李子金关于三角函数造表法的研究

LI ZIJIN'S RESEARCH ON THE METHODS OF MAKING TRIGONOMETRIC FUNCTION TABLE

该文从两个正余弦函数经验公式、两个正弦函数内插法计算公式、公式计算精度的比较和改进、分析与评价等4个方面介绍清初数学家李子金在三角函数选表法方面所做出的独特成果及研究过程。最后指出：李子金研究新的三角函数选表公式的目的是便于进行天文观测和生产实践中应用；他的数学研究思想体现了“中西会通”时代数学研究的特点；两个经验公式体现了我国传统数学中“衰分术”思想，形式新颖；“四差通用法”公式，独立于朱世杰，把中国内插法计算公式推进到四次；三角函数造表法公式，既与早期西洋传入的按“六宗率”、“三要法”计算风格不同，又与后期引进的幂级数计算公式有别，具有独特风格。

This paper briefs Li Zijin's unique research achievements in the methods of making a trigonometric function table and the process of research from the following four aspects: Two sine cosine function empirical formulae; Two sine function interpolation calculating formulae ; Comparison of the accuracy of several calculating formulae and their improved process ; Analysis and remarks. Finally it points out: Li's purpose of developing new calculating formulae for the trigonometric function table is to facilitate astronomical observations and their applications in production; His mathematical thoughts reflect the characteristics of mathematical research during the Chinese-Western Mixing periodi The two empirical formulae, which are novel in form, reflect the Cui Fen Shu in traditional Chinese mathematical thoughts; Si Cha Tong Yong Fa, independent of Zhu Sijie's, pushes the interpolation calculating formulae to the fourth power in China; Calculating formulae of the trigonometric function table are different from not only the calculating style according to Liu Zong Lu and Sun Yao Fa, both imported in the earlier period, but also the calculating formulae in power series imported in the later period.