DIAGRAMS AND EQUATIONS

In order to explain the rudiments of the procedure I chose to present a basic problem from which the procedure of the Celestial Source is removed²²⁵. This problem is the problem one and the procedure of Celestial Source of this problem is presented in part. III.A.1. Here one finds the translation of the second procedure for setting up equation, the procedure of Section of Pieces [of Areas].

Problem One.

Suppose there is one piece of square field, inside of which there is a circular pond full of water, while outside a land of thirteen mu seven fen and a half is counted. One does not record the diameter of the inside circle and the side of the outer square. One only says that

225 In the Yigu yanduan the two procedures are not autonomous. This point will be discuss later in Part. IV. D

101 [the distances] from the edge²²⁶ of the outer field reaching the edge of the inside pond [made] on the four sides are twenty bu each.

One asks how long the diameter of the inside circle and the sides of the outer square are.

The answer says: The side of the outer square field is sixty bu. The diameter of the inside pond is twenty bu.

[…]

One looks for this according to the Section of Pieces [of Areas]. From the genuine area (真積), one subtracts four pieces of the square of the reaching bu (四段至步冪) to make the dividend (實). Four times the reaching bu (四之至步) makes the joint (從). Two fen and a half is the constant divisor (常法).

227

The meaning says: From the genuine area, to subtract (內減) four pieces of the square of the reaching bu is to subtract (減去) four corners. [Taking] two fen and a half as the constant divisor, is that for each bu of the inside [part] full [of water], one [takes] off (却) seven fen and a half, outside there are two fen and a half.

Problem one, description:

First let’s remind of the statement of the problem in modern terms and then we will proceed to the explanation.

226 楞, leng.

227 j1-4: subtract. c1-4: joint. abcd: two fen five li.

102 Let a be the distance from the middle of the side of the square to the pond, 20bu; let A be the area of the square field (S) less the area of the circular pond (C), 13mu 17fen, or 3300bu;

and x be the diameter of the pond. One looks for x, the diameter.

If one transcribes into modern terms the first paragraph inaugurating the procedure Section of Pieces [of Areas], the interpretation that would naturally come to mind is the following: A is “the genuine area”, that is to say the area (the field less the pond) given originally in the statement and a is “the reaching bu”, that is to say the distance express in the unit of bu starting from the middle of the side of the square and reaching the diameter of the pond. A – 4a² is the “dividend”, that is to say the constant term. 4ax is the “joint divisor”, that is to say the term in x and 0.25x² is the “constant divisor”, that is to say the term in x square. And the equation would be: A – 4a² = 4ax + 0.25x².

But this is forgetting that equality and the unknown are never stated in the procedure. One only sees three separated elements I artificially schematize as the following:

A – 4a² Shi, “dividend”.

4a Cong, “joint”.

0.25 Chang fa, “constant divisor”

We also have to notice that in this paragraph the way to compute “dividend” and the

“joint” is given, but the “constant divisor” is directly given and, from this first sentence, one do not know where these coefficients come from. This sentence presents the coefficients of the equation as final results without justification. One the basis of this sentence only, it is difficult to understand how and why to transform the data given in the statement into equation. This example illustrates the difficulty of associating an equation, as we define it nowadays, to this procedure.

In the Chinese text, one in fact first reads the procedure leading to tabular settings, and the three coefficients are given always in the same way: the dividend, the joint, the constant divisor or corner. Those are names of fixed positions on the counting support related to the algorithm of division and of root extraction, as we saw in part III. The rank of the dividend is filled with the quantity which will be diminished or augmented until its exhaustion. Once the numbers are set on the support, one can proceed to “opening the

103 square”, that is to extract the square root. The expression kai ping fang, 開’平方, appears in the procedure of the Sections of Areas in pb.22; 62; 63.

The most interesting part of the procedure is that it links the situation described in the statement and the setting up on the support with a diagram. The discourse inserted inside and around the diagram gives explanations for the considered numbers. This caption incites to read the diagram as assemblage of pieces representing the terms of the equation.

As we will see now in the example, this assemblage conceptualized as “stacked areas”. That is the reader has to visualize a double reading of the diagram as sort of piled areas. The areas are not as pieces of puzzles which are put next to each other. This point also constitutes a key point which opposes the diagram in the Chinese tradition to the Euclidian one²²⁸. In the present case, the apex are never marked and taken into account. A diagram is considered here as a set of plain surfaces on which one has to operate.

To understand the diagram of the problem one, for example, one has to start with the data given in the statement and which are represented by Li Ye in the very first diagram illustrating the statement. One knows the distance a and the area A [figure.1.1], with this one can identify a square field in which there are squares of side a. So on the given area A, one starts with constructing four squares with the given distance a, what corresponds to 4a², and which is a constant [Figure.1.2]. The purpose is to express the known area in term of what is unknown. When these squares are removed, the area that remains can be read as an expression of the terms of the unknown. We have thus A – 4a², and the green cross-shaped area represents 4ax+ x² [figure. 1.3]. That is why Li Ye writes “Subtracting four pieces of the square of the reaching bu from the genuine area is to subtract four corners”. But this area does not correspond to the area given in the statement because the area of the circular pond still has to be removed. This area equals₄³x², that is to subtract an area of 0.75x², from the square in the center: x² - ₄³x². That is to remove one circle, and the area that remains corresponds to 0.25x² [in green, figure.1.4]. That is why Li Ye writes “Taking two fen and a half as the constant divisor, is that on one bu of the inside (part) full (of water), one removes seven fen and a half, outside there are two fen and a half”. Thus, one read the diagram as: A – 4a² = 4ax + 0.25x².

Figure 1.1

228 [Chemla Karine, 2001], p. 18. [Volkov Alexei, 2007].

104 Figure 1.2

Figure 1.3

Figure 1.4

What in shows with the pb.1 is valid also for the other problems. The diagram can be interpreted as a superposition of areas showing that a known area is equivalent to the same area expressed according to the unknown. That is to say, the diagram is the object of two readings; one translates the diagram into constant, the other into unknown. The diagram is in fact the figure taken by the quadratic equation. This double reading is the expression of the equality. The global area corresponds to the constant term, and to draw it is equivalent to write that this value is equal to the squares representing the term in x and x². To trace a diagram is to express the equality, and to see the terms of the equation. This is why Li Ye

105 writes in pb. 23, 25, 27 that the terms are “自見”, zi jian, literally “self visible” or better “self evident”²²⁹.

To read the diagram and the “meaning” accompagnying it indicates how the data of the problem can be transformed. To draw the diagram that is to trace the provenance of the terms of the equation. The computation of the different coefficients which are on the support is in fact followed in parallel with the construction of the diagram. The reader can trace back the provenance of constant divisor, which was given just as a final result on the description of the counting support. The role of the diagram is for this part heuristic, because the procedure is used to discover the equation which is verified by the unknown.

This research is made through transformations areas, reducing the known area given in the statement to an assemblage of pieces of areas which can be interpreted in terms of polynomials.

The diagram comments the provenance of the area which is used for computation (A- 4a²) and wich is placed as dividend, and the provenance of the joint (4ax) and the constant divisor (0.25x²). By legitimizing the provenance of the areas, it consequently confirms also the validity of the procedure. This reasoning appears also as a justification of the choice of the values (A- 4a²; 4ax; 0.25x²) for the settings on the counting support. A same diagram verifies the validity of a given procedure, which relates to the computation of coefficients presented on the counting support. The diagram is at the same time a interpretation, a rewording, of the data of the statement of the problem and a way to visualize the equation which is verified by the unknown. And it provides verification on how the data of the statement are transformed into an equation. So its value is also demonstrative. This means that the diagrams are used in a context of argumentation, they do not appear only as mere illustrations. They are a way to master the nature of the transformation of areas entering the reasoning.

The procedure Sections of Pieces [of Areas] has for function to reword the condition of the statement into terms of configuration which put into light the link between the known and unknown quantities of the problem. Then it proceeds to decomposing and gathering necessary for visualizing the fundamental identity proposed as resolution – identity between on one side an area numerically known and an assemblage of areas expressed with the unknown. Then the procedure of Section of Pieces [of Areas] establishes the link between a diagram and an arithmetical resolution. The solution of a problem is articulated around the relation between a diagram showing an identity and a tabular setting, whose modality of use is well known at this time. And this relation of correspondence is playing the role of equation.

The study of the relation between problems will give more elements concerning the demonstrative value of the diagram. But before reaching this point, I want to focus on two

229 Pb. 23: “求之自見隅從”, “the joint and the corner one looks for are self evident”.

106 important aspects and specificities of the diagrams in the Yigu yanduan. I mentioned that areas are marked instead of sides, and that operations are required and I mentioned geometrical demonstration of the extraction of the square root. I want first to give more details on these points in order to understand how to read diagrams.

The next example will illustrate more clearly how operations on diagrams provide the origin and justification of the construction of coefficient.

107 5.3 TRANSFORMATION OF DIAGRAMS.

Several comments written by Li Ye indicate that the diagrams are supposed to be reproduced by the reader. They are not merely object to be read, but also object to be drawn.

The diagram of the problem 45 is provided with the following comment: 若稍有偏側, 則不 能用也”But if [the drawing] is slightly distorted or leaning, then one cannot use it”.

Figure 8. pb.45

It seems that carfefulness is an important issue. Indeed, Li Ye comments the diagram of the pb 61: 此圖內二分合畫作極細形狀, “The two fen inside this diagram have to be drawn in an extremely thin shape”,

Figure 9. pb.61

I do not know why such a precision is required, and I do not understand Li Ye felt the necessity to write these sentences in these two problems²³⁰. Never the less one guesses that there is a practice of construction. Diagrams are supposed to be drawn, and their

230 Also Pb. 19: 今求方斜, 故其圖須細分之. “ Now, one looks for the diagonal of the square, therefore, this diagram requires thiny part”.

108 elements are supposed to be object of transformations. I mentioned previously that areas have to be “assembled”, “gathered”, “decomposed”. Some vocabulary in the “meaning”

reminds of manipulations: 貼, tie, “to past” (pb. 34²³¹) or 疊, die, “to stack”, “to pile up” (pb.

19; 20; 22; 24 ; 26; 52; 53; 58; 59²³²). But it is difficult to imagine that 64 figures are materialy manipulated, cut into pieces and piled together. It is a fortiori quite difficult to perform manipulations of “negative” areas. I thus wonder how the reader is supposed to draw and proceed at the same time to the decomposition and assemblage of pieces of areas.

The reading of pb.21 will provide some clues.

Problem twenty one.

Suppose there are three pieces of squares fields. [Added] together the area counts four thousand seven hundred seventy bu. One only says that the sides of the squares are mutually comparable²³³ and the sides of the three squares summed up together yields one hundred eight bu.

One asks how long the sides of the three squares each are.

[…]

234

[…]

One looks for this according to the section of pieces [of areas]. Place the quantity of the sum [of the area of the three fields]. What results once divided by three is the side of the middle square. Self [multiply] this to make the square. Triple this further and subtract this from the area to make the dividend. There is no joint. The constant divisor is two bu.

231 Pb. 34 : 八个從步內, 貼入八个斜至步冪. “Inside the eitght bu of the joint, one pasts eight squares of the bu of the reaching diagonal”

232 For example pb.20: 於從步上, 疊用了六百二十五个池徑冪. “On the bu of the joint, one stackes six hundred twenty five squares of the diameter of the pond”.

233 方方相較 : the difference between the side of the small square and the side of middle square equals the difference of the side of the big square and the side of the middle square.

234 A: big square. B: middle square. C: small square.

109

235

The meaning says: from the bu of the area, one subtracts three squares of the middle square.

Outside there are two squares. Therefore, it yields two bu, the constant divisor.

Problem twenty one, description.

Let a, b and c be the respective sides of the squares A, B, C. Their sum equal to 108 bu; let the sum of A + B + C = 4770bu; and c-b = b-a = x. One looks for x, the difference between the sides of the squares.

A+B+C – 3b² dividend

Ø joint

2 Constant divisor

235 j1-3: subtract. q: to go to. l: to come to. k: empty. f1, f2: square.

110 The equation that comes to mind is: A+B+C – 3b² = 2x²

First, one has to interpret the data of the problem. One has two data: an area equal to A+B+C and a distance equal to a+b+c. As c-b = b-a, on infers that

3 a b c

  b. One will start the procedures with expressing each of the aera according to b. See [Figure 21.1]

That is for B : B = b²

For A : b²(2bxx²). To make A, one removes from b² a gnomon made of two rectangles stacked on one square. These two rectangles translates what is unknown: their length is b, and their width is x. Or in other term b²  A 2bxx²

For C: b²2bxx². To make C, one add to b² a gnomon made of two rectangles, whose length is b and width is x. To this another square of side x is added at the corner to complete the area.

Therefore, each of the squares was expressed according to the constant and the unknown identified in the statement.

Second, to construct the constant term, one wants to remove 3 squares of side b. That is A+B+C - 3b². Li Ye writes “from the bu of the area, one subtracts three squares of the middle square”. On the diagram represented by Li Ye, two of the squares are marked by the character jian, 減. One starts with removing these two squares, one from B and one from C.

See [Figure 21.2]. The problem is now to remove the third square of side b. To remove this third square, that is in fact to remove A + 2bx -x². If one re-assemble the elements together, and recompose the diagram, one obtains the [Figure 21.3]. That is a square of side c, from which was removed b² once (this area is marked by “void”, kong, by Li Ye), on which is

“stacked” a square of side a in the middle, with a gnomon made of two rectangles of length b and width x, from the latter, a square of side x was one removed (See construction of A according to b). Once one removed this third square, it remains at two of the corners, two squares of side x. This is why Li Ye writes: “outside there are two squares”. See [Figure 21.4].

We have thus represented A+B+C = b² + (b² - 2bx +x²) + (b² + 2bx + x²) and transformed this into A+B+C – 3b² = 2x²

111 Figure 21.1

figure 21.2

Figure 21.3

112 Figure 21.4

The interesting point of this problem is that Li Ye chose to represent two steps of the procedure with two diagrams. But we have a total of three diagrams in this problem.

1) The diagram illustrating the statement

2) A diagram whose shape is identical to the first one and which operates a transition from the data of statement to the first step of the procedure. It shows the two areas which are removed and that the small square will be the object of next operation.

This small square is marked by the character qu, 去, “to go”.

3) A third diagram which shows the destination of the small square, marked by the character lai, 來, “to come”. The areas that were subtracted are represented by dotted lines. The third diagram is the result of imaginary manipulations: it vizualise the equation and justify the origin of its terms.

We can see here how three diagrams tell the story of an algorithm for setting up a equation. The diagram illustrating the statement is a part of it. If one gathers this evidence, with the fact that diagrams have to reproduced, one sees that the reader is supposed to draw the diagram and to imagine the manipulations which lead from the first diagram to the final one. There might be some “mental manipulations”. I define “mental” or “imaginary manipulation” as a way to visualize and follow transformations on a drawn geometrical figure. This shows that the textual part of the Section of Pieces [of Areas] testifies of a important non discursive practice. Reading the text is small part of the work of the reader;

the main part of his activity is a practice dealing with diagrams.

Here we also see that the work by Li Ye on diagrams shows an interesting continuity and

Here we also see that the work by Li Ye on diagrams shows an interesting continuity and

在文檔中 exploring the features of algebra in medieval China:the case of yigu yanduan (頁 101-164)