Chinese mathematics and western mathematics integrated in the Tongwen Suanzhi

It has been widely accepted that the Tongwen Suanzhi (1614) was translated by Matteo Ricci and Li Zhizao from Christophorus Clavius's Epitome Arithmaticae Practicae (1583). However, on closer inspection, it becomes apparent that the Tongwen Suanzhi includes a great deal of supplementary material taken by Li from traditional Chinese mathematical texts, including, as Li himself confirms, the Nine Chapters. This paper therefore seeks to demonstrate that in contrast to previous assumptions, when receiving western mathematical texts, far from acting as passive receivers, Chinese scholars in fact drew upon their local traditions to interpret the newly received western materials. In this paper we provide a list of traditional Chinese mathematical works that Li Zhizao may have referred to, as well as the western texts that he was influenced by. Further, by showing how Li combined these two mathematical traditions, we demonstrate that the Tongwen Suanzhi was not merely a work of translation, but in fact represents an integration of western and traditional Chinese mathematics.