Generic procedure

The two examples above are different from each other. In this section I will try to give the different steps of the algorithm fitting to all (or at least to the largest majority) problems.

The purpose is to gather the common points of a maximum of problems¹⁴².

First one has to choose an unknown number and on the basis of the condition given in the statement, one has to find the equation that governs the unknown. The problems are solved after setting up an equation whose chosen quantity for unknown is the root (only one the root, always positive, is given), this by means of computation on polynomials. But the procedure to solve the equation is never given.

I chose to decompose the procedure into eight steps corresponding to the eight “ritual”

sentences composing the discourse of each of the problems. These sentences give rhetorically a list of operations leading the construction of mathematical expressions and imply manipulations performed on a counting support. These manipulations are never described, the reader is supposed to be acquainted with them. Among the eight sentences, three mains steps can be underlined:

(1) A first mathematical expression corresponding to the area of one of the figures named in the statement is computed.

(2) A second mathematical expression corresponding to the other figure is after computed.

The second expression is subtracted from the first one, or rarely added (pb.21; 23 to 30; 38;

43; 46; 63).

(3) The expression resulting from this operation is equal to the area given in the statement in term of constant. They are cancelled from each other to give the equation.

Here follow the “sub-steps” of the procedure accompanied with the Chinese expression starting each sentence:

1.a) 立天元一, li tian yuan yi, “set up one Celestial Source”. Chose the unknown and define it. Li Ye never discusses the choice of the unknown and the unknown will always be the root of the equation, and this will always be positive and unique.

1.b) Express one of the segments of the diagram in term of unknown and constant to obtain an expression of first degree. The constant is always the constant given in the

142 I do not think that the term “general” fits to this part. It evoks immediately the mathematical process of generalization, therefore I use a less connoted term, “generic”, as a synonym of “common”.

67 statement expressed through abbreviation. It is either added or subtracted from the unknown and sometime multiplied.

1.c) 自之, zi zhi, “this by itself”;自增乘, zi zeng cheng, “to augment by self-multiplying”

(pb. 1; 2; 3; 5; 7; 8; 17; 18; 19; 20; 22; 24; 25; 29), 自乘, zi cheng, “multiply by itself”

(pb. 26; 31; 32; 45; 47)¹⁴³. Square this expression to obtain another expression of second degree which translates an area composing the surface named in the statement. This first polynomial is placed on the top of the counting support.

2.a) 再立天元, zai li tianyuan, “set up again the Celestial Source”; or 再置天元, zai zhi tianyuan (pb. 38; 39; 46; 48; 49; 50; 52; 53; 54; 46); 又立天元, you li tian yuan (pb. 23 ; 25; 34; 43); 又置天元 , you zhi tianyuan (pb. 40; 47; 51); 又以天元, you yi tian yuan (pb. 2; 3); 用天元, yong tianyuan (pb. 55); 次立天元 , ce li tianyuan (pb. 59),. (The expression is different from 1.a, there are no character 一). Use the unknown again.

2.b) Use the unknown and the constant, if needed, to compute the expression of the other area composing the surface of the diagram. This is the second polynomial.

2.c) 減頭位, jian tou wei, “subtract from the top position”; 減田積, jian tian ji (pb. 3; 4), 內減, nei jian (pb. 32 ; 33), 相減, xiang jian (pb. 6), 加入, jia ru (pb. 21 ; 38), 併入頭位, bing ru tou wei (pb. 23; 24; 26; 27; 28; 29; 30; 46), 添入頭位, tian ru tou wei (pb.23), 併 下三位, bing xia san wei (pb. 63). Subtract or add the second (or other) polynomial to the first to obtain an expression in term of unknown equal to the area or to several times the area given in term of constant in the statement. This third polynomial, derived from the two first ones, is placed on the left of the counting support.

3.a) 然後列真積, ranrou lie zhen ji, “after, place the genuine area”. Place the quantity corresponding to the constant given in the statement on the counting support and make it equal to the third polynomial.

3.b) 與左相消, yu zuo xiangxiao, “with what is on the left, eliminate from one another”.

The expression of the area in constant term and the expression of the same area in term of unknown (the third polynomial) area eliminated from each other. This is the equation. The different terms of the equation can be negative or positive depending on the way of performing the subtraction. Either the constant term is subtracted from the polynomial; either the polynomial is subtracted from the constant term. It seems the choice of doing one way or the other is random, while this is not the case concerning

143 I indicate first the most common expression used in the Yigu yanduan, then add the occurrences of other expressions with the number of problems. If the number of one of the problem is not in the list, it means that the common expression is applied.

68 the Sections of Areas. Li Rui insists on this point in his commentary of the problem 1, noticing that the different ways lead to different signs, and as long as the signs are correct, the subtraction can be performed in one direction or the other: “If, according to the method “eliminating form one another”, one subtracts the quantity which is sent to the left from quantity that follows, then, in that case, one obtains a positive dividend, a negative joint and a negative corner. If one subtracts the quantity that follows from the quantity which is sent to the left, then the positive or negative are exchanged in comparison with this [above]. What one obtains is a negative dividend, a positive joint, and a positive corner” (pb 1¹⁴⁴).

144 According to Li Rui, this peculiarity is the main difference between the procedure of the Celestial Source and the procedure of borrowing the root. And this point constitutes the crucial point of disagreement between Li Rui and the editor of the Siku quanshu. I will not treat this material in the present study.

69 4.2 MANIPULATIONS ON COUNTING SUPPORT.

Li Ye does not describe the algorithms of operations with polynomials. There is no description on how to perform addition, subtraction, multiplication, division and extraction of root with counting rods in the Yigu yanduan, whether it is operations concerning constants, polynomials or constant with polynomials. The tabular settings represented in the discourse shows the configuration of counting rods only at the step of the result of the algorithm. The list of manipulation leading to this result is not represented, neither described. The reader is supposed to be familiar with these basic operations. But from the discourse and the tabular setting, we can gather some clues concerning those algorithms.

Other earlier mathematical treatises will also provide complement of information. The description of algorithms will help to understand the concept of equation and later it will help to understand the practices of the procedure of section of area.