Names of positions

We will now have a look to the vocabulary used by Li Ye to name positions on the counting support. Several positions on the support are mentioned and a polynomial is attached to each of the position.

Different verbs are used to mean “to place”¹⁵⁰:

 The character 立, li, “to set up”, appears in the procedure of the Celestial Source only.

It is never used in other situation and it is systematically used in each problem in the same sentence inaugurating the procedure of the Celestial Source. I do not know if li means “to place” something on the counting support to represent the unknown, or if its meaning is more abstract, like “to conceive” the unknown.

 The character 列, lie, “to place”, “to arrange”, appears in the procedure of Celestial Source and in old procedure. It is only used to recommend placing the constant area given in the statement on the counting support.

 The character 置, zhi, “to put”, appears in the procedure of Celestial Source, Sections of Pieces [of Areas] and old procedure too. In the old procedure, it refers to placing

149 (1) is from pb.11b; (2) and (4) from pb.22; (3) from pb. 4; (5) from pb.26.

150 Some verbs are clearly indicating manipulation, tui,推 “to regress”, jin, 進. “to progress”; but this vocabulary appears only in the procedure of section of area and in the old procedure. Tui appears in pb.56, 59 and 60, jin in pb. 10 and 11.

73 the constant divisor on the counting support. In the section of area, it is to place the quantity which will be used to make the dividend. And, in the procedure of Celestial Source, it is sometimes used when one places the tian yuan on the support the second time. In fact, this character is used in any situation different from the two other mentioned above.

First, each time one wants to construct a polynomial, one has to place an object (bamboo rod?) to represent the unknown; either the object or the place of the object, or both, is named “tian yuan”. As for each of the problem, one has to create two polynomials, the tian yuan is placed two times on the counting support.

Secondly, a top position (tou wei, 頭位) is mentioned, where the first polynomial is placed and kept for further later operations. There is no mention of place for the second polynomial. The later is immediately used in subtraction or addition; therefore, one does not need to preserve it. After, the tian yuan was placed further may be this polynomial was placed on the top too, right under or next to the first one in order to be subtracted or added.

May be we can imagine that for addition and subtraction, two expressions were placed on the counting support one above, or next to, the other. Coefficients of same degree of polynomials were on the same column or row facing each other then added or subtracted. It is suggested by three of the problems. In pb 21, 43 and 62, coefficient of three expressions have to be added together. The first one is placed above (shang wei, 上位), the second on the next position (ci wei, 次位 in pb 21, 62) or middle position (zhong wei, 中位 in pb 43) and the third below (xia wei, 下位) and the 3 positions (san wei,三位 ) are added¹⁵¹.

151One noticed that the names of position, 上, 下, 中, 右, 天元 coincide with the names of fixed positions on the table which is used to play Go (wei qi, 圍其) with red or white and black or blue token (chou, 籌) [He Yunpo ,何云波, 2001]. [Martzloff Jean-Claude, 1998], p. 259, also noticed that tian yuan is also the name of the central point of the grid on which the game is played. The tian yuan area on the grid is a circular area delimited at its north, south, east and west by intersections named 中腹, zhong fu. The square surrounding the tian yuan circle is named 地, di, “earth”. There are nineteen positions on the support on intersections of lines of the grid where the token are placed, in the paragraph of the Jing Zhai gu jin tu I translated in supplement, nineteen positions were listed as positions on a support. The same table was also used for yi-jing divination with sticks since Tang dynasty at least [Liu Shancheng, 刘善承, 2007]. This support was a very common object for literates, and the ability of playing Go was listed among the basic skills noble people should study [Bottermans Jack, 2008].

74 Thirdly, a left position (左) is mentioned, where the third polynomial resulting of subtraction of the first and second polynomial is placed. Then the constant expression of the area is also placed on the counting support and subtracted from what is on the left. There is no information on the position of the constant and of the final equation on the support.

Three times an “earth position” (di, 地) is mentioned. In pb 19 and 43, quantities for intermediate operations are placed there. In the pb.19 mentioned in our previous chapter on uses of data, the earth position is the position where the constant distance given in the statement is transformed into another constant which will be used to construct polynomials.

In pb.43, the constant circular area given in the statement is multiplied in order to compute the area with another value of π. Both times this position is mentioned in the case of the transformation of the data of the problem. Those are extra operations which are not concerned with polynomials. In pb 37, in the section of area, this character indicates the bottom of the geometrical diagram. But one cannot deduce that “earth position” is always below. The classical representation of sky and earth on divination support places the sky as a central circle surrounded by the earth which has the shape of square.

The different rows of the tabular setting for quadratic equation are not given names in the procedure of Celestial Source. But for the 6 cases presenting linear equation, that are pb.38, 44, 48, 56, 59, 60, the sentence 下法, 上實, xia fa, shang shi, “below is the divisor, above is the dividend”, follows the tabular setting of the equation. The constant term is above and is occupying a place named shi, and the unknown is below and occupy a place named fa. These terms are translated as “dividend” and “divisor” because of the strong analogy with the algorithm of the division as we will see now.

75 4.2.3 Arithmetic of numbers and polynomials.

To our knowledge there are no available ancient mathematical Chinese treatise describing the elementary operations on polynomials and how those operations were performed on a counting support. The Yigu yanduan does not provides descriptions of algorithms, neither on constants, nor on polynomials. But several famous more ancient mathematical treatises provide algorithms on operations on constants.

The chapter I of the Nine Chapters on Mathematical Procedures, 九章算術¹⁵², Jiu zhang suan shu, presents a systematic treatment of arithmetic operations with fractions and the chapter IV, an algorithm on extraction of square and cubic roots. The geometrical inspiration of the algorithm of the extraction of root is explained by Liu Hui (劉徽) in his commentary (263 AD). The Nine Chapters had an extremely important influence on the development of Chinese mathematics, as Li Ye testifies it in his preface: “When it comes to a mathematics book, regardless the mathematician's school, The Nine Chapters (九章) is commonly traced back to as the root. Meanwhile, Liu Hui (劉徽) and Li Chun-feng (李淳風)'s notes and comments on The Nine Chapters (九章) make the mathematics even more perspicuous”¹⁵³. The dating of the book is disputable, but according to the present available material [Li Yan, Du Shiran, 1987] concluded that it was completed around the first century AD¹⁵⁴. After the Nine Chapters, other mathematicians proposed algorithm of root extraction based on the same geometrical idea, but the setting of the computation became slightly different showing an evolution concerning the algorithm¹⁵⁵.

One of the other treatises presenting such algorithms on constants is the Sunzi suanjing, 孫子算徑, The Mathematical Classic of Sun Zi, written around 400 AD¹⁵⁶. The treatise has three chapters. Its first part describes the multiplication and division process and illustrates them with detailed examples. The second part presents the method of calculating with fractions and extracting square roots. The last part collects some problems in arithmetic. This work was often use as a central source of reference for the analysis of the evolution of arithmetic¹⁵⁷. We also already mentioned that polynomial appear directly in the 13^th century with the place value notation describe in the previous section. So there is a

152 We will shorten the title “Nine Chapters on Mathematical Procedures” into “Nine Chapters”.

153其撰者成書者，無慮百家，然皆以「九章」為祖。而劉徽、李淳風又加注釋，而此道益明。

154 More detail on the different thesis on the history of composition of the Nine Chapters by Guo Shuchun in [Chemla Karine, Guo Shuchun, 2004] Chapter B, p. 43-46.

155 [Chemla Karine, 1982], p.7.7.

156 [Qian Baocong, 錢寶琮, 1963].

The composition of the book cannot be earlier than 280AD and no later than 473 AD according to [Lam Lay-Yong, Ang Tian Se, 2004], p.27. See also [Jean-Claude Martzoff, 1997] pp. 136-138.

157 This treatise was the object of two studies: [Lam Lay-Yong, Ang Tian Se, 2004], [Qian Baocong, 錢寶琮, 1963], and several historians studied its algorithm in some chapters: [Li Yan, Du Shiran, 1987],[Guo Shuchun, 郭書春, 2010], [Chemla Karine, Guo Shuchun, 2004] among others.

76 gap of nine centuries between our witnesses. [Chemla Karine, 1996]¹⁵⁸ shows that, despite some transformations of the notation, there is a remarkable persistency of tabular settings since the Han dynasty. What the Sunzi suanjing tells us about algorithms and tabular settings, and other treatises like the Yang Xiahou suanjing and the Zhang Qiujian suanjing¹⁵⁹ reproduced, is in adequacy with the graphs of numbers used by Li Ye. All are organising mathematical objects systematically around vertical and horizontal settings. The stability of this dispositive is in fact induced by the respect of the rules for placing numbers. The stability is due to the organisation of practices around procedures.

On the basis of these two works, The Nine Chapters and the Sunzi suanjing, [Chemla Karine, 1982] examines the techniques available to Li Ye for establishing equation in the Ceyuan haijing. In the chapter 7 of her PhD dissertation, she made a succinct presentation of the history of the different methods for the root extraction. This history led to see how arised the place value notation of equations. In the chapter 8, she examines the place value notation for polynomials. But as one has to infere the arithmetical operations on polynomials from Li Ye’s testimony, she detours through the algorithm on constants in the Sunzi suanjing to see if the reconstruction of Li Ye’s practice of operations is valid.

On the basis of these sources, I will give an account of operations on constants and polynomials in the Yigu yanduan. This part will underline the difference between polynomials and equations in the work of Li Ye.

i. Addition and Subtraction.

Neither the Sunzi suanjing nor the Nine Chapters give account of detailed procedures of addition or subtraction. But, on the basis of the algorithm of multiplication which contains additions, historians reconstructed how these operations were probably preformed¹⁶⁰. Even if we do not know how Li Ye exactly proceeded for these operations, we see that there are lots of occurrences of these in the Yigu yanduan. In each of the problems polynomials are added together or subtracted from one another.

158 [Chemla Karine, 1996], p. 118-125.

159 夏候楊算徑, Mathematical classic of Yang Xiahou, p. 558; 張邱建算徑, Mathematical classic of Zhang Qiujian, p.381-5. [Qian Baocong, 錢寶琮, 1963].

These two treatises, with the Sunzi suanjing, belong to the list of the “ten classics” (算徑十書, suanjing shi shu) required as textbooks for studying mathematics prescribed by the government in the Sui and Tang dynasties (316-907 AD). This collection was the first time published in the 7^th century by Li Chunfeng and al. [Volkov Alexei, 2008], p. 63. [Siu Mankeung, Volkov Alexei, 1999], p. 90.

Also on this topic [Li Yan, Du Shiran, 1987], p.105-6.

160 [Lam Lay-Yong, Ang Tian Se, 2004], p.71.

77 Probably polynomials were placed “face to face”, and their coefficients of same degree were added or subtracted from each other, as it is suggested in pb. 21; 43 ; 63¹⁶¹. This is also what is suggested by [Chemla Karine, 1983], p. 8.5¹⁶².

The vocabulary of addition and subtraction is diversified. Its diversification seems to be according to the mathematical object (constant or polynomial) which is involved in the operation.

We have for the addition:

加 A, jia. “to add”. Names the operation to add A to the result of the previous operation (constant or polynomial). A is a constant.

AB 共, gong. “the sum”. A and B are added together, A and B are two constants computed in previous operations.

AB 和, he. “the sum”. A and B are added together, A and B are two constants given in the statement.

併 A, bing. “to add”. A constant or polynomial is added to A.

We have for the subtraction:

A 減 B, jian. “ to subtract”. B is subtracted from A.

相消, xiang xiao, we translate by “to eliminate from one another”. Subtraction made in one sense or the other ( A-B or B-A) of one expression of the area containing the unknown with the equivalent expression in constant term. In the Ceyuan haijing, it is the elimination of two polynomials representing the same object. Here the operation concerns a polynomial with a constant term.

A 差, cha. For example, 廣差, guang cha, “difference [between] the two widths [of rectangles]”. It names the difference between two constants of “same nature” given in the statement, like, between the two lengths of two different rectangles.

Li Ye makes a clear distinction between the different types of subtractions. In each of the problems, when one subtracts the second polynomial from the first one, the operation is never prescribed by another verb than jian and the operation is always done in the same direction. The expression xiangxiao appears only at the very last step of the procedure that is to subtract two equivalent expressions to make the equation.

161 See translation given in III.A.1

162 I will not deepen the questions of reconstruction of algorithm of addition and subtraction. For our purpose, it is better to concentrate on multiplication, division and root extraction.

78 ii. Multiplication.

In the Yigu yanduan, there are even less details concerning multiplication of polynomials. In fact, there are no cases of two different polynomials multiplying each other in the Yigu yanduan. But there are many cases of multiplication of polynomial by a constant or of a polynomial multiplied by itself.

The vocabulary of multiplication is:

自增乘 , zi zeng cheng, “to self-multiply by augmenting”. This expression appears strickly when the first polynomial is multiplied by itself. It appears only in chapter one and two in nineteen of the problems¹⁶³. Perhaps, the idea of “augmenting” (zeng) is implied by the fact that the expression multiplied by itself augments of on power or one row on the counting support. In the Nine Chapters, A 增 B names an addition¹⁶⁴. May be there was an allusion of the multiplication considered as an iteration of an addition. But I do not know how to link the different explanations.

自之, zi zhi, “this times itself” is used for the transformation of one polynomial into a polynomial of upper power by self multiplying. It names the same operation as zi zeng cheng. These characters are systematically used for the second polynomial, sometime for the first one.

乘, cheng, “to multiply” names the operation of multiplication in general. For most of the case, in the Yigu yanduan, it names the multiplication of a polynomial by a constant.

倍 A, pei, “to double”. That is to multiply a constant by two.

A 因, yin, “to multiply by A”. That is to multiply a polynomial by A, a constant of one digit.

This operation applies mainly to the second polynomial. The expression 三因之 or 四因之 is sometimes reduced to 三之, 四之, etc. This operation is sometimes followed by a division：

三因四而一.

In the case of a multiplication by a constant, the algorithm consists in multiplying successively all the coefficient by the constant. In the case of multiplication of a polynomial by itself, we can imagine that the algorithm is the same as two different polynomials multiplying each other, as it described in the chapter 8 of [Chemla Karine, 1983], p.8.5-7.

From her reading of the Ceyuan haijing, she reconstructed the operation of multiplication on the basis of the correlation between the description of the equation and the steps of

163 See part. III. A. 2

164 See Karine Chemla’s lexicon, [Chemla Karine, Guo Shuchun, 2004], p. 1030.

79 according to the number of digits. The procedure in the Sunzi suanjing prescribes to multiply the number placed below, digit after digit, by the greatest digit of the multiplier and to put the intermediate results in the middle row where they are added progressively. On the contrary, the division sets the dividend in the middle, the divisor below on the left, and the digits of the results are put at the upper row, following a decreasing order. Each of the digits will multiply the number situated below, and the intermediate results are progressively removed from the middle row. That is the reason why Sunzi said that the two procedures are contrary of one another¹⁶⁷.

I will not develop further the description of the algorithm of multiplication. I want to concentrate on a peculiar point concerning multiplication. There is another allusion to multiplication in the procedure of Celestial Source. Sometimes Li Ye mentions denominator while one does not see any fractions, and this denominator is used to perform a multiplication, like in the following example (pb11a)¹⁶⁸:

[…]再立天元方面. 以自之, 又就分母四之得 為四池積. 以減頭位得 為四段如積,

寄左. 然後列真積, 又就分四之得二萬四千八百一十六步. 與左相消得 .

165 [Chemla Karine, 1982]. Description of equation in ch.7. The study of the steps of computation is based on the reading of problem V-12 of the Ceyuan haijing.

166 We send the reader back to the reading of [Chemla Karine, 1982], p. 8.6-7 for the verification and comparison with the Sunzi suanjing.

167 Li Ye also opposes the two operations in the paragraph we translated from the Jing Zhai gu jin tu. See supplements.

168 This operation is found also in pb. 12; 14; 15; 16; 17; 18; 19; 20; 22; 23; 25; 26; 27; 28; 29; 30; 43; 50; 52;

53; 54; 56; 57; 58; 61; 63; 64.

80 Here I translate: “Set up again the Celestial Source, the side of the square. This times itself and, by using of the denominator, quadrupling this yields 0

4

yuan as four areas of the pond.

Subtracting from what is on the top position yields 12288 twenty four thousand eight hundred sixteen bu. With what is on the left, eliminating them

from one another yields mathematical Chinese tradition, fractions are treated like a couple of numbers evolving in a peculiar way according to arithmetical operations and they have no special representations¹⁶⁹. In the Yigu yanduan, there are no peculiar settings for fractions. Only pb.43 shows some operations on fractions and Li Ye gives detailed explanations on how to operate on them. But the fractions are never presented in tabular settings. Li Ye operates on denominator and numerator separately, and quantities are conceived as a relation between

from one another yields mathematical Chinese tradition, fractions are treated like a couple of numbers evolving in a peculiar way according to arithmetical operations and they have no special representations¹⁶⁹. In the Yigu yanduan, there are no peculiar settings for fractions. Only pb.43 shows some operations on fractions and Li Ye gives detailed explanations on how to operate on them. But the fractions are never presented in tabular settings. Li Ye operates on denominator and numerator separately, and quantities are conceived as a relation between