VI.46 Arthur Cayley
b. Richmond, England, 1821; d. Cambridge, England, 1895
Algebra; geometry; mathematical astronomy
At the beginning of his career in the 1840s, Cayley laid down subjects that informed much of his later research. The novelties of his very first undergraduate paper, “On a theorem in the geometry of position” (1841), are the now-standard notation for DETERMINANTS [III.15] of arrays set between vertical lines and the introduction of the Cayley-Menger determinant. Following HAMILTON’S [VI.37] discovery of the QUATERNIONS [III.76] (1843), Cayley expressed rotations in three-dimensional space via the succinctly expressed mapping χ → q–1xq, a result that led him to the Cayley-Klein parameters. He outlined the nonassociative system of the octaves (CAYLEY NUMBERS [III.76]), the intersection of curves (the Cayley-Bacharach theorem), and a dual curve called the Cayleyan. In major papers, he described a theory of multilinear determinants and ELLIPTIC FUNCTIONS [V.31] as doubly infinite products. In concert with George Salmon he investigated the famous twenty-seven lines that lie in a cubic surface. The most important studies among his juvenilia, though, were his first steps in invariant theory (1845, 1846), the field in which his reputation was made.
Between 1849 and 1863, years spent as a qualified London barrister, Cayley broadened his range, but unlike other gentlemen of science who roamed across a multitude of subjects, he restricted his activity exclusively to mathematics. This was mostly pure mathematics. He generalized PERMUTATION GROUPS [III.68] using the calculus of operations as a basis, and he saw that not only were matrices useful as a notational device, but they also constituted a study in their own right. Not generally an excitable person, at the point of discovery he declared the Cayley-Hamilton theorem as “very remarkable” and generations of mathematicians have shared his delight. Matrix algebra was used in his solution of the Cayley-Hermite problem, which required a description of those linear transformations that leave a bilinear form invariant. A special case of the solution gives rise to the Cayley orthogonal transform (I – T) (I + T)-1. The links between quaternions, matrices, and group theory that he observed in the 1850s are indicative of his concern for the organization of mathematics.
In the 1850s, Cayley set in motion his famous memoirs on quantics, a term he coined for algebraic forms, now referred to as multilinear homogeneous algebraic forms. He discovered Cayley’s formula for the general form of covariants of binary forms and Cayley’s law for counting them. In the Sixth Memoir (1859), he demonstrated that EUCLIDEAN GEOMETRY [I.3 §6.2] was part of PROJECTIVE GEOMETRY [I.3 §6.7] rather than the converse. The idea of a projective metric (Cayley’s absolute) was seen by KLEIN [VI.57] in the 1870s as the unifying conceptual idea for classifying non-Euclidean geometries.
For twenty-five years, from 1858, he was the editor of the Monthly Notices of the Royal Astronomical Society. In astronomy he contributed to the theory of elliptic planetary motion, calculatory work that demanded an assiduous attention to detail. His work on the lunar theory was noteworthy, and in one long calculation he helped to settle an Anglo-French controversy by verifying the correct value for the secular acceleration of the Moon, which had been established by John Couch Adams in 1853.
Cayley returned to the academic world in 1863 as the founding Sadleirian Professor of Pure Mathematics at Cambridge. In 1868 Paul Gordan startled invariant theorists by proving that invariants and covariants of a binary quantic could be expressed in terms of a finite basis. This contradicted an earlier result of Cayley’s but, undaunted, he completed his series with a listing of the irreducible invariants and covariants of the binary form of order 5 (the binary quintic), and their connecting syzygies.
Many developments in pure mathematics can be traced back to his minor notes of the 1870s and 1880s, including the theory of knots, fractals, dynamic programming, and group theory (the well-known Cayley’s theorem). In graph theory, the number of distinct labeled trees with n nodes being nn-2 is known as Cayley’s graph theorem. He brought his theoretical knowledge of graphical trees to bear on the problem of counting isomers in organic chemistry, thus prompting questions about the actual existence of certain chemical compounds that have since been discovered in many instances by chemists. In the last decade of his life, Cayley set about the task that gave him an important line of contact with today’s mathematicians: the publication of his Collected Mathematical Papers in thirteen large volumes by Cambridge University Press.
Further Reading
Crilly, T. 2006. Arthur Cayley: Mathematician Laureate of the Victorian Age. Baltimore, MD: Johns Hopkins University Press.