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The Chinese Roots of Linear Algebra.

The Chinese Roots of Linear Algebra. By ROGER HART. Baltimore: JOHNS HOPKINS UNIVERSITY PRESS, 2011. Pp. xiii + 286. $65.

This book is about f[]ngcheng [TEXT NOT REPRODUCIBLE IN ASCII] litt a procedure in ancient Chinese mathematics for solving parallel equations with multiple unknowns. Roger Hart reconstructs f[]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[]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[]ngji[]sh[]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[]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 historiographic import of the subject and an outline of the remaining chapters, all of which help the reader navigate the work. The final chapter restates the author's conclusions and elaborates a number of questions for further research. At the core of the book are five chapters treating three highly technical topics:

1. ying buzu 'excess and deficit', or systems of two conditions in two unknowns, a simpler precursor to the full f[]ngcheng procedure;

2. recovery of f[]ngcheng itself and explication of it in detail;

3. the "well problem" (for which there is no Chinese name): solution of a problem with n conditions in n + 1 unknowns, which the author shows is the earliest evidence of determinantal calculation, predating the seventeenth-century work of Leibniz and Seki.

Each is handled with all the necessary attention to texts, transcription of mathematical content, and interpretation. Much helpful matter surrounds this core. Chapter two presents a concise, readable introduction to the modern mathematical ideas themselves, as well as an introduction to traditional Chinese mathematics; chapter three reviews the textual sources and their commentaries. There is also an extensive bibliography of pre-modern Chinese mathematical texts, and that is in addition to the regular bibliography of the book's sources. A number of problems and their solutions are transcribed, diagrammed, and annotated in detail, and an appendix offers a resume of similar examples from the (non-general) solutions of the third-century Diophantus of Alexandria. The index is adequate if a little thin. As of this writing (September 2011), Hart has begun setting up a database of pre-modern Chinese linear algebra problems, solutions, and texts; it is on line at

So much for material content. The historical import of this book is to illuminate one of the antecedents of modern mathematics in the Sinosphere. The immediate incarnation of the modern mathematical synthesis goes back to a quickening and cross-fertilization that took place around the decades when (I do not jest) coffee made its way into Europe on a broad scale. But it was not the first such synthesis and it was fed by many strains of tradition and transmission. Now, there is no question of finding vectors, affine transformations, decompositions, or other elements of the developed field of modern linear algebra in the records of f[]ngcheng. The chief ancient discoveries of which Hart provides evidence, other than augmented matrices themselves, are Gaussian elimination (c. second century c.E.) and calculation of determinants (c. 1025 and 1661 for two different kinds of solution). We would, of course, like very much to know how Leibniz and Seki may have been influenced by older Chinese learning. Hart is apparently at work on more focused investigation of that problem and here he makes no wild claims, beyond raising the question and providing a part of the hard evidence on which it must eventually be decided. He does observe that there is nothing to show that the techniques he has reconstructed actually originated in China, merely that they have come down to us in Chinese compilations. In particular, nothing links the techniques intrinsically to any specific language or culture. Hart adds, provocatively, that it is not even clear that the people documenting them fully understood them.

This last observation raises the subject of the Chinese use of written symbols in the special context of mathematics and their relation to literacy. Mathematical quantities in the Jiuzhang suanshu are represented as numerals rather than words for numbers. That is, they are expressed not with the Chinese characters for ordinary words (si [TEXT NOT REPRODUCIBLE IN ASCII] 'four', liu [TEXT NOT REPRODUCIBLE IN ASCII] 'six', ba [TEXT NOT REPRODUCIBLE IN ASCII] 'eight', etc.), but rather in "rod-numeral" format ([TEXT NOT REPRODUCIBLE IN ASCII] '4', [TEXT NOT REPRODUCIBLE IN ASCII] '6', [TEXT NOT REPRODUCIBLE IN ASCII] '8', etc.), which derives from the physical chou [TEXT NOT REPRODUCIBLE IN ASCII] 'counting rods' used in ancient times and which even survives in certain popular contexts today. Some abstract concepts are written in characters, representing ordinary Chinese words in application to mathematics:

shi [TEXT NOT REPRODUCIBLE IN ASCII] 'seed, substance'[right arrow] 'dividend in division', perhaps 'upper quantity'

fa [TEXT NOT REPRODUCIBLE IN ASCII] 'method'[right arrow] 'divisor in division', perhaps 'lower quantity'

mu [TEXT NOT REPRODUCIBLE IN ASCII] 'mother'[right arrow] 'primary quantity'

zi [TEXT NOT REPRODUCIBLE IN ASCII] 'child'[right arrow] 'dependent quantity'

(In Han phonology, all four words would have been checked syllables and each pair is in a single tone, but it is hard to find immediate significance in that fact.)

The Western reader may find it strange to see whole Chinese words in places where modern mathematics uses symbols and letters. But in the earlier West--in Greek texts and in Boethius, for example--mathematics was written out in prose just as it was in traditional China. And even after symbols began to dominate European mathematics, the Roman and Greek letters conventionally used for naming variables and functions normally originated as abbreviations of real European words. Transparent conventions of this kind include i from integer, n from number, and f from function; many others can be identified with a little study. (More opaque perhaps are [summation] and [integral] from Latin summa 'sum', it originally from [pi][epsilon]pi[mu][epsilon]tpoc 'perimeter', x for an arbitrary variable from Arabic sifr 'zero', the source of our cipher and earlier transliterated xifr.) Although these symbol-letters can be used abstractly, taking on meanings other than those named here, they came into being as abbreviations of actual words. A few abstract logical symbols also have origins as letters of the alphabet, emptied entirely of their linguistic content and then put to work as true abstractions. One such is u for 'union (of sets)', spun around to become the various set operators [intersection] 'intersection', [subset] '(strict) subset', and [contains] '(strict) superset' (and cf. logical [disjunction] and [and]). Another is [THETA] for 'tight bound' in asymptotic theory; it apparently represents no word but recalls 0 'bounded above' and [OMEGA] 'bounded below', both standing for German Ordnung 'order [of growth of a function]', the bar in the middle of [THETA]being iconic for 'bounded on both sides'. [Introduced without explanation in Donald E. Knuth, "Big Omicron and Big Omega and Big Theta," ACM SIGACT News 8 (1976): 18-24; see p. 20.] The abstract, symbolic garb of modern mathematics and logic is rather new in the history of ideas. The choice of symbols usual in the modern mathematical synthesiS situates its linguistic origin in European civilization.

What about f[]ngcheng? Hart argues (p. 50 and elsewhere) that it was originally carried out using counters on a board, to aid first visualizing the relationship among unknowns in two dimensions and then cross-multiplying values in different parts of the board, so to arrive at general solutions of systems of linear equations. He observes that the earlier Chinese sources for f[]ngcheng lack diagrams--most odd for a visual and manual method--and are written in difficult literary language. He says:
  The translation of.f[]ngcheng calculations into classical
  Chinese, modern English prose, or modern mathematical terminology
  renders these practices almost incomprehensible. (p. 191)

So he proposes that they were transmitted in written form by Chinese literati who lacked a full grasp of the mathematics but codified it in the kind of elegant language that their own social class valued. Hart considers f[]ngcheng to have been fundamentally a hands-on process invented and practiced by people who not only lacked abstract mathematical notation but were actually illiterate. He illustrates the procedures he has reconstructed with some 150 diagrams of counting tables, as well as in modern matrix and algebraic notation--and he considers the use of illustrations the key to his successful reconstruction. If so, it was a fine inspiration; it enables the reader to follow the presentation with much more clarity and ease than otherwise.

Learning in the modern world is a wide-ranging synthesis, and a non-European origin for any particular idea is not surprising. But it is sobering to think that two techniques for solving matrix calculations, both considered standard today, could have been developed by non-literate people moving counting rods around with their hands. And that is where the hard evidence, together with some deductions that seem reasonable to this reviewer, now points.

Overall this book shows very careful work and clear presentation. If I have any single complaint, it is that individual Chinese words are not adequately indexed--but perhaps the author's on-line materials will remedy that. 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.

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Author:Branner, David Prager
Publication:The Journal of the American Oriental Society
Article Type:Book review
Date:Oct 1, 2011
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