CAYLEY, ARTHUR

born Aug. 16, 1821, Richmond, Surrey, Eng. died Jan. 26, 1895, Cambridge, Cambridgeshire English mathematician who played a leading role in founding the modern British school of pure mathematics. Cayley was born during a periodic visit of his family to England from Russia, where his father was engaged in trade. Cayley's remarkable mathematical ability became apparent by his skill as a child in doing complex calculations for amusement. On the advice of school authorities, his father, who had by then settled in England, enrolled him in May 1839 at Trinity College, University of Cambridge, where he mastered Greek, French, German, and Italian and distinguished himself in mathematics. Following his graduation in 1842, he obtained a three-year appointment at Trinity, which allowed him to begin work on the problems in mathematics that occupied his attention for the next 50 years. During this extremely productive period, he also began a lifelong interest in mountaineering, painting, and travel. Because no positions in mathematics were open to him when his term ended in 1845, he entered Lincoln's Inn, London, to prepare for a legal career. Admitted to the bar in 1849, Cayley earned just enough by practicing law during the next 14 years to allow him to pursue his interest in mathematics. During this time Cayley wrote his brilliant mathematical papers. In 1850 he met James Joseph Sylvester, a fellow lawyer and mathematician, and the two thenceforth spent much time in enthusiastic collaboration. Cayley's work treated nearly every subject of pure mathematics. The concept that the order of points formed by intersecting lines is always invariant, regardless of spatial transformations, is an application of the theory of algebraic invariance, which he originated and developed with encouragement from Sylvester. This concept is of importance in working out spacetime relationships in physics. Cayley's development of the geometry of spaces of any number of dimensions is also significant in conceptualizing four dimensions (spacetime) in relativity and in going beyond the dependence on points and lines as elements by which geometric space is constructed. Cayley also developed the algebra of matrices, which are arrays of numbers in rows and columns, in which the order and direction of multiplication determines the quantitative result. This tool was used by the German physicist Werner Heisenberg in 1925 for his work in quantum mechanics. Cayley also prepared the way for the idea that Euclidean and non-Euclidean geometries are special cases of the same kind of geometry. He did this by devising a means of uniting projective geometry, which is dependent upon invariant properties of figures, and metrical geometry, which is dependent upon sizes of angles and lengths of lines. He also prepared two reportson theoretical dynamics and on the mean motion of the Moonfor the British Association for the Advancement of Science (1857, 1862). Cayley practiced law until 1863, when he was elected to the new Sadlerian chair of pure mathematics at Cambridge. He married Susan Moline the same year. From the time of his arrival at Cambridge until his death he was constantly engaged in mathematical investigation. He was also influential in assisting women to be admitted as students for the first time. His lectures at Cambridge attracted very few students; among them, however, was A.R. Forsyth, who succeeded him in the Sadlerian chair and, by introducing the new theory of functions that had been making progress in France and Germany, helped to bring English mathematics back into the mainstream of European trends. In 188182 Cayley lectured at Johns Hopkins University in Baltimore on Abelian functionsa means of combining numbers such that the result of mathematical treatment is independent of the order. At Johns Hopkins he again met his friend Sylvester, who had become professor there in 1876. Cayley was the recipient of nearly every academic distinction that can be conferred upon an eminent man of science: honorary degrees from several universities, election as fellow or foreign corresponding member of the academies of several countries, and the Copley Medal in 1883 from the Royal Society of London. At various times he was president of the Cambridge Philosophical Society, of the London Mathematical Society, and of the Royal Astronomical Society. Additional reading The basic biography is by A.R. Forsyth in vol. 8 of The Collected Mathematical Papers of Arthur Cayley (1895). E.T. Bell, Men of Mathematics, ch. 21 (1937, reissued 1986), studies Cayley and Sylvester.