Zhang Dunren's Contribution to the Development of the “Dayan Zongshu Shu”

In the history of solving congruence equations, Zhang Dunren (张敦仁, 1754—1834) of the mid Qing Dynasty played a crucial role bridging the past and the future. Almost a contemporary of the German mathematician Gauss, Zhang proposed the qiuyi shu (求一术, method for finding unity) based on the Southern Song mathematician Qin Jiushao (秦九韶)'s dayan zongshu shu (大衍总数术, general dayan method). Zhang also observed that the qiuyi shu originated from the problem of wu buzhi shu (物不知数, things not knowing their numbers) in the Sunzi suanjing (孙子算经, Mathematical Classic of Sunzi), and the great mathematical practice of ancient Chinese astronomy and calendrical science for solving the problem of the shangyuan jinian (上元积年, accumulation of years of the upper element). Zhang simplified and proceduralized Qin's methods from the aspects of “transforming question numbers into fixed numbers”, “calculating multiplication rates”, and “calculating the shangyuan jinian”, creating the dayan zongshu shu, a pearl of pre-modern Chinese mathematics. This method in turn served as the foundation for the subsequent work of Luo Tengfeng (骆腾凤), Shi Yuechun (时曰醇), Huang Zongxian (黄宗宪), and others.