Arthur Cayley is undoubtedly the best known English mathematician of the 19th century, and this well-written biography is the first we have of him. Tony Crilly has spent twenty years researching and writing it, and it is full of shrewd observation and careful analysis.

Perhaps the hardest thing for a modern reader to understand is the nature of Cambridge and of intellectual life in Britain in the period, but this is necessary, for Cayley was a Cambridge product and after a period as a fairly successful lawyer was a Cambridge professor for the second half of his life. On the one hand, Cambridge dominated the small world of English universities, and mathematics was central to the education it provided, but on the other its attitude to research was frequently negligent. Teaching at Cambridge was dominated by the examination system, the so-called Tripos, and success in these exams could lead to a job for life. To graduate top, and become the so-called Senior Wrangler, brought you instant fame and opened all doors to your future career, so not surprisingly competition was intense. Teaching, however, was in the hands of private tutors. Neither the colleges not the university could do much about it, and successive reforms broke with little intended effect on the rocks of the Tripos until the whole system of ranking students in order of merit was abandoned in 1906, eleven years after Cayley's death. And indeed Cayley's requirements as a lecturer, like the other Cambridge professors, were slight: just one course of lectures a year, and they were not closely tied to syllabus of the Tripos exams. This was not a position from which to bring about reform.

The Tripos encouraged rote learning and technical dexterity; down the century it gave the impression that mathematics was all already known, and Cayley's teaching style in due course was dry and did little to encourage a spirit of originality. But he was, of course, a highly original and productive mathematician, and here another baleful influence of Cambridge set in. This was a widespread utilitarianism, amounting to a philistinism about pure mathematics. One can almost argue that Cayley's greatest achievement was simply to defy this attitude and to win thorough to general admiration for the way he honoured his lifelong calling to his subject. If Cambridge encouraged research at all, and it did with growing success as the century wore on, it was in the physical sciences and their applications. But it is one thing if William Thomson, Lord Kelvin, regretted that Cayley was not to be drawn into 'useful' work, another if John Couch Adams did. For a start, Thomson was up in bustling Glasgow, and for another he had many distinguished discoveries to his credit. But Adams, who had let the discovery of Neptune slip through British fingers, accomplished work of value but little of distinction in the course of 33 years as the Lowndean Professor of astronomy and geometry at Cambridge itself.

In these circumstances it is hardly surprising that Cayley changed Cambridge very little. He graduated as Senior Wrangler in 1842 and obtained a Fellowship at his college, Trinity, which lasted him for a number of years. But then no position was available for him, and in 1846 he moved to London and took up the law, helped, of course, by his reputation as a Senior Wrangler. He was passed over for positions in Scottish universities that went to people with a greater interest in physics, and only got back to Cambridge in 1863, when one of the regular reforms culminated in the creation of a new chair for which he was pre-eminently suited. But all the time he was publishing new results, and it was while he was a lawyer that he got to know his famous contemporary and colleague James Joseph Sylvester and began his monumental study of invariants. (And let me say that, if you think you will enjoy this book you will probably also enjoy Karen Hunger Parshall's James Joseph Sylvester: Jewish Mathematician in a Victorian World .)

Invariant theory proved to be a lifelong interest, encompassing not only the series of ten 'Memoirs on Quantics' but numerous related issues, and Crilly deals very sensitively with this major aspect of Cayley's work. Cayley was a calculator — he even employed human computers, as they were called, to help him with the work. He had a formidable skill, much commented on by those who knew him, for marshalling huge formulae. Not only would this make for tedious reading today, it is arguable that this style of work was responsible for its mixed quality. Discoveries were made, an area of research discovered, but mistakes were made that marred it, and both Gordan and, later but spectacularly, Hilbert dug much deeper. Crilly is interesting on the way both Cayley and Sylvester relied too naively on assumptions of algebraic independence among the objects they studied and on reasoning that was generic at best. It would seem that the British remained longer than Continental mathematicians in a world of theorems that admit exceptions, rather than entering a world of proven results, explicit conditions of validity, conjectures and counter-examples.

Almost all of Cayley's work is discussed in these pages, as is almost all of his life, and curiously we learn more about him as a child than as a husband and father. He does seem to have been a kind man, a principled Victorian without being stern, a little remote and by no means a gifted communicator since he found it difficult to realise that his readers and listeners generally lacked his mathematical ability. He helped to improve the education of women at Cambridge, and he was diligent behind the scenes, but there are almost no documents to take us into his home. As for the work, Crilly has decided to describe it with a minimum of detail. I felt a desire for more information at times, for good examples to illustrate how the ideas hung together and even what the mistakes might have been. After all, Cayley chose to speak most eloquently through his work, and if I may contrive a mountaineering metaphor (Cayley was an enthusiastic climber, another Cambridge trait) Crilly describes the quantics, syzygies, semi-invariants and the like as distant peaks we may see, not as climbs and paths we are invited to follow. But perhaps that is the right decision, given the nature of Cayley's lasting contribution. He produced very little mathematics one wants to read today, and too much that is slight, insufficiently pursued and insufficiently accurate. Rather, his achievement was to be, for almost all his long career, the very model for the English of a pure mathematician, the first of international stature the country had produced. It is one of the achievements of this graceful book that Crilly does not describe Cayley anachronistically and with our expectations of what a mathematician should be, but presents him as he surely must be seen historically, as the mathematician laureate of the Victorian age.

Jeremy Gray is Professor of the History of Mathematics at the Centre for the History of the Mathematical Sciences of the Open University, in Milton Keynes, UK. He is the author of many books on the history of mathematics.