清代算家的勾股恒等式证明与应用述略

A Study of Proof and the Application of Gougu Identities during Qing Dynasty

勾股定理与勾股恒等式是中国传统数学重要的基础知识。清代算家在勾股恒等式的增补、证明及应用方面做出了丰硕的成果。这一方面目前尚未见到专题的研究。文章重点考察勾股恒等式的证明,兼及应用,指出吴嘉善"勾股比例表"给出的20式使得勾股恒等式形成系统,而各式证明的依据均见于赵爽"勾股圆方图注",还指出了勾股恒等式的应用与《数理精蕴》的关系。由此可以比较全面地理解清代算家勾股和较术的成果。

The Gougu rule(Pythagorean theorem) and identities are an important part of the rudiments of ancient mathematics in China. Books by mathematicians during the Qing Dynasty(1644-1911) that were devoted to the amplification, proof and application of identities indicate a wealth of results that have not been specially studied. This paper lays emphasis on the investigation on examples for proof as well as the application of identities appearing in the above books. This paper points out that Gougu bili biao(Gougu Table of Proportion, 1863) by Wu Jiashan(1820-1885) provides a complete set with a total of 20 identities. All of the principles needed for identity proving can be found in Gougu yuanfang tuzhu(Illustrated Comments on Right Angled Triangle) by Zhao Shuang early in the 3 rdcentury. Furthermore, the paper points out the relation between the application of the identities and Shuli jingyun(Essential Principles of mathematics, 1723). Based on such historical mathematical materials,a better and more comprehensive understanding of Gougu identities can be gained.