Abstract

Extending the zengcheng kaifangfa 增乘開方法 for square and cube roots, Jia Xian 賈憲 (11th cent.) introduced the triangle for binomial expansions, or binomial coefficients for extractions of nth roots. Using tianyuanshu 天元術 and the zengcheng kaifangfa for polynomial equations in thirteenth-century China, the theory of equations became the most important development in the history of Chinese mathematics. Interpolations by difference sequences were introduced in the Shoushili 授時曆 (1281). Zhu Shijie 朱世傑 completed the Suanxue qimeng 算學啓蒙 (1299) and Siyuan yujian 四元玉鑑 (1303). In the latter, he showed that the triangle for binomial expansions gives rise to the fundamental structures for solving equations and the theory of finite series. Along with these, we discuss the contributions to the theory of equations and finite series in Joseon 朝鮮 (1392–1910) Korea. We deal with the contributions of Hong Jeong-ha 洪正夏 (1684–1727) in his Gu-il jib 九一集 (1713–1724) and then those of Lee Sang-hyeog 李尙爀 (b. 1810) in his Igsan 翼算 (1868).