The purpose of this presentation is to investigate Chinese “non-official” mathematics of the first millennium AD on the basis of the extant medieval sources. It is known that from the mid-first millennium onwards, a number of mathematical texts were used as textbooks in state educational institutions concerned mainly with administrative affairs and calendar making. But during the Song dynasty (960-1279), there existed mathematical texts not related to official education. For instance, Yang Hui, 楊輝, Qin Jiushao, 秦九韶, Li Ye, 李冶, and Zhu Shijie, 朱世杰 composed texts which were not dealing with economical and astronomical matters and were never used for state examinations. These authors referred to more ancient texts. The goal of this paper is to make the medieval treatises “speak” about their genealogy.

To do so, I propose to focus on one mathematical procedure discussed by these authors. Two of them, namely, Li Ye and Yang Hui presented the same mathematical procedure, tiaoduan (條段), “section of pieces [of areas]”, that was used to set up algebraic equations by means of geometry. Li Ye claimed to have found his inspiration in the Yiguji益古集 (Collection increasing the ancient [knowledge], the 11th c.) and Yang Hui quoted Liu Yi’s劉益Yigu genyuan議古根源 (Discussion on the origin of the ancient [ideas and procedures], the 10-11th c.). Both treatises are lost; one just knows from their titles that they refer to some “ancient” (古) sources. Besides, Li Ye presented an “old procedure” (舊術) used for solution of problems as opposed to the “new” one. One has here elements for building a genealogy of methods, and I will attempt to explain the meaning of the terms “ancient” and “old” by comparing the procedures presented in the these treatises. The comparative method will help to identify which parts of the texts and which elements of the procedures are to be dated of the 13th century, of the 10th century, or of the first millennium. In doing so, I hope to be able to reconstruct some features of “non-official” mathematics of the first millennium that until now has been perceived solely on the basis of the collection of treatises Suan jing shi shu 算經十書used during the Tang dynasty for educational purposes.