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Illustrating The Nine Chapters on the Mathematical Art: Their Use in a College Mathematics History Classroom – The Text and Its Contents

Joel K. Haack (University of Northern Iowa)

The Nine Chapters on the Mathematical Art is widely acclaimed as "the supreme classical Chinese mathematical work" [Shen, v], "a kind of 'mathematical bible'" [Dauben 227, quoting Martzloff 127, who quotes Wang Ling's 1956 thesis 16], the "most influential of all Chinese mathematical texts" [Swetz, 8], and "the most influential of all Chinese mathematical texts ... which occupies a similar position in Chinese mathematics to that of Euclid's Elements in Western mathematics" [Joseph 135]. Present-day versions of the Nine Chapters are based on one that was compiled by Zhang Cang and Geng Shouchang, most likely around 100 BCE, according to the preface of Liu Hui to his important commentary in the 3rd century CE. Shen's edition includes the commentaries by Liu and by Li Chunfeng and others (7th century). Shen [1] and Martzloff [134] list additional commentaries from the 5th, 6th, 7th, 13th, and 19th centuries.

The transliteration of the title of this work (simplified Chinese: 九章算术; traditional Chinese: 九章算術; pinyin: Jiǔzhāng Suànshù [Wikipedia]) is variously given as Chiu Chang Suan Shu [Swetz and Joseph], Jiuzhang Suanshu [Shen], and Jiu zhang suan shu [Dauben]; here it will be convenient, as in Shen, to simply call it the Nine Chapters. The publication of Shen's translation in 1999 has made this work readily available in the English-speaking world.

Joseph Dauben notes that the title of the work has been translated into English in a number of ways in addition to the one given above. These include Arithmetic in Nine Sections; Computational Prescriptions in Nine Chapters; Nine Categories of Mathematical Methods; and The Nine Chapters on Mathematical Procedures [227]. The last of these is a translation itself from a 2004 French translation of the Nine Chapters by Karine Chemla and Guo Shuchun. We had the opportunity to meet and talk with Guo Shuchun at the Institute for the History of Natural Sciences, part of the Chinese Academy of Science in Beijing.

Figure 3. Guo Shuchun displays his and Karine Chemla’s translation from Chinese into French of the Nine Chapters; behind him is Tina Straley, then Executive Director of the MAA (photo by the author).

Unlike Euclid's Elements, the Nine Chapters is not organized as an axiomatic presentation of theorems and proofs; rather it is a collection of problems organized by topic and by the algorithms used for their solution. It is intended to be practical and pedagogical [Shen, vii] and was one of the works studied for official mathematics examinations for civil servants in the Tang Dynasty [Dauben 227].

The importance of the Nine Chapters is recognized among contemporary Chinese mathematicians and historians; Figures 4-6 are photographs of presentations that we saw in China that mentioned the Nine Chapters.

Figure 4. Dianzhou Zhang, of East Central Normal University in Shanghai, shows a slide from a lecture, "Mathematical Exchange Between China and the United States," which began with background on ancient Chinese mathematics (photo by the author).

Figure 5. Guo Sharong of Inner Mongolia Normal University shows a slide from a lecture, "Some New Thoughts on Chinese Mathematics During the 13th and 14th Centuries – The Construction of Mathematical Models." Some problems in the work of the 13th-century mathematician Li Ye, namely Yuan Cheng Tu Shi (Illustration of a Circle Town), can be traced to the Nine Chapters (photo by the author).

Figure 6. A lecture by Feng Lisheng of Tsinghua University, "From Counting Rods to Abacus: Traditional Chinese Counting Techniques," included a discussion of the use of counting rods to perform basic arithmetic operations in the Nine Chapters (photo by the author).

Table of Contents of The Nine Chapters on the Mathematical Art

Now, we turn to the mathematical content of the Nine Chapters. The titles of the nine chapters, with a brief summary of the content of each, are as follows [Haack]:

Chapter 1. Field Measurement. This chapter includes problems to find the areas of rectangles, triangles, trapezoids, circles, and related regions. Arithmetic techniques are developed to carry out computations with fractional quantities.

Chapter 2. Millet and Rice. This chapter includes problems of proportions and unit prices. The Rule of Three (similar to cross multiplication) for solving proportions is seen as an extension of the work with fractions in the first chapter.

Chapter 3. Distribution by Proportion. This chapter extends the problems solved via proportion in the previous chapter.

Chapter 4. Short Width. This chapter seeks the side or diameter of a region from known areas and volumes. Arithmetic results developed include algorithms to extract square roots and cube roots by hand.

Chapter 5. Construction Consultations. This chapter seeks the volumes of a number of solids that occur in construction problems. Formulas for the solution of such problems are developed, often in several different ways.

Chapter 6. Fair Levies. Further extensions of the proportion problems of Chapters 2 and 3 are developed here. There is also a consideration of the sums of arithmetic progressions. The kinds of word problems that appear include work problems and distance and rate problems.

Chapter 7. Excess and Deficit. The chapter title refers to a technique to solve two linear equations in two unknowns [double false position]. Again, a variety of word problems are solved using this technique.

Chapter 8. Rectangular Arrays. Problems of agricultural yields, the sale of animals, and a variety of other problems, all leading to systems of linear equations, are solved by a method that is known in the West as Gaussian elimination. The problems lead to systems ranging from two equations in two unknowns to six equations in six unknowns, and one problem leads to an indeterminate system of five equations in six unknowns.

Chapter 9. Right-angled Triangles. Problems in surveying and in the lengths of various line segments are solved using the Gougu Rule, the Chinese version of the theorem known in the West as the Pythagorean Theorem. The problems demonstrate familiarity with Pythagorean triples.

Joel K. Haack (University of Northern Iowa), "Illustrating The Nine Chapters on the Mathematical Art: Their Use in a College Mathematics History Classroom – The Text and Its Contents," Convergence (April 2017)