*The Babylonian Theorem: The Mathematical Journey to Pythagoras and Euclid*, Peter S. Rudman, 2010. 248 pp. illustrations, bibliography, index, and appendices. $26 hardcover, ISBN 71-1-59102-773-7. Prometheus Books, 59 John Glenn Dr., Amherst, NY 1422-2119. www.prometheusbooks.com

Perhaps the most controversial mathematical artifact in existence is Plimpton 322, a cuneiform tablet from the Old Babylonian Period (2000-1600 BCE) containing columns of numbers that have been interpreted as Pythagorean triples. The existence of such evidence casts doubt on the Greek claim for priority in understanding and using the mathematical proposition that relates the lengths of the three sides of a right triangle and is commonly known as the “Pythagorean Theorem.” Recent scholarship in the history of mathematics has affirmed that several ancient peoples, including the Hindus, the Chinese, and the Babylonians, knew and used the “Pythagorean Theorem” centuries before Pythagoras existed.

The content and form of the Plimpton tablet opened speculation on the mathematical techniques necessary for its compilation. Scholars agree that the Babylonian work implied algebraic formulation. Supporting this theory are collections of algebraic problems from the Old Babylonian Period revealing valid solution procedures. The work of Otto Neugebauer (1899–1990) in the 1930s showed that the ancient Babylonians did propose and solve right triangle problems, as well as obtain solutions for quadratic equations. In this work, they used algebraic techniques. This conclusion gave strength to the theory that much of the mathematical knowledge of the Egyptians and Greeks evolved from Babylonian efforts. In more recent years, scholarship on ancient Babylonian mathematics has greatly advanced due to the efforts of such individuals as Jens Høyrup, Eleanor Robson, and Jorän Friberg. If the Babylonians possessed strong algebraic traditions, what was this algebra like? Most investigators now feel that it was a geometrically conceived algebra where an expression such as \(A\times B\) could be visualized as a rectangle with area \(AB\).

Peter Rudman has taken this theory and explored it in some depth, tracing the advance of mathematical concepts and achievements from Old Babylonia to later societies. Rudman is not an academic historian of mathematics. He is a physicist and teacher who has used his scientific background to produce a compelling theory of mathematical understanding and transmission that is the subject of *The Babylonian Theorem. *He states the “Babylonian Theorem” itself as follows.

**Babylonian Theorem**: For any right triangle \((a,b,c),\) it is possible to construct another right triangle with sides: \(\sqrt{4ab},\) \(a-b\), \(a+b.\)

Starting with this theorem, the author uses a series of demonstrations in geometric algebra to derive the “Pythagorean Theorem,” develop square root algorithms, estimate the value of π, and obtain solutions for sets of linear equations and quadratic equations. Much is speculation and conjecture, really experimentation; but it is very good exploration, an adventure in thought. His demonstrations are convincing, offering nice examples of how the history of mathematics can be investigated. I found the reading of this book both interesting and worthwhile. It is a lovely, thought-provoking work. Brief exercises for the reader to solve, called “Fun Questions,” are interspersed throughout the text. The book is highly recommended for both secondary school and academic library acquisition as well as for personal reading.

See also the MAA Review by Jeremy J. Gray.