99 Variations on a proof

we see how they generated, and were informed by the use of mathematics. Important scientific issues indicate the problems with traditional views, touching on the messy political and religious contexts. The characters involved are Tycho Brahe with critical observations, Kepler, who worked out the planetary orbits, and Galileo who observed and calculated and convinced people of a sun-centred universe. This is a ‘good read’ with both popular stories and serious content. By the early seventeenth century the actors were learning to adapt old methods to novel situations and invent new mathematics. Thus William Oughtred set out a more down-to-earth approach to learning, Girard Desargues founded projective geometry, Pierre de Fermat developed number theory, and René Descartes formulated his rational philosophy, science and mathematics. This last section is well-structured and interesting, but quite difficult for the less experienced; the authors are expecting the reader to do some serious work here. The final chapter acts as an overview, a reflection on the content and ambitions of the first thirteen chapters. One can approach the context of historical accounts as parts of a dialogue: whowere they writing to?What were they writing for (or about)? Andwe can also ask of the present volume, ‘What (or who) is this book for?’ The private scholar? The individual or college setting up a new course? But we must remember; this is not just a ‘history’ book. This book is a resource. It describes an optional course that was written for an undergraduate mathematics programme. From the introduction, we have: ‘We hope that [the book] will provide a rich introduction not only to the history of mathematics, but to mathematics itself ’ (pp 2–3). Despite the challenges, it succeeds admirably, and is highly recommended.