*Pythagoras’ Revenge: A Mathematical Mystery*, Arturo Sangalli, 2009, 224 pp., 10 line illus., 1 map, cloth, $24.95, ISBN 978-0-691-04955-7, Princeton University Press, 41 William St., Princeton, NJ 08540.

I have been interested in the Pythagoreans ever since I wrote a paper on them for my Comparative Religions course during my senior year of high school. I was much disheartened later to find out that most of the information on them that I had read in Bertrand Russell’s *History of Western Philosophy** *was obtained from sources that could be considered less than reliable. Arturo Sangalli has chosen to present much of this information and other choice snippets of mathematics in the form of a “mathematical mystery”. The author quite clearly states that the Pythagoras story is “part fact, part fiction”. Sangalli then goes on to present a fictionalized account which mixes the rediscovery of an ancient manuscript explaining the main tenets of Pythagoreanism with the reincarnation of Pythagoras himself. Topics of ancient scholarship, textual criticism, and cultural imperialism are well covered. In this way, the author hopes to bring the confluence of archeology, mathematics, and philosophy to a much wider audience.

In one chapter the author employs the fictional ancient character of Lysis to present the basic principles of Pythagoreanism . He does a fine job of explaining the enigmatic aphorisms of the group. For example, the true meaning of the saying “Do not poke the fire with a sword” is that one should not provoke a man in anger. The history of the Pythagorean theorem is discussed in a later chapter. Contributions of the Babylonians, Indians, and Chinese are mentioned in passing. While the author refers directly to the *Sulba Sutras*, he does not mention the clay tablet known as Plimpton 322. The story of Pythagoras’ sacrifice of an ox upon proof of the theorem is related but no attempt is made to show how this story contradicts his beliefs and practices of vegetarianism. Plato’s diagram from the *Meno *and the Platonic solids are discussed. The author makes reference to Garfield’s proof of the Pythagorean theorem, and the work of Galileo, Kepler, and Einstein are discussed in passing. Detailed discussion of figurate numbers, the harmonic mean and the mathematical ratios of the musical scale are noteworthy. His discussion of the ratio of diagonal to side of a square culminates in a proof of their incommensurability using the idea of odd and even numbers such as the Pythagoreans would have demonstrated it.

One main theme of the story is the conflict between randomness and the determinism of Pythagoras’ “music of the spheres”. Pythagoras’ revenge arises from the fact that numbers “increasingly reign over the human-made world that is life in a modern society.” Additional mathematical topics discussed include: the 15 puzzle; Rubik’s cube; permutations; the cap distribution problem; Euclid’s *Elements*; generation of random binary sequences; pi series and Turing machines. Five appendices contain discussions of derangements, the infinite number of primes, random sequences, visual proof of the Pythagorean Theorem, and perfect and figured numbers.

By choosing to write a fictionalized account, Sangalli has given himself free reign on his subject. It is obvious that the author is trying to cash in on the current popular obsession with the *DaVinci Code*. Princeton University Press refers to the *Code* in its press release. Many of the more esoteric references such as the Naassene sect and the Neopitagorica Basilica really do exist. While I am usually wary of such blurring between fact and fiction, I can only imagine some student becoming as fascinated by these tales as I was and going on to do some real research into these subjects. For that reason alone I can give the book a reserved recommendation.

James F. Kiernan, Adjunct Professor, Brooklyn College, CUNY, New York

See also the MAA Review by Michele Intermont.