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Chinese Roots of Linear Algebra

by Roger Hart Johns Hopkins University Press
Pub Date:
01/2011
ISBN:
9780801897559
Format:
Hbk 304 pages
Price:
AU$135.00 NZ$139.13
Product Status: Available in Approx 14 days
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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.

Preface
1. Introduction
Overview of This Book
Historiographic Issues
Outline of the Chapters
2. Preliminaries
Chinese Conventions
Chinese Mathematics
Modern Mathematical Terminology
3. The Sources: Written Records of Early Chinese Mathematics
Practices and Texts in Early Chinese Mathematics
The Book of Computation
The Nine Chapters on the Mathematical Arts
4. Excess and Deficit
Excess and Deficit Problems in the Book of Computation
"Excess and Deficit," Chapter 7 of the Nine Chapters
5. Fangcheng, Chapter 8 of the Nine Chapters
The Fangcheng Procedure
Procedure for Positive and Negative Numbers
Conclusions
6. The Fangcheng Procedure in Modern Mathematical Terms
Conspectus of Fangcheng Problems in the Nine Chapters
Elimination
Back Substitution
Is the Fangcheng Procedure Integer-Preserving?
Conclusions
7. The Well Problem
Traditional Solutions to the Well Problem
The Earliest Extant Record of a Determinantal Calculation
The Earliest Extant Record of a Determinantal Solution
Conclusions
8. Evidence of Early Determinantal Solutions
The Classification of Problems
Five Problems from the Nine Chapters
Conclusions
9. Conclusions
The Early History of Linear Algebra
Questions for Further Research
Methodological Issues
Significance and Implications
Appendix A: Examples of Similar Problems
Examples from Diophantus's Arithmetica
Examples from ModernWorks on Linear Algebra
Appendix B: Chinese Mathematical Treatises
Bibliographies of Chinese Mathematical Treatises
Mathematical Treatises Listed in Chinese Bibliographies
Appendix C: Outlines of Proofs
Bibliography of Primary and Secondary Sources
Index

"It seems likely that Hart's thoughtful, meticulous book will be the precursor to much fruitful study not only of pre-modern Chinese mathematics but also the roles of literacy and notation in its transmission."

Roger Hart is Director of the Confucius Institute and an associate professor of history at Texas Southern University. He has received fellowships from ACLS, NEH, and Mellon. Previous appointments include Seoul National University, University of Texas at Austin, University of Chicago, Institute for Advanced Study, Stanford, UC Berkeley, and Harvard. He is author of The Chinese Roots of Linear Algebra, also published by Johns Hopkins.