Different cultures of computation in seventh century China from the viewpoint of square root extraction

Abstract

The aim of this paper is to bring to light a previously unknown geometrical method for extracting the square root in seventh century China. In order to achieve this goal, a seventh century commentary by the scholar Jia Gongyan, 賈公彥, on a Confucian canon, the Rites of Zhou Dynasty [Zhouli 周禮], is analysed. This is compared with the commentary by his contemporary Li Chunfeng, 李淳風, which is referred to in another mathematical book, the Mathematical Procedures of the Five Canons, [Wujing Suanshu 五經筭術]. Although these two scholars probably knew each other, they used very different methods to solve the same problem in relation to square root extraction. It is argued that the differences mainly lie in two aspects: firstly, Jia Gongyan mostly made use of geometry while Li Chunfeng used counting rods; secondly, the two methods had different geometrical interpretations. Given the fact that the method of square root extraction Jia Gongyan uses is one among many other methods he employed in mathematics, and it has the same features as the others; moreover, other commentators on the Confucian Canons use similar mathematical methods, this paper closes with a general discussion on mathematical cultures. It is suggested that there were three elements to mathematical practice in seventh century China: geometry, counting rods, and written texts. The interplay and structure between the three elements is seen to influence mathematical practices.