Most of us know about Euclid the geometer, and about the Euclidean geometry named for him. If pressed, we might remember his most famous work, the *Elements*, which actually includes far more than just geometry. But for most of us, it ends there: we have never heard of the *Data*, the *Optics*, or (even more unlikely), the *Phaenomena*. So it's very good to have a new edition of the latter as the latest volume in the *History of Mathematics/Sources* series published by the American Mathematical Society and the London Mathematical Society.

*Phaenomena* means things that are apparent or observed; in the context of Greek science, it refers to observational astronomy, the attempt to describe (and, later, to quantify) the observed motion of the stars and planets. Euclid's book predates Ptolemy by some three or four centuries, so it gives us an unusual glimpse of early Greek astronomy. That it has survived at all probably owes much to the prestige of its author; many other books of this sort seem to have been superceded by Ptolemy's *Almagest* and therefore stopped being copied. Books that weren't copied didn't survive; sometimes we know their titles, but more often we know nothing at all about them.

The *Phaenomena* deals with what we might describe as the "easy" part of observational astronomy: the motion of the stars and of the sun. The major goal seems to be to study when stars rise above the horizon and when they set below it. In particular, this includes studying the length of the day. There is nothing here about the planets and their motions.

Euclid's approach is almost purely mathematical: once he has described the basic model for how the "cosmos" works, he sets out to prove a sequence of theorems about how various arcs move above and below the horizon. The text pressuposes quite a lot of knowledge of spherical geometry ("spherics"); thus, it does not seem to be aimed at beginners. To my untrained eyes, it does not seem to "get" anywhere... though that is hard to judge without more knowledge of the state of astronomy in Euclid's time.

Modern readers are often surprised to note the absence of quantitative issues in the *Elements.* In that book, no lengths, areas, or angles are measured; they are just compared. The *Phaenomena* is the same way: one wants Euclid to tell us how to decide how many hours are in a day, but that never comes. All we learn is about which times are equal and which are greater or smaller than others.

As the authors note, this is the first English translation of Euclid's *Phaenomena*. The edition and translation are beautifully done. The text is not easy to read, but then, it probably never *was* easy to read. The notes and introduction are very helpful. Anyone interested in the history of Greek science will want to have a copy. So will anyone who is curious about the history of how the Greeks deployed mathematical models to understand astronomical phenomena.

This is actually a new printing of a book first published by Garland in its *Sources and Studies in the History and Philosophy of Classical Science* series. The authors have provided a few new references, but the main body of the text is the same.

My students often seem to feel that since the Greeks worked with a geocentric model of the cosmos, they must have been benighted ignorant souls, incapable of understanding the "obvious" fact that the earth orbits around the sun. Perhaps this is a result of too many pop-science books and articles that wax eloquent about the supposedly profound intellectual impact of the Copernican revolution. In any case, working through this book would be a great antidote to such chronological arrogance. The model deployed here is elegant, effective, and (as far as it goes) perfectly true to the facts. And the mathematics is anything but simple-minded.

Fernando Q. Gouvêa loves reading old books.