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Summary Article: Cayley, Arthur (1821-1895)
from The Hutchinson Dictionary of Scientific Biography

Place: United Kingdom, England

Subject: biography, maths and statistics

English mathematician who was responsible for the formulation of the theory of algebraic invariants. A prolific writer of scholarly papers, he also developed the study of n-dimensional geometry, introducing the concept of the ‘absolute’, and devised the theory of matrices.

Cayley was born in Richmond, Surrey, on 16 August 1821, the son of a merchant and his wife who were visiting England from their home in St Petersburg, Russia. Cayley spent the first eight years of his life in Russia, and then attended a small private school in London, before moving to King's College School there. He entered Trinity College, Cambridge, as a ‘pensioner’ to study mathematics and became a scholar in 1840. He graduated with distinction in 1842. Awarded a fellowship at the college, he took up law at Lincoln's Inn in 1846 instead, prevented from remaining at Cambridge through his reluctance to take up religious orders - at that time a compulsory qualification. Cayley was called to the Bar in 1849 and worked as a barrister for many years before, in 1863, he was elected to the newly established Sadlerian Chair of Pure Mathematics at Cambridge. He occupied the post until he died in Cambridge on 26 January 1895.

Cayley published about 900 mathematical notes and papers on nearly every pure mathematical subject, as well as on theoretical dynamics and astronomy. Some 300 of these papers were published during his 14 years at the Bar, and for part of that time he worked in collaboration with James Joseph Sylvester, another lawyer dividing his time between law and mathematics. Together they founded the algebraic theory of invariants (although in their later lives they drifted apart, until Cayley lectured at Johns Hopkins University, Baltimore, 1881-82 at Sylvester's invitation).

The beginnings of a theory of algebraic invariants may be traced first in the work of Joseph Lagrange, who investigated binary quadratic form in 1773. Later, in 1801, Karl Gauss studied binary ternary forms. A final impetus was provided by George Boole, who, in a paper published in 1841, showed that all discriminants - special functions of the roots of an equation, expressible in terms of the coefficients - displayed the property of invariance. Two years later, Cayley himself published two papers on invariants; the first was on the theory of linear transformations. In the second paper he examined the idea of covariance, setting out to find ‘all the derivatives of any number of functions which have the property of preserving their form unaltered after any linear transformations of the variables’. He was the first mathematician to state the problem of algebraic invariance in general terms, and his work immediately attracted a lot of interest from other mathematicians.

Over the next 35 years he wrote ten papers on what he called ‘quantics’ (which later mathematicians refer to as ‘form’) in which he gave a lively account of the theory as it was being developed. He used the term ‘irreducible invariant’ and defined it as an invariant that cannot be expressed rationally and integrally in terms of invariants of the same quantic(s) but of a degree lower in the coefficients than its own. At the same time he acknowledged that there are many circumstances in which irreducible invariants and covariants are limited. (His system was eventually simplified and generalized by David Hilbert.)

Cayley developed a theory of metrical geometry that could be identified with the non-Euclidean geometry of such mathematicians as Nikolai Lobachevski, János Bolyai, and Bernhard Riemann. His geometry was the geometry of n dimensions. He introduced the concept of the ‘absolute’ into geometry, which links projective geometry with non-Euclidean geometry, and together with Felix Klein, distinguished between ‘hyperbolic’ and ‘elliptic’ geometry - a distinction that was of great historical significance. When Cayley's ‘absolute’ was real, his distance function was that of hyperbolic geometry, and when ‘absolute’ was imaginary, the formulae reduced to Riemann's elliptic geometry.

Cayley also created a theory of matrices that did not need repeated reference to the equations from which their elements were taken, and established the principles for forming general algebraic functions of matrices. He went on to derive many important theorems of matrix theory. He claimed to have arrived at the theory of matrices via determinants, but he always made great use of geometrical analogies in his algebraic and analytical work.

He also laid down in general terms the elements of a study of ‘hyperspace’, and in 1860 devised a system of six homogeneous coordinates of a line. These are now more often known as Plücker's line coordinates because the same ideas were independently published - five years later - by Julius Plücker (whose assistant was Cayley's former collaborator, Felix Klein).

Cayley wrote on almost every contemporary subject in mathematics, but completed only one full-length book. He clarified many of the theorems of algebraic geometry that had previously been only hinted at, and he was one of the first to realize how many different areas of mathematics were drawn together by the theory of groups. Awarded both the Royal Medal (1859) and the Copley Medal (1881) of the Royal Society, generally in demand for both his legal and his administrative skills, Cayley played a great part in bring mathematics in England back into the mainstream and in founding the modern British school of pure mathematics.

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