EASTM 20 (2003): 45-72

The Westernization of : 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 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

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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, 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.

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Rules of Arithmetic for Surveying; 1275).4 In the early fourteenth century, * 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, , 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 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.

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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.

The Development of Yang Hui's Method

From the mid-seventeenth century onward, instead, several mathematicians began to present explanations of Yang Hui's method. The first of them was Fang Zhongtong JJ r:p ~ ( 1633-1698), who collected some duoji problems in his Shudu yan f& /!r 1rr (Expansion of Numbers and Magnitudes; 1661 ). One of these problems, called the method of "Seeking the Sum of Succeeding Numbers" II (shunjia qiuji fa Ill$! ;/JD >l< ffi T:t), i.e., [ i, is as follows: i =I

Supposing the last term of the series is 15 , what is the sum of the series? Method: The product of the last two numbers, 14 and 15, is 210. Dividing it by 2, we obtain l05. The sum of 105 and the last line, I 5, is 120, which is the sum of the series. Another method: Multiplying the last line 15 by the foll ow­ ing line 16, we obtain 240. Dividing it by 2, the result, 120, is the answer we want. I I

To explain this method, Fang Zhongtong gives the following diagrams (fig. 4):

Figure 4. Fang Zhongtong's diagrams for the method of "Seeking the Sum of Succeeding Numbers" 12

~in- ,.y ~t< I e. • • • • ... • • f - i :- , =-· ·-:~: ::l:; • • • • o Cl 0 I ~ -t •1 •('.'Pfl0c()~•, • • ClnOO , O . .... 0001.>ojo . -.. • . . 0 :.i O Q' Qjf') ~ "d ..\'.. r'r

11 Shudu yan (Siku 4u:mshu ed.), I I .4a-h. 12 Shut/11 Yan, I 1.4h.

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After the diagrams, Fang comments:

[Fang Zhong]tong says (tongyue IB1 El), the product or the two numbers is the area of the rectangle; in this way, the Lri­ angle is changed into a rectangle. Dividing the product by 2, we turn the reclanglc back into a triangle. For instance, if the last line is 7, then the product of 6 and 7 is the sum of the rectangle jia-yi-bing, which does not contain the last term [7]; the product of 7 and 8 is the sum of ding-wu-ji, half of which is the sum of the pile. t3

Although the proof cannot be regarded as a complete demonstration, it explains Fang's method. In the late seventeenth century, in his Shuxue yao ~ ~ £,i; (Key to Mathe­ matics; 1681 ), Du Zhigeng H 92rl liJ~ gave the proofs of several formulas mostly found in Yang Hui's work. First, Du gave the proofs of the formulas for a square pile (pingfang dui zp: ;'j .tffi) and an isosceles triangular pile (sanjiao pingdui .=. ;1-i zp: .!#.). Then, he used these two formulas to obtain the proofs of the formulas for the sum of the blocks in a trapezoidal pile (fixing pingdui ;j-5(5 ~ zp: it) and a regular hexagonal pile (liuhian pingdui I\ i! zp: .ti). After that, he gave the proofs of the formulas for the sum of solid piles in a right triangular prism (qiandu), an equilateral triangular prism (fangdi gaodui 1j ~ i@i ±i), a right triangular tetrahedron (sanjiao gaodui -= ft] r'if/1 :tit.), an equilateral triangular tetrahedron (sanjiao ruimian clui = fr] ift jyj :ti), a right angular hexahedron (zhidi gaodui @[ 11£ r'i'/1 !f{J, and an equilateral triangular hexahedron (zhicli ruimian dui ill[ Jg, $ft [HJ ±i), always using the formulas for a rectangular paral­ lelepiped. Below is his proof of the formula for the sum of a square pile (see also rig. 5):

Supposing the perimeter of the square pile is 24, the method for seeking the sum of the pile is as follows: Dividing the perimeter by 4, we obtain 6, and then, adding I to 6, we ob­ tain 7; taking the square of'7, 49, we obtain the desired sum. The diagram on the lert is a square pile, and the one on the right is a square. Both the length and the width of the two are 7, but while the perimeter of the square is 28, the perimeter or the square pile is only 24. The reason for the difference is that the perimeter of the square is continuous, while the perimeter of the square pile is intermittent. t4

I I S/1111/11 \'ITl'~lliHHb In l>lil; in n11>dl'tll 1101:1111>11. l•I Sl,11111,· 1·1111 (Si~11 q11:insl111 l'd.), 11.1(,h 17a, prnpnsition :n.

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Figure 5. Du Zhigeng's method for the sum of a pilel5

To prove the formulas for the sum of the solid piles, Du Zhigeng simply placed several piles together to obtain a rectangular parallelepiped, and used the formula for the sum of the parallelepiped to obtain the sum of the solid piles. In his His­ tory of Chinese Mathematics, J.-C. Martzloff gives an example of Du's proof of the formula for a pyramidal pile (jangzhui duo 15 it ±~, called siyu duo [9 ~~ ±~ in Yang Hui's work).16 To provide a clearer view of Du's method, I consider the following example (see also fig. 6):

Supposing the side of the bottom of a right angular solid pile (qiandu gaoduo ~ ~ rl'l.J ¼1:) is 5 [for the right angular solid pile, the height of the pile is equal to the side of the base, so there is no need to mention the height of the pile], the method for seeking the sum of the pile is as follows: Multi­ ply the side plus one [5+ I] by the height [5], and multiply the product (30] by the width of the pile [5]; divide the re­ sult by 2, and we obtain the desired sum.17

IS Shuxue ww, 4.J7a. 16 Marlzlorl 1997. 303-304. 17 Sl,11111,· VCI//, 4.40a-h, proposilio11 44,

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Figure 5. Du Zhigeng's method for the sum of a pilel5

To prove the formulas for the sum of the solid piles, Du Zhigeng simply placed several piles together to obtain a rectangular parallelepiped, and used the formula for the sum of the parallelepiped to obtain the sum of the solid piles. In his His­ tory of Chinese Mathematics, J.-C. Martzloff gives an example of Du's proof of the formula for a pyramidal pile (jangzhui duo 15 it ±i, called siyu duo [9 ~~ ±s;l: in Yang Hui's work).16 To provide a clearer view of Du's method, I consider the following example (see also fig. 6):

Supposing the side of the bottom of a right angular solid pile (qiandu gaoduo ~ ~ ~ ~) is 5 [for the right angular solid pi le, the height of the pile is equal to the side of the base, so there is no need to mention the height of the pile], the method for seeking the sum of the pile is as follows: Multi­ ply the side plus one [5+ I] by the height [5], and multiply the product (30] by the width of the pile [5]; divide the re­ sult by 2, and we obtain the desired sum.17

15 Shuxue yao, 4.37a. 16 Mart:r.lolT 1997, )o:l-304. 17 Sl,11111,• V

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figurate numbers analogous to those of Du in Europe itself before the nineteenth century."21 The origin of Du's method becomes clearer if one considers Lhal, at the be­ ginning of lhe development of the duoji method, Shen Gua and Yanghui invented their procedures for calculating certain sums of piles (i.e., figurate numbers) based on volume theory. Statements that the volume of a right triangular prism is half the volume of a cube with the same base, and that the volume of a pyramid is one third the volume of a cube with the same base, arc found in the Jiuzhang suanshu fL ~ ~ f¥f (Mathematics in Nine Chapters; ca. second century BC) and were demonstrated by jij tt& in 263 AD. Yang Hui calculated the vol­ umes of the right triangular prism and the pyramid using these formulas in his Xiangjie jiuzhang suanshu. Du Zhigeng also collected these results together in chapter 4 of his Shuxue yao. Moreover, using a special example to illustrate a general theory is commonly used in traditional Chinese mathematics. There are sufficient grounds, therefore, to conclude that Du followed the traditional Chi­ nese method. In 1712, the Kangxi Emperor (1654-1722) requested Chen Houyao ~*~WI ( 1648-1722), He Guozong {iiJ ~ ff~ ('?-1766), Ming Antu fljJ '!i. Iii (?-1764 ?), and Mei Juecheng fa~ foc (1681-1763) to compile the Shulijingyun f)( lltl. *Ill ~ (Collected Basic Principles of Mathematics; 1723). Chapter 30 of this work deals with the duiduo Ji i5il: (i.e., the duoji) method and contains some formulas with their proofs. Although this has not been noticed in earlier studies on the Shuli jingyun, the method of demonstration is similar to Du Zhigeng's.22 The method used by Fang Zhongtong and Du Zhigeng is basically a tradi­ tional one, but Western mathematics undeniably had a distinct influence on the work of both. Although there were no proofs or explanations for the duoji method before the middle of the seventeenth century, by the second half of the seventeenth century some mathematicians not only gave their duoji procedures, bul also tried to explain the reasons for them. Can this simply be a coincidence? To answer this question, one should consider the development of mathematics in seventeenth-century China. At the end of the previous century, when the Jesu­ its introduced Western mathematics into China, many achievements of the great mathematicians of the Song period had been lost, and many earlier sources were no longer available. 23 In 1607, a Chinese version of the first six books of Euclid's Elements was published. Later, a few works on Western mathematical sciences were translated into Chinese, covering elementary materials on survey­ ing, geometrical models of the heavens and earth, and astronomical and hydro­ logical instruments. Some Chinese scholars actively studied and transmitted

21 Martzlolf 1997, 304. 22 Sec S/wli ji11g)'1m (ell. of 1723), 30. I 7a-6 I h. Fu Daiwic ( 1992) has also given a re­ construction of Yang llui's method in a similar way. 2 l (iuo I 1) 1>7.

Downloaded from Brill.com10/07/2021 08:10:01AM via free access Tian Miao: The Westernization of Chinese Mathematics 55 them.24 Even during the turbulent period of the Ming-Qing transition there were scholars attracted to Western Learning (xixue j7g ~!). Fang Zhongtong and Du 7.higeng belonged to this group. Both were familiar with the translated Western works, and tried to synthesize Chinese and Western mathematics into a unified system based on lhe Jiuzhang suanshu.25 Many scholars of this reriod worked lo integrate Western knowledge with Chinese traditional learning, and particularly Western mathematics with Chinese traditional mathematics. Some even argued that the ancient Chinese mathemati­ cians had already made discoveries and advances that were subsequenlly lost and later became known again through Western mathematics. However, while making such claims, these scholars also transformed traditional Chinese mathematics. Fang Zhongtong and Du Zhigeng, like the compilers of the Shuli jingyun, were not entirely satisfied with the approach used in traditional mathematical works. Accordingly, they not only tried Lo present their methods, but also lo explain the reasons for them. 26 For example, the Shuxue yew has definitions, propositions, and proofs. Its author, Du Zhigeng, also tried to use deductive reasoning by repeatedly referring lo results that had already been established, inspired in doing so by the formal models of the Elements. However, unlike the Elements, the propositions (ze ~IJ) consist not of theorems or constructions, but of algorithmic procedures.27 Al the end of eighteenth century, Wang Lai 1-:[ * (1768- I 813), one of the main mathematicians of the Qing period, gave the first general formula for lhe triangular pile in his Dijian shuli ~ * ~ llll. (Mathematical Theory of Increases hy Degrees; 1799):

24 On the influence of Western mathematical works upon the development of Chinese 111a1hcmaties in the late Ming and the early Qing period, see Engclfriet 1998, 356-371, \83-40 I; Jami 1991. 25 Fang Zhongtong's father, Fang Yizhi )j J;) ~ (1611-1671), was acquainted with 1:ran<,ois Sambiasi (Bi Fangji ~ 1J '/ff, l582-l649) and Johann Adam Schall von Bell ("1;111g Ruowang o/~j :f',= 'i.'E_, 1591-1666). Fang Zhongtong wrote the .lihe yue )L {□f t'l (Synopsis of the Elements), while Du Zhigeng wrote the .lihe lunyue IL fiif r;iflj %ti (Sum- 111;1ry of the t.'lements; 1700). See Engclfrict 1998, loc. cit. '.1(> Ancient Chinese mathematical works largely consist only of problems, answers, :iml sometimes procedures. Explanations of the procedures were either absent or ex- 11r111cly synthetic. However, both Fang Zhongtong and Du Zhigeng interspersed their wn1 k with paragraphs beginning with tong yue iffi B ("[Fang Zhong]tong says") or jie ,·11,· Wi I ·l ('"the explanation is"), in which they explained the general method concerned wit I, the partinilar prnhlcm under examination. -' 7 On the inrl11c11rc ol Western mathematics upon the work of Fang Zhongtong and I l11 /.l1ige11g, ~1T L1111i I')') I; E11gelfrie1 I ')')8, 389-405. On the transmission of Western 111allll'111:i1i,., i111111· Ka11gxi pe1i11d, Sl"L' lla11 1991.

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C" = m! "' n!(m-n)

Wang also linked the duoji method with combinatorial numbers. Immediately after him, Dong Youcheng I ffiE ~ (1791-1823) linked the sums of the duoji method to circle measurement (ceyuan i&U Iii) problems. However, neither Wang nor Dong gave any explanation or proof of their formulas or methods.28

The Development of Zhu Shijie's Method

Zhu Shijie's method, which develops the duoji problem in a purely abstract way, seems to have been entirely forgotten or lost during the period from 1350 to 1800. Its rediscovery was made by Li Rui * ~ (1769-1817). In the early nine­ teenth century, Ruan Yuan ~Jt 5c (1764-1849) found a manuscript copy of the Siyuan yujian. Several mathematicians of the time were interested in it, and Zhang Dunren ~.R ~ 1= (1754-1834) was among them. As Zhang could not un­ derstand the problems involved in the duoji method, he wrote to Li Rui asking him to study those problems. In 1816, Li Rui replied to Zhang with a complete explanation of Zhu Shijie's duoji method.29 Later, Li Rui, Shen Qinpei tt fX ~ (fl. 1801-1830), Luo Shilin ff-I ± ~ (1789-1853), and Dai Xu m ~.~ (1805- 1860) all produced full reconstructions and explications of the Siyuan yujian, but none of them gave a proper explanation of their method of calculating the sum of the piles-30 Li Shanlan, the real successor of Zhu Shijie, came to the stage fifty years later. In 1867, he published his Duoji bilei, the first comprehensive work on the duoji method. In this work, Li presented his famous Li Renshu $ 3: ,m Iden­ tity:3 I

28 Zhu Shijie already grasped the general rules of the triangular pile underlying the duoji formulas given in his Siyuan yujian, but did not give the general formula. Wang Lai did not read the Siyuan yujian but discovered this formula by himself. 29 See "Li Shangzhi zhuan" $ f.'d -Z. f$. (Biography of Li Shangzhi) in Ruan Yuan's Yanjing shi erji ~ *-fil '.¥'. = ~ (Second Collection from the Yanjing Studio; Zhonghuu shuju ed., 1993), 4.483. As Li Rui's letter is not extant, his explanation is unknown to us. 30 On the rediscovery and transmission of the Siyuan yujian in the Qing period, sec Du 1966; Tian I 999. 31 In this paper, I will use Dp,n as the symbol for the sum of the lirst n numbers of the p-th line of lhe Jia Xian lriangle, and dp,n as lhe symbol for lhe 11-lh 1111111hcr of the 1riangul.1r p-lh oblique line oflhe Jia Xian triangle.

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" I p+I ,i - i+I 2 I [,[-r(r + J) .. · (r + p-1)) =[,cf'[, --r(r +I) · · · (r + 2p -1) (I) i=I pi i=I r=I (2p)! where cf' is the i-th number of the p-th line of diagram 2 (for example, c~ is 4)

Diagram 2 Diagram3 I I

4 4 9 9 I I 11 16 36 16 26 66 26 and the formula for the sum of the finite power series (chengfang duo * 1f ti):

11 11 n-i+I \ 1 1 [,r >+ =[Af' [, --r(r+l)···(r+p) (2) ,=I i=I r=I (p-J)! where Af' (equivalent to the Eulerian Numbers) is the i-th number of the p-th line of diagram 3.32 The style and format of the Duoji hilei are remarkably different from those of the traditional Chinese mathematical works. The work consists of four chapters, each of which forms a complete system. Chapter I deals with the triangular pile and its variant forms; chapter 2 with the finite power series; chapter 3 with the triangular self-multiplying pile (sanjiao zicheng duo =- fil El * ti); and chapter 4 with the modified triangular pile (sanjiao hian duo = fil ~ ti). The Duoji hilei includes fifteen tables (similar to the Pascal Triangle), fifty-seven different piles, 124 formulas, and 112 demonstrations (cao ~ ). This arrangement was rare in ancient Chinese mathematical texts. Nonetheless, despite its innovative ar- 1;1ngcment, the Duoji hilei "still conformed to the major features of traditional ( 'hincse mathematics," namely, using a few examples to demonstrate a formula, without providing proper proofs.33 Li Shanlan demonstrates the formulas in the variant forms of triangular piles based on his formulas for the triangular piles, but docs not give rigorous demonstrations of his basic formulas. There is therefore 110 indication of his reconstruction of Zhu Shijie's method.34

12 On the prnol of these two formulas, sec Wang 1990. II lll)J'llg t•)<)t. 1·1 On t.i Shnnlan 's /)1wji /Jill'i, sec Wang 1990; Li I 983; 1-lorng 1991

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After the publication of the Duoji bilei, no leading Chinese mathematician paid attention to the study of Yang Hui's method. On the other hand, the study of Zhu Shijie's more abstract method became an important subject for many mathematicians. Among them, Liu Yicheng IU 1¥ fj_ made remarkable advances in the duoji method. Liu was a teacher of mathematics at the Guangfangyan guan !Ji ;j j§ ~g (Foreign Language School) and the Qiuzhi shuyuan 3J< ~ ~ ~ic (Academy of Qiuzhi).35 In 1898, he published his lianyi 'an suangao AA JfJi ~ ~ ffi (Mathematical Manuscript of Jianyi'an), a collection of questions and answers for the examinations at the Qiuzhi shuyuan. There are twenty-five problems con­ cerning the duoji method in this book. Liu gives many new formulas, including one for the sum of the product of any two triangular piles:

n I I [--r(r+l)···(r+ p-1)•--r(r+ l)···(r+q-1) = r=I (p-])! (q+l)! n+I p-r+I I (3) Elf L ---r (r+l)···(r+ p+q - 3) 1=1 r=I ( p + q - I)!

where

ifi :S min (p, q) or[; =0

Diagram 4

2 3 3 4 6 4

and b p.,. is the i-th number of the p-th horizontal line of the Jia Xian triangle (diagram 4 )_36

35 The Guangfangyan guan was inaugurated in July 1864. Mathematics was included in its curriculum. See Horng 1991, 91-93; Xiong 1994, 334-349. The Qiuzhi Shuyuan was a traditional academy in Shanghai from 1876 to 1904. A number of mathematicians graduated from it. See Tian 1992; Tian 1997, 22-24, 42-45, 69-78. 36 About the general proof of the formula, see Tian 1993.

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Liu Yicheng explained the formula as follows:

Every series of triangular piles may be rewritten as a sum of several other series of different heights (the number of the numbers contained in the finite series) in the following way:

D, .• = D 1•• (n, height) D2,n = D1,n + D2,n-1 D3,n = Dl,n + 2D2,n-1 + D3,n-2 D4,n = Dl,n + 3D2,n-1 + 3D3,n-2 + D4,n-3 Ds,11 =D1,n +4D2,n-1 +6D3,n-2 +4D4,n-3 +Ds,n-4

Briefly, we can write the above formulas as:

D 1•• I

D 2,n I I

D3•• I 2 l

D4 •• 1 3 3 l

D5,n 146 4 l

[ ... ] This diagram gives the multiples of (different piles) in one pile.37

By analyzing the formula for the sum of the squares of a fi­ nite series (formula!),[ ... ] and by the method of using lines to divide the product of two respective numbers into series, [ ... ] it is possible to obtain the following results:

(a) D2,11 • D2.n = D3,n + D3,n-1 D2,n • D3,n = D 4,n + 2D 4.n-1 D2.n • D 4,n = Ds,n + 3D5,n-1

(b) D3,n • D3.n = Ds,n + 4Ds,n-l + Ds,n-2 D3,n • D4,n = D6,n + 6D6,n-l + 3D6,n-2 D3,il • Ds,n = D7.n + 8D7,n-1 + 6D7.n-2

37 In lhc dil11ram, I, 4, 6, 4, I arc the five multiples of D~.n •

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(c) D4.n • D4.11 = D1.n + 9D1.n-1 + 9D1.n-2 + D7.n-3 D 4.n • D5.11 = D87.n + 12D8.n-l + 8D8.11-2 + 4D7,n-3 D 4.11 • D6,n = D9.n + 15D9,n-l + 30D9,n-2 + I 0D9.n-3

By analyzing the second and th'ird groups of formulas above, it follows that the index (zhishu rn ft for a pile as D,.,,,., m is the index) of the product of the two piles is equal to the sum of the index of the two multipliers minus I. For in­ stance, if the indexes of the two piles are n and m, the index for their product is n+m-1. Then, writing the multiples in the diagram above by pairs, multiply the respective numbers in two lines. If the number of the numbers in one line is higher than the number of the numbers in the corresponding line, the numbers in the longer line that have no corresponding number in the shorter line are eliminated, and the products are nothing but the coefficients in the formulas (a), (b), and (c). Tn the diagram above, if the index of one pile is n, ... the numbers in the line of multiples are

(n-l)(n-2) (n - l)(n - 2)(n - 3) 1, n-1, ----- 2 2•3

If the index of the other pile is m, the numbers in the line of multiples for it are

(m- l)(m-2) (m-l)(m-2)(m-3) I, m-1, ------2 2•3

For instance, if n is 3 and m is 4, then the two formulas are changed into I, 2, I and I, 3, 3, I. Therefore, the index of the product is 4+3-1, which is 6, and the multiples are I, 6,

and 3, which is just the same as the product of D3.n and D4,n _38

38 Jianyi'a11 suangao (ed. of 1900), 3.2b-4a, year of gengyin /ji Ji[. The index for .n the mul­ D3,n is 3, and the multipliers for it are, I, 2, I, while the index for D4 is 4, and tipliers for it I, 3, 3, I. Following the procedure Liu Yicheng gave above, the index for

the prod·uct is 3+4-1, which is 6. As the second I in the line or multipliers for D4,, has no respective number in the former, so, it can he elimin.1tcu. And the new multiplier is Ix I = I , 2 x 3 = 6 , Ix 3 = 3 . Therefore, the new pih: rnn he wrillen us D,,.,, + 6Do ... _, + 3D,,.,, _1, which is nothing hut D," · /),,,.

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This provides a clear description of Liu Yicheng's method. There is still one point that deserves attention, however-namely, what Liu Yicheng means when he mentions the method of using lines to divide a pile into a series of piles. The example that follows helps to clarify this point. In the autumn of 1885, in the second problem of the text, Liu Yicheng gave a proof of Li Shanlan's formula for the finite power series. When n is equal to 4, then the square of n is 16. Liu Yicheng arranged 16 into a diagram as follows:

Diagram 5

l L ;!/YI 1 1/1/i atI 1/i/l

In diagram 5, Liu Yicheng pointed out that the numbers on the left side of line

L form a D 2,4 , while the numbers on the right side of line L form a D2,4 . Therefore,

4 3 2 4 = L, i + Li = d 2.4 + d 2,3 i=I i=I and

JI 2 II n-1 [r = [1 + [i = D2.11 + D211-1 r=I i=I i=I

As

3 2 r = r • r = rD2,r + rD2 ,r-l

.1'> Liu Yid1c11g gave a proof of this formula al the Qiuzhi shuyuan cxaminution proh­ llilll in I l!lt'i. <>n lhc proof of the for11111lu, sec Tian 199].

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1 40 fl- r = D J.11 +4D J.11-I + D J 11-2 r=I Later, Liu Yichcng provided demonstrations for

and

He then pointed out that other formulas for the finite power series could be dem­ onstrated in the same way. As his proof of Lhe square of square series is only based on case studies, from the point of view of Western mathematics the conclu­ sion obtained from this induclion is not complete. Since Liu Yicheng's method of demonstration is not different from the one used by Du Zhigeng, it seems that during the intervening two hundred years there had been virtually no develop­ ment in the method of demonstration. However, Liu Yicheng's method deserves attention for two reasons. First, in his solution of the duoji problem, Liu Yicheng follows a traditional Chinese method. In the example of the finite power series, Liu cuts the number 16 into Lwo triangular piles, and uses the triangular pile formula to obtain the formula (2). This is the method he applies throughout the Jianyi'an suangao. Culling a figure into several pieces wilh various shapes and using these to find the volume or area of the figure was a method commonly used in Chinese geome­ try. Induction also was commonly used in Chinese mathematics. Besides this, in his Siyuan yujian, Zhu Shijie had used lines to cut the Jia Xian triangle into a system or piles, so Liu Yicheng's method consists in an extension of Zhu Shijie's method.4 1 Second, it is very likely thal Liu Yicheng's method was commonly in use in Lhc late Qing period. Liu Yichcng was acquainted with Li Shanlan's work. In J 866, Liu went lo Shanghai, where he met Li and learned mathematics from him. Afterwards, Li wrote to Hua Hengfang W. r/i1 75 (1833-1902), praising Liu as one of the outstanding mathematicians of the younger generation.42 Liu cited Li's Duoji hi lei in his Jianyi 'an suangao, so the work of Li must have had a consider­ able impacl on Liu's work. It is quite possible that Li's method is essentially the same as Liu's. Since Li was acquainted with Luo Shilin and Dai Xu, moreover,

40 .lianyi 'an .1·1wngao, 2. 6a-8a, year of yiyo11 6 1JLj. 41 On the dcmonslralion or this formula and for dl:lails ;100111 Ilic i/1111;; sl11dy of l.iu Yichcng, sec Tian I 1)1.J:l. ·12 S1·1· l.i Sh;111l;111's IL·lll'i' 10 II11a lkngfang in Yan l 111JO, 478,

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Luo and Dai may also have used the same approach in their studies of the duoji method.

The duoji Method after 1890

After the transmission of Western mathematics to China in the late Ming and the early Qing period, a second wave of strong influence occurred after the 1850s with the translation of several Western mathematical books into Chinese. These included Elias Loomis's Elements of Analytical Geometry and of the Differential and Integral Calculus (I 851, translated in 1859), Augustus De Morgan's Ele­ ments of Algebra (I 835, translated in 1859), and William Wallace's articles on "Algebra" and "Fluxions" in the eighth edition of the Encyclopaedia Britannica (1852-60, translated in 1873 and l 874 ). Through these and other works, Chinese mathematicians became acquainted with symbolic algebra, calculus, and other branches of Western mathematics. However, until 1890 there was hardly any Western influence on the basic duoji method; and although Liu Yicheng in 1885 used algebra to demonstrate the formula (2) above, he essentially used a Lradi­ tional Chinese method. However, after 1890, Liu Yicheng began to seek the algebraic expressions of basic duoji formulas, and Laughl his method to his students. In 1890, Liu Yicheng gave four algebraic expressions for the formula for Lhe triangular pile:

" p(p+l)·· ·(p+r-2) Dp_n=E (r-1)!

"r(r- l)• •· (r _ i - 1) p(p-l)···(p-i+I) D,,.,,=~ ·1 • 1-I I . ("-I)'I .

p(p+l) ···(p+n-1) D,,_,,= I n. and

n(n + 1)- ··(n+ p-1) D =----- ''·" pl - - -

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He also gave derivations for these formulas in his Jianyi'an suangao. Although his method of demonstration was not complete, it provided a basis for further study.43 In 1898, Cui Chaoqing -gt !j[Jl /,t (1860-1943) published his Duoji yide il if - 1~ (One Achievement in the duoji Method). A student of Liu Yicheng, Cui was an active mathematician and a teacher of mathematics. In the preface to his work, he wrote:

I have studied the Siyuan yujian and the Duoji bilei for a long time, and I understand the formulas and methods in these two books very well. However, I cannot find the foun­ dation of the duoji method in these or any other mathemati­ cal books. Recently, I have produced an algebraic derivation for the formulas for the first three piles of the triangular pile series. I showed my work to some friends, and they all think it is worth publishing. As all the other formulas can be de­ rived from the formula for the triangular pile, it is easy to find the principles of other formulas based on my study.

n(n+ I) For example: If p=2, the formula for the triangular pile is D 2,n - --- 2 As: nxn = lxn i.e. lx(l+n-1) lxn lx(2+n-2) lxn lx(3+n-3)

lxn lx(n-1+1) lxn lxn

cutting the diagram inlo two diagrams, one is as follows:

!xi i.e. I + lx2 + 2 + lx3 + 3

+ lx(n-1) + 11-I

+ lxn + 11

and the other is as follows:

lx(n-1) i.e. n-1

43 Jic111vi'm1 .1·11a11,11ao, J. I 5a- I 7a, year of ,111•11,11yi11.

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+ lx(n-2) + n-2 + Ix(n-3) + n-3

+ !X2 + 2 + !XI + I

Then

f/ X II = l+l + 2+2 + 3+3

+ (n-J)+(n-1) + n

Therefore

~ n- +n = l+I + 2+2 + 3+3

+ n-l+n-1 + n+n

Finally,

n(n + I) D1.11 = 2

<:ui Chaoqing's demonstration may be summarized in the following way. As , 11-=11X11 =n+n+11+• · +11 " = [i+(11-i) i=l

II II =[i+[n-i I I I l

,, II I coc }) I }__:i I I I I

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n 2 +n=2ti i=I

Therefore

ti= n(n+I) i=I 2

Cui Chaoqing gave demonstrations for the other two formulas in the same way.44 There is no new achievement in Cui Chaoqing's work. Chinese mathemati­ cians already knew the formulas given in his book from a much earlier time. However, Cui is the first Chinese mathematician who thought that the basic for­ mulas fur Lhc duoji method needed to be demonstrated. Before him, major mathematicians including Wang Lai, Li Shanlan, and Liu Yicheng accepted those formulas as a fact without any explanation. In addition, Cui's statement that his friends thought his work Lo be significant may be read as an indication thal Chi­ nese mathematicians had by then accepted the more rigorous methodology of Western mathematics. In 1903, Zhang Xi 5ft ii, another student of Liu Yicheng, published his Duiduo shu .ll tl f1fLj (The duoji Method). Very little is known about Zhang's life and work. Besides the Duiduo shu, he also published a book entitled Qiewuzhai suangao ~ '1'2f ~ ~ lf

p(p+1) p(p+l)···(p+n-2) D = I +p+---+···------"·" 2! (11 -1)! was written as shown in fig. 7.

Figure 7. Zhang Xi's duoji formufa46 • ~i/ j • L ~1"jJ(_, __ .J ijprjj"

44 /)11oji yid,· (Gujin suanxuc congshu ctl .. 190)}, la-5h. 45 /)110 ~ is the right part ol lhc l'11arac1cr ,/1111 J't': (pile). 4<> l>11id1111 .1·hu, I h.

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Zhang Xi's expression of the duoji formulas is the same as Liu Yicheng's first expression for the formula for the triangular pile. After that, he gives two axioms, one

and

Dp.n-1 = Dp.n -dn

the other

dn =Dp.n-1

Then he uses the two axioms to demonstrate the following formula:

D p-1,n = D p.n - D p,n-1 Zhang I)

The whole text of Zhang Xi followed a similar format, by which he presented about fifty formulas and their algebraic demonstrations. The following is an ex­ ample taken from his book.

As

Dp-l,11 = D,,,n -D ,,.n-1

D p-1,n-l = D p,n-1 - D p,n-2 then

D p-1,n - D p-1,n-l = D p-2,n = D p,n - 2D p,n-1 + D p,n-2

Therefore

D p-2,n = D p.fl - 2D p,n-1 + D p,n-2

D p-2.n-1 = D ,,,n-1 - 2D p,n-2 + D p,n-3 and so

D p-2,n - D p-2,n-l = D p-3,n = D p,n - 3D p.n-1 + 3D p,n-2 - D p,n-3 [... l

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Consequently:

1 ·n +i(i+l)D - ···+(-l)"- D Zhang2) D p-i,n D = p,n - l p,n-1 ! p.n-2 p.l 2

Let i =-I

i(i + 1) Zhang3) D p+i.n = D p,n + iD p ,n-1 + --2,- D p,n-2 + ... + D p,1

and also:

A i(i + l) ···(i+11 -2) A i(i+l) 7Jrnng4 ) = iip,n +iApn I +--upn 2 + .. ·+------u,, I iip-1',n .- 2! .- (11 - I)! '

Let p=O,

. i(i+I) i(i+l)··· (i+n-2) I),.. = 1+1+---+ .. ·+------l,n 2! (n-1)!

(Here, Zhang Xi gives the proof of the general formula for U1c triangular pile.)

A . i(i+l) i(i+l) .. ·(i+n-3) L> · = 1+1+---+ .. ·+------1,n-l 2! (n-2 )!

Then:

p(p+l)···(p+n-2) /),.. -/),.. I=/),.. I =------1,n 1,11 - ,- .n (n-1 )!

Therefore,

,_,,_, D,,_, _,, = D,,,,, -ii;_,_2D,,,,,_, +ii;_2,3D,,,,,_ 2 -···+(-1)' D1 Zhan~71 fl p+l,11 = fl P," + /),. i-1,2 fl p,11-/ + !),. i-2,3 fl p,11-2 +' .. +fl p, 11 - I

and

D,,,,, = D,,+;,,, -Ai-l,2Dp+i,11-I +A;-2,JDp+/,11-2 - .. ·+(- !)' D,,.,,,,_,

D,,,,, =cDp-i,11 +t:.,-1.2Dp,11-I +t:.,-2,3Dp-i,11-2 + .. ·+( I)' !),, ' ·" 1

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This first group of formulas in the Duiduo shu shows that Zhang Xi fully applied an algebraic or, more generally speaking, Western method.47 In 1909, Zhou Da f,!rJ ts (1878- ?) published his Duoji xinyi :l:l ffi Jilfr ~ (New Meaning of the duoji Method). The expression of the duoji formulas in this work differs from Zhang Xi's. For example, the first formula given by Zhou Da is as follows:

. _ x(x+l) x(x+l)(x+2) A x(x+l)· .. (x+n) j(x)-A0 +A1 ---+A2 -----+···+ ; ------2! 3! (n+l)!

In this formula, the sum of a pile was written as a function. Zhou Da found the A; in a kind of method of undetermined coefficients, already introduced into China earlier in the Chinese version of De Morgan's Elements ofAlgebra (1859).48

Conclusion

After the end of the Second Opium War in I 860, some Chinese high officials and scholars were impressed by the destructive power of Western weapons. As the skill in the manufacture of firearms required a sound knowledge of mathematics, they encouraged its study. Therefore, some mathematicians-including Li Shanlan, Liu Yicheng, Cui Chaoqing, Zhang Xi, and Zhou Da-became teachers of mathematics and wrote books on the subject. These works clearly show that some branches of traditional mathematics were westernized soon after the trans­ mission to China of symbolic algebra, calculus, and other branches of Western mathematics. The development of the duoji method followed a different course. As late as 1885, Liu Yicheng used the traditional Chinese method in his study of the duoji problem; however, he knew the Western method as he had proofread the Chinese versions of Wallace's "Fluxions" and "Algebra." Like him, several mathemati­ cians mentioned above-including Fang Zhongtong, Du Zhigeng, the compilers ur the Shuli jingyun, and Li Shanlan (the translator of Loomis's Elements ofAna­ l1•tical Geometry and De Morgan's Elements of Algebra)-were all familiar with Western mathematics, but used traditional methods in their studies. One reason

47 Herc, Zhang Xi did not give the general proof but only a simple induction. How­ •:vcr, he gave a symholi<.: algcbrai<.: proof of the formula. This method also shows that the < 'hincsc mathcmati<.:ians or his time were not familiar with the method of complete indue- 11011 •IK Sec n111(ii .1i11ri (Fuhui shuangxiuguan suangao iJ,T,\ ~ 11~ Mt j,r{ 1,1: tffi ed., 1909), I :1 - I h; and /\kxalllkr Wylie and f ,i Shan Ian's Chinese translation or De Morgan's Dai.1·/111 1 1 1111· / l: IU~ '~ Wil·1111·111s ol /\lg1·hra; I 8'i )), I

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for this may be that no works on combinatorics had been translated into Chinese at that time, so the Chinese malhematicians were unfamiliar with the branch of Western mathematics corresponding to the duoji method. Another reason may be that, although Li Shanlan and Liu Yicheng knew Western mathematics, both had studied traditional mathematics at an early age, so their mathematical thinking remained a traditional one and they were content with their case-by-case demon­ stration and methodology. Their students, however, were different. They learned Western mathematics at a young age and wanted to find the foundation of the duoji method. As they did not find explanations in traditional sources or in their teachers' works, they tried to find a foundation for the duoji method in the Western method. Despite individual differences, their approach was essentially the same. Thus, Cui Chao­ qing, Zhang Xi, and Zhou Da presented the proofs for the basic duoji method using Western mathematics. At the same time, the first two rejected their teacher's method without comment. In conclusion, the development of the duoji method presents an example of how the study of a subject of traditional Chinese mathematics was westernized. At first, following the Western model, Chinese mathematicians changed the style of their works, but continued to use the traditional Chinese approach. Gradually, they used the algebraic method and gave symbolic expressions of the duoji for­ mulas, but at first Western algebra for them was only a calculation technique. Finally, after the 1890s, Chinese mathematicians not only used Western algebra in establishing new duoji formulas, but also employed it as a method of demon­ stration. Thus, by the end of the nineteenth century, the westernization of their duoji method was achieved.

References

Du Shiran H E ~- 1966. "Zhu Shijie yanjiu" * t±t ~ Wf ~ (A study on Zhu Shijie). In Qian Baocong f~ Jif f]f-, ed., Songyuan shuxueshi lunwenji * ft ~ q'. ~ iB X ~ (Collected studies on the history of mathematics in the Song and Yuan periods), 166-209. Beijing: Kexue chubanshe.

Engelfriet, Peter M. 1998. Euclid in China: The Genesis of the First Chinese Translation of Euclid's Elements, Books I-VI (Jihe yuanben, Beijing, 1607) and its Reception up to 1723. Leiden: E. J. Brill. Fu Daiwie fr# 7( ~ - 1992. "Zhongsuanshi duoji shu yuanliu xinlun: 'Shanggong' yu 'Shaoguang' liangtiao xiansuo de lishi yanhua" rt, 1;!: ':1: ~ ffi f1ILJ i'J);i 1/ii $Jr ~ii'li - 'lffi J.jJ J W r Y il/1 1 w,i i!Xi M~ '.t l'l'-J /fl iJ.: iLiJ 1t (A new view of the origins anti Jevclopmcnt of the duoji methoJ in the history of Chi­ nese mathcmalks: The hislorical transformalion of its two trends lhcgunl

Downloaded from Brill.com10/07/2021 08:10:01AM via free access Tian Miao: The Westernization of Chinese Mathematics 71

by the "Discussing Works" and "Small Width" chapters [of the Jiuzhang suanshu]). In Fu Daiwie, ed., Yi shikong li de zhishi zuizhu ~ Bi @ _!I!. ~ ~D i.R ~~(The pursuit of knowledge in time and space), 69-113. Taipei: Dongda lushu gongsi. Guo Shuchun ~-ts~- 1997. Zhongguo guclai shuxue 41 00 ti' {i; me q: (An­ cient Chinese mathematics). Beijing: Shangwu yinshuguan.

Han Qi ~ f~. 1991. "Kangxi shidai chuanru de xifang shuxue ji qi dui Zhongguo shuxue de yingxiang" /.J ~~ □ t {i; ff A ~ ~ ;5 me ~ & ;It: Jl1 41 00 ~ '.cf'. i't-J Jj ll[a] (The transmission of Western mathematics in the Kangxi reign­ period and its influence on Chinese mathematics). Ph.D. diss., Beijing: In­ stitute for the History of Natural Sciences.

Horng Wannsheng. 1991. "Li Shanlan: The Impact of Western Mathematics in China during the Lale 19th Century." Ph.D. diss., City University of New York.

Jami, Catherine. 1991. "Scholars and Mathematical Knowledge During the Late Ming and Early Qing." Historia Scientiarum 42: 99-109.

Li Yan and Du Shiran. 1987. Chinese Mathematics: A Concise History. Trans­ lated by Hohn N. Crossley and Anthony W.-C. Lun. Oxford: Oxford Sci­ ence Publications.

Li Zhaohua +~IS$. 1983. "Li Shanlan duoji shu yanjiu" :$ ~ ~ ±h): f_R 7f;.: liH 31: (A study of Li Shanlan's duoji method). Tianjin shifan claxue xuebao 7(. 7$ jijj fl'[* '.cf'. q: t~ (Tianjin Normal University journal) 1983: 65-78.

Martzloff, Jean-Claude. 1997. A History of Chinese Mathematics. Translated by Stephen S. Wilson. Berlin: Springer-Verlag. Originally published as His­ toire des mathematiques chi noises (Paris: Masson, 1988).

Needham, Joseph. I 959. Science and Civilisation in China. Vol. 3: Mathematics and the Sciences of the Heavens and the Earth, with the collaboration of Wang Ling. Cambridge: Cambridge University Press. Qian Baocong ii if ffr;, ed. 1964. Zhongguo shuxue shi 41 ~ ~ * 1:_ (A his­ tory of Chinese mathematics). Beijing: Kexue chubanshe.

Tian Miao m~- 1992. "Qingmo shuxuejia yu shuxue jiaoyujia Liu Yicheng" ffll" ;;k ~ "r': % !.:c; ~ q: fJ( r=f * -Y:IJ ~ f]i (Liu Yicheng, a mathematician and teacher or mathematics al the end of the Qing period). In Li Di * @, ed., Sh11.rneshi yaniiu wenji fill ·y: ~ Jiff J·~ X ~ (Collecte

Downloaded from Brill.com10/07/2021 08:10:01AM via free access 72 EASTM 20 (2003)

---. 1993. "Liu Yicheng duoji shu yanjiu" Y:iJ $ ;f£ ti :fH # lirf ~ (A study of Liu Yicheng's duoji method). In Li Di * @,ed., Shuxueshi yanjiu wenji ~ '.:jc'.: .?t: fiJI 1i'. :X ~ (Collected studies on the history of mathematics), 5: 70-81. Huhehaote: Neimenggu daxue chubanshe; Taipei: Jiuzhang chuban­ she.

--- . 1997. "Qingmo shuyuan de shuxue de jiaoyu yanjiu" Y~ * ~ 11% liJ !~ ~ fj( W tiff J~ (Mathematics education in the traditional academies of the late Qing dynasty). Ph.D. diss., Beijing: Institute for the History of Natural Sciences.

---. 1999. "Siyuan yujian de Qingdai banben ji jialing sicao de jiaokan yan­ jiu" [9 7t :::Ii ~ B1 yrlf i-1:: hN ;$: R. 1itl % ~ ~ 01 t~ &f)J ~ff ~ (On the dif­ ferent versions of the Siyuan yujian and the "Four examples" in the Qing dynasty). Ziran kexueshi yanjiu § ?.t -l4 $ .?t: ftff ~ (Studies in the history or natural sciences) I 8. 1: 36-47. Wang Yusheng :E Ni} 1:.. 1990. "Li Shanlan yanjiu" * ~ ~ tiff~ (A study on Li Shanlan). In Mei Rorigzhao fi: ;;_ ~R-, ed., Mingqing shuxueshi lunwenji l:l)j trlf ~ '$ ~ i-B x :it: (Collected studies on the history of mathematics in the Ming and Qing periods), 334-408. Nanjing: Jiangsu jiaoyu chubanshe.

Xiong Yuezhi fl~ Fl z . 1994. Xixue dongjian yu wan Qing shehui ~ ~ * y'fly ':3j B~ miITJ ,f; (The eastward transmission of Western learning and late Qing society). Shanghai: Shanghai renmin chubanshe.

Yan Dunjie F ~J ~- 1990. "Li Shanlan nianpu dingzhcng ji buyi" :$ ~ ~ if. i~ iT :iE :& H m(Corrections and addenda to Li Shanglan's chronological biography). In Mei Rongzhao fij ~ ~#,, ed., Mingqing shuxueshi lunwenji l:l)j mt:!I: ~ .?t: if; x ~ (Collected studies on the history of mathematics in the Ming and Qing periods), 473-478. Nanjing: Jiangsu jiaoyu chubanshe.

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