Why did Liu Hui fail to derive the volume of a sphere?

AbstractUsing the problem of deriving the volume of a sphere as its central focus, this paper tries to show the importance of different “heuristics” in Liu Hui and Zu Geng's ideas and theories of geometry. Rather than dismissing Liu's failure as due to inadequate time or effort, it argues that this failure was inherent in Liu's own heuristic, a powerful pattern of reasoning that enabled Liu to solve many geometrical problems, but also restrained him from finding the volume of a sphere. Zu Geng's heuristic, on the other hand, revealed its strength in problems concerning the sphere, although this does not imply that it could cover a wider range of geometrical problems than Liu's approach. Thus, directly beyond the problems concerned with the sphere, the central purpose of this paper is to use Liu's and Zu's heuristics (or patterns of geometrical reasoning) as guidelines in reconstructing and elucidating at least part of the historical structure of ancient Chinese geometry. From this perspective, light can also be thrown upon the geometrical reasonings of later figures such as Wang Xiaotong, Shen Kuo, Mei Wending, Jiao Xun, Li Huang, and Xu Youren. Thus, instead of using the classification scheme of 20th-century Western mathematics, the present approach preserves the historical context of these respective heuristics while organizing the different historical trends of Chinese geometrical reasoning into interconnected and consistent patterns