The Chinese Roots of Linear Algebra

3. The Sources: Written Records of Early Chinese Mathematics

Chapter 3 The Sources: Written Records of Early Chinese Mathematics In this chapter, I will present a preliminary overview of the textual records for Chinese mathematics. The chapter addresses three matters: 1. The first section, “Practices and Texts in Early Chinese Mathematics,” distinguishes between the practices of fangcheng adepts and the texts written by the literati who recorded those practices. 2. The next, “The Book of Computation,” presents a brief summary of the earliest extant record of Chinese mathematics, a funerary object buried in approximately 186 B.C.E. 3. The final section, and by far the longest, “The Nine Chapters on the Mathematical Arts,” examines the earliest transmitted Chinese text devoted to mathematics, focusing on the differing editions, the commentary and subcommentary , and modern critical studies. The chapter will focus primarily on the Nine Chapters, simply because there are no extant materials on the fangcheng adepts, and very few related to the Book of Computation. Practices and Texts in Early Chinese Mathematics We know virtually nothing about the fangcheng practitioners themselves: we do not even know their names; we have no records of any writings they produced; they were most likely illiterate, but if they were able to write, their writings would likely have been denounced by the literati as vulgar; and thus it is unlikely that anything they wrote would have been collected by the literati or preserved in their libraries. We do, however, have some prefaces, commentaries, and sometimes biographical information about the literati who compiled treatises on Chinese mathematical arts. We also have similar compilations of other arts, such as medicine and astronomy, which provide further relevant information. We must first draw a distinction between the textual tradition and fangcheng practice. The literati who translated fangcheng practices into narrative form were 27 28 3 The Sources: Written Records of Early Chinese Mathematics hardly disinterested ethnographers. Although sometimes attributing credit to their unnamed sources, more often they were highly critical of the arcane fangcheng procedures, which they apparently failed to understand. For example, even Liu Hui " (fl. 263 C.E.), who wrote the earliest extant commentary on the Nine Chapters and is often considered the most eminent mathematician of imperial China, apparently did not understand that the complexity of the fangcheng procedure in the Nine Chapters was necessary in order to avoid even more complicated calculations with fractions (see chapter 6), as is suggested by the derision he expresses toward fangcheng practitioners: Those who are clumsy in the essential principles vainly follow this original [fangcheng] procedure, some placing counting rods so numerous that they fill a carpet, seemingly so fond of complexity as to easily make mistakes . They seem to be unaware of the error [in their approach], and on the contrary, desire by the use of more [counting rods] to be highly esteemed. Therefore, of their calculations, all are ignorant of the establishment of understanding ; instead, they are specialized to an extreme. # $   %   ##  (JZSS, juan 8, 19a; Chemla and Guo 2004, 650–51; Shen, Lun, and Crossley 1999, 426.) The criticisms expressed above are fairly typical of the views of the literati who compiled texts on the mathematical arts. The Book of Computation The earliest extant Chinese mathematical treatise is the Book of Computation (Suan shu shu ), unearthed in 1984 from Han Dynasty Tomb No. 247, at Zhangjia Mountain  in the Jingzhou district of the city of Jingzhou , in Hubei Province. The tomb contained over 1,200 bamboo strips from several treatises.1 It is believed that the tomb dates from about 186 B.C.E.; but the mathematics it contains is estimated to date as much as 300 years earlier. The Book of Computation has been reconstructed from these bamboo strips (Fig. 3.1): all of the strips constituting the Book of Computation had become immersed in silt; about half of the bamboo strips were no longer bound together by thread and had become dislodged from their original position; some of the strips have deteriorated, and portions are no longer legible. The strips were pieced together on the basis of their relative position and their neighboring strips, as well as the content and topic of the strips. There are no fangcheng problems in the Book of Computation, and the term fangcheng is not mentioned.2 1 Other treatises unearthed include Er nian lü ling & ', Zou yan shu , Gai lü (, Mai shu , and Yin shu . 2 This paragraph summarizes material in the introduction to Peng 2001, 1–35 (in Chinese). For studies in English, see Dauben 2008 and...