This enjoyable book on the mathematics of ancient Egypt and Mesopotamia, with some things to say about the mathematics in Euclid’s *Elements*, could be read with pleasure by advanced high school students and used to teach a range of mathematical ideas and approaches to solving problems. It should appeal to anyone with a taste for the history of mathematics. I liked the brisk style in which it is written; the exposition is very clear and the problems are attractive.

Rudman engages at one point or another with most of the people who have written about these subjects, and he does so in a no-nonsense fashion that respects the great difficulties we have in interpreting the sources. It is easy enough to give modern readings of the ancient texts, and many writers have done so, offering a variety of methods for solving the problems that the texts leave unanswered. How did the Egyptian scribe find the numbers in the 2 : *n* table in the Rhind papyrus? How were the numbers in Plimpton 322 discovered? Could the Mesopotamians *prove* the Pythagorean theorem? Unfortunately, as Rudman cheerfully indicates, there is not much evidence to anchor this or that method in the distant past. The likelihood is that we shall never know for sure, and we can only make plausible guesses; Rudman offers his guesses with due caution, and gives his reasons for agreeing or disagreeing with others. This open-ended approach is rather inviting. His views on Mesopotamian mathematics are interestingly close to the geometric interpretations of Jens Høyrup and Eleanor Robson, who have written about these matters with the benefit of a thoroughly linguistic training.

It is not clear if Rudman has the ability to read the ancient scripts or if he relies on the work of others, which weakens his disagreement with Høyrup in Chapter 4 about what the words in the tablets mean. Attention to what the tablets actually say has contributed to an advance in our understanding of how they were written, and if this stops the history of ancient mathematics from being a game for mathematicians that is a small price to play for better knowledge.

This leads to the only significant weakness in the book: the care addressed to interpretation is not matched by a care about the sources themselves. Several tablets pass by with their strange modern names (VAT 8389,YBC 4608, TMS 1) but nowhere are these names explained, or any account of where the tablets come from provided, and, as Høyrup (2002) and Eleanor Robson (2008) have shown, we have quite a lot of fairly good evidence that helps us understand how and why the texts were written. To that extent this book contributes to the feeling that ancient mathematics is to be approached as a series of meta-problems: here is some mathematics, find the method that the original author used. This is a pity, not just because it shuts out relevant scholarship, but because some of the archaeological approaches and findings would make the subject even more interesting.

It would also have been better if the texts discussed here were linked to the fine source book edited by Victor Katz (2007), which is exactly the book that anyone excited by Rudman’s book should turn to next.

**References**

Høyrup, J. 2002. *Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and its Kin.* Springer.

Katz, V. (ed.) 2007. *The Mathematics of **Egypt**, **Mesopotamia**, **China**, **India**, and Islam: A Sourcebook*, Princeton University Press.

Robson, E. 2008. *Mathematics in Ancient **Iraq*, Princeton University Press.

Jeremy Gray is Professor of the History of Mathematics at the Centre for the History of the Mathematical Sciences of the Open University, in Milton Keynes, UK. He is the author of many books on the history of mathematics, including, most recently, *Plato's Ghost*.