  We have carried out a wide range of calculations for the $S=1/2$ Heisenberg
model with nearest- and second-neighbor interactions on a two-dimensional
lattice which describes the geometry of the vanadium ions in the spin-gap
system CaV$_4$O$_9$. The methods used were convergent high-order perturbation
expansions (``Ising'' and ``Plaquette'' expansions at $T=0$, as well as
high-temperature expansions) for quantities such as the uniform susceptibility,
sublattice magnetization, and triplet elementary excitation spectrum.
Comparison with the data for CaV$_4$O$_9$ indicates that its magnetic
properties are well described by nearest-neighbor exchange of about 200K in
conjunction with second-neighbor exchange of about 100K.
