  The small angle approximation often fails to explain experimental data, does
not even predict if a plane pendulum's period increases or decreases with
increasing amplitude. We make a perturbation ansatz for the Conserved Energy
Surfaces of a one-dimensional, parity-symmetric, anharmonic oscillator. A
simple, novel algorithm produces the equations of motion and the period of
oscillation to arbitrary precision. The Jacobian elliptic functions appear as a
special case. Thrift experiment combined with recursive data analysis provides
experimental verification of well-known predictions. Development of the
quantum/classical analogy enables comparison of time-independent perturbation
theories. Many of the useful notions herein generalize to integrable and
non-integrable systems in higher dimensions.
