Chinese Mathematics in the Thirteenth Century: The Shu Chiu-Chang of China

One studies Chinese mathematics primarily as one of the ways of apprehending the whole mind of a civilization that was thoroughly interconnected within itself, in terms of intellectual concepts, social organization, and aesthetic expression. The extent to which thirteenth-century Chinese mathematics anticipated modern or Western results is not of real relevance. As Nathan Sivin states in his Foreword, "Ideas which...were perceived merely as the outdated and misguided backdrop of 'modern' anticipations must now be evaluated as seriously as the latter, for they played no less important a role in defining the ancient scientist's conception of the natural world—and thus the direction and style of his investigations."

That said, a modern Westerner can hardly suppress pointing out the two most important ways in which Ch'in Chiu-shao advanced the mathematics of his time and place: he stated the "Chinese remainder theorem" for indeterminate equations of the first degree, which is more general than Gauss's rule of 1801; and his algorithm for solving equations of higher degree (including the tenth degree) is identical with Horner's (1819). These two results also entailed methodological advances: Ch'in's account of his method for solving indeterminate problems is the first generally stated mathematical formulation in the Chinese literature; and his work on equations of higher degree was wholly speculative, or experimental, going well beyond the traditional bounds of Chinese mathematics—the solution of purely practical problems.

One further comparative historical note may not be out of order. Libbrecht writes, "Chinese mathematics forms part of medieval mathematics, of the algorithmic phase we find in all civilized countries at that time. In reading Ch'in's text, I tried to place it within this algorithmic mathematical conception, which was the preamble to modern algebraic logistic." Implementing this approach, the author compares the treatment of indeterminate equations during this period in India, Islam, and Europe and finds that Ch'in's techniques were unprecedented. This alone should demonstrate the importance of this study to universal mathematical history.

The essence of the book remains its insight into Chinese thought and life, as revealed by the general concepts that emerge and interrelate and by the practical mathematical problems posed by Ch'in that tie into the everyday realities of his time. It is especially this last aspect that makes the book useful to China scholars generally.

This is the first volume in The MIT East Asian Science Series, of which Nathan Sivin is general editor.