  Analytic description of the evolution of cosmic ray showers is dominated by
the Greisen equation nearly five decades old. We present an alternative
approach with several advantages. Among the new features are a prediction of
the differential distribution, replacing Greisen's form which fails to be
positive definite. Explicit comparison with Monte Carlo simulations shows
excellent agreement after a few radiation lengths of development. We find a
clear connection between Monte Carlo adjustment of Greisen's form and
underlying physics, and present a concise derivation with all steps explicit.
We also reconstruct the steps needed to reproduce Greisen's approximate
formula, which appears not to have been published previously.
