A monumental accomplishment in the history of nonWestern 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 Chineselanguage 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 determinantalstyle 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.""
