*Arthur Cayley: Mathematician Laureate of the Victorian Age*, Tony Crilly, 2006, xxiii + 609 pp., hardcover, $69.95, ISBN 0-8018-8011-4, The Johns Hopkins University Press, 2715 North Charles Street, Baltimore, MD, 21218-4363. Tel: 1-800-537-5487 or https://www.press.jhu.edu/.

In his introduction for *Arthur Cayley: Mathematician Laureate of the Victorian Age*, Tony Crilly asks the question, “Why is he not more widely known?” (p. xiv). Crilly goes on to make the point that in “the Victorian scientific establishment,…to be a mathematician was to be ‘like Cayley’” (p. xiv). Yet it feels that very early in his work, Crilly wishes to tell the detailed story of Cayley’s contributions and importance as a great mathematician. This book is rich in detail and provides an account of Cayley’s life, from his early international upbringing; as a young, promising mathematics student; then as a rising barrister; and finally, as mathematician of great acclaim. Crilly’s thorough, well-written biography begins beautifully with a glimpse into a Victorian family scene – Cayley’s family – and in this opening moment of the book, Crilly reveals in a subtle way the esteem he holds for Arthur Cayley.

As a student of the history of mathematics and a proponent of using history in teaching mathematics, I believe this book will serve as a resource for personal study for several reasons. First, I found the two appendices invaluable. Appendix A describes (in chart form, alphabetical by last name) Arthur Cayley’s social circle. Persons from all arcs of Cayley’s circle are included here: persons of influence, colleagues, contemporary professionals, students – even those who may be considered enemies. The chart identifies the person; their age as of 1863 (when Cayley was named Sadleirian Professor), if alive; the dates of their affiliation with or influence on Cayley; and a brief description of the connection with him. As an added characteristic of the affiliation, the entry in the chart is bolded for each person considered “intimates in Cayley’s larger community of scholars and friends” (p. 443). This is a wonderful resource – to be able to quickly cross-reference colleagues and mentors and others influenced by and who influenced Cayley. In researching some of the more involved developments in mathematics, the human dimension of the people, circumstances, and connections is often necessary to understand and appreciate the development itself. With Appendix A, Crilly provides an accessible reference to appreciate the connections which were significant to Cayley and his work. To accompany the information provided in Appendix A, Crilly includes a chronology of Cayley’s life before Part I of the book. Although this would be better placed either with Appendix A or as another appendix, the chronology does afford easy access to the significant events in Cayley’s life. Second, Appendix B contributes a glossary of terms, which permitted me to keep up with the mathematical content without having to access another resource.

Finally, the account of Cayley’s mathematical work in the book is extensive. Though many of Cayley’s contributions extend beyond the scope of school level 9 – 14 mathematics, most are of great value to teachers at those levels. In addition to describing his “remarkable matrix algebra” (p. 226), the book introduced me to Cayley’s role in the development of other topics, including invariant theory, elliptic functions, and group theory. I was fascinated by the influence of Cayley’s early experience with chemistry, which “played a part in suggesting and enriching some important mathematical problem areas in his own research of the 1870s and 1880s” (p. 21), including the development of Cayley’s “point of view of invariant theory” (p. 194).

I highly recommend *Arthur Cayley: Mathematician Laureate of the Victorian Age* as a valuable addition to one’s personal or institutional library. There is everything to enjoy about this book: the writing, the content, the essential tribute to Cayley’s life and contributions. In every part and chapter I was engaged by the expert way in which Crilly illuminated the prolific mathematician as an equally admirable man. Appropriately, Lord Kelvin observed Cayley as “one of the makers of mathematics” (p. 431) and further noted that he came “to the honour of knowing him personally, [and] the awe was evaporated by the sunshine of his genial kindness; the admiration has remained unabated to this day, and his friendship has been one of the valued possessions of my life” (p. 431).

Kathleen M. Clark, Assistant Professor, The Florida State University

See also the MAA Review by Jeremy Gray.