The Almagest: Introduction to the Mathematics of the Heavens

The Almagest is one of those books that is much talked about but rarely read. More often than not, people refer to it as the source of the bad old geocentric astronomy, gloriously surpassed by Copernicus’s heliocentric system. This book is intended to help those few brave people who would like to get to know this amazing book.

What Ptolemy sets out to do is to create a mathematical theory of the motion of the stars (easy) and planets (very hard). In this context, the “planets” are the seven objects visible in the sky that move (after we subtract out the daily rotation): Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn. The ultimate goal is to be able to predict where a certain planet will be (or has been). This was both a practical problem (for casting horoscopes, predicting eclipses, and so on) and a mathematical challenge.

From the practical point of view, Ptolemy’s theory is a huge success, accurately predicting planetary positions. As a mathematical theory, it is very impressive but not immune to criticism. Ptolemy’s main tool is uniform circular motion, which is certainly easy to use and understand, but the planets do not really move in uniform circles, so a variety of tricks have to be used: circles whose center travels along another circle, circles whose center is not the center of the earth, motion that is uniform only when observed from a point that is not the center of the circle. This makes the theory quite complicated.

In order to make all this yield usable numerical tables, Ptolemy uses the “chord” function: given an arc of a circle, its chord is the length of the corresponding inscribed segment. As is well known, the chord is twice the sine of half the angle, so the table of chords is essentially a sine table. Much of the first book of the Almagest is devoted to constructing the table of chords, making it the oldest surviving trigonometrical treatise.

Teaching all this is quite hard. The Almagest is long and technical, after all. There was a whole tradition of writing “epitomes of the Almagest,” essentially guides or summaries. Springer recently republished Pedersen’s A Survey of the Almagest, a modern example of this genre.

This book presents an alternative. Instead of a summary or survey, it presents us with translations of selected portions of the Almagest, chosen to be teachable (if not easy). The book has 246 pages only, including many pages of commentary and annotation. (Compare that to Toomer’s translation of the whole book, which has xvii+693 pages.) As the blurb by James Evans says “Nothing will ever make mastering the astronomy of Ptolemy’s Almagest an easy task, but this book of selections, commentary and supporting material will at least make it a fair fight.”

But does anyone really want to teach the Almagest? It seems that St. John’s College still does, and perhaps some other colleges in the “great books” tradition do. Beyond that, the likely location for teaching it would be courses in Ancient Science and Astronomy, which I’m sure exist, but not in great number. In any case, for anyone who wants to know the Almagest first hand, this book may well be the best available resource.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.