*The History of Mathematics: A Brief Course*, 2nd edition, Roger Cooke, 2005, xviii+607 pp., $105, ISBN 0-471-44459-6. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ 07030-5774.http://www.wiley.com/

The author begins the Preface with an explanation for readers of the first edition, and I think it is worthwhile to read for those considering this text: “The present volume constitutes such an extensive rewriting of the original that it amounts to a considerable stretch in the meaning of the phrase second edition. Although parts of the first edition have been retained, I have completely changed the order of presentation of the material…. In this second edition, after a general survey of mathematics and mathematical practice in Part 1, the primary division is by subject matter: numbers, geometry, algebra, analysis, mathematical inference.”

It is this last comment that contains what is different about this book. Instead of considering time periods or geographical regions and the mathematics of each, Cooke looks at each mathematics topic and follows it through the time periods and various cultures. For example, Part 5 considers algebra, and it looks at problems that lead to algebra in Egypt, Mesopotamia, India, and China, and continues through equations in Diophantus, solving equations in China, India, the Muslim world, and Europe, and finishes with the theory of equations and algebraic structures. I must say that I enjoyed this approach as a nice change of pace from other history of mathematics texts. One can see the parallel development between cultures, or how one culture and time period influenced other, later cultures.

All the familiar material that one would expect in a history of mathematics book is here. However, there are enough new insights presented to make it interesting even for one who has read many such books. Cooke’s text reads like a great story that integrates many historical trends. This book is more “text oriented” than “formula oriented,” so an instructor using this as a text will need to fill in some of the mathematics. It is important to note that Cooke calls his exercises “Questions and Problems.” There are many open ended questions that are great for starting a discussion or assigning as a short writing assignments, but are not as appropriate for “typical” homework assignments. Many of the problems are quite challenging, so the instructor will need to chose carefully based on the level of student in the class.

This book should engage students, and Chapters 1-4 contain a great deal of material that students usually have to find outside their textbook: background history, relationship of mathematics to the broader culture, and a great deal of information on women in mathematics. Once the students are taken in by the story, it will be the instructor’s job to elaborate on the historical calculations and proofs. For an experienced instructor in a history of mathematics class, this is an ideal situation; both the instructor and the text get to do their jobs.

Gary S. Stoudt, Professor of Mathematics, Indiana University of Pennsylvania