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Biography of Arthur Cayley (1821-1895)

By Prof. Steven H. Weintraub

Dept. of Mathematics

Lehigh University

Arthur Cayley (16 August 1821-26 January 1895) was one of the preeminent British mathematicians of the 19th century. His mathematical talent was recognized early. He entered Trinity College, Cambridge, as a student in 1838, at the age of 17, and upon his graduation he became a fellow at Trinity College. Upon the end of this fellowship Cayley pursued a career as a lawyer, but his first love was always for mathematics. In 1863 he was appointed the Sadleirian Professor of Pure Mathematics at Cambridge, a position he held until his death at the age of 73.

Cayley published 966 mathematical papers and one book during the course of his career, and his collected works (not including the book) comprise 13 volumes. His first published papers were written while he was a student, and he remained mathematically active up until his death. He received many awards for his work, including election to the Royal Society of London in 1852, the Copley Medal (the Royal Society's highest honor) in 1882, election as President of the British Association for the Advancement of Science in 1883, and honorary degrees from the universities of Oxford, Dublin, Edinburgh, Gottingen, Heidelberg, Leiden, and Bologna. From 1868 to 1870 he served as President of the London Mathematical Society (LMS), and in 1884 he was awarded the De Morgan medal by the LMS for his mathematical work.

Cayley's work spans a very wide range of fields within mathematics. These include many areas of algebra, including group theory, matrix theory, quadratic forms, and especially algebraic geometry; the theory of elliptic functions (the subject of his book "An Elementary Treatise on Elliptic Functions") and theta functions; definite integrals and differential equations; and, in the areas of applied mathematics, mechanics, dynamics and mathematical astronomy. His major papers were interspersed with a wide variety of notes on more minor topics. As a sampling of the work for which he is best known, there are: Cayley tables and Cayley's theorem in group theory (Every group is isomorphic to a subgroup of a symmetric group); the Cayley-Salmon theorem in algebraic geometry (A general cubic surface admits exactly 27 lines); and the Cayley-Hamilton theorem in matrix theory (Every matrix satisfies its characteristic polynomial).

One of the chief topics of Cayley's mathematical investigations was invariant theory, a branch of algebra. (Cayley coined the term "quantics" for his investigations here, though that term has not stuck.) Invariant theory was intensely studied throughout Europe in the 19th century, and Cayley was the leader of the British school of invariant theorists. Other mathematicans who worked in this field were the eminent mathematicians J. J. Sylvester (a close friend and colleague of Cayley's) and George Salmon, mentioned above. Cayley's approach to invariant theory involved prodigious algebraic calculations, some of which he performed himself and for some of which he obtained the help of others, including James Cockle and Robert Harley, mathematicians in their own right, and other lesser known figures. Cockle and Harley were well known in their own day (each of them having been elected to the Royal Society upon Cayley's nomination), but their work has since passed into obscurity.

The documents in this collection are related to Cayley's work in invariant theory.