By comparison with his contemporaries, James Joseph Sylvester (1814-1897), one of the dominant figures of nineteenth-century British mathematics, had an unconventional career. He twice crossed the Atlantic in pursuit of academic employment, he held chairs in London, Woolwich and Oxford, and for more than a decade he was engaged in the practice of law. But while Sylvester may have spent many years physically separated from the mainstream of British and European mathematics, his letter-writing ensured that he remained very much in touch with its developments.

Sylvester, like many Victorian mathematicians, was a prolific correspondent--although not in quite the same league as Charles Dodgson (Lewis Carroll) who wrote more than 98,000 letters in the last 37 years of his life! As Karen Hunger Parshall shows, Sylvester's correspondence makes for informative reading. From a collection of some 1200 letters of Sylvester and his circle Parshall has selected 140 "to cover Sylvester's life and work as completely and as substantively as the imperfect historical record allows." [p.viii]

Starting with Sylvester's application for the chair of Natural Philosophy at University College London (UCL) in 1837, and ending with a letter to the Editor of *The National Observer* written in 1896, the selected correspondence charts the course of Sylvester's professional life. It rapidly becomes evident that Sylvester was not a disciplined correspondent; rather, his letters ebb and flow with the tide of his ideas. When he was at his most creative mathematically, so too was he most expansive with his pen. And this characteristic is mirrored in the structure of the book with individual chapters reflecting specific phases in his career as opposed to different decades of his life.

Thirty-two of Sylvester's correspondents are brought together in this volume although almost a third of the letters come from Arthur Cayley, Sylvester's friend for almost fifty years. Cayley had saved over 500 of his letters from Sylvester and 36 of these, together with six from Cayley to Sylvester, are published here. While many of them dwell on the development of invariant theory, excursions into number theory and group theory are recorded too. Although those selected span almost the entire duration of the friendship, the majority of them come from Sylvester's most productive years, 1850-1854 and 1876-1883.

Other mathematicians represented in the correspondence include George Boole, Michel Chasles, Felix Klein, Paul Gordan, Charles Hermite, George Salmon and William Spottiswoode. In the case of some correspondents, such as Chasles, they are represented by a single letter, while in the case of others, such as Hermite, there are a number of letters written over a significant period of time. There are also groups of letters that focus on a single incident or particular mathematical concern, such as those written by Salmon to Sylvester in 1852 which dwell upon the interconnections between geometry and algebra and in which "Salmon, the geometer, both learns from and instructs Sylvester, the algebraist". [p.42]

Sylvester's correspondence was not only concerned with mathematics per se. The letters also lay witness to his success and failure in the academic marketplace in England and the USA. Starting from UCL they can be used to track him to his short-lived (and somewhat notorious) stay at the University of Virginia, his failure to win a post at Columbia, on through Lincoln's Inn to his position at Woolwich, his appointment at Johns Hopkins, and finally his return to England to take up the Savilian chair of geometry at Oxford. And not only do they provide a descriptor of the actual events but they are peppered with delightful details, such as Sylvester's insistence upon having his salary at Johns Hopkins paid in gold.

In sum the letters make for intriguing reading and all the more so for Parshall's excellent editorial input. A short contextualising essay prefaces each chapter, and obscurities in the text-of which there are many, both literary and mathematical-are fully explained in detailed and well-informed footnotes which are complemented by an extensive bibliography. The book provides not just a portrait of Sylvester but it also digs deep into the heart of nineteenth-century Western mathematics teaching and research. Some of the mathematics discussed in the letters is challenging, particularly to twentieth-century eyes, but Parshall's commentary ensures that the thread is never lost, and the book is accessible to anyone with an interest in Sylvester or his period. If there is a criticism, it is that there are no illustrations. It would have been very nice to know what some of the other characters looked like!

June Barrow-Green j.e.barrow-green@open.ac.uk is a Research Fellow in the Pure Mathematics department of The Open University, Milton Keynes, UK, where she is working on 19th and 20th century British mathematics. She is also on the Council of the British Society for the History of Mathematics.