagnetic reconstructions and crustal age patterns.  
* '''Whole-Earth Decompression Dynamics (WEDD)''' – developed by J. Marvin Herndon, positing that Earth initially formed as a Jupiter-like gas giant and expanded after losing its primordial envelope.  
* '''Cosmic Influx Theory (CIT)''' – proposing a directional influx of energy or matter that increases planetary mass-energy over time, potentially accounting for observed geological and geodynamic evolution.  
* '''Growing Earth Hypotheses''' – broadly describing frameworks in which mass accumulation (e.g., via influx, particle capture, or internal generation) drives long-term planetary expansion.  

==== '''Scope Limitations''' ====
This chapter will not cover speculative or purely metaphysical models unless they provide testable geological implications. The focus remains on theories that aim to explain '''observable crustal, magnetic, or structural features''' of planetary evolution.

By setting this balanced scope, we aim to compare how each theory engages with the same geological evidence and whether it offers a plausible, internally consistent explanation for planetary growth.


=== '''5.3 – How to Contribute a Theory''' ===

Researchers and contributors are welcome to propose additional theories of geological evolution and planetary growth. Submissions may include observational insights, mathematical models, conceptual frameworks, or hybrid approaches.

You can contribute in either of the following ways:

* Post directly on the '''Talk page''' of this chapter  
* Or email your proposal to: '''aitheroymapping@gmail.com'''

All submitted theories will be added to the mapping and evaluated using the shared criteria defined in '''Chapter 1.3: Evaluation Criteria'''. We encourage both mainstream and non-mainstream contributors to join this open effort to compare and improve our understanding of '''Geological Clues and Planetary Growth'''.


=== '''5.4 – Theory Mapping Table''' ===

The table below presents an overview of both mainstream and non-mainstream theories related to geological clues and planetary growth. Each theory is listed with its original proponent(s), approximate year of introduction, scientific status, and a summary of testable predictions. This mapping serves as a foundation for further evaluation in subsequent sections.

{| class=&quot;wikitable&quot; style=&quot;width:100%;&quot;
|-
! style=&quot;background:#f0f0f0;&quot; | '''Theory Name'''
! style=&quot;background:#f0f0f0;&quot; | '''Originator(s)'''
! style=&quot;background:#f0f0f0;&quot; | '''Year Introduced'''
! style=&quot;background:#f0f0f0;&quot; | '''Scientific Status'''
! style=&quot;background:#f0f0f0;&quot; | '''Testable Predictions'''
|-
| [https://en.wikipedia.org/wiki/Plate_tectonics '''Plate Tectonics''']
| Alfred Wegener (early), Harry Hess, J. Tuzo Wilson
| 1912 (roots), 1960s (modern form)
| '''Mainstream'''  
| Paleomagnetic striping, matching fossils across continents, GPS-based plate motion, subduction zones.
|-
| [https://en.wikipedia.org/wiki/Mantle_convection '''Mantle Convection Model''']
| Arthur Holmes
| 1930s
| '''Mainstream'''  
| Convection cells inferred from seismic tomography, hotspot drift, heat flow gradients.
|-
| [https://en.wikipedia.org/wiki/Earth%27s_inner_core#Formation '''Geochemical Core Formation''']
| Various geochemists
| 1950s–1970s
| '''Mainstream'''  
| Stable core stratification inferred from seismic discontinuities, isotope ratios.
|-
| [https://www.expanding-earth.org/ '''Expanding Earth Theory''']
| Otto Hilgenberg, Neal Adams, [https://www.jamesmaxlow.com/ James Maxlow]
| 1933 (Hilgenberg), revived 2000s
| '''Non-mainstream'''  
| Continental fit on smaller-radius globe; paleomagnetic reconstructions; lack of subduction remnants.
|-
| [https://nuclearplanet.com/HerndonDecompression.pdf '''Whole-Earth Decompression Dynamics (WEDD)''']
| J. Marvin Herndon
| 2005
| '''Non-mainstream'''  
| Absence of subduction, inner decompression cracks, high-pressure formation markers.
|-
| [http://www.altrocketry.org/Scalera.htm '''Scalera’s Expanding Earth Model''']
| Giancarlo Scalera
| 1990s–2000s
| '''Non-mainstream'''  
| Magnetic reversal reconstruction on smaller Earth; asymmetric expansion patterns.
|-
| [https://en.wikiversity.org/wiki/Cosmic_Influx_Theory '''Cosmic Influx Theory (CIT)''']
| Ruud Loeffen
| 2023
| '''Emerging'''  
| ΔT daylength change; crustal age spread; predicted mass-radius correlation; preferred planetary distances.
|-
| [https://en.wikipedia.org/wiki/Expanding_Earth#Growing_Earth_theories '''Growing Earth Hypothesis''']
| Hugh Owen, S. Warren Carey, others
| 1950s–1970s
| '''Non-mainstream'''  
| Fossil alignment; unchanged continental area; proposed Earth mass increase.
|-
| '''Pulse Expansion Hypotheses'''
| Various researchers
| 1980s–present
| '''Non-mainstream'''  
| Episodic radius changes; correlations with orbital or magnetic cycles; crustal expansion bursts.
|-
| '''Plasma Core Hypothesis'''
| Fringe physics theorists
| 2000s (unverified)
| '''Non-mainstream'''  
| Intermittent core energy output; seismic anomalies; geoelectric fluctuations.
|}
TGD proposes that Earth’s growth results from discrete cosmic expansion phases. This framework supports an expanding Earth scenario in which sudden increases in planetary radius coincide with biological and geological transitions, all embedded within a higher-dimensional space-time structure (''M''⁴ × ''CP''₂). See section 8.8.18.


==== '''5.5 Evaluation Criteria (Unified Framework)''' ====
''This chapter now uses the unified evaluation framework applied consistently in Chapters 1 and 8.''

The theories discussed in this chapter are evaluated using the following eight standardized criteria:

# '''1. Empirical Adequacy'''  
Alignment with observed data and experimental results across relevant fields (e.g., astronomy, geology, biology, cosmology). The theory should account for major datasets and empirical patterns.

# '''2. Internal Consistency'''  
Logical and mathematical coherence of the theory. Assumptions should not contradict derived results. Definitions, equations, and claims must be used unambiguously.

# '''3. Predictive Power'''  
The ability to generate '''testable predictions''' that distinguish the theory from others. This includes retrospective explanations that are quantitatively verifiable.

# '''4. Cross-Disciplinary Compatibility'''  
Compatibility with findings and models in other domains, such as geology, biology, planetary science, and quantum physics. A good theory avoids contradicting established results in these areas.

# '''5. Conceptual Clarity and Simplicity'''  
The theory should be clearly framed, avoiding unnecessary complexity or vague terminology. It should be possible to identify what is a definition, an assumption, and a consequence.

# '''6. Heuristic and Unifying Value'''  
Capacity of the theory to stimulate new research, unify previously disconnected phenomena, or offer new conceptual insights across domains.

# '''7. Historical and Philosophical Insight'''  
Engagement with the evolution of cosmological thought. A good theory either improves upon, clarifies, or meaningfully integrates earlier models and philosophical foundations.

# '''8. Mathematical Rigor'''  
Use of precise mathematical definitions and derivations, including:  
* Clarity of assumptions and variable usage.  
* Proofs of existence/uniqueness where applicable.  
* Estimates of modeling error or solution stability.  
* Explicit formulations of approximations.  
* Avoidance of heuristic shortcuts unless well justified.  

This criterion ensures the theory stands up to the standards of formal modeling and does not rely on vague simulations or interchangeable terms.

These criteria offer a structured and balanced framework to evaluate the scientific value and coherence of diverse geological theories. The results of this comparative evaluation are summarized in the table below and explored in more detail in the next subsection ('''5.6 – AI Evaluation Based on Shared Criteria''').

=== '''5.6 – AI Evaluation Based on Shared Criteria''' ===

This subsection presents an AI-assisted comparative evaluation of geological and planetary growth theories based on the shared set of criteria defined in Chapter 1.3. The goal is to identify patterns, strengths, and limitations in how each theory aligns with geological evidence and broader physical frameworks. Each criterion is discussed in its own section to highlight how different models address (or fail to address) key aspects of geological evolution.

This qualitative synthesis draws from published literature, observational datasets, and conceptual consistency across theories.

==== '''Empirical Adequacy''' ====
'''Plate Tectonics''' shows strong alignment with surface features, paleomagnetic striping, and GPS-based motion.  
'''Scalera’s Model''', '''Expanding Earth''', and '''CIT''' provide reconstructions consistent with seafloor age data and ΔT daylength trends, though they are not widely integrated in mainstream datasets.  
'''WEDD''' and the '''Growing Earth Hypothesis''' partially align with crustal expansion indicators but lack full empirical coverage.  
'''Plasma Core''' and '''Pulse Expansion''' theories have low empirical foundation at present.

==== '''Internal Consistency''' ====
'''Plate Tectonics''' and the '''Geochemical Core Model''' are internally robust within standard physics.  
'''CIT''' displays strong internal logic, especially in combining influx-driven mass growth with observable rotational and structural consequences.  
'''Scalera’s reconstruction''' is consistent within its own geometric framework.  
'''WEDD''' and '''Pulse Expansion''' models require assumptions that remain unverified.  
'''Plasma Core''' models lack formal mathematical structure.

==== '''Explanatory Power''' ====
'''Plate Tectonics''' excels in explaining seafloor spreading, mountain formation, and subduction.  
'''CIT''' adds explanatory strength by accounting for ΔT, mass increase, and crustal evolution.  
'''Expanding Earth''' models unify fossil distribution and seafloor geometry but struggle with subduction absence.  
'''WEDD''' provides a bold narrative but lacks detailed explanation for observed crustal segmentation.  
'''Plasma Core''' and '''Pulse Expansion''' remain too speculative to evaluate fully.

==== '''Predictive Strength''' ====
'''CIT''' makes several testable predictions: preferred planetary radii, orbital periods, and ΔT trends.  
'''Plate Tectonics''' makes strong regional predictions (e.g., fault activity), though it assumes constant planetary volume.  
'''Scalera’s Model''' predicts asymmetric expansion signatures in paleomagnetic maps.  
'''WEDD''' and '''Growing Earth''' models do not consistently yield unique predictive outcomes.  
'''Pulse Expansion''' suggests episodic changes but lacks precise timing predictions.

==== '''Simplicity''' ====
'''Geochemical Core Formation''' and '''Plate Tectonics''' are relatively parsimonious within accepted physical theory.  
'''CIT''' uses a unifying influx mechanism rather than complex mantle cycles, offering simplicity at the foundational level.  
'''WEDD''' and '''Plasma Core''' involve complex and speculative internal dynamics.  
'''Expanding Earth''' models vary in complexity depending on geometric assumptions.

==== '''Compatibility with Cosmology''' ====
'''CIT''' explicitly integrates cosmological influx and aligns with universal mass-energy considerations.  
'''Plate Tectonics''' is agnostic toward cosmology, focusing on geophysical processes.  
'''Expanding Earth''', '''WEDD''', and '''Pulse Expansion''' lack direct ties to cosmological models, though some propose external triggers (e.g., loss of atmosphere, orbital shifts).  
'''Plasma Core''' remains disconnected from broader physics frameworks.

==== '''Philosophical and Foundational Clarity''' ====
'''CIT''' is transparent in its foundational assumptions: directional influx, non-conservation of planetary mass, and ontological realism.  
'''Plate Tectonics''' assumes mass conservation and passive mantle behavior.  
'''WEDD''' introduces assumptions about planetary origin that are difficult to validate.  
'''Pulse Expansion''' and '''Plasma Core''' are underdefined and raise open questions about mechanism and ontology.

In summary, this AI-assisted evaluation reveals a diverse landscape of geological theories with varying strengths. While mainstream models such as Plate Tectonics show strong empirical grounding and institutional support, emerging frameworks like the Cosmic Influx Theory (CIT) offer alternative interpretations that integrate geological, rotational, and cosmological observations. Several non-mainstream theories demonstrate internal coherence and partial explanatory overlap, but often lack predictive clarity or cosmological linkage. These differences will be further contextualized in the next subsection ('''5.7 – Helicopter View – Scientific Reception and Emerging Tensions''').


=== '''5.7 – Helicopter View – Scientific Reception and Emerging Tensions''' ===

The field of geological and planetary growth theories reflects a complex interplay between long-standing institutional consensus and a growing body of alternative interpretations. Mainstream geology—centered around Plate Tectonics and mantle convection—has dominated textbooks, academic curricula, and funding structures since the mid-20th century. These models are widely supported by seismic, geodetic, and oceanic crustal data, and have been reinforced by satellite-based GPS measurements.

However, the assumption of a constant Earth radius and conserved planetary mass has increasingly been questioned by independent researchers, especially in light of paleomagnetic reconstructions, crustal age symmetry, and ΔT daylength data. Theories such as the '''Expanding Earth model''', '''Scalera’s asymmetric reconstructions''', '''WEDD''', and the '''Cosmic Influx Theory (CIT)''' propose dynamic planetary evolution, suggesting a potential increase in planetary mass and/or volume over geological time.

These proposals have often met with institutional resistance or dismissal. Alternative models are frequently labeled as “non-scientific” or “fringe,” not necessarily because of a lack of internal coherence or observational support, but due to their incompatibility with foundational assumptions in current geological frameworks. In particular, models that question subduction or posit mass accretion are seen as challenging the core structure of mainstream tectonics.

Nonetheless, there are signs of growing interest in re-evaluating some of these assumptions. Peer-reviewed publications by researchers such as Scalera, Herndon, and Maxlow have brought new visibility to expansion-based models. Online academic platforms (e.g., ResearchGate, Academia.edu, Zenodo) have enabled independent researchers to circulate testable hypotheses outside of traditional peer-review bottlenecks.

The '''Cosmic Influx Theory (CIT)''', in particular, offers a potentially unifying framework by linking geological evolution to cosmological energy influx. Its integration of ΔT measurements, crustal spreading observations, and mass-radius correlations allows it to make both local and systemic predictions—bridging Earth science and astrophysics.

Scientific reception remains polarized. While mainstream geophysics continues to emphasize plate mechanics within a closed mass system, the increasing availability of open data and simulation tools has enabled a new generation of researchers to explore models once considered out of bounds.

This epistemic tension reflects a broader issue in the philosophy of science: whether consensus and institutional stability should take precedence over open theoretical competition. As this chapter has shown, some alternative theories demonstrate sufficient internal logic and empirical grounding to warrant renewed attention and evaluation.

'''This evolving landscape underscores the importance of maintaining an open, evidence-based dialogue as new data and frameworks challenge established geological paradigms.'''


=== '''5.8 – Next Steps''' ===

This chapter has provided a structured comparison of geological and planetary growth theories, ranging from well-established models such as Plate Tectonics to emerging frameworks like the Cosmic Influx Theory (CIT). By applying consistent evaluation criteria, we aimed to highlight both the strengths and open questions associated with each theory.

Future work may proceed along several complementary paths:

* '''Data Re-evaluation and Re-analysis''' – Using modern crustal age maps, paleomagnetic reconstructions, and ΔT daylength datasets to test claims made by both mainstream and non-mainstream theories.

* '''Simulation and Modeling''' – Developing open-source tools or visual reconstructions to compare predictions of radius growth, seafloor spreading patterns, and tectonic dynamics under different theoretical assumptions.

* '''Interdisciplinary Integration''' – Encouraging collaboration between geophysicists, cosmologists, and planetary scientists to explore whether planetary mass evolution can be coherently linked with astrophysical influx, orbital behavior, or thermodynamic energy flows.

* '''Theory Contributions and Dialogue''' – Welcoming new contributions via the Talk page or email (see 5.3), including refined versions of existing theories or hybrid proposals that draw from multiple traditions.

* '''Educational Use and Critical Inquiry''' – Making this chapter a foundation for open discussion, coursework, or critical reading groups focused on understanding the evidence and assumptions underlying planetary evolution.

While institutional resistance to expansion-based or influx-driven models may persist, open-access publishing and AI-assisted review now allow for broader scrutiny and comparison. We encourage readers to explore the models presented here with a critical yet open mind and to consider how future geological discoveries may challenge prevailing frameworks.

This concludes Chapter 5. The next chapter will shift focus to biological and paleontological evidence for planetary evolution.


'''◀ [[AI-Assisted Evaluation of Cosmological Theories/Chapter 4: Planetary Formation, Disk Structures, and System Evolution|Previous]] | [[AI-Assisted Evaluation of Cosmological Theories|Main Page]] | [[AI-Assisted Evaluation of Cosmological Theories/Chapter 6: Biological and Paleontological Clues|Next ▶]]'''</text>
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      <text bytes="22309" sha1="2zk1wf4cbfiydlnfiqtb0yojrgvh2d1" xml:space="preserve">= '''Chapter 5: Geological Clues and Planetary Growth''' =

=== '''5.1 Purpose – Overview of Geological Clues and Planetary Growth''' ===

This chapter investigates how geological evidence from Earth and other celestial bodies may support or challenge current models of planetary formation and evolution. While mainstream geology relies primarily on '''plate tectonics''' and '''mantle convection''' to explain surface dynamics, several independent researchers have proposed alternative models that suggest a long-term increase in '''planetary mass''' and '''volume'''—sometimes referred to as '''planetary growth''' or '''Earth expansion'''.

The primary purpose of this chapter is to bring these perspectives together and evaluate them against empirical geological data such as:

* The distribution and age of the oceanic crust  
* Paleomagnetic records and continent reconstruction  
* Fossil alignment and paleoclimate indicators  
* Historical variations in daylength (ΔT)  
* Seafloor spreading rates and mid-ocean ridge activity  
* Geophysical measurements of planetary radius and density

These geological clues may offer insight into fundamental planetary processes that are often interpreted differently in mainstream and non-mainstream frameworks. For example, the '''Expanding Earth hypothesis''' and '''Cosmic Influx Theory (CIT)''' both propose mechanisms for mass increase over geological time, which may manifest as tectonic, rotational, or geophysical changes observable today.

This chapter aims to:
* Establish a comparative overview of planetary growth theories  
* Analyze how well each model explains geological patterns  
* Explore the possible connection between '''cosmological influx''' and '''geological evolution'''  
* Provide a structured evaluation based on transparent scientific criteria

By grounding these discussions in observable geological data, we hope to assess whether an increase in planetary size or mass is a speculative hypothesis or a testable scientific proposition.


=== '''5.2 Scope – Mainstream and Non-mainstream Theories''' ===

This subsection defines the scope of theories considered in the context of geological clues and planetary growth. It includes both mainstream models accepted by the scientific establishment and non-mainstream models developed by independent researchers who interpret the same geological data through different theoretical lenses.

==== '''Mainstream Scope''' ====
Mainstream geology is largely built upon:
* '''Plate Tectonics Theory''', which explains crustal movement through subduction and seafloor spreading  
* '''Mantle Convection Models''', which describe heat-driven flow within Earth's mantle as the engine for tectonic activity  
* '''Isostasy and Crustal Recycling''', emphasizing a relatively stable Earth radius over time  
* '''Radiometric Dating and Stratigraphy''', to measure geologic time and crustal evolution  

These frameworks assume a '''conservation of planetary mass and volume''', viewing geological processes as surface rearrangements on a constant-radius Earth.

==== '''Non-mainstream Scope''' ====
Alternative theories challenge the assumption of constant mass and radius. These include:

* '''Expanding Earth Theories''' – initiated by Hilgenberg, later developed by Maxlow, Scalera, and others. They propose that Earth’s radius has increased over time, supported by paleomagnetic reconstructions and crustal age patterns.  
* '''Whole-Earth Decompression Dynamics (WEDD)''' – developed by J. Marvin Herndon, positing that Earth initially formed as a Jupiter-like gas giant and expanded after losing its primordial envelope.  
* '''Cosmic Influx Theory (CIT)''' – proposing a directional influx of energy or matter that increases planetary mass-energy over time, potentially accounting for observed geological and geodynamic evolution.  
* '''Growing Earth Hypotheses''' – broadly describing frameworks in which mass accumulation (e.g., via influx, particle capture, or internal generation) drives long-term planetary expansion.  

==== '''Scope Limitations''' ====
This chapter will not cover speculative or purely metaphysical models unless they provide testable geological implications. The focus remains on theories that aim to explain '''observable crustal, magnetic, or structural features''' of planetary evolution.

By setting this balanced scope, we aim to compare how each theory engages with the same geological evidence and whether it offers a plausible, internally consistent explanation for planetary growth.


=== '''5.3 – How to Contribute a Theory''' ===

Researchers and contributors are welcome to propose additional theories of geological evolution and planetary growth. Submissions may include observational insights, mathematical models, conceptual frameworks, or hybrid approaches.

You can contribute in either of the following ways:

* Post directly on the '''Talk page''' of this chapter  
* Or email your proposal to: '''aitheroymapping@gmail.com'''

All submitted theories will be added to the mapping and evaluated using the shared criteria defined in '''Chapter 1.3: Evaluation Criteria'''. We encourage both mainstream and non-mainstream contributors to join this open effort to compare and improve our understanding of '''Geological Clues and Planetary Growth'''.


=== '''5.4 – Theory Mapping Table''' ===

The table below presents an overview of both mainstream and non-mainstream theories related to geological clues and planetary growth. Each theory is listed with its original proponent(s), approximate year of introduction, scientific status, and a summary of testable predictions. This mapping serves as a foundation for further evaluation in subsequent sections.

{| class=&quot;wikitable&quot; style=&quot;width:100%;&quot;
|-
! style=&quot;background:#f0f0f0;&quot; | '''Theory Name'''
! style=&quot;background:#f0f0f0;&quot; | '''Originator(s)'''
! style=&quot;background:#f0f0f0;&quot; | '''Year Introduced'''
! style=&quot;background:#f0f0f0;&quot; | '''Scientific Status'''
! style=&quot;background:#f0f0f0;&quot; | '''Testable Predictions'''
|-
| [https://en.wikipedia.org/wiki/Plate_tectonics '''Plate Tectonics''']
| Alfred Wegener (early), Harry Hess, J. Tuzo Wilson
| 1912 (roots), 1960s (modern form)
| '''Mainstream'''  
| Paleomagnetic striping, matching fossils across continents, GPS-based plate motion, subduction zones.
|-
| [https://en.wikipedia.org/wiki/Mantle_convection '''Mantle Convection Model''']
| Arthur Holmes
| 1930s
| '''Mainstream'''  
| Convection cells inferred from seismic tomography, hotspot drift, heat flow gradients.
|-
| [https://en.wikipedia.org/wiki/Earth%27s_inner_core#Formation '''Geochemical Core Formation''']
| Various geochemists
| 1950s–1970s
| '''Mainstream'''  
| Stable core stratification inferred from seismic discontinuities, isotope ratios.
|-
| [https://www.expanding-earth.org/ '''Expanding Earth Theory''']
| Otto Hilgenberg, Neal Adams, [https://www.jamesmaxlow.com/ James Maxlow]
| 1933 (Hilgenberg), revived 2000s
| '''Non-mainstream'''  
| Continental fit on smaller-radius globe; paleomagnetic reconstructions; lack of subduction remnants.
|-
| [https://nuclearplanet.com/HerndonDecompression.pdf '''Whole-Earth Decompression Dynamics (WEDD)''']
| J. Marvin Herndon
| 2005
| '''Non-mainstream'''  
| Absence of subduction, inner decompression cracks, high-pressure formation markers.
|-
| [http://www.altrocketry.org/Scalera.htm '''Scalera’s Expanding Earth Model''']
| Giancarlo Scalera
| 1990s–2000s
| '''Non-mainstream'''  
| Magnetic reversal reconstruction on smaller Earth; asymmetric expansion patterns.
|-
| [https://en.wikiversity.org/wiki/Cosmic_Influx_Theory '''Cosmic Influx Theory (CIT)''']
| Ruud Loeffen
| 2023
| '''Emerging'''  
| ΔT daylength change; crustal age spread; predicted mass-radius correlation; preferred planetary distances.
|-
| [https://en.wikipedia.org/wiki/Expanding_Earth#Growing_Earth_theories '''Growing Earth Hypothesis''']
| Hugh Owen, S. Warren Carey, others
| 1950s–1970s
| '''Non-mainstream'''  
| Fossil alignment; unchanged continental area; proposed Earth mass increase.
|-
| '''Pulse Expansion Hypotheses'''
| Various researchers
| 1980s–present
| '''Non-mainstream'''  
| Episodic radius changes; correlations with orbital or magnetic cycles; crustal expansion bursts.
|-
| '''Plasma Core Hypothesis'''
| Fringe physics theorists
| 2000s (unverified)
| '''Non-mainstream'''  
| Intermittent core energy output; seismic anomalies; geoelectric fluctuations.
|- 
| style=&quot;background-color:#f2f2f2;&quot; | Topological GeometroDynamics (TGD) 
|| style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream 
|| style=&quot;background-color:#f2f2f2;&quot; | Earth's radius increases during stepwise cosmic expansion events 
|| style=&quot;background-color:#f2f2f2;&quot; | Expanding Earth explained via geometric evolution of space-time surfaces in (''M''⁴ × ''CP''₂) 
|| style=&quot;background-color:#f2f2f2;&quot; | Suggests correlation between Earth's growth and sudden emergence of multicellular life
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://tgdtheory.fi/public_html/articles/preCE.pdf Expanding Earth and Sudden Biological Emergence – TGD]

|}

==== '''5.5 Evaluation Criteria (Unified Framework)''' ====
''This chapter now uses the unified evaluation framework applied consistently in Chapters 1 and 8.''

The theories discussed in this chapter are evaluated using the following eight standardized criteria:

# '''1. Empirical Adequacy'''  
Alignment with observed data and experimental results across relevant fields (e.g., astronomy, geology, biology, cosmology). The theory should account for major datasets and empirical patterns.

# '''2. Internal Consistency'''  
Logical and mathematical coherence of the theory. Assumptions should not contradict derived results. Definitions, equations, and claims must be used unambiguously.

# '''3. Predictive Power'''  
The ability to generate '''testable predictions''' that distinguish the theory from others. This includes retrospective explanations that are quantitatively verifiable.

# '''4. Cross-Disciplinary Compatibility'''  
Compatibility with findings and models in other domains, such as geology, biology, planetary science, and quantum physics. A good theory avoids contradicting established results in these areas.

# '''5. Conceptual Clarity and Simplicity'''  
The theory should be clearly framed, avoiding unnecessary complexity or vague terminology. It should be possible to identify what is a definition, an assumption, and a consequence.

# '''6. Heuristic and Unifying Value'''  
Capacity of the theory to stimulate new research, unify previously disconnected phenomena, or offer new conceptual insights across domains.

# '''7. Historical and Philosophical Insight'''  
Engagement with the evolution of cosmological thought. A good theory either improves upon, clarifies, or meaningfully integrates earlier models and philosophical foundations.

# '''8. Mathematical Rigor'''  
Use of precise mathematical definitions and derivations, including:  
* Clarity of assumptions and variable usage.  
* Proofs of existence/uniqueness where applicable.  
* Estimates of modeling error or solution stability.  
* Explicit formulations of approximations.  
* Avoidance of heuristic shortcuts unless well justified.  

This criterion ensures the theory stands up to the standards of formal modeling and does not rely on vague simulations or interchangeable terms.

These criteria offer a structured and balanced framework to evaluate the scientific value and coherence of diverse geological theories. The results of this comparative evaluation are summarized in the table below and explored in more detail in the next subsection ('''5.6 – AI Evaluation Based on Shared Criteria''').

=== '''5.6 – AI Evaluation Based on Shared Criteria''' ===

This subsection presents an AI-assisted comparative evaluation of geological and planetary growth theories based on the shared set of criteria defined in Chapter 1.3. The goal is to identify patterns, strengths, and limitations in how each theory aligns with geological evidence and broader physical frameworks. Each criterion is discussed in its own section to highlight how different models address (or fail to address) key aspects of geological evolution.

This qualitative synthesis draws from published literature, observational datasets, and conceptual consistency across theories.

==== '''Empirical Adequacy''' ====
'''Plate Tectonics''' shows strong alignment with surface features, paleomagnetic striping, and GPS-based motion.  
'''Scalera’s Model''', '''Expanding Earth''', and '''CIT''' provide reconstructions consistent with seafloor age data and ΔT daylength trends, though they are not widely integrated in mainstream datasets.  
'''WEDD''' and the '''Growing Earth Hypothesis''' partially align with crustal expansion indicators but lack full empirical coverage.  
'''Plasma Core''' and '''Pulse Expansion''' theories have low empirical foundation at present.

==== '''Internal Consistency''' ====
'''Plate Tectonics''' and the '''Geochemical Core Model''' are internally robust within standard physics.  
'''CIT''' displays strong internal logic, especially in combining influx-driven mass growth with observable rotational and structural consequences.  
'''Scalera’s reconstruction''' is consistent within its own geometric framework.  
'''WEDD''' and '''Pulse Expansion''' models require assumptions that remain unverified.  
'''Plasma Core''' models lack formal mathematical structure.

==== '''Explanatory Power''' ====
'''Plate Tectonics''' excels in explaining seafloor spreading, mountain formation, and subduction.  
'''CIT''' adds explanatory strength by accounting for ΔT, mass increase, and crustal evolution.  
'''Expanding Earth''' models unify fossil distribution and seafloor geometry but struggle with subduction absence.  
'''WEDD''' provides a bold narrative but lacks detailed explanation for observed crustal segmentation.  
'''Plasma Core''' and '''Pulse Expansion''' remain too speculative to evaluate fully.

==== '''Predictive Strength''' ====
'''CIT''' makes several testable predictions: preferred planetary radii, orbital periods, and ΔT trends.  
'''Plate Tectonics''' makes strong regional predictions (e.g., fault activity), though it assumes constant planetary volume.  
'''Scalera’s Model''' predicts asymmetric expansion signatures in paleomagnetic maps.  
'''WEDD''' and '''Growing Earth''' models do not consistently yield unique predictive outcomes.  
'''Pulse Expansion''' suggests episodic changes but lacks precise timing predictions.

==== '''Simplicity''' ====
'''Geochemical Core Formation''' and '''Plate Tectonics''' are relatively parsimonious within accepted physical theory.  
'''CIT''' uses a unifying influx mechanism rather than complex mantle cycles, offering simplicity at the foundational level.  
'''WEDD''' and '''Plasma Core''' involve complex and speculative internal dynamics.  
'''Expanding Earth''' models vary in complexity depending on geometric assumptions.

==== '''Compatibility with Cosmology''' ====
'''CIT''' explicitly integrates cosmological influx and aligns with universal mass-energy considerations.  
'''Plate Tectonics''' is agnostic toward cosmology, focusing on geophysical processes.  
'''Expanding Earth''', '''WEDD''', and '''Pulse Expansion''' lack direct ties to cosmological models, though some propose external triggers (e.g., loss of atmosphere, orbital shifts).  
'''Plasma Core''' remains disconnected from broader physics frameworks.

==== '''Philosophical and Foundational Clarity''' ====
'''CIT''' is transparent in its foundational assumptions: directional influx, non-conservation of planetary mass, and ontological realism.  
'''Plate Tectonics''' assumes mass conservation and passive mantle behavior.  
'''WEDD''' introduces assumptions about planetary origin that are difficult to validate.  
'''Pulse Expansion''' and '''Plasma Core''' are underdefined and raise open questions about mechanism and ontology.

In summary, this AI-assisted evaluation reveals a diverse landscape of geological theories with varying strengths. While mainstream models such as Plate Tectonics show strong empirical grounding and institutional support, emerging frameworks like the Cosmic Influx Theory (CIT) offer alternative interpretations that integrate geological, rotational, and cosmological observations. Several non-mainstream theories demonstrate internal coherence and partial explanatory overlap, but often lack predictive clarity or cosmological linkage. These differences will be further contextualized in the next subsection ('''5.7 – Helicopter View – Scientific Reception and Emerging Tensions''').


=== '''5.7 – Helicopter View – Scientific Reception and Emerging Tensions''' ===

The field of geological and planetary growth theories reflects a complex interplay between long-standing institutional consensus and a growing body of alternative interpretations. Mainstream geology—centered around Plate Tectonics and mantle convection—has dominated textbooks, academic curricula, and funding structures since the mid-20th century. These models are widely supported by seismic, geodetic, and oceanic crustal data, and have been reinforced by satellite-based GPS measurements.

However, the assumption of a constant Earth radius and conserved planetary mass has increasingly been questioned by independent researchers, especially in light of paleomagnetic reconstructions, crustal age symmetry, and ΔT daylength data. Theories such as the '''Expanding Earth model''', '''Scalera’s asymmetric reconstructions''', '''WEDD''', and the '''Cosmic Influx Theory (CIT)''' propose dynamic planetary evolution, suggesting a potential increase in planetary mass and/or volume over geological time.

These proposals have often met with institutional resistance or dismissal. Alternative models are frequently labeled as “non-scientific” or “fringe,” not necessarily because of a lack of internal coherence or observational support, but due to their incompatibility with foundational assumptions in current geological frameworks. In particular, models that question subduction or posit mass accretion are seen as challenging the core structure of mainstream tectonics.

Nonetheless, there are signs of growing interest in re-evaluating some of these assumptions. Peer-reviewed publications by researchers such as Scalera, Herndon, and Maxlow have brought new visibility to expansion-based models. Online academic platforms (e.g., ResearchGate, Academia.edu, Zenodo) have enabled independent researchers to circulate testable hypotheses outside of traditional peer-review bottlenecks.

The '''Cosmic Influx Theory (CIT)''', in particular, offers a potentially unifying framework by linking geological evolution to cosmological energy influx. Its integration of ΔT measurements, crustal spreading observations, and mass-radius correlations allows it to make both local and systemic predictions—bridging Earth science and astrophysics.

Scientific reception remains polarized. While mainstream geophysics continues to emphasize plate mechanics within a closed mass system, the increasing availability of open data and simulation tools has enabled a new generation of researchers to explore models once considered out of bounds.

This epistemic tension reflects a broader issue in the philosophy of science: whether consensus and institutional stability should take precedence over open theoretical competition. As this chapter has shown, some alternative theories demonstrate sufficient internal logic and empirical grounding to warrant renewed attention and evaluation.

'''This evolving landscape underscores the importance of maintaining an open, evidence-based dialogue as new data and frameworks challenge established geological paradigms.'''


=== '''5.8 – Next Steps''' ===

This chapter has provided a structured comparison of geological and planetary growth theories, ranging from well-established models such as Plate Tectonics to emerging frameworks like the Cosmic Influx Theory (CIT). By applying consistent evaluation criteria, we aimed to highlight both the strengths and open questions associated with each theory.

Future work may proceed along several complementary paths:

* '''Data Re-evaluation and Re-analysis''' – Using modern crustal age maps, paleomagnetic reconstructions, and ΔT daylength datasets to test claims made by both mainstream and non-mainstream theories.

* '''Simulation and Modeling''' – Developing open-source tools or visual reconstructions to compare predictions of radius growth, seafloor spreading patterns, and tectonic dynamics under different theoretical assumptions.

* '''Interdisciplinary Integration''' – Encouraging collaboration between geophysicists, cosmologists, and planetary scientists to explore whether planetary mass evolution can be coherently linked with astrophysical influx, orbital behavior, or thermodynamic energy flows.

* '''Theory Contributions and Dialogue''' – Welcoming new contributions via the Talk page or email (see 5.3), including refined versions of existing theories or hybrid proposals that draw from multiple traditions.

* '''Educational Use and Critical Inquiry''' – Making this chapter a foundation for open discussion, coursework, or critical reading groups focused on understanding the evidence and assumptions underlying planetary evolution.

While institutional resistance to expansion-based or influx-driven models may persist, open-access publishing and AI-assisted review now allow for broader scrutiny and comparison. We encourage readers to explore the models presented here with a critical yet open mind and to consider how future geological discoveries may challenge prevailing frameworks.

This concludes Chapter 5. The next chapter will shift focus to biological and paleontological evidence for planetary evolution.


'''◀ [[AI-Assisted Evaluation of Cosmological Theories/Chapter 4: Planetary Formation, Disk Structures, and System Evolution|Previous]] | [[AI-Assisted Evaluation of Cosmological Theories|Main Page]] | [[AI-Assisted Evaluation of Cosmological Theories/Chapter 6: Biological and Paleontological Clues|Next ▶]]'''</text>
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      <text bytes="22305" sha1="hrtu5hjksp03goo1gzepefp3xir56c9" xml:space="preserve">= '''Chapter 5: Geological Clues and Planetary Growth''' =

=== '''5.1 Purpose – Overview of Geological Clues and Planetary Growth''' ===

This chapter investigates how geological evidence from Earth and other celestial bodies may support or challenge current models of planetary formation and evolution. While mainstream geology relies primarily on '''plate tectonics''' and '''mantle convection''' to explain surface dynamics, several independent researchers have proposed alternative models that suggest a long-term increase in '''planetary mass''' and '''volume'''—sometimes referred to as '''planetary growth''' or '''Earth expansion'''.

The primary purpose of this chapter is to bring these perspectives together and evaluate them against empirical geological data such as:

* The distribution and age of the oceanic crust  
* Paleomagnetic records and continent reconstruction  
* Fossil alignment and paleoclimate indicators  
* Historical variations in daylength (ΔT)  
* Seafloor spreading rates and mid-ocean ridge activity  
* Geophysical measurements of planetary radius and density

These geological clues may offer insight into fundamental planetary processes that are often interpreted differently in mainstream and non-mainstream frameworks. For example, the '''Expanding Earth hypothesis''' and '''Cosmic Influx Theory (CIT)''' both propose mechanisms for mass increase over geological time, which may manifest as tectonic, rotational, or geophysical changes observable today.

This chapter aims to:
* Establish a comparative overview of planetary growth theories  
* Analyze how well each model explains geological patterns  
* Explore the possible connection between '''cosmological influx''' and '''geological evolution'''  
* Provide a structured evaluation based on transparent scientific criteria

By grounding these discussions in observable geological data, we hope to assess whether an increase in planetary size or mass is a speculative hypothesis or a testable scientific proposition.


=== '''5.2 Scope – Mainstream and Non-mainstream Theories''' ===

This subsection defines the scope of theories considered in the context of geological clues and planetary growth. It includes both mainstream models accepted by the scientific establishment and non-mainstream models developed by independent researchers who interpret the same geological data through different theoretical lenses.

==== '''Mainstream Scope''' ====
Mainstream geology is largely built upon:
* '''Plate Tectonics Theory''', which explains crustal movement through subduction and seafloor spreading  
* '''Mantle Convection Models''', which describe heat-driven flow within Earth's mantle as the engine for tectonic activity  
* '''Isostasy and Crustal Recycling''', emphasizing a relatively stable Earth radius over time  
* '''Radiometric Dating and Stratigraphy''', to measure geologic time and crustal evolution  

These frameworks assume a '''conservation of planetary mass and volume''', viewing geological processes as surface rearrangements on a constant-radius Earth.

==== '''Non-mainstream Scope''' ====
Alternative theories challenge the assumption of constant mass and radius. These include:

* '''Expanding Earth Theories''' – initiated by Hilgenberg, later developed by Maxlow, Scalera, and others. They propose that Earth’s radius has increased over time, supported by paleomagnetic reconstructions and crustal age patterns.  
* '''Whole-Earth Decompression Dynamics (WEDD)''' – developed by J. Marvin Herndon, positing that Earth initially formed as a Jupiter-like gas giant and expanded after losing its primordial envelope.  
* '''Cosmic Influx Theory (CIT)''' – proposing a directional influx of energy or matter that increases planetary mass-energy over time, potentially accounting for observed geological and geodynamic evolution.  
* '''Growing Earth Hypotheses''' – broadly describing frameworks in which mass accumulation (e.g., via influx, particle capture, or internal generation) drives long-term planetary expansion.  

==== '''Scope Limitations''' ====
This chapter will not cover speculative or purely metaphysical models unless they provide testable geological implications. The focus remains on theories that aim to explain '''observable crustal, magnetic, or structural features''' of planetary evolution.

By setting this balanced scope, we aim to compare how each theory engages with the same geological evidence and whether it offers a plausible, internally consistent explanation for planetary growth.


=== '''5.3 – How to Contribute a Theory''' ===

Researchers and contributors are welcome to propose additional theories of geological evolution and planetary growth. Submissions may include observational insights, mathematical models, conceptual frameworks, or hybrid approaches.

You can contribute in either of the following ways:

* Post directly on the '''Talk page''' of this chapter  
* Or email your proposal to: '''aitheroymapping@gmail.com'''

All submitted theories will be added to the mapping and evaluated using the shared criteria defined in '''Chapter 1.3: Evaluation Criteria'''. We encourage both mainstream and non-mainstream contributors to join this open effort to compare and improve our understanding of '''Geological Clues and Planetary Growth'''.


=== '''5.4 – Theory Mapping Table''' ===

The table below presents an overview of both mainstream and non-mainstream theories related to geological clues and planetary growth. Each theory is listed with its original proponent(s), approximate year of introduction, scientific status, and a summary of testable predictions. This mapping serves as a foundation for further evaluation in subsequent sections.

{| class=&quot;wikitable&quot; style=&quot;width:100%;&quot;
|-
! style=&quot;background:#f0f0f0;&quot; | '''Theory Name'''
! style=&quot;background:#f0f0f0;&quot; | '''Originator(s)'''
! style=&quot;background:#f0f0f0;&quot; | '''Year Introduced'''
! style=&quot;background:#f0f0f0;&quot; | '''Scientific Status'''
! style=&quot;background:#f0f0f0;&quot; | '''Testable Predictions'''
|-
| [https://en.wikipedia.org/wiki/Plate_tectonics '''Plate Tectonics''']
| Alfred Wegener (early), Harry Hess, J. Tuzo Wilson
| 1912 (roots), 1960s (modern form)
| '''Mainstream'''  
| Paleomagnetic striping, matching fossils across continents, GPS-based plate motion, subduction zones.
|-
| [https://en.wikipedia.org/wiki/Mantle_convection '''Mantle Convection Model''']
| Arthur Holmes
| 1930s
| '''Mainstream'''  
| Convection cells inferred from seismic tomography, hotspot drift, heat flow gradients.
|-
| [https://en.wikipedia.org/wiki/Earth%27s_inner_core#Formation '''Geochemical Core Formation''']
| Various geochemists
| 1950s–1970s
| '''Mainstream'''  
| Stable core stratification inferred from seismic discontinuities, isotope ratios.
|-
| [https://www.expanding-earth.org/ '''Expanding Earth Theory''']
| Otto Hilgenberg, Neal Adams, [https://www.jamesmaxlow.com/ James Maxlow]
| 1933 (Hilgenberg), revived 2000s
| '''Non-mainstream'''  
| Continental fit on smaller-radius globe; paleomagnetic reconstructions; lack of subduction remnants.
|-
| [https://nuclearplanet.com/HerndonDecompression.pdf '''Whole-Earth Decompression Dynamics (WEDD)''']
| J. Marvin Herndon
| 2005
| '''Non-mainstream'''  
| Absence of subduction, inner decompression cracks, high-pressure formation markers.
|-
| [http://www.altrocketry.org/Scalera.htm '''Scalera’s Expanding Earth Model''']
| Giancarlo Scalera
| 1990s–2000s
| '''Non-mainstream'''  
| Magnetic reversal reconstruction on smaller Earth; asymmetric expansion patterns.
|-
| [https://en.wikiversity.org/wiki/Cosmic_Influx_Theory '''Cosmic Influx Theory (CIT)''']
| Ruud Loeffen
| 2023
| '''Emerging'''  
| ΔT daylength change; crustal age spread; predicted mass-radius correlation; preferred planetary distances.
|-
| [https://en.wikipedia.org/wiki/Expanding_Earth#Growing_Earth_theories '''Growing Earth Hypothesis''']
| Hugh Owen, S. Warren Carey, others
| 1950s–1970s
| '''Non-mainstream'''  
| Fossil alignment; unchanged continental area; proposed Earth mass increase.
|-
| '''Pulse Expansion Hypotheses'''
| Various researchers
| 1980s–present
| '''Non-mainstream'''  
| Episodic radius changes; correlations with orbital or magnetic cycles; crustal expansion bursts.
|-
| '''Plasma Core Hypothesis'''
| Fringe physics theorists
| 2000s (unverified)
| '''Non-mainstream'''  
| Intermittent core energy output; seismic anomalies; geoelectric fluctuations.
|- 
| style=&quot;background-color:#f2f2f2;&quot; | Topological GeometroDynamics (TGD) 
| style=&quot;background-color:#f2f2f2;&quot; | Nonmainstream 
| style=&quot;background-color:#f2f2f2;&quot; | Earth's radius increases during stepwise cosmic expansion events 
| style=&quot;background-color:#f2f2f2;&quot; | Expanding Earth explained via geometric evolution of space-time surfaces in (''M''⁴ × ''CP''₂) 
| style=&quot;background-color:#f2f2f2;&quot; | Suggests correlation between Earth's growth and sudden emergence of multicellular life
|-
| colspan=&quot;5&quot; style=&quot;background-color:#f2f2f2;&quot; | Related link: [https://tgdtheory.fi/public_html/articles/preCE.pdf Expanding Earth and Sudden Biological Emergence – TGD]

|}

==== '''5.5 Evaluation Criteria (Unified Framework)''' ====
''This chapter now uses the unified evaluation framework applied consistently in Chapters 1 and 8.''

The theories discussed in this chapter are evaluated using the following eight standardized criteria:

# '''1. Empirical Adequacy'''  
Alignment with observed data and experimental results across relevant fields (e.g., astronomy, geology, biology, cosmology). The theory should account for major datasets and empirical patterns.

# '''2. Internal Consistency'''  
Logical and mathematical coherence of the theory. Assumptions should not contradict derived results. Definitions, equations, and claims must be used unambiguously.

# '''3. Predictive Power'''  
The ability to generate '''testable predictions''' that distinguish the theory from others. This includes retrospective explanations that are quantitatively verifiable.

# '''4. Cross-Disciplinary Compatibility'''  
Compatibility with findings and models in other domains, such as geology, biology, planetary science, and quantum physics. A good theory avoids contradicting established results in these areas.

# '''5. Conceptual Clarity and Simplicity'''  
The theory should be clearly framed, avoiding unnecessary complexity or vague terminology. It should be possible to identify what is a definition, an assumption, and a consequence.

# '''6. Heuristic and Unifying Value'''  
Capacity of the theory to stimulate new research, unify previously disconnected phenomena, or offer new conceptual insights across domains.

# '''7. Historical and Philosophical Insight'''  
Engagement with the evolution of cosmological thought. A good theory either improves upon, clarifies, or meaningfully integrates earlier models and philosophical foundations.

# '''8. Mathematical Rigor'''  
Use of precise mathematical definitions and derivations, including:  
* Clarity of assumptions and variable usage.  
* Proofs of existence/uniqueness where applicable.  
* Estimates of modeling error or solution stability.  
* Explicit formulations of approximations.  
* Avoidance of heuristic shortcuts unless well justified.  

This criterion ensures the theory stands up to the standards of formal modeling and does not rely on vague simulations or interchangeable terms.

These criteria offer a structured and balanced framework to evaluate the scientific value and coherence of diverse geological theories. The results of this comparative evaluation are summarized in the table below and explored in more detail in the next subsection ('''5.6 – AI Evaluation Based on Shared Criteria''').

=== '''5.6 – AI Evaluation Based on Shared Criteria''' ===

This subsection presents an AI-assisted comparative evaluation of geological and planetary growth theories based on the shared set of criteria defined in Chapter 1.3. The goal is to identify patterns, strengths, and limitations in how each theory aligns with geological evidence and broader physical frameworks. Each criterion is discussed in its own section to highlight how different models address (or fail to address) key aspects of geological evolution.

This qualitative synthesis draws from published literature, observational datasets, and conceptual consistency across theories.

==== '''Empirical Adequacy''' ====
'''Plate Tectonics''' shows strong alignment with surface features, paleomagnetic striping, and GPS-based motion.  
'''Scalera’s Model''', '''Expanding Earth''', and '''CIT''' provide reconstructions consistent with seafloor age data and ΔT daylength trends, though they are not widely integrated in mainstream datasets.  
'''WEDD''' and the '''Growing Earth Hypothesis''' partially align with crustal expansion indicators but lack full empirical coverage.  
'''Plasma Core''' and '''Pulse Expansion''' theories have low empirical foundation at present.

==== '''Internal Consistency''' ====
'''Plate Tectonics''' and the '''Geochemical Core Model''' are internally robust within standard physics.  
'''CIT''' displays strong internal logic, especially in combining influx-driven mass growth with observable rotational and structural consequences.  
'''Scalera’s reconstruction''' is consistent within its own geometric framework.  
'''WEDD''' and '''Pulse Expansion''' models require assumptions that remain unverified.  
'''Plasma Core''' models lack formal mathematical structure.

==== '''Explanatory Power''' ====
'''Plate Tectonics''' excels in explaining seafloor spreading, mountain formation, and subduction.  
'''CIT''' adds explanatory strength by accounting for ΔT, mass increase, and crustal evolution.  
'''Expanding Earth''' models unify fossil distribution and seafloor geometry but struggle with subduction absence.  
'''WEDD''' provides a bold narrative but lacks detailed explanation for observed crustal segmentation.  
'''Plasma Core''' and '''Pulse Expansion''' remain too speculative to evaluate fully.

==== '''Predictive Strength''' ====
'''CIT''' makes several testable predictions: preferred planetary radii, orbital periods, and ΔT trends.  
'''Plate Tectonics''' makes strong regional predictions (e.g., fault activity), though it assumes constant planetary volume.  
'''Scalera’s Model''' predicts asymmetric expansion signatures in paleomagnetic maps.  
'''WEDD''' and '''Growing Earth''' models do not consistently yield unique predictive outcomes.  
'''Pulse Expansion''' suggests episodic changes but lacks precise timing predictions.

==== '''Simplicity''' ====
'''Geochemical Core Formation''' and '''Plate Tectonics''' are relatively parsimonious within accepted physical theory.  
'''CIT''' uses a unifying influx mechanism rather than complex mantle cycles, offering simplicity at the foundational level.  
'''WEDD''' and '''Plasma Core''' involve complex and speculative internal dynamics.  
'''Expanding Earth''' models vary in complexity depending on geometric assumptions.

==== '''Compatibility with Cosmology''' ====
'''CIT''' explicitly integrates cosmological influx and aligns with universal mass-energy considerations.  
'''Plate Tectonics''' is agnostic toward cosmology, focusing on geophysical processes.  
'''Expanding Earth''', '''WEDD''', and '''Pulse Expansion''' lack direct ties to cosmological models, though some propose external triggers (e.g., loss of atmosphere, orbital shifts).  
'''Plasma Core''' remains disconnected from broader physics frameworks.

==== '''Philosophical and Foundational Clarity''' ====
'''CIT''' is transparent in its foundational assumptions: directional influx, non-conservation of planetary mass, and ontological realism.  
'''Plate Tectonics''' assumes mass conservation and passive mantle behavior.  
'''WEDD''' introduces assumptions about planetary origin that are difficult to validate.  
'''Pulse Expansion''' and '''Plasma Core''' are underdefined and raise open questions about mechanism and ontology.

In summary, this AI-assisted evaluation reveals a diverse landscape of geological theories with varying strengths. While mainstream models such as Plate Tectonics show strong empirical grounding and institutional support, emerging frameworks like the Cosmic Influx Theory (CIT) offer alternative interpretations that integrate geological, rotational, and cosmological observations. Several non-mainstream theories demonstrate internal coherence and partial explanatory overlap, but often lack predictive clarity or cosmological linkage. These differences will be further contextualized in the next subsection ('''5.7 – Helicopter View – Scientific Reception and Emerging Tensions''').


=== '''5.7 – Helicopter View – Scientific Reception and Emerging Tensions''' ===

The field of geological and planetary growth theories reflects a complex interplay between long-standing institutional consensus and a growing body of alternative interpretations. Mainstream geology—centered around Plate Tectonics and mantle convection—has dominated textbooks, academic curricula, and funding structures since the mid-20th century. These models are widely supported by seismic, geodetic, and oceanic crustal data, and have been reinforced by satellite-based GPS measurements.

However, the assumption of a constant Earth radius and conserved planetary mass has increasingly been questioned by independent researchers, especially in light of paleomagnetic reconstructions, crustal age symmetry, and ΔT daylength data. Theories such as the '''Expanding Earth model''', '''Scalera’s asymmetric reconstructions''', '''WEDD''', and the '''Cosmic Influx Theory (CIT)''' propose dynamic planetary evolution, suggesting a potential increase in planetary mass and/or volume over geological time.

These proposals have often met with institutional resistance or dismissal. Alternative models are frequently labeled as “non-scientific” or “fringe,” not necessarily because of a lack of internal coherence or observational support, but due to their incompatibility with foundational assumptions in current geological frameworks. In particular, models that question subduction or posit mass accretion are seen as challenging the core structure of mainstream tectonics.

Nonetheless, there are signs of growing interest in re-evaluating some of these assumptions. Peer-reviewed publications by researchers such as Scalera, Herndon, and Maxlow have brought new visibility to expansion-based models. Online academic platforms (e.g., ResearchGate, Academia.edu, Zenodo) have enabled independent researchers to circulate testable hypotheses outside of traditional peer-review bottlenecks.

The '''Cosmic Influx Theory (CIT)''', in particular, offers a potentially unifying framework by linking geological evolution to cosmological energy influx. Its integration of ΔT measurements, crustal spreading observations, and mass-radius correlations allows it to make both local and systemic predictions—bridging Earth science and astrophysics.

Scientific reception remains polarized. While mainstream geophysics continues to emphasize plate mechanics within a closed mass system, the increasing availability of open data and simulation tools has enabled a new generation of researchers to explore models once considered out of bounds.

This epistemic tension reflects a broader issue in the philosophy of science: whether consensus and institutional stability should take precedence over open theoretical competition. As this chapter has shown, some alternative theories demonstrate sufficient internal logic and empirical grounding to warrant renewed attention and evaluation.

'''This evolving landscape underscores the importance of maintaining an open, evidence-based dialogue as new data and frameworks challenge established geological paradigms.'''


=== '''5.8 – Next Steps''' ===

This chapter has provided a structured comparison of geological and planetary growth theories, ranging from well-established models such as Plate Tectonics to emerging frameworks like the Cosmic Influx Theory (CIT). By applying consistent evaluation criteria, we aimed to highlight both the strengths and open questions associated with each theory.

Future work may proceed along several complementary paths:

* '''Data Re-evaluation and Re-analysis''' – Using modern crustal age maps, paleomagnetic reconstructions, and ΔT daylength datasets to test claims made by both mainstream and non-mainstream theories.

* '''Simulation and Modeling''' – Developing open-source tools or visual reconstructions to compare predictions of radius growth, seafloor spreading patterns, and tectonic dynamics under different theoretical assumptions.

* '''Interdisciplinary Integration''' – Encouraging collaboration between geophysicists, cosmologists, and planetary scientists to explore whether planetary mass evolution can be coherently linked with astrophysical influx, orbital behavior, or thermodynamic energy flows.

* '''Theory Contributions and Dialogue''' – Welcoming new contributions via the Talk page or email (see 5.3), including refined versions of existing theories or hybrid proposals that draw from multiple traditions.

* '''Educational Use and Critical Inquiry''' – Making this chapter a foundation for open discussion, coursework, or critical reading groups focused on understanding the evidence and assumptions underlying planetary evolution.

While institutional resistance to expansion-based or influx-driven models may persist, open-access publishing and AI-assisted review now allow for broader scrutiny and comparison. We encourage readers to explore the models presented here with a critical yet open mind and to consider how future geological discoveries may challenge prevailing frameworks.

This concludes Chapter 5. The next chapter will shift focus to biological and paleontological evidence for planetary evolution.


'''◀ [[AI-Assisted Evaluation of Cosmological Theories/Chapter 4: Planetary Formation, Disk Structures, and System Evolution|Previous]] | [[AI-Assisted Evaluation of Cosmological Theories|Main Page]] | [[AI-Assisted Evaluation of Cosmological Theories/Chapter 6: Biological and Paleontological Clues|Next ▶]]'''</text>
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      <text bytes="20002" sha1="85lhl3glyllqdm3zu3un2li5t3fe7uu" xml:space="preserve">= '''Chapter 6: Biological and Paleontological Clues''' =

== 6.1 Purpose – Biological and Paleontological Clues ==

The purpose of this chapter is to explore how biological and paleontological evidence—traditionally associated with evolutionary biology and Earth history—can also provide valuable insights for evaluating cosmological theories. While most cosmological models are constructed from astronomical observations and physical laws, the long-term patterns of life on Earth may reflect deeper planetary and cosmic influences.

Life does not evolve in isolation. It depends on planetary conditions shaped by cosmological processes: the stability of Earth’s orbit, solar radiation levels, planetary mass and gravity, atmospheric composition, and even plate tectonics. Likewise, fossil records reveal striking transitions—such as mass extinctions and bursts of biodiversity—that may correspond to larger planetary or cosmic events.

This chapter invites contributors to examine how such clues may support or challenge competing cosmological frameworks. For example:
* The theory of Earth Expansion suggests a changing planetary radius that could influence evolution and extinction patterns.
* Mainstream astrophysics connects biological timelines to solar evolution and asteroid impacts.
* Non-mainstream theories—such as Cosmic Influx Theory—propose directional energy flows that might influence planetary life indirectly through geophysical changes.

By including biological and fossil-based data, we broaden the evidential base for cosmological inquiry and invite interdisciplinary reflection on the origins, transformations, and long-term stability of life in a cosmic context.


== 6.2 Scope – Mainstream and Non-mainstream Theories ==

This chapter includes both mainstream and non-mainstream theories that offer explanations linking biological and paleontological clues to broader cosmological processes.

Mainstream scientific perspectives generally accept that life on Earth is deeply influenced by astronomical and planetary factors:
* **Astrophysical models** explain how solar evolution affects planetary climates and habitability zones.
* **Impact theories** (e.g., Chicxulub asteroid) connect mass extinctions to cosmic events.
* **Plate tectonics and continental drift** shape ecosystems, evolution, and extinction patterns over geological time.

However, growing evidence—including rapid climate shifts, tectonic anomalies, and sudden biodiversity changes—has also stimulated interest in alternative frameworks that integrate Earth’s biological history into larger cosmic dynamics.

This chapter therefore also considers non-mainstream perspectives such as:
* **Earth Expansion Theories**, which posit that changes in Earth’s size and gravity could affect species evolution and extinction timing.
* **Directed Panspermia**, which explores whether life arrived from or was influenced by external cosmic agents.
* **Spiral Cosmology**, which links galactic motion to cycles of evolution and extinction.
* **Cosmic Influx Theory (CIT)**, which proposes a continuous directional influx of energy or particles from space. In CIT, this influx contributes not only to gravity but also to increasing planetary mass and internal energy—potentially influencing geological activity and, indirectly, evolutionary pressures and extinction events.

The scope of this chapter is thus interdisciplinary and inclusive. It encourages readers and contributors to explore the potential interplay between life’s history on Earth and the larger processes governing planetary and cosmic evolution. All theories must propose a link—direct or indirect—between biological/paleontological patterns and cosmological mechanisms.


== 6.3 How to Contribute a Theory ==

Researchers and contributors are welcome to propose additional theories that relate biological evolution and fossil records to planetary or cosmological models. Submissions may include observational insights, mathematical models, conceptual frameworks, or hybrid approaches.

You can contribute in either of the following ways:

* Post directly on the Talk page of this chapter  
* Or email your proposal to: aitheroymapping@gmail.com  

All submitted theories will be added to the mapping and evaluated using the shared criteria defined in Chapter 1.3: Evaluation Criteria. We encourage both mainstream and non-mainstream contributors to join this open effort to compare and improve our understanding of Biological and Paleontological Clues in relation to cosmological theories.


== 6.4 Theory Mapping Table ==

This table presents a comparative overview of theories that attempt to link biological and paleontological clues to planetary or cosmological models. Both mainstream and non-mainstream approaches are included. Each theory is followed by a reference line shaded identically to the theory row, maintaining visual pairing for clarity.

{| class=&quot;wikitable&quot; style=&quot;width:100%; text-align:center;&quot;
|-
! style=&quot;background-color:#f2f2f2;&quot; | '''Theory Name'''
! style=&quot;background-color:#f2f2f2;&quot; | '''Type of Framework'''
! style=&quot;background-color:#f2f2f2;&quot; | '''Biological Focus'''
! style=&quot;background-color:#f2f2f2;&quot; | '''Paleontological Signals'''
! style=&quot;background-color:#f2f2f2;&quot; | '''Cosmological Link'''
! style=&quot;background-color:#f2f2f2;&quot; | '''Temporal Scope'''

|-
| style=&quot;background-color:#ffffff;&quot; | '''Darwinian Evolution'''
| style=&quot;background-color:#ffffff;&quot; | Mainstream
| style=&quot;background-color:#ffffff;&quot; | Natural selection, gradual evolution
| style=&quot;background-color:#ffffff;&quot; | Transitional fossils, biodiversity patterns
| style=&quot;background-color:#ffffff;&quot; | Linked indirectly via planetary stability and solar radiation
| style=&quot;background-color:#ffffff;&quot; | Phanerozoic

|-
| colspan=&quot;6&quot; style=&quot;background-color:#ffffff; text-align:left;&quot; | Related link: [https://en.wikipedia.org/wiki/Darwinism Darwinian Evolution]

|-
| style=&quot;background-color:#e6e6e6;&quot; | '''Earth Expansion Theory'''
| style=&quot;background-color:#e6e6e6;&quot; | Non-mainstream
| style=&quot;background-color:#e6e6e6;&quot; | Species adaptation to gravity and surface change
| style=&quot;background-color:#e6e6e6;&quot; | Apparent size changes in fossils; shifting habitats
| style=&quot;background-color:#e6e6e6;&quot; | Changing gravity and surface area affect evolutionary pressures
| style=&quot;background-color:#e6e6e6;&quot; | Precambrian to present

|-
| colspan=&quot;6&quot; style=&quot;background-color:#e6e6e6; text-align:left;&quot; | Related link: [https://en.wikipedia.org/wiki/Expanding_Earth Expanding Earth Theory]

|-
| style=&quot;background-color:#ffffff;&quot; | '''Cosmic Influx Theory (CIT)'''
| style=&quot;background-color:#ffffff;&quot; | Non-mainstream
| style=&quot;background-color:#ffffff;&quot; | Life adapts to increasing internal energy and mass
| style=&quot;background-color:#ffffff;&quot; | Evolutionary leaps, extinction patterns linked to influx variations
| style=&quot;background-color:#ffffff;&quot; | Energy influx affects planetary expansion, heat, and tectonics
| style=&quot;background-color:#ffffff;&quot; | Full Earth history

|-
| colspan=&quot;6&quot; style=&quot;background-color:#ffffff; text-align:left;&quot; | Related link: [https://en.wikiversity.org/wiki/Cosmic_Influx_Theory Cosmic Influx Theory (CIT)]

|-
| style=&quot;background-color:#e6e6e6;&quot; | '''Punctuated Equilibrium'''
| style=&quot;background-color:#e6e6e6;&quot; | Mainstream
| style=&quot;background-color:#e6e6e6;&quot; | Rapid evolutionary jumps
| style=&quot;background-color:#e6e6e6;&quot; | Sudden appearances and disappearances of species
| style=&quot;background-color:#e6e6e6;&quot; | Possibly linked to environmental or cosmic disturbance cycles
| style=&quot;background-color:#e6e6e6;&quot; | Phanerozoic

|-
| colspan=&quot;6&quot; style=&quot;background-color:#e6e6e6; text-align:left;&quot; | Related link: [https://en.wikipedia.org/wiki/Punctuated_equilibrium Punctuated Equilibrium]

|-
| style=&quot;background-color:#ffffff;&quot; | '''Milankovitch Cycles and Solar Forcing'''
| style=&quot;background-color:#ffffff;&quot; | Mainstream
| style=&quot;background-color:#ffffff;&quot; | Climate-driven species migration and extinction
| style=&quot;background-color:#ffffff;&quot; | Ice age cycles, faunal shifts
| style=&quot;background-color:#ffffff;&quot; | Orbital and solar variations affecting Earth's climate and ecosystems
| style=&quot;background-color:#ffffff;&quot; | Quaternary and earlier

|-
| colspan=&quot;6&quot; style=&quot;background-color:#ffffff; text-align:left;&quot; | Related link: [https://en.wikipedia.org/wiki/Milankovitch_cycles Milankovitch Cycles]

|-
| style=&quot;background-color:#e6e6e6;&quot; | '''Galactic Biodiversity Cycle'''
| style=&quot;background-color:#e6e6e6;&quot; | Hybrid
| style=&quot;background-color:#e6e6e6;&quot; | Periodic evolution and extinction
| style=&quot;background-color:#e6e6e6;&quot; | 62-million-year biodiversity cycles
| style=&quot;background-color:#e6e6e6;&quot; | Solar system’s oscillation through galactic plane
| style=&quot;background-color:#e6e6e6;&quot; | Phanerozoic

|-
| colspan=&quot;6&quot; style=&quot;background-color:#e6e6e6; text-align:left;&quot; | Related link: [https://www.nature.com/articles/nature03339 Rohde &amp; Muller (2005), Nature]

|-
| style=&quot;background-color:#ffffff;&quot; | '''Directed Panspermia'''
| style=&quot;background-color:#ffffff;&quot; | Non-mainstream
| style=&quot;background-color:#ffffff;&quot; | Origin of life on Earth
| style=&quot;background-color:#ffffff;&quot; | Sudden appearance of complex life
| style=&quot;background-color:#ffffff;&quot; | Life seeded from extraterrestrial or engineered sources
| style=&quot;background-color:#ffffff;&quot; | Origin and early biosphere

|-
| colspan=&quot;6&quot; style=&quot;background-color:#ffffff; text-align:left;&quot; | Related link: [https://en.wikipedia.org/wiki/Directed_panspermia Directed Panspermia]

|-
| style=&quot;background-color:#e6e6e6;&quot; | '''Spiral Cosmology'''
| style=&quot;background-color:#e6e6e6;&quot; | Non-mainstream
| style=&quot;background-color:#e6e6e6;&quot; | Biological evolution tied to spiral wave cycles
| style=&quot;background-color:#e6e6e6;&quot; | Periodic mass extinctions and evolutionary resets
| style=&quot;background-color:#e6e6e6;&quot; | Cosmic motion through structured spiral fields
| style=&quot;background-color:#e6e6e6;&quot; | Full Earth history

|-
| colspan=&quot;6&quot; style=&quot;background-color:#e6e6e6; text-align:left;&quot; | Related link: [https://zenodo.org/records/10675169 Spiral Cosmology on Zenodo]

|-
| style=&quot;background-color:#ffffff;&quot; | '''Geomagnetic Reversal Hypothesis'''
| style=&quot;background-color:#ffffff;&quot; | Hybrid
| style=&quot;background-color:#ffffff;&quot; | Mutation events, migration pressures
| style=&quot;background-color:#ffffff;&quot; | Extinction correlations with magnetic field reversals
| style=&quot;background-color:#ffffff;&quot; | Field reversals affect radiation shielding and atmospheric conditions
| style=&quot;background-color:#ffffff;&quot; | Mesozoic–Cenozoic

|-
| colspan=&quot;6&quot; style=&quot;background-color:#ffffff; text-align:left;&quot; | Related link: [https://en.wikipedia.org/wiki/Geomagnetic_reversal Geomagnetic Reversal]

|-
| style=&quot;background-color:#e6e6e6;&quot; | '''Plasma Cosmology (Biological Interpretation)'''
| style=&quot;background-color:#e6e6e6;&quot; | Non-mainstream
| style=&quot;background-color:#e6e6e6;&quot; | Life shaped by electrical and plasma environment
| style=&quot;background-color:#e6e6e6;&quot; | Catastrophic reorganization of biosphere
| style=&quot;background-color:#e6e6e6;&quot; | Cosmic plasma discharges interacting with Earth
| style=&quot;background-color:#e6e6e6;&quot; | Ancient mythological and early human history

|-
| colspan=&quot;6&quot; style=&quot;background-color:#e6e6e6; text-align:left;&quot; | Related link: [https://www.holoscience.com Plasma Cosmology – Holoscience]

|-
| style=&quot;background-color:#ffffff;&quot; | '''Anthropic Cosmology / Fine-Tuned Earth'''
| style=&quot;background-color:#ffffff;&quot; | Hybrid
| style=&quot;background-color:#ffffff;&quot; | Life’s emergence depends on cosmological constants
| style=&quot;background-color:#ffffff;&quot; | Life appears &quot;fine-tuned&quot; to planetary conditions
| style=&quot;background-color:#ffffff;&quot; | Universe structured to permit life; observer bias
| style=&quot;background-color:#ffffff;&quot; | Origin of life to present

|-
| colspan=&quot;6&quot; style=&quot;background-color:#ffffff; text-align:left;&quot; | Related link: [https://en.wikipedia.org/wiki/Anthropic_principle Anthropic Principle]
|-
| style=&quot;background-color:#e6e6e6;&quot; | '''Structured Loitering and Coil Theory'''
| style=&quot;background-color:#e6e6e6;&quot; | Non-mainstream
| style=&quot;background-color:#e6e6e6;&quot; | Biological structuring reflects energy organization
| style=&quot;background-color:#e6e6e6;&quot; | Behavioral complexity models matter formation
| style=&quot;background-color:#e6e6e6;&quot; | Loitering and coiling patterns in energy and communication
| style=&quot;background-color:#e6e6e6;&quot; | Birdsong, writing systems, neurological signals
|-
| colspan=&quot;6&quot; style=&quot;background-color:#e6e6e6; text-align:left;&quot; | Related link: [https://www.academia.edu/119030837/Structured_Loitering_in_Communication_May_Inform_Genesis_of_Structure_and_Gravity_from_Organized_Energy Ken Shiovitz – Structured Loitering and Coil Theory (Academia.edu)]

|}
TGD proposes a deep connection between biological evolution and space-time structure, introducing a hierarchy of quantum phases labeled by effective Planck constants. These phases are embedded in a higher-dimensional geometry (''M''⁴ × ''CP''₂) and are suggested to influence neural processes and evolutionary transitions. See section 8.8.18.


==== '''6.5 – Evaluation Criteria (Unified Framework)''' ====
''This chapter now uses the unified evaluation framework applied consistently in Chapters 1 and 8.''

The theories discussed in this chapter are evaluated using the following eight standardized criteria:

# '''1. Empirical Adequacy'''  
Alignment with observed data and experimental results across relevant fields (e.g., astronomy, geology, biology, cosmology). The theory should account for major datasets and empirical patterns.

# '''2. Internal Consistency'''  
Logical and mathematical coherence of the theory. Assumptions should not contradict derived results. Definitions, equations, and claims must be used unambiguously.

# '''3. Predictive Power'''  
The ability to generate '''testable predictions''' that distinguish the theory from others. This includes retrospective explanations that are quantitatively verifiable.

# '''4. Cross-Disciplinary Compatibility'''  
Compatibility with findings and models in other domains, such as geology, biology, planetary science, and quantum physics. A good theory avoids contradicting established results in these areas.

# '''5. Conceptual Clarity and Simplicity'''  
The theory should be clearly framed, avoiding unnecessary complexity or vague terminology. It should be possible to identify what is a definition, an assumption, and a consequence.

# '''6. Heuristic and Unifying Value'''  
Capacity of the theory to stimulate new research, unify previously disconnected phenomena, or offer new conceptual insights across domains.

# '''7. Historical and Philosophical Insight'''  
Engagement with the evolution of cosmological thought. A good theory either improves upon, clarifies, or meaningfully integrates earlier models and philosophical foundations.

# '''8. Mathematical Rigor'''  
Use of precise mathematical definitions and derivations, including:  
* Clarity of assumptions and variable usage.  
* Proofs of existence/uniqueness where applicable.  
* Estimates of modeling error or solution stability.  
* Explicit formulations of approximations.  
* Avoidance of heuristic shortcuts unless well justified.  

This criterion ensures the theory stands up to the standards of formal modeling and does not rely on vague simulations or interchangeable terms.

=== '''6.7 Helicopter View – Scientific Reception and Emerging Tensions''' ===

Theories that link biological and paleontological data to cosmological or planetary dynamics operate at the intersection of multiple scientific domains. This multidisciplinary position creates both opportunities and tensions within the broader scientific landscape.

Mainstream theories such as '''Darwinian Evolution''', '''Punctuated Equilibrium''', and '''Milankovitch Cycles''' are firmly embedded in the academic consensus. They benefit from robust empirical datasets, widespread institutional support, and established explanatory frameworks. However, they often remain siloed within evolutionary biology or climate science and rarely engage with cosmological perspectives. This creates a blind spot in linking planetary evolution to long-term cosmic cycles.

Alternative models such as the '''Galactic Biodiversity Cycle''', '''Spiral Cosmology''', and '''Cosmic Influx Theory (CIT)''' attempt to bridge that gap by embedding biological evolution within larger cosmic structures or energy dynamics. These frameworks challenge conventional timelines and causality, often facing skepticism due to their speculative or integrative nature. While not rejected outright, they are typically excluded from mainstream research funding and journals unless framed cautiously.

Theories like '''Earth Expansion''' and '''Plasma Cosmology''' face more pronounced resistance. Despite offering explanatory models for fossil distribution and extinction patterns, they challenge core geophysical assumptions and are often dismissed as fringe. Nonetheless, recent geological data and renewed interest in Earth’s deep-time history have created space for limited re-evaluation of such ideas.

Conceptual frameworks like '''Directed Panspermia''' and '''Anthropic Cosmology''' are received with ambivalence. They do not easily integrate into empirical research programs but continue to provoke philosophical and theoretical interest—especially regarding life’s origins and the conditions necessary for biological complexity.

A more recent contribution, '''Structured Loitering and Coil Theory''', introduces a novel analogy between '''biological behavior''' (such as birdsong structuring and writing systems) and '''energy organization processes''' that may underlie gravity and matter formation. While lacking formal mathematical foundations, this theory exemplifies an increasingly common approach: using '''patterns in life systems''' to inspire new ways of thinking about fundamental physics. Its reception highlights the '''epistemological tension''' between metaphor-driven insight and formalized scientific modeling. The theory occupies an interesting niche at the '''intersection of behavioral biology, cognitive structure, and cosmology''', and may stimulate future dialogue across traditionally separated disciplines.

In summary, there is an increasing recognition that '''life on Earth cannot be fully understood in isolation from the planet’s geodynamic context and possibly even from its cosmic environment.''' However, the institutional and disciplinary boundaries of science still limit serious engagement with non-mainstream models. Tensions persist between the desire for interdisciplinary synthesis and the need to protect methodological rigor. The challenge is to maintain open inquiry while demanding conceptual clarity and empirical grounding.

== 6.8 Next Steps ==

This chapter has outlined a wide range of theories that attempt to connect biological and paleontological clues to planetary and cosmological processes. The mapping and evaluation provide a foundation for interdisciplinary dialogue, bridging evolutionary biology, geology, and cosmology.

Future contributions may include additional theories, updated datasets (e.g., biodiversity cycles, extinction timelines), or new interdisciplinary models. Researchers are encouraged to refine existing frameworks or introduce novel mechanisms that link life’s history on Earth to cosmic dynamics.

We invite further participation through the Talk page or by email submission. All contributions will be evaluated using the shared criteria defined in Chapter 1.3 to ensure clarity and comparability across chapters.


'''◀ [[AI-Assisted Evaluation of Cosmological Theories/Chapter 5: Geological Clues and Planetary Growth|Previous]] | [[AI-Assisted Evaluation of Cosmological Theories|Main Page]] | [[AI-Assisted Evaluation of Cosmological Theories/Chapter 7: Cosmological Parameters and Universal Constants|Next ▶]]'''</text>
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      <text bytes="29500" sha1="4ycfv1kklcm0yfhso5q75wpt6qof39b" xml:space="preserve">= '''Chapter 7: Cosmological Parameters and Universal Constants''' =

=== '''7.1 – Purpose – Overview Cosmological Parameters and Universal Constants''' ===

This subsection introduces the key cosmological parameters and universal constants that underpin modern models of the universe. These values are foundational for describing the large-scale structure, expansion history, and composition of the cosmos. By defining and constraining these constants, researchers are able to test, compare, and refine competing cosmological theories.

The most widely used parameters include:

'''Hubble Constant (H₀)''' – The current rate of cosmic expansion, typically expressed in km/s/Mpc. It plays a central role in determining the age and scale of the universe.

'''Matter Density Parameter (Ωₘ)''' – The fraction of the critical density contributed by all forms of matter (baryonic and dark matter).

'''Dark Energy Density Parameter (Ω_Λ)''' – The fraction of the critical density attributed to dark energy, often associated with the cosmological constant (Λ).

'''Spectral Index (nₛ)''' – Characterizes the scale dependence of primordial density fluctuations.

'''Amplitude of Scalar Fluctuations (Aₛ)''' – Sets the overall level of fluctuations in the early universe, which seeded the formation of galaxies and clusters.

'''Optical Depth (τ)''' – Relates to the reionization history of the universe, affecting how the cosmic microwave background (CMB) photons have been scattered.

'''σ₈''' – A measure of the clustering strength of matter on scales of 8 h⁻¹ Mpc, used in large-scale structure studies.

These parameters are complemented by a set of universal physical constants that appear across all physical theories:

'''Gravitational Constant (G)'''

'''Speed of Light (c)'''

'''Planck’s Constant (h)'''

'''Boltzmann Constant (k_B)'''

'''Cosmological Constant (Λ)''' – interpreted as vacuum energy or dark energy density

'''Fine-Structure Constant (α)'''

Together, these parameters serve as the empirical backbone for cosmological models, allowing them to be calibrated against observational data. Most mainstream models—such as the ΛCDM model—use a minimal set of six to seven parameters, yet tensions persist between different measurement techniques. For example, the value of H₀ inferred from the cosmic microwave background (CMB) differs significantly from values measured using Type Ia supernovae, leading to the well-known '''Hubble tension'''. Similarly, discrepancies in measurements of σ₈ suggest the possibility of unknown physical processes or the need for alternative frameworks.

This chapter evaluates how different cosmological theories define, derive, or reinterpret these parameters. It also considers whether a theory proposes a new universal constant, modifies existing ones, or offers an explanation for observed tensions. The role of these constants in the internal logic and predictive success of a theory is an essential component of the comparative evaluation.

=== '''7.2 – Scope – Mainstream and Non-mainstream Theories''' ===

This subsection outlines the scope of Chapter 7 in terms of the cosmological theories being considered, with specific attention to how each theory engages with cosmological parameters and universal constants. Both mainstream and non-mainstream theories are included in the evaluation to ensure a broad, inclusive, and critically informed comparison.

==== '''Mainstream Cosmological Frameworks''' ====

Mainstream cosmological models are those widely accepted within the current scientific consensus and supported by large-scale observational programs such as the Planck satellite, the Sloan Digital Sky Survey (SDSS), and the Dark Energy Survey (DES). The primary framework is the:

'''ΛCDM Model (Lambda Cold Dark Matter)'''
This model is based on general relativity and assumes a universe dominated by cold dark matter (CDM) and a cosmological constant (Λ) representing dark energy. It uses a standard set of six parameters to fit observational data:
– H₀ (Hubble constant)
– Ωₘ (matter density)
– Ω_Λ (dark energy density)
– nₛ (spectral index)
– Aₛ (amplitude of fluctuations)
– τ (optical depth)

ΛCDM has achieved notable empirical success, including accurate fits to the cosmic microwave background (CMB) power spectrum and the large-scale distribution of galaxies. However, it faces unresolved tensions such as the disparity in H₀ values derived from early- and late-universe measurements, and differences in predicted versus observed values of σ₈ (matter clustering amplitude).

Other general relativity–based extensions or modifications, such as quintessence models (dynamic dark energy), modified gravity (e.g., f(R) theories), or models invoking additional neutrino species, are sometimes treated as extensions of ΛCDM rather than fundamentally separate theories.

==== '''Non-Mainstream and Emerging Theories''' ====

This chapter also includes theories that are not currently part of mainstream consensus but that offer new approaches to cosmological constants or parameter derivation. These may include:

'''Alternative Gravity Theories''' – including MOND (Modified Newtonian Dynamics), TeVeS, conformal gravity, and entropic gravity. These models often aim to explain galactic rotation curves or cosmological structure without invoking dark matter.

'''Emergent and Flow-based Models''' – including theories where cosmic structure arises from energy influxes or universal flows (e.g., Cosmic Influx Theory, Spiral Cosmology). These models may propose new constants (such as influx-based scaling parameters) or reinterpret standard ones.

'''Continuous Creation or Steady-State Frameworks''' – These posit a continuous generation of matter or energy, potentially affecting the interpretation of Hubble expansion, cosmic microwave background, and the age of the universe.

'''Topological or Geometric Reinterpretations''' – Some proposals modify the dimensionality or topology of space-time itself, redefining how constants such as G or c emerge from underlying structures.

==== '''Comparative Perspective''' ====

By evaluating both mainstream and non-mainstream approaches on equal terms, this chapter aims to provide a comparative framework grounded in empirical adequacy, predictive capacity, and internal coherence. The inclusion of non-standard models is not intended to imply equivalence of scientific status, but rather to highlight the diversity of ideas that reinterpret or challenge conventional uses of cosmological parameters.

The theories evaluated here must define how they handle:

The derivation or reinterpretation of cosmological parameters

The role and meaning of universal constants in their framework

Their position relative to known tensions (e.g., Hubble tension, σ₈ tension)

This scope ensures that all models are critically evaluated using the same criteria, while allowing room for genuinely novel approaches to cosmology.

== '''7.3 How to Contribute a Theory ''' ==

Researchers and contributors are welcome to propose additional theories that relate biological evolution and fossil records to planetary or cosmological models. Submissions may include observational insights, mathematical models, conceptual frameworks, or hybrid approaches.

You can contribute in either of the following ways:

* Post directly on the Talk page of this chapter  
* Or email your proposal to: aitheroymapping@gmail.com  

All submitted theories will be added to the mapping and evaluated using the shared criteria defined in Chapter 1.3: Evaluation Criteria. We encourage both mainstream and non-mainstream contributors to join this open effort to compare and improve our understanding of Biological and Paleontological Clues in relation to cosmological theories.

=== '''7.4 – Theory Mapping Table''' ===

This table presents a structured comparison of diverse cosmological theories based on how they define or reinterpret key cosmological parameters and universal constants. The table includes both mainstream and non-mainstream paradigms, encouraging open comparative analysis grounded in scientific rigor.

{| class=&quot;wikitable&quot; style=&quot;width:100%; text-align:left;&quot;
! style=&quot;width:15%;&quot; | '''Theory Name'''
! style=&quot;width:15%;&quot; | '''H₀ Treatment'''
! style=&quot;width:15%;&quot; | '''Ω Parameters'''
! style=&quot;width:15%;&quot; | '''Role of Λ / Dark Energy'''
! style=&quot;width:20%;&quot; | '''Treatment of G, c, and other constants'''
! style=&quot;width:20%;&quot; | '''New Parameters or Constants Proposed'''

|-
| '''ΛCDM (Standard Model)''' || Empirically fitted (Planck: 67.8 ± 0.9) || Ωₘ, Ω_Λ, Ω_r defined from observation || Constant Λ; vacuum energy interpretation || G and c are fixed universal constants || None
|-style=&quot;background:#f9f9f9;&quot; 
| '''Modified Gravity (f(R))''' || H₀ typically fitted or adjusted || Ωₘ redefined dynamically || Λ may be replaced by geometric terms || G may vary with scale or curvature || Function f(R), effective gravitational coupling
|-
| '''MOND / TeVeS''' || H₀ usually inherited from observations || Ωₘ reinterpreted (no dark matter) || Λ often excluded; not essential || G modified below a₀; new scale-invariant dynamics || a₀ (critical acceleration)
|-style=&quot;background:#f9f9f9;&quot; 
| '''Emergent Gravity (Verlinde)''' || Predicts emergent H₀ value || Ω parameters secondary to entropy/gravity || Dark energy as emergent entropic phenomenon || G and Λ derived from holographic principles || Entropic scaling parameter
|-
| '''Cosmic Influx Theory (CIT)''' || Derives H₀ = 67.8 via VRMS and (γ−1) || Ω values not directly defined || Λ absorbed in influx framework || G = (γ−1)/4π; κ and κ_CIT used as scaling factors || κ_CIT, κ, VRMS
|-style=&quot;background:#f9f9f9;&quot; 
| '''Spiral Cosmology''' || H₀ seen as emergent from cosmic motion || Ω reinterpreted in spiraling energy flows || Λ not needed; structure from motion || c and G may emerge from wavefront geometry || Spiral frequency and structure constants
|-
| '''Topological Field Framework (Hall)''' || May reinterpret H₀ as boundary effect || Ωₘ redefined from field topology || Λ as a topological effect or unnecessary || G emerges from higher-dimensional field curvature || Topological curvature parameters
|-style=&quot;background:#f9f9f9;&quot; 
| '''Conformal Gravity (Mannheim–Kazanas)''' || Derives H₀ from conformally invariant field equations || Ω values fitted without dark matter || Λ not fundamental; dynamic curvature replaces Λ || G emerges from conformal symmetry conditions || Conformal coupling constant, γ₀
|-
| '''Quasi-Steady State Cosmology (QSSC)''' || H₀ varies over long cycles || Ω values evolve over cosmic epochs || Λ varies cyclically with creation field || G and c are constant, but new field equations applied || Creation field C, periodic Λ(t)
|-style=&quot;background:#f9f9f9;&quot; 
| '''Dynamic Universe (Tuomo Suntola)''' || H₀ derived from zero-energy balance || Ω reinterpreted geometrically || Λ not needed; curvature and motion suffice || G and c emerge from dynamic 4-sphere geometry || Zero-energy condition; time-proportional scaling
|}
TGD proposes that fundamental constants, including Newton’s gravitational constant, emerge from geometric parameters such as the radius of ''CP''₂ in the 8-dimensional embedding space (''M''⁴ × ''CP''₂). Coupling constants are interpreted as discrete critical values linked to number-theoretical invariants. See section 8.8.18.

==== '''7.5 Evaluation Criteria (Unified Framework)''' ====
''This chapter now uses the unified evaluation framework applied consistently in Chapters 1 and 8.''

The theories discussed in this chapter are evaluated using the following eight standardized criteria:

# '''1. Empirical Adequacy'''  
Alignment with observed data and experimental results across relevant fields (e.g., astronomy, geology, biology, cosmology). The theory should account for major datasets and empirical patterns.

# '''2. Internal Consistency'''  
Logical and mathematical coherence of the theory. Assumptions should not contradict derived results. Definitions, equations, and claims must be used unambiguously.

# '''3. Predictive Power'''  
The ability to generate '''testable predictions''' that distinguish the theory from others. This includes retrospective explanations that are quantitatively verifiable.

# '''4. Cross-Disciplinary Compatibility'''  
Compatibility with findings and models in other domains, such as geology, biology, planetary science, and quantum physics. A good theory avoids contradicting established results in these areas.

# '''5. Conceptual Clarity and Simplicity'''  
The theory should be clearly framed, avoiding unnecessary complexity or vague terminology. It should be possible to identify what is a definition, an assumption, and a consequence.

# '''6. Heuristic and Unifying Value'''  
Capacity of the theory to stimulate new research, unify previously disconnected phenomena, or offer new conceptual insights across domains.

# '''7. Historical and Philosophical Insight'''  
Engagement with the evolution of cosmological thought. A good theory either improves upon, clarifies, or meaningfully integrates earlier models and philosophical foundations.

# '''8. Mathematical Rigor'''  
Use of precise mathematical definitions and derivations, including:  
* Clarity of assumptions and variable usage.  
* Proofs of existence/uniqueness where applicable.  
* Estimates of modeling error or solution stability.  
* Explicit formulations of approximations.  
* Avoidance of heuristic shortcuts unless well justified.  

This criterion ensures the theory stands up to the standards of formal modeling and does not rely on vague simulations or interchangeable terms.

=== '''7.6 – AI Evaluation Based on Shared Criteria''' ===

This subsection presents an AI-assisted comparative evaluation of cosmological theories, based on the shared criteria defined in [[AI-Assisted_Evaluation_of_Cosmological_Theories/Chapter_1:_Introduction_and_Evaluation_Criteria#1.3:_List_of_Evaluation_Criteria|Chapter 1.3: List of Evaluation Criteria]]. The goal is to clarify how each theory treats the fundamental cosmological parameters and universal constants discussed in this chapter. The evaluation highlights '''internal coherence''', '''empirical alignment''', '''explanatory ambition''', and '''potential testability'''.

No star ratings are assigned here. Instead, each theory is qualitatively assessed across the criteria through short descriptive paragraphs.

'''ΛCDM (Standard Model)''' offers '''strong empirical alignment''' with CMB, supernova, and large-scale structure data. Its '''internal structure''' is based on general relativity and well-developed mathematical models, making it '''logically robust'''. However, it provides '''limited explanation''' for the specific values of constants such as H₀ or Λ, which are treated as '''empirical inputs'''. Despite this, it demonstrates '''predictive success''' across multiple observations. The model is '''conceptually economical''', relying on a '''minimal parameter set''', and is '''deeply integrated''' into the broader framework of modern physics. Known tensions (e.g., H₀ discrepancy) invite refinement or reinterpretation but do not undermine its overall compatibility with data. It remains '''broadly testable''' through increasingly precise cosmological measurements.

'''Modified Gravity (f(R))''' theories attempt to explain dark energy and acceleration through '''extensions of the Einstein-Hilbert action'''. They offer a '''reinterpretation of cosmic parameters''' in terms of curvature dynamics rather than exotic energy components. These models vary widely in structure and '''observational fit''', depending on the chosen function f(R), but many reproduce key cosmological features. Their '''mathematical framework''' is internally consistent, and they contribute meaningfully to the debate on the '''nature of gravity'''. While their explanatory power is promising, some formulations require '''additional parameters or fine-tuning'''. Compatibility with '''structure formation data''' and '''gravitational lensing tests''' remains an important area for empirical evaluation.

'''MOND / TeVeS''' seeks to account for galactic rotation curves without invoking dark matter, introducing a '''critical acceleration scale''' as a new constant. It performs '''very well at galactic scales''', offering a direct fit to rotation data. However, it faces '''challenges''' when applied to cosmological phenomena such as the '''cosmic microwave background''' and '''large-scale structure'''. The theory’s internal dynamics are '''logically constructed''' and '''mathematically consistent''' within their range of application. Its '''predictive capacity''' is constrained to specific regimes, and it introduces '''conceptual simplicity''' with '''minimal new constants'''. '''Empirical tensions''' with high-redshift data suggest the need for further development or '''hybrid models'''.

'''Emergent Gravity (Verlinde)''' offers an '''innovative theoretical framework''' in which gravity and inertia '''emerge from entropic and information-theoretic principles'''. This model reframes fundamental constants as '''emergent properties''' of underlying entropy distributions. While its internal structure is '''elegant''' and '''conceptually coherent''', '''empirical validation''' remains limited. It has provided '''tentative fits''' to certain lensing effects, but '''full cosmological modeling''' is still in development. The theory’s '''explanatory reach''' is broad, particularly in redefining Λ and inertia, though specific '''quantitative predictions''' are still rare. It is '''testable in principle''', and future '''observational probes''' could refine or challenge its foundational assumptions.

'''Cosmic Influx Theory (CIT)''' proposes that the '''gravitational constant''' and '''Hubble parameter''' derive from a '''continuous influx of energy into matter'''. It uses the '''relativistic factor (γ−1)''' and a '''universal root mean square velocity (VRMS)''' to derive G and H₀, aligning with '''Planck observations''' without parameter fitting. The theory is '''internally consistent''' and offers '''strong explanatory coherence''' for the emergence of constants. Although it does not define all standard cosmological parameters, its introduction of '''scaling constants''' (κ and κ_CIT) provides a '''novel interpretation''' of structure and '''mass-energy evolution'''. The theory is '''open to falsification''' through predictions on '''orbital dynamics''' and '''preferred planetary distances''', and invites new observational research.

'''Spiral Cosmology''' describes the evolution of the universe through '''organized spiraling flows''' of energy and matter. It offers a '''dynamic interpretation''' of cosmic structure and attempts to '''derive constants''' from the geometry of spiral motion. Its explanatory framework is '''creative and unified''', though primarily '''qualitative''' at this stage. '''Observational alignment''' is still under development, and '''quantitative predictions''' remain limited. The model maintains '''internal coherence''' and offers '''potential compatibility''' with both '''geometric''' and '''wave-based models''' of the universe. It presents a '''unique lens''' on cosmic evolution and may be tested through '''pattern recognition''' in galactic distributions and background structure.

'''Topological Field Framework (Hall)''' attempts to explain the '''emergence of physical constants''' and structure from the '''topology of underlying fields'''. It replaces '''fixed constants''' with '''dynamic relationships''' determined by higher-dimensional field configurations. This theory is '''conceptually ambitious''' and '''mathematically coherent''' within its proposed framework. While it currently lacks '''detailed empirical predictions''', it offers a '''strong explanatory model''' for the origin of constants such as G and c. The framework’s future success depends on its ability to '''generate testable consequences''' and interface with '''observational cosmology'''.

'''Conformal Gravity (Mannheim–Kazanas)''' modifies general relativity using '''conformal symmetry''' and has had success fitting '''galactic rotation curves''' without dark matter. It interprets '''cosmic acceleration''' and '''gravitational potentials''' without requiring Λ or exotic matter components. The theory’s structure is '''mathematically rigorous''' and '''geometrically elegant'''. While it fits some astrophysical data well, its '''compatibility with the CMB''' and '''early universe''' remains under scrutiny. It provides a '''deeper explanation''' for gravitational behavior, but requires additional work to extend to full cosmological predictions. It is '''empirically testable''' and offers a '''valuable challenge''' to conventional dark matter models.

'''Quasi-Steady State Cosmology (QSSC)''' revives and modifies the steady-state model by introducing a '''continuous creation field (C-field)''' and '''cyclic cosmological evolution'''. It offers an '''alternative interpretation''' of redshift and cosmic expansion that challenges the Big Bang model. The theory '''aligns with some observational patterns''' but struggles to reproduce others, such as the '''CMB spectrum''' and '''light element abundances'''. Its internal logic is '''self-consistent''' within its paradigm and introduces an '''original approach''' to '''matter generation''' and '''cosmic cycles'''. QSSC provides '''testable ideas''', particularly related to '''periodicity''' and '''structural evolution''', but remains '''outside the mainstream''' due to its divergence from accepted data interpretations.

'''Dynamic Universe (Tuomo Suntola)''' introduces a '''zero-energy cosmological model''' in which space, time, and matter evolve from a '''dynamically balanced 4-sphere geometry'''. This framework offers a '''coherent mathematical structure''' with an explicit '''derivation of H₀''' and other constants from '''curvature and motion''', rather than treating them as empirical inputs. It avoids the need for '''inflation''', '''dark energy''', or '''dark matter''' by reinterpreting cosmic expansion. The model’s '''internal consistency''' is strong, and its '''elegance''' lies in its '''unified geometric logic'''. '''Testable consequences''' emerge through its '''time-scaling formulations''' and '''reconstruction of historical cosmic dynamics''', providing a fertile ground for further research.

=== '''7.7 – Helicopter View – Scientific Reception and Emerging Tensions''' ===

This subsection offers a broader reflection on how the theories presented in Chapter 7 are received within the scientific community, and how ongoing observational tensions and theoretical gaps have shaped the landscape of cosmological research. While the '''ΛCDM model''' remains the dominant paradigm, persistent anomalies and the rise of innovative frameworks have stimulated growing interest in alternatives that offer novel interpretations of cosmological parameters and universal constants.

The '''ΛCDM model''' has achieved widespread acceptance due to its empirical success in matching high-precision data from the Planck satellite, baryon acoustic oscillations, and Type Ia supernovae. Its internal consistency and predictive utility are widely recognized. However, the model also leaves several foundational questions unresolved. Chief among these is the origin and nature of the '''cosmological constant (Λ)''', which contributes over 68% of the energy density of the universe and yet remains theoretically unexplained. Furthermore, the persistent disparity between early-universe (CMB-derived) and late-universe (distance ladder) measurements of the '''Hubble constant (H₀)''' has become known as the '''&quot;Hubble tension&quot;'''. Similar discrepancies related to the amplitude of matter clustering ('''σ₈''') have raised additional concerns about potential missing physics.

These tensions have opened a window for the exploration of '''non-mainstream cosmologies'''. '''Modified gravity models''' such as '''f(R)''' and '''conformal gravity''' offer geometric alternatives that attempt to reproduce cosmic acceleration and structure formation without invoking dark energy or dark matter. While these models often require additional functions or parameters, they are gaining attention for their potential to resolve observational anomalies through new mathematical structures.

Theories such as '''MOND''', '''TeVeS''', and '''emergent gravity''' have had mixed receptions. '''MOND''' has provided compelling fits to galactic rotation curves but lacks a fully developed cosmological extension. '''Emergent gravity''' introduces a radically different ontology—treating gravity and inertia as statistical phenomena—which appeals to theorists seeking deeper unifying principles. Nonetheless, these theories face strong skepticism from mainstream cosmologists due to limited empirical validation and difficulties in reproducing CMB anisotropies and structure growth data.

The inclusion of conceptual models such as '''Spiral Cosmology''', '''Cosmic Influx Theory (CIT)''', and the '''Dynamic Universe''' represents a growing movement within the scientific periphery to explore frameworks that derive cosmological constants from deeper principles—be they geometric, energetic, or topological. These models often operate outside the dominant institutional research programs, yet they reflect a longstanding scientific aspiration: to understand not only how the universe behaves, but why the constants that govern its behavior take the values they do.

Institutional acceptance of these models varies widely. Peer-reviewed publications remain limited for some, while others circulate primarily via academic repositories, conferences, or platforms such as '''Zenodo''', '''ResearchGate''', or '''Wikiversity'''. This reflects a broader tension in the scientific ecosystem: while mainstream paradigms are highly productive and empirically successful, they may not fully accommodate unconventional but potentially insightful hypotheses.

In this context, the '''AI-assisted comparative framework''' applied in this chapter aims to foster transparent, criteria-based evaluation of both accepted and emerging theories. By focusing on '''empirical adequacy''', '''internal logic''', and '''explanatory potential'''—rather than academic consensus alone—this approach supports a more inclusive and methodologically consistent cosmological discourse. 

=== '''7.8 – Next Steps''' ===

This final subsection outlines recommended next steps for researchers, contributors, and interested readers following the comparative evaluation of cosmological parameters and universal constants in Chapter 7.

First, contributors are encouraged to refine or expand the theory entries presented here by improving clarity, adding new references, or providing updated observational data. As ongoing missions such as the '''James Webb Space Telescope (JWST)''', '''Euclid''', and the '''Vera Rubin Observatory''' generate increasingly precise cosmological measurements, all models—mainstream and alternative—will face new opportunities for testing and validation. Particular attention should be paid to how each theory addresses current tensions, including the '''Hubble tension''', the '''σ₈ clustering discrepancy''', and the unexplained nature of '''Λ (dark energy)'''.

Second, AI-assisted comparative tools may be developed further to support large-scale cross-analysis of parameter predictions and internal consistencies. This could include algorithmic comparison of theory-derived parameter sets against datasets from Planck, SDSS, DESI, or gravitational wave observatories. Readers or institutions with expertise in computational cosmology are invited to explore these possibilities and report outcomes.

Third, alternative frameworks that derive constants from first principles—whether geometric, entropic, influx-based, or topological—may benefit from collaborative dialogue. Theories such as '''CIT''', '''Spiral Cosmology''', and the '''Dynamic Universe''' propose unorthodox but structured ways to reinterpret physical constants. These models can be further explored, modified, or combined in search of converging predictions.

Finally, this chapter should remain a living document. As the boundaries of cosmology continue to evolve, Wikiversity provides an open-access platform for iterative updates, interdisciplinary integration, and transparent evaluation. Contributors from all scientific backgrounds are welcome to propose revisions, submit new models, or refine the criteria used here. The comparative structure established in Chapter 7 will also serve as a foundation for deeper synthesis in [[AI-Assisted_Evaluation_of_Cosmological_Theories/Chapter_8:_Critical_Evaluation_and_Future_Research_Paths|Chapter 8]], where star ratings, meta-evaluations, and future research priorities will be formally proposed.

'''◀ [[AI-Assisted Evaluation of Cosmological Theories/Chapter 6: Biological and Paleontological Clues|Previous]] | [[AI-Assisted Evaluation of Cosmological Theories|Main Page]] | [[AI-Assisted Evaluation of Cosmological Theories/Chapter 8: Critical Evaluation and Future Research Paths|Next ▶]]'''</text>
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      <text bytes="45040" sha1="skevsovnnde8ih9nxxg103d2wbly7wp" xml:space="preserve">== '''Chapter 8: Methods, Tools, and AI-Assisted Evaluation''' =

== '''8.1 Purpose''' ==

This chapter describes the methodological foundations of the project, including the role of AI in evaluating theories, how contributors interact with the system, and how assessments are recorded, displayed, and updated.

== '''8.2 Toolchain and Workflow''' ==

AI tools like ChatGPT are used to analyze, compare, and refine theories

Tables and frameworks are generated collaboratively using open formats

Contributors submit input via email or editing suggestions

Ratings and evaluations are dynamically updated based on ongoing analysis

== '''8.3 AI-Based Rating System: Motivation and Procedure''' ==

To support comparative evaluation without personal or institutional bias, this project uses an AI-based rating system. ChatGPT acts as a neutral evaluator, analyzing each theory across clearly defined criteria in '''8.5 Understanding the Star Ratings'''.

This system is designed to be:

Transparent – Each rating is justified through AI’s large-scale reference analysis.

Dynamic – Contributors may submit additional materials to request re-evaluation.

Consistent – All evaluations are performed by the same AI logic, eliminating personal bias.

Contributors may ask ChatGPT to re-read specific articles, datasets, or theoretical arguments. If new insights are found, ratings will be updated and transparently noted.

This approach represents a shift toward evidence-driven, large-scale comparative review, using AI not as a gatekeeper but as a tool to synthesize and validate.

== '''8.4 Open Participation''' ==

This chapter is also where future documentation of the workflow and collaborative mechanisms will be expanded. Users who submit theories are encouraged to:

Describe their framework in terms of assumptions, predictions, and compatibility

Suggest how their model could be tested or falsified

Provide references or original materials for AI evaluation

All analysis is open, and contributors may propose improvements at any time.

== '''8.5 Understanding the Star Ratings''' ==

The AI Evaluation Table below rates theories across seven scientific criteria defined in '''Chapter 1.3'''. Here we provide a full explanation of what each criterion means and how it is applied.

'''8.5.1. Empirical Adequacy'''  
Does the theory fit known observations and experimental data? High scores require support from astronomy, cosmology, geology, or lab-based physics. Theories that contradict established measurements or lack empirical grounding score lower.

'''8.5.2. Internal Consistency'''  
Are the theory’s assumptions, mathematics, and logic self-coherent? A consistent theory does not contain contradictions, undefined steps, or ad hoc assumptions.

'''8.5.3. Predictive Power'''  
Does the theory make clear, testable predictions that distinguish it from others? Theories that anticipate new phenomena or retrodict known data gain higher ratings.

'''8.5.4. Cross-Disciplinary Compatibility'''  
Is the theory consistent with findings from other scientific fields, such as geology, chemistry, biology, or planetary science? The more compatible it is, the higher the score.

'''8.5.5. Conceptual Clarity and Simplicity'''  
Is the theory logically simple and intuitively understandable, without unnecessary complexity? This criterion rewards elegance, not oversimplification.

'''8.5.6. Heuristic Value'''  
Does the theory stimulate new questions, research directions, or rethinking of existing problems? A high score reflects creative scientific potential.

'''8.5.7. Historical and Philosophical Insight'''  
Does the theory connect meaningfully to the historical development of science or reflect philosophical depth? Theories grounded in tradition or conceptual evolution are valued here.

'''8.5.8. Mathematical Rigor'''  
Does the theory provide clear mathematical formulations, derivations, and quantitative predictions? Theories are valued for their use of equations to express core principles, ensure internal consistency, and generate testable results.


Each theory receives a rating from ★☆☆☆☆ to ★★★★★ per criterion. The total score (max 40) gives a general measure of its scientific coherence and reach. The reasoning behind the scores is available under each theory (subsection 8.8) or can be requested in more detail.


=== '''8.6 – Comparative Table of AI Ratings (Updated with Criterion 8: Mathematical Rigor)''' ===

''Note: In July 2025, an eighth evaluation criterion was added: '''Mathematical Rigor'''. The total score is now out of 40 stars instead of 35.  ''
''All ratings are expressed in whole stars (★), without fractional values, to ensure clarity in display and consistency with the visual format of this table.''  
''Theories are listed in the same order as in Section 8.8.''

{| class=&quot;wikitable&quot;
! '''Theory''' !! '''EA''' !! '''IC''' !! '''PP''' !! '''CC''' !! '''CS''' !! '''HV''' !! '''HP''' !! '''MR''' !! '''Total (★/40)'''
|-
| '''8.8.1 General Relativity''' || ★★★★★ || ★★★★★ || ★★★★★ || ★★★★☆ || ★★★☆☆ || ★★★★★ || ★★★★★ || ★★★★★ || '''37'''
|-
| '''8.8.2 Newtonian Gravity''' || ★★★★☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || ★★★★★ || ★★★★☆ || ★★★★★ || ★★★★☆ || '''33'''
|-
| '''8.8.3 MOND''' || ★★★★☆ || ★★★☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || ★★★☆☆ || '''26'''
|-
| '''8.8.4 Emergent Gravity''' || ★★★☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || '''24'''
|-
| '''8.8.5 Big Bang''' || ★★★★☆ || ★★★★★ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || ★★★★★ || ★★★★☆ || ★★★★★ || '''34'''
|-
| '''8.8.6 Steady State Theory''' || ★★☆☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★★★☆ || ★★★☆☆ || ★★★★☆ || ★★★☆☆ || '''23'''
|-
| '''8.8.7 Big Crunch''' || ★★☆☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★★☆☆ || ★★★★☆ || ★★★☆☆ || '''22'''
|-
| '''8.8.8 Big Bounce''' || ★★☆☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || '''22'''
|-
| '''8.8.9 Cosmic Influx Theory (CIT)''' || ★★★★☆ || ★★★★☆ || ★★★★☆ || ★★★★☆ || ★★★★☆ || ★★★★★ || ★★★★☆ || ★★★★☆ || '''33'''
|-
| '''8.8.10 Spiral Cosmology''' || ★★☆☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★★★☆ || ★★★★☆ || ★★☆☆☆ || '''21'''
|-
| '''8.8.11 Topological Field Framework''' || ★★☆☆☆ || ★★★★☆ || ★★★☆☆ || ★★★★☆ || ★★★☆☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || '''27'''
|-
| '''8.8.12 Conformal Gravity''' || ★★★☆☆ || ★★★★☆ || ★★★☆☆ || ★★★☆☆ || ★★★☆☆ || ★★★★☆ || ★★★★☆ || ★★★★☆ || '''28'''
|-
| '''8.8.13 Quasi-Steady State Cosmology''' || ★★☆☆☆ || ★★★☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★★★☆ || ★★★★☆ || ★★☆☆☆ || '''23'''
|-
| '''8.8.14 Dynamic Universe''' || ★★★☆☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || ★★★★☆ || ★★★★☆ || ★★★★☆ || ★★★★☆ || '''30'''
|-
| '''8.8.15 Multiverse''' || ★★☆☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || '''23'''
|-
| '''8.8.16 Theory of Universality''' || ★☆☆☆☆ || ★★☆☆☆ || ★☆☆☆☆ || ★☆☆☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★★☆☆ || ★★☆☆☆ || '''16'''
|-
| '''8.8.17 Structured Loitering''' || ★★☆☆☆ || ★★☆☆☆ || ★☆☆☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★★☆☆ || ★☆☆☆☆ || '''16'''


&lt;!-- put an empty line here --&gt;

|}

''Legend:''  
''EA = Empirical Adequacy IC = Internal Consistency PP = Predictive Power CC = Cross-Disciplinary Compatibility  ''
''CS = Conceptual Simplicity HV = Heuristic Value HP = Historical/Philosophical Insight MR = Mathematical Rigor''

== '''8.7 Requesting Detailed Justification''' ==

“Somebody” interested in the full motivation for a theory’s score—criterion by criterion—can obtain this by:

1. Visiting the '''discussion page''' of this Wikiversity chapter.
  
2. Mentioning the theory by name and explicitly asking for the detailed breakdown.
  
3. Receiving an AI-generated or authored comment elaborating why each star rating was assigned. 

These ratings are transparent and intended to support critical evaluation and ongoing refinement of both established and alternative cosmological theories. Constructive feedback, new data, or arguments may lead to adjustments in the scores, provided clear reasoning is presented.

If a contributor or author does not agree with the evaluation or prefers not to have their theory represented under this framework, they may request that all related content and ratings for that theory be removed. This ensures that participation remains voluntary and respectful of intellectual ownership.

'''Mainstream theories are subject to the same critical standard.'''  
If someone presents a reasoned objection to the current star ratings of a widely accepted theory (e.g. General Relativity, Big Bang), their explanation will be reviewed. If the argument is well-founded, ChatGPT may generate a revised evaluation. Both the original and alternative viewpoints can be documented transparently if needed.

If you want to submit a '''Rating Rebuttal''', please use the template provided on the '''[[Talk:AI-Assisted Evaluation of Cosmological Theories|Discussion page]]'''.

== '''8.8 Detailed Evaluations of Theories''' ==

This section will be expanded progressively as new theories are added. Each evaluation is generated in collaboration with ChatGPT, based on a standardized set of seven criteria defined in Chapter 1. This approach allows for clear, consistent, and rapid assessment of both mainstream and non-mainstream cosmological theories within minutes.

=== '''8.8.1 General Relativity – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/General_relativity General Relativity]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Matches a wide range of observations: gravitational lensing, GPS corrections, perihelion precession, black hole dynamics, and gravitational waves. Supported by multiple experiments. || ★★★★★
|-
| '''Internal Consistency''' || Highly consistent within its differential geometric framework. Built upon Einstein's field equations with tensor calculus. Few internal contradictions, though extensions (e.g. quantum gravity) face challenges. || ★★★★★
|-
| '''Predictive Power''' || Successfully predicted gravitational redshift, time dilation, frame-dragging, and gravitational waves. Continues to guide observations in astrophysics. || ★★★★★
|-
| '''Cross-Disciplinary Compatibility''' || Compatible with cosmology and astrophysics. Some tension with quantum theory. Less integrated with planetary geology or biology. || ★★★★☆
|-
| '''Conceptual Clarity and Simplicity''' || Complex mathematical structure makes it less intuitive. Conceptually abstract (spacetime curvature, geodesics). Clarity improves with education, but simplicity is low. || ★★★☆☆
|-
| '''Heuristic Value''' || Inspired vast developments in cosmology, black hole theory, and relativistic astrophysics. Foundation for modern gravitational physics. || ★★★★★
|-
| '''Historical and Philosophical Insight''' || Deep philosophical implications about space, time, and causality. Represents a major shift from Newtonian absolute space. Influenced 20th-century philosophy of science. || ★★★★★
|-
| '''Mathematical Rigor''' || Built upon advanced mathematics: Riemannian geometry, Einstein field equations, tensor calculus. Equations are precise, formal, and deeply embedded in differential geometry. || ★★★★★
|}

'''Total: 39/40'''

=== '''8.8.2 Newtonian Gravity – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation Newton's Law of Universal Gravitation]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Accurately describes gravitational interactions for most macroscopic systems (planets, satellites, projectiles) under low-speed, weak-field conditions. Deviates in extreme conditions (e.g. near black holes). || ★★★★☆
|-
| '''Internal Consistency''' || Mathematically self-consistent with inverse-square law and Newton's laws of motion. Assumes instantaneous action at a distance, which conflicts with relativity. || ★★★★☆
|-
| '''Predictive Power''' || Predicts planetary orbits, tides, escape velocities, and Keplerian motion. Fails for relativistic effects (e.g. Mercury’s precession, gravitational lensing). || ★★★★☆
|-
| '''Cross-Disciplinary Compatibility''' || Well-integrated in classical mechanics, astronomy, and engineering. Less compatible with modern cosmology or relativistic frameworks. || ★★★☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Simple and intuitive: gravity as a force between masses. Easily grasped and widely taught. || ★★★★★
|-
| '''Heuristic Value''' || Inspired centuries of scientific discovery and classical mechanics. Still used in teaching and engineering. Limited in modern theoretical development. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Founded classical physics. Major leap in scientific method and mathematical modeling. Paved the way for Enlightenment-era science. || ★★★★★
|-
| '''Mathematical Rigor''' || Clear and elegant use of calculus and vector algebra (e.g. \(\displaystyle F = G \frac{m_1 m_2}{r^2}\)). Highly accessible and historically groundbreaking, but lacks deeper geometric or relativistic structures. || ★★★★☆
|}

'''Total: 33/40'''

=== '''8.8.3 MOND (Modified Newtonian Dynamics) – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics Modified Newtonian Dynamics (MOND)]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Fits galactic rotation curves without invoking dark matter. Matches Tully-Fisher relation. Performance weakens at cluster and cosmological scales. || ★★★★☆
|-
| '''Internal Consistency''' || Original formulation is non-relativistic and phenomenological. Several relativistic extensions (e.g. TeVeS) exist but introduce complexity and fine-tuning. || ★★★☆☆
|-
| '''Predictive Power''' || Predicts galaxy dynamics from baryonic matter alone. Less predictive at larger scales or in non-galactic contexts. || ★★★☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Primarily astrophysical. Some tension with cosmology, structure formation, and gravitational lensing. Not aligned with particle physics. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Simple at galactic scale: modifies acceleration below a threshold \(a_0\). Extensions are less intuitive. || ★★★★☆
|-
| '''Heuristic Value''' || Challenges dark matter paradigm and motivates alternate gravity models. Sparked theoretical and observational debate. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Offers a conceptual challenge to Newton/Einstein gravity. Philosophically provocative, but limited historical lineage. || ★★★☆☆
|-
| '''Mathematical Rigor''' || Employs interpolating functions and modified Poisson equations. Relativistic extensions (e.g. TeVeS) involve tensor-vector-scalar frameworks. Rigorous in parts, but lacks unified formalism. || ★★★☆☆
|}

'''Total: 28/40'''

=== '''8.8.4 Emergent Gravity – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Entropic_gravity Emergent / Entropic Gravity]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Attempts to account for galactic rotation without dark matter by linking gravity to entropy and information. Some results match MOND-like behavior, but broad observational support remains limited. || ★★★☆☆
|-
| '''Internal Consistency''' || Theoretical framework draws from thermodynamics, holography, and information theory. Conceptually coherent, but not fully developed as a unified physical model. || ★★★☆☆
|-
| '''Predictive Power''' || Offers qualitative insights but lacks precise predictive capabilities in most contexts. No wide adoption for simulations or system modeling. || ★★☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Integrates ideas from black hole thermodynamics, quantum information, and spacetime geometry. Weak integration with observational astronomy or geology. || ★★★☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Abstract and speculative. Concepts like entropic forces and holographic screens are not intuitive for most readers. || ★★☆☆☆
|-
| '''Heuristic Value''' || Stimulates discussion about the nature of spacetime and gravity. Has inspired new theoretical directions in quantum gravity and information theory. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Philosophically intriguing: redefines gravity as emergent rather than fundamental. Links to ideas from Bekenstein and Hawking. || ★★★★☆
|-
| '''Mathematical Rigor''' || Relies on concepts from thermodynamics (e.g. entropy gradients), statistical mechanics, and quantum gravity. Uses integral relations and variational principles, but lacks a standardized set of equations for general use. || ★★★☆☆
|}

'''Total: 26/40'''

=== '''8.8.5 Big Bang – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Big_Bang Big Bang Theory]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Strongly supported by CMB, redshift-distance relation, and light element abundances. However, JWST data revealing early, mature galaxies challenges the predicted timeline of structure formation. || ★★★★☆
|-
| '''Internal Consistency''' || Internally coherent within the ΛCDM framework. Logical structure is intact, though singularity and quantum-gravity transitions remain unresolved. || ★★★★★
|-
| '''Predictive Power''' || Successfully predicted CMB and nucleosynthesis. But recent observations (e.g. JWST galaxies at z &gt; 10) were not anticipated, requiring post-hoc model adjustments. || ★★★★☆
|-
| '''Cross-Disciplinary Compatibility''' || Strong integration with general relativity, quantum field theory, and thermodynamics. Limited links to geology or planetary evolution. || ★★★★☆
|-
| '''Conceptual Clarity and Simplicity''' || Conceptually complex genesis: singularity, inflation, and multiple postulated fields (inflaton, dark energy, dark matter). Relies on hypothetical constructs. || ★★★☆☆
|-
| '''Heuristic Value''' || Stimulated massive observational and theoretical advances. Has driven decades of mission planning and cosmological interpretation. || ★★★★★
|-
| '''Historical and Philosophical Insight''' || Represented a paradigm shift from steady-state thinking. Raises deep ontological and metaphysical questions (e.g. time’s origin, creation ex nihilo). || ★★★★☆
|-
| '''Mathematical Rigor''' || Grounded in Einstein’s field equations, Friedmann dynamics, and thermodynamics. Offers well-defined, highly developed formalism. || ★★★★★
|}

'''Total: 34/40'''

=== '''8.8.6 Steady State Theory – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Steady_state_model Steady State Cosmology]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Early successes with radio‐source counts, but contradicted by the cosmic microwave background, evolving galaxy populations, and quasar statistics. || ★★☆☆☆
|-
| '''Internal Consistency''' || Mathematically self-consistent within the “perfect cosmological principle,” requiring continuous matter creation at a fixed rate. Logical but invokes an ad-hoc creation field (C-field). || ★★★☆☆
|-
| '''Predictive Power''' || Predicted constant large-scale density and specific radio‐source number counts; few successful novel predictions beyond its initial scope. || ★★☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Limited overlap with modern astrophysics and particle physics; conflicts with nucleosynthesis and CMB observations. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Intuitively simple: Universe is homogeneous in space **and** time, avoiding an initial singularity. Minimal parameter set. || ★★★★☆
|-
| '''Heuristic Value''' || Historically spurred observational tests that ultimately favored Big Bang models; now mainly of pedagogical interest. || ★★★☆☆
|-
| '''Historical and Philosophical Insight''' || Embodies the perfect cosmological principle and continuous-creation idea, provoking debates on temporality and cosmological assumptions. || ★★★★☆
|-
| '''Mathematical Rigor''' || Employs Friedmann-like solutions with a creation term; uses relativistic field equations but lacks the richer formal development of ΛCDM or GR extensions. || ★★★☆☆
|}

'''Total: 23/40'''

=== '''8.8.7 Big Crunch – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Big_Crunch Big Crunch]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Once considered viable if cosmic density exceeded the critical value. Current observations (accelerating expansion, dark energy) contradict its key assumptions. || ★★☆☆☆
|-
| '''Internal Consistency''' || Theoretically consistent as a time-reversed Big Bang within general relativity. Requires high matter density and no (or reversing) dark energy. || ★★★☆☆
|-
| '''Predictive Power''' || Predicts a decelerating expansion turning to collapse. Testable in principle, but not supported by current data. || ★★☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Compatible with thermodynamic and relativistic models of entropy and time symmetry, but unsupported by astronomical data. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Intuitively mirrors Big Bang, offering closure and symmetry. Simple in concept, but difficult to reconcile with observed acceleration. || ★★★☆☆
|-
| '''Heuristic Value''' || Motivated theoretical discussion on cosmological fate and cyclic models. Limited influence in current cosmology. || ★★★☆☆
|-
| '''Historical and Philosophical Insight''' || Explores cosmological temporality and finitude. Once a philosophically compelling counterbalance to eternal expansion. || ★★★★☆
|-
| '''Mathematical Rigor''' || Based on time-reversible solutions to Friedmann equations and relativistic cosmology. Rigorous within GR but not extended in modern frameworks. || ★★★☆☆
|}

'''Total: 23/40'''

=== '''8.8.8 Big Bounce – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Big_Bounce Big Bounce]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || No direct observational evidence yet. Some loop quantum cosmology models suggest signatures in the CMB, but these remain speculative. || ★★☆☆☆
|-
| '''Internal Consistency''' || Offers a logically coherent alternative to singularity-based models. Dependent on quantum gravity frameworks (e.g. loop quantum gravity) that remain under development. || ★★★☆☆
|-
| '''Predictive Power''' || Provides potential testable differences in early universe structure and CMB fluctuations. Predictions are still uncertain and model-dependent. || ★★☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Bridges general relativity with quantum mechanics. Limited overlap with geology or observational astronomy. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Complex and abstract. The idea of a cyclical universe is conceptually appealing, but quantum corrections are not intuitive. || ★★☆☆☆
|-
| '''Heuristic Value''' || Stimulates exploration of singularity resolution and quantum gravity cosmologies. Encourages investigation of pre-Big Bang conditions. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Resonates with ancient cyclical cosmologies and philosophical ideas of eternal recurrence. Reframes the question of origins. || ★★★★☆
|-
| '''Mathematical Rigor''' || Based on extensions of Friedmann equations using loop quantum corrections or other quantum gravity approaches. Some models are mathematically formal, but the field is still unsettled. || ★★★☆☆
|}

'''Total: 24/40'''

=== '''8.8.9 Cosmic Influx Theory (CIT) – AI Rating Summary''' ===

''Related link:'' [https://en.wikiversity.org/wiki/Cosmic_Influx_Theory Cosmic Influx Theory]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Strong match with observed planetary structuring, VRMS-based system modeling, and geological trends like daylength and expansion. Supported by exoplanet data and disk morphology (e.g. HD 163296). || ★★★★☆
|-
| '''Internal Consistency''' || Equations and constants (e.g. κ₍CIT₎, (γ−1)/4π) are logically coherent. Internal derivations remain consistent across cosmological and planetary domains. || ★★★★☆
|-
| '''Predictive Power''' || Offers specific predictions (e.g. Trappist-1 preferred distance and orbital period). Some predictions still await observational confirmation. || ★★★★☆
|-
| '''Cross-Disciplinary Compatibility''' || Integrates cosmology, geology, biology, and observational astronomy. Compatible with expanding Earth, daylength data, and ring formation physics. || ★★★★☆
|-
| '''Conceptual Clarity and Simplicity''' || Introduces new yet intuitive ideas like influx and preferred distances. Avoids abstract constructs like dark matter/energy. || ★★★★☆
|-
| '''Heuristic Value''' || Inspires re-evaluation of mainstream assumptions, links to overlooked or discarded theories (e.g. Le Sage, expansion tectonics). || ★★★★★
|-
| '''Historical and Philosophical Insight''' || Reconnects with early gravitational push models and continuous creation ideas, offering philosophical alternatives to entropy-based models. || ★★★★☆
|-
| '''Mathematical Rigor''' || Provides original equations (e.g. for \(D_{\text{pref}}\), \(G = (\gamma - 1)/4\pi\), \(\kappa = v_{\text{RMS}}^2 / c^4\)) and consistent dimensional analysis. Excel-based datasets link math to observations. Lacks field-theoretic formalism. || ★★★★☆
|}

'''Total: 34/40'''

=== '''8.8.10 Spiral Cosmology – AI Rating Summary''' ===

''Related link:'' [https://www.academia.edu/103005946/Cosmology_as_Spiral_Evolution Spiral Cosmology on Academia.edu]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Offers visual and structural explanations for spiral galaxy morphology and cosmic rotation patterns. Less directly tied to quantitative data or tested predictions. || ★★☆☆☆
|-
| '''Internal Consistency''' || Conceptually consistent in proposing self-similar spiral evolution at multiple scales, but lacks a developed dynamic or energetic framework. || ★★☆☆☆
|-
| '''Predictive Power''' || Suggests qualitative evolutionary stages and possible cyclic features, but does not provide specific numerical predictions. || ★★☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Makes symbolic and philosophical connections across cosmology, biology, and culture. Scientific integration with physical fields is minimal. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Emphasizes intuitive and visual structures (e.g. spirals), which are accessible but may oversimplify physical complexity. || ★★★☆☆
|-
| '''Heuristic Value''' || Stimulates reflection on cosmic structure, symmetry, and recursion. Encourages reinterpretation of known forms (e.g. galaxies, DNA, hurricanes). || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Reconnects with ancient and Renaissance cosmologies linking form and function across scales. Offers metaphysical resonance. || ★★★★☆
|-
| '''Mathematical Rigor''' || Utilizes geometric symbolism (e.g. spiral ratios, golden mean) but lacks physical equations or dynamical systems modeling. No quantitative derivations. || ★★☆☆☆
|}

'''Total: 23/40'''

=== '''8.8.11 Topological Field Framework – AI Rating Summary''' ===

''Related link:'' [https://www.researchgate.net/publication/393122856_A_Topological_Field_Framework_for_Particle_Mass_Gauge_Interactions_and_Emergent_Gravity A Topological Field Framework on ResearchGate]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Proposes a geometric origin of particle properties and fundamental constants, but lacks direct comparison with observational or experimental datasets. Empirical testing is suggested but not yet demonstrated. || ★★☆☆☆
|-
| '''Internal Consistency''' || Builds a coherent and logically structured model grounded in topology and pressure gradients. Concepts are well integrated, with minimal contradictions. || ★★★★☆
|-
| '''Predictive Power''' || Offers a pathway to derive constants such as G, c, and Λ from topology and boundary conditions. However, quantitative predictions are still under development or pending validation. || ★★★☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Bridges gravitational theory, particle physics, and topology. Aligns with ideas in quantum geometry, though not yet embedded in mainstream formulations. || ★★★★☆
|-
| '''Conceptual Clarity and Simplicity''' || Dense but conceptually focused. Uses field and pressure analogies to unify gravity and gauge interactions, though the abstract nature may limit accessibility. || ★★★☆☆
|-
| '''Heuristic Value''' || Provides an original and stimulating approach to unify physical constants via geometry. Encourages rethinking of foundational assumptions in physics. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Revives geometrical unification ideals from early 20th-century physics and connects them to modern field-based ontology. Philosophically grounded. || ★★★★☆
|-
| '''Mathematical Rigor''' || Employs differential geometry, topological mapping, and field pressure modeling. Mathematical structure is present, but derivations are at a conceptual stage. || ★★★☆☆
|}

'''Total: 27/40'''

=== '''8.8.12 Conformal Gravity – AI Rating Summary''' ===

''Related link:'' [https://ui.adsabs.harvard.edu/abs/1989ApJ...342..635M Exact Vacuum Solution to Conformal Weyl Gravity and Galactic Rotation Curves]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Successfully models galactic rotation curves and lensing effects without invoking dark matter. However, challenges remain regarding early-universe phenomena and the CMB. || ★★★☆☆
|-
| '''Internal Consistency''' || Mathematically self-consistent within its conformal symmetry framework. Gravitational dynamics are derived cleanly from a fourth-order field equation. || ★★★★☆
|-
| '''Predictive Power''' || Accurately predicts galaxy-scale observations. However, extrapolations to cosmological scales require further development and empirical testing. || ★★★☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Intersects with quantum field theory through conformal invariance and offers alternatives to ΛCDM. Limited overlap with standard model particle physics. || ★★★☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Builds from a well-defined symmetry principle (conformal invariance). Some complexity arises due to fourth-order derivatives and unfamiliar constructs. || ★★★☆☆
|-
| '''Heuristic Value''' || Encourages reconsideration of gravitational assumptions and dark matter. Inspires new theoretical directions and alternative metrics. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Connects with earlier Weyl geometry and symmetry-based models. Philosophically significant as a symmetry-driven alternative to general relativity. || ★★★★☆
|-
| '''Mathematical Rigor''' || Employs rigorous derivations based on conformal symmetry and higher-order field equations. Technically sophisticated. || ★★★★☆
|}

'''Total: 28/40'''

=== '''8.8.13 Quasi-Steady State Cosmology (QSSC) – AI Rating Summary''' ===

''Related link:'' [https://www.ias.ac.in/article/fulltext/pram/053/06/1093-1104 The Quasi-Steady State Cosmology: Theory and Observations – Pramana Journal]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Offers explanations for redshift, large-scale structure, and quasar distributions. However, it faces difficulties matching the observed CMB spectrum and primordial element abundances. || ★★☆☆☆
|-
| '''Internal Consistency''' || Theoretical foundations are logically constructed around a C-field and creation events. Internally coherent but relies on non-standard mechanisms not universally accepted. || ★★★☆☆
|-
| '''Predictive Power''' || Makes unique predictions about cosmic cycles, matter creation, and galaxy evolution. Some predictions remain qualitative or are difficult to test. || ★★★☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Attempts to integrate cosmology with continuous creation physics. However, the C-field concept is not aligned with standard field theory or particle physics. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Proposes an intuitive cyclical model of cosmic evolution. Some components, such as the C-field, are abstract and complex to formalize. || ★★★☆☆
|-
| '''Heuristic Value''' || Offers a provocative alternative to the Big Bang. Stimulates re-evaluation of singularity-based models and encourages cyclic interpretations. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Extends the steady-state philosophy of Hoyle. Challenges the singular origin narrative with philosophical depth and continuity. || ★★★★☆
|-
| '''Mathematical Rigor''' || Uses mathematical models for cyclic expansion and C-field dynamics. However, these are not widely adopted or fully developed in mainstream literature. || ★★☆☆☆
|}

'''Total: 23/40'''

=== '''8.8.14 Dynamic Universe (Tuomo Suntola) – AI Rating Summary''' ===

''Related link:'' [https://physicsfoundations.org/suntola/dynamic-universe Dynamic Universe – Physics Foundations Society]  
''Additional source:'' [https://www.academia.edu/37149633/The_Dynamic_Universe_Toward_a_unified_picture_of_physical_reality The Dynamic Universe on Academia.edu]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Provides testable predictions for cosmological redshift, Hubble parameter, and time dilation without invoking dark energy. Several predictions align well with observational data, though not yet widely confirmed. || ★★★☆☆
|-
| '''Internal Consistency''' || The model is highly self-consistent, based on a single zero-energy principle and evolving 4-sphere geometry. Internally coherent and logically derived. || ★★★★☆
|-
| '''Predictive Power''' || Predicts cosmological parameters from first principles, including a time-evolving H₀ and expansion behavior. Some predictions differ from ΛCDM but remain observationally accessible. || ★★★★☆
|-
| '''Cross-Disciplinary Compatibility''' || Attempts unification of relativity, cosmology, and quantum phenomena under a geometric framework. Still under integration with conventional physics models. || ★★★☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Built on a single geometric principle with minimal assumptions. Clear in its physical logic, though unfamiliar to those trained in standard GR or QFT. || ★★★★☆
|-
| '''Heuristic Value''' || Inspires a rethinking of time, space, and energy conservation. Offers a conceptually elegant reformulation of cosmic dynamics. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Echoes Machian ideas and Einstein’s early search for balance models. Challenges the notion of spacetime curvature as fundamental. || ★★★★☆
|-
| '''Mathematical Rigor''' || Employs rigorous geometric derivations and differential equations. Mathematical structure is well-developed, though outside conventional formalisms. || ★★★★☆
|}

'''Total: 30/40'''


=== '''8.8.15 Multiverse Theory – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Multiverse Multiverse – Wikipedia]  
''Additional source:'' [https://www.scientificamerican.com/article/does-the-multiverse-really-exist/ Scientific American – Does the Multiverse Really Exist?]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Lacks direct observational support. No known experiment can confirm or refute the existence of other universes. Some indirect inferences arise from inflationary models or quantum theory, but none are conclusive. || ★★☆☆☆
|-
| '''Internal Consistency''' || Varies by formulation. Quantum Many-Worlds, eternal inflation, and string theory landscapes are logically coherent within their frameworks, but depend on assumptions not testable in this universe. || ★★★☆☆
|-
| '''Predictive Power''' || Extremely weak. Most formulations make no concrete predictions for our universe that differ from single-universe models. Some argue it &quot;predicts&quot; everything, making it unfalsifiable. || ★★☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Connects to quantum theory, inflationary cosmology, string theory, and philosophy. However, lacks integration with empirical fields like astronomy, geology, or planetary science. || ★★★☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Conceptually speculative and often confusing. Raises questions about what counts as &quot;real&quot; and how probability works across unobservable universes. Not parsimonious. || ★★☆☆☆
|-
| '''Heuristic Value''' || High inspiration in theoretical physics and cosmology. Stimulates debate on fine-tuning, origins, and the limits of science. Has generated many models and new lines of thought. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Reflects an evolution of thinking about cosmological plurality. Raises deep metaphysical questions about causality, determinism, and reality. || ★★★★☆
|-
| '''Mathematical Rigor''' || Strong in some versions (e.g. string theory landscapes, quantum branches), but speculative assumptions extend beyond tested mathematics. || ★★★☆☆
|}

'''Total: 23/40'''

=== '''8.8.16 Theory of Universality – AI Rating Summary''' ===

''Primary source:'' [https://www.iosrjournals.org/iosr-jap/papers/Vol11-issue2/Series-3/D1102031953.pdf K.S. Narayana – Theory of Universality (IOSR-JAP, 2019)]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Some known constants (e.g. Hubble parameter, proton lifetime) are reproduced numerically, but without grounding in observational datasets or statistical validation. No empirical testing methodology is provided. || ★☆☆☆☆
|-
| '''Internal Consistency''' || The derivations follow internal logic, but some assumptions—like matter exceeding light speed or redefining constants—conflict with established physics and dimensional coherence. || ★★☆☆☆
|-
| '''Predictive Power''' || No distinct or testable predictions are formulated. The theory remains mostly retrospective in deriving values, not prospective in forecasting new results. || ★☆☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Conflicts with key principles in general relativity, quantum theory, and thermodynamics. It does not integrate with empirical fields or build bridges across scientific domains. || ★☆☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || The theory aims at unification but uses complex constructs with speculative interpretations. Lacks parsimony and clear explanatory logic. || ★★☆☆☆
|-
| '''Heuristic Value''' || Inspires rethinking of fundamental constants and offers alternative pathways to explore force unification. Useful for stimulating discussion despite lack of empirical grounding. || ★★★☆☆
|-
| '''Historical and Philosophical Insight''' || Emphasizes classical ideals of universal unity. Raises metaphysical questions about origins and physical laws. Philosophically bold, though scientifically unorthodox. || ★★★☆☆
|-
| '''Mathematical Rigor''' || Includes numerous equations and derivations, but lacks clarity in notation, dimensional checks, and formal structure. Relies on fitting numerical coincidences. || ★★☆☆☆
|}

'''Total: 16/40'''

=== '''8.8.17 Structured Loitering – AI Rating Summary''' ===

''Primary source:'' [https://www.academia.edu/119030837/Structured_Loitering_in_Communication_May_Inform_Genesis_of_Structure_and_Gravity_from_Organized_Energy Structured Loitering in Communication… by Kenneth Shiovitz (Academia.edu)]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Presents original experimental attempts to detect mass changes associated with energy structure interactions using LED light and sealed systems. While exploratory and not yet independently verified, the study provides observational data aligned with the theory’s core hypothesis. || ★★☆☆☆

|-
| '''Internal Consistency''' || Offers a conceptual link between “structured loitering” in communication and physics, but lacks a coherent mathematical or mechanistic framework that unambiguously connects the concepts. || ★★☆☆☆
|-
| '''Predictive Power''' || Does not propose testable or falsifiable predictions; the model remains speculative and descriptive. || ★☆☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Attempts interdisciplinary fusion—information theory, biology (birdsong/human writing), evolution, and physics—but lacks integration with standard physics or biological models. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Presents an original metaphor (“loitering”) but the idea remains vague; lacks clear definitions and relies on analogical reasoning rather than structured argumentation. || ★★☆☆☆
|-
| '''Heuristic Value''' || Stimulating and creative: suggests novel perspectives on emergence and complexity, potentially inspiring further exploration of informational origins of gravity and structure. || ★★★☆☆
|-
| '''Historical and Philosophical Insight''' || Offers a philosophical reflection on how structure and gravity might derive from organized energy patterns and communication processes, echoing information-centric worldviews. || ★★★☆☆
|-
| '''Mathematical Rigor''' || Contains no formal equations, models, or derivations; the discussion remains qualitative and metaphorical. || ★☆☆☆☆
|}

'''Total: 16/40'''

== '''8.9 Proposed theories that will be evaluated soon''' ==

- TGD from Matti Pitkänen 

'''◀ [[AI-Assisted_Evaluation_of_Cosmological_Theories/Chapter_7:_Cosmological_Parameters_and_Universal_Constants|Previous]] | [[AI-Assisted_Evaluation_of_Cosmological_Theories|Main Page]] | [[AI-Assisted_Evaluation_of_Cosmological_Theories/Chapter_9:_Documentation_and_Transparency|Next ▶]]'''</text>
      <sha1>skevsovnnde8ih9nxxg103d2wbly7wp</sha1>
    </revision>
    <revision>
      <id>2721163</id>
      <parentid>2721162</parentid>
      <timestamp>2025-07-10T04:20:10Z</timestamp>
      <contributor>
        <username>Ruud Loeffen</username>
        <id>2998353</id>
      </contributor>
      <comment>/* 8.8.17 Structured Loitering – AI Rating Summary */ add 8.8.18 Topological GeometroDynamics (TGD) – AI Rating Summary</comment>
      <origin>2721163</origin>
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      <text bytes="45663" sha1="tkxejeejugdntjtha69tyko9gamatgb" xml:space="preserve">== '''Chapter 8: Methods, Tools, and AI-Assisted Evaluation''' =

== '''8.1 Purpose''' ==

This chapter describes the methodological foundations of the project, including the role of AI in evaluating theories, how contributors interact with the system, and how assessments are recorded, displayed, and updated.

== '''8.2 Toolchain and Workflow''' ==

AI tools like ChatGPT are used to analyze, compare, and refine theories

Tables and frameworks are generated collaboratively using open formats

Contributors submit input via email or editing suggestions

Ratings and evaluations are dynamically updated based on ongoing analysis

== '''8.3 AI-Based Rating System: Motivation and Procedure''' ==

To support comparative evaluation without personal or institutional bias, this project uses an AI-based rating system. ChatGPT acts as a neutral evaluator, analyzing each theory across clearly defined criteria in '''8.5 Understanding the Star Ratings'''.

This system is designed to be:

Transparent – Each rating is justified through AI’s large-scale reference analysis.

Dynamic – Contributors may submit additional materials to request re-evaluation.

Consistent – All evaluations are performed by the same AI logic, eliminating personal bias.

Contributors may ask ChatGPT to re-read specific articles, datasets, or theoretical arguments. If new insights are found, ratings will be updated and transparently noted.

This approach represents a shift toward evidence-driven, large-scale comparative review, using AI not as a gatekeeper but as a tool to synthesize and validate.

== '''8.4 Open Participation''' ==

This chapter is also where future documentation of the workflow and collaborative mechanisms will be expanded. Users who submit theories are encouraged to:

Describe their framework in terms of assumptions, predictions, and compatibility

Suggest how their model could be tested or falsified

Provide references or original materials for AI evaluation

All analysis is open, and contributors may propose improvements at any time.

== '''8.5 Understanding the Star Ratings''' ==

The AI Evaluation Table below rates theories across seven scientific criteria defined in '''Chapter 1.3'''. Here we provide a full explanation of what each criterion means and how it is applied.

'''8.5.1. Empirical Adequacy'''  
Does the theory fit known observations and experimental data? High scores require support from astronomy, cosmology, geology, or lab-based physics. Theories that contradict established measurements or lack empirical grounding score lower.

'''8.5.2. Internal Consistency'''  
Are the theory’s assumptions, mathematics, and logic self-coherent? A consistent theory does not contain contradictions, undefined steps, or ad hoc assumptions.

'''8.5.3. Predictive Power'''  
Does the theory make clear, testable predictions that distinguish it from others? Theories that anticipate new phenomena or retrodict known data gain higher ratings.

'''8.5.4. Cross-Disciplinary Compatibility'''  
Is the theory consistent with findings from other scientific fields, such as geology, chemistry, biology, or planetary science? The more compatible it is, the higher the score.

'''8.5.5. Conceptual Clarity and Simplicity'''  
Is the theory logically simple and intuitively understandable, without unnecessary complexity? This criterion rewards elegance, not oversimplification.

'''8.5.6. Heuristic Value'''  
Does the theory stimulate new questions, research directions, or rethinking of existing problems? A high score reflects creative scientific potential.

'''8.5.7. Historical and Philosophical Insight'''  
Does the theory connect meaningfully to the historical development of science or reflect philosophical depth? Theories grounded in tradition or conceptual evolution are valued here.

'''8.5.8. Mathematical Rigor'''  
Does the theory provide clear mathematical formulations, derivations, and quantitative predictions? Theories are valued for their use of equations to express core principles, ensure internal consistency, and generate testable results.


Each theory receives a rating from ★☆☆☆☆ to ★★★★★ per criterion. The total score (max 40) gives a general measure of its scientific coherence and reach. The reasoning behind the scores is available under each theory (subsection 8.8) or can be requested in more detail.


=== '''8.6 – Comparative Table of AI Ratings (Updated with Criterion 8: Mathematical Rigor)''' ===

''Note: In July 2025, an eighth evaluation criterion was added: '''Mathematical Rigor'''. The total score is now out of 40 stars instead of 35.  ''
''All ratings are expressed in whole stars (★), without fractional values, to ensure clarity in display and consistency with the visual format of this table.''  
''Theories are listed in the same order as in Section 8.8.''

{| class=&quot;wikitable&quot;
! '''Theory''' !! '''EA''' !! '''IC''' !! '''PP''' !! '''CC''' !! '''CS''' !! '''HV''' !! '''HP''' !! '''MR''' !! '''Total (★/40)'''
|-
| '''8.8.1 General Relativity''' || ★★★★★ || ★★★★★ || ★★★★★ || ★★★★☆ || ★★★☆☆ || ★★★★★ || ★★★★★ || ★★★★★ || '''37'''
|-
| '''8.8.2 Newtonian Gravity''' || ★★★★☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || ★★★★★ || ★★★★☆ || ★★★★★ || ★★★★☆ || '''33'''
|-
| '''8.8.3 MOND''' || ★★★★☆ || ★★★☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || ★★★☆☆ || '''26'''
|-
| '''8.8.4 Emergent Gravity''' || ★★★☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || '''24'''
|-
| '''8.8.5 Big Bang''' || ★★★★☆ || ★★★★★ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || ★★★★★ || ★★★★☆ || ★★★★★ || '''34'''
|-
| '''8.8.6 Steady State Theory''' || ★★☆☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★★★☆ || ★★★☆☆ || ★★★★☆ || ★★★☆☆ || '''23'''
|-
| '''8.8.7 Big Crunch''' || ★★☆☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★★☆☆ || ★★★★☆ || ★★★☆☆ || '''22'''
|-
| '''8.8.8 Big Bounce''' || ★★☆☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || '''22'''
|-
| '''8.8.9 Cosmic Influx Theory (CIT)''' || ★★★★☆ || ★★★★☆ || ★★★★☆ || ★★★★☆ || ★★★★☆ || ★★★★★ || ★★★★☆ || ★★★★☆ || '''33'''
|-
| '''8.8.10 Spiral Cosmology''' || ★★☆☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★★★☆ || ★★★★☆ || ★★☆☆☆ || '''21'''
|-
| '''8.8.11 Topological Field Framework''' || ★★☆☆☆ || ★★★★☆ || ★★★☆☆ || ★★★★☆ || ★★★☆☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || '''27'''
|-
| '''8.8.12 Conformal Gravity''' || ★★★☆☆ || ★★★★☆ || ★★★☆☆ || ★★★☆☆ || ★★★☆☆ || ★★★★☆ || ★★★★☆ || ★★★★☆ || '''28'''
|-
| '''8.8.13 Quasi-Steady State Cosmology''' || ★★☆☆☆ || ★★★☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★★★☆ || ★★★★☆ || ★★☆☆☆ || '''23'''
|-
| '''8.8.14 Dynamic Universe''' || ★★★☆☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || ★★★★☆ || ★★★★☆ || ★★★★☆ || ★★★★☆ || '''30'''
|-
| '''8.8.15 Multiverse''' || ★★☆☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || '''23'''
|-
| '''8.8.16 Theory of Universality''' || ★☆☆☆☆ || ★★☆☆☆ || ★☆☆☆☆ || ★☆☆☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★★☆☆ || ★★☆☆☆ || '''16'''
|-
| '''8.8.17 Structured Loitering''' || ★★☆☆☆ || ★★☆☆☆ || ★☆☆☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★★☆☆ || ★☆☆☆☆ || '''16'''


&lt;!-- put an empty line here --&gt;

|}

''Legend:''  
''EA = Empirical Adequacy IC = Internal Consistency PP = Predictive Power CC = Cross-Disciplinary Compatibility  ''
''CS = Conceptual Simplicity HV = Heuristic Value HP = Historical/Philosophical Insight MR = Mathematical Rigor''

== '''8.7 Requesting Detailed Justification''' ==

“Somebody” interested in the full motivation for a theory’s score—criterion by criterion—can obtain this by:

1. Visiting the '''discussion page''' of this Wikiversity chapter.
  
2. Mentioning the theory by name and explicitly asking for the detailed breakdown.
  
3. Receiving an AI-generated or authored comment elaborating why each star rating was assigned. 

These ratings are transparent and intended to support critical evaluation and ongoing refinement of both established and alternative cosmological theories. Constructive feedback, new data, or arguments may lead to adjustments in the scores, provided clear reasoning is presented.

If a contributor or author does not agree with the evaluation or prefers not to have their theory represented under this framework, they may request that all related content and ratings for that theory be removed. This ensures that participation remains voluntary and respectful of intellectual ownership.

'''Mainstream theories are subject to the same critical standard.'''  
If someone presents a reasoned objection to the current star ratings of a widely accepted theory (e.g. General Relativity, Big Bang), their explanation will be reviewed. If the argument is well-founded, ChatGPT may generate a revised evaluation. Both the original and alternative viewpoints can be documented transparently if needed.

If you want to submit a '''Rating Rebuttal''', please use the template provided on the '''[[Talk:AI-Assisted Evaluation of Cosmological Theories|Discussion page]]'''.

== '''8.8 Detailed Evaluations of Theories''' ==

This section will be expanded progressively as new theories are added. Each evaluation is generated in collaboration with ChatGPT, based on a standardized set of seven criteria defined in Chapter 1. This approach allows for clear, consistent, and rapid assessment of both mainstream and non-mainstream cosmological theories within minutes.

=== '''8.8.1 General Relativity – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/General_relativity General Relativity]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Matches a wide range of observations: gravitational lensing, GPS corrections, perihelion precession, black hole dynamics, and gravitational waves. Supported by multiple experiments. || ★★★★★
|-
| '''Internal Consistency''' || Highly consistent within its differential geometric framework. Built upon Einstein's field equations with tensor calculus. Few internal contradictions, though extensions (e.g. quantum gravity) face challenges. || ★★★★★
|-
| '''Predictive Power''' || Successfully predicted gravitational redshift, time dilation, frame-dragging, and gravitational waves. Continues to guide observations in astrophysics. || ★★★★★
|-
| '''Cross-Disciplinary Compatibility''' || Compatible with cosmology and astrophysics. Some tension with quantum theory. Less integrated with planetary geology or biology. || ★★★★☆
|-
| '''Conceptual Clarity and Simplicity''' || Complex mathematical structure makes it less intuitive. Conceptually abstract (spacetime curvature, geodesics). Clarity improves with education, but simplicity is low. || ★★★☆☆
|-
| '''Heuristic Value''' || Inspired vast developments in cosmology, black hole theory, and relativistic astrophysics. Foundation for modern gravitational physics. || ★★★★★
|-
| '''Historical and Philosophical Insight''' || Deep philosophical implications about space, time, and causality. Represents a major shift from Newtonian absolute space. Influenced 20th-century philosophy of science. || ★★★★★
|-
| '''Mathematical Rigor''' || Built upon advanced mathematics: Riemannian geometry, Einstein field equations, tensor calculus. Equations are precise, formal, and deeply embedded in differential geometry. || ★★★★★
|}

'''Total: 39/40'''

=== '''8.8.2 Newtonian Gravity – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation Newton's Law of Universal Gravitation]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Accurately describes gravitational interactions for most macroscopic systems (planets, satellites, projectiles) under low-speed, weak-field conditions. Deviates in extreme conditions (e.g. near black holes). || ★★★★☆
|-
| '''Internal Consistency''' || Mathematically self-consistent with inverse-square law and Newton's laws of motion. Assumes instantaneous action at a distance, which conflicts with relativity. || ★★★★☆
|-
| '''Predictive Power''' || Predicts planetary orbits, tides, escape velocities, and Keplerian motion. Fails for relativistic effects (e.g. Mercury’s precession, gravitational lensing). || ★★★★☆
|-
| '''Cross-Disciplinary Compatibility''' || Well-integrated in classical mechanics, astronomy, and engineering. Less compatible with modern cosmology or relativistic frameworks. || ★★★☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Simple and intuitive: gravity as a force between masses. Easily grasped and widely taught. || ★★★★★
|-
| '''Heuristic Value''' || Inspired centuries of scientific discovery and classical mechanics. Still used in teaching and engineering. Limited in modern theoretical development. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Founded classical physics. Major leap in scientific method and mathematical modeling. Paved the way for Enlightenment-era science. || ★★★★★
|-
| '''Mathematical Rigor''' || Clear and elegant use of calculus and vector algebra (e.g. \(\displaystyle F = G \frac{m_1 m_2}{r^2}\)). Highly accessible and historically groundbreaking, but lacks deeper geometric or relativistic structures. || ★★★★☆
|}

'''Total: 33/40'''

=== '''8.8.3 MOND (Modified Newtonian Dynamics) – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics Modified Newtonian Dynamics (MOND)]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Fits galactic rotation curves without invoking dark matter. Matches Tully-Fisher relation. Performance weakens at cluster and cosmological scales. || ★★★★☆
|-
| '''Internal Consistency''' || Original formulation is non-relativistic and phenomenological. Several relativistic extensions (e.g. TeVeS) exist but introduce complexity and fine-tuning. || ★★★☆☆
|-
| '''Predictive Power''' || Predicts galaxy dynamics from baryonic matter alone. Less predictive at larger scales or in non-galactic contexts. || ★★★☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Primarily astrophysical. Some tension with cosmology, structure formation, and gravitational lensing. Not aligned with particle physics. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Simple at galactic scale: modifies acceleration below a threshold \(a_0\). Extensions are less intuitive. || ★★★★☆
|-
| '''Heuristic Value''' || Challenges dark matter paradigm and motivates alternate gravity models. Sparked theoretical and observational debate. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Offers a conceptual challenge to Newton/Einstein gravity. Philosophically provocative, but limited historical lineage. || ★★★☆☆
|-
| '''Mathematical Rigor''' || Employs interpolating functions and modified Poisson equations. Relativistic extensions (e.g. TeVeS) involve tensor-vector-scalar frameworks. Rigorous in parts, but lacks unified formalism. || ★★★☆☆
|}

'''Total: 28/40'''

=== '''8.8.4 Emergent Gravity – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Entropic_gravity Emergent / Entropic Gravity]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Attempts to account for galactic rotation without dark matter by linking gravity to entropy and information. Some results match MOND-like behavior, but broad observational support remains limited. || ★★★☆☆
|-
| '''Internal Consistency''' || Theoretical framework draws from thermodynamics, holography, and information theory. Conceptually coherent, but not fully developed as a unified physical model. || ★★★☆☆
|-
| '''Predictive Power''' || Offers qualitative insights but lacks precise predictive capabilities in most contexts. No wide adoption for simulations or system modeling. || ★★☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Integrates ideas from black hole thermodynamics, quantum information, and spacetime geometry. Weak integration with observational astronomy or geology. || ★★★☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Abstract and speculative. Concepts like entropic forces and holographic screens are not intuitive for most readers. || ★★☆☆☆
|-
| '''Heuristic Value''' || Stimulates discussion about the nature of spacetime and gravity. Has inspired new theoretical directions in quantum gravity and information theory. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Philosophically intriguing: redefines gravity as emergent rather than fundamental. Links to ideas from Bekenstein and Hawking. || ★★★★☆
|-
| '''Mathematical Rigor''' || Relies on concepts from thermodynamics (e.g. entropy gradients), statistical mechanics, and quantum gravity. Uses integral relations and variational principles, but lacks a standardized set of equations for general use. || ★★★☆☆
|}

'''Total: 26/40'''

=== '''8.8.5 Big Bang – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Big_Bang Big Bang Theory]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Strongly supported by CMB, redshift-distance relation, and light element abundances. However, JWST data revealing early, mature galaxies challenges the predicted timeline of structure formation. || ★★★★☆
|-
| '''Internal Consistency''' || Internally coherent within the ΛCDM framework. Logical structure is intact, though singularity and quantum-gravity transitions remain unresolved. || ★★★★★
|-
| '''Predictive Power''' || Successfully predicted CMB and nucleosynthesis. But recent observations (e.g. JWST galaxies at z &gt; 10) were not anticipated, requiring post-hoc model adjustments. || ★★★★☆
|-
| '''Cross-Disciplinary Compatibility''' || Strong integration with general relativity, quantum field theory, and thermodynamics. Limited links to geology or planetary evolution. || ★★★★☆
|-
| '''Conceptual Clarity and Simplicity''' || Conceptually complex genesis: singularity, inflation, and multiple postulated fields (inflaton, dark energy, dark matter). Relies on hypothetical constructs. || ★★★☆☆
|-
| '''Heuristic Value''' || Stimulated massive observational and theoretical advances. Has driven decades of mission planning and cosmological interpretation. || ★★★★★
|-
| '''Historical and Philosophical Insight''' || Represented a paradigm shift from steady-state thinking. Raises deep ontological and metaphysical questions (e.g. time’s origin, creation ex nihilo). || ★★★★☆
|-
| '''Mathematical Rigor''' || Grounded in Einstein’s field equations, Friedmann dynamics, and thermodynamics. Offers well-defined, highly developed formalism. || ★★★★★
|}

'''Total: 34/40'''

=== '''8.8.6 Steady State Theory – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Steady_state_model Steady State Cosmology]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Early successes with radio‐source counts, but contradicted by the cosmic microwave background, evolving galaxy populations, and quasar statistics. || ★★☆☆☆
|-
| '''Internal Consistency''' || Mathematically self-consistent within the “perfect cosmological principle,” requiring continuous matter creation at a fixed rate. Logical but invokes an ad-hoc creation field (C-field). || ★★★☆☆
|-
| '''Predictive Power''' || Predicted constant large-scale density and specific radio‐source number counts; few successful novel predictions beyond its initial scope. || ★★☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Limited overlap with modern astrophysics and particle physics; conflicts with nucleosynthesis and CMB observations. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Intuitively simple: Universe is homogeneous in space **and** time, avoiding an initial singularity. Minimal parameter set. || ★★★★☆
|-
| '''Heuristic Value''' || Historically spurred observational tests that ultimately favored Big Bang models; now mainly of pedagogical interest. || ★★★☆☆
|-
| '''Historical and Philosophical Insight''' || Embodies the perfect cosmological principle and continuous-creation idea, provoking debates on temporality and cosmological assumptions. || ★★★★☆
|-
| '''Mathematical Rigor''' || Employs Friedmann-like solutions with a creation term; uses relativistic field equations but lacks the richer formal development of ΛCDM or GR extensions. || ★★★☆☆
|}

'''Total: 23/40'''

=== '''8.8.7 Big Crunch – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Big_Crunch Big Crunch]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Once considered viable if cosmic density exceeded the critical value. Current observations (accelerating expansion, dark energy) contradict its key assumptions. || ★★☆☆☆
|-
| '''Internal Consistency''' || Theoretically consistent as a time-reversed Big Bang within general relativity. Requires high matter density and no (or reversing) dark energy. || ★★★☆☆
|-
| '''Predictive Power''' || Predicts a decelerating expansion turning to collapse. Testable in principle, but not supported by current data. || ★★☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Compatible with thermodynamic and relativistic models of entropy and time symmetry, but unsupported by astronomical data. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Intuitively mirrors Big Bang, offering closure and symmetry. Simple in concept, but difficult to reconcile with observed acceleration. || ★★★☆☆
|-
| '''Heuristic Value''' || Motivated theoretical discussion on cosmological fate and cyclic models. Limited influence in current cosmology. || ★★★☆☆
|-
| '''Historical and Philosophical Insight''' || Explores cosmological temporality and finitude. Once a philosophically compelling counterbalance to eternal expansion. || ★★★★☆
|-
| '''Mathematical Rigor''' || Based on time-reversible solutions to Friedmann equations and relativistic cosmology. Rigorous within GR but not extended in modern frameworks. || ★★★☆☆
|}

'''Total: 23/40'''

=== '''8.8.8 Big Bounce – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Big_Bounce Big Bounce]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || No direct observational evidence yet. Some loop quantum cosmology models suggest signatures in the CMB, but these remain speculative. || ★★☆☆☆
|-
| '''Internal Consistency''' || Offers a logically coherent alternative to singularity-based models. Dependent on quantum gravity frameworks (e.g. loop quantum gravity) that remain under development. || ★★★☆☆
|-
| '''Predictive Power''' || Provides potential testable differences in early universe structure and CMB fluctuations. Predictions are still uncertain and model-dependent. || ★★☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Bridges general relativity with quantum mechanics. Limited overlap with geology or observational astronomy. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Complex and abstract. The idea of a cyclical universe is conceptually appealing, but quantum corrections are not intuitive. || ★★☆☆☆
|-
| '''Heuristic Value''' || Stimulates exploration of singularity resolution and quantum gravity cosmologies. Encourages investigation of pre-Big Bang conditions. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Resonates with ancient cyclical cosmologies and philosophical ideas of eternal recurrence. Reframes the question of origins. || ★★★★☆
|-
| '''Mathematical Rigor''' || Based on extensions of Friedmann equations using loop quantum corrections or other quantum gravity approaches. Some models are mathematically formal, but the field is still unsettled. || ★★★☆☆
|}

'''Total: 24/40'''

=== '''8.8.9 Cosmic Influx Theory (CIT) – AI Rating Summary''' ===

''Related link:'' [https://en.wikiversity.org/wiki/Cosmic_Influx_Theory Cosmic Influx Theory]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Strong match with observed planetary structuring, VRMS-based system modeling, and geological trends like daylength and expansion. Supported by exoplanet data and disk morphology (e.g. HD 163296). || ★★★★☆
|-
| '''Internal Consistency''' || Equations and constants (e.g. κ₍CIT₎, (γ−1)/4π) are logically coherent. Internal derivations remain consistent across cosmological and planetary domains. || ★★★★☆
|-
| '''Predictive Power''' || Offers specific predictions (e.g. Trappist-1 preferred distance and orbital period). Some predictions still await observational confirmation. || ★★★★☆
|-
| '''Cross-Disciplinary Compatibility''' || Integrates cosmology, geology, biology, and observational astronomy. Compatible with expanding Earth, daylength data, and ring formation physics. || ★★★★☆
|-
| '''Conceptual Clarity and Simplicity''' || Introduces new yet intuitive ideas like influx and preferred distances. Avoids abstract constructs like dark matter/energy. || ★★★★☆
|-
| '''Heuristic Value''' || Inspires re-evaluation of mainstream assumptions, links to overlooked or discarded theories (e.g. Le Sage, expansion tectonics). || ★★★★★
|-
| '''Historical and Philosophical Insight''' || Reconnects with early gravitational push models and continuous creation ideas, offering philosophical alternatives to entropy-based models. || ★★★★☆
|-
| '''Mathematical Rigor''' || Provides original equations (e.g. for \(D_{\text{pref}}\), \(G = (\gamma - 1)/4\pi\), \(\kappa = v_{\text{RMS}}^2 / c^4\)) and consistent dimensional analysis. Excel-based datasets link math to observations. Lacks field-theoretic formalism. || ★★★★☆
|}

'''Total: 34/40'''

=== '''8.8.10 Spiral Cosmology – AI Rating Summary''' ===

''Related link:'' [https://www.academia.edu/103005946/Cosmology_as_Spiral_Evolution Spiral Cosmology on Academia.edu]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Offers visual and structural explanations for spiral galaxy morphology and cosmic rotation patterns. Less directly tied to quantitative data or tested predictions. || ★★☆☆☆
|-
| '''Internal Consistency''' || Conceptually consistent in proposing self-similar spiral evolution at multiple scales, but lacks a developed dynamic or energetic framework. || ★★☆☆☆
|-
| '''Predictive Power''' || Suggests qualitative evolutionary stages and possible cyclic features, but does not provide specific numerical predictions. || ★★☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Makes symbolic and philosophical connections across cosmology, biology, and culture. Scientific integration with physical fields is minimal. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Emphasizes intuitive and visual structures (e.g. spirals), which are accessible but may oversimplify physical complexity. || ★★★☆☆
|-
| '''Heuristic Value''' || Stimulates reflection on cosmic structure, symmetry, and recursion. Encourages reinterpretation of known forms (e.g. galaxies, DNA, hurricanes). || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Reconnects with ancient and Renaissance cosmologies linking form and function across scales. Offers metaphysical resonance. || ★★★★☆
|-
| '''Mathematical Rigor''' || Utilizes geometric symbolism (e.g. spiral ratios, golden mean) but lacks physical equations or dynamical systems modeling. No quantitative derivations. || ★★☆☆☆
|}

'''Total: 23/40'''

=== '''8.8.11 Topological Field Framework – AI Rating Summary''' ===

''Related link:'' [https://www.researchgate.net/publication/393122856_A_Topological_Field_Framework_for_Particle_Mass_Gauge_Interactions_and_Emergent_Gravity A Topological Field Framework on ResearchGate]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Proposes a geometric origin of particle properties and fundamental constants, but lacks direct comparison with observational or experimental datasets. Empirical testing is suggested but not yet demonstrated. || ★★☆☆☆
|-
| '''Internal Consistency''' || Builds a coherent and logically structured model grounded in topology and pressure gradients. Concepts are well integrated, with minimal contradictions. || ★★★★☆
|-
| '''Predictive Power''' || Offers a pathway to derive constants such as G, c, and Λ from topology and boundary conditions. However, quantitative predictions are still under development or pending validation. || ★★★☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Bridges gravitational theory, particle physics, and topology. Aligns with ideas in quantum geometry, though not yet embedded in mainstream formulations. || ★★★★☆
|-
| '''Conceptual Clarity and Simplicity''' || Dense but conceptually focused. Uses field and pressure analogies to unify gravity and gauge interactions, though the abstract nature may limit accessibility. || ★★★☆☆
|-
| '''Heuristic Value''' || Provides an original and stimulating approach to unify physical constants via geometry. Encourages rethinking of foundational assumptions in physics. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Revives geometrical unification ideals from early 20th-century physics and connects them to modern field-based ontology. Philosophically grounded. || ★★★★☆
|-
| '''Mathematical Rigor''' || Employs differential geometry, topological mapping, and field pressure modeling. Mathematical structure is present, but derivations are at a conceptual stage. || ★★★☆☆
|}

'''Total: 27/40'''

=== '''8.8.12 Conformal Gravity – AI Rating Summary''' ===

''Related link:'' [https://ui.adsabs.harvard.edu/abs/1989ApJ...342..635M Exact Vacuum Solution to Conformal Weyl Gravity and Galactic Rotation Curves]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Successfully models galactic rotation curves and lensing effects without invoking dark matter. However, challenges remain regarding early-universe phenomena and the CMB. || ★★★☆☆
|-
| '''Internal Consistency''' || Mathematically self-consistent within its conformal symmetry framework. Gravitational dynamics are derived cleanly from a fourth-order field equation. || ★★★★☆
|-
| '''Predictive Power''' || Accurately predicts galaxy-scale observations. However, extrapolations to cosmological scales require further development and empirical testing. || ★★★☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Intersects with quantum field theory through conformal invariance and offers alternatives to ΛCDM. Limited overlap with standard model particle physics. || ★★★☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Builds from a well-defined symmetry principle (conformal invariance). Some complexity arises due to fourth-order derivatives and unfamiliar constructs. || ★★★☆☆
|-
| '''Heuristic Value''' || Encourages reconsideration of gravitational assumptions and dark matter. Inspires new theoretical directions and alternative metrics. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Connects with earlier Weyl geometry and symmetry-based models. Philosophically significant as a symmetry-driven alternative to general relativity. || ★★★★☆
|-
| '''Mathematical Rigor''' || Employs rigorous derivations based on conformal symmetry and higher-order field equations. Technically sophisticated. || ★★★★☆
|}

'''Total: 28/40'''

=== '''8.8.13 Quasi-Steady State Cosmology (QSSC) – AI Rating Summary''' ===

''Related link:'' [https://www.ias.ac.in/article/fulltext/pram/053/06/1093-1104 The Quasi-Steady State Cosmology: Theory and Observations – Pramana Journal]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Offers explanations for redshift, large-scale structure, and quasar distributions. However, it faces difficulties matching the observed CMB spectrum and primordial element abundances. || ★★☆☆☆
|-
| '''Internal Consistency''' || Theoretical foundations are logically constructed around a C-field and creation events. Internally coherent but relies on non-standard mechanisms not universally accepted. || ★★★☆☆
|-
| '''Predictive Power''' || Makes unique predictions about cosmic cycles, matter creation, and galaxy evolution. Some predictions remain qualitative or are difficult to test. || ★★★☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Attempts to integrate cosmology with continuous creation physics. However, the C-field concept is not aligned with standard field theory or particle physics. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Proposes an intuitive cyclical model of cosmic evolution. Some components, such as the C-field, are abstract and complex to formalize. || ★★★☆☆
|-
| '''Heuristic Value''' || Offers a provocative alternative to the Big Bang. Stimulates re-evaluation of singularity-based models and encourages cyclic interpretations. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Extends the steady-state philosophy of Hoyle. Challenges the singular origin narrative with philosophical depth and continuity. || ★★★★☆
|-
| '''Mathematical Rigor''' || Uses mathematical models for cyclic expansion and C-field dynamics. However, these are not widely adopted or fully developed in mainstream literature. || ★★☆☆☆
|}

'''Total: 23/40'''

=== '''8.8.14 Dynamic Universe (Tuomo Suntola) – AI Rating Summary''' ===

''Related link:'' [https://physicsfoundations.org/suntola/dynamic-universe Dynamic Universe – Physics Foundations Society]  
''Additional source:'' [https://www.academia.edu/37149633/The_Dynamic_Universe_Toward_a_unified_picture_of_physical_reality The Dynamic Universe on Academia.edu]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Provides testable predictions for cosmological redshift, Hubble parameter, and time dilation without invoking dark energy. Several predictions align well with observational data, though not yet widely confirmed. || ★★★☆☆
|-
| '''Internal Consistency''' || The model is highly self-consistent, based on a single zero-energy principle and evolving 4-sphere geometry. Internally coherent and logically derived. || ★★★★☆
|-
| '''Predictive Power''' || Predicts cosmological parameters from first principles, including a time-evolving H₀ and expansion behavior. Some predictions differ from ΛCDM but remain observationally accessible. || ★★★★☆
|-
| '''Cross-Disciplinary Compatibility''' || Attempts unification of relativity, cosmology, and quantum phenomena under a geometric framework. Still under integration with conventional physics models. || ★★★☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Built on a single geometric principle with minimal assumptions. Clear in its physical logic, though unfamiliar to those trained in standard GR or QFT. || ★★★★☆
|-
| '''Heuristic Value''' || Inspires a rethinking of time, space, and energy conservation. Offers a conceptually elegant reformulation of cosmic dynamics. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Echoes Machian ideas and Einstein’s early search for balance models. Challenges the notion of spacetime curvature as fundamental. || ★★★★☆
|-
| '''Mathematical Rigor''' || Employs rigorous geometric derivations and differential equations. Mathematical structure is well-developed, though outside conventional formalisms. || ★★★★☆
|}

'''Total: 30/40'''


=== '''8.8.15 Multiverse Theory – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Multiverse Multiverse – Wikipedia]  
''Additional source:'' [https://www.scientificamerican.com/article/does-the-multiverse-really-exist/ Scientific American – Does the Multiverse Really Exist?]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Lacks direct observational support. No known experiment can confirm or refute the existence of other universes. Some indirect inferences arise from inflationary models or quantum theory, but none are conclusive. || ★★☆☆☆
|-
| '''Internal Consistency''' || Varies by formulation. Quantum Many-Worlds, eternal inflation, and string theory landscapes are logically coherent within their frameworks, but depend on assumptions not testable in this universe. || ★★★☆☆
|-
| '''Predictive Power''' || Extremely weak. Most formulations make no concrete predictions for our universe that differ from single-universe models. Some argue it &quot;predicts&quot; everything, making it unfalsifiable. || ★★☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Connects to quantum theory, inflationary cosmology, string theory, and philosophy. However, lacks integration with empirical fields like astronomy, geology, or planetary science. || ★★★☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Conceptually speculative and often confusing. Raises questions about what counts as &quot;real&quot; and how probability works across unobservable universes. Not parsimonious. || ★★☆☆☆
|-
| '''Heuristic Value''' || High inspiration in theoretical physics and cosmology. Stimulates debate on fine-tuning, origins, and the limits of science. Has generated many models and new lines of thought. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Reflects an evolution of thinking about cosmological plurality. Raises deep metaphysical questions about causality, determinism, and reality. || ★★★★☆
|-
| '''Mathematical Rigor''' || Strong in some versions (e.g. string theory landscapes, quantum branches), but speculative assumptions extend beyond tested mathematics. || ★★★☆☆
|}

'''Total: 23/40'''

=== '''8.8.16 Theory of Universality – AI Rating Summary''' ===

''Primary source:'' [https://www.iosrjournals.org/iosr-jap/papers/Vol11-issue2/Series-3/D1102031953.pdf K.S. Narayana – Theory of Universality (IOSR-JAP, 2019)]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Some known constants (e.g. Hubble parameter, proton lifetime) are reproduced numerically, but without grounding in observational datasets or statistical validation. No empirical testing methodology is provided. || ★☆☆☆☆
|-
| '''Internal Consistency''' || The derivations follow internal logic, but some assumptions—like matter exceeding light speed or redefining constants—conflict with established physics and dimensional coherence. || ★★☆☆☆
|-
| '''Predictive Power''' || No distinct or testable predictions are formulated. The theory remains mostly retrospective in deriving values, not prospective in forecasting new results. || ★☆☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Conflicts with key principles in general relativity, quantum theory, and thermodynamics. It does not integrate with empirical fields or build bridges across scientific domains. || ★☆☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || The theory aims at unification but uses complex constructs with speculative interpretations. Lacks parsimony and clear explanatory logic. || ★★☆☆☆
|-
| '''Heuristic Value''' || Inspires rethinking of fundamental constants and offers alternative pathways to explore force unification. Useful for stimulating discussion despite lack of empirical grounding. || ★★★☆☆
|-
| '''Historical and Philosophical Insight''' || Emphasizes classical ideals of universal unity. Raises metaphysical questions about origins and physical laws. Philosophically bold, though scientifically unorthodox. || ★★★☆☆
|-
| '''Mathematical Rigor''' || Includes numerous equations and derivations, but lacks clarity in notation, dimensional checks, and formal structure. Relies on fitting numerical coincidences. || ★★☆☆☆
|}

'''Total: 16/40'''

=== '''8.8.18 Topological GeometroDynamics (TGD) – AI Rating Summary''' ===
''Primary sources:''
[https://tgdtheory.fi/public_html/articles/TGDtheory2024I.pdf TGD as it is towards end of 2024 – Part I]
[https://tgdtheory.fi/public_html/articles/TGDtheory2024II.pdf TGD as it is towards end of 2024 – Part II]

&lt;table class=&quot;wikitable&quot;&gt; &lt;tr&gt; &lt;th&gt;'''Criterion'''&lt;/th&gt; &lt;th&gt;'''Description'''&lt;/th&gt; &lt;th&gt;'''Rating'''&lt;/th&gt; &lt;/tr&gt; &lt;tr&gt; &lt;td&gt;'''Empirical Adequacy'''&lt;/td&gt; &lt;td&gt;TGD is rich in theoretical proposals—such as predictions about coupling constants, p-adic physics, and quantized criticality—but lacks direct empirical validation. Claims are often qualitative or philosophical, with few testable predictions implemented or tested.&lt;/td&gt; &lt;td&gt;★★☆☆☆&lt;/td&gt; &lt;/tr&gt; &lt;tr&gt; &lt;td&gt;'''Internal Consistency'''&lt;/td&gt; &lt;td&gt;The theory presents a highly consistent internal structure based on an 8-dimensional embedding space (M⁴ × CP₂), number theory, and a unified view of interactions. The coherence across domains (cosmology, quantum physics, biology) is impressive.&lt;/td&gt; &lt;td&gt;★★★★☆&lt;/td&gt; &lt;/tr&gt; &lt;tr&gt; &lt;td&gt;'''Predictive Power'''&lt;/td&gt; &lt;td&gt;TGD makes bold theoretical predictions about particle masses, cosmological evolution, and even consciousness; however, these are often indirect, qualitative, or mathematically untested in standard frameworks.&lt;/td&gt; &lt;td&gt;★★☆☆☆&lt;/td&gt; &lt;/tr&gt; &lt;tr&gt; &lt;td&gt;'''Cross-Disciplinary Compatibility'''&lt;/td&gt; &lt;td&gt;Integrates gravitation, particle physics, cosmology, biology, and neuroscience under a single geometric and number-theoretical framework. Strong interdisciplinary ambition, though not well integrated with conventional theories in these domains.&lt;/td&gt; &lt;td&gt;★★★★☆&lt;/td&gt; &lt;/tr&gt; &lt;tr&gt; &lt;td&gt;'''Conceptual Clarity and Simplicity'''&lt;/td&gt; &lt;td&gt;Conceptually intricate and difficult for non-specialists. TGD proposes radically new metaphysical and mathematical foundations that are not easily intuitive or accessible.&lt;/td&gt; &lt;td&gt;★★☆☆☆&lt;/td&gt; &lt;/tr&gt; &lt;tr&gt; &lt;td&gt;'''Heuristic Value'''&lt;/td&gt; &lt;td&gt;Inspires novel questions about the unification of forces, the role of number theory, and consciousness in physics. Has stimulated a wide body of internal development and speculative exploration.&lt;/td&gt; &lt;td&gt;★★★★☆&lt;/td&gt; &lt;/tr&gt; &lt;tr&gt; &lt;td&gt;'''Historical and Philosophical Insight'''&lt;/td&gt; &lt;td&gt;Offers a deep philosophical stance on geometry, consciousness, time, and mathematical existence. Rich in insight about foundational structures and conceptual shifts in physics.&lt;/td&gt; &lt;td&gt;★★★★☆&lt;/td&gt; &lt;/tr&gt; &lt;tr&gt; &lt;td&gt;'''Mathematical Rigor'''&lt;/td&gt; &lt;td&gt;Employs advanced mathematics: higher-dimensional manifolds, p-adics, number theory, and embedding techniques. Formal structure is evident but not presented using conventional field-theoretic rigor or peer-reviewed derivations.&lt;/td&gt; &lt;td&gt;★★★★☆&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt;
'''Total: 26/40'''

== '''8.9 Proposed theories that will be evaluated soon''' ==

- TGD from Matti Pitkänen 

'''◀ [[AI-Assisted_Evaluation_of_Cosmological_Theories/Chapter_7:_Cosmological_Parameters_and_Universal_Constants|Previous]] | [[AI-Assisted_Evaluation_of_Cosmological_Theories|Main Page]] | [[AI-Assisted_Evaluation_of_Cosmological_Theories/Chapter_9:_Documentation_and_Transparency|Next ▶]]'''</text>
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        <username>Ruud Loeffen</username>
        <id>2998353</id>
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      <text bytes="45881" sha1="j4b044ls9r9fuoex34zpu5r8mtu5gcd" xml:space="preserve">== '''Chapter 8: Methods, Tools, and AI-Assisted Evaluation''' =

== '''8.1 Purpose''' ==

This chapter describes the methodological foundations of the project, including the role of AI in evaluating theories, how contributors interact with the system, and how assessments are recorded, displayed, and updated.

== '''8.2 Toolchain and Workflow''' ==

AI tools like ChatGPT are used to analyze, compare, and refine theories

Tables and frameworks are generated collaboratively using open formats

Contributors submit input via email or editing suggestions

Ratings and evaluations are dynamically updated based on ongoing analysis

== '''8.3 AI-Based Rating System: Motivation and Procedure''' ==

To support comparative evaluation without personal or institutional bias, this project uses an AI-based rating system. ChatGPT acts as a neutral evaluator, analyzing each theory across clearly defined criteria in '''8.5 Understanding the Star Ratings'''.

This system is designed to be:

Transparent – Each rating is justified through AI’s large-scale reference analysis.

Dynamic – Contributors may submit additional materials to request re-evaluation.

Consistent – All evaluations are performed by the same AI logic, eliminating personal bias.

Contributors may ask ChatGPT to re-read specific articles, datasets, or theoretical arguments. If new insights are found, ratings will be updated and transparently noted.

This approach represents a shift toward evidence-driven, large-scale comparative review, using AI not as a gatekeeper but as a tool to synthesize and validate.

== '''8.4 Open Participation''' ==

This chapter is also where future documentation of the workflow and collaborative mechanisms will be expanded. Users who submit theories are encouraged to:

Describe their framework in terms of assumptions, predictions, and compatibility

Suggest how their model could be tested or falsified

Provide references or original materials for AI evaluation

All analysis is open, and contributors may propose improvements at any time.

== '''8.5 Understanding the Star Ratings''' ==

The AI Evaluation Table below rates theories across seven scientific criteria defined in '''Chapter 1.3'''. Here we provide a full explanation of what each criterion means and how it is applied.

'''8.5.1. Empirical Adequacy'''  
Does the theory fit known observations and experimental data? High scores require support from astronomy, cosmology, geology, or lab-based physics. Theories that contradict established measurements or lack empirical grounding score lower.

'''8.5.2. Internal Consistency'''  
Are the theory’s assumptions, mathematics, and logic self-coherent? A consistent theory does not contain contradictions, undefined steps, or ad hoc assumptions.

'''8.5.3. Predictive Power'''  
Does the theory make clear, testable predictions that distinguish it from others? Theories that anticipate new phenomena or retrodict known data gain higher ratings.

'''8.5.4. Cross-Disciplinary Compatibility'''  
Is the theory consistent with findings from other scientific fields, such as geology, chemistry, biology, or planetary science? The more compatible it is, the higher the score.

'''8.5.5. Conceptual Clarity and Simplicity'''  
Is the theory logically simple and intuitively understandable, without unnecessary complexity? This criterion rewards elegance, not oversimplification.

'''8.5.6. Heuristic Value'''  
Does the theory stimulate new questions, research directions, or rethinking of existing problems? A high score reflects creative scientific potential.

'''8.5.7. Historical and Philosophical Insight'''  
Does the theory connect meaningfully to the historical development of science or reflect philosophical depth? Theories grounded in tradition or conceptual evolution are valued here.

'''8.5.8. Mathematical Rigor'''  
Does the theory provide clear mathematical formulations, derivations, and quantitative predictions? Theories are valued for their use of equations to express core principles, ensure internal consistency, and generate testable results.


Each theory receives a rating from ★☆☆☆☆ to ★★★★★ per criterion. The total score (max 40) gives a general measure of its scientific coherence and reach. The reasoning behind the scores is available under each theory (subsection 8.8) or can be requested in more detail.


=== '''8.6 – Comparative Table of AI Ratings (Updated with Criterion 8: Mathematical Rigor)''' ===

''Note: In July 2025, an eighth evaluation criterion was added: '''Mathematical Rigor'''. The total score is now out of 40 stars instead of 35.  ''
''All ratings are expressed in whole stars (★), without fractional values, to ensure clarity in display and consistency with the visual format of this table.''  
''Theories are listed in the same order as in Section 8.8.''

{| class=&quot;wikitable&quot;
! '''Theory''' !! '''EA''' !! '''IC''' !! '''PP''' !! '''CC''' !! '''CS''' !! '''HV''' !! '''HP''' !! '''MR''' !! '''Total (★/40)'''
|-
| '''8.8.1 General Relativity''' || ★★★★★ || ★★★★★ || ★★★★★ || ★★★★☆ || ★★★☆☆ || ★★★★★ || ★★★★★ || ★★★★★ || '''37'''
|-
| '''8.8.2 Newtonian Gravity''' || ★★★★☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || ★★★★★ || ★★★★☆ || ★★★★★ || ★★★★☆ || '''33'''
|-
| '''8.8.3 MOND''' || ★★★★☆ || ★★★☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || ★★★☆☆ || '''26'''
|-
| '''8.8.4 Emergent Gravity''' || ★★★☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || '''24'''
|-
| '''8.8.5 Big Bang''' || ★★★★☆ || ★★★★★ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || ★★★★★ || ★★★★☆ || ★★★★★ || '''34'''
|-
| '''8.8.6 Steady State Theory''' || ★★☆☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★★★☆ || ★★★☆☆ || ★★★★☆ || ★★★☆☆ || '''23'''
|-
| '''8.8.7 Big Crunch''' || ★★☆☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★★☆☆ || ★★★★☆ || ★★★☆☆ || '''22'''
|-
| '''8.8.8 Big Bounce''' || ★★☆☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || '''22'''
|-
| '''8.8.9 Cosmic Influx Theory (CIT)''' || ★★★★☆ || ★★★★☆ || ★★★★☆ || ★★★★☆ || ★★★★☆ || ★★★★★ || ★★★★☆ || ★★★★☆ || '''33'''
|-
| '''8.8.10 Spiral Cosmology''' || ★★☆☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★★★☆ || ★★★★☆ || ★★☆☆☆ || '''21'''
|-
| '''8.8.11 Topological Field Framework''' || ★★☆☆☆ || ★★★★☆ || ★★★☆☆ || ★★★★☆ || ★★★☆☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || '''27'''
|-
| '''8.8.12 Conformal Gravity''' || ★★★☆☆ || ★★★★☆ || ★★★☆☆ || ★★★☆☆ || ★★★☆☆ || ★★★★☆ || ★★★★☆ || ★★★★☆ || '''28'''
|-
| '''8.8.13 Quasi-Steady State Cosmology''' || ★★☆☆☆ || ★★★☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★★★☆ || ★★★★☆ || ★★☆☆☆ || '''23'''
|-
| '''8.8.14 Dynamic Universe''' || ★★★☆☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || ★★★★☆ || ★★★★☆ || ★★★★☆ || ★★★★☆ || '''30'''
|-
| '''8.8.15 Multiverse''' || ★★☆☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★☆☆☆ || ★★★★☆ || ★★★★☆ || ★★★☆☆ || '''23'''
|-
| '''8.8.16 Theory of Universality''' || ★☆☆☆☆ || ★★☆☆☆ || ★☆☆☆☆ || ★☆☆☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★★☆☆ || ★★☆☆☆ || '''16'''
|-
| '''8.8.17 Structured Loitering''' || ★★☆☆☆ || ★★☆☆☆ || ★☆☆☆☆ || ★★☆☆☆ || ★★☆☆☆ || ★★★☆☆ || ★★★☆☆ || ★☆☆☆☆ || '''16'''
|-
| '''8.8.18 Topological GeometroDynamics (TGD)''' || ★★☆☆☆ || ★★★★☆ || ★★☆☆☆ || ★★★★☆ || ★★☆☆☆ || ★★★★☆ || ★★★★☆ || ★★★★☆ || '''26'''


&lt;!-- put an empty line here --&gt;

|}

''Legend:''  
''EA = Empirical Adequacy IC = Internal Consistency PP = Predictive Power CC = Cross-Disciplinary Compatibility  ''
''CS = Conceptual Simplicity HV = Heuristic Value HP = Historical/Philosophical Insight MR = Mathematical Rigor''

== '''8.7 Requesting Detailed Justification''' ==

“Somebody” interested in the full motivation for a theory’s score—criterion by criterion—can obtain this by:

1. Visiting the '''discussion page''' of this Wikiversity chapter.
  
2. Mentioning the theory by name and explicitly asking for the detailed breakdown.
  
3. Receiving an AI-generated or authored comment elaborating why each star rating was assigned. 

These ratings are transparent and intended to support critical evaluation and ongoing refinement of both established and alternative cosmological theories. Constructive feedback, new data, or arguments may lead to adjustments in the scores, provided clear reasoning is presented.

If a contributor or author does not agree with the evaluation or prefers not to have their theory represented under this framework, they may request that all related content and ratings for that theory be removed. This ensures that participation remains voluntary and respectful of intellectual ownership.

'''Mainstream theories are subject to the same critical standard.'''  
If someone presents a reasoned objection to the current star ratings of a widely accepted theory (e.g. General Relativity, Big Bang), their explanation will be reviewed. If the argument is well-founded, ChatGPT may generate a revised evaluation. Both the original and alternative viewpoints can be documented transparently if needed.

If you want to submit a '''Rating Rebuttal''', please use the template provided on the '''[[Talk:AI-Assisted Evaluation of Cosmological Theories|Discussion page]]'''.

== '''8.8 Detailed Evaluations of Theories''' ==

This section will be expanded progressively as new theories are added. Each evaluation is generated in collaboration with ChatGPT, based on a standardized set of seven criteria defined in Chapter 1. This approach allows for clear, consistent, and rapid assessment of both mainstream and non-mainstream cosmological theories within minutes.

=== '''8.8.1 General Relativity – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/General_relativity General Relativity]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Matches a wide range of observations: gravitational lensing, GPS corrections, perihelion precession, black hole dynamics, and gravitational waves. Supported by multiple experiments. || ★★★★★
|-
| '''Internal Consistency''' || Highly consistent within its differential geometric framework. Built upon Einstein's field equations with tensor calculus. Few internal contradictions, though extensions (e.g. quantum gravity) face challenges. || ★★★★★
|-
| '''Predictive Power''' || Successfully predicted gravitational redshift, time dilation, frame-dragging, and gravitational waves. Continues to guide observations in astrophysics. || ★★★★★
|-
| '''Cross-Disciplinary Compatibility''' || Compatible with cosmology and astrophysics. Some tension with quantum theory. Less integrated with planetary geology or biology. || ★★★★☆
|-
| '''Conceptual Clarity and Simplicity''' || Complex mathematical structure makes it less intuitive. Conceptually abstract (spacetime curvature, geodesics). Clarity improves with education, but simplicity is low. || ★★★☆☆
|-
| '''Heuristic Value''' || Inspired vast developments in cosmology, black hole theory, and relativistic astrophysics. Foundation for modern gravitational physics. || ★★★★★
|-
| '''Historical and Philosophical Insight''' || Deep philosophical implications about space, time, and causality. Represents a major shift from Newtonian absolute space. Influenced 20th-century philosophy of science. || ★★★★★
|-
| '''Mathematical Rigor''' || Built upon advanced mathematics: Riemannian geometry, Einstein field equations, tensor calculus. Equations are precise, formal, and deeply embedded in differential geometry. || ★★★★★
|}

'''Total: 39/40'''

=== '''8.8.2 Newtonian Gravity – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation Newton's Law of Universal Gravitation]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Accurately describes gravitational interactions for most macroscopic systems (planets, satellites, projectiles) under low-speed, weak-field conditions. Deviates in extreme conditions (e.g. near black holes). || ★★★★☆
|-
| '''Internal Consistency''' || Mathematically self-consistent with inverse-square law and Newton's laws of motion. Assumes instantaneous action at a distance, which conflicts with relativity. || ★★★★☆
|-
| '''Predictive Power''' || Predicts planetary orbits, tides, escape velocities, and Keplerian motion. Fails for relativistic effects (e.g. Mercury’s precession, gravitational lensing). || ★★★★☆
|-
| '''Cross-Disciplinary Compatibility''' || Well-integrated in classical mechanics, astronomy, and engineering. Less compatible with modern cosmology or relativistic frameworks. || ★★★☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Simple and intuitive: gravity as a force between masses. Easily grasped and widely taught. || ★★★★★
|-
| '''Heuristic Value''' || Inspired centuries of scientific discovery and classical mechanics. Still used in teaching and engineering. Limited in modern theoretical development. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Founded classical physics. Major leap in scientific method and mathematical modeling. Paved the way for Enlightenment-era science. || ★★★★★
|-
| '''Mathematical Rigor''' || Clear and elegant use of calculus and vector algebra (e.g. \(\displaystyle F = G \frac{m_1 m_2}{r^2}\)). Highly accessible and historically groundbreaking, but lacks deeper geometric or relativistic structures. || ★★★★☆
|}

'''Total: 33/40'''

=== '''8.8.3 MOND (Modified Newtonian Dynamics) – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics Modified Newtonian Dynamics (MOND)]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Fits galactic rotation curves without invoking dark matter. Matches Tully-Fisher relation. Performance weakens at cluster and cosmological scales. || ★★★★☆
|-
| '''Internal Consistency''' || Original formulation is non-relativistic and phenomenological. Several relativistic extensions (e.g. TeVeS) exist but introduce complexity and fine-tuning. || ★★★☆☆
|-
| '''Predictive Power''' || Predicts galaxy dynamics from baryonic matter alone. Less predictive at larger scales or in non-galactic contexts. || ★★★☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Primarily astrophysical. Some tension with cosmology, structure formation, and gravitational lensing. Not aligned with particle physics. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Simple at galactic scale: modifies acceleration below a threshold \(a_0\). Extensions are less intuitive. || ★★★★☆
|-
| '''Heuristic Value''' || Challenges dark matter paradigm and motivates alternate gravity models. Sparked theoretical and observational debate. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Offers a conceptual challenge to Newton/Einstein gravity. Philosophically provocative, but limited historical lineage. || ★★★☆☆
|-
| '''Mathematical Rigor''' || Employs interpolating functions and modified Poisson equations. Relativistic extensions (e.g. TeVeS) involve tensor-vector-scalar frameworks. Rigorous in parts, but lacks unified formalism. || ★★★☆☆
|}

'''Total: 28/40'''

=== '''8.8.4 Emergent Gravity – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Entropic_gravity Emergent / Entropic Gravity]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Attempts to account for galactic rotation without dark matter by linking gravity to entropy and information. Some results match MOND-like behavior, but broad observational support remains limited. || ★★★☆☆
|-
| '''Internal Consistency''' || Theoretical framework draws from thermodynamics, holography, and information theory. Conceptually coherent, but not fully developed as a unified physical model. || ★★★☆☆
|-
| '''Predictive Power''' || Offers qualitative insights but lacks precise predictive capabilities in most contexts. No wide adoption for simulations or system modeling. || ★★☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Integrates ideas from black hole thermodynamics, quantum information, and spacetime geometry. Weak integration with observational astronomy or geology. || ★★★☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Abstract and speculative. Concepts like entropic forces and holographic screens are not intuitive for most readers. || ★★☆☆☆
|-
| '''Heuristic Value''' || Stimulates discussion about the nature of spacetime and gravity. Has inspired new theoretical directions in quantum gravity and information theory. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Philosophically intriguing: redefines gravity as emergent rather than fundamental. Links to ideas from Bekenstein and Hawking. || ★★★★☆
|-
| '''Mathematical Rigor''' || Relies on concepts from thermodynamics (e.g. entropy gradients), statistical mechanics, and quantum gravity. Uses integral relations and variational principles, but lacks a standardized set of equations for general use. || ★★★☆☆
|}

'''Total: 26/40'''

=== '''8.8.5 Big Bang – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Big_Bang Big Bang Theory]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Strongly supported by CMB, redshift-distance relation, and light element abundances. However, JWST data revealing early, mature galaxies challenges the predicted timeline of structure formation. || ★★★★☆
|-
| '''Internal Consistency''' || Internally coherent within the ΛCDM framework. Logical structure is intact, though singularity and quantum-gravity transitions remain unresolved. || ★★★★★
|-
| '''Predictive Power''' || Successfully predicted CMB and nucleosynthesis. But recent observations (e.g. JWST galaxies at z &gt; 10) were not anticipated, requiring post-hoc model adjustments. || ★★★★☆
|-
| '''Cross-Disciplinary Compatibility''' || Strong integration with general relativity, quantum field theory, and thermodynamics. Limited links to geology or planetary evolution. || ★★★★☆
|-
| '''Conceptual Clarity and Simplicity''' || Conceptually complex genesis: singularity, inflation, and multiple postulated fields (inflaton, dark energy, dark matter). Relies on hypothetical constructs. || ★★★☆☆
|-
| '''Heuristic Value''' || Stimulated massive observational and theoretical advances. Has driven decades of mission planning and cosmological interpretation. || ★★★★★
|-
| '''Historical and Philosophical Insight''' || Represented a paradigm shift from steady-state thinking. Raises deep ontological and metaphysical questions (e.g. time’s origin, creation ex nihilo). || ★★★★☆
|-
| '''Mathematical Rigor''' || Grounded in Einstein’s field equations, Friedmann dynamics, and thermodynamics. Offers well-defined, highly developed formalism. || ★★★★★
|}

'''Total: 34/40'''

=== '''8.8.6 Steady State Theory – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Steady_state_model Steady State Cosmology]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Early successes with radio‐source counts, but contradicted by the cosmic microwave background, evolving galaxy populations, and quasar statistics. || ★★☆☆☆
|-
| '''Internal Consistency''' || Mathematically self-consistent within the “perfect cosmological principle,” requiring continuous matter creation at a fixed rate. Logical but invokes an ad-hoc creation field (C-field). || ★★★☆☆
|-
| '''Predictive Power''' || Predicted constant large-scale density and specific radio‐source number counts; few successful novel predictions beyond its initial scope. || ★★☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Limited overlap with modern astrophysics and particle physics; conflicts with nucleosynthesis and CMB observations. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Intuitively simple: Universe is homogeneous in space **and** time, avoiding an initial singularity. Minimal parameter set. || ★★★★☆
|-
| '''Heuristic Value''' || Historically spurred observational tests that ultimately favored Big Bang models; now mainly of pedagogical interest. || ★★★☆☆
|-
| '''Historical and Philosophical Insight''' || Embodies the perfect cosmological principle and continuous-creation idea, provoking debates on temporality and cosmological assumptions. || ★★★★☆
|-
| '''Mathematical Rigor''' || Employs Friedmann-like solutions with a creation term; uses relativistic field equations but lacks the richer formal development of ΛCDM or GR extensions. || ★★★☆☆
|}

'''Total: 23/40'''

=== '''8.8.7 Big Crunch – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Big_Crunch Big Crunch]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Once considered viable if cosmic density exceeded the critical value. Current observations (accelerating expansion, dark energy) contradict its key assumptions. || ★★☆☆☆
|-
| '''Internal Consistency''' || Theoretically consistent as a time-reversed Big Bang within general relativity. Requires high matter density and no (or reversing) dark energy. || ★★★☆☆
|-
| '''Predictive Power''' || Predicts a decelerating expansion turning to collapse. Testable in principle, but not supported by current data. || ★★☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Compatible with thermodynamic and relativistic models of entropy and time symmetry, but unsupported by astronomical data. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Intuitively mirrors Big Bang, offering closure and symmetry. Simple in concept, but difficult to reconcile with observed acceleration. || ★★★☆☆
|-
| '''Heuristic Value''' || Motivated theoretical discussion on cosmological fate and cyclic models. Limited influence in current cosmology. || ★★★☆☆
|-
| '''Historical and Philosophical Insight''' || Explores cosmological temporality and finitude. Once a philosophically compelling counterbalance to eternal expansion. || ★★★★☆
|-
| '''Mathematical Rigor''' || Based on time-reversible solutions to Friedmann equations and relativistic cosmology. Rigorous within GR but not extended in modern frameworks. || ★★★☆☆
|}

'''Total: 23/40'''

=== '''8.8.8 Big Bounce – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Big_Bounce Big Bounce]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || No direct observational evidence yet. Some loop quantum cosmology models suggest signatures in the CMB, but these remain speculative. || ★★☆☆☆
|-
| '''Internal Consistency''' || Offers a logically coherent alternative to singularity-based models. Dependent on quantum gravity frameworks (e.g. loop quantum gravity) that remain under development. || ★★★☆☆
|-
| '''Predictive Power''' || Provides potential testable differences in early universe structure and CMB fluctuations. Predictions are still uncertain and model-dependent. || ★★☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Bridges general relativity with quantum mechanics. Limited overlap with geology or observational astronomy. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Complex and abstract. The idea of a cyclical universe is conceptually appealing, but quantum corrections are not intuitive. || ★★☆☆☆
|-
| '''Heuristic Value''' || Stimulates exploration of singularity resolution and quantum gravity cosmologies. Encourages investigation of pre-Big Bang conditions. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Resonates with ancient cyclical cosmologies and philosophical ideas of eternal recurrence. Reframes the question of origins. || ★★★★☆
|-
| '''Mathematical Rigor''' || Based on extensions of Friedmann equations using loop quantum corrections or other quantum gravity approaches. Some models are mathematically formal, but the field is still unsettled. || ★★★☆☆
|}

'''Total: 24/40'''

=== '''8.8.9 Cosmic Influx Theory (CIT) – AI Rating Summary''' ===

''Related link:'' [https://en.wikiversity.org/wiki/Cosmic_Influx_Theory Cosmic Influx Theory]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Strong match with observed planetary structuring, VRMS-based system modeling, and geological trends like daylength and expansion. Supported by exoplanet data and disk morphology (e.g. HD 163296). || ★★★★☆
|-
| '''Internal Consistency''' || Equations and constants (e.g. κ₍CIT₎, (γ−1)/4π) are logically coherent. Internal derivations remain consistent across cosmological and planetary domains. || ★★★★☆
|-
| '''Predictive Power''' || Offers specific predictions (e.g. Trappist-1 preferred distance and orbital period). Some predictions still await observational confirmation. || ★★★★☆
|-
| '''Cross-Disciplinary Compatibility''' || Integrates cosmology, geology, biology, and observational astronomy. Compatible with expanding Earth, daylength data, and ring formation physics. || ★★★★☆
|-
| '''Conceptual Clarity and Simplicity''' || Introduces new yet intuitive ideas like influx and preferred distances. Avoids abstract constructs like dark matter/energy. || ★★★★☆
|-
| '''Heuristic Value''' || Inspires re-evaluation of mainstream assumptions, links to overlooked or discarded theories (e.g. Le Sage, expansion tectonics). || ★★★★★
|-
| '''Historical and Philosophical Insight''' || Reconnects with early gravitational push models and continuous creation ideas, offering philosophical alternatives to entropy-based models. || ★★★★☆
|-
| '''Mathematical Rigor''' || Provides original equations (e.g. for \(D_{\text{pref}}\), \(G = (\gamma - 1)/4\pi\), \(\kappa = v_{\text{RMS}}^2 / c^4\)) and consistent dimensional analysis. Excel-based datasets link math to observations. Lacks field-theoretic formalism. || ★★★★☆
|}

'''Total: 34/40'''

=== '''8.8.10 Spiral Cosmology – AI Rating Summary''' ===

''Related link:'' [https://www.academia.edu/103005946/Cosmology_as_Spiral_Evolution Spiral Cosmology on Academia.edu]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Offers visual and structural explanations for spiral galaxy morphology and cosmic rotation patterns. Less directly tied to quantitative data or tested predictions. || ★★☆☆☆
|-
| '''Internal Consistency''' || Conceptually consistent in proposing self-similar spiral evolution at multiple scales, but lacks a developed dynamic or energetic framework. || ★★☆☆☆
|-
| '''Predictive Power''' || Suggests qualitative evolutionary stages and possible cyclic features, but does not provide specific numerical predictions. || ★★☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Makes symbolic and philosophical connections across cosmology, biology, and culture. Scientific integration with physical fields is minimal. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Emphasizes intuitive and visual structures (e.g. spirals), which are accessible but may oversimplify physical complexity. || ★★★☆☆
|-
| '''Heuristic Value''' || Stimulates reflection on cosmic structure, symmetry, and recursion. Encourages reinterpretation of known forms (e.g. galaxies, DNA, hurricanes). || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Reconnects with ancient and Renaissance cosmologies linking form and function across scales. Offers metaphysical resonance. || ★★★★☆
|-
| '''Mathematical Rigor''' || Utilizes geometric symbolism (e.g. spiral ratios, golden mean) but lacks physical equations or dynamical systems modeling. No quantitative derivations. || ★★☆☆☆
|}

'''Total: 23/40'''

=== '''8.8.11 Topological Field Framework – AI Rating Summary''' ===

''Related link:'' [https://www.researchgate.net/publication/393122856_A_Topological_Field_Framework_for_Particle_Mass_Gauge_Interactions_and_Emergent_Gravity A Topological Field Framework on ResearchGate]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Proposes a geometric origin of particle properties and fundamental constants, but lacks direct comparison with observational or experimental datasets. Empirical testing is suggested but not yet demonstrated. || ★★☆☆☆
|-
| '''Internal Consistency''' || Builds a coherent and logically structured model grounded in topology and pressure gradients. Concepts are well integrated, with minimal contradictions. || ★★★★☆
|-
| '''Predictive Power''' || Offers a pathway to derive constants such as G, c, and Λ from topology and boundary conditions. However, quantitative predictions are still under development or pending validation. || ★★★☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Bridges gravitational theory, particle physics, and topology. Aligns with ideas in quantum geometry, though not yet embedded in mainstream formulations. || ★★★★☆
|-
| '''Conceptual Clarity and Simplicity''' || Dense but conceptually focused. Uses field and pressure analogies to unify gravity and gauge interactions, though the abstract nature may limit accessibility. || ★★★☆☆
|-
| '''Heuristic Value''' || Provides an original and stimulating approach to unify physical constants via geometry. Encourages rethinking of foundational assumptions in physics. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Revives geometrical unification ideals from early 20th-century physics and connects them to modern field-based ontology. Philosophically grounded. || ★★★★☆
|-
| '''Mathematical Rigor''' || Employs differential geometry, topological mapping, and field pressure modeling. Mathematical structure is present, but derivations are at a conceptual stage. || ★★★☆☆
|}

'''Total: 27/40'''

=== '''8.8.12 Conformal Gravity – AI Rating Summary''' ===

''Related link:'' [https://ui.adsabs.harvard.edu/abs/1989ApJ...342..635M Exact Vacuum Solution to Conformal Weyl Gravity and Galactic Rotation Curves]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Successfully models galactic rotation curves and lensing effects without invoking dark matter. However, challenges remain regarding early-universe phenomena and the CMB. || ★★★☆☆
|-
| '''Internal Consistency''' || Mathematically self-consistent within its conformal symmetry framework. Gravitational dynamics are derived cleanly from a fourth-order field equation. || ★★★★☆
|-
| '''Predictive Power''' || Accurately predicts galaxy-scale observations. However, extrapolations to cosmological scales require further development and empirical testing. || ★★★☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Intersects with quantum field theory through conformal invariance and offers alternatives to ΛCDM. Limited overlap with standard model particle physics. || ★★★☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Builds from a well-defined symmetry principle (conformal invariance). Some complexity arises due to fourth-order derivatives and unfamiliar constructs. || ★★★☆☆
|-
| '''Heuristic Value''' || Encourages reconsideration of gravitational assumptions and dark matter. Inspires new theoretical directions and alternative metrics. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Connects with earlier Weyl geometry and symmetry-based models. Philosophically significant as a symmetry-driven alternative to general relativity. || ★★★★☆
|-
| '''Mathematical Rigor''' || Employs rigorous derivations based on conformal symmetry and higher-order field equations. Technically sophisticated. || ★★★★☆
|}

'''Total: 28/40'''

=== '''8.8.13 Quasi-Steady State Cosmology (QSSC) – AI Rating Summary''' ===

''Related link:'' [https://www.ias.ac.in/article/fulltext/pram/053/06/1093-1104 The Quasi-Steady State Cosmology: Theory and Observations – Pramana Journal]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Offers explanations for redshift, large-scale structure, and quasar distributions. However, it faces difficulties matching the observed CMB spectrum and primordial element abundances. || ★★☆☆☆
|-
| '''Internal Consistency''' || Theoretical foundations are logically constructed around a C-field and creation events. Internally coherent but relies on non-standard mechanisms not universally accepted. || ★★★☆☆
|-
| '''Predictive Power''' || Makes unique predictions about cosmic cycles, matter creation, and galaxy evolution. Some predictions remain qualitative or are difficult to test. || ★★★☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Attempts to integrate cosmology with continuous creation physics. However, the C-field concept is not aligned with standard field theory or particle physics. || ★★☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Proposes an intuitive cyclical model of cosmic evolution. Some components, such as the C-field, are abstract and complex to formalize. || ★★★☆☆
|-
| '''Heuristic Value''' || Offers a provocative alternative to the Big Bang. Stimulates re-evaluation of singularity-based models and encourages cyclic interpretations. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Extends the steady-state philosophy of Hoyle. Challenges the singular origin narrative with philosophical depth and continuity. || ★★★★☆
|-
| '''Mathematical Rigor''' || Uses mathematical models for cyclic expansion and C-field dynamics. However, these are not widely adopted or fully developed in mainstream literature. || ★★☆☆☆
|}

'''Total: 23/40'''

=== '''8.8.14 Dynamic Universe (Tuomo Suntola) – AI Rating Summary''' ===

''Related link:'' [https://physicsfoundations.org/suntola/dynamic-universe Dynamic Universe – Physics Foundations Society]  
''Additional source:'' [https://www.academia.edu/37149633/The_Dynamic_Universe_Toward_a_unified_picture_of_physical_reality The Dynamic Universe on Academia.edu]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Provides testable predictions for cosmological redshift, Hubble parameter, and time dilation without invoking dark energy. Several predictions align well with observational data, though not yet widely confirmed. || ★★★☆☆
|-
| '''Internal Consistency''' || The model is highly self-consistent, based on a single zero-energy principle and evolving 4-sphere geometry. Internally coherent and logically derived. || ★★★★☆
|-
| '''Predictive Power''' || Predicts cosmological parameters from first principles, including a time-evolving H₀ and expansion behavior. Some predictions differ from ΛCDM but remain observationally accessible. || ★★★★☆
|-
| '''Cross-Disciplinary Compatibility''' || Attempts unification of relativity, cosmology, and quantum phenomena under a geometric framework. Still under integration with conventional physics models. || ★★★☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Built on a single geometric principle with minimal assumptions. Clear in its physical logic, though unfamiliar to those trained in standard GR or QFT. || ★★★★☆
|-
| '''Heuristic Value''' || Inspires a rethinking of time, space, and energy conservation. Offers a conceptually elegant reformulation of cosmic dynamics. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Echoes Machian ideas and Einstein’s early search for balance models. Challenges the notion of spacetime curvature as fundamental. || ★★★★☆
|-
| '''Mathematical Rigor''' || Employs rigorous geometric derivations and differential equations. Mathematical structure is well-developed, though outside conventional formalisms. || ★★★★☆
|}

'''Total: 30/40'''


=== '''8.8.15 Multiverse Theory – AI Rating Summary''' ===

''Related link:'' [https://en.wikipedia.org/wiki/Multiverse Multiverse – Wikipedia]  
''Additional source:'' [https://www.scientificamerican.com/article/does-the-multiverse-really-exist/ Scientific American – Does the Multiverse Really Exist?]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Lacks direct observational support. No known experiment can confirm or refute the existence of other universes. Some indirect inferences arise from inflationary models or quantum theory, but none are conclusive. || ★★☆☆☆
|-
| '''Internal Consistency''' || Varies by formulation. Quantum Many-Worlds, eternal inflation, and string theory landscapes are logically coherent within their frameworks, but depend on assumptions not testable in this universe. || ★★★☆☆
|-
| '''Predictive Power''' || Extremely weak. Most formulations make no concrete predictions for our universe that differ from single-universe models. Some argue it &quot;predicts&quot; everything, making it unfalsifiable. || ★★☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Connects to quantum theory, inflationary cosmology, string theory, and philosophy. However, lacks integration with empirical fields like astronomy, geology, or planetary science. || ★★★☆☆
|-
| '''Conceptual Clarity and Simplicity''' || Conceptually speculative and often confusing. Raises questions about what counts as &quot;real&quot; and how probability works across unobservable universes. Not parsimonious. || ★★☆☆☆
|-
| '''Heuristic Value''' || High inspiration in theoretical physics and cosmology. Stimulates debate on fine-tuning, origins, and the limits of science. Has generated many models and new lines of thought. || ★★★★☆
|-
| '''Historical and Philosophical Insight''' || Reflects an evolution of thinking about cosmological plurality. Raises deep metaphysical questions about causality, determinism, and reality. || ★★★★☆
|-
| '''Mathematical Rigor''' || Strong in some versions (e.g. string theory landscapes, quantum branches), but speculative assumptions extend beyond tested mathematics. || ★★★☆☆
|}

'''Total: 23/40'''

=== '''8.8.16 Theory of Universality – AI Rating Summary''' ===

''Primary source:'' [https://www.iosrjournals.org/iosr-jap/papers/Vol11-issue2/Series-3/D1102031953.pdf K.S. Narayana – Theory of Universality (IOSR-JAP, 2019)]

{| class=&quot;wikitable&quot;
! '''Criterion''' !! '''Description''' !! '''Rating'''
|-
| '''Empirical Adequacy''' || Some known constants (e.g. Hubble parameter, proton lifetime) are reproduced numerically, but without grounding in observational datasets or statistical validation. No empirical testing methodology is provided. || ★☆☆☆☆
|-
| '''Internal Consistency''' || The derivations follow internal logic, but some assumptions—like matter exceeding light speed or redefining constants—conflict with established physics and dimensional coherence. || ★★☆☆☆
|-
| '''Predictive Power''' || No distinct or testable predictions are formulated. The theory remains mostly retrospective in deriving values, not prospective in forecasting new results. || ★☆☆☆☆
|-
| '''Cross-Disciplinary Compatibility''' || Conflicts with key principles in general relativity, quantum theory, and thermodynamics. It does not integrate with empirical fields or build bridges across scientific domains. || ★☆☆☆☆
|-
| '''Conceptual Clarity and Simplicity''' || The theory aims at unification but uses complex constructs with speculative interpretations. Lacks parsimony and clear explanatory logic. || ★★☆☆☆
|-
| '''Heuristic Value''' || Inspires rethinking of fundamental constants and offers alternative pathways to explore force unification. Useful for stimulating discussion despite lack of empirical grounding. || ★★★☆☆
|-
| '''Historical and Philosophical Insight''' || Emphasizes classical ideals of universal unity. Raises metaphysical questions about origins and physical laws. Philosophically bold, though scientifically unorthodox. || ★★★☆☆
|-
| '''Mathematical Rigor''' || Includes numerous equations and derivations, but lacks clarity in notation, dimensional checks, and formal structure. Relies on fitting numerical coincidences. || ★★☆☆☆
|}

'''Total: 16/40'''

=== '''8.8.18 Topological GeometroDynamics (TGD) – AI Rating Summary''' ===
''Primary sources:''
[https://tgdtheory.fi/public_html/articles/TGDtheory2024I.pdf TGD as it is towards end of 2024 – Part I]
[https://tgdtheory.fi/public_html/articles/TGDtheory2024II.pdf TGD as it is towards end of 2024 – Part II]

&lt;table class=&quot;wikitable&quot;&gt; &lt;tr&gt; &lt;th&gt;'''Criterion'''&lt;/th&gt; &lt;th&gt;'''Description'''&lt;/th&gt; &lt;th&gt;'''Rating'''&lt;/th&gt; &lt;/tr&gt; &lt;tr&gt; &lt;td&gt;'''Empirical Adequacy'''&lt;/td&gt; &lt;td&gt;TGD is rich in theoretical proposals—such as predictions about coupling constants, p-adic physics, and quantized criticality—but lacks direct empirical validation. Claims are often qualitative or philosophical, with few testable predictions implemented or tested.&lt;/td&gt; &lt;td&gt;★★☆☆☆&lt;/td&gt; &lt;/tr&gt; &lt;tr&gt; &lt;td&gt;'''Internal Consistency'''&lt;/td&gt; &lt;td&gt;The theory presents a highly consistent internal structure based on an 8-dimensional embedding space (M⁴ × CP₂), number theory, and a unified view of interactions. The coherence across domains (cosmology, quantum physics, biology) is impressive.&lt;/td&gt; &lt;td&gt;★★★★☆&lt;/td&gt; &lt;/tr&gt; &lt;tr&gt; &lt;td&gt;'''Predictive Power'''&lt;/td&gt; &lt;td&gt;TGD makes bold theoretical predictions about particle masses, cosmological evolution, and even consciousness; however, these are often indirect, qualitative, or mathematically untested in standard frameworks.&lt;/td&gt; &lt;td&gt;★★☆☆☆&lt;/td&gt; &lt;/tr&gt; &lt;tr&gt; &lt;td&gt;'''Cross-Disciplinary Compatibility'''&lt;/td&gt; &lt;td&gt;Integrates gravitation, particle physics, cosmology, biology, and neuroscience under a single geometric and number-theoretical framework. Strong interdisciplinary ambition, though not well integrated with conventional theories in these domains.&lt;/td&gt; &lt;td&gt;★★★★☆&lt;/td&gt; &lt;/tr&gt; &lt;tr&gt; &lt;td&gt;'''Conceptual Clarity and Simplicity'''&lt;/td&gt; &lt;td&gt;Conceptually intricate and difficult for non-specialists. TGD proposes radically new metaphysical and mathematical foundations that are not easily intuitive or accessible.&lt;/td&gt; &lt;td&gt;★★☆☆☆&lt;/td&gt; &lt;/tr&gt; &lt;tr&gt; &lt;td&gt;'''Heuristic Value'''&lt;/td&gt; &lt;td&gt;Inspires novel questions about the unification of forces, the role of number theory, and consciousness in physics. Has stimulated a wide body of internal development and speculative exploration.&lt;/td&gt; &lt;td&gt;★★★★☆&lt;/td&gt; &lt;/tr&gt; &lt;tr&gt; &lt;td&gt;'''Historical and Philosophical Insight'''&lt;/td&gt; &lt;td&gt;Offers a deep philosophical stance on geometry, consciousness, time, and mathematical existence. Rich in insight about foundational structures and conceptual shifts in physics.&lt;/td&gt; &lt;td&gt;★★★★☆&lt;/td&gt; &lt;/tr&gt; &lt;tr&gt; &lt;td&gt;'''Mathematical Rigor'''&lt;/td&gt; &lt;td&gt;Employs advanced mathematics: higher-dimensional manifolds, p-adics, number theory, and embedding techniques. Formal structure is evident but not presented using conventional field-theoretic rigor or peer-reviewed derivations.&lt;/td&gt; &lt;td&gt;★★★★☆&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt;
'''Total: 26/40'''

== '''8.9 Proposed theories that will be evaluated soon''' ==

- TGD from Matti Pitkänen 

'''◀ [[AI-Assisted_Evaluation_of_Cosmological_Theories/Chapter_7:_Cosmological_Parameters_and_Universal_Constants|Previous]] | [[AI-Assisted_Evaluation_of_Cosmological_Theories|Main Page]] | [[AI-Assisted_Evaluation_of_Cosmological_Theories/Chapter_9:_Documentation_and_Transparency|Next ▶]]'''</text>
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    <title>Social Victorians/Pleydell-Bouverie Skelmersdale Wedding</title>
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      <text bytes="27478" sha1="tnqsg76a96wdvgg5r37h9ctljlfdo1w" xml:space="preserve">=Event=
Wedding of Lady Wilma Pleydell-Bouverie and Lord Skelmersdale

==Overview==

==Logistics==
* 15 August 1889, Thursday
===Staff and Vendors===
*The Hon. and Rev. Bertrand Pleydell-Bouverie, rector of Pewsey, Wilts, uncle of thy bride, assisted by the Hon. and Rev. Lord William Cecil (brother-in-law of the bridegroom), of Bishops Hatfield, Herts., conducted the ceremony.

==Related Events==
* Funerals on both sides of this wedding, so the people invited were &quot;the nearest relatives and most intimate friends&quot;
* Reception hosted by the Earl and Countess of Radnor, Longford Castle (bride's parents)
* Honeymoon at &quot;Pewsey Rectory, the residence of the bride’s uncle&quot;
==Who Was Present==
===Bride and Bridesmaids===
====Bride====
Lady Wilma's wedding dress&lt;blockquote&gt;was of cream-white satin duchesse, trimmed with handsome Brussels lace flounces, which, along with he Brussels lace veil, were the gifts of her mother. Her veil was arranged over a small spray of orange flowers, and fastened by diamond ornaments.&lt;/blockquote&gt;Lady Wilma's bouquet&lt;blockquote&gt;of exquisite blooms was the gift of the non-commissioned officers of Lord Skelmersdale’s troop.&lt;/blockquote&gt;Lady Wilma's going-away dress was&lt;blockquote&gt;grey and white cloth trimmed with grey and silver passementerie, and Virot hat to match. [Madame Virot was &quot;one of the two most famous hat designers in Paris during the second half of the 19th century.&quot;&lt;ref&gt;{{Cite journal|date=2025-06-30|title=Madame Virot|url=https://en.wikipedia.org/w/index.php?title=Madame_Virot&amp;oldid=1298040198|journal=Wikipedia|language=en}}&lt;/ref&gt;)&lt;/blockquote&gt;

====Bridesmaids====

# Lady Maud Bootle Wilbraham (sister of the groom)
# Lady Mary Edith Bootle Wilbraham (sister of the groom)
# Miss Florence Hankey
# Miss Kathleen Hankey
# Lady Evelyn Parker
# The Hon. Winifred Sturt

The bridesmaids wore&lt;blockquote&gt;cream-white costumes, trimmed with gold embroidery and chiffons, and Virot hats to match. The bridegroom's gifts were a handsome brooch in the form of a moonstone heart set round with diamonds, surmounted with a diamond coronet, and “goodyears” bouquet of choice red roses in foliage.&lt;/blockquote&gt;

===Groom and Best Man===

* Captain the Hon. Lionel Byng, Royal Horse Guards, best man

===People Who Attended===

# The Hon. and Rev. Beatrice Pleydell-Bouverie
# Mr. and Mrs. Frank Harris (could this be the writer?)
==What People Wore==
# 
==Gifts==
===Tenants, Employees and Local Organizations===

==== For the Bride ====

* 140 Members of the Ladies String Band and Chorus — Diamond feather.
* Miss Pincoff and her little Pupils — Photograph in frame.
* Old Family Servants — Silver toilet set.
* Under Gardeners at Longford Castle — Glass epergne with hanging baskets and mirrored base.
* Employés on the Longford Estate — Black marble clock.
* The Tenants on the Longford Estate — Gold bracelet with five large diamonds set round.
* The Mayor and Corporation of Folkestone — Silver inkstand.
* The Farm Tenants on the Folkestone Estate — Silver inkbottle in silver plate.

==== For the Groom ====

* The Lathom Household and Workpeople on the Lathom Estate — Silver salver.
* Dalton Farm Workmen — Silver and ivory paper cutter.
* The Lathom Schools — Silver sauce boat.
* The Lathom Almswomen — Silver muffineers.
* The Tenants on the Lathom Estate — Large silver gilt bowl on ebony stand.
* Officers of the Royal Horse Guards — Large two-handled silver salver, with inscription.

===Unusual or Interesting Gifts===

* Several pictures, prints or paintings, but not very many: Three prints in France, Picture, Painting in frame, Crayon drawing of Lady Radnor, Water colour sketch
* Books: Silver Prayer Book, Silver-bound Prayer Book, ‘‘Follies and Fashions of our Grandfathers,” and Dickens’s &quot;Christmas Carols” [sic], Book, Book, Tennyson's poems
* Black lace flounces
* Two framed autotypes
* George III. cheese scoop
* Old gold and agati snuff box
* Old Canterbury cross
* Old arabic clasp
* Worked cushion and chair backs
* Silver coffee-pot and carved cocoa-nut mounted in silver
* Silver and bog quaigh [shallow 2-handle cup, Scots]
* Silver mug penwiper
* Tobacco pouch

===Furniture===

* Painted and worked screen, Worked fire screen
* Small inlaid table, lnlaid table, Old Dutch card table, Nest of tables, Dining-room table
* Marqueterie china cabinet
* Carved oak bracket cupboard with engraved brass plaques of Longford Castle
* Bureau, Bureau
* Silver and glass liqueur stand
* Arm-chair

===Brand Names===

* Bottle and tray (Benares work [— Benares ware is from what is now Varanasi, India])
* Spode china jar
* Chippendale table
* Two Downton chairs
* Valoris jar [probably Vallauris?]

==Anthology==
From the ''Lady's Pictorial'':&lt;blockquote&gt;On Thursday, the 15th inst., at Britford Church, near Longford Castle, Salisbury, the marriage took place of Lord Skelmersdale, of the Royal Horse Guards, eldest son of the Right Hon. the Earl of Lathom, and Lady Wilma Pleydell-Bouverie, only daughter of the Earl of Radnor. The bride was given away by her father, and Captain the Hon. Lionel Byng, Royal Horse Guards, acted as best man. The service was fully choral. The ceremony was performed by the Hon. and Rev. Bertrand Pleydell-Bouverie, rector of Pewsey, Wilts, uncle of thy bride, assisted by the Hon. and Rev. Lord William Cecil (brother-in-law of the bridegroom), of Bishops Hatfield, Herts. The bride's dress was of cream-white satin duchesse, trimmed with handsome Brussels lace flounces, which, along with he Brussels lace veil, were the gifts of her mother. Her veil was arranged over a small spray of orange flowers, and fastened by diamond ornaments. The six bridesmaids were the Ladies Maud and Mary Edith Bootle Wilbraham, sisters of the bridegroom; Misses Florence and Kathleen Hankey, Lady Evelyn Parker, and the Hon. Winifred Sturt. They were attired in cream-white costumes, trimmed with gold embroidery and chiffons, and Virot hats to match. The bridegroom's gifts were a handsome brooch in the form of a moonstone heart set round with diamonds, surmounted with a diamond coronet, and “goodyears” bouquet of choice red roses in foliage. The bridal bouquet of exquisite blooms was the gift of the non-commissioned officers of Lord Skelmersdale’s troop. Owing to both families being in mourning, none but the nearest relatives and most intimate friends were invited to the ceremony, and were afterwards received by the Earl and Countess of Radnor at Longford Castle. In the afternoon Lord and Lady Skelmersdale left for Pewsey Rectory, the residence of the bride’s uncle, where the honeymoon will be spent, the going-away dress being grey and white cloth trimmed with grey and silver passementerie, and Virot hat to match. The numerous presents included —&lt;p&gt;

From the Prince and Princess of Wales, a diamond and catseye bangle; the Duchess of Fife and the Princesses Victoria and Maud of Wales, diamond and moonstone brooch; Lord Skelmersdale, half hoop diamond ring, pearl and diamond ring, diawond and ruby safety brooch, diamond aigrette, silver card case, and silver model violn; the Earl and Countess of Radnor, diamond and and tiara, moonstone and diamond brooch, large single stone diamond St. And ew’s [Andrew’s] Cross brooch, wedding veil and flounces of Brussels lace, and despatch bag: the Earl of Lathom, large diamond star; Countess of Lathom, emerald and diamond necklet, emerald and pearl bracelet, emerald and diamond bangle and sapphire half-hoop ring; Viscount Folkestone, blue enamel pearl and diamond brooch, with blue enamel diamond watch pendant; Hon. Stuart Pleydell-Bouverie, pearl bonnet pins, water-colour sketch, and small silver frame; the Ladies Bootle Wilbraham, the Hon Randle [?] and the Hon. Reginald Bootle Wilbraham, pearl and coral ring; Right Hon. Henry Chaplin, M.P., large diamond pendant; the Hon. Mr and Mrs. Duncombe[,] R. Bouverie, diamond and pearl spray; Earl and Countess Stanhope, silver and crocodile leather clock; Countess of Macclestield, china coffee set; Countess of Crawford, five o'clock tea service; Lord and Lady William Nevill, tortoiseshell and gold paper knife; the Mayor and Corporatlom of Folkestone, silver inkstand; tenants on the Longford estate, gold bracelet set round with five large diamonds; employés on the Longford estate, black [?] marble clock; old family servants, silver toilet set; farm tenants on the Folkestone estate, silver inkbottle in silver plate; the under-gardeners of Longford estate, glass flower stand; Marquis and Marchioness of Salisbury, silver candlesticks; Lady Ampthill, silver teapot, cream jug, and sugar basin; Earl and Countess of Derby, silver teapot, kettle, and stand; Earl of Morley, ivory and silver paper cutter; Sir Algernon and Lady Borthwick, palr of glass liqueur bottles with silver necks; Viscount and Viscountess Wolmer, silver mounted matchbox; Sir Francis and Lady Montefiore, small silver tray; Viscount and Viscountess Cranborne, stick, &amp;c.&lt;ref&gt;&quot;Skelmersdale — Pleydell-Bouverie.&quot; ''Lady's Pictorial'' 24 August 1889, Saturday: 30 [of 52], Col. 1a–b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005980/18890824/106/0030. Same print title, p. 264.&lt;/ref&gt;&lt;/blockquote&gt;From the ''Wiltshire County Mirror'':&lt;p&gt;

The list of gifts is set as an unordered list here instead of as separate paragraphs, as it is in the original text, to save space and clarify.&lt;blockquote&gt;As we were unable last week to obtain a full list of the wedding presents sent to Lord Skelmersdale and the Lady Wilma Pleydell-Bouverie, we now produce it as follows: —

* Their Royal Highnesses The Prince and Princess of Wales — Diamond and cat’s-eye bangle.
* Their Roval Highnesses the Duchess of Fife and the Princesses Victoria and Maud of Wales — Diamond and moonstone brooch.
* Lord Skelmersdale — Half-hoop diamond ring, pearl and diamond ring, diamond and ruby safety brooch, diamond aigrette, silver card case, and silver model violin.
* The Earl and Countess of Radnor — Diamond and pearl tiara.
* The Countess of Radnor — Moonstone and diamond brooch, large single stone diamond St. Andrew's Cross brooch, wedding veil and flounces of Brussels lace, and dispatch box.
* The Earl of Lathom — Large diamond star.
* The Countess of Lathom — Emerald and diamond necklet, emerald and pearl bracelet, emerald and diamond bangle, and sapphire half-hoop ring.
* Viscount Folkestone — Blue enamel and pearl diamond brooch, with blue enamel and diamond watch pendant.
* The Hon. Stuart Bouverie — Pearl bonnet pins, water-coloured sketch, and small silver frame.
* The Ladies Wilbraham and the Hon. Randle and the Hon. Reginald Wilbraham  — Pearl and coral ring.
* The Right Hon. Henry Chapin, M.P. — Large diamond pendant.
* The Hon. Mr. and Mrs. Duncombe P. Bouverie — Diamond and pearl spray.
* Lady June Ellice — Half-hoop emerald ring and three Dresden China baskets.
* Mr. Ellice — Sappire and diamond ring and cheque for £100.
* Colonel and Lady Margaret Maitland —Pearl half-hoop ring.
* Lady Anne Speirs — Moonstone and diamond brooch.
* Mr. and Mrs. Ernest Chaplin — Diamond crescent.
* Mr. and Lady Edith Cotes — Cat's-eye and diamond bracelet.
* The Right Hon. Mr. and Hon. Mrs. Edward P. Bouverie —Topaz and pearl bracelet.
* Mr. and Mrs. Cecil Chaplin — Diamond marguerite brooch.
* The Hon. Kenelm P. Bouverie — Diamond and moonstone bangle.
* Lord Alington — Diamond horse-shoe brooch.
* Lady Jane Von Koughuet — Gold pencil bangle.
* Mr. Charles Clifford — Gold and lapislazale [? lapis lazuli] brooch. 
* Miss Florence and Miss Kathleen S. Hankey — Double-heart pearl brooch.
* Lady Harriet Wentworth — Diamond moonstone and pearl brooch.
* Mrs. Bradford Atkinson—Diamond and ruby button brooch.
* Sir Thomas Lucas — Diamond bracelet.
* Lady Caroline Kerrison — Pearl and diamond horse-shoe brooch.
* Lady Cecil Scott Montague and Lady Margaret Kerr — Pearl brooch.
* Lady Revelstoke — Diamond lyre brooch.
* Mr. Charles Rogers — Starstone and diamond safety pin brooch.
* Miss Chappell — Diamond ‘‘merrythought&quot; brooch.
* Miss Emily Mildmay — Moonstone and diamond star brooch.
* Mr. and Mrs. Mildmay — Diamond and sapphire brooch.
* Mr. Philip H. Coxe — Pearl and diamond bangle.
* Miss Mabel Sands — Torquoise [sic] brooch.
* Mr. and Mrs. W. H. Leslie — Diamond and sapphire bracelet.
* Mr. and Mrs. Coleridge Kennard — Pearl and diamond “Swan&quot; brooch.
* Mr. and Lady Emily Van de Weyer — Pearl and diamond and sapphire and diamond bracelets.
* 140 Members of the Ladies String Band and Chorus — Diamond feather.
* Mr. Westrow Hulse — Diamond and pearl cleff brooch.
* Mr. and Mrs. Arthur Walter — Gold chain bracelet.
* Lady Gertrude and the Hon. Christopher Pleydell-Bouverie — Silver and inlaid wood inkstand.
* The Hon. and Rev. Beatrice and Lady Constance Pleydell-Bouverie — White China dessert service and velvet and silver-framed mirror.
* The Hon. Mark Pleydell-Bouverie — Silver hand glass.
* The Earl of Verulom — Silver hand candlestick.
* The Dowager Countess of Craven — Pair of silver candlesticks.
* The Earl and Countess Cadogan — Three large silver sugar sifters.
* The Marchioness of Bath — Silver basket.
* The Marchioness of Lothian — Two silver trays.
* Lady Mary Hervey — Silver smelling bottle.
* Mrs. Frank Harris — Silver frame.
* Mr. Frank Harris — Silver hair-pin box.
* The Countess of Haddington — Silver smelling bottle.
* The Countess of Hopetown [Hopetoun?] — Two silver frames.
* Lady Evelyn Parker — Silver and glass bottle and picture.
* Lady Eleanor Heneage — Small silver cruet stand.
* Lady Nina Balfour — Silver match-box case.
* Mr. Leo Frank Schuster — Silver basket.
* Lady Mary Parker — Ivory and silver paper cutter.
* Sir Edward and Lady Hulse —Silver bowl.
* The Earl and Countess of Stanhope — Silver and crocodile leather clock.
* Mr. Adam Kennard — Silver and mother-o’-pearl opera glasses with handle.
* Mr. Alan MacKinnow — Small silver violin.
* Mr. Ward Cook — Silver basin.
* Miss Esther Maitland — Silver pin tray.
* Mrs. William Whitbread — Silver tray.
* Mr. Hamilton Hulse — Old silver box.
* Lady Ventry — Silver pen and pencil.
* Mr. and Mrs. Charles Tudway — Silver mustard pot.
* Mrs. Sidmouth Skipworth — Silver heart-shaped box.
* Mr. Reginald Skipworth — Silver box.
* Mr. Henry Newton — Silver lozenge box.
* Mr. and Mrs. William Malcolm — Silver bowl.
* The Hon. Eda Campbell — Old Dutch silver boat.
* Mr. and Mrs. Bowles — Silver bottle and tray (Benares work [— Benares ware is from what is now Varanasi, India]).
* The Hon. Lady and Miss Filmer — Cut glass bottle with silver top.
* Miss Lauman — Silver Prayer Book.
* Mr. J. E. Martin (60th Rifles) — Silver button-hook and shoe-horn.
* The Misses M. and B. Kennard — Small round silver box.
* Mr. E. O. Pleydell-Bouverie — Silver inkstand.
* Mrs. C. B. Ewart — Silver hand mirror.
* Mr. Ernest Albert Sandeman — Silver box.
* Miss Mabel Gage — Silver scent bottle.
* The Countess Nelson — Silver heart-shaped boxes.
* Mrs. Napier Sturt — Silver seal.
* Mr. Wingfield — Silvor-topped china bottles.
* The Dean of Salisbury— Silver-bound Prayer Book.
* Miss Mary and Mr. Hugh Grimston — Silver box.
* Miss Nina Ellice — Old silver box, stone top.
* Mr. Gilmon and Mr. Chaplin (60th Rifles) — Silver dish.
* The [[Social Victorians/People/Bourke|Hon. Mrs. Algernon Bourke]] — Miniature silver kettle.
* Mrs. Francis Fletcher — Old silver dagger.
* Mr. Charles G. Barnett — Silver cup.
* Mrs. and Miss Ward —Glass smelling bottle, silver top, with coronet and monogram engraved thereon.
* Dr. Aiken — Old silver model windmill.
* Mrs. Aiken — Old silver model and spinning wheel.
* Viscountess Evelyn — Silver scent bottle.
* The Hon. Robert Hardinge — Silver lamp.
* Captain Leonard Russell — lvory and silver paper cutter.
* Miss Bertha and Messrs. Frank, Charles, and Vere Chaplin — Silver pen and pencil.
* Miss Grenfield [sic] and Miss M. Grenfell — Silver and tortoise-shell inkpot.
* Mr. John Jeffery and the Misses Jeffery — Lozenge shaped silver box.
* Mrs. Guy Stephenson — Silver needle case.
* Mr. Bankes — Old silver cup.
* Miss Nunn — Old silver Norwegian spoon.
* Mr. and Mrs. Whitmore — Old silver cream jug.
* Mr. and Mrs. T. Bloom — Silver button-hook.
* Mr. and Mrs. Scott-Gatty — Silver paper clip.
* The Marchioness of Blandford — Old silver spoon.
* Mr. P. Egerton Warbutton — Silver and tortoiseshell box.
* The Hon. Lionel Holland — Glass silver gum bottle.
* Mr. and Mrs. Charles Leslie — Silver purse.
* Lady Mary Agar — Silver match-box case.
* The Countess of Normanton — Silver sugar basin and tongs.
* Mr. and Mrs. Albert Sanderman — Silver box.
* Mr. W. Radford — Silver salt cellars.
* Lord and Lady Penzance — Travelling clock.
* The Marchioness of Bristol — Gold pencil case.
* The Hon. Winifred Sturt — Card case with watch and monogram.
* Mrs. Crowe — Black lace flounces.
* Mr. Reginald Lucas — ‘‘Follies and Fashions of our Grandfathers,” and Dickens’s &quot;Christmas Carols.”
* Miss Alice Riley — China vases.
* Lord Dorchester — Pair of Ormolu candlesticks.
* Mr. and Mrs. Hankey — White ostrich feather fan.
* Mr. and Mrs. Lionel Barry — Black lace fan.
* The Hon. Mrs. William Lowther — Painted and worked screen.
* The Misses Lowther — Small inlaid table.
* Mr. Cecil Kennard — Standard lamp and shade.
* Lady Stephenson — Two framed autotypes.
* Lieutenant Mark Ken, R.N. — Painted gauze fan.
* The Misses Rachael and Katherine Stephenson — Three prints in France.
* Mr. Edward Ellice — Pair of gold-topped scent bottles.
* Dr., Mrs., and Misses Bowles — Set of Carlsbad glass tumblers in case.
* Mr. and Mrs. Lionel Bulteel — Clock.
* Miss Margaret Whitbread — Photograph frame.
* Mr. and Mrs. Percy Wormald — Travelling clock.
* Mrs. Chappell — Old brocade cushion.
* Mr. Arthur Chapell — OId French violin bow.
* Miss Wingfield — Two small Dresden china boxes.
* The Viscountess Barrington — Marqueterie china cabinet.
* The Hon. Florence Barrington — Old china teapot.
* Mrs. Hussey — Water-coloured sketch.
* Miss A. Bouverie — Pair of flower glasses in iron stand.
* Miss Pincoff and her little Pupils — Photograph in frame.
* Mr. and Mrs. Henry Bowles — Venetian glass vase and candlesticks.
* Major and the Hon. Mrs. Stirling — George III. cheese scoop.
* The Countess of Macclesfield — China coffee set.
* Miss Lily and Miss Blanche Fane — Spode china jar.
* Miss Eleanor Ewart — Red silk umbrella.
* Miss Ethel and Miss Clare Buchanan — Old brocade photograph frame.
* Mrs. Wingfield — Old Florentine casket.
* Mr. Herbert Praed — Chippendale table.
* The Rev G. W. Murray — Book.
* The Rev. O. and Mrs. Stephens — Tall blue vase.
* The Hon. Mrs. Gage — Photograph frame.
* Lady Evelyn Parker — Picture.
* The Countess of Crawford — Five o'clock tea service.
* Mr. and Mrs. Hulbert — Two Downton chairs.
* Miss Constance Fletcher — Small white china lamp.
* Mrs. Kendal Grimstone — [[Social Victorians/Victorian Things#Doily|Lace d'oyleys]].
* Mr. and Mrs. Harold Boulton — Brass clock.
* Mr. and Mrs. E. H. Hulse — White lace painted fan.
* Mr. Maurice Wingfield — Old gold and agati snuff box.
* Elizabeth, Countess of Wilton, and Mr. Pryor — Gold topped scent bottle.
* Miss Kathleen Cockburn Bouverie and Miss Evelyn Pleydell-Bouverie — Small gold box.
* Major and Lady Gwendoline Little — White lace fan.
* Miss Muriel and Miss Sibell Chaplin —Tortoiseshell bonnet comb. [Col. 2c–3a]
* Lady Churchill and Lady Sybil Knox — Pair of old Dresden china flower pots.
* Mrs. George Mildway — Worked fire screen.
* Col. Fitzgerald — Old Canterbury cross.
* The Rev. W. H. McKnight — Reading lamp.
* Miss Victoria Studly Westoby — Embroidered blotter.
* Miss C. St. John Mildmay — Old arabic clasp.
* Mrs. Philip Bouverie — Gold Etin [sic étui?] case.
* Lord and Lady William Neville — Gold and tortoiseshell paper-knife.
* Miss Mary Ellice — Painting in frame.
* Mr. Wells—lnlaid table.
* Mrs. W. Radford and the Misses Radford — Worked cushion and chair backs.
* Dr. Robson Roose — Book.
* Mr. Norman — Crayon drawing of Lady Radnor.
* Mr. and Mrs. Highman — Painted handkerchief case.
* Mr. Adey — Carved oak bracket cupboard with engraved brass plaques of Longford Castle.
* Mrs. H. Fulton — lvory and silver handled paper knife.
* Mrs. Custance — Silver salt cellars.
* Mr. Snook — Oak and silver tankard.
* Mr. Horder — Large white china nautilus shell.
* Mr. and Mrs. Wilkes — Brass and crystal table lamp.
* Mrs. Sefton — Dresden china gilt cup and saucer.
* Mr. Surrey (Folkestone) — Two china figures.
* Mrs. Charles Warman — Toilet pincushion with monogram and coronet.
* Old Family Servants — Silver toilet set.
* Under Gardeners at Longford Castle — Glass epergne with hanging baskets and mirrored base.
* Employés on the Longford Estate — Black marble clock.
* The Tenants on the Longford Estate — Gold bracelet with five large diamonds set round.
* The Mayor and Corporation of Folkestone — Silver inkstand.
* The Farm Tenants on the Folkestone Estate — Silver inkbottle in silver plate.
The Bridegroom's Presents

* Lady Wilma Pleydell Bouverie — Pearl and diamond pin, and Morocco and silver card case.
* The Earl and Countess of Lathom — Fitted travelling bag.
* The ladies Bootle Wilbraham and the Hon. Reginald Bootle Wilbraham — Silver cream jug and pearl studs.
* The Hon. Mrs. Wilbraham — Silver coffee-pot, hot milk jug, and stand.
* Lord and Lady William Cecil — Three silver entrèe dishes.
* The Hon. Mrs. Edward Wilbrabam — Pair of silver bottles.
* Lady Ampthill — Silver tea-pot, sugar basin, and cream jug.
* Mr. and the Hon. Mrs. Bateman — Bureau.
* Col. and the Hon. Mrs. Burges — Silver bowl.
* The Hon. Rose Wilbraham — Silver salt cellars.
* Mr. Arthur Brooke — Silver frame.
* The Hon. Stuart Pleydell-Bouverie — Water colour sketch.
* Sir Henry and Lady Loch — Old Dutch card table.
* Lord Ampthill and the Hon. Theo. Russell — Silver mounted flask.
* Lord and Lady Stanley of Preston — Pair of silver dishes.
* The Hon. Mr. and Mrs. Francis Villiers — Apostle spoons.
* The Hon. Mr. and Mrs. George Villiers — Silver coffee pot and hot milk jug.
* The Marquis and Marchioness of Salisbury — Silver candlesticks.
* The Earl and Countess of Coventry — Silver mustard pot.
* The Earl and Countess of Crawford — Silver-mounted liqueur glasses.
* The Earl and Countess of Derby — Silver tea-pot, kettle, and stand.
* The Earl of Morley — lvory and silver paper cutter.
* Dowager Countess of Morley — Silver match-box.
* The Earl of Meath — Valoris jar.
* Mr. Octavius Brooke — Silver mug.
* Mr. F. H. Helbert — Silver cigarette lighter.
* Mrs. Alban Gibbs — Silver sugar bowl.
* Herr Gustav Mauskopf — Hock glasses and jug.
* Lady Campbell (of Garscube) — Silver bottle.
* Miss C. Campbell — Silver lamp.
* Mrs. Kennedy —lvory and silver paper cutter.
* The Rev. J. and Mrs. Thornton — lnkstand and candlesticks.
* The Hon. Robert Boyle —Walnut and brass clock.
* Mr. Augustus Yorke — lvory and silver pepper grinder.
* Mr. and Mrs. James Lowther — Copper writing set.
* Mr. C. W. Darby Gritfith — Pink china five o'clock tea set.
* Mr. C. B. Balfour — Silver mug penwiper [sic].
* Sir Algernon and Lady Borthwick — Pair of glass liqueur bottles with silver necks.
* Miss Villiers Lister — Walking stick.
* Miss F. and Miss K. Hankey — Red and blue leather blotting book.
* The Viscount and Lady Maud Wolmer — Silver-mounted match box.
* Mrs. Stronge — Silver (Walrus) paper weight.
* Miss Lauman — Silver match box case.
* Canon and Mrs. Blundell — Silver coffee-pot and carved cocoa-nut mounted in silver.
* The Hon. John Scott Montague — Match box.
* Lord Poltimore — Twelve silver-mounted liqueur glasses.
* Lord Hugh Cecil — Lemon-squeezer.
* Mr. Leopo!d De Rothschild — Silver candlesticks.
* Mrs. Frederick Villiers — Large silver-topped scent bottle.
* Lady Emma Talbot — Pair of silver candlesticks.
* The Misses E. and M. Burges — Silver match box.
* Miss Agnes Bateman — Bureau.
* Mr. and Mrs. F. Egerton Warburton —Silver string box.
* Sir Francis and Lady Montifiore — Small silver tray.
* Mrs. Arthur Lascelles — Nest of tables.
* Mrs. Goodwin — Patchwork cushion.
* The Rev. J. Lomax — Set of salt cellars.
* Mr. Ernest Farquhar — Carved ivory and silver button-hook.
* Mr. Guyr Burges — Travelling watch.
* Major and Mrs. Barrington Foote — Suite de Cheminée.
* Mr. R. Egerton Warburton — Silver toby jug.
* Mr. C. Kindersley — Copper gipsy kettle.
* The Hon. Sidney Agar — Gold and tortoiseshell walking stick.
* Mr. F. C. Peplor — Silver mounted claret jug.
* Lord and Lady Dynever — Ivory and silver paper-knife.
* Mr. Reginald Peel —Silver sugar sifter.
* Mr. Evelyn Cecil — Walnut and silver writing set.
* Lord Alcester — Champagne jugs.
* The Earl and Countess of Lytton — Dining-room table.
* Miss Hand — Small silver toast racks.
* Mr. Holland — Walking stick.
* Lady Margaret Jenkins — White china lamp.
* Mr. M. J. Hughes — Silver and glass liqueur stand.
* The Hon. Aubrey FitzClarence — Silver pepper castor.
* Colonel and Hon. Mrs. Burges — Large silver bowl.
* Mr. Pascoe Glyn — Silver platter.
* Mr. T. Brooke — Ivory and silver paper-knife.
* Captain Stewart — Silver and bog quaigh [shallow 2-handle cup, Scots].
* The Hon. Victor Stanley — Silver salt cellars.
* Viscount and Viscountess Cranborne — Stick.
* Captain Austruther Thomson — Gold headed stick.
* The Rev. H. Montagu and Mrs. Villiers — Tennyson's poems.
* Comte and Comtesse di Castéja — Garniture de Cheminée.
* Field Marshal Sir Patrick and Lady Grant — Silver-topped walking-stick.
* The Hon. E and Lady Alice Stanley — Arm-chair.
* The Earl of Arran — Cigar case.
* Major Wickham — Glass and silver inkstand.
* Mrs. Charles Wilson — Silver sugar castor.
* Lady Hood — Gold box.
* The Countess of Galloway — Siiver cigarette lamp.
* Mr. Roper — Silver dish.
* The Earl and Countess of Jersey — Pair of field glasses.
* Mr. and Lady Isabel Larnach — Silver asparagus tongs.
* Mr. Ernest Villiers — Opera glasses.
* Mr. J. Metcalfe — Tobacco pouch.
* The Lathom Household and Workpeople on the Lathom Estate — Silver salver.
* Dalton Farm Workmen — Silver and ivory paper cutter.
* The Lathom Schools — Silver sauce boat.
* The Lathom Almswomen — Silver muffineers.
* The Tenants on the Lathom Estate — Large silver gilt bowl on ebony stand.
* Officers of the Royal Horse Guards — Large two-handled silver salver, with inscription.&lt;ref&gt;&quot;Marriage of Lady Wilma Pleydell-Bouverie and Lord Skelmersdale.&quot; ''Wiltshire County Mirror'' 23 August 1889, Friday: 6 [of 8], Col. 2a–3c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004828/18890823/080/0006. Print title: ''Wiltshire County Mirror and Express'', p. 6.&lt;/ref&gt;&lt;/blockquote&gt;

== Notes and Questions ==
# 

==References==
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      <text bytes="27629" sha1="d7bcgamw08k12ljjg6or8lonugcgaqa" xml:space="preserve">=Event=
Wedding of Lady Wilma Pleydell-Bouverie and Lord Skelmersdale

==Overview==

==Logistics==
* 15 August 1889, Thursday
===Staff and Vendors===
*&quot;The Hon. and Rev. Bertrand Pleydell-Bouverie, rector of Pewsey, Wilts, uncle of the bride, assisted by the Hon. and Rev. Lord William Cecil (brother-in-law of the bridegroom), of Bishops Hatfield, Herts,&quot;&lt;ref name=&quot;:0&quot; /&gt; () conducted the ceremony.

==Related Events==
* Funerals on both sides of this wedding, so the people invited were &quot;the nearest relatives and most intimate friends&quot;&lt;ref name=&quot;:0&quot; /&gt; ()
* Reception hosted by the Earl and Countess of Radnor, Longford Castle (bride's parents)
* Honeymoon at &quot;Pewsey Rectory, the residence of the bride’s uncle&quot;&lt;ref name=&quot;:0&quot; /&gt; ()
==Who Was Present==
===Bride and Bridesmaids===
====Bride====
Lady Wilma's wedding dress&lt;blockquote&gt;was of cream-white satin duchesse, trimmed with handsome Brussels lace flounces, which, along with he Brussels lace veil, were the gifts of her mother. Her veil was arranged over a small spray of orange flowers, and fastened by diamond ornaments.&lt;ref name=&quot;:0&quot; /&gt; ()&lt;/blockquote&gt;Lady Wilma's bouquet&lt;blockquote&gt;of exquisite blooms was the gift of the non-commissioned officers of Lord Skelmersdale’s troop.&lt;ref name=&quot;:0&quot; /&gt; ()&lt;/blockquote&gt;Lady Wilma's going-away dress was&lt;blockquote&gt;grey and white cloth trimmed with grey and silver passementerie, and Virot hat to match.&lt;ref name=&quot;:0&quot; /&gt; () [Madame Virot was &quot;one of the two most famous hat designers in Paris during the second half of the 19th century.&quot;&lt;ref&gt;{{Cite journal|date=2025-06-30|title=Madame Virot|url=https://en.wikipedia.org/w/index.php?title=Madame_Virot&amp;oldid=1298040198|journal=Wikipedia|language=en}}&lt;/ref&gt;)&lt;/blockquote&gt;

====Bridesmaids====

# Lady Maud Bootle Wilbraham (sister of the groom)
# Lady Mary Edith Bootle Wilbraham (sister of the groom)
# Miss Florence Hankey
# Miss Kathleen Hankey
# Lady Evelyn Parker
# The Hon. Winifred Sturt

The bridesmaids wore&lt;blockquote&gt;cream-white costumes, trimmed with gold embroidery and chiffons, and Virot hats to match. The bridegroom's gifts were a handsome brooch in the form of a moonstone heart set round with diamonds, surmounted with a diamond coronet, and “goodyears” bouquet of choice red roses in foliage.&lt;ref name=&quot;:0&quot; /&gt; ()&lt;/blockquote&gt;

===Groom and Best Man===

* Captain the Hon. Lionel Byng, Royal Horse Guards, best man

===People Who Attended===

# The Hon. and Rev. Beatrice Pleydell-Bouverie
# Mr. and Mrs. Frank Harris (could this be the writer?)
==What People Wore==
# 
==Gifts==
===Tenants, Employees and Local Organizations===

==== For the Bride ====

* 140 Members of the Ladies String Band and Chorus — Diamond feather.
* Miss Pincoff and her little Pupils — Photograph in frame.
* Old Family Servants — Silver toilet set.
* Under Gardeners at Longford Castle — Glass epergne with hanging baskets and mirrored base.
* Employés on the Longford Estate — Black marble clock.
* The Tenants on the Longford Estate — Gold bracelet with five large diamonds set round.
* The Mayor and Corporation of Folkestone — Silver inkstand.
* The Farm Tenants on the Folkestone Estate — Silver inkbottle in silver plate.

==== For the Groom ====

* The Lathom Household and Workpeople on the Lathom Estate — Silver salver.
* Dalton Farm Workmen — Silver and ivory paper cutter.
* The Lathom Schools — Silver sauce boat.
* The Lathom Almswomen — Silver muffineers.
* The Tenants on the Lathom Estate — Large silver gilt bowl on ebony stand.
* Officers of the Royal Horse Guards — Large two-handled silver salver, with inscription.

===Unusual or Interesting Gifts===

* Several pictures, prints or paintings, but not very many: Three prints in France, Picture, Painting in frame, Crayon drawing of Lady Radnor, Water colour sketch
* Books: Silver Prayer Book, Silver-bound Prayer Book, ‘‘Follies and Fashions of our Grandfathers,” and Dickens’s &quot;Christmas Carols” [sic], Book, Book, Tennyson's poems
* Black lace flounces
* Two framed autotypes
* George III. cheese scoop
* Old gold and agati snuff box
* Old Canterbury cross
* Old arabic clasp
* Worked cushion and chair backs
* Silver coffee-pot and carved cocoa-nut mounted in silver
* Silver and bog quaigh [shallow 2-handle cup, Scots]
* Silver mug penwiper
* Tobacco pouch

===Furniture===

* Painted and worked screen, Worked fire screen
* Small inlaid table, lnlaid table, Old Dutch card table, Nest of tables, Dining-room table
* Marqueterie china cabinet
* Carved oak bracket cupboard with engraved brass plaques of Longford Castle
* Bureau, Bureau
* Silver and glass liqueur stand
* Arm-chair

===Brand Names===

* Bottle and tray (Benares work [— Benares ware is from what is now Varanasi, India])
* Spode china jar
* Chippendale table
* Two Downton chairs
* Valoris jar [probably Vallauris?]

==Anthology==
From the ''Lady's Pictorial'':&lt;blockquote&gt;On Thursday, the 15th inst., at Britford Church, near Longford Castle, Salisbury, the marriage took place of Lord Skelmersdale, of the Royal Horse Guards, eldest son of the Right Hon. the Earl of Lathom, and Lady Wilma Pleydell-Bouverie, only daughter of the Earl of Radnor. The bride was given away by her father, and Captain the Hon. Lionel Byng, Royal Horse Guards, acted as best man. The service was fully choral. The ceremony was performed by the Hon. and Rev. Bertrand Pleydell-Bouverie, rector of Pewsey, Wilts, uncle of the bride, assisted by the Hon. and Rev. Lord William Cecil (brother-in-law of the bridegroom), of Bishops Hatfield, Herts. The bride's dress was of cream-white satin duchesse, trimmed with handsome Brussels lace flounces, which, along with he Brussels lace veil, were the gifts of her mother. Her veil was arranged over a small spray of orange flowers, and fastened by diamond ornaments. The six bridesmaids were the Ladies Maud and Mary Edith Bootle Wilbraham, sisters of the bridegroom; Misses Florence and Kathleen Hankey, Lady Evelyn Parker, and the Hon. Winifred Sturt. They were attired in cream-white costumes, trimmed with gold embroidery and chiffons, and Virot hats to match. The bridegroom's gifts were a handsome brooch in the form of a moonstone heart set round with diamonds, surmounted with a diamond coronet, and “goodyears” bouquet of choice red roses in foliage. The bridal bouquet of exquisite blooms was the gift of the non-commissioned officers of Lord Skelmersdale’s troop. Owing to both families being in mourning, none but the nearest relatives and most intimate friends were invited to the ceremony, and were afterwards received by the Earl and Countess of Radnor at Longford Castle. In the afternoon Lord and Lady Skelmersdale left for Pewsey Rectory, the residence of the bride’s uncle, where the honeymoon will be spent, the going-away dress being grey and white cloth trimmed with grey and silver passementerie, and Virot hat to match. The numerous presents included —&lt;p&gt;

From the Prince and Princess of Wales, a diamond and catseye bangle; the Duchess of Fife and the Princesses Victoria and Maud of Wales, diamond and moonstone brooch; Lord Skelmersdale, half hoop diamond ring, pearl and diamond ring, diawond and ruby safety brooch, diamond aigrette, silver card case, and silver model violn; the Earl and Countess of Radnor, diamond and and tiara, moonstone and diamond brooch, large single stone diamond St. And ew’s [Andrew’s] Cross brooch, wedding veil and flounces of Brussels lace, and despatch bag: the Earl of Lathom, large diamond star; Countess of Lathom, emerald and diamond necklet, emerald and pearl bracelet, emerald and diamond bangle and sapphire half-hoop ring; Viscount Folkestone, blue enamel pearl and diamond brooch, with blue enamel diamond watch pendant; Hon. Stuart Pleydell-Bouverie, pearl bonnet pins, water-colour sketch, and small silver frame; the Ladies Bootle Wilbraham, the Hon Randle [?] and the Hon. Reginald Bootle Wilbraham, pearl and coral ring; Right Hon. Henry Chaplin, M.P., large diamond pendant; the Hon. Mr and Mrs. Duncombe[,] R. Bouverie, diamond and pearl spray; Earl and Countess Stanhope, silver and crocodile leather clock; Countess of Macclestield, china coffee set; Countess of Crawford, five o'clock tea service; Lord and Lady William Nevill, tortoiseshell and gold paper knife; the Mayor and Corporatlom of Folkestone, silver inkstand; tenants on the Longford estate, gold bracelet set round with five large diamonds; employés on the Longford estate, black [?] marble clock; old family servants, silver toilet set; farm tenants on the Folkestone estate, silver inkbottle in silver plate; the under-gardeners of Longford estate, glass flower stand; Marquis and Marchioness of Salisbury, silver candlesticks; Lady Ampthill, silver teapot, cream jug, and sugar basin; Earl and Countess of Derby, silver teapot, kettle, and stand; Earl of Morley, ivory and silver paper cutter; Sir Algernon and Lady Borthwick, palr of glass liqueur bottles with silver necks; Viscount and Viscountess Wolmer, silver mounted matchbox; Sir Francis and Lady Montefiore, small silver tray; Viscount and Viscountess Cranborne, stick, &amp;c.&lt;ref name=&quot;:0&quot;&gt;&quot;Skelmersdale — Pleydell-Bouverie.&quot; ''Lady's Pictorial'' 24 August 1889, Saturday: 30 [of 52], Col. 1a–b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005980/18890824/106/0030. Same print title, p. 264.&lt;/ref&gt;&lt;/blockquote&gt;From the ''Wiltshire County Mirror'':&lt;p&gt;

The list of gifts is set as an unordered list here instead of as separate paragraphs, as it is in the original text, to save space and clarify.&lt;blockquote&gt;As we were unable last week to obtain a full list of the wedding presents sent to Lord Skelmersdale and the Lady Wilma Pleydell-Bouverie, we now produce it as follows: —

* Their Royal Highnesses The Prince and Princess of Wales — Diamond and cat’s-eye bangle.
* Their Roval Highnesses the Duchess of Fife and the Princesses Victoria and Maud of Wales — Diamond and moonstone brooch.
* Lord Skelmersdale — Half-hoop diamond ring, pearl and diamond ring, diamond and ruby safety brooch, diamond aigrette, silver card case, and silver model violin.
* The Earl and Countess of Radnor — Diamond and pearl tiara.
* The Countess of Radnor — Moonstone and diamond brooch, large single stone diamond St. Andrew's Cross brooch, wedding veil and flounces of Brussels lace, and dispatch box.
* The Earl of Lathom — Large diamond star.
* The Countess of Lathom — Emerald and diamond necklet, emerald and pearl bracelet, emerald and diamond bangle, and sapphire half-hoop ring.
* Viscount Folkestone — Blue enamel and pearl diamond brooch, with blue enamel and diamond watch pendant.
* The Hon. Stuart Bouverie — Pearl bonnet pins, water-coloured sketch, and small silver frame.
* The Ladies Wilbraham and the Hon. Randle and the Hon. Reginald Wilbraham  — Pearl and coral ring.
* The Right Hon. Henry Chapin, M.P. — Large diamond pendant.
* The Hon. Mr. and Mrs. Duncombe P. Bouverie — Diamond and pearl spray.
* Lady June Ellice — Half-hoop emerald ring and three Dresden China baskets.
* Mr. Ellice — Sappire and diamond ring and cheque for £100.
* Colonel and Lady Margaret Maitland —Pearl half-hoop ring.
* Lady Anne Speirs — Moonstone and diamond brooch.
* Mr. and Mrs. Ernest Chaplin — Diamond crescent.
* Mr. and Lady Edith Cotes — Cat's-eye and diamond bracelet.
* The Right Hon. Mr. and Hon. Mrs. Edward P. Bouverie —Topaz and pearl bracelet.
* Mr. and Mrs. Cecil Chaplin — Diamond marguerite brooch.
* The Hon. Kenelm P. Bouverie — Diamond and moonstone bangle.
* Lord Alington — Diamond horse-shoe brooch.
* Lady Jane Von Koughuet — Gold pencil bangle.
* Mr. Charles Clifford — Gold and lapislazale [? lapis lazuli] brooch. 
* Miss Florence and Miss Kathleen S. Hankey — Double-heart pearl brooch.
* Lady Harriet Wentworth — Diamond moonstone and pearl brooch.
* Mrs. Bradford Atkinson—Diamond and ruby button brooch.
* Sir Thomas Lucas — Diamond bracelet.
* Lady Caroline Kerrison — Pearl and diamond horse-shoe brooch.
* Lady Cecil Scott Montague and Lady Margaret Kerr — Pearl brooch.
* Lady Revelstoke — Diamond lyre brooch.
* Mr. Charles Rogers — Starstone and diamond safety pin brooch.
* Miss Chappell — Diamond ‘‘merrythought&quot; brooch.
* Miss Emily Mildmay — Moonstone and diamond star brooch.
* Mr. and Mrs. Mildmay — Diamond and sapphire brooch.
* Mr. Philip H. Coxe — Pearl and diamond bangle.
* Miss Mabel Sands — Torquoise [sic] brooch.
* Mr. and Mrs. W. H. Leslie — Diamond and sapphire bracelet.
* Mr. and Mrs. Coleridge Kennard — Pearl and diamond “Swan&quot; brooch.
* Mr. and Lady Emily Van de Weyer — Pearl and diamond and sapphire and diamond bracelets.
* 140 Members of the Ladies String Band and Chorus — Diamond feather.
* Mr. Westrow Hulse — Diamond and pearl cleff brooch.
* Mr. and Mrs. Arthur Walter — Gold chain bracelet.
* Lady Gertrude and the Hon. Christopher Pleydell-Bouverie — Silver and inlaid wood inkstand.
* The Hon. and Rev. Beatrice and Lady Constance Pleydell-Bouverie — White China dessert service and velvet and silver-framed mirror.
* The Hon. Mark Pleydell-Bouverie — Silver hand glass.
* The Earl of Verulom — Silver hand candlestick.
* The Dowager Countess of Craven — Pair of silver candlesticks.
* The Earl and Countess Cadogan — Three large silver sugar sifters.
* The Marchioness of Bath — Silver basket.
* The Marchioness of Lothian — Two silver trays.
* Lady Mary Hervey — Silver smelling bottle.
* Mrs. Frank Harris — Silver frame.
* Mr. Frank Harris — Silver hair-pin box.
* The Countess of Haddington — Silver smelling bottle.
* The Countess of Hopetown [Hopetoun?] — Two silver frames.
* Lady Evelyn Parker — Silver and glass bottle and picture.
* Lady Eleanor Heneage — Small silver cruet stand.
* Lady Nina Balfour — Silver match-box case.
* Mr. Leo Frank Schuster — Silver basket.
* Lady Mary Parker — Ivory and silver paper cutter.
* Sir Edward and Lady Hulse —Silver bowl.
* The Earl and Countess of Stanhope — Silver and crocodile leather clock.
* Mr. Adam Kennard — Silver and mother-o’-pearl opera glasses with handle.
* Mr. Alan MacKinnow — Small silver violin.
* Mr. Ward Cook — Silver basin.
* Miss Esther Maitland — Silver pin tray.
* Mrs. William Whitbread — Silver tray.
* Mr. Hamilton Hulse — Old silver box.
* Lady Ventry — Silver pen and pencil.
* Mr. and Mrs. Charles Tudway — Silver mustard pot.
* Mrs. Sidmouth Skipworth — Silver heart-shaped box.
* Mr. Reginald Skipworth — Silver box.
* Mr. Henry Newton — Silver lozenge box.
* Mr. and Mrs. William Malcolm — Silver bowl.
* The Hon. Eda Campbell — Old Dutch silver boat.
* Mr. and Mrs. Bowles — Silver bottle and tray (Benares work [— Benares ware is from what is now Varanasi, India]).
* The Hon. Lady and Miss Filmer — Cut glass bottle with silver top.
* Miss Lauman — Silver Prayer Book.
* Mr. J. E. Martin (60th Rifles) — Silver button-hook and shoe-horn.
* The Misses M. and B. Kennard — Small round silver box.
* Mr. E. O. Pleydell-Bouverie — Silver inkstand.
* Mrs. C. B. Ewart — Silver hand mirror.
* Mr. Ernest Albert Sandeman — Silver box.
* Miss Mabel Gage — Silver scent bottle.
* The Countess Nelson — Silver heart-shaped boxes.
* Mrs. Napier Sturt — Silver seal.
* Mr. Wingfield — Silvor-topped china bottles.
* The Dean of Salisbury— Silver-bound Prayer Book.
* Miss Mary and Mr. Hugh Grimston — Silver box.
* Miss Nina Ellice — Old silver box, stone top.
* Mr. Gilmon and Mr. Chaplin (60th Rifles) — Silver dish.
* The [[Social Victorians/People/Bourke|Hon. Mrs. Algernon Bourke]] — Miniature silver kettle.
* Mrs. Francis Fletcher — Old silver dagger.
* Mr. Charles G. Barnett — Silver cup.
* Mrs. and Miss Ward —Glass smelling bottle, silver top, with coronet and monogram engraved thereon.
* Dr. Aiken — Old silver model windmill.
* Mrs. Aiken — Old silver model and spinning wheel.
* Viscountess Evelyn — Silver scent bottle.
* The Hon. Robert Hardinge — Silver lamp.
* Captain Leonard Russell — lvory and silver paper cutter.
* Miss Bertha and Messrs. Frank, Charles, and Vere Chaplin — Silver pen and pencil.
* Miss Grenfield [sic] and Miss M. Grenfell — Silver and tortoise-shell inkpot.
* Mr. John Jeffery and the Misses Jeffery — Lozenge shaped silver box.
* Mrs. Guy Stephenson — Silver needle case.
* Mr. Bankes — Old silver cup.
* Miss Nunn — Old silver Norwegian spoon.
* Mr. and Mrs. Whitmore — Old silver cream jug.
* Mr. and Mrs. T. Bloom — Silver button-hook.
* Mr. and Mrs. Scott-Gatty — Silver paper clip.
* The Marchioness of Blandford — Old silver spoon.
* Mr. P. Egerton Warbutton — Silver and tortoiseshell box.
* The Hon. Lionel Holland — Glass silver gum bottle.
* Mr. and Mrs. Charles Leslie — Silver purse.
* Lady Mary Agar — Silver match-box case.
* The Countess of Normanton — Silver sugar basin and tongs.
* Mr. and Mrs. Albert Sanderman — Silver box.
* Mr. W. Radford — Silver salt cellars.
* Lord and Lady Penzance — Travelling clock.
* The Marchioness of Bristol — Gold pencil case.
* The Hon. Winifred Sturt — Card case with watch and monogram.
* Mrs. Crowe — Black lace flounces.
* Mr. Reginald Lucas — ‘‘Follies and Fashions of our Grandfathers,” and Dickens’s &quot;Christmas Carols.”
* Miss Alice Riley — China vases.
* Lord Dorchester — Pair of Ormolu candlesticks.
* Mr. and Mrs. Hankey — White ostrich feather fan.
* Mr. and Mrs. Lionel Barry — Black lace fan.
* The Hon. Mrs. William Lowther — Painted and worked screen.
* The Misses Lowther — Small inlaid table.
* Mr. Cecil Kennard — Standard lamp and shade.
* Lady Stephenson — Two framed autotypes.
* Lieutenant Mark Ken, R.N. — Painted gauze fan.
* The Misses Rachael and Katherine Stephenson — Three prints in France.
* Mr. Edward Ellice — Pair of gold-topped scent bottles.
* Dr., Mrs., and Misses Bowles — Set of Carlsbad glass tumblers in case.
* Mr. and Mrs. Lionel Bulteel — Clock.
* Miss Margaret Whitbread — Photograph frame.
* Mr. and Mrs. Percy Wormald — Travelling clock.
* Mrs. Chappell — Old brocade cushion.
* Mr. Arthur Chapell — OId French violin bow.
* Miss Wingfield — Two small Dresden china boxes.
* The Viscountess Barrington — Marqueterie china cabinet.
* The Hon. Florence Barrington — Old china teapot.
* Mrs. Hussey — Water-coloured sketch.
* Miss A. Bouverie — Pair of flower glasses in iron stand.
* Miss Pincoff and her little Pupils — Photograph in frame.
* Mr. and Mrs. Henry Bowles — Venetian glass vase and candlesticks.
* Major and the Hon. Mrs. Stirling — George III. cheese scoop.
* The Countess of Macclesfield — China coffee set.
* Miss Lily and Miss Blanche Fane — Spode china jar.
* Miss Eleanor Ewart — Red silk umbrella.
* Miss Ethel and Miss Clare Buchanan — Old brocade photograph frame.
* Mrs. Wingfield — Old Florentine casket.
* Mr. Herbert Praed — Chippendale table.
* The Rev G. W. Murray — Book.
* The Rev. O. and Mrs. Stephens — Tall blue vase.
* The Hon. Mrs. Gage — Photograph frame.
* Lady Evelyn Parker — Picture.
* The Countess of Crawford — Five o'clock tea service.
* Mr. and Mrs. Hulbert — Two Downton chairs.
* Miss Constance Fletcher — Small white china lamp.
* Mrs. Kendal Grimstone — [[Social Victorians/Victorian Things#Doily|Lace d'oyleys]].
* Mr. and Mrs. Harold Boulton — Brass clock.
* Mr. and Mrs. E. H. Hulse — White lace painted fan.
* Mr. Maurice Wingfield — Old gold and agati snuff box.
* Elizabeth, Countess of Wilton, and Mr. Pryor — Gold topped scent bottle.
* Miss Kathleen Cockburn Bouverie and Miss Evelyn Pleydell-Bouverie — Small gold box.
* Major and Lady Gwendoline Little — White lace fan.
* Miss Muriel and Miss Sibell Chaplin —Tortoiseshell bonnet comb. [Col. 2c–3a]
* Lady Churchill and Lady Sybil Knox — Pair of old Dresden china flower pots.
* Mrs. George Mildway — Worked fire screen.
* Col. Fitzgerald — Old Canterbury cross.
* The Rev. W. H. McKnight — Reading lamp.
* Miss Victoria Studly Westoby — Embroidered blotter.
* Miss C. St. John Mildmay — Old arabic clasp.
* Mrs. Philip Bouverie — Gold Etin [sic étui?] case.
* Lord and Lady William Neville — Gold and tortoiseshell paper-knife.
* Miss Mary Ellice — Painting in frame.
* Mr. Wells—lnlaid table.
* Mrs. W. Radford and the Misses Radford — Worked cushion and chair backs.
* Dr. Robson Roose — Book.
* Mr. Norman — Crayon drawing of Lady Radnor.
* Mr. and Mrs. Highman — Painted handkerchief case.
* Mr. Adey — Carved oak bracket cupboard with engraved brass plaques of Longford Castle.
* Mrs. H. Fulton — lvory and silver handled paper knife.
* Mrs. Custance — Silver salt cellars.
* Mr. Snook — Oak and silver tankard.
* Mr. Horder — Large white china nautilus shell.
* Mr. and Mrs. Wilkes — Brass and crystal table lamp.
* Mrs. Sefton — Dresden china gilt cup and saucer.
* Mr. Surrey (Folkestone) — Two china figures.
* Mrs. Charles Warman — Toilet pincushion with monogram and coronet.
* Old Family Servants — Silver toilet set.
* Under Gardeners at Longford Castle — Glass epergne with hanging baskets and mirrored base.
* Employés on the Longford Estate — Black marble clock.
* The Tenants on the Longford Estate — Gold bracelet with five large diamonds set round.
* The Mayor and Corporation of Folkestone — Silver inkstand.
* The Farm Tenants on the Folkestone Estate — Silver inkbottle in silver plate.
The Bridegroom's Presents

* Lady Wilma Pleydell Bouverie — Pearl and diamond pin, and Morocco and silver card case.
* The Earl and Countess of Lathom — Fitted travelling bag.
* The ladies Bootle Wilbraham and the Hon. Reginald Bootle Wilbraham — Silver cream jug and pearl studs.
* The Hon. Mrs. Wilbraham — Silver coffee-pot, hot milk jug, and stand.
* Lord and Lady William Cecil — Three silver entrèe dishes.
* The Hon. Mrs. Edward Wilbrabam — Pair of silver bottles.
* Lady Ampthill — Silver tea-pot, sugar basin, and cream jug.
* Mr. and the Hon. Mrs. Bateman — Bureau.
* Col. and the Hon. Mrs. Burges — Silver bowl.
* The Hon. Rose Wilbraham — Silver salt cellars.
* Mr. Arthur Brooke — Silver frame.
* The Hon. Stuart Pleydell-Bouverie — Water colour sketch.
* Sir Henry and Lady Loch — Old Dutch card table.
* Lord Ampthill and the Hon. Theo. Russell — Silver mounted flask.
* Lord and Lady Stanley of Preston — Pair of silver dishes.
* The Hon. Mr. and Mrs. Francis Villiers — Apostle spoons.
* The Hon. Mr. and Mrs. George Villiers — Silver coffee pot and hot milk jug.
* The Marquis and Marchioness of Salisbury — Silver candlesticks.
* The Earl and Countess of Coventry — Silver mustard pot.
* The Earl and Countess of Crawford — Silver-mounted liqueur glasses.
* The Earl and Countess of Derby — Silver tea-pot, kettle, and stand.
* The Earl of Morley — lvory and silver paper cutter.
* Dowager Countess of Morley — Silver match-box.
* The Earl of Meath — Valoris jar.
* Mr. Octavius Brooke — Silver mug.
* Mr. F. H. Helbert — Silver cigarette lighter.
* Mrs. Alban Gibbs — Silver sugar bowl.
* Herr Gustav Mauskopf — Hock glasses and jug.
* Lady Campbell (of Garscube) — Silver bottle.
* Miss C. Campbell — Silver lamp.
* Mrs. Kennedy —lvory and silver paper cutter.
* The Rev. J. and Mrs. Thornton — lnkstand and candlesticks.
* The Hon. Robert Boyle —Walnut and brass clock.
* Mr. Augustus Yorke — lvory and silver pepper grinder.
* Mr. and Mrs. James Lowther — Copper writing set.
* Mr. C. W. Darby Gritfith — Pink china five o'clock tea set.
* Mr. C. B. Balfour — Silver mug penwiper [sic].
* Sir Algernon and Lady Borthwick — Pair of glass liqueur bottles with silver necks.
* Miss Villiers Lister — Walking stick.
* Miss F. and Miss K. Hankey — Red and blue leather blotting book.
* The Viscount and Lady Maud Wolmer — Silver-mounted match box.
* Mrs. Stronge — Silver (Walrus) paper weight.
* Miss Lauman — Silver match box case.
* Canon and Mrs. Blundell — Silver coffee-pot and carved cocoa-nut mounted in silver.
* The Hon. John Scott Montague — Match box.
* Lord Poltimore — Twelve silver-mounted liqueur glasses.
* Lord Hugh Cecil — Lemon-squeezer.
* Mr. Leopo!d De Rothschild — Silver candlesticks.
* Mrs. Frederick Villiers — Large silver-topped scent bottle.
* Lady Emma Talbot — Pair of silver candlesticks.
* The Misses E. and M. Burges — Silver match box.
* Miss Agnes Bateman — Bureau.
* Mr. and Mrs. F. Egerton Warburton —Silver string box.
* Sir Francis and Lady Montifiore — Small silver tray.
* Mrs. Arthur Lascelles — Nest of tables.
* Mrs. Goodwin — Patchwork cushion.
* The Rev. J. Lomax — Set of salt cellars.
* Mr. Ernest Farquhar — Carved ivory and silver button-hook.
* Mr. Guyr Burges — Travelling watch.
* Major and Mrs. Barrington Foote — Suite de Cheminée.
* Mr. R. Egerton Warburton — Silver toby jug.
* Mr. C. Kindersley — Copper gipsy kettle.
* The Hon. Sidney Agar — Gold and tortoiseshell walking stick.
* Mr. F. C. Peplor — Silver mounted claret jug.
* Lord and Lady Dynever — Ivory and silver paper-knife.
* Mr. Reginald Peel —Silver sugar sifter.
* Mr. Evelyn Cecil — Walnut and silver writing set.
* Lord Alcester — Champagne jugs.
* The Earl and Countess of Lytton — Dining-room table.
* Miss Hand — Small silver toast racks.
* Mr. Holland — Walking stick.
* Lady Margaret Jenkins — White china lamp.
* Mr. M. J. Hughes — Silver and glass liqueur stand.
* The Hon. Aubrey FitzClarence — Silver pepper castor.
* Colonel and Hon. Mrs. Burges — Large silver bowl.
* Mr. Pascoe Glyn — Silver platter.
* Mr. T. Brooke — Ivory and silver paper-knife.
* Captain Stewart — Silver and bog quaigh [shallow 2-handle cup, Scots].
* The Hon. Victor Stanley — Silver salt cellars.
* Viscount and Viscountess Cranborne — Stick.
* Captain Austruther Thomson — Gold headed stick.
* The Rev. H. Montagu and Mrs. Villiers — Tennyson's poems.
* Comte and Comtesse di Castéja — Garniture de Cheminée.
* Field Marshal Sir Patrick and Lady Grant — Silver-topped walking-stick.
* The Hon. E and Lady Alice Stanley — Arm-chair.
* The Earl of Arran — Cigar case.
* Major Wickham — Glass and silver inkstand.
* Mrs. Charles Wilson — Silver sugar castor.
* Lady Hood — Gold box.
* The Countess of Galloway — Siiver cigarette lamp.
* Mr. Roper — Silver dish.
* The Earl and Countess of Jersey — Pair of field glasses.
* Mr. and Lady Isabel Larnach — Silver asparagus tongs.
* Mr. Ernest Villiers — Opera glasses.
* Mr. J. Metcalfe — Tobacco pouch.
* The Lathom Household and Workpeople on the Lathom Estate — Silver salver.
* Dalton Farm Workmen — Silver and ivory paper cutter.
* The Lathom Schools — Silver sauce boat.
* The Lathom Almswomen — Silver muffineers.
* The Tenants on the Lathom Estate — Large silver gilt bowl on ebony stand.
* Officers of the Royal Horse Guards — Large two-handled silver salver, with inscription.&lt;ref&gt;&quot;Marriage of Lady Wilma Pleydell-Bouverie and Lord Skelmersdale.&quot; ''Wiltshire County Mirror'' 23 August 1889, Friday: 6 [of 8], Col. 2a–3c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004828/18890823/080/0006. Print title: ''Wiltshire County Mirror and Express'', p. 6.&lt;/ref&gt;&lt;/blockquote&gt;

== Notes and Questions ==
# 

==References==
{{reflist}}</text>
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    <title>Constructive Neural Networks</title>
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      <comment>Adding course plan sections (WIP)</comment>
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|title=Constructive Neural Networks
|description=Neural networks that dynamically grow their own structure.
|department=Computer Science
|course=Machine Learning
|level=Intermediate
}}

== Introduction ==
This learning project aims to provide an introduction to constructive algorithms for [[w:Artificial Neural Network|artificial neural networks]], which combine to produce '''constructive neural networks''', and present ongoing research in the development of constructive algorithms for [[w:Transformer (machine learning)|transformer-based neural networks]].

Artificial neural network (ANN) researchers first succeeded in training multilayered perceptrons using [[w:Backpropagation|error back-propagation]] in the 1980s. ANNs with multiple neurons and layers raised the challenge of choosing the number of neurons and their arrangement. [[w:Deep learning|Deep neural networks]] have demonstrated that more layers (tens or hundreds), more parameters (billions) and architectural tricks ([[w:Residual neural network|residual connections]], [[w:Attention (machine learning)|attention]], etc) can significantly increase the model capabilities. But the final architecture is still often the result of a manual search, and these deep neural networks have a fixed architecture defined at initialisation. 

Constructive algorithms were developed to '''dynamically grow their architecture''' as they learn. A constructive neural network (CNN) is the combination of a constructive algorithm with an ANN schema. Together these must be designed to decide when  construct (triggers) and what to construct (components, connections, parameter values). Training neural networks with constructive algorithms has potential advantages:
* Automated architecture design
* Efficient training from small size
* Continual learning without catastrophic forgetting

Constructive algorithms had some limited early success in the 1980s and 1990s; however, there are also significant challenges to developing constructive algorithms. Constructive algorithms have been largely absent from recent transformer-based architectures that have been the source of many enormous advances in AI natural language processing (text and speech), vision, and multi-modal understanding. Nevertheless, there is some developments that can be interpreted as constructive algorithms and many opportunities to explore potential applications.

This learning project has two major sections:
# Exercises to reproduce some historical constructive algorithms and theory on the general components of constructive algorithms. 
# Research on the development of constructive algorithms for applications to modern transformer-based neural networks.

== Learning Objectives ==
After completing this learning project, you will be able to:
* Reproduce foundational constructive neural networks using Python.
* Understand fundamental components of a constructive algorithm and neural network schema.
* Participate in current research applying constructive algorithms to transformer-based neural networks.

== Prerequisites ==
To get the most out of this project, you should have a basic understanding of:
* [[w:Artificial neural network|Artificial Neural Networks (ANNs)]]: Concepts like neurons, weights, biases, and activation functions.
* Familiarity with libraries like [[w:NumPy|NumPy]] and [[w:PyTorch|PyTorch]] will be helpful for the practical exercises.
* [[w:Algorithm design|Algorithm design]]: Different techniques for efficient algorithm design and implementation.

== Course Pages ==

The development of this course is a work in progress.

The content will include examinations of historical constructive neural networks, including reproductions and training in PyTorch. A technical component will examine the challenges of implementing and training neural networks that have changing structure. Theoretical components will generalise the examined constructive neural networks into some standard features for designing constructive algorithms. Constructive algorithms will then be applied to basic transformer-based neural networks.

=== History 1: Dynamic Node Creation (Ash, 1989) ===

Expected completion date: 13 July 2025

Dynamic Node Creation is one of the first and simplest constructive neural networks. The constructive algorithm adds neurons to the hidden layer of a multi-layered perception trained with backprop. This section will cover paper and develop Python code to train an MLP with dynamic node creation.

Ash, T (1989). Dynamic Node Creation in Backpropagation Networks. Connection Science, 1(4), 365–375.
https://doi.org/10.1080/09540098908915647

=== Technical 1: Constructive Algorithms in Practice ===

Expected completion date: 3 August 2025

A practical problem of constructive neural networks is that the architecture is not static. Many machine learning tools and frameworks have been optimised for predetermined computational graphs. This section examines approaches to efficiently train a constructive neural network and different approaches to managing memory.

=== History 2: Cascade Correlation (Fahlman &amp; Lebiere, 1989) === 

Expected completion date: 20 July 2025

Cascade-correlation is a slightly better known and more complex constructive neural network. The constructive algorithm adds neurons just before the output with input connections from all prior neurons and freezes the previous weights.

Fahlman, S &amp; Lebiere, C (1989). The Cascade-Correlation Learning Architecture. Advances in Neural Information Processing Systems, Vol 2, Ed. D. Touretzky. Morgan-Kaufmann.
https://proceedings.neurips.cc/paper_files/paper/1989/hash/69adc1e107f7f7d035d7baf04342e1ca-Abstract.html

=== History 3. Growing Neural Gas (Friztke, 1994) ===

Expected completion date: 27 July 2025

Growing Neural Gas is an unsupervised constructive neural network that extends the work of Neural Gas and Kohonen Self-Organizing Maps.

Fritzke, B (1994). A Growing Neural Gas Network Learns Topologies. Advances in Neural Information Processing Systems, Vol 7, Eds. G. Tesauro, D. Touretzky &amp; T. Leen. MIT Press.
https://proceedings.neurips.cc/paper_files/paper/1994/hash/d56b9fc4b0f1be8871f5e1c40c0067e7-Abstract.html

=== Theory 1: Basic Components of Constructive Neural Networks ===

Expected completion date: 3 August 2025

Given a short exploration of early constructive neural networks and constructive algorithms, we can identify the basic components that are common.

Constructive algorithms have standard elements:
# Triggers for construction
# Processes to select new component-locations
# Functions for calculated new parameters

Constructive neural networks can have algorithms that are tightly-coupled with particular architectures or neuron models, but not always. This section explores options for applying prior constructive algorithms to different artificial neural networks.

=== Theory 2: Constructing Transformers ===

Expected completion date: 10 August 2025

This section will present an initial examination of the transformer architecture and the potential applications of constructive algorithms.

=== Research 1: TBA ===

Expected completion date TBA.

=== Research 2: TBA ===

Expected date TBA.

=== Research 3: TBA ===

Expected date TBA.</text>
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== Licensing ==
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== Licensing ==
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