*The Pythagorean Theorem : A 4,000 Year History*, Eli Maor, 2007, xvi + 259 pp.: sidebars;9 color plates; 8 appendices; chronology; index , ISBN-13: 978-0-691-12526-8, Princeton University Press, $24.95. www.press.princeton.edu

The Pythagorean Theorem is the fourth in a series of books on historical topics by Eli Maor, Loyola University. The book is intended for the “reader with an interest in the history of mathematics” having sufficient knowledge of high school mathematics and some calculus. Some topics that require rigorous mathematical development are relegated to the appendices. The sidebars are varied in intent and seem to be afterthoughts which sometimes appear only marginally related to the chapters to which they are associated.

The first third of the book is devoted to the history of mathematics reflecting applications of the Pythagorean theorem up to the time of Euler. The source of much of this material is drawn from the works D. E. Smith and Howard Eves with an occasional reference to more modern authors such as George Joseph or William Dunham. For the sake of historical completion much mathematics is included that seems to have little to do with the actual subject. In contrast, the relevant work of Hippocrates of Chios and Heron and Ptolemy of Alexandria is relegated to later chapters. The first sidebar is devoted to demonstrating that the Egyptians had no knowledge of the Pythagorean Theorem. Yet, D. E. Smith hints at the existence of a Kahun (Berlin) papyrus containing Pythagorean triples. This is a typical example of how the author fails to consult current scholarship. The book is rife with the prejudices of older historians of mathematics including the often-repeated Greek fear of infinity and the tendency to repeat apocryphal stories as if they had some merit. I couldn’t help notice that Knorr’s *Evolution of the Euclidean Elements* (1975) was unfortunately missing from the bibliography.

The book then takes a break from the history of mathematics, per se, to discuss Elisha Loomis’s *The Pythagorean Proposition* (1907). This work contains 371 proofs of the Pythagorean Theorem. Maor reviews the more important proofs and adds a few new ones. The origin of the distance formula is clarified in Chapter 9 but only after having previously mentioned it without indicating its relationship to the Pythagorean Theorem. That connection might be lost to the general reader. The author resumes his historical investigation in Chapter 10 (p.145). Maor then shows the influence of the theorem on such modern topics as rectilinear coordinates, quaternions, vectors, non-Euclidean and differential geometries, and the theory of relativity. The sidebar of Chapter 14 on relativity contains 4 brainteasers that have nothing to do with relativity; 2 of these really belong to an earlier chapter containing Asian mathematics. The final two chapters attempt to sum up the universality of the theorem. Since the appendices are intended for rigorous mathematical expositions, I feel that the general reader would benefit if the appendices contained more of the difficult mathematics contained in the first 16 chapters.

There is much to be commended in this book. However, I would not recommend it as an introduction to the history of mathematics. Many of the references need updating to include current scholarship. The sequencing of some topics could be improved. I’m sure this is not the author’s fault, but the diagrams are frequently placed inconveniently on the backside of the page after the relevant exposition. I look forward to the second, improved, edition of this book.

Jim Kiernan, Adjunct Professor, Brooklyn College, New York City

See also the MAA Review by Amy Shell-Gellasch.