  Among various analytic approximations for the growth of density fluctuations
in the expanding Universe, Zel'dovich approximation and its extensions in
Lagrangian scheme are known to be accurate even in mildly non-linear regime.
The aim of this paper is to investigate the reason why these Zel'dovich-type
approximations work accurately beyond the linear regime from the following two
points of view: (1) Dimensionality of the system and (2) the Lagrangian scheme
on which the Zel'dovich approximation is grounded. In order to examine the
dimensionality, we introduce a model with spheroidal mass distribution. In
order to examine the Lagrangian scheme, we introduce the Pad\'e approximation
in Eulerian scheme. We clarify which of these aspects supports the unusual
accuracy of the Zel'dovich-type approximations. We also give an implication for
more accurate approximation method beyond the Zel'dovich-type approximations.
