When I first got Peter Rudman’s *How Mathematics Happened: The First 50,000 Years*, I was leery. What would a retired professor of solid-state physics with no publishing record in the history of mathematics say in a book about the history of mathematics — especially the history of ancient mathematics, a much contested area of research within the math history community? The back cover states that the author, “interprets the archeological record so that it is *now* possible to understand and to appreciate the work of the ancients” [reviewer’s italics]. In particular, he sets out to explain Plimpton 322 and the Egyptian use of unit fractions. My concern was that either he was a crack-pot, or he would simply regurgitate old theories. What new interpretation could he give that no researcher trained in the field had not already given or debunked?

Rudman makes several good points and presents some interesting theories, but he also makes many unsubstantiated generalizations and assumptions. For example, the author gives a rather convincing argument that the Babylonian base-60 number system was not a “choice”, as some history texts try to argue, but a natural result of generations upon generations of people using a natural base-10 system for counting and a base-6 system of land measurement. The result is that the Babylonian base-60 system is in fact an alternating base-10 and base-6 replacement system. He goes on to say that their system is inconvenient due to the large amount of memorization that would be required to do multiplication. This assumes, of course, that they would memorize, and downplays the fact that there are a large number of tablets from the Old Babylonian era containing multiplication tables. Throughout the book, he makes generalizations based on our culture, and only peripherally engages the fact that these cultures were vastly different from ours, and that mathematics was conducted by a few select individuals for very specific purposes.

As another example, he comments repeatedly on the transmission of mathematics between Babylon and Egypt, without addressing how such transmission would occur, or more importantly, the vast differences in their mathematical practices that argue against early transmission. Finally, the author makes the unfortunate choice of delving into pedagogy and educational politics in the last few pages of the book. This is not only unnecessary, it leaves the reader with a negative last impression. It would have been best if he had ended off on a high mathematical note.

*How Mathematics Happened* is an object lesson in both the glories of critical reasoning and the pitfalls and dangers of historical research. Rudman provides some thought-provoking arguments and interpretations that bear consideration. A teacher of a critical reasoning or history course could make good use of both the strengths and weaknesses of this book. In general, it is an interesting read, but the author’s arguments must be taken with a grain of salt.

Amy Shell-Gellasch is a Faculty Fellow at Pacific Lutheran University in Tacoma, WA. She is actively involved with the MAA and its History of Mathematics SIGMAA as chairperson to several committees. She enjoys researching and promoting the use of history in the teaching of mathematics through editing books and organizing meetings. She received her bachelor’s degree from the University of Michigan in 1989, her master’s degree from Oakland University in Rochester, Michigan in 1995, and her doctor of arts degree from the University of Illinois at Chicago in 2000.