An odd pair with a passion for all things pure

Peter Neumann examines the achievements and the lives of a Victorian mathematical duo with a shared love of algebra

In E. T. Bell's much-read, much-loved and much-despised book Men of Mathematics (New York 1937), Arthur Cayley (1821-95) and James Joseph Sylvester (1814-97) share a chapter whose heading is "Invariant Twins". In most mathematicians' minds they go together. They were friends and collaborators, mathematicians in the broad sense of being interested in most of their subject - most of pure mathematics at least - but their main contributions were in algebra, and it is as algebraists that they are remembered.

More different twins could not be conceived. Sylvester was Jewish, excitable, often in trouble, but, when given the opportunity, a leader. Cayley was very English, quiet, never in trouble and had few leadership qualities. Still, they had much in common.

Sylvester's Jewishness and mathematical precocity marked him out for teasing and bullying at school. His fiery temperament led to fights and suspensions. He nevertheless reached St John's College, Cambridge, and was 2nd Wrangler in 1837. At that time, Jews could sit the examinations but were banned from taking degrees or fellowships. He went to London, where University College had been set up without religious restrictions and, aged 23, was appointed professor of natural philosophy. During his three and a half years there the experimental side of his teaching grew ever more burdensome and he sought an academic post in mathematics. He found one at the University of Virginia. This was a disastrous episode in his career, however, and after run-ins with his students and with the University Council he resigned after just three months.

There followed 18 months of unemployment in the US, after which he returned to London where, in due course, he became a successful actuary and a barrister. His job with Equity and Law Life Assurance Society provided security but did not leave him the time he wanted to prosecute mathematical researches, and in 1855 he became professor of mathematics at the Royal Military Academy, Woolwich. Although relations between Sylvester and his military employers were never easy, he remained there for nearly 15 years until he was pensioned off prematurely at the age of 56.

But he was soon appointed professor at the newly founded Johns Hopkins University in Baltimore. There he was paid $6,000 (£1,250 at the time) a year in gold, far more than could be earned in England, and was in an environment where he could establish a research school, found a mathematical research journal and focus his energies on mathematical research. From 1876 to 1883 he was in his element.

In 1884, he returned to England to become Savilian professor of geometry at Oxford University. Here, he found mathematics moribund and his attempts to stimulate research were foiled by disappointingly uninterested and unambitious colleagues. After eight more years he gave up his duties and retired, depressed, to live out the final few years of his life in London.

Cayley's more humdrum life can be summarised more briefly. He was born into a prosperous middle-class family and he too had his mathematical precocity recognised at school. He went to Trinity College, Cambridge, graduated 1st Wrangler in 1842 and became a fellow of Trinity that same year. Since he was unwilling to take Holy Orders, his fellowship was of limited duration.

Besides, he was not a natural teacher and found his college duties irksome.

He entered Lincoln's Inn in 1846, using his fellowship stipend and dividend to support himself while he trained for the bar, to which he was called in 1849. As a successful conveyancer he lived a comfortable life and still had time for research and prolific writing on mathematics.

In 1863, aged 42, Cayley was elected to the Sadleirian professorship of mathematics at Cambridge. He married, raised a family, researched, contributed a little to university administration and politics, and lived out the even course of his life, apparently never moved to found a research school as Sylvester had done in Baltimore.

Cayley and Sylvester met in 1847. They were pupil barristers at the time and it seems likely that it was mathematics, not law, that brought these two very different personalities together and algebra that kept them together. Old-style algebra. The theory of equations. Equations that describe interesting geometrical objects, such as lines, circles, ellipses, parabolas, hyperbolas and their higher-dimensional (and higher-degree) analogues. Those equations are relations between the co-ordinates of points. What happens if we change to a new system of co-ordinates? The equations will change, but since the geometrical objects remain the same there must be some quantities derivable from the original equations that remain unaltered.

Cayley and Sylvester were by no means the first to investigate these algebraic invariants. They were, however, the two mathematicians who developed Invariant Theory into a major 19th-century preoccupation. Their collaboration was quite remarkable. They discussed mathematics face to face, they wrote notes and letters to each other, they worked incessantly at the details of their theory. And always they maintained ownership, each of his own ideas. Sylvester, the more prickly and the more colourful of the two, was also the more possessive and the one who insisted on drawing the boundaries. This extraordinary relationship lasted out their lives. They worked together for nearly 50 years and not a single jointly authored work resulted. How refreshingly different from our own times.

Looking back from our time to theirs, what do we see as their main achievements? There are various contributions by Cayley to matrix theory, group theory and geometry, mainly at the foundations and none going very deep. There are various contributions by Sylvester to matrix theory, number theory and combinatorics. But, sadly, little of their great creation of Invariant Theory has survived. In a very British way they were obsessed with detail and calculation. Their view of the forest was limited by the trees. Already in 1868 Paul Gordan in Germany had used the more powerful methods of "modern algebra" to prove that all the invariants of an equation in two variables can be derived from just finitely many of them, a theorem that Cayley and Sylvester were seeking but for which their computational methods were inadequate. In 1893, David Hilbert used even more general methods to prove this finiteness theorem for invariants of equations in any number of variables - another result in which Cayley and Sylvester had invested much fruitless thought and calculation. The new German approach produced one success after another. Although, as Hilbert himself stressed, those successes owed much to the data that Cayley and Sylvester had accumulated, the methods went far beyond those of the British mathematicians and eventually replaced "old-style algebra" completely.

Ironically, part of what lies behind the German successes was Group Theory, the study of symmetry. Cayley is known for his attempts to produce an abstract theory of groups. Those attempts were far from successful, but they did inspire others. Moreover, neither Cayley nor Sylvester ever came to grips with the deeper theory of groups that was being developed in France and Germany, and which was a natural and powerful tool in the German version of Invariant Theory that carried the day.

Why then should we now be interested in these two men? These two books (which are uncommonly like their subjects - Sylvester sharp, exciting and highly focused; Cayley somewhat more benign and fuzzy, giving much detail about minor figures far removed from the main story; both discursive, extensive and meticulously researched) are significant contributions to the history of mathematics.

In their day, Cayley and Sylvester were as influential as anyone. Their contributions to mathematics were highly regarded. They received recognition in, and indeed honours from, many countries. To understand the development of 20th-century ideas, we must understand where they came from and, as historians, we cannot focus merely on what survived, ignoring the mathematics out of which it grew. Therefore, Cayley and Sylvester remain as important today as in their own time. In 1895, Cayley's Times obituarist wrote that "few, we believe, would demur to placing Professor Cayley in the first rank of mathematicians of any time and any country". Parshall writes of Sylvester, "Iin his time and in his place, he was both a leader and a pathbreaker". That is what these two books are about.

Peter Neumann is fellow and tutor, The Queen's College, Oxford.