# The Chinese Roots of Linear Algebra

Publication Year: 2011

Published by: The Johns Hopkins University Press

#### Cover

#### Title Page

#### Copyright

#### Contents

Download PDF (59.0 KB)

pp. vii-viii

#### Preface

Download PDF (103.8 KB)

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...

#### 1. Introduction

Download PDF (135.2 KB)

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...

#### 2. Preliminaries

Download PDF (172.6 KB)

pp. 11-26

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

#### 3. The Sources: Written Records of Early Chinese Mathematics

Download PDF (698.4 KB)

pp. 27-43

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

#### 4. Excess and Deficit

Download PDF (378.2 KB)

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...

#### 5. *Fangcheng*, Chapter 8 of the Nine Chapters

Download PDF (524.1 KB)

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*...

#### 6. The *Fangcheng* Procedure in Modern Mathematical Terms

Download PDF (226.1 KB)

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...

#### 7. The Well Problem

Download PDF (1.1 MB)

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...

#### 8. Evidence of Early Determinantal Solutions

Download PDF (378.6 KB)

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...

#### 9. Conclusions

Download PDF (136.1 KB)

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...

#### Appendix A: Examples of Similar Problems

Download PDF (146.8 KB)

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,...

#### Appendix B: Chinese Mathematical Treatises

Download PDF (1.8 MB)

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....

#### Appendix C: Outlines of Proofs

Download PDF (125.6 KB)

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

Download PDF (1.1 MB)

pp. 262-278

#### Index

Download PDF (717.6 KB)

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