The Pythagorean Theorem: A 4,000-Year History

by Eli Maor

Overview

By any measure, the Pythagorean theorem is the most famous statement in all of mathematics. In this book, Eli Maor reveals the full story of this ubiquitous geometric theorem. Maor shows that the theorem, although attributed to Pythagoras, was known to the Babylonians more than a thousand years earlier. Pythagoras may have been the first to prove it, but his proof—if indeed he had one—is lost to us. The theorem itself, however, is central to almost every branch of science, pure or applied. Maor brings to life many of the characters that played a role in the development of the Pythagorean theorem, providing a fascinating backdrop to perhaps our oldest enduring mathematical legacy.

Product Details

ISBN-13:	9780691148236
Publisher:	Princeton University Press
Publication date:	08/15/2010
Series:	Princeton Science Library , #28
Edition description:	Reprint
Pages:	280
Product dimensions:	6.10(w) x 9.10(h) x 0.70(d)
Age Range:	14 - 17 Years

About the Author

Eli Maor is the author of Venus in Transit, Trigonometric Delights, To Infinity and Beyond, and e: The Story of a Number (all Princeton). He teaches the history of mathematics at Loyola University in Chicago and at the Graham School of General Education at the University of Chicago.

Table of Contents

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
A. How did the Babylonians Approximate? 219
B. Pythagorean Triples 221
C. Sums of Two Squares 223
D. A Proof that is Irrational 227
E. Archimedes' Formula for Circumscribing Polygons 229
F. Proof of some Formulas from Chapter 7 231
G. Deriving the Equation x^2/3 + y^2/3 = 1 235
H. Solutions to Brainteasers 237
Chronology 241
Bibliography 247
Illustrations Credits 251
Index 253

What People are Saying About This

Levy

Eli Maor has brought four thousand years of history back to life, all based on the Pythagorean theorem but still giving the times a distinctly human look. This book is designed for readers who are inspired, or who want to be inspired, by the numbers that Eli uses to tell his story. Readers will learn about the mathematics of the time, but more important, they will understand the people and the ideas of that period. A monumental effort.
— David H. Levy, National Sharing the Sky Foundation

Langer

Eli Maor states that the Pythagorean theorem 'is arguably the most frequently used theorem in all of mathematics.' He then supports this claim by taking his reader on a journey from the earliest evidence of knowledge of the theorem to Einstein's theory of relativity and Wiles's proof of Fermat's last theorem, from the Babylonians around 1800 BCE to the end of the twentieth century. I think that the reader who makes the journey with Maor will be convinced beyond a reasonable doubt. He is the first author who has sifted through all the mathematics, history of mathematics, and physics books and collected for us just the material directly related to the Pythagorean theorem.
— Robert W. Langer, Professor Emeritus, University of Wisconsin, Eau Claire

Dava Sobel

At last, a popular book that isn't afraid to print a mathematical formula in all its symbolic glory! Thanks to Eli Maor for proving—in his delightful, playful way—the eternal importance of a three-sided idea as old as humankind.
— Dava Sobel, author of "Longitude"

Nahin

There's a lot more to the Pythagorean theorem than a² + b² = c², and you'll find it all in Eli Maor's new book. Destined to become a classic, this book is written with Maor's usual high level of skill, scholarship, and attention to detail. He's also got a sense of humor that will please a range of readers. As we used to say in the 1950s, 'Miss it and be square!'
— Paul J. Nahin, author of "Chases and Escapes" and "Dr. Euler's Fabulous Formula"

From the Publisher

"At last, a popular book that isn't afraid to print a mathematical formula in all its symbolic glory! Thanks to Eli Maor for proving—in his delightful, playful way—the eternal importance of a three-sided idea as old as humankind."—Dava Sobel, author of Longitude

"Eli Maor has brought four thousand years of history back to life, all based on the Pythagorean theorem but still giving the times a distinctly human look. This book is designed for readers who are inspired, or who want to be inspired, by the numbers that Eli uses to tell his story. Readers will learn about the mathematics of the time, but more important, they will understand the people and the ideas of that period. A monumental effort."—David H. Levy, National Sharing the Sky Foundation

"There's a lot more to the Pythagorean theorem than a² + b² = c², and you'll find it all in Eli Maor's new book. Destined to become a classic, this book is written with Maor's usual high level of skill, scholarship, and attention to detail. He's also got a sense of humor that will please a range of readers. As we used to say in the 1950s, 'Miss it and be square!'"—Paul J. Nahin, author of Chases and Escapes and Dr. Euler's Fabulous Formula

"Eli Maor states that the Pythagorean theorem 'is arguably the most frequently used theorem in all of mathematics.' He then supports this claim by taking his reader on a journey from the earliest evidence of knowledge of the theorem to Einstein's theory of relativity and Wiles's proof of Fermat's last theorem, from the Babylonians around 1800 BCE to the end of the twentieth century. I think that the reader who makes the journey with Maor will be convinced beyond a reasonable doubt. He is the first author who has sifted through all the mathematics, history of mathematics, and physics books and collected for us just the material directly related to the Pythagorean theorem."—Robert W. Langer, Professor Emeritus, University of Wisconsin, Eau Claire

Editorial Reviews

This excellent biography of the theorem is like a history of thought written in lines and circles, moving from ancient clay tablets to Einstein's blackboards. . . . There is something intoxicating about seeing one truth revealed in so many ways. It all makes for hours of glorious mathematical distraction.—-Ben Longstaff, New Scientist


[The Pythagorean Theorem] is aimed at the reader with an interest in the history of mathematics. It should also appeal to most well-educated people...It is a story based on a theme and guided by a timeline...As a popular account of important ideas and their development, the book should be read by anyone with a good education. It deserves to succeed.—-Peter M. Neumann, Times Higher Education Supplement


Based on this recent book, Maor just keeps getting better. Already recognized for his excellent books on infinity, the number e, and trigonometry, Maor offers this new work as a comprehensive overview of the Pythagorean Theorem...If one has never read a book by Eli Maor, this book is a great place to start.—-J. Johnson, Choice


Maor expertly tells the story of how this simple theorem known to schoolchildren is part and parcel of much of mathematics itself. . . . Even mathematically savvy readers will gain insights into the inner workings and beauty of mathematics.—-Amy Shell-Gellasch, MAA Reviews


Maor's book is a concise history of the Pythagorean theorem, including the mathematicians, cultures, and people influenced by it. The work is well written and supported by several proofs and exampled from Chinese, Arabic, and European sources the document how these unique cultures came to understand and apply the Pythagorean theorem. [The book] provides thoughtful commentary on the historical connections this fascinating theorem has to many cultures and people.—-Michael C. Fish, Mathematics Teacher


This book will make for good supplementary reading for high school students, high school teachers, and those with a general interest in mathematics. . . . The author's enthusiasm for his subject is evident throughout the book.—-James J. Tattersull, Mathematical Reviews


This book goes beyond the theorem and its proofs to set it beautifully in the context of its time and subsequent history.—-Eric S. Rosenthal, Mathematics Magazine


This is an excellent book on the history of the Pythagorean Theorem. . . . This book is suitable to any student who has basic knowledge of calculus but the layperson will also find it interesting. . . . Maor has an exceptional method of writing very technical mathematics in a seamlessly way.—-Kuldeep, Mathematics and My Diary


All in all, this affordable book, as with Maor's previous titles, is rollicking good fun and highly recommended to anyone with even the slightest interest in the history of mathematics.—-Francis A, Grabowski, European Legacy


The Pythagorean Theorem is rich in information, careful in its presentation, and at times personal in its approach. . . . The variety of its topics and the engaging way they are presented make The Pythagorean Theorem a pleasure to read.—-Cecil Rousseau, College Math Journal


"At last, a popular book that isn't afraid to print a mathematical formula in all its symbolic glory! Thanks to Eli Maor for proving—in his delightful, playful way—the eternal importance of a three-sided idea as old as humankind."—Dava Sobel, author of Longitude


"Eli Maor has brought four thousand years of history back to life, all based on the Pythagorean theorem but still giving the times a distinctly human look. This book is designed for readers who are inspired, or who want to be inspired, by the numbers that Eli uses to tell his story. Readers will learn about the mathematics of the time, but more important, they will understand the people and the ideas of that period. A monumental effort."—David H. Levy, National Sharing the Sky Foundation


"There's a lot more to the Pythagorean theorem than a² + b² = c², and you'll find it all in Eli Maor's new book. Destined to become a classic, this book is written with Maor's usual high level of skill, scholarship, and attention to detail. He's also got a sense of humor that will please a range of readers. As we used to say in the 1950s, 'Miss it and be square!'"—Paul J. Nahin, author of Chases and Escapes and Dr. Euler's Fabulous Formula


"Eli Maor states that the Pythagorean theorem 'is arguably the most frequently used theorem in all of mathematics.' He then supports this claim by taking his reader on a journey from the earliest evidence of knowledge of the theorem to Einstein's theory of relativity and Wiles's proof of Fermat's last theorem, from the Babylonians around 1800 BCE to the end of the twentieth century. I think that the reader who makes the journey with Maor will be convinced beyond a reasonable doubt. He is the first author who has sifted through all the mathematics, history of mathematics, and physics books and collected for us just the material directly related to the Pythagorean theorem."—Robert W. Langer, Professor Emeritus, University of Wisconsin, Eau Claire

From the Publisher

The Pythagorean Theorem is rich in information, careful in its presentation, and at times personal in its approach. . . . The variety of its topics and the engaging way they are presented make The Pythagorean Theorem a pleasure to read.

College Math Journal - Cecil Rousseau

This is an excellent book on the history of the Pythagorean Theorem. . . . This book is suitable to any student who has basic knowledge of calculus but the layperson will also find it interesting. . . . Maor has an exceptional method of writing very technical mathematics in a seamlessly way.

Mathematics and My Diary - Kuldeep

This book goes beyond the theorem and its proofs to set it beautifully in the context of its time and subsequent history.

Mathematics Magazine - Eric S. Rosenthal

This book will make for good supplementary reading for high school students, high school teachers, and those with a general interest in mathematics. . . . The author's enthusiasm for his subject is evident throughout the book.

Mathematical Reviews - James J. Tattersull

Maor's book is a concise history of the Pythagorean theorem, including the mathematicians, cultures, and people influenced by it. The work is well written and supported by several proofs and exampled from Chinese, Arabic, and European sources the document how these unique cultures came to understand and apply the Pythagorean theorem. [The book] provides thoughtful commentary on the historical connections this fascinating theorem has to many cultures and people.

Mathematics Teacher - Michael C. Fish

Maor expertly tells the story of how this simple theorem known to schoolchildren is part and parcel of much of mathematics itself. . . . Even mathematically savvy readers will gain insights into the inner workings and beauty of mathematics.

MAA Reviews - Amy Shell-Gellasch

Based on this recent book, Maor just keeps getting better. Already recognized for his excellent books on infinity, the number e, and trigonometry, Maor offers this new work as a comprehensive overview of the Pythagorean Theorem...If one has never read a book by Eli Maor, this book is a great place to start.

Choice - J. Johnson

[The Pythagorean Theorem] is aimed at the reader with an interest in the history of mathematics. It should also appeal to most well-educated people...It is a story based on a theme and guided by a timeline...As a popular account of important ideas and their development, the book should be read by anyone with a good education. It deserves to succeed.

Times Higher Education Supplement - Peter M. Neumann

This excellent biography of the theorem is like a history of thought written in lines and circles, moving from ancient clay tablets to Einstein's blackboards. . . . There is something intoxicating about seeing one truth revealed in so many ways. It all makes for hours of glorious mathematical distraction.

New Scientist - Ben Longstaff

All in all, this affordable book, as with Maor's previous titles, is rollicking good fun and highly recommended to anyone with even the slightest interest in the history of mathematics.

European Legacy - Francis A

The Pythagorean Theorem is rich in information, careful in its presentation, and at times personal in its approach. . . . The variety of its topics and the engaging way they are presented make The Pythagorean Theorem a pleasure to read.
— Cecil Rousseau

College Math Journal

All in all, this affordable book, as with Maor's previous titles, is rollicking good fun and highly recommended to anyone with even the slightest interest in the history of mathematics.
— Francis A, Grabowski

European Legacy

This is an excellent book on the history of the Pythagorean Theorem. . . . This book is suitable to any student who has basic knowledge of calculus but the layperson will also find it interesting. . . . Maor has an exceptional method of writing very technical mathematics in a seamlessly way.
— Kuldeep

Mathematics and My Diary

This book goes beyond the theorem and its proofs to set it beautifully in the context of its time and subsequent history.
— Eric S. Rosenthal

Mathematics Magazine

This book will make for good supplementary reading for high school students, high school teachers, and those with a general interest in mathematics. . . . The author's enthusiasm for his subject is evident throughout the book.
— James J. Tattersull

Mathematical Reviews

Maor's book is a concise history of the Pythagorean theorem, including the mathematicians, cultures, and people influenced by it. The work is well written and supported by several proofs and exampled from Chinese, Arabic, and European sources the document how these unique cultures came to understand and apply the Pythagorean theorem. [The book] provides thoughtful commentary on the historical connections this fascinating theorem has to many cultures and people.
— Michael C. Fish

Mathematics Teacher

Maor expertly tells the story of how this simple theorem known to schoolchildren is part and parcel of much of mathematics itself. . . . Even mathematically savvy readers will gain insights into the inner workings and beauty of mathematics.
— Amy Shell-Gellasch

MAA Reviews

Based on this recent book, Maor just keeps getting better. Already recognized for his excellent books on infinity, the number e, and trigonometry, Maor offers this new work as a comprehensive overview of the Pythagorean Theorem...If one has never read a book by Eli Maor, this book is a great place to start.
— J. Johnson

Choice

[The Pythagorean Theorem] is aimed at the reader with an interest in the history of mathematics. It should also appeal to most well-educated people...It is a story based on a theme and guided by a timeline...As a popular account of important ideas and their development, the book should be read by anyone with a good education. It deserves to succeed.
— Peter M. Neumann

Times Higher Education Supplement

This excellent biography of the theorem is like a history of thought written in lines and circles, moving from ancient clay tablets to Einstein's blackboards. . . . There is something intoxicating about seeing one truth revealed in so many ways. It all makes for hours of glorious mathematical distraction.
— Ben Longstaff

New Scientist