Treatment of polynomials

Before describing the modification made by editors to polynomials and equation, I will first briefly give examples of how were presented these objects.

In the 13^th century China, an equation is presented as a tabular setting of numerical coefficients presented vertically. This kind of representation is not new. In the Chinese antiquity, one also sets up the coefficient of linear equation in vertical columns⁸⁴. Each line was corresponding to the coefficients attached to the same unknown and the line below contained the constant term. The semantic of this tabular array is immanent and markers of position are not required. The development of a symbolism to determine the signification of respective positions on the support in the 13^th century is a rupture with the antiquity⁸⁵. The use of the characters 太, Tai, or 元, Yuan, on the side of the array defines the signification of the other numbers relatively to the position marked.

Here, follow several examples of polynomial as they can be seen the Zhibuzu zhai cong shu. The following tabular setting has the character tai on the upper rank to mark the

82 See translation of pb.38 in supplement.

83 See translations of pb. 38, 53, 54, 56, 57, 64. These problems are all provided with commentaries and addendum by the editor of the Siku quanshu to understand the procedure of section of area

84 [Chemla Karine, 1996]

85 [Breard Andrea, 2000]

36 constant term. is read in modern terms as: 2700 + 252x + 5.87x². The fhas the

character yuan for the term in x. is read in modern terms as: 0x + 4x².

As it is a place value system, a same sign set at two different positions receives two

different meanings. For example, is read 14 + 1x and is read 3780 + 228x + 1x². In this example, the number 1 at the lower row indicates two different powers of x⁸⁶.

In Yigu yanduan, polynomials and equations appear represented by a configuration of numbers whose signification is given by their position (wei, 位) on a matrix like array. The term wei is used in different ways to refer to positions where a whole polynomial stands (頭 位, tou wei, top position) or to name one of its row (上位, shang wei, 下位, xia wei, 中位, zhong wei) inside of column where the polynomial is written. The meaning is provided by the position inside a column. The column indicates the polynomial or equation, and the row, the unknown or indeterminate. The role of the positions is to express the power of the unknown or of the indeterminate.

In fact two place-value systems are in use: a first one to write numbers⁸⁷, and a second one to write polynomials and equations. On the two or three rows containing the coefficient of the polynomials, units are in the column of units and tens in the column of tens. Li Rui describes this system in its commentary of problem one: “On three ranks, upper, middle and lower, one strictly sets up and places the bu of each rank at the positions corresponding to each other. On the left of the bu are the tens, the hundreds and the thousands and the ten thousands. On the right of the bu are the tenths, the hundredths, the thousandths and the ten thousandths. When, under the [mathematical] expression⁸⁸, one marks down the character bu, then the positions marked down it that of the bu. The positions in the rank above and below corresponding to this character bu are also bu. In the case [the character bu] is not written, hence, the final position on the right side is the bu. If on each rank, the last positions are not exactly right one above the other, then the last position of the left side of first the rank is the [position of the] bu. The positions one above

86 A more detailed description is given in part III. B.1. The difference between polynomials and equations is given in part. III. C.

87 See Part III. B.1 Writing numbers.

88 式, shi.

37 the other on each rank corresponding to the last position are also the position of the bu⁸⁹“.

This way of setting up quantities is clearly testified by the examples above extracted from Li Rui’s edition.

But in the different available editions of the Yigu yanduan, the two Siku quanshu and the one by Li Rui, one notices also some differences in writing of polynomials and equations.

In the entire edition of the text in the two Siku quanshu, polynomials are never aligned, while it is the case in the edition by Li Rui. The editor of the Siku quanshu did not pay much attention to placing the numbers at a very strict position, that is: the units, tens, etc. in a same column. In the Siku Quanshu, not only the mathematical expressions are not aligned, but sometimes they are cut and written in two columns (see Figure1, siku), while this never happens in Li Rui edition

Figure 1. siku

Another difference is noticeable between the two editions. In the edition by Li Rui of the Yigu yanduan, negative coefficients are distinguished from the positive ones by a

diagonal stroke, like in the following example: (we read: 53325 – 465x – 0.47x²). The same notation for negative can be seen in the edition by Li Rui of the Ceyuan haijing. This observation led historians to the following conclusion: “Such a symbol is met with in both

89Pb.1: 上中下三層從戴而列每層步位皆上下相當步之左為十百千萬步,之右為分釐豪絲.式下注有步字者

便以所注之位為步. 其上下層與此步字相當之位亦為步也.其不注者則以右方尾位為步.若上下層尾位不正 相當則以偏在左方一層之尾位為步.其上下層與此尾位相當之位亦為步也. Concerning the character bu, see part III.1

38 works of Li Yeh. If we are not sure whether he was the first Chinese who used the symbol, yet his works are perhaps the oldest writings wherein it was made use of, that are transmitted to our time⁹⁰”. Mikami Yoshio, John Hoe, Lam Lay Yong. Ho Peng Yoke and Mei Rongzhao all agree on this point, and according to them, Li Ye himself “indicated negative quantities by drawing an oblique line over the final digit of the number concerned⁹¹”.

But surprisingly, in the two editions of Siku quanshu of the Yigu yanduan I could consult, there is no trace of such notation, while, on the contrary, there are diagonal strokes for the same editions of the Ceyuan haijing, and in the Ming edition of the same text too.

One wonders why there is such disparity in the editions of Li Ye’s works in the Siku quanshu.

Li Rui himself added a commentary in the problem.1 concerning negative and positive: “In the mathematical expressions of the original edition (元本算式), positive or negative are not differentiated. According to Shen Gua, in the Dream Pool Essays⁹², “in arithmetic, one uses red and black rod sticks to differentiate the negative quantity from the positive one⁹³”. And again, in the Mathematical Treatise in Nine Sections⁹⁴ by Qin Jiushao, in the diagram of the extraction of the root in the fourth roll, “the negative expressions are drawn in black, while positive expressions are drawn in vermilion⁹⁵”, both are conform to the explanations by Liu Hui in the Nine Chapters on the Mathematical Art⁹⁶ who says that “red is for the positive expressions, while black is for the negative expressions⁹⁷”. According to this, one knows that, at this period, mathematical expressions were probably be drawn in red or black in order to differentiate them. But the copyists altered this [notation]. Now, following the example given by The Sea Mirror of the Circle Measurements, for every negative expression, one draws an oblique stroke to record it, so that all the positions of the expressions (算位) are easy to differentiate⁹⁸”. Indeed the different editions I consulted showed that Li Rui added himself a sign for negative quantities in the zhibu zhai congshu edition of the Yigu yanduan, and this sign was not in the editions he is collating. But this sign was visible in the materials he is using for his edition of the Ceyuan haijing.

It is impossible to know why there was such difference between the different editions of the works of Li Ye. Either the positive and negative were written in different colors, as Li Rui suggests it, and due to technical or economical constraints of edition, the red zhang, which is the title as it appears in the Yongle Dadian. The title Shu shu jiu zhang is found in the Siku quanshu, which title might be a reference to an older edition used by the editor, Dai Zhen.

95 in 欽定四庫全書, 數書九章, 卷 4, 27.

39 color was not used any more, and/or the material used by Li Rui for the editions of Ceyuan haijing and Yigu yanduan are issued from different older editions which were not related to each other. But obviously, at least we can conclude that Li Rui does not see any color in the material he is using for his edition.

Some interpret this disparity in writing signs of quantities in China as a disparity between the northern and southern Song⁹⁹-which besides proves the lack of exchanges between the north and the south of China. Mathematicians of the south would use red colours for positive numbers and black for negative, while mathematicians of the north were using a diagonal stroke. Following this hypothesis, it is difficult to imagine that Li Ye used two different notations, including one from southern China. Although there are high chances that colors were used, we have to admit that do not know how signs were recorded in the original document produced by Li Ye, whether is it the Yigu yanduan or the Ceyuan haijing.

Indeed one also notices also many mistakes concerning numbers in tabular settings in the edition of the Siku quanshu: some numbers 1 are missing (pb. 10, 11, 24, 46) or digits are mistaken (pb.27: 1 instead of 2, pb.30: 21 instead of 43), some zeros are missing (pb.3 twice, pb.6 twice, pb.22 and pb.48 twice), or the characters 步, bu, 太, tai and 元, yuan are sometime missing in polynomials (pb.14, 53, 56, 59, 60, 61) or tai is written instead of yuan in one of the polynomial of pb.23. Sometime the character bu is written directly in the sentence following the polynomial as if it was not part of the polynomial.

Li Rui insisted on the importance of these characters. Li Rui described the way Li Ye writes polynomial in his commentary of the problem one: “For all the mathematical expressions (算式, suan shi), the genuine area (真積, zhen ji) is named tai ji (the Great ultimate, 太極¹⁰⁰), then on its side one writes down the character tai (太). The empty quantity (虛數, xu shu) is named tian yuan (Celestial Source), and on its side one writes down the character yuan (元). One rank under the rank of tai, is the rank of yuan, and one rank under the rank of yuan is the rank of the square, which is self-multiplied. If the character tai is

99 [Chemla Karine, 1982] mentions previous studies on this topic 4.4-5

100 Common English translations of the cosmological Taiji are the "Supreme Ultimate" by [Le Blanc Charles, 1985] and [Zhang Dainian, Ryden Edmund, 2002] or "Great Ultimate" by [Chen Ellen, 1989], [Robinet Isabelle, 2008]; but other versions are "Great Absolute", or "Supreme Polarity" by [Adler Joseph, 1999]. More ancient translation are “Extreme limit”, “great extreme” according to [Mikami Yoshio, 1913] or "Supreme Pole"

[Needham Joseph, Ronan Colin, 1978].

Here I follow Isabelle Robinet explanation. Taiji is understood to be the highest conceivable principle, that from which existence flows. This is very similar to the Taoist idea "reversal is the movement of the Dao". The

"supreme ultimate" creates yang and yin: movement generates yang; when its activity reaches its limit, it

becomes tranquil. Through tranquility the supreme ultimate generates yin. When tranquility has reached its limit, there is a return to movement. Movement and tranquility, in alternation, become each the source of the other.

The distinction between the yin and yang is determined and the two forms (that is, the yin and yang) stand revealed. By the transformations of the yang and the union of the yin, the 5 elements (Qi) of water, fire, wood, metal and earth are produced. These 5 Qi become diffused, which creates harmony. Once there is harmony the 4 seasons can occur. Yin and yang produced all things, and these in their turn produce and reproduce, this makes these processes never ending.

40 written down, the character yuan is not written, and if the character yuan is written down, the character tai is not written”¹⁰¹.

Indeed, as Li Rui mentioned it, Li Ye added character tai, 太, “extreme”, on the right on the upper rank to indicate the constant term. Sometimes the character yuan, 元,

“source”, is written instead to indicate the coefficient of the first power of the unknown.

Those characters are carefully written in Li Rui’s edition, while the editions of the Siku quanshu are no so meticulous. Like it was the case for the writing of negative and positives quantities, this demonstrates once again that the commentaries by Li Rui concerning the writing of mathematical expressions have to be interpreted as corrections he is adding and not as descriptions of what he is seeing in his source materials. The edition by Li Rui can be interpreted as an attempt to reconstruct how the polynomials are supposed to be like according to him. And this shows the care of Li Rui in writing mathematical expressions and that the tabular settings we are seeing in the edition of the Zhibuzu zhai congshu are in fact partial construction made by Li Rui.

These corrections made by Li Rui have for consequences that the polynomials look clearer according to the criteria of a modern reader. They are presented in quite clear arrays, negative are clearly different from positive coefficients and it is easy to differentiate decimal quantities. It gives a better visibility of the polynomials, and therefore an easier access to reading for us. These corrections added to a virulent debate comparing the two procedures, Borrowing the root and Celestial Source, put the light on this first procedure. The procedure of the Celestial Source was the object attention of the commentators, especially of Li Rui.

This led to think that the Yigu yanduan has for main topic the procedure of the Celestial Source. And, as the practice of reproducing diagram is non discursive, as commentaries or editorial notes to diagrams are rare, and as the geometrical procedure is not a problem for

101Pb.1: “太即真數.此即四十步併一池徑.銳案:凡算式真積曰太極, 旁記太字.虛數曰天元,旁記元字.太之下

一層為元.元之下一層為元自乘冪.記太字則不記元字.記元字則不記太字. 其太元俱不記者,則以上方一層 為太也”.

But the terminology used by Li Rui in this commentary quoted above deserves few remarks, because one observed some differences between the vocabulary used by Li Ye and the one used by Li Rui. First, the expression “suan shi” (算式) is never used by Li Ye. This term appear only in the commentary by Li Rui. Li Ye used only the term shi (式) to name the tabular setting. The term zhen ji, “genuine area” (真積), appear sin almost every problems of the Yigu yanduan, but it does not seem to name the same object in the commentary by Li Rui and in the text by Li Ye. Li Ye uses it to name the area given in the statement of the problem, expressed with a constant, and which will be cancelled from the polynomial expressing the same area in term of unknown at the end of the procedure. While the commentators, Li Rui and also the editor of the Siku quanshu, use this term to name any constant term in any of the polynomials, which are set up at the first row of the column of the tabular setting. Li Ye opposes this “genuine area” to an “empty area” (xu ji. 虛積) . The expression “empty area”

appears only twice in the procedure of the Celestial Source, that is in pb.1 and 2, to name the area expressed in term of unknown and equivalent to the area given in constant term in the statement of the problem. In the procedure of Celestial Source, Li Ye keeps the term of zhen and xu only to qualify the last polynomial, while Li Rui uses the term xu, “empty”, to name any coefficients (數, shu) of the first or second power of the unknown in every of the polynomials. Li Ye is talking about specific expressions of areas, while Li Rui applied the same character to coefficients.

41 the commentators, the procedure of Section of Pieces [of Areas] was discarded. The part IV of this study will show that this procedure was the emerging part of an iceberg.

I want to end this chapter on an opening remark. This remark will be usefull to understand the differentiation between polynomials and equations in part III. We notice that in none of editions consulted, the two characters are never written down together, and that those two characters are never written in the last tabular setting, wether it is in the Siku quanshu or in Li Rui’s edition.

42