A prelude to finite elements: The fruitful problem of the brachistochrone

When did fi nite elements begin? It is difficult to trace the origins of fi nite-element methods because of a basic problem in defi ning precisely what constitutes a “fi nite-element method.” To most mathematicians, it is a method of piecewise polynomial approximation. Its origins are therefore frequently traced to the appendix of a paper by Courant [1], in which piecewise-linear approximations of the Dirichlet problem over a network of triangles was discussed. The “Interpretation of Finite Diff erences” by Polya [2] is regarded as embodying piecewisepolynomial approximation aspects of fi nite elements.