Copyright © 2008 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY All Rights Reserved ISBN 978- 0- 691- 12745-3 British Library C atal oging- in-P ublication Data is available Library of Congress Cataloging-in-Publication Data Fukagawa, Hidetoshi, 1943– Sacred mathematics : Japanese temple geometry / Fukagawa Hidetoshi, Tony Rothman. p. cm. Includes bibliographical references. ISBN 978-0-691-12745-3 (alk. paper) 1. Mathematics, Japanese—History. 2. Mathematics—Japan— History—Tokugawa period, 1600–1868. 3. Mathematics—Problems, exercises, etc. 4. Yamaguchi, Kazu, d. 1850. I. Rothman, Tony. II. Title. QA27.J3F849 2008 510.952—dc22 2007061031 This book has been composed in New Baskerville Printed on acid- free paper. ∞ press.princeton.edu Printed in the United States of America 1 3 5 7 9 10 8 6 4 2 Frontispiece: People enjoying mathematics. From the 1826 Sanp¯o Binran, or Handbook of Mathematics. (Courtesy Naoi Isao.) To the Memory of Dan Pedoe Who fi rst introduced sangaku to the world Fukagawa Hidetoshi To Freeman Dyson For long friendship Tony Rothman Contents Foreword by Freeman Dyson ix Preface by Fukagawa Hidetoshi xiii Preface by Tony Rothman xv Ac know ledg ments xix What Do I Need to Know to Read This Book? xxi Notation xxv 1 Japan and Temple Geometry 1 2 The Chinese Foundation of Japa nese Mathematics 27 3 Japan ese Mathematics and Mathematicians of the Edo Period 59 4 Easier Temple Geometry Problems 89 5 Harder Temple Geometry Problems 145 6 Still Harder Temple Geometry Problems 191 7 The Travel Diary of Mathematician Yamaguchi Kanzan 243 8 East and West 283 9 The Mysterious Enri 301 10 Introduction to Inversion 313 For Further Reading 337 Index 341 Foreword by Freeman Dyson T his is a book about a special kind of geometry that was invented and widely practiced in Japan during the centuries when Japan was isolated from Western infl uences. Japa nese geometry is a mixture of art and math- ematics. The experts communicated with one another by means of sangaku, which are wooden tablets painted with geometrical fi gures and displayed in Shinto shrines and Buddhist temples. Each tablet states a theorem or a problem. It is a challenge to other experts to prove the theorem or to solve the problem. It is a work of art as well as a mathematical statement. Sangaku are perishable, and the majority of them have decayed and disappeared during the last two centuries, but enough of them have survived to fi ll a book with examples of this unique Japan ese blend of exact science and ex- quisite artistry. Each chapter of the book is full of interesting details, but for me the most novel and illuminating chapters are 1 and 7. Chapter 1 describes the historical development of sangaku, with emphasis on Japan’s “peculiar insti- tution,” the samurai class who had originally been in de pen dent warriors but who settled down in the seventeenth century to become a local aristoc- racy of well- educated offi cials and administrators. It was the samurai class that supplied mathematicians to create the sangaku and work out the prob- lems. It is remarkable that sangaku are found in all parts of Japan, including remote places far away from cities. The reason for this is that samurai w ere spread out all over the country and maintained good communications even with remote regions. Samurai ran schools in which their children became literate and learned mathematics. Samurai combined the roles which in medieval Eu rope were played separately by monks and feudal lords. They were scholars and teachers as well as administrators. Chapter 7 is my favorite chapter, the crown jewel of the book. It contains extracts from the travel diary of Yamaguchi Kanzan, a mathematician who made six long journeys through Japan between 1817 and 1828, recording details of the sangaku and their creators that he found on his travels. The