Nine Chapters on the Art of Mathematics

The Nine Chapters on the Art of Mathematics is the oldest and best known of the Chinese “mathematical classics.” It seems to have originated sometime in the first two centuries BCE as a sort of compendium of known mathematical techniques. It comes to us with a commentary by Liu Hui (around 263 CE) and further elaboration by a team led by Li Chunfeng (604–672 CE), created in the process of assembling the ten mathematical classics.

The Nine Chapters cover the basic algorithms of Chinese mathematics, starting from “Rectangular Fields” (measurement of areas) and “Grains” (problems involving proportionality and the rule of three). The last two chapters treat quite elaborate — and interesting — material: chapter eight, on “Rectangular Arrays,” teaches how to solve systems of linear equations by a system equivalent to Gaussian elimination, while chapter nine, on “Base-Height” deals with the Pythagorean theorem and its applications.

While the original Nine Chapters contained problems and methods for their solution, Liu Hui’s commentary adds explanations and justifications and so offers insight into the sophisticated mathematical thinking that undergirds the more algorithmic material. It becomes clear that Liu Hui deserves to be on the list of the world’s great mathematicians.

This edition of the Chinese text with translations was published in 2013 and seems not to have been much noticed by the American mathematical community. It should be! The book appears as part of the Library of Chinese Classics, a project that aims to produce new bilingual editions of many older texts. (I don’t know whether other mathematical texts are planned; people interested in Chinese mathematics should keep watch.) It is beautifully done, on nice paper with a good binding, spread over three reasonably-sized volumes.

After an introduction to the series of Chinese Classics, we get two introductions to this particular book. The first is in Chinese, so the best I can do is conjecture that it is Guo Shuchun’s introduction to the Chinese text. Then comes an introduction by Joseph Dauben and Xu Yibao, discussing the history and contents of the Nine Chapters, briefly surveying existing translations into Western languages (most notably the masterful Chinese-French edition by Guo Shuchun and Karine Chemla), and explaining the approach they have chosen for their translation and commentary.

These preliminaries taken care of, we get the actual text on facing pages, Chinese on the left and English on the right. The text includes all three historical layers: the original Nine Chapters, Liu Hui’s comments, and the later comments by Li Chunfeng. The Chinese text is actually presented twice: in the original, and in a modern Chinese translation. To distinguish the different layers in the English translation and commentary, the short original text is printed in all caps, while the commentary paragraphs are introduced by an identifier to tell the reader whether it is Liu Hui or Li Chunfeng that is writing. There are footnotes and also commentary by Dauben and Xu. This layout is somewhat complex and requires the reader to pay attention, but I think it works better than using typeface changes to distinguish the various layers, which is the strategy employed in the older translation by Shen, Crossley, and Lun (Oxford, 1999).

As Jeff Chen points out in his review of The Emperor’s New Mathematics, there has been growing interest in the history of Chinese mathematics, both in China and in other countries. This book joins a growing list of titles (see Chen’s review) that includes both primary source material and studies. I hope to see more of these in the future.

Since the book was published in China, it can be hard to find, so let me attempt to aid interested readers by pointing to the proper Amazon.com page. For a new edition of this quality and in three volumes, the price is a bargain.

To summarize: this is a beautifully made, well-translated and annotated edition of one of the most important primary sources for the history of Chinese mathematics. Guo, Dauben, and Xu deserve our thanks.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. He likes books, especially old ones.