The Chinese Roots of Linear Algebra. By ROGER HART. Baltimore: JOHNS HOPKINS UNIVERSITY PRESS, 2011. Pp. xiii + 286. $65.
This book is about f[a.bar]ngcheng [TEXT NOT REPRODUCIBLE IN ASCII] litt a procedure in ancient Chinese mathematics for solving parallel equations with multiple unknowns. Roger Hart reconstructs f[a.bar]ngcheng in terms of modern linear or matrix algebra and shows that it involves methods associated with Leibniz (1646-1716), Seki Takakazu [TEXT NOT REPRODUCIBLE IN ASCII] (1642-1708), and Gauss (1777-1855), but long predating those men. The material is dense and made more intractable by the fact that the original texts exist only as fragments in late collections or in reconstructions later still. But the author, who holds higher degrees in mathematics and the intellectual history of Western learning in China, has produced a really meticulous display of philology and mathematical reconstruction. He succeeds in cleaning up a rat's nest of a subject and laying it open to the reader in a clear and precise form, without overreaching himself for attractive conclusions. Hart does offer much to think about, however--among other things, that Gaussian elimination and general solutions of systems of linear equations are ultimately of non-literate origin and attested in China some seventeen centuries before Gauss.
Our primary source for f[a.bar]ngcheng is the Jiutzhang suanshu [TEXT NOT REPRODUCIBLE IN ASCII] (Nine Chapters on the Mathematical Arts) of second-century C.E. date. It survives in thirteenth-century fragments, materials collected in the fifteenth-century Yongle dadian [TEXT NOT REPRODUCIBLE IN ASCII] (itself now also fragmentary), and an imperfect reconstruction by Dai Zhen [TEXT NOT REPRODUCIBLE IN ASCII] (1724-77) based on those sources. It has been much studied and discussed since Did Zhen's time and especially in the past century, but never before with the detail of Hart's book. Excavated materials (from Zh[a.bar]ngji[a.bar]sh[a.bar]n [TEXT NOT REPRODUCIBLE IN ASCII] in 1984) show us that the mathematics of parallel equations with two unknowns was being practiced in the second century B.C.E., although "f[a.bar]ngcheng" is not seen there. The term, meaning perhaps "[manipulating related] quantities on a board," survives in modern language to mean "equation" itself.
Hart's work is thorough, thoughtful, and effectively organized. The main conclusions are summarized in an introduction, together with a review of the...