From the extraction of square root to the concept of equation

We saw previously that the names of position for the division are: quotient, 商, on the first row, dividend,實, on the second, and divisor, 法, on the last row. These are the names associated with the position on the counting support.

商 實 法

And those names were precisely attributed by Li Ye to the different terms of a linear equation in the procedure of Celestial Source. In pb. 38; 44; 48; 56; 59; 60; one reads 下法 上實, xia fa shang shi, “the divisor is below and the dividend is above”. Those names are also systematically used in the procedure of Section of Pieces [of Areas]¹⁷⁷. Now, to justify the appellation of shi and fa in equations and why one reads the expression “to divide by extraction of the square root” in pb.23, we will give an account of the procedure of root extraction.

Sunzi explains the method of extracting the square root with two examples (Ch. 2, pb.19-20). We will follow the first example of Ch2, pb.19 and decompose the procedure into 17 steps following a syntaxical decoupage of the text. Then I will remind of the main points of comparison with the algorithm of division which were already observed by historians¹⁷⁸. The algorithm develops simultaneously a process of computation and a description of the setting up of its realisation on the counting support. There is an immediate parallel with the algorithm of division. The process consists in reducing progressively the dividend while one computes number by number the quotient¹⁷⁹.

176 Sunzi suanjing, pp.63-65 and p. 194, see [Lam Lay-Yong, Ang Tian Se, 2004], p.79-91. Xiahou Yang suanjing, see [Qian Baocong, 錢寶琮, 1963], p.558. Nine Chapters, ch.1 [Chemla Karine, Guo Shuchun, 2004], pp.131-134 .

177 See Part. IV. A.

178 For other explanations on the extraction of square or cube root see [Li Yan, Du Shiran, 1987], pp. 118-121.

[Chemla Karine, Guo Shuchun, 2004], p.322-330, [Lam Lay Yong, Ang Tian Se, 2004], ch.4.

179Zhang Qiujian uses the same setting for the computation but a different terminology for the division: fang fa (方法) and lian fa (廉法). The expression lian fa is frequently used in the old procedure in the Yigu yanduan.

85 The Ch2 (中卷), pb.19 of the Sunzi suanjing states¹⁸⁰:

今有積，二十三萬四千五百六十七步。

問：為方幾何？

答曰：四百八十四步九百六十八分步之三百一十一。

Suppose there is an area of two hundred thirty four thousand five hundred sixty seven bu.

(234,567 bu)

Problem: how much makes the side of the square.

Answer: four hundred eighty four bu, three hundred eleven parts out nine hundred sixty eight bu. ( 311

484968bu)

術曰:置積二十三萬四千五百六十七步，為實，

The procedure says: One puts down the area two hundred thirty four thousand five hundred sixty seven bu, as dividend (shi).

2 3 4 5 6 7

shi (step 1)

次借一算為下法，步之超一位至百而止。

Next one borrows one rod as lower divisor (xia fa). [From the place of the] bu (i.e. units) one over passes one place (chao yi wei) to reach the hundred¹⁸¹ and stop”.

2 3 4 5 6 7

180 Text from: http://ctext.org/sunzi-suan-jing/zh. I removed the editors’ commentaries. The literal translation is mine. The representation of the counting support is a hypothetical reconstruction by Lam Lay Yong in [Lam Lay Yong, Ang Tian Se, 2004].

We have to keep in mind that we do not know how these materials were like.

181 In this last sentence, the word “hundred” was replaced by “ten thousand” by Lam Lay Yong. She justifies this correction in [Lam Lay yong, Ang Tian Se, 2004] p.95. She justifies this correction with the reading of steps 7 and 12. The reader is asked to move this rod back towards the right two places a time so that ultimately it is once again below the unit of the shi. In her explanation of the algorithm of root extraction in the Nine Chapters, Karine Chemla shows that the borrowed rod (jie fa) is first place in the position of the units. This rod is after towards the left, from 10² to 10², until it reaches the farest position under the dividend. That is 10²ⁿ if the first number of the root is 10ⁿ. From these n “jumps”, one deduces the first number of the root, named quotient.

[Chemla Karine, Guo Shuchun, 2004], p. 326. Athough the algoritms 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 correction to text requires discussion. [Chemla Karine, 1994], p. 17 in her comparison of the algorithm of root extrantion from the Zhang Qiujian suanjing and the one by Kushyar ibn Labban, explains the role of the

“borrowed rod”. She mentions that this rod has different roles in Chinese algorithms of root extraction.

86 上商置四百于實之上，

One puts down four hundred [as] quotient (shang) above the dividend, 4

After, one puts down forty thousand below the dividend (shi), and above the lower divisor (xia fa), and one calls it square divisor (fang fa).

4

[Once] the removal is completed, one doubles the square divisor (fang fa).

182According to Lam Lay Yong, the determination of the digit 4 for the hundreds of the root is through trial and error. It is the largest possible digit which leaves a non-negative numeral for the dividend. She formulates the same hypothesis for steps 8 and 13. But this cannot be confirmed.

183Having obtained the digit for the hundreds of the root, the same digit is also placed in the row immediately below the shi and the same column as that of the single rod of the xia fa. It represents 40,000 for the fang fa.

184 In her lexicon to the Nine Chapters, Karine Chemla translates ming as “to name”. This terms applies when the operation of the root extraction is not finished, after one has just determined the number of the unit of the root: the result is produced by “naming” the number whose root is suched for. [Chemla Karine, Guo Shuchun, 2004], p. 963. See also [Li Jimin, 1990], p. 150.

185Sunzi uses the quotient to multiply the square divisor. The place value of the product corresponds with the digit of the square divisor so that the product, 16, is subtracted from the 23 of the dividend. The subtraction of these two numbers leaves 7 in place of 23, so the dividend is now 74,567.

87

One shifts (tui) the square divisor (fang fa) [to the right] by one [place] and one shifts again the lower divisor (xia fa).

After, one puts down quotient (shang) in the upper [position], eighty, next to the previous quotient (shang).

One also puts down eight hundred below the square divisor (fang fa) and above the lower divisor (xia fa) and one calls it edge divisor (lian fa).

4 8

186The single rod of the low divisor (xia fa) in the hundreds’place indicates the determination of the tens for the root.

187Having obtained the number for the tens of the root, the same number is placed in the row immediately below the fang fa and in the same column as that of the single rod of the xia fa.

88 方廉各命上商八十以除實，

One [will multiply] each of the square (fang) and edge (lian) [divisors], one names (ming) eighty the quotient (shang) in the upper [position], one removes this from the dividend (shi).

4 8

One shifts the square divisor (fang fa) [to the right] by one [place] and one shifts again the lower diviser (xia fa).

188The digit of the tens of the shang multiplies the value in the fang fa. The product, 64, is subtracted from the 74 of the shi. The subtraction of the two numbers leaves 10 in place of 74, so the shi is now 10,567. Next, the tens of the shang multiplies the digit of the lian fa. The product, 64, is subtracted from the 105 of the shi. The subtraction of the two numbers leaves 41 in place of 105, so the shi is now 4,167.

89

After, one puts down the quotient (shang) in the upper [position], four, next to the previous one.

One also puts down four below the square divisor (fang fa) and above the lower divisor (xia fa) and calls it corner divisor (yu fa).

4 8 4

One [will multiply] each of the square (fang), side (lian) and corner (yu), one names four the quotient (shang) in the upper [position], and one removes from the dividend (shi).

4 8 4

189The single rod of the xia fa in the units’place indicates that determination of the units’digit for the root.

90

The quotient (shang) in the upper [position] is four hundred eighty four, the divisor (fa) in the lower [position] yields nine hundred sixty eight, the remainder (bu jin, lit. “not exhausted”) is three hundred eleven. It makes that the side of the square is four hundred eighty four bu, three hundred eleven parts out nine hundred sixty eight bu. ( 311

484968bu).

190The digit for the units of the shang multiplies the digits of the fang fa, which includes the lian fa. Following the method of multiplication, 4 first multiplies 9 to give 36 and this is subtracted from 41 above to leave 567 for the shi. Next 4 multiplies 6 to give 24 and this is subtracted from the 56 above to leave 327 for the shi. The digit for the units of the shang also multiplies the digit of the yu fa. The product, 16, is subtracted from the 27 of the shi. The subtraction of the two numbers leaves 11 in place of 27, so the shi is now 311.

191This step is alike steps 6 and 11.

91 This procedure has remarkable similarity with the procedures for division. The parallesism appear with the following observations: The dividend is called shi and the number whose root is extracted is called shi. This is the first number to be put on the support, and its digits set the values of the places for the other digits on the support. The divisor termed fa is moved from right to left such that its first digit from the left is placed below the first or second digit of the dividend. The quantity added at the place of fang fa, is used like a divisor, and the quantity placed at the rank of shi is treated like a dividend. The operation brings back the extraction of the root to a division.

The algorithm of root extraction in the Nine Chapters, follows the same principles but the setting is slightly different¹⁹². There is an evolution of the algorithm in which appears a place value notation for the equation associated to these extractions¹⁹³. Karine Chemla Karine gives the following interpretation concerning the parallel between the division and the extraction of square root: the fact according to which terms like “dividend”

and “divisor” names the positions corresponding to one of the successive steps of computation has two functions. First, it allows reproducing the same list of operation in a iterative way. Secondly, it allows modelling the extraction of the square root on the model of division¹⁹⁴. Not only the same name of position is used, but also the way of using the position during the succession of operation is done in a same way. Names and management of positions are key points for the correlation of the two procedures. In other publication, Karine Chemla¹⁹⁵ shows that there is a work which consists in exploration of relations between operations of root extraction and division, and its expression is a revision of the different ways of computing and naming positions. She concludes that the work which she analyses in the Nine Chapters is in fact perpetuated at the 13^th century.

The elaboration of the procedure of division lead to a general procedure which mechanically extract the square root of a number. This method is not only used as an algorithm; it also provides the basis for the development of further procedure solving quadratic equations. The configuration necessary for conveying the meaning of these equations is inextricably expressed in the positions occupied by the rod numerals on the support. This justify the use of the same terms of divisor to name position on the support for division, extracting the root and finally the term of the equation as they are placed on the same emplacement. Once the equation is set up on the support, one has just to apply a procedure like the one described above to solve it. The development of the algorithm of root extraction leads the concept and solution of equation.

Just as the development of the square root method was based on the knowledge of division, the concept of polynomial equation is derived from the algorithm of the extraction of the square root. The conception of the equation is algorithmic: it is an operation with two

192 [Chemla Karine, 1983]. p. 7.7. [Chemla Karine, Guo Shuchun, 2004], p. 324-26.

193 [Chemla Karine, 1994a]

194 [Chemla Karine, Guo Shuchun, 2004], p. 327.

195 [Chemla Karine, 1993]

92 different terms, a dividend and one or two divisors, which is solved by the algorithm of the extraction of the square root. What we in fact identify as equations in the representation of tabular settings is an opposition between a dividend (constant term) and other coefficients.

This peculiarity of the conception of equation is due to essential role played by the counting support and the way to articulate the different algorithms (division, etc.) together.

I noticed that in all the available editions of the Yigu yanduan, the last mathematical expression never contains any character tai, 太, or yuan, 元. That is to say, that the

“unknown”, which is represented in the polynomials, is absent from the equation. The reason is that once the procedure of setting up equation is over, the last expression obtained will not be the object of further operations¹⁹⁶. The number of ranks is the only pertinent information for extracting the positive root. Once one obtained the equation, one can forget the marks on the right. There is thus a distinction between mathematical expressions for polynomials and mathematical expressions for equations. The configuration can design either an equation or a polynomial, but by adding the characterTai or Yuan to the column, Li Ye makes precisely what we name “polynomial” and the absence of sign is precisely the mark of the equation¹⁹⁷. On this point, I disagree with Lam Lay Yong, “These columnar arrays of numbers do not differentiate between mere algebraic expressions and equations¹⁹⁸”, and Li Yan and Du Shi-ran “the various configurations can be regarded as either equations or polynomials¹⁹⁹”.

The distinction between a certain concept of polynomial and a concept of equation is in fact an important feature of the Yigu yanduan. The investigation on what is an equation is precisely the central purpose of the Section of Pieces [of Areas], as one will see now.

196 [Karine Chemla, 1983], note (a) in chapter 8.3

197 Li Rui ends one of his commentaries to pb.1 the following way: “In the case of neither of the [characters]

tai and yuan are written down, then one takes the upper rank as the rank of tai” (其太元俱不記者,則以上方一 層為太也). Altough Li Rui did not write any Tai or Yuan to the last mathematical expressions, it seems from this commentary that he considers the last expression (i.e. the equation) to be read like the other ones (i.e. the polynomial). Another commentary discussing the comparison between the procedure of Celestial Source and the procedure of Borrowing the Root in pb. 1 suggests that Li Rui does not see any equations in the Yigu yanduan. See translation of pb.1 in supplement.

198 [Lam Lay yong, 1984]. p.245.

199 [Li Yan, Du shi-ran, 1987]. p.138.

93