The Chinese Roots of Linear Algebra This page intentionally left blank Roger Hart The Chinese Roots of Linear Algebra The Johns Hopkins University Press Baltimore © 2011 The Johns Hopkins University Press All rights reserved. Published 2011 Printed in the United States of America on acid-free paper 987654321 The Johns Hopkins University Press 2715 North Charles Street Baltimore, Maryland 21218-4363 www.press.jhu.edu Library of Congress Control Number: 2010924546 ISBN 13: 978-0-8018-9755-9 ISBN: 0-8018-9755-6 A catalog record for this book is available from the British Library. Special discounts are available for bulk purchases of this book. For more information, please contact Special Sales at 410-516-6936 or [email protected]. The Johns Hopkins University Press uses environmentally friendly book materials, including recycled text paper that is composed of at least 30 percent post-consumer waste, whenever possible. All of our book papers are acid-free, and our jackets and covers are printed on paper with recycled content. With love, gratitude, and affection to my daughter, Nikki Janan Hart, and to my parents, Walter G. Hart, Jr. and Charlene Hart This page intentionally left blank Contents Preface ............................................................ ix 1 Introduction ...................................................... 1 Overview of This Book .............................................. 2 Historiographic Issues .............................................. 4 Outline of the Chapters ............................................. 9 2 Preliminaries ...................................................... 11 Chinese Conventions . .............................................. 11 Chinese Mathematics .............................................. 13 Modern Mathematical Terminology . ................................ 16 3 The Sources: Written Records of Early Chinese Mathematics ......... 27 Practices and Texts in Early Chinese Mathematics . ................... 27 The Book of Computation ........................................... 28 The Nine Chapters on the Mathematical Arts ......................... 30 4 Excess and Deficit ................................................. 44 Excess and Deficit Problems in the Book of Computation .............. 44 “Excess and Deficit,” Chapter 7 of the Nine Chapters .................. 57 5 Fangcheng, Chapter 8 of the Nine Chapters .......................... 67 The Fangcheng Procedure .......................................... 67 Procedure for Positive and Negative Numbers ........................ 81 Conclusions . ...................................................... 85 6 The Fangcheng Procedure in Modern Mathematical Terms .......... 86 Conspectus of Fangcheng Problems in the Nine Chapters .............. 87 Elimination . ...................................................... 88 Back Substitution .................................................. 93 Is the Fangcheng Procedure Integer-Preserving? . .................... 99 Conclusions . ...................................................... 109 vii viii Contents 7 The Well Problem .................................................. 111 Traditional Solutions to the Well Problem ............................ 112 The Earliest Extant Record of a Determinantal Calculation . .......... 125 The Earliest Extant Record of a Determinantal Solution ............... 132 Conclusions . ...................................................... 149 8 Evidence of Early Determinantal Solutions.......................... 151 The Classification of Problems . ..................................... 151 Five Problems from the Nine Chapters ............................... 156 Conclusions . ...................................................... 179 9 Conclusions ....................................................... 180 The Early History of Linear Algebra . ................................ 180 Questions for Further Research . ..................................... 188 Methodological Issues .............................................. 189 Significance and Implications . ..................................... 191 Appendix A: Examples of Similar Problems .............................. 193 Examples from Diophantus’s Arithmetica ............................ 194 Examples from Modern Works on Linear Algebra . ................... 211 Appendix B: Chinese Mathematical Treatises ............................ 213 Bibliographies of Chinese Mathematical Treatises . ................... 213 Mathematical Treatises Listed in Chinese Bibliographies .............. 230 Appendix C: Outlines of Proofs .......................................... 255 Bibliography of Primary and Secondary Sources ......................... 262 Index .................................................................. 279 Preface My inquiry into linear algebra in China began in the context of my research on the introduction of Euclid’s Elements into China in 1607 by the Italian Jesuit Matteo Ricci (1552–1610) and his most important Chinese collaborator, the official Xu Guangqi (1562–1633). Chinese mathematics of the time, Xu had argued in his prefaces and introductions, was in a state of decline, and all that remained of it was vulgar and corrupt. Western mathematics was in every way superior, he asserted, and in the end the loss of Chinese mathematics was no more re- grettable than discarding “tattered sandals.” Xu’s pronouncements have been so persuasive that they have, at least until recently, been accepted for the most part by historians—Chinese and Western alike—as fact. But as I studied the Chinese mathematics of the period, I became increasingly skeptical of Xu’s claims. The most interesting evidence against his claims was to be found in Chinese developments in linear algebra—arguably the most sophis- ticated and recognizably “modern” mathematics of this period in China. General solutions to systems of n linear equations in n unknowns had not been known in Europe at the time, and their importance was not lost on the Jesuits and their col- laborators. For they copied, without attribution, one by one, linear algebra prob- lems from the very Chinese mathematical treatises they had denounced as vulgar and corrupt. They then included the problems in a work titled Guide to Calcula- tion (Tong wen suan zhi , 1613), which they presented as a translation of German Jesuit Christopher Clavius’s (1538–1612) Epitome arithmeticae practicae (1583). Chinese readers of this “translation” had no way of knowing that none of these problems on linear algebra had been known in Europe at the time. I sought, then, to trace the specific Chinese sources from which the Jesuits and their collaborators had copied their problems. I was particularly interested in one problem, which I will call the “well problem,” in which the value assigned to the depth of a well is explained as resulting from the calculation 2 × 3 × 4 × 5 × 6 + 1. I was perplexed by this unexplained and apparently ad hoc calculation, and at first supposed that it was merely a numerological coincidence that multiplying the elements of the diagonal of a matrix together, and then adding one, would result in the value for the depth of the well. The Jesuits and their collaborators ix x Preface offered no explanation for this calculation, and indeed had evidently not noticed that an explanation might be necessary. The well problem had originated in the Nine Chapters on the Mathematical Arts (Jiuzhang suanshu ,c.1st cen- tury C.E.), the earliest transmitted text on Chinese mathematics. But in examin- ing the earliest extant version of the Nine Chapters, I found no mention of this perplexing calculation for finding the depth of the well. It was included, however, in a more popular text from the fifteenth century, Wu Jing’s (n.d.) Complete Compendium of Mathematical Arts of the Nine Chapters, with Detailed Commen- tary, Arranged by Category (Jiuzhang xiang zhu bilei suanfa da quan , preface dated 1450, printed in 1488). However, an explanation for this calculation was not to be found in Wu Jing’s Complete Compendium either. It happened, at that time, that I had been accorded the great privilege of spending a year at the Institute for Advanced Study in Princeton. The Institute is a special place where one is encouraged to believe that something perplex- ing deserves serious inquiry. Such freedom, as I hope this book demonstrates, is essential to academic research: in any investigation, at each fork in the road, if one ignores what might be crucial evidence, one risks taking the wrong path, only, perhaps, to wind up further and further from the truth. The puzzle of the well problem proved to be one such crucial fork. For eventually, I noticed that the calculation I had stumbled upon was not merely numerological coincidence, but what we would call in modern terms the determinant of the matrix of coeffi- cients.1 This interested me. Not quite sure what I was looking for, or why, I began to look for general solutions to the well problem. After more than a little work, I found that the solutions to the problem could be found more easily by deter- minantal calculations than by the usual elimination procedures. What was more interesting still was the extraordinary symmetry of these solutions. And although determinantal solutions are impracticably difficult for matrices with 5 equations in 5 unknowns, it was only the symmetric placement of numerous zeros in the well problem that permitted a determinantal solution. Then I noticed something that would permanently change the way I looked at