牛顿定理的一种双重证明，它给出了代数方程的系数与其根的幂和之间的关系
A double demonstration of a theorem of Newton, which gives a relation between the coefficient of an algebraic equation and the sums of the powers of its roots
摘要：
翻译自拉丁原文，“Demonstratio gemina theoretical Neutoniani，quo traditur relatio inter coefficientes cuiusvis aequationis algebraicae et summas potestatum radicum eiusdem”（1747年）。Enestrom指数中的E153。本文给出了牛顿恒等式的两个证明，牛顿恒等式用多项式的系数表示根的幂和。第一个证明需要对数的导数。第二个证明使用归纳法和以下事实：在次数为$n$的多项式中，$x^{n-k}$的系数等于$k$根乘以$(-1)^k$的乘积之和。
英文摘要：
Translation from the Latin original, "Demonstratio gemina theorematis Neutoniani, quo traditur relatio inter coefficientes cuiusvis aequationis algebraicae et summas potestatum radicum eiusdem" (1747). E153 in the Enestrom index. In this paper Euler gives two proofs of Newton's identities, which express the sums of powers of the roots of a polynomial in terms of its coefficients. The first proof takes the derivative of a logarithm. The second proof uses induction and the fact that in a polynomial of degree $n$, the coefficient of $x^{n-k}$ is equal to the sum of the products of $k$ roots, times $(-1)^k$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
