Abstract
The mathematics of 12th–14th-century China is known for its beautiful algebraic texts. Unfortunately, information concerning their context of transmission and instruction is scarce. One interesting pattern is that many of the texts share a predisposition for tabular setting and several of these texts refer to the same algebraic procedure named tian yuan 天元 (Celestial Source) used to set up polynomial equations. The setting of these equations on a counting surface is the result of a specificity of using counting rods for the algorithm of division . Precisely, the role of division for setting up and solving equations is fundamental to the algorithm. This chapter presents an excerpt borrowed from Li Ye’s 李冶 Yigu yanduan 益古演段 (the Development of Pieces [of Area according to the Collection] Augmenting the Ancient [Knowledge], 1259). It presents first a basic example of the Celestial Source procedure, then attempts reconstitution of polynomials on the counting surface and ends with comparative observations related to the chapter on the Bījagaṇitāvataṃsa (BGA) written by Nārāyaṇa in 14th-century India . The description of the algorithm for setting up a quadratic equation is interesting from a comparative perspective. The way in which lists of operations are ordered shows that Indian and Chinese authors had different interests, addressed different difficulties and understood mathematical concepts differently, while referring to division, using tabular setting and “model” equations.
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- 1.
Pollet and Ying (2017).
- 2.
“Development (演) of Pieces (段) [of Areas] [according to] [the collection] Augmenting (益) the Ancient (古) [Knowledge].” It is a presentation of a geometrical procedure named “Development by the Section of Pieces [of Areas],” based on an ancient treatise titled Yiguji 益古集 (Collection Augmenting the Ancient [knowledge]) of the 11th century (for a translation of terms see Pollet (2012) and Pollet and Ying (2017)).
- 3.
Biographies of Li Ye can be found in English in Mikami (1913, p. 80); Ho (1973, pp. 313–320); Lam (1984, pp. 237–239); Li and Du (1987, p. 114) and in Chinese in Mei (1966, p. 107). His life has been the object of several notices since the Yuan dynasty, with the first being written in 1370 in Yuan Shi 元史 (Official History of the Yuan, chapter 160) and the last being written in 1799 in Chouren Zhuan 疇人傳 (Biographies of Astronomers [and Mathematicians]) by Ruan Yuan 阮元 (1764–1849). I will not consider further this material in the present work.
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In the present case, the text is composed of a mathematical discourse made of sentences written with Chinese characters and tabular mathematical expressions, of diagrams with their legends and of commentaries. Here I discuss what I referred to as mathematical discourse. The two other items composing the text are the objects of other publications.
- 7.
There are many publications on this procedure. The reader can refer to Pollet (2012, 2014); Pollet and Ying (2017); Chemla and Guo (2004, pp. 314–322); Chemla (1982, Chap. 8); Li and Du (1987, pp. 17–19); Lam and Ang (2004, pp. 63–71); and Martzloff (1987, pp. 229–249) for publications in Western languages, among others. Here I just indicate the main thread of the procedure for the purpose of comparison with Indian documents discussed in Chapter “Reading Algorithms in Sanskrit” of the present volume.
- 8.
This approximates the value of π as 3. Arguably, this value does not mean that the author was not aware of the fact that π is not equal to three; rather, the value is the result of a process which calculates the areas of circular figures from corresponding square figures. Thus, the four circular areas are equivalent to three squares with sides equal to their diameter. This arithmetical transformation is convenient for the geometrical transformation of areas. To represent four circles, three squares may be drawn instead. Liu Hui 劉徽 (3rd century AD) gives π ≈ 3.1416 and Zu Chongzhi’s 祖沖之 (AD 429–500) approximation was between 3.1415926 and 3.1415927.
- 9.
For the signification of the absence or presence of the character tai see Pollet and Ying (2017).
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I mark the center of the counting surface with a black dot. The representation of the surface is only partial for the reason of economy of space.
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See Pollet and Ying (2017) for the distinctions between “to eliminate” (xiang xiao) and “to subtract” (jian).
- 12.
The oblique lines at the coefficient mark the coefficient as negative. These were added by Qing dynasty editor, Li Rui. It is possible that in the original text the coefficients were not negative, but the disposition of rods was supposed to be interpreted as 0.25x2 + 80x = 1700. See Pollet (2014) and Pollet and Ying (2017).
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Different descriptions of the algorithms of division and root extraction from different sources are given in Western languages by Wang Ling and Needham (1955); Chemla and Guo (2004, pp. 314–322); Chemla (1982, Chap. 8); Li and Du (1987, pp. 7–19); Lam and Ang (2004, Chap. 3, Sect. 4); and Martzloff (1987, pp. 229–249), among others. We will not discuss the evolution and details of different algorithms and their historiography.
- 14.
Sunzi suanjing belongs to the Shi bu suanjing 十部算經 (Ten Books of Mathematical Classic) used in education during the Sui (581–618) and Tang (618–907) dynasties. Wang (1964) suggested that it was composed between 280 and 473. Its earliest surviving edition is dated from the Southern Song dynasty, 1127–1279 (Li and Du 1987, pp. 92–93; Qian 1963; Lam and Ang 2004, pp. 63–71).
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There is also a meaningful difference between algorithms. The root extraction uses a supplementary rod named the “borrowed rod” (jie suan). In her explanation of the algorithm of root extraction in the Nine Chapters, Chemla shows that the borrowed rod is first placed in the position of the units. This rod is later moved toward the left, from 102n to 102n+2, until it reaches the leftmost position under the dividend, that is, 102n if the first digit of the root is a · 10n, 1 ≤ a ≤ 9. One deduces what is the power corresponding to the first digit of the root, named “the quotient” (Chemla and Shuchun 2004, p. 326). Since the algorithm of the Nine Chapters is slightly different from the one of the Sunzi suanjing, the interpretation of the role of the “borrowed rod,” its position and modifications made in the latter text, require discussion. In her comparison of the algorithm of root extraction from the Zhang Qiujian suanjing and the one by Kushyar ibn Labban, Chemla explains the role of the “borrowed rod.” She mentions that this rod has different roles in Chinese algorithms of root extraction (Chemla 1994, p. 17)
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My forthcoming publication, “The Empty and the Full,” is dedicated to it. See also Lam and Ang (2004).
References
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[Both edited in the Wen yuan ge Siku quanshu 文淵閣四庫全書 (Complete library of the Four Treasuries, [copy preserved in] Wenyuan Pavilion), original edition of 1789 stored in National Palace Museum, Taiwan.]
Wen jin ge Siku quanshu 文津閣四庫全書 (Complete library of the Four Treasuries, [copy preserved in] Wenjin Pavilion), Beijing: Shangwu, 2005, vol. 799 [a reprint of the original edition of 1789].
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Appendices
Appendix
Problem 1 of Chap. 1 of Yigu yanduan. The commentaries and the second procedure were removed from the text below. To read their translations, see Pollet and Ying (2017). The present version of the Chinese text is based on the edition made by the mathematician Li Rui (LR) in 1798. This edition was carefully compared to two manuscripts of 1782 published in the imperial encyclopedia, the Siku quanshu (The Complete Library of the Four Treasuries 四庫全書), namely the Wenjing (WJG) and Wenyan (WYG) editions. See Pollet (2014) for details of the comparison. The numbering of the sentences is mine—(1.2) means sentence number 2 of problem 1—it is added in order to observe a correspondence between Chinese text and translation.
(1.1) 第一問
今有方田一段, 內有圓池水占, 之外計地一十 三畝七分半. 竝不記內圓外方.只云從外田楞 至內池楞, 四邊各二十步.
問內圓外方各多少?
荅曰: 外田方六十步, 內池徑二十步.
(1.2) 法曰: 立天元一為內池徑. 加倍至步得
為田方面.
(1.3) 以自增 乘得
為方積, 於頭.
(1.4) 再立天元一為內池徑. 以自之, 又三因, 四而一得
為池積.
(1.5) 以減頭位, 得
為一段虛積, 寄左 (Fig. 17).
The captions read: “(a) 至水二十步; (b) 方田六十步.” For translation, see Fig. 18
The captions read: “(a) the bu reaching the water are 20; (b) the side of the field is 60 bu.”
(1.6) 然後列真積. 以畝法.通之, 得三千三百步. 與左相消
(1.7) 得
開平方, 得二十步, 為圓池徑也. 倍至步, 加池徑, 即外方面也.
Translation
(1.1) Let us suppose there is one piece of square field, inside which there is a circular pond. Outside the [area] occupied by water, one counts thirteen mu, seven fen and a half [of fen] of land. Moreover, there is no record of the [dimensions] of the inner circle and the outer square. It is only said that [the distance] from the edge of the outer field to the edge of the inside pond on [all] four sides is twenty bu.
One asks how much are [the diameter of] the inner circle and [the side of] the outer square.
The answer says: the side of the outer field is sixty bu. The diameter of the inside pond is twenty bu.
(1.2) The method says: set up one Celestial Source as the diameter of the inside pond. Adding twice the reaching bu yields
as the side of the field.
(1.3) Augmenting this by self-multiplying yields
as the area of the square, which is sent to the top.
(1.4) Set up again one Celestial Source as the diameter of the inside pond. This times itself and multiplied further by three then divided by four yields
as the area of the pond.
(1.5) Subtracting this from the top position yields
as a piece of the empty area, which is sent to the left.
(1.6) Next, place the real area. With the divisor of mu, making this communicate yields three thousand and three hundred bu.
(1.7) With what is on the left, eliminating from one another yields
.
Opening the square yields twenty bu as diameter of the circular pond. Adding twice the reaching bu to the diameter of the pond gives the side of the outer square.
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Pollet, CV. (2018). Interpreting Algorithms Written in Chinese and Attempting the Reconstitution of Tabular Setting: Some Elements of Comparative History. In: Volkov, A., Freiman, V. (eds) Computations and Computing Devices in Mathematics Education Before the Advent of Electronic Calculators. Mathematics Education in the Digital Era, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-73396-8_8
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