  We compute the Casimir interaction energy between two perfectly conducting,
concentric cylinders, using the mode-by-mode summation technique. Then we
compare it with the approximate results obtained using the proximity theorem
and a semiclassical approximation based on classical periodic orbits. We show
that the proximity theorem with a particular choice for the effective area
coincides with the semiclassical approximation and reproduces the exact result
far beyond its expected range of validity. We also compute the force between
slightly eccentric cylinders and discuss the advantages of using a cylindrical
geometry to measure the Casimir force.
