The Chinese Roots of Linear Algebra

Preface

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 regrettable 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 sophisticated 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 collaborators . For they copied, without attribution, one by one, linear algebra problems from the very Chinese mathematical treatises they had denounced as vulgar and corrupt. They then included the problems in a work titled Guide to Calculation (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 century C.E.), the earliest transmitted text on Chinese mathematics. But in examining 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 Commentary , 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 perplexing 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 coefficients .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 determinantal 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...