The solution of a memorable problem by a special artifice of calculation

TL;DR

This paper discusses a historical mathematical problem involving optimizing an integral of a function over a curve, using Euler's methods and polar coordinates, as originally presented in 1810.

Contribution

It revisits Euler's classical problem and demonstrates the application of the Euler-Lagrange equation with polar coordinates for optimization.

Findings

Derivation of the optimal curve using Euler-Lagrange equation

Application of polar coordinates in solving the problem

Historical insight into 19th-century calculus techniques

Abstract

E731 in the Enestrom index. Originally published as "Solutio problematis ob singularia calculi artificia memorabilis", Memoires de l'academie des sciences de St-Petersbourg 2 (1810), 3-9. For $z$ the distance from the origin, and $v$ a given function of $z$, Euler wants to find a curve $s$ such that the integral of $z$ over $s$ is a maximum or a minimum. He starts with the Euler-Lagrange equation, and does a lot of manipulations with polar coordinates.