Geometrical Figures and Generality in Ancient China and Beyond: Liu Hui and Zhao Shuang, Plato and Thabit ibn Qurra

Abstract

ArgumentThis paper argues that there was a shift in China in the nature, and use, of geometrical figures between around the beginning of the Common Era and the third century. Moreover, I suggest that the emphasis mathematicians in ancient China placed on generality as a guiding theoretical value may account for this shift. To make this point, I first give a new interpretation of a text often discussed, which is part of the opening section of The Gnomon of the Zhou (first century B.C.E. or C.E.). This interpretation suggests that the argument presented in this text for establishing the so-called “Pythagorean theorem” is based upon a certain kind of drawing. Secondly, I contrast this passage with Chinese texts from the third century on the same topic, but relying on a completely different type of drawing. What commands the difference in the kinds of drawing is that the latter drawings are “more general” than the former, in a sense to be made explicit. This paper hence aims at making a contribution to the study of geometrical figures in ancient China.Commenting on one of the latter figures, one of the authors of the third century, Liu Hui, describes how various algorithms emerge out of the same transformation of one particular figure. His remarks provide grounds for commenting on the link between the general and the particular, in relation to figures and how algorithms rely on them, as the issue was perceived by the practitioners themselves.The particular figure in question and its transformation are exactly what we find in the Meno, though in relation to a different mathematical issue. The link of that very figure to the one that is perceived as its “generalization” for several algorithms, including the so-called “Pythagorean theorem,” is made not only in Liu Hui, but also by Thabit ibn Qurra (ninth century C.E.), in a letter where he explicitly addresses the purpose of generalizing the reasoning of the Meno. This parallel offers an appropriate basis to highlight differences in terms of conception and use of figures.

Changes and Continuities in the Use of Diagrams Tu in Chinese Mathematical Writings (Third Century to Fourteenth Century) [I]

“…Such a hypothesis could also explain why the tu 圖, to which Liu Hui's commentary refers, were not handed down with the text: like counting rods and blocks, they were material objects that were used with the texts, but they were not included on the page. 35 As for why the 1213 edition of The Gnomon of the Zhou 35 Chemla (2005) suggested that there was a shift in the nature of tu 圖, between the beginning of the Common era and the third century, new types of tu 圖 being introduced at some point during this time span and co-existing with tu 圖 of the earlier type. According to the conclusions reached in this other publication, all these tu 圖 were material objects.…”

“…Moreover, the 6 It is singular by comparison to the practices attested to in other mathematical traditions. I dealt with this issue mainly in Chemla (1994a;2005). I summarized some ideas and discussed the impact of this question on the issue of critical edition of diagrams in chapter A and the introduction to chapter 9, in Chemla and Guo (2004: 34-36, 661-701).…”

“…17 In fact, the opening section of The Gnomon of the Zhou is formulated in such a way that it seems to indirectly testify to the fact that a kind of geometrical diagram is taken into consideration and progressively modified. Indeed, the section refers to a material configuration composed of two squares, and it then makes use of several verbs that seemingly prescribe material transformations of the configuration, which change it into the square of the hypotenuse (Chemla 2005). However, as I said above, the text does not make any explicit reference to the diagram.…”

“…I suggested that the descriptions of the lay out and the execution of the computations were more and more precise as time elapsed. 17 I refer the reader to Chemla (2005), where I argued in favor of this interpretation. In this article, I give the argumentation on the basis of which I support Li Jimin 李繼閔's restoring of the graphic features of the diagrams underlying the opening section of The Gnomon of the Zhou, and I translate the relevant piece of text.…”

Looking at it from Asia: the Processes that Shaped the Sources of History of Science

“…Both reveal similar 57 On the question of the generality of figures in ancient China, see (Chemla 2005). features, in my view for the same reasons. Both reveal similar 57 On the question of the generality of figures in ancient China, see (Chemla 2005). features, in my view for the same reasons.…”