  We study numerically, using a one-dimensional Heisenberg model, the
spin-Peierls transition in the linear Cu$^{2+}$ spin-1/2 chains in the
inorganic compound CuGeO$_3$ which has been recently observed experimentally.
We suggest that the magnetic susceptibility, the temperature dependence of the
spin gap and the spin-Peierls transition temperature of this material can be
reasonably described by including nearest and next nearest neighbor
antiferromagnetic interactions along the chain. We estimate that the nearest
neighbor exchange parameter J is approximately $160\:\rm K$, and that the next
nearest neighbor exchange parameter is approximately $0.36\:\rm J$.
