The equation (2.18) was derived by Euler around 1740. His original derivation was very different from the one we gave, and relied on discretization. The idea is to approximate a general curve by a piecewise linear one passing through points, as in Figure 2.8. The problem of optimizing the locations of these points is finite-dimensional, hence it can be studied using standard tools. The equation (2.18) is obtained in the limit as .

An alternative way to arrive at the same result, free of geometric considerations and relying on analysis alone, was proposed by Lagrange in 1755 (when he was only 19 years old). Lagrange described his argument in a letter to Euler, who was quite impressed by it and subsequently coined the term ``calculus of variations" for Lagrange's method. Even though the problem formulation and the solution given by Lagrange differ from the modern treatment in the notation and other details, the main ideas behind our derivation of the Euler-Lagrange equation are essentially contained in his work.