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 tradition of pure mathematics and spent 35 years as a professor at Trinity College, Cambridge. He is known for putting forward what is now called the Cayley–Hamilton theorem, which states that every square matrix satisfies its own characteristic polynomial, and he proved it for 2×2 and 3×3 matrices. He also introduced the modern definition of an abstract group—a set with a binary operation that follows specific rules—distinguishing it from Évariste Galois’ earlier idea of permutation groups. In group theory, his name appears in Cayley tables, Cayley graphs, and Cayley’s theorem, while in combinatorics, Cayley’s formula bears his name.

Cayley was born in Richmond, London, to Henry Cayley, a merchant who had settled in Saint Petersburg, Russia, and who was a distant cousin of the aeronautical engineer George Cayley. His mother was Maria Antonia Doughty, whose father’s name suggests English roots despite some writers describing her as Russian. His brother, Charles Bagot Cayley, became a linguist. Arthur spent his first eight years in Saint Petersburg before his family moved permanently to Blackheath, London, in 1829, where he attended a private school. At 14, he entered King’s College School. There, his talent for difficult math problems became clear, and the school’s master advised his father to send him to Cambridge rather than train him for the family business.

At 17, Cayley began his studies at Trinity College, Cambridge. He showed skill not only in mathematics but also in Greek, French, German, and Italian. The Cambridge Mathematical Journal had recently been founded, and at age 20 Cayley contributed three papers 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. After earning his M.A. and winning a fellowship through competitive exams, he stayed at Cambridge for four more years, tutoring a few students but mainly writing 28 memoirs for the Mathematical Journal.

Because his fellowship had a limited term, Cayley needed a career and chose law, entering Lincoln’s Inn in London at age 24. He specialized in conveyancing. During his legal training, he traveled to Dublin to hear William Rowan Hamilton lecture on quaternions. His friend J. J. Sylvester, an actuary in London, often walked with him around the courts of Lincoln’s Inn, discussing invariants and covariants. Over fourteen years in law, Cayley produced between two and three hundred mathematical papers.

Around 1860, Cambridge created the Sadleirian professorship in pure mathematics, and Cayley, then 42, became its first holder. His duties were to teach and advance pure mathematics. He left a profitable legal practice for a modest salary but never regretted the change, as it let him focus on mathematics. He married, settled in Cambridge, and enjoyed a happy home life—something Sylvester later envied. Initially, the Sadleirian professor lectured only one term per year, but a university 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 typically lectured on whatever he was currently researching.

Cayley published a long series of memoirs covering all areas of pure mathematics and served as a referee for mathematical papers for many societies. In 1872, he became an honorary fellow of Trinity College, and three years later a paid ordinary fellow. Around that time, friends commissioned a portrait, and Maxwell wrote an address praising Cayley’s major works, including his work on *n*-dimensional analytical geometry, determinants, matrices, skew surfaces, and cubic scrolls. Beyond algebra, Cayley made key 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 he founded the algebro-geometric theory of ruled surfaces. In combinatorics, he used generating functions to count the number of trees on labeled vertices. In 1876, he published a *Treatise on Elliptic Functions*. He also supported the movement for university education for women at Cambridge.