  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 $(-1)^k$.
