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{{Short description|English mathematician (1821–1895)}}
{{For|Sir Arthur Cayley, Baronet|Cayley baronets}}
{{Use dmy dates|date=December 2016}}
{{Use British English|date=December 2016}}
{{Infobox scientist
| name              = Arthur Cayley
| honorific_suffix  = {{post-nominals|country=GBR|FRS|size=100%}}
| image             = Arthur Cayley.jpg
| image_size        = 240px
| caption           = 
| birth_date        = {{birth date|1821|8|16|df=y}}
| birth_place       = [[Richmond, London|Richmond]], [[Surrey]], England
| death_date        = {{death date and age|1895|1|26|1821|8|16|df=y}}
| death_place       = [[Cambridge]], England
| alma_mater        = {{nowrap|[[Trinity College, Cambridge]] (BA, 1842)}}
| academic_advisors = {{ubl|[[George Peacock (mathematician)|George Peacock]]|[[William Hopkins]]}}
| notable_students  = {{ubl|[[H. F. Baker]]|[[Andrew Forsyth]]|[[Charlotte Scott]]}}
| known_for         = {{ubl|[[Algebraic geometry]]| [[Group theory]]| [[Cayley–Hamilton theorem]] | [[Cayley–Dickson construction]]}}
| awards            = {{ubl|[[Smith's Prize]] (1842)|[[De Morgan Medal]] (1884)|[[Royal Medal]] (1859)|[[Copley Medal]] (1882)}}
| footnotes         = 
| field             = [[Mathematics]]
| work_institution  = [[Trinity College, Cambridge]]
| education         = [[King's College School]]
}}

'''Arthur Cayley''' {{post-nominals|country=GBR|FRS}} ({{IPAc-en|ˈ|k|eɪ|l|i}}; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of [[pure mathematics]], and was a professor at [[Trinity College, Cambridge]] for 35 years.

He postulated what is now known as the [[Cayley–Hamilton theorem]]—that every [[square matrix]] is a root of its own [[characteristic polynomial]], and verified it for matrices of order 2 and 3.<ref>See Cayley (1858) "A Memoir on the Theory of Matrices", ''Philosophical Transactions of the Royal Society of London'', '''148''' :  24 : "I have verified the theorem, in the next simplest case, of a matrix of the order 3, ... but I have not thought it necessary to undertake the labour of a formal proof of the theorem in the general case of a matrix of any degree."</ref> He was the first to define the concept of an abstract [[group (mathematics)|group]], a set with a [[Binary function|binary]] operation satisfying certain laws,<ref>Cayley (1854) [https://books.google.com/books?id=_LYConosISUC&pg=PA40 "On the theory of groups, as depending on the symbolic equation θ<sup>n</sup> = 1,"] ''Philosophical Magazine'', 4th series, '''7''' (42) : 40–47.  However, see also the criticism of this definition in:  [http://www-history.mcs.st-and.ac.uk/HistTopics/Abstract_groups.html MacTutor:  The abstract group concept].</ref> as opposed to [[Évariste Galois]]' concept of [[permutation group]]s. In group theory, [[Cayley table]]s, [[Cayley graph]]s, and [[Cayley's theorem]] are named in his honour, as well as [[Cayley's formula]] in combinatorics.

==Early life==
Arthur Cayley was born in [[Richmond, London]], England, on 16 August 1821. His father, Henry Cayley, was a distant cousin of [[George Cayley]], the [[aeronautics]] engineer innovator, and descended from an ancient [[Yorkshire]] family. He settled in [[Saint Petersburg]], Russia, as a [[merchant]]. His mother was Maria Antonia Doughty, daughter of William Doughty. According to some writers she was Russian, but her father's name indicates an English origin. His brother was the linguist [[Charles Bagot Cayley]]. Arthur spent his first eight years in Saint Petersburg. In 1829 his parents were settled permanently at [[Blackheath, London]], where Arthur attended a private school. 

At age 14, he was sent to [[King's College School]]. The young Cayley enjoyed complex maths problems, and the school's master observed indications of his mathematical genius. He advised the father to educate his son not for his own business, as he had intended, but at the [[University of Cambridge]].

==Education==
At the age of 17 Cayley began residence at [[Trinity College, Cambridge]], where he excelled in Greek, French, German, and Italian, as well as [[mathematics]]. The cause of the [[Analytical Society]] had now triumphed, and the ''[[Cambridge Mathematical Journal]]'' had been instituted by Gregory and [[Robert Leslie Ellis]]. To this journal, at the age of twenty, Cayley contributed three papers, on subjects that had been suggested by reading the ''Mécanique analytique'' of [[Joseph Louis Lagrange]] and some of the works of [[Laplace]].

Cayley's tutor at Cambridge was [[George Peacock (mathematician)|George Peacock]] and his private coach was [[William Hopkins]]. He finished his undergraduate course by winning the place of [[Senior Wrangler]], and the first [[Smith's prize]].<ref>{{acad|id=CLY838A|name=Cayley, Arthur}}</ref> His next step was to take the M.A. degree, and win a Fellowship by competitive examination. He continued to reside at Cambridge University for four years; during which time he took some pupils, but his main work was the preparation of 28 memoirs to the '' Mathematical Journal''.

==Law career==
Because of the limited tenure of his fellowship it was necessary to choose a profession; like [[Augustus De Morgan|De Morgan]], Cayley chose law, and was admitted to [[Lincoln's Inn, London]] on 20 April 1846 at the age of 24.<ref>{{Cite book|url=https://archive.org/details/VOL218001893CHAPELREGISTERS/mode/2up|title=The Records of the Honorable Society of Lincoln's Inn Vol II, Admission Register 1420 - 1893|publisher=Lincoln's Inn|year=1896|location=London|pages=226}}</ref> He made a specialty of [[conveyancing]]. It was while he was a pupil at the [[bar examination]] that he went to [[Dublin]] to hear [[William Rowan Hamilton]]'s lectures on [[quaternion]]s.<ref name="dnb">{{cite DNBSupp|wstitle=Cayley, Arthur |first=Andrew Russell|last=Forsyth}}</ref>

His friend [[James Joseph Sylvester|J. J. Sylvester]], his senior by five years at Cambridge, was then an [[actuary]], resident in London; they used to walk together round the courts of [[Lincoln's Inn]], discussing the [[theory of invariants]] and covariants. During these fourteen years, Cayley produced between two and three hundred papers.<ref name="dnb"/>

==Professorship==
Around 1860, Cambridge University's [[Lucasian Professor of Mathematics]] ([[Isaac Newton|Newton]]'s chair) was supplemented by the new [[Sadleirian Professor of Pure Mathematics|Sadleirian]] professorship, using funds bequeathed by Lady Sadleir, with the 42-year-old Cayley as its first holder. His duties were ''"to explain and teach the principles of pure mathematics and to apply himself to the advancement of that science."'' He gave up a lucrative legal practice for a modest salary, but never regretted the exchange, since it allowed him to devote his energies to the pursuit that he liked best. He at once married and settled down in Cambridge, and (unlike Hamilton) enjoyed a home life of great happiness. Sylvester, his friend from his bachelor days, once expressed his envy of Cayley's peaceful family life, whereas the unmarried Sylvester had to fight the world all his days.

At first the Sadleirian professor was paid to lecture over one of the terms of the academic year, but the university financial reform of 1886 freed funds to extend his lectures to two terms. For many years his courses were attended only by a few students who had finished their examination preparation, but after the reform the attendance numbered about fifteen. He generally lectured on his current research topic.

As for his duty to the advancement of mathematical science, he published a long and fruitful series of memoirs ranging over all of pure mathematics. He also became the standing referee on the merits of mathematical papers to many societies both at home and abroad.

In 1872, he was made an honorary fellow of Trinity College, and three years later an ordinary fellow, a paid position. About this time his friends subscribed for a presentation portrait. [[James Clerk Maxwell|Maxwell]] wrote an address praising Cayley's principal works, including his Chapters on the Analytical Geometry of <math>n</math> dimensions; On the theory of [[Determinant]]s; Memoir on the theory of Matrices; Memoirs on skew surfaces, otherwise Scrolls; and On the delineation of a Cubic Scroll.<ref>[[s:To the Committee of the Cayley Portrait Fund|"To the Committee of the Cayley Portrait Fund"]], 1874</ref>

In addition to his work on [[algebra]], Cayley made fundamental contributions to [[algebraic geometry]]. Cayley and [[George Salmon|Salmon]] discovered the 27 lines on a [[cubic surface]]. Cayley constructed the [[Chow variety]] of all curves in projective 3-space.<ref>A. Cayley, Collected Mathematical Papers, Cambridge (1891), v. 4, 446−455. W. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry, Cambridge (1952), v. 2, p. 388.</ref> He founded the algebro-geometric theory of [[ruled surface]]s. His contributions to [[combinatorics]] include counting the ''n<sup>n</sup>''<sup>–2</sup> [[Tree (graph theory)|trees]] on ''n'' labeled vertices by the pioneering use of [[Generating function|generating functions]].

In 1876, he published a ''Treatise on [[Elliptic Functions]]''. He took great interest in the movement for the university education of women. At Cambridge the first women's colleges were Girton and Newnham. In the early days of [[Girton College]] he gave direct help in teaching, and for some years he was chairman of the council of [[Newnham College]], in the progress of which he took the keenest interest to the last.

In 1881, he received from the [[Johns Hopkins University]], [[Baltimore]], where Sylvester was then professor of mathematics, an invitation to deliver a course of lectures. He accepted the invitation, and lectured at Baltimore during the first five months of 1882 on the subject of the ''Abelian and Theta Functions''.

In 1893, Cayley became a foreign member of the [[Royal Netherlands Academy of Arts and Sciences]].<ref>{{cite web|url=http://www.dwc.knaw.nl/biografie/pmknaw/?pagetype=authorDetail&aId=PE00004572 |title=A. Cayley (1821 - 1895) |publisher=Royal Netherlands Academy of Arts and Sciences |access-date=19 April 2016}}</ref>

==British Association presidency==
In 1883, Cayley was President of the [[British Association for the Advancement of Science]]. The meeting was held at Southport, in the north of England. As the President's address is one of the great popular events of the meeting, and brings out an audience of general culture, it is usually made as little technical as possible. {{harvtxt|Cayley|1996}}  took for his subject the Progress of Pure Mathematics.

==The ''Collected Papers''==
In 1889, the [[Cambridge University Press]] began the publication of his collected papers, which he appreciated very much. He edited seven of the  quarto volumes himself, though suffering from a painful internal malady. He died 26 January 1895 at age 73. His funeral at Trinity Chapel was attended by the leading scientists of Britain, with official representatives from as far as Russia and America.

The remainder of his papers were edited by [[Andrew Forsyth]], his successor as Sadleirian professor, in total thirteen quarto volumes and 967 papers. His work continues in frequent use, cited in more than 200 mathematical papers in the 21st century alone.

Cayley retained to the last his fondness for novel-reading and for travelling. He also took special pleasure in paintings and architecture, and he practiced [[Watercolor painting|water-colour painting]], which he found useful sometimes in making mathematical diagrams.

==Legacy==
Cayley is buried in the [[Mill Road Cemetery, Cambridge|Mill Road cemetery]], Cambridge.

An 1874 portrait of Cayley by [[Lowes Cato Dickinson]] and an 1884 portrait by [[William Longmaid]] are in the collection of [[Trinity College, Cambridge]].<ref>{{cite web|title=Trinity College, University of Cambridge|url=https://www.bbc.co.uk/arts/yourpaintings/paintings/search/located_at/trinity-college-cambridge-5846_locations|publisher=BBC Your Paintings|access-date=12 February 2018|archive-url=https://archive.today/20140511164255/http://www.bbc.co.uk/arts/yourpaintings/paintings/search/located_at/trinity-college-cambridge-5846_locations|archive-date=11 May 2014|url-status=dead|df=dmy-all}}</ref>

A number of mathematical terms are named after him:
{{div col|colwidth=20em}}
* [[Cayley's theorem]]
* [[Cayley–Hamilton theorem]] in [[linear algebra]]
* [[Cayley–Bacharach theorem]]
* [[Grassmann–Cayley algebra]]
* [[Cayley–Menger determinant]]
* [[Cayley diagram]]s &ndash; used for finding [[cognate linkage]]s in mechanical engineering
* [[Cayley–Dickson construction]]
* [[Octonion|Cayley algebra]] (Octonion)
* [[Cayley graph]]
* [[Cayley numbers]]
* [[Cayley's sextic]]
* [[Cayley table]]
* [[Cayley–Purser algorithm]]
* [[Cayley's formula]]
* [[Cayley–Klein metric]]
* [[Klein model|Cayley–Klein model]] of [[hyperbolic geometry]]
* [[Cayley's Ω process]]
* [[Cayley surface (disambiguation)|Cayley surface]]
* [[Cayley transform]]
* [[Cayley's nodal cubic surface]]
* [[Cayley's ruled cubic surface]]
* The crater [[Cayley (crater)|Cayley]] on the [[Moon]] (and consequently the Cayley Formation, a geological unit named after the crater)
* [[Cayley's mousetrap]] — a card game
* [[Cayleyan]]
* [[Chasles–Cayley–Brill formula]]
* [[Hyperdeterminant]]
* [[Quippian]]
* [[Tetrahedroid]]
{{div col end}}

==Bibliography==
*{{Citation | last1=Cayley | first1=Arthur | author1-link=Arthur Cayley | title=An elementary treatise on elliptic functions | orig-year=1876 | url=https://archive.org/details/anelementarytre01caylgoog | publisher=Cornell University Library | isbn=978-1-112-28006-1 | mr=0124532 | year=2009}}
*{{Citation | last1=Cayley | first1=Arthur | author1-link=Arthur Cayley | title=The Collected Mathematical Papers | orig-year=1889 | url=http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABS3153  | publisher=[[Cambridge University Press]] | series=Cambridge Library Collection – Mathematics | isbn=978-1-108-00507-4 | id=[https://archive.org/search.php?query=The_collected_mathematical_papers_of_Arthur_Cayley archive] | year=2009 | volume=14 volumes}}
*{{Citation | last1=Cayley | first1=Arthur | author1-link=Arthur Cayley | title=The principles of book-keeping by double entry|publisher=Cambridge University Press|year=1894|url=https://archive.org/details/principlesofbook00caylrich}}

== See also ==

* [[List of things named after Arthur Cayley]]

==References==
{{reflist}}

==Sources==
*{{Citation | last1=Cayley | first1=Arthur | author1-link=Arthur Cayley | editor1-last=Ewald | editor1-first=William | title=From Kant to Hilbert: a source book in the foundations of mathematics. Vol. I, II | orig-year=1883 | url=https://books.google.com/books?id=Yil-EmrsT2wC | publisher=The Clarendon Press [[Oxford University Press]] | series=Oxford Science Publications | isbn=978-0-19-853271-2 | mr=1465678 |id=[https://archive.org/stream/collmathpapers11caylrich#page/n451/mode/2up Reprinted in collected mathematical papers volume 11] | year=1996 | chapter=Presidential address to the British Association | pages=542–573}}
*{{Citation | last1=Crilly | first1=Tony | title=A Victorian Mathematician: Arthur Cayley (1821–1895) | jstor=3618297 | publisher=[[The Mathematical Association]] | year=1995 | journal=[[The Mathematical Gazette]] | issn=0025-5572 | volume=79 | issue=485 | pages=259–262 | doi=10.2307/3618297| s2cid=188006684 }}
*{{Citation | last1=Crilly | first1=Tony | title=Arthur Cayley. Mathematician laureate of the Victorian age  | url=https://books.google.com/books?isbn=0801880114 | publisher=[[Johns Hopkins University Press]] | isbn=978-0-8018-8011-7 | mr=2284396 | year=2006}}
*{{Citation | last1=Macfarlane | first1=Alexander | author-link=Alexander Macfarlane| title=Lectures on Ten British Mathematicians of the Nineteenth Century | orig-year=1916 | url=https://archive.org/details/lecturesontenbri00macf | publisher=[[Cornell University Library]] | series=Mathematical monographs | isbn=978-1-112-28306-2 | year=2009 | volume=17}} ([https://web.archive.org/web/20170729051819/http://library.beau.org/gutenberg/etext06/tbmms10p.pdf complete text] at [[Project Gutenberg]])

==External links==
{{commons category}}
{{wikisource author}}
* {{MacTutor Biography|id=Cayley}}
* {{MathGenealogy|id=7824}}
* {{ScienceWorldBiography | urlname=Cayley | title=Cayley, Arthur (1821–1895)}}
* [http://digital.lib.lehigh.edu/cdm4/remain_viewer.php?DMTHUMB=1&ptr=3581&view=de Arthur Cayley Letters] to [[Robert Harley (mathematician)|Robert Harley]], 1859–1863. Available online through Lehigh University's [http://digital.lib.lehigh.edu/remain/index.html I Remain: A Digital Archive of Letters, Manuscripts, and Ephemera].
* {{cite journal|author= Salmon, George|author-link=George Salmon|title=Science Worthies. XXII.—Arthur Cayley|journal=Nature|volume=28|date=20 September 1883|pages=481–485|url=https://books.google.com/books?id=e30CAAAAIAAJ&pg=PA481|doi=10.1038/028481a0|doi-access=free}}
* {{cite journal|author=Scott, Charlotte Angas|author-link=Charlotte Scott|title=Arthur Cayley. Born August 16th, 1821. Died January 26th, 1895|journal=Bull. Amer. Math. Soc.|year=1895|volume=1|issue=6|pages=133–141|mr=1557369|doi=10.1090/s0002-9904-1895-00261-x|doi-access=free}}

{{Copley Medallists 1851-1900}}
{{De Morgan Medallists}}

{{Authority control}}

[[File:PD-icon.svg|12px|class=noviewer|link=|alt=Public Domain]]&nbsp;''This article incorporates text from the 1916 ''Lectures on Ten British Mathematicians of the Nineteenth Century'' by [[Alexander Macfarlane]], which is in the [[public domain]]''.

{{DEFAULTSORT:Cayley, Arthur}}
[[Category:1821 births]]
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[[Category:19th-century English mathematicians]]
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[[Category:Graph theorists]]
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[[Category:Historical treatment of octonions]]