  We give an elementary geometric re-proof of a formula discovered by Michel
Brion as well as two variants thereof. A subset of R^n gives rise to a formal
Laurent series with monomials corresponding to lattice points in the set. Under
suitable hypotheses, these series represent rational functions. We will prove
formulae relating the rational function of a lattice polytope P to the sum of
rational functions corresponding to the supporting cones subtended at the
vertices of P. The exposition should be suitable for everyone with a little
background in topology.
