  A rational polytope is the convex hull of a finite set of points in $\R^d$
with rational coordinates. Given a rational polytope $P \subseteq \R^d$,
Ehrhart proved that, for $t\in\Z_{\ge 0}$, the function $#(tP \cap \Z^d)$
agrees with a quasi-polynomial $L_P(t)$, called the Ehrhart quasi-polynomial.
The Ehrhart quasi-polynomial can be regarded as a discrete version of the
volume of a polytope. We use that analogy to derive a new proof of Ehrhart's
theorem. This proof also allows us to quickly prove two other facts about
Ehrhart quasi-polynomials: McMullen's theorem about the periodicity of the
individual coefficients of the quasi-polynomial and the Ehrhart-Macdonald
theorem on reciprocity.
