Up until fairly recently, mathematicians (and, for that matter, classicists) interested in learning about Greek mathematics had few good places to begin. There was Heath's massive two-volume History of Greek Mathematics, of course. But Heath's book is old-fashioned in many ways. It focuses almost exclusively on the "great texts", paying no attention at all to mathematics in everyday life or to social context. In addition, it inevitably does not reflect current ideas about the texts it does study. After all, Heath's history was written in the 1920s, and much has happened in the field since then.
One could go beyond Heath and look at the more specialized literature, but that is a big and daunting task. An introductory book was needed, and Cuomo's Ancient Mathematics admirably fills that gap.
Let's begin by stating the obvious: the title is just wrong. This book is not about ancient mathematics as a whole; rather it is about the mathematics of ancient Greece, the hellenistic period, and the Roman period, ranging from around 500 BC to 500 AD. There is nothing here about the mathematics of Egypt, Mesopotamia, India, China, or other cultures.
As a survey of Greco-Roman mathematics, however, the book is very good. Cuomo divides the book into four main periods: Early Greek, Hellenistic, Graeco-Roman, and Late Antiquity. For each period she writes two chapters. The first discusses "The Evidence", and the second attacks "The Questions". This is a very good way to proceed, because it allows a clear focus on how we know what (we think) we know, and highlights the fact that the questions have by no means all been answered. Also good is the emphasis Cuomo places on the cultural context of mathematical activity. Her collection of "the evidence" goes beyond the great texts of high-level "pure" mathematics to look at everyday mathematics and at the mathematics of science.
The one regret I have about the book is the way Cuomo quotes ancient texts. In a laudable attempt to preserve their spirit and meaning, she quotes them straight, without any annotation or much explanation. Many readers, however, will find them hard to digest. It would have been helpful to follow the quote with a more leisurely paraphrase, if possible one that attempted to motivate the argument. (Reviel Netz frequently provides such paraphrases in his work.) Paraphrasing is risky, since one can seriously misrepresent the originals — Heath, in particular, has been accused of doing this — but this is an argument for doing it well, not for not doing it at all.
All in all, this is a great addition to the literature. Historians will have quibbles with points of interpretation, of course, but that's par for the course. Mathematicians interested in learning about "Greek mathematics" will do well to start here.
Fernando Q. Gouvêa is Professor of Mathematics at Colby College and the co-author, with William P. Berlinghoff, of Math through the Ages. He somehow finds time to also be the editor of MAA Reviews.
1. Early Greek Mathematics: The Evidence
1.1. Material Evidence
1.2 .Historians, Playwrights and Lawyers
2. Early Greek Mathematics; The Questions
2.1. The Problem of Political Mathematics
3. Hellenistic Mathematics: The Evidence
3.1. Material Evidence
3.2. Non-Mathematical Authors - The Rest of the World
3.3. Non-Mathematical Authors - The Philosophers
3.4. Little People
4. Hellenistic Mathematics: The Questions
4.1. The Problem of the Real Euclid
4.2. The Problem of the Birth of a Mathematical Community
5. Graeco-Roman Mathematics: The Evidence
5.1. Material Evidence
5.4. The Other Romans
5.5. The Other Greeks
6. Graeco-Roman Mathematics: The Questions
6.1. The Problem of Greek vs. Roman Mathematics
6.2. The Problem of Pure vs. Applied Mathematics
7. Late Ancient Mathematics: The Evidence
7.1. Material Evidence
7.5. The Philosophers
7.6. The Rest of the World
8. Late Ancient Mathematics: The Questions
8.1. The Problem of Divine Mathematics
8.2. The Problem of Ancient Histories of Ancient Mathematics