*Aristarchus of Samos The Ancient Copernicus* is an unabridged Dover reprint of the classic work by Sir Thomas Heath, 1861–1940, originally published by the Oxford University Press in 1913. There are, of course, digitized copies of the original 1913 edition available online, for example at the Internet Archive/Million Book Project, with a direct link at Wilborhall.org, the old Brown University History of Mathematics department. The author, as is well known to historians of mathematics, was a distinguished scholar of ancient Greek mathematics and Fellow of Trinity College, Cambridge. His subject in this book, Aristarchus of Samos, circa 310–230 B.C., lived between the times of Euclid, circa 325–265 B.C., and Archimedes, circa 287–212 B.C. To give this a bit more temporal context, this would have been a few decades after the death of Aristotle, 384–322 B.C.

The book is 425 pages and consists of two main parts. The first is a survey of Greek astronomical thought beginning with astronomical references in the works of Homer and Hesiod, down through the pre-Socratic philosophers from Thales through Pythagoras and ultimately down to Plato and Aristotle. In the ancient Greek world, astronomy was considered a part of mathematics, which in turn was considered a part of philosophy. As well as advancing specific philosophies, these thinkers dabbled in cosmological speculations. The purpose of this part of the book’s narrative is to set the stage for the world views that prevailed in Aristarchus’s time and place his work in historical context, as well as to give a historical argument supporting Aristarchus as the originator of the heliocentric theory.

I found this history of ideas rather striking, evolving over a period of four centuries from Thales’s view that the earth was not only flat, but circular, floating on water like a cork, to the Pythagoreans’ spherical earth rotating around a “Central Fire,” and finally to Aristarchus’s “Copernican” view, 1700 years too early. A particulary interesting historical question is why Aristarchus’s heliocentric theory did not gain traction. In fact there are no extant works of Aristarchus that would attest to him being the originator of heliocentrism. The main historical evidence is doxographic, that is, his work is mentioned by other writers of antiquity. Most notably, Archimedes in his famous work, *The Sand Reckoner*, mentions Aristarchus as the originator of the heliocentric view.

The second part of the book is Heath’s English translation of the surviving original Greek treatise by Aristarchus, *On the Sizes and Distances of the Sun and Moon*. Curiously, the treatise is geocentric and makes no mention of heliocentrism, a fact that Heath explains by speculating that it may have been a very early work of the author or possibly that for the mathematical problems solved, neither heliocentrism or geocentrism is material, and geocentrism simply may have been more familiar and less distracting than heliocentrism. The Greek text itself consists of 26 pages and comprises 18 propositions. The basic mathematical problem of the treatise is the determination of the (relative) diameters of the sun and moon as well as the distance to those bodies from the earth. Since in Aristarchus’s time, the diameter of the earth was unknown, these calculations use the diameter of the earth as the basic unit of distance. Aristarchus’s solution to this problem concludes, among other things, that the distance of the sun from the earth is between eighteen and twenty times the distance from the earth to the moon (Proposition 7).

This book is very carefully written and is a serious work of historical scholarship. There are copious references along with an extensive bibliography and index. The treatise itself is of course primary source material. This book will appeal to mathematicians interested in the history of mathematics and I can conceive of several possible uses for undergraduates or courses in the History of Mathematics. The book is a masterful example of “history as argument” and students could benefit from learning that the history of ideas is often murky, full of disagreement and even contention. If time limits reading the entire book, many of the individual chapters are more or less independent and selected passages could be assigned. There is also the mathematics of the treatise itself. Since the original Greek text is included along with English translation, it could be used as helpful practice material for students or instructors wanting to learn a little ancient Greek.

A point of mathematical interest, and the real tour de force of the treatise, is that it solves what is essentially a trigonometric problem by means of pure geometry. This was before trigonometry was invented by Hipparchus a century later. As a preface to the translation Heath includes commentary on the history and editions of the treatise. There is also a very illuminating exposition of Aristarchus’s arguments in modern trigonometrical form.

This book will be of interest to mathematicians interested in the history of the subject as well as to those in physics, astronomy, philosophy and the history of science in general.

Steven Deckelman is a professor of mathematics at the University of Wisconsin-Stout, where he has been since 1997. He received his Ph.D from the University of Wisconsin-Madison in 1994 for a thesis in several complex variables written under Patrick Ahern. Some of his interests include complex analysis, mathematical biology and the history of mathematics.