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A Case Study of the Duoji Method and Its Development

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A Case Study of the Duoji Method and Its Development EASTM 20 (2003): 45-72 The Westernization of Chinese Mathematics: A Case Study of the duoji Method and its Development Tian Miao /Tian Miao is Associate Professor of History of Science at the Institute for the History of Natural Sciences, Chinese Academy of Science. Her research covers the history of mathematics in seventeenth- to nineteenth-century China. Recent publications include "Qingmo shuxue jiaoyu dui Zhongguo shuxue zhiyehua de yinxiang" m5K Wl + #1!.. ~ 31:1 c:p 00 Wl + ~R ~ 1t El~ :irJ Ufa] (The Impact of the Development of Mathematics Education on the Professionalization of Chi­ nese Mathematicians in Late Qing China) (Ziran kex.ueshi yanjiu El ~ 'f4 + _§e_ jiff ~ (Studies in the History of Natural Sciences), /998), "Qingmo shuxue jiaoshi de goucheng de tedian" m5K Wl + #1!.. jffi ITT ,tt] fflG ~ ,9- (A Study on 1he Formation of Mathematical Teachers in the Late Qing Dynasty) (Zhongguo keji shiliao c:p 00 N t:5Z _§e_ fl (Historical Materials of Science and Technology), 1998), "Jiegenfang, Tian yuan and Daishu: Algebra in Qing China" (Historia Scientiarum, /999), "Siyuan yujian de Qingdai banben Ji Jialing sicao de jiao­ /.:an yanjiu" iz:g 5t .=E ~ ITT mft fJ§_ * Ez ® 1;- iz:g "J/i- ITT ~ WI iiff Ji: (Textual Criticism Research on the Different Versions of the Siyuan Yujian and the /'roof~ for the Jialing sicao during the Qing Dynasty) (Ziran kexueshi yanjiu, /999).J * * * The duoji ±~ fl (lit., "summing piles") method of calculating the sum of a given pile is a major subject in traditional Chinese mathematics. After the sixteenth n.:ntury, Chinese mathematics failed to keep pace with Western mathematics, but Ilic duoji method is one in which the late Qing mathematicians made advances over their Western colleagues. Most historians of Chinese mathematics have only paid attention to its development prior to 1870, particularly following its devel­ opment Lo Li Shanlan's * * fjJ (1811-1882) Duoji bilei ±i fl tt ~ (On the Summation of the Finite Series; 1867). However, mathematicians after Li Downloaded from Brill.com10/07/2021 08:10:01AM via free access 46 EASTM 20 (2003) Shanlan also made contributions to this field, and in this context the impact of Western mathematics deserves a closer investigation. 1 The Origins of the duoji Method In the eleventh century, Shen Gua YX fE (1031-I095) gave Lhe formula to calcu­ late the number of objects contained in a pile with interstices (xiji It~ ffl; fig. I ).2 In modern notation, the formula can be written as I n v ==-[(2b+d)a +(2d +b)c]+-(c-a) 6 6 This is the first duoji formula in the history of Chinese mathematics. Figure 1. Shen Goa's formula for the number of objects contained in a pile with interstices3 - - - •·-- -- Later, in the second half of the thirteenth century, Yang Hui t~ *-'- collected sev­ eral formulas lo calculate the number of objects contained in a pile. These formu­ las are found in two of his works, the Xiangjie jiuzhang suanshu ~f /iR fL ~ ~ f1~j (Detailed Explanation of the Computation Methods in the Nine Chapters; 1261) and the Tianmu bilei chengchu jiefa EE ~ tt. ~Ji * ~ tJf $. (Practical I On the history of duoji method and the achievement of Chinese mathematicians, see Qian 1964, 187-205, 327-329; Needham 1959, 137-139; Du 1966; Li and Du 1987, 149- 161, 245-251; Martzloff 1997, 302-306, 341-352; Li 1983. I am grateful to Professor Guo Shirong for discussing with me many details of the duoji method, and to Professor Joseph Dauben and Mr John Moffett for improving the English style and langu:1ge or this article. 2 M<'ngxi hitm1 4} (.fi '·(I: ,i~ (nrush Talks from the Dream Brook; ed. ol I (i1.'i ), I X.2b- 4h. -1 ()ian 1%4, 187. Downloaded from Brill.com10/07/2021 08:10:01AM via free access Tian Miao: The Westernization of Chinese Mathematics 47 Rules of Arithmetic for Surveying; 1275).4 In the early fourteenth century, Zhu Shijie * it!: ?f; gave several more formulas in his Siyuan yujian ~ JC _:E I,\= (Precious Mirror of the Four Elements; 1303), specifically for the triangular pile (sanjiao duo = ~ ±511:) and the rectangular pile (sijiao duo ~ ~ ±511:) series.5 Although Shen Gua, Yang Hui, and Zhu Shijie did not explain in detail how they derived their formulas, evidence found in their works shows that both Shen and Yang solved the duoji problem in terms of the theory of volumes. This ap­ proach is simply noted as "Yang Hui's method" in this paper. Yang's formulas for the duoji method are in the "Shanggong" ~ JjJ ("Discussing Works") chapter of the Xiangjie jiuzhang suanshu, following formulas to calculate the volumes of the right triangular prism (qiandu II ~), the pyramid (yangma \Wi ,lt ),6 the tel- 4 Xiangjie jiuzhang suanshu (Yijiatang congshu ed.), 70b-78b; Tianmu bi lei chengchu jiefa, in Yang Hui suanfa ti Wlli ~ $ (Mathematical Methods of Yang Hui; Yijiatang congshu ed.), I Ob- l 8a. 5 Triangular pile series: l,p, p(p+ll, .... p(p+l)···(p+i-1) 2 i! Rectangular pile series: l, ... i(i+l)· ··(i+p-2)(2i+p-2) , . .. p! In the Siyuan yujian, Zhu Shijie gave the method to calculate the sum of the finite triangular series with p equal to I to 5. He named the first series (p= I), the "straw pile" (jiaocao duo ~ Ifi- .tfil:; see the sixth problem in the "Jiaocao xingduan" ~ "1if- ID rn chapter); the second series (p=2), the "next pile" (luoyi xing duo 1j;. - If; t,!il') and the "triangular pile" (sanjiao duo; see the first problem in the "Guoduo diecang" ~ t~ ~ ~ chapter); the third series (p=3), the "next to the triangular pile" (sanjiao luoyi xing duo =: fCJ m - ID ±~) and the "disperse stars pile" (sanxing xing duo ff{ £ ID .tfil:; see the second and Lhe sixth problem in Lhe "Jiaocao xingduan" chapter); the fourth series (p=4), Lhc "next to the disperse stars pil~" (sanxing geng luoyi xing duo fllj( £ ~ ~ - ID 1~) and the " triangular disperse stars pile" (sanjiao sanxing xing duo =: P:J fl& £ % ±Jt see 1he l1flh problem in the "Rux.iang zhaoshu" t/U {t jH ~ chapter); and the fifth series (p=5), the "next lo the triangular disperse stars pile" (sanjiao sanxing geng luoyi xing duo :_ ffJ flt'{ £ ~ 7/4- - :if; t,!f; see the sixth problem in the "Guoduo dicchang" chapter). Zhu Shijie also gave the formula for the rectangular pile and the "next to the rectangular pile" (sijiao luoyi xin,~ duo ~! flJ W{. - % ±ff; see the third and thirteenth problem of '"Gumluo Jiecang" chapter). The terms used by Zhu Shijie indicate that he knew the relation among the five piles. Furthermore, Zhu Shijic gave fig. 3 at the very beginning of I he Siyuw1 y11jian aJJing the Jiagonals to the Jia Xian triangle, which divided the diagram i1110 a series or triangular piles. Therefore, historians of Chinese mathematics believe that 1/.h11 Shijie knew the slrncture or a triangular pile anJ formula for the sum of the finite .\/111iirw and .l"ijirw sci ics. Sec Du 1966. <, Tlw \'1111g111r1 is a pyramid wilh a n.:ctangular hase anJ one edge perpendicular to the hasc. Downloaded from Brill.com10/07/2021 08:10:01AM via free access 48 EASTM 20 (2003) rahedral wedge (bienao '.$;:: Ill), the yanchu ~ ~ (a wedge with a trapezoidal base and both sides sloping), and others. However, Yang did not provide dia­ grams showing the shapes of his piles, nor proofs of his formulas (for the Chinese text, see fig. 2). Figure 2. Yang Hui's original text for the duoji method? Zhu Shij ie also gave no explanations, but historians of Chinese mathematics have shown that he originated a new approach to the duoji method. Zhu related the p­ th parallel leftward-oblique line of the Jia Xian fl ~ triangle (a tabulation sys­ tem for unlocking binomial coefficients) to his formulas for the duoji method. This approach, simply noted as "Zhu Shijie's method" in this paper, enabled him to represent the duoji method in a purely abstract way without reference to solid shapes8 (see fig. 3 and diagram 1). 7 Xiangjie jiuz.hang suanshu (Yijiatang congshu ed.), 76u-b. 8 Siyuan yujicm (Baifutang congshu ed.), "Jiaocao xingduun," 2.28b-30b; "Jinnji jiao­ cnn" ~ .Ji 3'.t~. 2.30h-32h; "Guoduo dicchang," 3.1 u-lJu. Downloaded from Brill.com10/07/2021 08:10:01AM via free access Tian Miao: The Westernization of Chinese Mathematics 49 Figure 3. Zhu Shijie's representation of the duoji method9 i+,9 JllJ •n I • Diagram 1 p=l p=2 2 I p=3 3 1 p=4 6 I p=S I p=6 20%. ~/1 p:=7 The duoji method was also mentioned in works published after the fourteenth l'cntury, including Wang Wensu's .:E X * Suanxue baojian • ~ W I:.: (Pre­ cious Mirror of Mathematics; 1524) and Cheng Dawei's ~ ""fr. {ll. Suanfa tong­ ;:m1g t-?: it. Mc '.;1~ (General Source of Computational Methods; 1592). IO How- '' Zhu Shijic, Siy1w11 y11jia11 (13.iifutung congshu ed .), "Jigu kaifang huiyao zhitu," I h. 10 Su,111.rn,• haojian (munuscript rcproc..luccc..l in Zho111UflW ki'Xut' ji.1·hu dianji tonRhui q I Ii&! H ,'fl: H: /1~i .uli •>• 9211- I OOh; Sumi/ii 1,ml(z1ml( (ed. of 1716), IU 2h- I 8u. Downloaded from Brill.com10/07/2021 08:10:01AM via free access 50 EASTM 20 (2003) ever, until the first half of the seventeenth century, Chinese mathematicians made virtually no progress in this area, and most of them merely repeated the formulas given by Yang Hui without providing further explanations or proofs.
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