ABC of MathematiciansAuthor(s): Michael HoltSource: Mathematics in School, Vol. 5, No. 1 (Jan., 1976), p. 15Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211492 .

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by Michael Holt

James Joseph Sylvester (1814-97) worked for more than fifty years on the theory of determinants. Determinants crop up in the solution of simultaneous equations. We shall show a simple example later. He was a prolific inventor of concepts and names. He coined strange names such as "Hessian", "Jacobean", "discriminant" and "umbral notation" (denoting quantities which were "mere shadows"). So much so that he was known as the mathematical Adam. He was of stocky build with a flam- ing mass of red hair. He was the epitome of the Romantic artist, being lively, passionate, stimulating, and decidedly excitable. Indeed, had he not become a mathematician, he would doubt- less have been a more than passable poet. His name is usually linked with that pillar of commonsense, Arthur Cayley, another great mathematician, with whom Sylvester enjoyed a close yet occasionally stormy Gilbert-and-Sullivan friendship.

Sylvester, the youngest of several brothers and sisters, was born of Jewish parents on 3 September 1814 in London. Little is known of his childhood nor of why, as a good Jew, he should have adopted the Christian name of a Pope! His mathematical genius showed itself early; by the age of 14 he was already working at the University of London where he studied under the great Augustus De Morgan (of De Morgan's Rule fame in logic). In 1829 he entered the Royal Institution at Liverpool where he stayed two years. He won the prize in mathematics. So far ahead of his fellow students was he, he was put in a class by himself. His years at Liverpool, however, were not happy ones, largely because of his fellow students' persecution of him for his blatant Jewishness (and, no doubt, because of most English- men's natural distaste for "cleverness").

He entered St. John's College, Cambridge University, at 17 and became second wrangler in the mathematical tripos. Because he was Jewish and refused to subscribe to the Thirty- Nine Articles of Faith he was barred from teaching at Cambridge.

In 1838, at 24, he got his first job, as Professor of Natural Philosophy-that is, mostly of physics-at University College, London, where his old master De Morgan was now one of his colleagues. Teaching science did not appeal to Sylvester at all and after some two years he abandoned it. Nevertheless he had in the meanwhile been elected a Fellow of the Royal Society at the unusually early age of 25.

With high hopes he sailed the Atlantic to become Professor of Mathematics at the University of Virginia in 1841. The post

turned out to be a ghastly mistake which he endured for three months before resigning. Harvard and Columbia Universities both turned him down, so he returned to England, and having had his fill of teaching, he became an actuary and a lawyer for a life insurance company. He served in this office from 1845 to 1855. But it was a hard grind and nearly the death of his creative energies. So he took pupils. And one of them was to become a household word throughout the world. Her name was Florence Nightingale. When Sylvester left the company, Florence Nightingale left for the Crimean War.

Before this, Sylvester had entered the Inner Temple and in 1850 had been called to the Bar. Here he met Arthur Cayley, who revived in him his life-long love of mathematics. The two friends used to tramp round the Courts of Lincoln's Inn. When Sylvester moved, they would meet halfway between their lodgings. Both were bachelors at the time. Sylvester never married.

Their discussions often centred on the theory of invariants (things that don't change), which Boole had by and large initiated (see Boole). Put in its simplest terms, what Sylvester and Cayley talked about were the invariants of equations. Take the quadratic ax'+2bx+c= 0. Now if b2-ac =O 0 then the quadratic has equal roots. Problem: To write out an equation such that b2 does indeed equal ac (that is, b2-ac = 0). Check that its roots are equal.

Now say you "transform" the quadratic by substituting for x the expression (q-sy)/(ry-p). it comes out that the roots of this new equation are also equal if (b'-ac) times the factor (ps-qr)2 is zero. As you see, the (b'-ac) bit-for which Sylvester coined the word "discriminant"-turns up unchanged; it is an invariant.

Of course, what Sylvester accomplished was far deeper than this summary could hope to indicate; it led on to his major work on determinants. Our expression (b2-ac) above in determinant form looks like this:

Ib a Ic bi

In 1854, at the age of 40, Sylvester applied for the professor- ship of mathematics at the Royal Military Academy, Woolwich. He was turned down. However, the successful candidate died next year and Sylvester was appointed. His university pay included the right to graze a horse, cow or goat on the common -none of which Sylvester kept! He held this post for sixteen years until he was retired at the early age of 56 in 1870. Even then he had to fight for his due pension rights.

In the meantime he had been made foreign correspondent of the French Academy of Sciences. He spent his retirement in London returning to his early love, poetry.

Then in 1876 he was invited to take the mathematics chair at the Johns Hopkins University, in the United States. The next seven years were probably the happiest and least troubled in his career. He was a born teacher and obviously inspired his students. Here he initiated research in pure mathematics in the United States and founded the American Journal of Mathematics.

In his eightieth year he was invited to become Savilian Pro- fessor of Geometry at Oxford University. He returned to England gratefully, homesick for his land, and much to the regret of his American colleagues.

Old as he was, he had something new to spring on his advanced students-a brand new theory of "differential in- variants". He lightened his inaugural lecture with a poem related to the mathematics involved, called "To A Missing Member of A Family of Terms in An Algebraic Formula". For Sylvester, Music and Mathematics were closely related and he looked forward to an intellect "glorified in some future ... Beethoven- Gauss".

He was, above all, a lover of life. Then in 1893, at 79, his eyesight began to fail. He retired from Oxford and went to live, alone and dejected, first in London, then in Tunbridge Wells. Yet even at 82 he recaptured some of his old fire and worked on Goldbach's famous conjecture that every even number is the sum of two primes.

In March 1897 he suffered a stroke and lost his powers of speech. He died, at 83, on 15 March 1897.

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