We calculate lower bounds on the size of sigmoidal neural networks
that approximate continuous functions. In particular, we show that
for the approximation of polynomials the network size has to grow
as $\Omega((\log k)^{1/4})$ where $k$ is the degree of the polynomials.
This bound is valid for any input dimension, i.e. independently of
the number of variables. The result is obtained by introducing a new
method employing upper bounds on the Vapnik-Chervonenkis dimension
for proving lower bounds on the size of networks that approximate
continuous functions.