圖 AS DIAGRAMS

Chemla Karine agrues that at the third century tu, 圖, were material objects, cut in paper with square-grid, and worked in specific ways²²⁰. Their areas were marked (not their points) by character or colors. Areas were cut into pieces and rearrangerd. But the meaning of the term, tu, changes before the 13^th century, and becomes an illustration inserted in books.

She distinguished several traditions regarding the nature of the tu in 13^th century, concerning Qin Jiushao and Yang Hui, for example. The question I want to tackle before describing the procedure of Section of Pieces [of Areas] is the definition of the term tu in the

217 [Martzloff Jean-Claude, 1987], p. 143.

218 This translation was reproduced in [Dauben Joseph, 2007], p. 328. This problem 8 is also presented in [Kong Guoping, 1987], [Kong Guoping, 1999] and [Kong Guoping, 2000].

219 [Lam Lay yong, 1983], p. 249.

220 [Chemla Karine, 2010]; [Chemla Karine, 2001]

98 Yigu yanduan. What are the objects referred as tu in the treatise? How does Li Ye refer to visual artifact in the Yigu yanduan? What should we identify as diagram?

The character tu, 圖,appears very few times in the Yigu yanduan, while the character shi, 式, is used more than twenty times in both procedures to name either the configuration of rods on the counting support or the figure in the Section of Pieces [of Areas]. The character tu appears only five times in the discourse by Li Ye. It appears in pb. 45 and 61 to recommend to draw carefully two of the diagrams of the Sections of Areas²²¹; in pb.22 to indicate that one of diagrams is the diagram for the old procedure and in pb.64, to mention that the problem contains exceptionally three diagrams. Each time this character is referring to visual artifacts inserted inside the text, but differentated from the discourse: diagrams.

Here tu refers to geometrical figure²²². But the fifth occurrence is notable.

The fifth occurrence of the term tu is particulary interesting in pb. 63. (Figure 5.

Pb.63):

Figure 5. pb.63

I translate this part as: “A diagram is provided on the left:

One diameter of the inside circle: ⁰

1 tai

One side of the small square: ⁶⁰

1 tai

One side of the big square: ¹¹⁰

1 tai

One diameter of the big circle: ¹⁶⁰

1 tai “

221 See part. IV. C.

222 In part IV.C I will discuss the status of these geometrical figures.

99 In this example, the four polynomials are presented separately in a list of two columns instead of being inserted inside a sentence. And this time, Li Ye names this configuration tu, 圖, “diagram”. Usually the character shi 式 is used to refer to the configuration.

For example (pb.11a): 以自之得下式 . “This times itself yields the following pattern […]”

In the available editions of Yigu yanduan and in Ceyuan haijing as Karine Chemla observed it²²³, the mathematical expressions are always written inside the space of the column containing the text, just like any part of a sentence. They are introduced by the character de, 得, “to yield” and after interpreted with the character wei, 為, “as”. They are not represented like independent drawings and many times Li Ye names this pictogram representation shi, 式, “pattern”, “configuration”, (see Figure 6 pb.11). There is continuity in the written text between the discourse and the configuration of numbers. Li Ye integrates the configuration to the written text as if it were a simple number, while the configuration itself extends the sentence as being inserted in the column. There are no such relations like picture/caption. Therefore configuration cannot be considered as an illustration²²⁴.

223 [Chemla Karine, 1996] She shows that there were different practices of representation, and that the transcription of tabular settings was not uniformed at the Sung-Yuan period. Li Ye distinguishes himself from contemporary mathematicians by the way he elaborates a transfer of the mathematical activity to the paper. That is, he develops a way to represent polynomials and equations proper to the written work and different from the ancient practices of manipulation of rods. The way of writing mathematical expressions by Li Ye was interpreted as a symbolisation of the object he is treating

She has shown that diverse texts manifest very differently positional notations from the Han to the Yuan dynasties. Such matrix arrays are continually used, but there are variations in their transcriptions. This variation shows an evolution toward autonomy of work on paper. Despite these variations, the organisation of the data is remarkably stable and the management of the operations on the support is lead by strict imperatives. This is a practice which testifies of a transition movement of the mathematical activity from the support to a paper based work and of development of symbolization.

224To understand the peculiarity of the Li Ye’s ways of writing polynomial, one can refer to other contemporary mathematicians. For example, Qin Jiushao seems closer to a pictorial configuration. On the contrary to Li Ye, Qin Jiushao inserts diverse state of the support as illustration. The discourse and its illustrations are discriminated, the discourse being sometimes a caption to the illustration. We also notice that Qin Jiushao refers to the tabular setting by using the character tu, 圖, “diagram” (Figure 7, Ch.2. p.21).

Figure 7, ch.2. p.21

100 Figure 6. pb.11

The configuration of these numbers introduced by Li Ye seemly transcribes some states of this support. But we cannot consider this configuration as a pure simple transcript of the different aspects or steps of manipulation of rods on the support. The configuration, as Li Ye presented it, is not a picture of the support. It is a step in putting down symbolization.

Li Ye makes clearly a distinction between configuration as visual artifact and configuration as part of the discourse. The character tu refers to the first one. I will show later how the reader is supposed to understand this object.