Order of problems (part 1)

Every historian who studied the Yigu yanduan noticed that the problems were classified according the geometrical shape proposed in the statement. But if this classification seems clear for the first twenty problems, the remaining part of the book contains dark areas. Here is the list of the type of geometrical figures following the order of problems:

Table 2: The geometrical shapes in the statement of problem Chapter 1 Pb.1-10 A circular pond in the centre of a square field

Pb.11-20 A square pond in the centre of a circular field Pb.21 3 squares fields of different size

Pb.22 A square field with a triangular pond at one corner Chapter 2 Pb.23-29 A square and a circular field next to each other

Pb.30 2 circulars fields

Pb.31 A rectangular field with a circular pond in the centre Pb.32-37 A circular field with a rectangular pond in the centre Pb.38 Two rectangular fields next to each other

Pb.39-42 A rectangular field with a circular pond in the centre Chapter 3 Pb.43 3 circular fields of different sizes with different value of π

Pb.44 A trapezoidal field

Pb.45 A square field with a square pond in the centre Pb.46 A square and a circular field next to each other

103 a: square field. b: square pond.

44 Pb.47 A rectangular field with a square pond in the centre

Pb.48 A square field with a rectangular pond in the centre Pb.49-52 A square field with a square pond in the centre Pb.53-54 A square field with a rectangular pond in the centre Pb.55-56 A circular field with a circular pond in the centre Pb.57-58 A circular field with a rectangular pond in the centre

Pb.59 A square field with a circular pond in the centre which has a square field in its centre

Pb.60 A circular field with a square pond in the centre which has a circular field in its centre

Pb.61 A square field with a circular pond at one corner Pb.62 A square field with a square pond at one corner

Pb.63 A big circular field, a big square field and a small square field with a circular pond in its centre

Pb.64 A square field with a concentric pond in the centre

At first sight, problems present statements and solutions independent from each other.

Although the chapter 2 and, more especially, the chapter 3 seem to present quite different problems, problems are ordered according to the statement of their geometrical shapes.

This appears clearly in the chapter 1: the problems 1 to 10 are concerning circular ponds inside square fields, while the problems 11 to 20 present the opposite: square ponds inside circular fields. In the chapter 3, while we have, for example, a square field with a circular pond in the centre which has a square field in its centre in pb.59, the following problem proposed an opposite figure: a circular field with a square pond in the centre which has a circular field in its centre. All problems concerning the same kind of geometrical shape are grouped together. The statements are regrouped in categories constructed according to geometrical shapes and then classified. But the more we pursue the reading of the Yigu yanduan, the less the order of problems is clear. It seems that when we arrive at the chapter 3, we have just a random list of all the possible figures, like diverse variations on the same topic: a field and a pond.

Inside the categorization of shapes of figures, I can identify another classification of problems. Each statement is composed of two data: an area and a distance. Each of the statements first situates the two figures one according the other, and then the area resulting of the difference between the areas of the two figures is given¹⁰⁴. After, one distance is given resulting from combination of operations on two segments, respectively one from the field and one from the pond. The two other segments are the ones asked by the problem and given as answers.

104 For few of the problems, there are three figures. In those cases, the sum of their areas if given, or the sum of two of area less the third one is given.

45 For example, in pb.12, one reads: “只云從外田楞通內方方面六十八步”, “One only says [the distance] from the outer edge (楞) of the field [going] through (通) the inside of the side of the square is sixty eight bu”. This distance is represented by a on the following diagram:

And the problems gives the side and the diameter as answers.

If one tries to express the distance given in the statement according to the two other distances required by the problem, the distance in the statement can be transcribed as the diameter of the outer circular field to which is added the side of the inner square pond and whose sum is divided by 2. We can transcribe this in modern terms as: (diameter + side)/2.

Here follows a list (Table 3) of the transcription of the distances given in each statement:

Table 3: The distance given in the statement Chapter 1 Pb.1 (Side – diameter)/2

Pb.2 (Side + diameter)/2 Pb.3 (Diagonal – diameter)/2 Pb.4 (Diagonal + diameter)/2 Pb.5 Perimeter - circumference Pb.6 Side = circumference Pb.7 Side - circumference Pb.8 Perimeter + circumference

Pb.9 Perimeter + circumference + (side – diameter)/2 Pb.10 Perimeter + circumference + (diagonal – diameter)/2 Pb.11a

Pb.11b

(Diameter – side)/2 Diameter- diagonal Pb.12 (Diameter + side)/2 Pb.13 (Diameter – diagonal)/2 Pb.14 (Diameter + diagonal)/2 Pb.15 Circumference - perimeter Pb.16 Circumference = perimeter Pb.17 Perimeter - diameter Pb.18 Circumference + perimeter

Pb.19 Circumference + perimeter + (diameter– side)/2

46 Pb.20 Circumference + perimeter + (diameter – diagonal)/2

Pb.21 Side of big square – side of middle square = side of middle square – side of small square

Pb.22 Diagonal – bisectrix of the triangular pond Chapter 2 Pb.23 Side - diameter

Chapter 3 Pb.43 Diameter of middle circle = Diameter of small circle + 9.

Diameter of big circle = diameter of middle circle + 9.

Pb.44 2 different segments of the same length.

Pb.45 Distance from one corner of the outer square to the opposite corner of the inner square

Pb.46 Diagonal + diameter Pb.47 (Length – diagonal)/2.

(Width – diagonal)/2

47

Pb.55 Circumference A + circumference B + (Diameter A/2 – diameter B) Pb.56 Diameter A/2 + diameter B

Pb.61 Diagonal – diameter – segment of diagonal Pb.62 Diagonal – side – segment of diagonal Pb.63 In small square: (Side – diameter)/2.

Side of small square + 50 = side of big square. that the statements are presented as a list of exploration of possibilities of construction of this distance, and that the same sequence of construction is observed for each of the categories of geometrical shape. If one considers the order of problems according to their statement, there is first a regrouping of problems according their geometrical shape; that is a grouping according to areas. And then the groups of problem are ordered according the construction of the second data, the distance. But in my table above, I transcribe this distance according to other datas involved in the procedure. This is means that it is the reading of the procedure which let appears this transcription and this order. I will show in a later chapter that the variations around this construction are justified by the procedure of the Section of Pieces of [Areas] and that is testify of a specific practice of ordering problems.

48 3.2 Diagram and statement

I will now turn to the question of practice of diagrams. Each of the statement is followed by a diagram. Some data of the statements are reported inside the diagram. This kind of diagram seems to illustrate and to summarize the data of the problem, generally naming the square field and the pond (if this is the case), indicating distances that are already known. Sometime the results that are expected are already written down. These notations are not systematic and vary from one problem to the other.

Looking at it closely, one notices that the majority of these diagrams contain only the data of one distance, and this distance is named according to the following system of abbreviation. Areas and segments are given in the same unit without differentiation, the bu, 步. There is nothing equivalent to our square units for areas. These values are always expressed in natural language in the statement. They are after referred through abbreviations. For example in the first problem 從外田楞至內池楞四邊二十步, “[the distance] from the edge of the outer field reaching the edge of the inside pond is twenty bu for each side” is reduced to 至步, “the reaching bu”. The abbreviation is reported as a caption in the diagram and is used to name the segment in question in the different procedures.

The distance which appears drawn or in a caption in the diagram is always the one which is involved in the construction of the first polynomial in the two procedures. I notice in [table 4] that another large quantity of diagrams contains no caption on data at all, but the distance given in statement is always drawn (see example pb.49 above). The diagrams containing answers are only giving one of the several answers that are asked. The answer which is given is the one that will be used to deduce the other answers. Curiously a small part of the diagrams contains data which are neither in the statement nor in the answer.

These concerns only perimeters or circumferences for which the side or diameter is given instead. Perimeter and circumference can be deduced from the side and the diameter given in the diagram. I can notice from these observations that the data given in the diagram in the statement are the one on the basis of which the other data are deduced, and these quantity are used to set up the algorithm.

Table 4. Type of data contained in diagram in the statement

problem total

One of the answer 1; 7; 9; 10; 11a; 13; 14; 19; 20; 61; 62 11

49 first polynomial. The diagram seems not only to illustrate and summarize the statement; it also seems to play another role. By representing other objects required by the procdedures, it seems to represent the first step of the algorithm. So the question is: are these first diagrams only illustrating data of wording or are they linked the procedures? If those diagrams are related to procedure, how are they linked with the second diagram presented in the Section of Pieces [of Areas] in a process of transformation? The study of the pb.21 given as an example in the chapter on the procedure of Section of Pieces [of Areas] ¹⁰⁵ will provide some clues on this practice.