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Twins in invariant theory
"The lives of Cayley and Sylvester should be written simultaneously if that were possible. Each is a perfect foil to the other, and the life of each supplies what is lacking in that of the other . Cayley's life was serene; Sylvester spent much of his spirit and energy, fighting the world... Yet these two mathematicians became close friends and inspired one another to some of the best work that either of them did, in the theories of invariants and matrices"— E.T. BELL
ARTHUR CAYLEY was born (August 16, 1821) at Richmond, England but spent the first eight years of his boyhood in St. Petersburg, Russia where his father was a successful merchant. This contributed to Cayley's later fluency in French.
Cayley entered King's College Senior School when only 14, although the mandatory age was 16. The first manifestations of superior talent were similar to those of Gauss (1777-1855). , namely an amazing skill in numerical calculations.
His teachers recognised his ability and recommended that the boy should make mathematics his career, to which the merchant father objected strongly. The latter was finally won over by the Principal of the school and gave his blessing and money for his son to study at Cambridge (1839-42).
By the end of his third year at Cambridge, the head examiner put Cayley ``in a class by himself, above the first.'' He was senior wrangler and won the Smith's prize. In October 1842, he was elected Fellow of Trinity College. He was tutor for three years, spending most of his time in research.
His first paper, published in 1841, grew out of his study of the work of French mathematicians Lagrange and Laplace. He published four papers the second year and thirteen the third year of his mathematical tripos. In 1846, Cayley left Cambridge No position as mathematician was open to him unless he could agree to the formality of holy orders.
So Cayley entered Lincoln's Inn to prepare himself for a legal career. For the 14 mortal years since 1849 he was at the Bar, he made an ample living; during this period of servitude.He wrote around 300 mathematical papers, many of which are now classic during those 14 years at the Bar.
Sylvester entered Cayley's life during this legal phase; he was called to the bar in November 1950. Many years later, Sylvester paid a glowing tribute to Cayley. as one. ``Who first opened my eyes and purged them of dross so that they could see and accept the higher mysteries of our common mathematical faith (Oxford inaugural lecture, 1885). They had derived theorems through the stimulus of conversations in the intervals between transacting legal business. In 1863 Cambridge University established the new chair of pure mathematics and offered the post to Cayley. He held this post until his death (January 26, 1895).
Although he made less money than he had at law, Cayley did not regret the change. He devoted himself entirely to mathematical research and university administration; in the latter field, his sound business training, legal expertise and impersonal judgement proved invaluable. For example, owing to Cayley's weighty, persuasive influence women were at last admitted as students, breaking the monkish seclusion of medieval Cambrdige.
His contributions
Cayley's fame rests on his contributions to invariant theory. Although Canchy, Jacobi and Eisenstein have a claim in this respect, none of them had proposed any general method for finding such invariant expressions. Cayley came up in 1845 with his pathbreaking memoir, ``On the Theory of Linear Transformations''.
Cayley found geometrical analogy of great assistance in his algebraic and analytical work; this resulted in his developing the subject of the geometry of `n' dimensions, which was then more mysterious than it seems to us today. He established the associative and distributive laws and the principles for forming general algebraic functions of matrices. Sixtyseven years after Cayley's invention, Werner Heinsberg in 1925 recognised in the algebra of matrices exactly the tool he needed for his revolutionary work in quantum mechanics.
His output of papers was prodigious, numbering nearly a thousand, since republished in 13 large quarto volumes. He published only one book ``Treatise on Elliptic Functions'' (1876). Hermite compared him with the French mathematician Canchy. . ``Esse quam videri'' was his motto. Cayley was the first English mathematician to achieve international recognition since Newton, and the first to open up Continental mathematics to an Anglophone audience. His achievements were memorably captured by Clerk Maxwell (1831-1879) in poetic words ``His soul, too large for vulgar space, in `n' dimensions flourished.'' (The Mathematical Intelligencer vol. 17, 1995)
(To be concluded)
R. Parthasarathy
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