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The Chinese Roots of Linear Algebra

Roger Hart

Publication Year: 2011

A monumental accomplishment in the history of non-Western mathematics, The Chinese Roots of Linear Algebra explains the fundamentally visual way Chinese mathematicians understood and solved mathematical problems. It argues convincingly that what the West "discovered" in the sixteenth and seventeenth centuries had already been known to the Chinese for 1,000 years. Accomplished historian and Chinese-language scholar Roger Hart examines Nine Chapters of Mathematical Arts—the classic ancient Chinese mathematics text—and the arcane art of fangcheng, one of the most significant branches of mathematics in Imperial China. Practiced between the first and seventeenth centuries by anonymous and most likely illiterate adepts, fangcheng involves manipulating counting rods on a counting board. It is essentially equivalent to the solution of systems of N equations in N unknowns in modern algebra, and its practice, Hart reveals, was visual and algorithmic. Fangcheng practitioners viewed problems in two dimensions as an array of numbers across counting boards. By "cross multiplying" these, they derived solutions of systems of linear equations that are not found in ancient Greek or early European mathematics. Doing so within a column equates to Gaussian elimination, while the same operation among individual entries produces determinantal-style solutions. Mathematicians and historians of mathematics and science will find in The Chinese Roots of Linear Algebra new ways to conceptualize the intellectual development of linear algebra.

Published by: The Johns Hopkins University Press

Title Page

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pp. vii-viii

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pp. ix-xiii

My inquiry into linear algebra in China began in the context of my research on the introduction of Euclidâ'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...

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1. Introduction

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pp. 1-10

In China, from about the first century C.E. through the seventeenth century, anonymous and most likely illiterate adepts practiced an arcane art termed fang-cheng (often translated into English as "matrices" or "rectangular arrays"). This art entailed procedures for manipulating counting rods arranged on a counting board,1 which enabled practitioners to produce answers to seemingly...

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2. Preliminaries

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pp. 11-26

This chapter presents background material necessary for understanding early Chinese linear algebra. It is divided into the following sections:...

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3. The Sources: Written Records of Early Chinese Mathematics

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pp. 27-43

In this chapter, I will present a preliminary overview of the textual records for Chinese mathematics. The chapter addresses three matters:...

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4. Excess and Deficit

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pp. 44-66

"Excess and deficit" problems, which seem to have been a precursor to fangcheng, are equivalent to systems of 2 conditions in 2 unknowns. They are found in the earliest surviving record of Chinese mathematics, the Book of Computation (c. 186 B.C.E.), and thus seem to predate fangcheng problems, which are more...

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5. Fangcheng, Chapter 8 of the Nine Chapters

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pp. 67-85

This chapter will focus on the fangcheng procedure, as presented in the Nine Chapters, to elucidate several key features: (1) fangcheng problems are displayed in two dimensions on the counting board; (2) entries are eliminated, in amanner similar to Gaussian elimination, by a form of "cross multiplication," bian cheng...

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6. The Fangcheng Procedure in Modern Mathematical Terms

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pp. 86-110

This chapter offers an analysis of the fangcheng procedure presented in chapter 8 of the Nine Chapters, which, as we saw in the preceding chapter, is presented there only in an abbreviated form. This chapter attempts to reconstruct the fangcheng procedure using modern mathematical terminology. The main goals here...

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7. The Well Problem

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pp. 111-150

This chapter focuses on the "well problem,"1 problem 13 from Fangcheng, chapter 8 of the Nine Chapters. This is arguably the most important of all the fangcheng problems.We will see that it is in commentaries on the well problem that we find the earliest written records--from anywhere in the world--of what we now call...

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8. Evidence of Early Determinantal Solutions

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pp. 151-179

The preceding chapter presented written records demonstrating a terminus ante quem of about 1025 for determinantal calculations and a terminus ante quem of 1661 for determinantal solutions for the "well problem." This chapter will examine evidence that determinantal methods may have been known as early as the...

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9. Conclusions

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pp. 180-192

The central thesis of this book is that it was the visualization in two dimensions of problems with n conditions in n unknowns that led to the discovery of solutions--solutions similar to those used today in modern linear algebra. This book attempts to reconstruct these mathematical practices, performed on the counting...

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Appendix A: Examples of Similar Problems

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pp. 193-212

Generally speaking, outside the Chinese sources analyzed in this book, there is comparatively little in the way of records of work on what we would now call systems of simultaneous linear equations before about 1678.1 There are some examples of systems of simultaneous linear equations in ancient Babylonia and Egypt,...

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Appendix B: Chinese Mathematical Treatises

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pp. 213-254

This appendix represents a preliminary attempt to assemble a comprehensive bibliography of all of the Chinese mathematical treatises that have been recorded in Chinese bibliographies throughout history....

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Appendix C: Outlines of Proofs

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pp. 255-261

This appendix presents brief justifications for several lemmas used in chapter 6. Because Sylvester's Identity is rarely included in current texts on linear algebra, for convenience I have summarized from standard sources a brief outline of a simple proof for the specific case of Sylvester's Identity used in chapter 6, Lemma...

Bibliography of Primary and Secondary Sources

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pp. 262-278


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pp. 279-286

E-ISBN-13: 9780801899584
E-ISBN-10: 0801899583
Print-ISBN-13: 9780801897559
Print-ISBN-10: 0801897556

Page Count: 304
Illustrations: 6 halftones, 5 line drawings
Publication Year: 2011