# Cayley, Arthur (DNB01)

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**CAYLEY**, ARTHUR (1821–1895), mathematician, the second son of Henry Cayley by his wife Maria Antonia Doughty, was born at Richmond in Surrey on 16 Aug. 1821. He entered Trinity College, Cambridge, in 1838, and became scholar of the college in 1840. In 1842 he graduated as senior wrangler, and was awarded the first Smith's prize immediately afterwards; and he was admitted to a Trinity fellowship on 3 Oct. in that year. He remained in Cambridge for a few years, giving himself up chiefly to mathematical research, and laying the foundation of several ranges of investigation which occupied him throughout his life. No congenial appointment, however, offered itself which was sufficient to keep him in residence; it thus became necessary to choose some profession. He selected law, left Cambridge in 1846, was admitted student of Lincoln's Inn on 20 April 1846, and was called to the bar on 3 May 1849. He devoted himself strictly to conveyancing; yet, instead of attempting to secure a large practice, he carefully limited the amount of work he would undertake. He made a distinct reputation by the excellence of his drafts, and it was asserted that, had he cared, he might have achieved a high legal position; but during the whole of his legal career he spent his jealously guarded leisure in the pursuit of mathematics.

Cayley remained at the bar for fourteen years. As an indication of his mathematical activity during this period, it may be sufficient to mention that he published more than two hundred mathematical papers, which include some of his most brilliant discoveries. A change made in the constitution of the Sadlerian foundation at Cambridge led to the establishment of the Sadlerian professorship of pure mathematics in that university; and on 10 June 1863 Cayley was elected into the professorship, an office which he held for the rest of his life. Henceforward he lived in the university, often taking an important share in its administration, but finding his greatest happiness in the discharge of his statutory duty 'to explain and teach the principles of pure mathematics, and to apply himself to the advancement of that science. Such a life naturally was of a quiet tenor, and Cayley did not possess the ambition of playing a prominent part in public life. Indeed, it was seldom that duties fell to him which brought him into popular notice; perhaps the most conspicuous exception was his presidency of the British Association in 1883. Scientific honours came to him in copious measure. He was made an honorary fellow of Trinity in 1872, and three years later was made an ordinary fellow once more, his first tenure having lapsed in 1852. He received honorary degrees from many bodies, among others from Oxford, Dublin, Edinburgh, Gottingen, Heidelberg, Leyden, and Bologna, as well as from his own university. From the Royal Society of London (of which he was elected fellow on 3 June 1852) he received a Royal medal in 1859 and the Copley medal in 1882, the latter being the highest honour which that body can bestow. In addition to membership of all the leading scientific societies of his own country, he was an honorary foreign member of the French Institute and of the academies of Berlin, Gottingen, St. Petersburg, Milan, Rome, Leyden, Upsala, and Hungary; and he accepted an invitation from the Johns Hopkins University, Baltimore, to deliver a special course of lectures there, discharging this office between December 1881 and June 1882. His life pursued an even scientific course, and his productive activity in mathematics was terminated only by his death, which occurred at Cambridge on 26 Jan. 1895. He is buried in the Mill Road cemetery, Cambridge. His portrait, painted by Mr. Lowes Dickinson in 1874, hangs in the dining hall of Trinity college; and a bust, by Mr. Henry Wiles, was placed in 1888 in the library of that college.

Cayley contributed to nearly every subject in the range of pure mathematics, and some of its branches owe their origin to him. Conspicuously among these may be cited the theory of invariants and covariants; the general establishment of hypergeometry on broad foundations, and specially the introduction of 'the absolute' into the discussion of metrical properties; the profound development of branches of algebra, which first were explained in a memoir on matrices; contributions to the theory of groups of operations; and advances in the theory of the solution of the quintic equation. Not less important were his contributions to the theory of analytical geometry, alike in regard to curves and to surfaces. There is hardly an important question in the whole range of either subject in the solution of which he has not had some share. Nor is it to the various theories in pure mathematics alone that he contributed. His services in the region of theoretical astronomy were of substantial importance; and in one instance he was enabled, by an elaborate piece of refined analysis, to take part in settling a controversy between his friend, John Couch Adams [q. v. Suppl.], and some French astronomers. Also, in framing any estimate of his work, account should be taken of the various papers he wrote upon theoretical dynamics, and in particular of two reports upon that subject presented to the British Association. It remains, of course, with the future to assign him his position among the masters of his science. By his contemporaries he was acknowledged one of the greatest mathematicians of his time.

As regards his publications, the body is to be found in the memoirs contributed, through more than fifty years, to various mathematical journals and to the proceedings of learned societies. His papers, amounting to more than nine hundred in number, have been collected and issued in a set of thirteen volumes, together with an index volume, by the Cambridge University Press (1889-98). Cayley himself published only one separate book, 'A Treatise on Elliptic Functions' (Cambridge, 1876; a second edition, with only slight changes, was published in 1895 after his death).

[Proceedings of the Royal Sec. vol. lviii. (1895), pp. i-xliii, reprinted as a preface to vol. viii. of the Collected Mathematical Papers, as just quoted. The exact dates and places of the publication of his memoirs are stated in connection with each paper contained in the thirteen volumes. Prefixed to vol. xi. is an excellent photograph of Cayley by Mr. A. G. Dew-Smith.]