Part II: Operations on Pieces of Areas

Movement is a key to understanding the diagrams of the Development of Pieces [of Areas]. Several different possible transformations can be observed in the diagrams of the Development of Pieces [of Areas]. Not only areas are moved virtually and cut into pieces but they are also filled and emptied. The relation between plain and empty areas lies at the core negative and positive coefficients. Indeed, the question is how to represent a negative coefficient by mean of areas. Through a comparison of Li Ye and Yang Hui’s works, it is possible to reconstruct some steps in the development of the elaboration of quadratic equation. It is also possible to distinguish several layers in the composition of the text transmitted by Li Ye. This section draws upon Problems 18 and 5 of the Development of Pieces [of Areas] and Problem 49 of Yang Hui’s Methods of Computation. Problems 18 and 5 offer cases from which the Old Procedure can be reconstructed and enable the interpretation of the Section of Pieces [of Areas] as containing several procedures depending on the different reading and cutting of the diagrams. In light of Problem 49 of Yang Hui’s Method of Computation, a clear evolution of concept of negative coefficient emerges. To understand the paths of these transmission and interpretation, it is first necessary to visualize the different operations of transformations as performed on diagram (illustrated by Problem 2).