The Chinese Roots of Linear Algebra

5. Fangcheng, Chapter 8 of the Nine Chapters

Chapter 5 Fangcheng, Chapter 8 of the Nine Chapters This chapter will focus on the fangcheng procedure, as presented in the Nine Chapters, to elucidate several key features: (1) fangcheng problems are displayed in two dimensions on the counting board; (2) entries are eliminated, in a manner similar to Gaussian elimination, by a form of “cross multiplication,” bian cheng , which involves “cross multiplying” an entire column by one entry; and (3) the approach to back substitution in the fangcheng procedure differs from the intuitive approach familiar from modern linear algebra. This chapter comprises two sections: 1. The first, “The Fangcheng Procedure,” examines in detail how fangcheng problems were solved on the counting board, including both elimination and back substitution. 2. The second, and more brief, section, “Procedure for Positive and Negative Numbers,” examines the rules presented in chapter 8 of the Nine Chapters for handling positive and negative numbers. The Fangcheng Procedure The original text of the Nine Chapters presents only this one “fangcheng procedure ” for solving all the problems in Chapter 8, supplemented by the “procedure for positive and negative [numbers]” to deal with negative numbers. Definitions of Fangcheng Before proceeding, however, we must first examine the precise meaning of the term fangcheng , which, as noted previously, is sometimes translated as “rectangular arrays” or “matrices.”1 Whereas the first character fang, which means 1 For a discussion of this issue, see Martzloff [1987] 2006, 250–51; Chemla and Guo 2004, 922–23. 67 68 5 Fangcheng, Chapter 8 of the Nine Chapters “rectangle” or “square,” is unambiguous, early Chinese sources provide different interpretations of the second character cheng: 1. The earliest extant commentary, by Liu Hui, dated 263 C.E.—possibly two centuries after the date of the compilation of the Nine Chapters2 —defines cheng as “measures,” citing the nonmathematical term kecheng , which means “collecting taxes according to tax rates.” Liu then defines fangcheng as a “rectangle of measures.” Liu’s commentary states: “Measure” (cheng) means “to find the measure” (kecheng). The collection of objects combines the heterogeneous; each [object] is displayed as having a number, and the sum together is stated as the constant term. Each column is formulated as the terms of a ratio: two objects with two measures, three objects with three measures, all are given measures in accord with the number of objects, and placed side-byside as columns. Therefore, it is called a “rectangular array” [lit., “rectangle of measures”].   4' &        (JZSS, juan 8, 1b). The term kecheng , however, is not a mathematical term—it appears nowhere else in the Nine Chapters, and in fact does not appear in any of the other mathematical texts collected in the Complete Collection of the Four Treasuries; outside of mathematics, kecheng is a term most commonly used for collecting taxes. 2. Li Ji’s  Nine Chapters on the Mathematical Arts, Pronunciations and Meanings (Jiuzhang suanshu yin yi , 11th century C.E.), an early commentary that offers definitions of important terms in the Nine Chapters, also glosses cheng as “measure,” again using a nonmathematical term, kelü  , commonly used for taxation. Li’s definition of fangcheng states, Fang means [on the] left and right. Cheng means terms of a ratio. Terms of a ratio [on the] left and right, combining together numerous objects, therefore [it] is called a “rectangular array.”  5  5    (JZSS, yin yi, 21b). The term that Li uses to explain cheng  is kelü  , which again is not a technical mathematical term, but instead is most commonly used to mean “rates of taxation” (shui lü ). 3. Yang Hui’s  Nine Chapters on the Mathematical Arts, with Detailed Explanations (Xiang jie jiuzhang suanfa  , 1261), defines cheng as a general term for measuring weight, height, and length. Detailed Explanations states, 2 On the dating of the Nine Chapters, see footnote 3 on page 30. The Fangcheng Procedure 69 What is called “rectangular” (fang) is the shape of the numbers; “measure ” (cheng) is the general term for [all forms of] measurement, also a method for equating weights, lengths, and volumes, especially referring to measuring clearly and distinctly the greater and lesser.      (YHJZ, 21a). In sum, Chinese explanations of the term fangcheng—ambiguous philological glosses from many centuries later—do not give us a definitive answer to the question of what the term meant at the time of the compilation of the Nine Chapters. For these reasons, fangcheng is sometimes left untranslated, as I have chosen to do in...