Arthur Cayley

Arthur Cayley (16 August 1821 – 26 January 1895) was an English mathematician whose work centered on algebra. He played a key role in establishing the modern British school of pure mathematics and spent 35 years as a professor at Trinity College, Cambridge. He is known for the Cayley–Hamilton theorem, which states that every square matrix satisfies its own characteristic polynomial; he proved this for 2×2 and 3×3 matrices. He also introduced the modern definition of an abstract group—a set with a binary operation obeying specific rules—distinct from Évariste Galois’ earlier idea of permutation groups. His name appears in group theory with Cayley tables, Cayley graphs, and Cayley’s theorem, and in combinatorics with Cayley’s formula.

Cayley was born in Richmond, London, on 16 August 1821. His father, Henry Cayley, a merchant based in Saint Petersburg, Russia, was a distant cousin of the aeronautical engineer George Cayley and came from an old Yorkshire family. His mother was Maria Antonia Doughty; though some writers describe her as Russian, her father’s name suggests English roots. His brother, Charles Bagot Cayley, became a linguist. Arthur spent his first eight years in Saint Petersburg. In 1829 his family settled permanently in Blackheath, London, where he attended a private school. At age 14 he entered King’s College School. He delighted in difficult math problems, and the school’s headmaster noticed his talent, advising his father to send him to Cambridge rather than train him for the family business.

At 17, Cayley began at Trinity College, Cambridge. He excelled not only in mathematics but also in Greek, French, German, and Italian. The Analytical Society had already succeeded in reforming Cambridge mathematics, and the *Cambridge Mathematical Journal* had been launched by Gregory and Robert Leslie Ellis. By age 20, Cayley had contributed three papers to that journal, inspired by reading Lagrange’s *Mécanique analytique* and works by Laplace. His tutor was George Peacock, and his private coach was William Hopkins. He finished his undergraduate studies as Senior Wrangler and won the first Smith’s prize. He then earned an M.A. and a fellowship through competitive examination. He remained at Cambridge for four more years, tutoring a few pupils but mainly writing 28 memoirs for the *Mathematical Journal*.

Because his fellowship had a limited term, Cayley needed a career. Like the mathematician De Morgan, he chose law, entering Lincoln’s Inn in London on 20 April 1846 at age 24. He specialized in conveyancing. While training for the bar, he traveled to Dublin to hear William Rowan Hamilton lecture on quaternions. His friend J. J. Sylvester, five years his senior at Cambridge, was then an actuary in London; the two would walk around Lincoln’s Inn’s courts discussing invariants and covariants. During these fourteen years as a lawyer, Cayley published between two and three hundred papers.

Around 1860, Cambridge created the Sadleirian professorship in pure mathematics, funded by a bequest from Lady Sadleir, to supplement the Lucasian chair. Cayley, then 42, became its first holder. His duties were to teach pure mathematics and advance the field. He left a profitable legal practice for a modest salary but never regretted it, as it let him focus on his true passion. He married, settled in Cambridge, and enjoyed a very happy home life—unlike Hamilton. Sylvester, his bachelor friend, envied Cayley’s peaceful family existence, remarking that he himself had to fight the world alone. Initially, the Sadleirian professor lectured only one term per year; a university financial reform in 1886 extended this to two terms. For many years only a few students who had finished their exams attended his lectures, but after the reform about fifteen came. He usually lectured on whatever he was currently researching. To advance mathematical science, he published a long series of memoirs covering all of pure mathematics. He also became a standing referee for mathematical papers for many societies at home and abroad. In 1872 he was made an honorary fellow of Trinity College, and three years later an ordinary fellow with a salary. Around that time, friends subscribed for a presentation portrait; Maxwell wrote an address praising Cayley’s major works, including his *Chapters on the Analytical Geometry of n Dimensions*, *On the Theory of Determinants*, *Memoir on the Theory of Matrices*, and memoirs on skew surfaces (scrolls) and cubic scrolls. Beyond algebra, Cayley made fundamental contributions to algebraic geometry. With Salmon, he discovered the 27 lines on a cubic surface. He constructed the Chow variety of all curves in projective 3-space and founded the algebro-geometric theory of ruled surfaces. In combinatorics, he used generating functions to count the n^(n−2) trees on n labeled vertices. In 1876 he published a *Treatise on Elliptic Functions*. He also took a strong interest in the movement for university education for women.