Transmission and Interactions Among Different Types of Geometrical Argumentations: From Jesuits in China to Nam Pyŏng-Gil in Korea

Abstract

Transmission and interactions among different types of geometrical argumentations constitute some of the most interesting stories in the history of East Asian science and mathematics. Since the early years of the seventeenth century, the Chinese began to learn European science and mathematics introduced by missionaries and incorporate these into translations and into their own works. At first, Euclid’s Elements, with its hypothetico-deductive structure, was translated in early 1600s. However, one of the most influential mathematical treatises in late imperial China and in contemporary Korea, the Shuli jingyun (Essential Principles of Mathematics, 1723), was composed as a synthesis of all the Chinese and European mathematical knowledge that was available to the Qing emperor Kangxi (r. 1662–1722) himself and his royal mathematicians. A section in this mathematical compendium is entitled “Jihe yuanben” (Elements of geometry), which does not refer to the first Chinese translation of Euclid’s Elements bearing the same Chinese title. It is actually taken from lecture notes written by the French Jesuits Jean-François Gerbillon and Joachim Bouvet when they taught mathematics to Kangxi in the 1690s. These notes were in turn based on the French geometry textbook Elémens de Géométrie by the Jesuit Ignace-Gaston Pardies.

The style of argumentation in Pardies’ text is to give quick and easy explanations that appeal not entirely to the rigour of logic but to the intuition of the reader, and this pedagogy was used by Gerbillon and Bouvet in their lecture notes, which was later compiled into the Shuli jingyun. Some cases can be found to show this style of argumentation. For the volume of pyramids, basically the argument is that a cube could be cut into three “pointed solids”, so the latter’s volume was one third of a cube, and then the volumes of all other pyramids and cones could be calculated by the same procedure, because pointed solids with equal base areas and equal heights would have equal volumes. For the relation between the surface area and the volume of the sphere, the reader is asked to imagine that the sphere is composed of “millions of tiny cones” whose bases are parts of the surface of the sphere, and whose heights are equal to the radius of the sphere.

The Shuli jingyun was transmitted to Korea shortly after its publication, and its influence can be seen on many cases, including the arguments on the volume of different kinds of pyramids and that of the sphere written in the Korean commentary for the Jiuzhang suanshu (Nine Chapters of Mathematical Art). The Korean commentator Nam Pyŏng-Gil intentionally replaced the traditional Chinese commentary on the two problems with his explanations that appeal mainly to intuition. This very style and its transmission is an interesting example of how mathematicians in pre-modern China and Korea chose their ways of composing texts and arguing mathematical propositions.

In this chapter, the McCune-Reischauer system and the Pinyin system are used for Romanising all Korean and Chinese words, respectively, except for the English articles in which authors’ names and titles have been Romanised in other ways. Koreans also use Chinese characters, so the Korean names, book titles or terminologies that are written in Chinese characters but originally used by Koreans are Romanised according to the McCune-Reischauer system for the Korean language.