by A. J. Crilly
© Oxford University Press 2004 All rights reserved
Cayley, Arthur (1821-1895), mathematician, was born on 16 August 1821 in Richmond, Surrey, the second son of five children of Henry Cayley (1768-1850), Russia merchant, and his wife, Maria Antonia Doughty (1794-1875). His parents lived in St Petersburg and Arthur was born on one of their summer visits to England. His grandfather John Cayley (1730-1795) served as consul-general in St Petersburg. Arthur Cayley was distantly related (fourth cousin) to Sir George Cayley FRS (1773-1857), inventor and aeronautical pioneer; his younger brother, Charles Cayley (1823-1883), became a noted translator and scholar. The family returned to England permanently in 1828. Cayley's father, aged sixty, established himself at 29 York Terrace, Regent's Park, became a director of the London Assurance Corporation, and was active in the reorganization of the Baltic exchange.
Education and first mathematics
After attending a private school at Blackheath, Cayley entered the senior department of King's College, London, at the unusually young age of fourteen. He consistently gained school prizes and in his final year won the chemistry prize in competition with specialist science students. He entered Trinity College, Cambridge, in October 1838 as a pensioner and continued his upward academic path by winning the college prizes. After the mathematical tripos examination in January 1842 he became senior wrangler and won a first Smith's prize, the result of a higher-level competitive examination taken by the most successful students in the tripos. He became a fellow of Trinity College the same year.
While an undergraduate Cayley published a paper on determinants, a subject connected with solving linear equations and one which was widely studied in the nineteenth century. Cayley introduced the array of coefficients within the now familiar vertical lines. His fresh choice of notation and his delight in coupling algebra with geometry indicates a rich vein in his work. Determinants became one of his favourite subjects and as a newly elected fellow of the Cambridge Philosophical Society he went on to describe n-dimensional determinants in a wide-ranging paper early in 1843. Although young, Cayley was familiar with the works of the European masters: Gabriel Cramer, étienne Bezout, Pierre Simon Laplace, Augustin Cauchy, and his 'illustrious' Carl Gustav Jacob Jacobi.
In 1844 Cayley discovered the work of George Boole on linear transformations, and in correspondence with him made the first steps in what was to become his best-known contribution to mathematics--invariant theory, as this subject became known in the 1850s. The general notion of an 'invariant' in mathematics is of a value or algebraic form dependent on variables subject to change but which itself does not change. For example, the area of a triangle in which a vertex is allowed to vary along a line parallel to its base is an invariant since it has the same value whatever the actual position of the vertex. Recognizing that a determinant was a special case of an invariant, Cayley introduced the term 'hyperdeterminant'. During the course of the nineteenth century there emerged two principal methods for generating invariants and it is significant that the seeds of both methods can be found in Cayley's early papers.
The year 1845 was an annus mirabilis in which Cayley published thirteen papers on a wide range of subjects. It was typical of his ability to recognize promising ideas that he was the first to write a paper on quaternions after their discovery by Sir William Rowan Hamilton in October 1843. Shortly afterwards, Cayley introduced the algebra of octonions as a generalization of quaternions. Invariant theory and quaternions became the two most intensively studied algebraic subjects in Britain during the second half of the century. Cayley made important contributions to each and was especially alive to the links between these two branches of mathematics.
In 1846 Cayley left Trinity College and entered Lincoln's Inn as a pupil of the celebrated conveyancing counsel Jonathan Henry Christie. He was called to the bar on 3 May 1849. During this period he maintained his research in mathematics and met his lifelong friend James Joseph Sylvester, who was also studying for the bar. In 1850 Cayley's father died, leaving him effectively head of the family. His brother lost money through financial speculation and his sisters, Sophia and Henrietta Caroline, were (and remained) unmarried. On 3 June 1852 Cayley was elected to the Royal Society and after only one year was proposed for a royal medal. Charles Darwin won one that year, but Cayley was awarded the medal in 1859.
From his days as a young fellow of Trinity College, Cayley had published regularly in the continental publications Journal des Mathématiques Pures et Appliquées and Journal für die Reine und Angewandte Mathematik. He performed the invaluable service of summarizing achievements of continental mathematicians by publishing extracts in the Cambridge Mathematical Journal and other English journals. In this vein he outlined the theory of groups instigated by évariste Galois about 1830 and, in a paper published in 1854, generalized it in the context of the calculus of operations. Cayley also published a major memoir on matrices in 1858. In this theory arrays of quantities called matrices are treated as single entities and the subject later became pivotal in the mathematics of the twentieth century. Other mathematicians had recognized that matrices were important, but Cayley felt the impetus to draw together the main ideas from which the theory could advance. He also discovered, independently of Hamilton, the striking theorem in the theory of matrices now known as the Cayley-Hamilton theorem.
Cayley's already wide knowledge of mathematics was matched by research qualities rarely found in one person. Apart from a sure grasp of algebraic principles and a refined geometrical intuition he was naturally inclined to the calculatory side of mathematics. This found expression in much of his algebra but especially in combinatorial mathematics (the subject then called 'tactic'). He showed great facility for dealing with generating functions and his expertise found applications in counting graphical structures. In the 1870s he applied these techniques to the enumeration of carbon/hydrogen molecules, results which were of considerable interest to chemists of that time. This same expertise in dealing with generating functions was fully utilized in invariant theory.
Work on invariant theory, 1850-1880
At the beginning of the 1850s Cayley entered a period of rich creativity. He had met Sylvester, and the two young mathematicians, in collaboration and in competition with each other, worked to establish invariant theory as a recognizable field of study. Cayley reworked his earlier ideas on hyperdeterminants and formalized the theory. In this, he focused on an invariant (the term replacing his earlier 'hyperdeterminant') and the more general notion of a covariant of a binary form (an algebraic form with two variables). He approached the subject through the study of differential operators in which he regarded invariants and covariants as those algebraic forms which were reduced to zero (annihilated) by two specific differential operators. The first paper read to the Royal Society on his assuming membership was an introductory memoir on quantics ('First memoir'). By introducing the term 'quantic', Cayley signalled his intention of formulating the theory afresh, synthesizing earlier results and promoting a new technical vocabulary. Some of the fanciful linguistic constructs employed by Cayley met with resistance from co-workers and have since passed into oblivion but others (for example, Sylvester's Hessian, Jacobian, invariant, covariant, annihilator) became established modern mathematical usage.
From 1854 to 1878 Cayley published a path-breaking series of memoirs on quantics in which he remodelled and extended his earlier work. He, Sylvester, and George Salmon from Dublin were dubbed the 'Invariant Trinity' by the mathematical community. In the period 1850-54, mathematicians from France, Italy, and Germany joined the study of invariant theory and an extensive body of knowledge soon developed.
Cayley's 'Second memoir' (1856) contained the algorithm by which invariants and covariants can be calculated. It is central to invariant theory and in subsequent memoirs Cayley put it to work in establishing miscellaneous results for binary forms. The importance of the 'Sixth memoir', published in 1859, overshadowed the computational gains of these earlier memoirs. In this paper he defined a notion of distance in projective geometry in terms of an absolute conic. This was a brilliant observation since projective geometry is a subject ostensibly free of metrical considerations such as size of angle and distance between points. Cayley's definition was later used as the basis for Felix Klein's schema for organizing non-Euclidean geometry (1872). Writing in 1859, Cayley had not realized the connection between the absolute and non-Euclidean geometry.
Marriage, teaching, and research
In 1857 Cayley became a fellow of the Royal Astronomical Society. He was a council member for thirty-five years and editor of the Monthly Notices, 1859-81, except during 1872-4 when he served as president. Cayley specialized in mathematical astronomy and carried out research in lunar studies, a dominant subject of research in the 1860s. One of his most notable pieces of work was verifying a controversial conclusion reached by John Couch Adams on the secular acceleration of the moon's mean motion. Typically, the confirmation of Adams's work came at the end of extensive algebraic calculations.
Like many of his generation from Oxford and Cambridge, Cayley was a keen alpinist and member of the Alpine Club. Occasionally his love of mountaineering and hillwalking found expression in papers on topological ideas.
By the mid-1850s Cayley's natural inclination was for a life in academe. He was a successful conveyancing barrister but he made no attempt to establish a large legal practice. Suitable positions in higher education were few in the middle years of the century. He unsuccessfully applied for the Lowndean chair at Cambridge and the chair of natural philosophy at Aberdeen, and suffered other disappointments before, on 10 June 1863, he was elected first Sadleirian professor at Cambridge. On 8 September 1863 Cayley married Susan Moline (c.1831-1923). They had two children, Henry (1870-1949) and Mary (1872-1950). Henry Cayley studied mathematics at Cambridge (twenty-fourth wrangler in 1890), but he felt in the shadow of his father and turned from mathematics to become an architect. Both Henry and Mary Cayley died childless.
In 1868 Cayley was surprised to learn that the German mathematician Paul Gordan had proved that all binary quantics admit a finite system of covariants. This contradicted Cayley's earlier work published in his 'Second memoir'. The German school had developed a powerful calculus based on one of Cayley's earlier discarded methods. Their calculus was expressed in a succinct notation but it is notable that Cayley did not switch to the new method despite its success. In the light of Gordan's results Cayley completed his work on the binary quintic polynomial using his own methods and published it in the 'Ninth memoir' (1871).
In 1876 Cayley published his only book: A Treatise on Elliptic Functions (2nd edn, 1895). Sylvester had been appointed to the Johns Hopkins University in the same year and Cayley visited him there for six months in 1882. On his return he was awarded the Royal Society's Copley medal. For virtually all his life, Cayley had been England's leading pure mathematician. In 1883 he was president of the British Association for the Advancement of Science. Against the trend of popular lectures for such meetings, Cayley delivered an address in the 'severely scientific class' (Forsyth, xxi), and, with his encyclopaedic knowledge of mathematics, surveyed the mathematical accomplishments of the period.
During Cayley's years at Cambridge his influence was limited by the overbearing dominance in the examination system of the mathematical tripos. Student success depended on being well drilled in examination skills, and undergraduates perceived no need to learn mathematics which was not part of the curriculum. Cayley's lectures were few in number and sparsely attended, but several of the best British mathematicians were influenced by him. William Kingdon Clifford made a special study of Cayley's topics, and James Whitbread Lee Glaisher, son of the celebrated balloonist and a specialist in number theory and elliptic functions, was a protégé. J. J. Thomson, discoverer of the electron, remembered attending one of Cayley's lectures where the audience of three sat round a table and his lot was to follow Cayley's algebraic manipulations upside down. Andrew Russell Forsyth (1858-1942) was perhaps the student whose mathematical interests coincided most with Cayley's. He became adept at the manipulations and expertise in invariant theory and adopted Cayley's particular approach to the subject. He succeeded Cayley as second Sadleirian professor in 1895.
Cayley's view on the beauty of mathematics permeated all his thinking. He never relinquished his appreciation of Descartes and his introduction of co-ordinates. His admiration of Euclid, and in particular Euclid's Elements, Book 5 (on the theory of proportion), was unbounded, but his conviction that Euclid should be studied undiluted hindered the reform of geometrical teaching in England.
Cayley continued mathematical production during his last years. Postgraduate research began too late for him to found a school of mathematics as might have been expected of the doyen of British pure mathematics. Cayley was Britain's outstanding pure mathematician of the nineteenth century. An algebraist, analyst, and geometer, he was able to link these vast domains of study. More than fifty concepts and theorems of mathematics bear his name. By the end of his life he was revered by mathematicians the world over. He had published almost one thousand mathematical papers. Shoals of academic honours came his way. In addition to the Copley medal (1882) he was awarded the honorary degree of ScD at Cambridge (1887) and made a member of the French Légion d'honneur. He died at his home, Garden House, Cambridge, on 26 January 1895. He was buried on 2 February 1895 in Mill Road cemetery, Cambridge, but the headstone has not survived.
A. J. CRILLY
Sources
A. R. F. [A. R. Forsyth], PRS, 58 (1895), i-xliv
J. D. North, 'Cayley, Arthur', DSB
The collected mathematical papers of Arthur Cayley, ed. A. Cayley and A. R. Forsyth, 14 vols. (1889-98)
G. Salmon, 'Science worthies: Arthur Cayley', Nature, 28 (1883), 481-5
T. Crilly, 'The rise of Cayley's invariant theory, 1841-62', Historia Mathematica, 13 (1986), 241-54
T. Crilly, 'The decline of Cayley's invariant theory, 1863-95', Historia Mathematica, 15 (1988), 332-47
K. H. Parshall, 'Towards a history of nineteenth-century invariant theory', The history of modern mathematics, ed. D. E. Rowe and J. McCleary, 1: Ideas and their reception (1989), 157-206, esp. 162-170
J. Foster, ed., Pedigrees of the county families of Yorkshire, 3 (1874)
A. Macfarlane, Lectures on ten British mathematicians of the nineteenth century (1916)
C. A. Bristed, Five years in an English university, 2 vols. (1852)
C. B. Boyer, History of analytical geometry (1956)
S. Rothblatt, The revolution of the dons: Cambridge and society in Victorian England (1968)
The Times (28 Jan 1895)
Manchester Guardian (6 Feb 1895)
Cambridge Review (7 Feb 1895)
m. cert.
d. cert.
CGPLA Eng. & Wales (1895)
Archives
CUL, papers
U. Reading L., papers relating to Hansen's lunar theory | Col. U., D. E. Smith MSS
CUL, letters to Lord Kelvin
CUL, corresp. with George Stokes
RAS, letters to Royal Astronomical Society
RS, corresp. with Sir J. F. W. Herschel
St John Cam., corresp. with James Sylvester
Trinity Cam., Boole-Cayley MSS
UCL, letters to Thomas Hirst
University of Göttingen, Klein-Cayley MSS
Likenesses
portrait, 1842 (The senior wrangler), repro. in V. A. Huber, The English universities, 3 vols. (1843)
L. Dickinson, oils, 1874, Trinity Cam.
W. H. Longmaid, oils, 1884, Trinity Cam.
Maull & Polyblank, photograph, NPG [see illus.]
H. Wiles, marble bust, Trinity Cam.; plaster model, Philosophical Library, Cambridge
photograph, Trinity Cam., Wren Library
portrait (of Cayley?), repro. in Scripta Mathematica, 6, 32-6
three photographs, RS
Wealth at death
£23,979 2s. 6d.: resworn probate, June 1895, CGPLA Eng. & Wales
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