When we approach ancient scientific texts, it is easy to make one of two opposite mistakes. On the one hand, we are often tempted to assume too quickly that we and the ancient author are “talking about the same thing”, even if we use different words. On the other, we can be so struck by the distance between the ancient mindset and worldview and our own that we see only the differences. In general, scientists and mathematicians are particularly susceptible to the first mistake, while humanists and historians sometimes fall into the second. The first error tends to produce modernizing translations in which Archimedes writes down formulas and find the sum of infinite series, while the second tends to cast doubt on the very possibility of translation.

This book is about how to translate ancient scientific and mathematical texts while still avoiding either extreme. Suppose, for example, that an ancient medical text says something about the heart. Should we use that word in the translation? If we do, aren’t we inevitably bringing in associated notions (pumping, circulation of blood) that had no place in ancient thought? If we don’t, aren’t we hiding something crucial from the reader?

The book is the proceedings volume for a symposium entitled “Writings of Early Scholars in the Ancient Near East, Egypt, and Greece: Zur Übertsetzbarkeit von Wissenschaftsspachen des Altertums,” held in Mainz in 2009. The contributions (in German and English) range from theoretical considerations of the role of language in scientific disciplines to discussions of the specific problems connected to translating medical, astronomical, and mathematical texts. For those interested in mathematics and its history, the most interesting articles will be the introductory overview by the editors and the last three articles, so we will focus on those.

The introduction usefully sets up the question and proposes a paradigm for translation: present the modern reader with both a literal translation and interpretive commentary. The commentary might have several layers, allowing for different approaches for the material.

The three articles about translating mathematical texts form the fifth section of the book. Jens Høyrup gives us “How to transfer the conceptual structure of Old Babylonian mathematics: solutions and inherent problems. With an Italian parallel.” This is a detailed discussion of his “conformal translation” of Old Babylonian mathematical texts. Høyrup has, of course, explained his method before (most notably in his *Lengths, Widths, Areas*), but it is useful to have a focused account of his decisions and the reasoning behind them.

Jim Ritter’s “Translating rational-practice texts” presents a couple of different ways to approach “procedural” texts, which explain how to solve a problem (or treat a disease) by giving an enumeration of steps. One of Ritter’s proposals is that these texts be “translated” into flowcharts to highlight their algorithmic character. The results are interesting, though I tend to see them as methods for understanding ancient texts rather than as “translations.”

Annette Imhausen’s “From the cave into reality: Mathematics and cultures” is the least satisfying of these essays, perhaps reflecting the fact that it was originally intended as introductory. It is very general and pays too much attention to the old controversy between Sabetai Unguru and older historians of Greek mathematics. This is an interesting story, but it must be discussed with far more care than is shown here. (There is a quick mention of the controversy in Høyrup’s article as well. One is struck by how much more balanced and irenic his discussion is.)

Some of the remaining papers are also useful to those interested in history. I found the two articles that deal with ancient translations of ancient texts particularly valuable, because they offer us a glimpse of how the ancient readers understood each other. I also enjoyed a couple of the medical and astronomical articles. Overall, this collection provides a good introduction to the issues of translation.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College.