前清中算家对理分中末线的研究

Chinese Mathematicians' Researches on the Golden Section in the Early Qing Dynasty

明末,《几何原本》中的黄金分割传入中国,其法理皆明,徐光启称之为理分中末线.论述了前清中算家对《几何原本》中理分中末线的研究:李子金另辟蹊径,从西方传入的切割线定理出发找到了一种解释;梅文鼎以"几何即勾股"的信念彻底解决了理分中末线的相关问题;杨作枚以中国古代数学传统的面积割补解释了理分中末线的作法,居然与《几何原本》中的解释殊途同归.三位中算家从不同角度对理分中末线进行研究,均是以股为勾2倍的勾股形作为解释基础,同时赋予黄金分割清晰多样的几何图解.虽然他们对理分中末线的应用程度有高下之分,但从解释黄金分割来说,可谓是异曲同工,均对传入中国的西方数学知识做出了创新性的工作.

During the late Ming Dynasty,Golden Section,which is named by Xu Guangqi（徐光启） as Lifenzhongmoxian（理分中末线）, was introduced into China by Euclid’s element, and furthermore, Chinese mathematicians in the early Qing Dynasty enriched Golden Section by clarifying its principles according to their understandings. Three Chinese mathematicians, Li Zijin（李子金）, Mei Wending（梅文鼎）, and Yang Zuomei（杨作枚）, studied Golden Section from different aspects. What underlies their understandings is that they all adopted the same principle, gougu （勾股） principle, as the basis for explaining Golden Section. At the same time, they illustrated Golden Section by using various clear diagrams. Therefore, although there is different among them in the depth of adopting Golden Section,it is beyond all doubt that the three arithmeticians have brought forth new ideas from the western arithmetic.