Mathematics is not what it used to be, at least for gaining honors in college. I did not know until reading this book that at the University of Cambridge in the early nineteenth century the *only* way to receive a bachelor’s degree with honors was to place in the Mathematical Tripos examination. I had the impression that skill in Greek and Latin, or perhaps theology, counted; but that was not so. Mathematics was the sole road to distinction. When you consider that a good degree from Cambridge gave a very large boost to a future career in the law, the Church of England, or education, this is either a tribute to the power of mathematics, or very strange.

The Tripos was an examination taken in January in a student’s third and last year at Cambridge. There were two parts, the bookwork in which the examinee demonstrated his knowledge of known theorems and their proofs, and a section of problems. In 1824 the test took twenty-eight hours over four days. This increased to thirty-three hours spread over a week in 1839, which persisted until better sense prevailed in 1873, when it was decided to eliminate, after three days, all except those who had done well enough to have the prospect of gaining high honors.

About 35% of graduates obtained honors, divided into classes: the Wranglers, the Senior Optimes, and the Junior Optimes. The term “wrangler” goes back to the fifteenth century, when to graduate students had to engage in oral debate (in Latin) with an official of the university. The examiner sat on a three-legged stool, whence “Tripos,” and those who argued, or wrangled, best were the most worthy graduates. The wranglers were ranked and to be First Wrangler was, for a time, a celebrity and someone from whom great things were expected.

Some wranglers never lived up to their promise, but very many did, such as Augustus De Morgan (fourth, 1827), J. J. Sylvester (second, 1837), George Green (fourth, 1837), G. G. Stokes (first, 1841), Arthur Cayley (first, 1842), and John Venn (sixth, 1857). Other wranglers, whose names members of the MAA will not recognize, went on to achieve eminence in non-mathematical fields. It is as we tell our students: mathematical training is valuable no matter what you do later.

High-ranked wranglers were offered posts as Fellows of their colleges. Fellows had few duties and received not much pay; though some stayed on for a lifetime, most left after a few years. To be a Fellow it was necessary to become an ordained priest of the Church of England and to remain umarried, two requirements that may have encouraged new Fellows to move on.

In the first half of the 1800s, almost all C. of E. priests were graduates of Oxford or Cambridge , not least because the two schools controlled the assignment of people to parishes. Cambridge graduates also became lawyers or educators. They did *not* go into trade. Engineers and military men were produced by other schools.

The subtitle of *Mr Hopkins’ Men* is “Cambridge Reform and British Mathematics in the 19th Century.” At the start on the 19th century, British mathematics lagged behind that on the European continent because it had not yet shaken off the stultifying influence of Newton’s notation for calculus. Continental mathematicians were able to do great things using Leibniz’s notation and, perhaps as a reaction, at Cambridge analysis was looked down upon in comparison with geometry. Analysis, the geometers said, gave results from mere symbol manipulation, whereas in geometry thinking was always required.

A group of Cambridge undergraduates, wiser than their elders, campaigned to progress to the *d*-ism of Leibniz from the *dot*age of Newton. The campaign succeeded in part because the composition and grading of the Tripos was in the hands of those who had recently done well in it.

To do well in the Tripos it was not enough to turn up on the appointed day with pencil in hand. Extra preparation was needed and William Hopkins (1793-1866) was one of many who made their livings, and quite good ones they were, by providing it. He was one of the most successful coaches — in 1854 seven of the top nine wranglers were his students.

Craik’s book provides all of the above information and much more. It is meant for non-mathematical readers — I saw only three equations, and there are no statements of Tripos problems — and provides a thorough picture of nineteenth century British mathematics, both at Cambridge and generally. There are brief biographies of many wranglers, both those celebrated and those unknown, and extended accounts of some of them. Professor Craik gives the story of John Adams and the discovery of Neptune, the unconventional life of George Green, and the interesting career of J. W. Colenso (second wrangler, 1837) a bishop who was found guilty of heresy. There are forty-two portraits of Hopkins’ top wranglers, mostly looking earnest. Professor Craik ranges widely and his book contains much of interest.

Woody Dudley took the Putnam Examination twice, but placed well down in the pack.