  This paper provides a quite simple method of Tonelli's calculus of variations
with positive definite and superlinear Lagrangians. The result complements the
classical literature of calculus of variations before Tonelli's modern
approach. Inspired by Euler's spirit, the proposed method employs finite
dimensional approximation of the exact action functional, whose minimizer is
easily found as a solution of Euler's discretization of the exact
Euler-Lagrange equation. The Euler-Cauchy polygonal line generated by the
approximate minimizer converges to an exact smooth minimizing curve. This
framework yields an elementary proof of the existence and regularity of
minimizers within the family of smooth curves and hence, with a minor
additional step, within the family of Lipschitz curves, without using modern
functional analysis on absolutely continuous curves and lower semicontinuity of
action functionals.
