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**The Chinese roots of linear algebra.**
*(English)*
Zbl 1215.01003

Baltimore, MD: Johns Hopkins University Press (ISBN 978-0-8018-9755-9/pbk). xiii, 286 p. (2011).

The present study analyses the history of the ancient Chinese mathematical techniques fangcheng (rectangular arrays) found in the oldest known authentic Chinese mathematical text, the Suanshu shu (Book of Computation), the Jiuzhang suanshu (the ‘Nine Chapters’) and many other later Chinese sources up to the 17th century, all tackled in a critical way. Taking avail of such a solid documentary basis, the study is carried out with an unprecedented degree of precision, erudition, and expertise, mathematical and sinological, superseding by far everything previously written on the subject by historians of Chinese mathematics. Moreover, an incredible wealth of figures and diagrams provides a welcomed help to readers which is all the more essential since these Chinese mathematics are essentially visual and are intended to be practised in two dimensions rather than in rhetorical or symbolic form. But above all, the conclusions obtained by the author challenge those previously admitted in a convincing way. Among these, the author proves that the gist of the fangcheng method is a counter-intuitive integer-preserving approach which avoid fractions within the context of a kind of back substitution different from that attested in linear algebra. Another very interesting result concerns the analysis of the problem 13 from chapter 9 of the Nine Chapters (the “well problem”) in terms of what the author calls ‘determinantal methods’. And in this respect, a new conclusion is obtained: contrary to what is generally admitted by historians of Chinese mathematics, on the ground that this problem contains 5 equations and 6 unknowns, the ancient Chinese believed that it has only one solution and not infinitely many. Still another important point is the observation made by the author of the importance of the influence of notion of cross multiplication which is first used in chapter 7 of the Nine Chapters, a chapter devoted to the methods of double false position. In a word, historians of mathematics have now at their disposal a new and good basis for future studies of the proto-history of linear algebra, given that, needless to say many other points still have to be investigated: for example, welcomed additions to the present study would be (1) the taking into consideration of the linear problems of the Suanxue qimeng (Introduction to Mathematics) (1299) and of the Siyuan yujian (Jade Mirror of the Four Unknowns) (1303), and (2) the fact that in the context of Japanese traditional mathematics, the notion of determinant developed by Seki Takakazu was prompted not exactly by the study of linear systems but rather by the problem of the elimination of unknowns between non-linear polynomial systems of equations (and the same is true in the case of Leibniz).

Reviewer: Jean-Claude Martzloff (Paris)

### MSC:

01A25 | History of Chinese mathematics |

01A35 | History of mathematics in Late Antiquity and medieval Europe |

15-03 | History of linear algebra |