《中西数学图说》勾股和较问题探微

A Probe into the Problems of Gougu Hejiao in Zhongxi Shuxue Tushuo

《中西数学图说》是晚明进士李笃培在西方数学传入中国的背景下,撰写的一部会通中西的数学著作。该书对勾股和较问题的研究,在晚明时期别具特色,颇有创获。通过对该书勾股和较问题的分析,本文指出,虽然《勾股义》与《同文算指》是《中西数学图说》勾股和较知识的直接来源,但该书并没有采用前两者仿照《几何原本》给出的证明形式,而是利用传统算书中的出入相补原理给予证明。与赵爽的“勾股圆方图注”文字相契合的勾股和较图形,是《中西数学图说》证明勾股和较问题的依据。在原有勾股和较问题的基础上,该书对所有勾股和较情形进行了系统的总结,并对其中需要论证的情况一一绘图证明,与清代中前期中算家在勾股和较领域所做的工作有很多重合和相似的地方。对勾股和较问题的几何证明和系统总结,体现了《中西数学图说》在西方数学的影响下,整理传统算学内容的努力。

Zhongxi shuxue tushuo中西数学图说(Illustration of Chinese and Western Mathematics)is a mathematical work written by Li Dupei,a scholar of the late Ming Dynasty,against the background of the introduction of western mathematics into China.The book's research on gougu hejiao勾股和较(sum and difference of three sides of right-angled triangles)was unique for the period and ground-breaking.Through an analysis of the issues tackled in the book,this paper points out that although the works Tongwen suanzhi同文算指and Gougu yi勾股义were the direct sources of knowledge on gougu hejiao in Li's work,he did not adopt the form of proof given by them that was from the Jihe yuanben几何原本(Chinese translation of Euclid's Elements),but instead uses the principle of churu xiangbu出入相补(out-in complementary)from Chinese traditional mathematics.Geometric figures,which are consistent with Zhao Shuang's赵爽Gougu yuanfang tuzhu勾股圆方图注,are the basis for the proof of gougu hejiao problems in Zhongxi shuxue tushuo.On the basis of the existing problems of gougu hejiao,Li makes a systematic summary of all the situations in which it is used,and lays out and proves one by one the situations that need to be demonstrated.The achievements of the work have many overlaps and similarities with the work done by mathematicians in the early and middle Qing dynasty on gougu hejiao.The geometric proof and systematic summarization of gougu hejiao reflect the efforts of Zhongxi shuxue tushuo to interpret the contents of traditional mathematics under the influence of western mathematics.