Seki Takakazu’s Method on the Remainder Problem

Abstract

It is known that a system of simultaneous congruences of first degree which will be henceforth called a remainder problem, first appeared in the Sunzi’s Arithmetical Canon (c.AD400). Later in China, remainder problems were discussed in many books, and some of these Chinese books were introduced into Japan by the seventeenth century. The eminent Japanese mathematician Seki Takakazu (c.1642–1708) investigated remainder problems adopting the term the art of cutting bamboo [jianguan shu] which is found in the Chinese book, Yang Hui’s Arts on Arithmetic [Yanghui Suanfa] (1275) by Yang Hui. Seki is supposed to have consulted the Chinese book, but Seki’s method is much more advanced than Yang Hui’s. Seki generalized the theory on the remainder problem and showed the procedure for the solution systematically. The aim of this paper is to analyze Seki’s method on the remainder problem in comparison with Chinese books, especially with the Mathematical Treatise in Nine Chapters [Shushu Jiuzhang] (1247) by Qin Jiushao.