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

This article aims at highlighting a radical change in the materiality of tu between the time when they were first mentioned in Chinese mathematical texts in the third century commentaries on Canons and the thirteenth century, from which there are abundant illustrations in treatises. Moreover, it intends to highlight that the meaning of the word tu 圖, as used in mathematical writings, greatly changed over the same time span. It argues that third century tu 圖 were material objects, cut in paper with squared-grid, and worked out in specific ways. They probably always displayed particular dimensions and only represented objects for plane geometry. Their areas, and not their points, were marked, and they were marked by characters or colors. Areas were cut into pieces and rearranged. Such is the contribution mathematical texts can offer for capturing the nature of tu for these early periods. In contrast to this, thirteenth century tu 圖 to which mathematical texts refer were included in the texts themselves and hence articulated with the discourse on the surface of the page. Moreover, the extension of what could be represented in a tu 圖 increased tremendously. However, as I show in part II of this paper, in the thirteenth century, several traditions must be distinguished, regarding the nature of tu 圖 and the way in which they were integrated into the text. Moreover, part II shows that despite this break in the nature of tu 圖 some thirteenth century mathematicians inherited ways of working with tu 圖 from earlier times. I argue that this occurred within the framework of a specific mathematical domain, that is, a given subtradition. The mathematicians operating within this framework brought into play the same markers (colors, characters) for areas and adapted the operations onto paper. However, what is most interesting is that they made use of these traditional ways of working with figures while bestowing new mathematical meanings upon them. This thus presents an interesting case of continuity and rupture within a given tradition. All these uses of figures, in their variety, are specific to China and differ from the way in which other traditions used figures in mathematics.