Part I: Text, Diagram and Equation

Despite their scarcity, diagrams play an important role in Chinese mathematics. In China, diagrams are indissociable from the history of elaboration of algebraic objects. To understand how diagrams and equations are related, recent eras offer a better point of departure than the Song–Yuan period. This chapter starts with a description of the algebraic procedure to set up quadratic equation, which is at least in part diagrammatic in origin. After the description of the procedure presented in Problem 1 of the Development of Pieces [of Areas], and its general history has been established, the difference between the concept of equation as found in the procedure and modern concepts can be discussed. The origin of these differences lies in the practice of executing divisions and extracting square roots with counting rods and/or diagrams, as has been show by experts in history of Chinese mathematics. Then, a description of the geometrical procedure is introduced. This description clarifies that (1) diagrams play an argumentative role and (2) they are, ontologically, literal expressions of the mathematical object (such as the quadratic equation). An example shows how to visualise diagrams as the movement of areas and not as static lines. The chapter contains two problems, Problem 1 exemplifies the basis and Problem 21 provides a clear picture of transformations. All coloured diagrams are modern and thus artificial representation. These diagrams attempt to show, as a cartoonist might, the different cinematic moments of a movement. Their weakness lies in their modernity. Modern objects always hide contemporary concerns and purposes, probably different from Li Ye or Yang Hui’s intentions. The reconstructed diagrams remain just a contemporary attempt to reconstruct ancient objects. While intended to represent movement by picturing it at separate instances, they have paradoxically stopped, like still-frames of a motion picture. However, the stop-motion animations are the best didactic tool to guide a reader through a lost ‘way of seeing’ mathematical object.