Arthur Cayley

Arthur Cayley FRS (/ˈkeɪli/; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics.

As a child, Cayley enjoyed solving complex maths problems for amusement. He entered Trinity College, Cambridge, where he excelled in Greek, French, German, and Italian, as well as mathematics. He worked as a lawyer for 14 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.^[1] He was the first to define the concept of a group in the modern way—as a set with a binary operation satisfying certain laws.^[2] Formerly, when mathematicians spoke of "groups", they had meant permutation groups. Cayley tables and Cayley graphs as well as Cayley's theorem are named in honour of Cayley.

Early years

Arthur Cayley was born in

Around 1860, Cambridge University's Lucasian professorship of pure mathematics (Newton's chair) was supplemented by the new 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 a ordinary fellow, a paid position. About this time his friends subscribed for a presentation portrait. Maxwell wrote an address praising Cayley's principlal works, including his Chapters on the Analytical Geometry of ${\displaystyle n}$ dimensions; On the theory of Determinants; Memoir on the theory of Matrices; Memoirs on skew surfaces, otherwise Scrolls; and On the delineation of a Cubic Scroll.^[6]

In addition to his work on algebra, Cayley made fundamental contributions to algebraic geometry. Cayley and Salmon discovered the 27 lines on a cubic surface. Cayley constructed the Chow variety of all curves in projective 3-space.^[7] He founded the algebro-geometric theory of ruled surfaces. His contributions to combinatorics include counting the n^n–2 trees on n labeled vertices by the pioneering use of generating functions.

In 1876 he published a Treatise on

, 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.^[8]

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. 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 74. 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 21^st 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 water-colour painting, which he found useful sometimes in making mathematical diagrams.

Legacy

Cayley is buried in the 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.^[9]

A number of mathematical terms are named after him:

• Cayley's theorem
• Cayley–Hamilton theorem in linear algebra
• Cayley–Bacharach theorem
• Grassmann–Cayley algebra
• Cayley–Menger determinant
• Cayley diagrams – used for finding cognate linkages
  in mechanical engineering
• Cayley–Dickson construction
• Cayley algebra (Octonion)
• Cayley graph
• Cayley numbers
• Cayley's sextic
• Cayley table
• Cayley–Purser algorithm
• Cayley's formula
• Cayley–Klein metric
• Cayley–Klein model of hyperbolic geometry
• Cayley's Ω process
• Cayley surface
• Cayley transform
• Cayley's nodal cubic surface
• Cayley's ruled cubic surface
• The 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

Bibliography

• Cayley, Arthur (2009) [1876], An elementary treatise on elliptic functions, Cornell University Library,
  MR 0124532
• Cayley, Arthur (2009) [1889], The Collected Mathematical Papers, Cambridge Library Collection – Mathematics, vol. 14 volumes,
  ISBN 978-1-108-00507-4, archive
• Cayley, Arthur (1894), The principles of book-keeping by double entry, Cambridge University Press

See also

• List of things named after Arthur Cayley

References

1. ^ 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."
2. ^ Cayley (1854) "On the theory of groups, as depending on the symbolic equation θⁿ = 1," Philosophical Magazine, 4th series, 7 (42) : 40–47. However, see also the criticism of this definition in: MacTutor: The abstract group concept.
3. ^ "Cayley, Arthur (CLY838A)". A Cambridge Alumni Database. University of Cambridge.
4. ^ The Records of the Honorable Society of Lincoln's Inn Vol II, Admission Register 1420 - 1893. London: Lincoln's Inn. 1896. p. 226.
5. ^ ^{a b} Forsyth, Andrew Russell (1901). "Cayley, Arthur" . In Lee, Sidney (ed.). Dictionary of National Biography (1st supplement). London: Smith, Elder & Co.
6. ^ "To the Committee of the Cayley Portrait Fund", 1874
7. ^ 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.
8. ^ "A. Cayley (1821 - 1895)". Royal Netherlands Academy of Arts and Sciences. Retrieved 19 April 2016.
9. ^ "Trinity College, University of Cambridge". BBC Your Paintings. Archived from the original on 11 May 2014. Retrieved 12 February 2018.

Sources

• Cayley, Arthur (1996) [1883], "Presidential address to the British Association", in Ewald, William (ed.), From Kant to Hilbert: a source book in the foundations of mathematics. Vol. I, II, Oxford Science Publications, The Clarendon Press
  MR 1465678, Reprinted in collected mathematical papers volume 11
• Crilly, Tony (1995), "A Victorian Mathematician: Arthur Cayley (1821–1895)",
  S2CID 188006684
• Crilly, Tony (2006), Arthur Cayley. Mathematician laureate of the Victorian age,
  MR 2284396
• ISBN 978-1-112-28306-2 (complete text at Project Gutenberg
  )

External links

Wikimedia Commons has media related to Arthur Cayley.

Wikisource has original works by or about:
Arthur Cayley

• O'Connor, John J.; Robertson, Edmund F., "Arthur Cayley", MacTutor History of Mathematics Archive, University of St Andrews
• Arthur Cayley at the Mathematics Genealogy Project
• ScienceWorld
  .
• Arthur Cayley Letters to Robert Harley, 1859–1863. Available online through Lehigh University's I Remain: A Digital Archive of Letters, Manuscripts, and Ephemera.
• doi:10.1038/028481a0
  .
• MR 1557369
  .

 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.