法尼亚诺关于双纽线弧长测量问题的研究

Fagnano’s Study on the Arc Length of the Lemniscate

椭圆积分不能用初等函数表示,最早出现在与椭圆弧长相关的问题中。法尼亚诺尝试利用圆锥曲线的性质以及等分第一象限双纽线弧长的方法测量双纽线的弧长,在研究的过程中得到了双纽线的倍弧长公式,这对欧拉有关椭圆积分的工作产生了重要影响。对法尼亚诺有关双纽线弧长工作的研究有助于更清晰地呈现椭圆积分的历史发展过程。

Elliptic integrals cannot be expressed as elementary functions and first appeared in problems related to the length of elliptic arcs.Fagnano tried to measure the arc length of the double button by using the properties of conic curves and the arc length of the biquitic first quadrant double button,and obtained the double arc length formula of the double button line in the process of research,which had an important impact on Euler’s work on elliptic integrals.The study of Fagnano’s work on the arc length of the double button line contributes to a clearer picture of the historical development of elliptic integrals.