**CAYLEY, ARTHUR** (1821–1895), English mathematician, was born at Richmond, in Surrey, on the 16th of August 1821, the second son of Henry Cayley, a Russian merchant, and Maria Antonia Doughty. His father, Henry Cayley, retired from business in 1829 and settled in Blackheath, where Arthur was sent to a private school kept by the Rev. G. B. F. Potticary; at the age of fourteen he was transferred to King’s College school, London. He soon showed that he was a boy of great capacity, and in particular that he was possessed of remarkable mathematical ability. On the advice of the school authorities he was entered at Trinity College, Cambridge, as a pensioner. He was there coached by William Hopkins of Peterhouse, was admitted a scholar of the college in May 1840, and graduated as senior wrangler in 1842, and obtained the first Smith’s Prize at the next examination. In 1842, also, he was elected a fellow of Trinity, and became a major fellow in 1845, the year in which he proceeded to the M.A. degree. He was assistant tutor of Trinity for three years. In 1846, having decided to adopt the law as a profession, he left Cambridge, entered at Lincoln’s Inn, and became a pupil of the conveyancer Mr Christie. He was called to the bar in 1849, and remained at the bar fourteen years, till 1863, when he was elected to the new Sadlerian chair of pure mathematics in the university of Cambridge. He settled at Cambridge in the same year, and married Susan, daughter of Robert Moline of Greenwich. He continued to reside in Cambridge and to hold the professorship till his death, which occurred on the 26th of January 1895. From the time he went first to Cambridge till his death he was constantly engaged in mathematical investigation. The number of his papers and memoirs, some of them of considerable length, exceeds 800; they were published, at the time they were composed, in various scientific journals in Europe and America, and are now embodied, through the enterprise of the syndics of the Cambridge University Press, in thirteen large quarto volumes. These form an enduring monument to his fame. He wrote upon nearly every subject of pure mathematics, and also upon theoretical dynamics and spherical and physical astronomy. He was quite as much a geometrician as he was an analyst. Among his most remarkable works may be mentioned his ten memoirs on quantics, commenced in 1854 and completed in 1878; his creation of the theory of matrices; his researches on the theory of groups; his memoir on abstract geometry, a subject which he created; his introduction into geometry of the “absolute”; his researches on the higher singularities of curves and surfaces; the classification of cubic curves; additions to the theories of rational transformation and correspondence; the theory of the twenty-seven lines that lie on a cubic surface; the theory of elliptic functions; the attraction of ellipsoids; the British Association Reports, 1857 and 1862, on recent progress in general and special theoretical dynamics, and on the secular acceleration of the moon’s mean motion. He is justly regarded as one of the greatest of mathematicians. Competent judges have compared him to Leonhard Euler for his range, analytical power and introduction of new and fertile theories. He was the recipient of nearly every academic distinction that can be conferred upon an eminent man of science. Amongst others may be noted honorary degrees by the universities of Oxford, Dublin, Edinburgh, Göttingen, Heidelberg, Leiden and Bologna. He was fellow or foreign corresponding member of the French Institute, the academies of Berlin, Göttingen, St Petersburg, Milan, Rome, Leiden, Upsala and Hungary; and he was nominated an officer of the Legion of Honour by President Carnot. At various times he was president of the Cambridge Philosophical Society, of the London Mathematical Society and of the Royal Astronomical Society. He was elected a fellow of the Royal Society in 1852, and received from that body a Royal medal in 1859 and the Copley medal in 1882. He also received the De Morgan medal from the London Mathematical Society, and the Huygens medal from Leiden. His nature was noble and generous, and the universal appreciation of this fact gave him great influence in his university. His portrait, by Lowes Dickinson, was placed in the hall of Trinity College in 1874, and his bust, by Henry Wiles, in the library of the same college in 1888. (P. A. M.)