  We study the Dirichlet spectrum of the Laplace operator on geodesic balls
centred at a pole of spherically symmetric manifolds. We first derive a
Hadamard--type formula for the dependence of the first eigenvalue $\lambda_{1}$
on the radius $r$ of the ball, which allows us to obtain lower and upper bounds
for $\lambda_{1}$ in specific cases. For the sphere and hyperbolic space, these
bounds are asymptotically sharp as $r$ approaches zero and we see that while in
two dimensions $\lambda_{1}$ is bounded from above by the first two terms in
the asymptotics for small $r$, for dimensions four and higher the reverse
inequality holds.
  In the general case we derive the asymptotic expansion of $\lambda_{1}$ for
small radius and determine the first three terms explicitly. For compact
manifolds we carry out similar calculations as the radius of the geodesic ball
approaches the diameter of the manifold. In the latter case we show that in
even dimensions there will always exist logarithmic terms in these expansions.
