高斯与双纽线的划分

Gauss and the Division of Lemniscate

摘要: 高斯继承法尼亚诺和欧拉对双纽线划分的研究，为了获得等分点满足的方程，在三角函数的启示下将双纽线积分反演，而方程的次数超过有实际意义的根的个数促使他将双纽线函数的定义域扩充至虚数，获得了历史上第一个双周期函数，为了解释这些周期，高斯考虑了复积分，先于柯西拥有了留数概念。

Abstract: Building upon the work of Fagnano and Euler on the division of the lemniscate, Gauss sought to derive equations satisfying the condition of equal division points. Inspired by trigonometric functions, he reversed the lemniscate integral. However, the degree of the resulting equation exceeded the number of practically meaningful roots, prompting him to extend the domain of the lemniscate function to include imaginary numbers. This yielded the first double-periodic function in history. To explain these periods, Gauss considered complex integration, thus anticipating Cauchy’s development of the residue concept.