Arthur Cayley

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Lore & Background

Arthur Cayley was born in Richmond, London, on 16 August 1821. His father, Henry Cayley, was a distant cousin of George Cayley, the aeronautics engineer, and settled in Saint Petersburg, Russia, as a merchant. His mother was Maria Antonia Doughty; according to some writers she was Russian, but her father's name indicates an English origin. Arthur spent his first eight years in Saint Petersburg, then in 1829 his family settled permanently at Blackheath, London, where he attended a private school. At age 14 he was sent to King's College School, where the school's master observed his mathematical genius and advised his father to educate him at the University of Cambridge rather than for business. At age 17 Cayley began residence at Trinity College, Cambridge, excelling in Greek, French, German, Italian, and mathematics. His tutor was George Peacock and his private coach was William Hopkins. He finished as Senior Wrangler and won the first Smith's prize. After taking his M.A. and winning a Fellowship, he resided at Cambridge for four years, preparing 28 memoirs for the Mathematical Journal. Because his fellowship had limited tenure, he chose a law career and was admitted to Lincoln's Inn, London on 20 April 1846. During fourteen years as a lawyer, he produced between two and three hundred papers, often discussing invariants with his friend J. J. Sylvester. Around 1860, Cayley became the first holder of the new Sadleirian professorship at Cambridge. He gave up his legal practice for a modest salary, married, and settled in Cambridge. He lectured on his current research, published a long series of memoirs, and served as referee for mathematical papers. In 1872 he was made an honorary fellow of Trinity College, and three years later an ordinary fellow. He also helped teach at Girton College and chaired the council of Newnham College. In 1881 he accepted an invitation from Johns Hopkins University to lecture in Baltimore for five months in 1882 on Abelian and Theta Functions.

Reader's Guide

Arthur Cayley's significance lies in his foundational contributions to modern algebra and algebraic geometry. He postulated the Cayley–Hamilton theorem, which states that every square matrix is a root of its own characteristic polynomial, and verified it for matrices of order 2 and 3. He was the first to define the concept of an abstract group—a set with a binary operation satisfying certain laws—distinct from Évariste Galois' permutation groups. This abstraction became central to group theory, with Cayley tables, Cayley graphs, and Cayley's theorem named after him. In combinatorics, Cayley's formula counts the number of trees on n labeled vertices. He also discovered, with Salmon, the 27 lines on a cubic surface, constructed the Chow variety of curves, and founded the algebro-geometric theory of ruled surfaces. His collected papers fill thirteen quarto volumes containing 967 papers, and his work continues to be cited in hundreds of mathematical papers in the 21st century. Cayley served as President of the British Association for the Advancement of Science in 1883, and his legacy includes the lunar crater Cayley and the Cayley Formation.

Did You Know?

• Cayley spent his first eight years in Saint Petersburg, Russia, because his father was a merchant there.
• At age 14, his schoolmaster at King's College School observed his mathematical genius and advised his father to send him to Cambridge instead of into business.
• Cayley produced between two and three hundred mathematical papers during his fourteen years working as a lawyer.
• He gave up a lucrative legal practice to become the first Sadleirian professor at Cambridge, taking a modest salary.
• Cayley's collected mathematical papers fill thirteen quarto volumes containing 967 papers.