  The large deflection of a circular thin plate under uniform external pressure
is a classic problem in solid mechanics, dated back to Von K{\'a}rm{\'a}n
\cite{Karman}. {This problem is reconsidered in this paper using an analytic
approximation method, namely the homotopy analysis method (HAM).} Convergent
series solutions are obtained for four types of boundary conditions with rather
high nonlinearity, even in the case of $w(0)/h>20$, where $w(0)/h$ denotes the
ratio of central deflection to plate thickness. Especially, we prove that the
previous perturbation methods for an arbitrary perturbation quantity (including
the Vincent's [2] and Chien's [3] methods) and the modified iteration method
[4] are only the special cases of the HAM. However, the HAM works well even
when the perturbation methods become invalid. All of these demonstrate the
validity and potential of the HAM for the Von K{\'a}rm{\'a}n's plate equations,
and show the superiority of the HAM over perturbation methods for highly
nonlinear problems
