The Nine Chapters on the Mathematical Art: Companion and Commentary (review)

The Nine Chapters on the Mathematical Art: Companion and Commentary, by Shen Kangshen, John N. Crossley, and Anthony W.-C. Lun, is the first full English translation of the Chinese classic Juizhang suanshu, which includes the Commentary by the third-century mathematician Liu Hui . (There is a German translation by K. Vogel, Neun Bücher arithmetischer Technik [Braunschweig, 1968], and a French translation is being prepared by Karine Chemla and Guo Shuchung.) This particular work is essential for an understanding of the Chinese tradition in mathematics, for in China it played a role very similar to that of Euclid's Elements of Geometry in premodern Europe. The differences between the Chinese and European traditions can thus, at least initially, be approached by noting the similarities and contrasts between the Nine Chapters and the Elements. Such a comparison is initiated in this volume through the extensive annotations provided by Shen, Crossley, and Lun. In addition, the authors also offer some comparisons with ancient Indian mathematics.

The basic text of the Juizhang suanshu, which is thought to have been written not later than 100 B.C.E., consists of nine chapters, each of which contains problems and their solutions together with brief indications of the methods used to arrive at the solutions. Each of the nine chapters is devoted to a specific kind of problem, and the problems are graded in order of difficulty. The first few problems in each chapter introduce basic methods, and these are then given more complex applications. The chapter headings are: "Field Measurement," "Millet and Rice," "Distribution by Proportion," "Short Width," "Construction Consultations," "Fair Levies," "Excess and Deficit," "Rectangular Arrays," and "Right Angled Triangles." These give some indication of the contexts of particular kinds of mathematical problems.

The "Field Measurement" chapter covers the calculation of areas and the theory of fractions. "Millet and Rice" is concerned with conversions between various units and focuses on the Rule of Three (the Jinyou Rule) and its applications. "Distribution by Proportion" continues the mathematical theme of the previous chapter covering direct and compound proportions. The chapter on "Short Width" contains problems where the area or volume is given and the length of a side is to be found. This leads to methods for extracting square and cube roots. Construction problems fall into three categories: finding the volumes of various solids; finding the side, altitude, or circumference of a solid whose volume is known; and finding the number of laborers required for a job if the daily work per capita and total work to be done are known. The chapter on "Fair Levies" elaborates on the further development of techniques for handling ratios and proportions. "Excess and Deficit" involves the solution of linear equations using the Rule of Double False Position, whereas the problems in "Rectangular Arrays" involve the solutions of systems of linear equations using a matrix-like array. Finally, the chapter on the right triangle concerns applications of the result that the European tradition has known as the Pythagorean Theorem.

In addition, this volume includes Lui Hui's Sea Island Mathematical Manual (Hai dao suanjing), a continuation of the last of the nine chapters that extends its methods to more complex problems connected with surveying. Lui Hui's original title for it was "The Rule of Double Differences," referring to the fact that it concerns surveying techniques using more than one gnomon.

The procedure followed in this volume is to introduce the material in each chapter with a discussion of the types of problems, identifying them in modern terms and then looking at how other countries have treated the same or similar material. This discussion is followed by a translation of the Nine Chapters text, together with Lui Hui's Commentary. In addition, the Tang dynasty commentary by Li Chunfeng , Liang Shu , Wang Zhenru , and others has been included. All of these are annotated by the...