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The Pythagorean Theorem: A 4,000-Year History

Eli Maor
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Amy Shell-Gellasch
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Those of you who have enjoyed Eli Maor’s other books, such as Trigonometric Delights, will surely enjoy his newest work, The Pythagorean Theorem: A 4,000-Year History. As the name suggests, Maor traces the history of the Pythagorean Theorem from the Babylonians to the present. However, this book is not so much a history of the theorem, as a look at how it has been explored and used through mathematical history.

Maor expertly tells the story of how this simple theorem known to schoolchildren is part and parcel of much of mathematics itself. He uses the Pythagorean Theorem to show how interconnected the various disciplines of mathematics are. After the obligatory brief discussion about Andrew Wiles, Maor briefly looks at the Babylonian, Egyptian and Chinese work with right triangles. He then looks at Pythagoras and his school.

Maor goes on to present the theorem’s appearance and implications in Euclid’s Elements, Viéte’s work on trigonometric functions, its use in differential triangles in the early calculus, and ultimately builds to its use in current studies of space-time and the shape of the universe. Al so, though this is not the focus of the book, Maor does present a few of the more than 400 proofs of the theorem, as well as interesting variations on some of these proofs.

I found the writing, as with all of Maor’s works, clear and concise as well as enjoyable. He has a talent for presenting technical topics to the lay audience at a fairly high level. He does this in such a way that the reader is simply carried along to higher levels painlessly and seamlessly. The organization of the text is such that more difficult material can be easily skipped without loss of continuity. Much of the more intricate mathematics is left to the appendices, while sidebars after many of the chapters provide interesting sidetracks.

I only have two criticisms. First, when discussing ancient Babylonian work on right triangles, Maor did not cite nor seem to use the work of the leading researcher in the field, Eleanor Robson.

Second, in his final chapter entitled “Afterthoughts”, Maor tries to make ties between the Pythagorean Theorem and other topics within and outside of mathematics. In discussing the connections within math, he points out the prevalence of the square root of x2 + y2 in integration tables, Bessel functions and metrics. However, since trigonometry is based on a right triangle imbedded in a circle, and many metrics are based on versions of the Euclidean distance formula, I felt that this chapter begged the question to some extent. It would have been more effective to present only the connections to subjects removed from mathematics, such as music.

That said, the above concerns in no way detract from the charm of this book. I would recommend it to anyone with an interest in the evolution of mathematics and its interconnectedness. Though written for the lay public, it would be most appropriate for those with some knowledge of calculus. This book is a good choice for college mathematics majors, and high school and college mathematics teachers would find it beneficial both personally and for its implications for teaching. Even mathematically savvy readers will gain insights into the inner workings and beauty of mathematics. I await Maor’s next work; he always tells a good story.

Amy Shell-Gellasch is a Faculty Fellow at Pacific Lutheran University in Tacoma, WA. She is actively involved with the MAA and its History of Mathematics SIGMAA as chairperson to several committees. She enjoys researching and promoting the use of history in the teaching of mathematics through editing books and organizing meetings. She received her bachelor’s degree from the University of Michigan in 1989, her master’s degree from Oakland University in Rochester, Michigan in 1995, and her doctor of arts degree from the University of Illinois at Chicago in 2000.

 List of Color Plates ix
Preface xi
Prologue: Cambridge, England, 1993 1
Chapter 1: Mesopotamia, 1800 bce 4
Sidebar 1: Did the Egyptians Know It? 13
Chapter 2: Pythagoras 17
Chapter 3: Euclid's Elements 32
Sidebar 2: The Pythagorean Theorem in Art, Poetry, and Prose 45
Chapter 4: Archimedes 50
Chapter 5: Translators and Commentators, 500-1500 ce 57
Chapter 6: François Viète Makes History 76
Chapter 7: From the Infinite to the Infinitesimal 82
Sidebar 3: A Remarkable Formula by Euler 94
Chapter 8: 371 Proofs, and Then Some 98
Sidebar 4: The Folding Bag 115
Sidebar 5: Einstein Meets Pythagoras 117
Sidebar 6: A Most Unusual Proof 119
Chapter 9: A Theme and Variations 123
Sidebar 7: A Pythagorean Curiosity 140
Sidebar 8: A Case of Overuse 142
Chapter 10: Strange Coordinates 145
Chapter 11: Notation, Notation, Notation 158
Chapter 12: From Flat Space to Curved Spacetime 168
Sidebar 9: A Case of Misuse 177
Chapter 13: Prelude to Relativity 181
Chapter 14: From Bern to Berlin, 1905-1915 188
Sidebar 10: Four Pythagorean Brainteasers 197
Chapter 15: But Is It Universal? 201
Chapter 16: Afterthoughts 208
Epilogue: Samos, 2005 213
Appendixes 219-237
Chronology 241
Bibliography 247
Illustrations Credits 251
Index 253


akirak's picture

This is an excellent book on the history of the Pythagorean Theorem. I learnt a great deal of history of mathematics. It is suitable for any student who has basic knowledge of calculus, but the layperson will also find it interesting.

There are a number of gems in the book which are not that well known in the mathematics community: "Hypotenuse" is derived from the Greek words hypo meaning "under" or "down" and teinen meaning "to stretch." The reason for this is that the hypotenuse of a right triangle in Euclid's Elements was always on the bottom. There are over 400 proofs of Pythagoras's Theorem. It was the French lawyer François Viète who first converted verbal algebra into symbolic algebra. Many more of these gems crop up throughout the book.

You will learn a lot from this book because it has been thoroughly researched and shows the different fields where Pythagoras's Theorem is used. The author has also made excellent use of illustrations so the layperson can understand without learning all the details. Maor has an exceptional method of writing very technical mathematics in a seamlessly way.

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