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The Continuation of Ancient Mathematics

Tina Su Lyn Lim and Donald B. Wagner
Nias Press
Publication Date: 
Number of Pages: 
[Reviewed by
Frank Swetz
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The Jiuzhang suanshu, also known as Nine Chapters of the Mathematical Art, is the first classic of ancient Chinese mathematics. Dated conservatively to the first century C.E., it is a bureaucratic handbook/text of the mathematics necessary to govern the early Chinese empire. Since its inception, the work has served as a source of instruction and reference for thousands of Chinese scholars. In the Nine Chapters’ passage through the centuries, it has been corrected, commented upon and annotated heavily.

Liu Hui (3rd Century C.E.), China’s greatest mathematician of antiquity, made the most profound annotations to the mathematical classic, but he also extended the material of the ninth chapter concerning applications of the right triangle. Eventually, this addition became a separate book, the Haidao suanjing, or Sea Island Mathematical Manual.

The Tang scholar Wang Xiaotong (ca.7th century C.E.) appears to have been involved with water conservancy: the construction of canals and  dikes to protect from flooding. He wrote an extension of the mathematics given in chapters five of the Jiuzhang, “Construction Consultations”, and chapter nine, “Right Triangles.” This work, consisting of twenty illustrative problems, is entitled Jigu suanshu, or Continuation of Ancient Mathematics. It was included in the “ten official canons”, the required mathematical curriculum for government officials. While some of these canons have been translated into English and studied by modern researchers, the Jigu suanshu has remained elusive until now, with the publication of The Continuation of Ancient Mathematics:Wang Xiaotong’s Jigu suanshu, Algebra and Geometry in 7th –Century China by Tina Su Lyn Lim and Donald B. Wagner. 

This book evolved from Lim’s M.Sc. thesis at the University of Copenhagen, mentored by Donald Wagner, a noted scholar in the history of Chinese science and mathematics. It is well conceived and organized. Part I supplies a general background about the mathematician/astronomer Wang Xiaotong and his involvement in state sponsored public works. Anthropologists would describe China as a hydraulic society, one is which national survival is contingent on water control. Part II discusses the specific mathematics employed in solving the problems and Part III translates and analyzes the problems.

The calculations in the text concern the assignment of corvée labor for various public work endeavors: the removal of earth and the construction of ramps and water retaining dikes. These discussions are strikingly similar to those found in collections of Babylonian problems, another hydraulic society. Justification for the solution procedures are given through geometric-algebraic demonstrations, dissection techniques and applications of “Horner’s method” to obtain roots. This method was well known by the Chinese since the appearance of the Jiuzhang.

If one is not familiar with “proof by dissection”, the examples given here, accompanied by illustrations, provide an excellent introduction to this technique. Ample supplementary instruction on the specifics of Chinese mathematics — rod numerals, computational algorithms, units of measure and the basic mathematical curriculum — is provided, allowing a novice Sino-researcher to proceed without difficulty. Useful “Appendices” are included.

In my examination of the text, I learned some hitherto unknown facts about the traditional examination system. This is a valuable contribution and a solid addition to our better understanding of Chinese mathematics and, indeed, the uses of mathematics in the ancient world.

Frank Swetz, Professor of Mathematics and Education, Emeritus, The Pennsylvania State University, is the author of several books on the history of mathematics. His research interests focus on societal impact on the development, and the teaching and learning, of mathematics.

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