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7. Conclusions

Liu Hui's explanations of the statements in the Jiuzhang suanshu are not based on an axiomatic system, so it is preferable not to call them "proofs"; I refer to them as "derivations" in analogy with the somewhat loose mathematical derivations which one find, for example, in modem engineering textbooks.

Liu Hui's derivations are based on assumptions, generally unstated, which he clearly considers to be obviously true. Among these assumptions are: that the volume of a box is the product of its dimensions, that the volume of the whole is the sum of the volumes of the parts; and the distributive law of algebra. Other assumptions, which we would not consider obvious, are that "the extreme of fineness . . . is without form," and a special case of Cavalieri's theorem (used in Liu Hui's treatment of curvilinear solids [Wagner 1978a]).

Liu Hui's standard of rigor is high. He is never guilty of circular reasoning; he never invokes numerological or mystical concepts; and in only the one case does he stray from considerations which we would accept as strictly mathematical.

We may imagine the frustration of many attempts to divide up a yangma into parts whose volumes are known. Not knowing Dehn's result, he must, in the beginning, have been convinced that it was possible to derive the volume of the yangma in the same way that the volumes of other objects are derived. It is surprising, and a clear indication of his standard of rigor, that he was not content to deal with the case of equal dimensions (Section 6.2), but found it necessary to deal explicitly with the general case. And it is an indication of his mathematical ability that he succeeded in finding a method of dealing with the problem.

0f course when I refer to Liu Hui here it is in a broad sense. Not only is there some doubt as to how much of Liu Hui's commentary was actually written by him; more important he was presumably only one in a long oral tradition of masters and disciples who gradually accumulated the mathematical results written down in his book.

In the pages above a number of weaknesses have been noted. Most important, Liu Hui's terminology and mode of exposition are not adequate to express his mathematical thought fully: it is clear that he is dealing with general cases, but he expresses his arguments in terms of specific cases. In his derivation of the volume of a xianchu the mathematical situation is extremely complex, and he cannot or will not structure his exposition in such a way as to deal adequately with this complexity. This is a weakness which the early Chinese mathematical tradition shares with the oral traditions of craftsmen everywhere (including modem computer programmers). In an oral exposition not everything need be made explicit; resort can be had instead to enlightening examples and "hand-waving." The herd work of stating precisely all assumptions and all the steps in an argument is surely one of the most important driving forces in the development of mathematics.

Another weakness is the failure to generalise. Liu Hui's conceptual framework was adequate, for example to deal with a much broader range of geometric solids than those which he actually considers in his commentary. Had he felt a need to push his methods to their inherent limits he would surely have contributed a great deal more to the mathematical tradition. Here we can see the double influence of the enormous prestige of the Jiuzhang suanshu: it provided a challenge and an inspiration; but it was often a strait jacket which confined the interests of mathematicians to certain specific problems.


I am grateful to Lis Brock-Berendsen, Albert Dien, Akira Fujieda, Olaf Schmidt, and Nathan Sivin for their guidance and criticism in connection with an earlier version of this paper; to Hui Dong Shin for writing the Chinese characters; and to Erik Skaaning for translating the German summary.

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