Pythagorean Triangles by Wacław Sierpiński, trans. Ambikeshwar Sharma, Yeshiva University, New York, 1962. Reprint, Dover, Mineola, NY, 2013. viii + 72 pp., ISBN 978-0-486-43278-6, $12.95 (paperback).

Who was the first mathematician you learned of in school? Do you remember the first time you heard the term theorem? For many students, the answer to both questions comes with the introduction of the Pythagorean theorem regarding right triangles. This foundational result may hold the record for the most proofs: Elisha Scott Loomis (profiled in the first year of the American Mathematical Monthly [6]) eventually compiled 370 in The Pythagorean Proposition [15]. His compendium (itself a classic, but unfortunately out of print) includes proofs by Gottfried Leibniz, U.S. President James Garfield, and Leonardo da Vinci (although that last attribution is questionable [12]). Loomis concludes the 1940 addenda with the understatement, “And the end is not yet.” Just in this Journal during the five years I served as editor, newly published proofs of the Pythagorean theorem included a Proof Without Words [10], one using continuity [13], and a long proof by a noted economist [2]. The ongoing fascination with the topic continues in this issue with two new framing proofs [1]; see that article also for a nice historical survey of proof techniques (briefer than Loomis’s book). A related discrete topic of number-theoretic interest is Pythagorean triples from right triangles whose sides all have integer lengths, i.e., positive integers x, y, z such that x2 + y2 = z2 (the square of the Major General’s hypotenuse). Many know the example (3, 4, 5), some (6, 8, 10) and (5, 12, 13), but there is a rich yet largely elementary theory here. These triples certainly predate Pythagoras, with several extant examples from ancient Babylonia, China, and India. Like the Pythagorean theorem, Pythagorean triples have been a mainstay of the MAA periodicals, from Dickson in the January 1894 inaugural issue of the Monthly [4] to recent articles in this Journal [7, 21]. A wonderful source for the theory of Pythagorean triples is still the book by Sierpiński under review here. Multiple times while I was fielding CMJ submissions, hopeful authors sent in Pythagorean triples results contained in this succinct book. One responded, “Thank you very much for the reference to Sierpiński’s book! I didn’t know that it existed!”—admirable excitement at a rejection notice. With this review, we hope to raise awareness of this enjoyable monograph. With its thorough and wide-ranging exploration of topics, elegant proofs, and informed history (especially concerning Fermat), Pythagorean Triangles could serve as a supplement in a number theory course or simply provide edifying mathematical pleasure reading.