The problem of axisymmetrical buckling of thin shallow circular spherical shells under uniform pressure with the edge either rigidly clamped or simply supported is studied. The Newton-spline function method is used to solve the non-linear differential equations of thin shallow circular spherical shells. Numerical results of upper and lower critical loads for shells with the value of geometric parameter lambda as great as 50 are obtained. The character of prebuckling and postbuckling and the problem of finding critical points is discussed. The problem is called the very large geometric parameter lambda buckling problem of a shallow circular spherical shell, whose results had been predicted by Budiansky. Here we confirm his result as being reasonable. Relations between the buckling model and the geometric parameter lambda which improve Karman and Tsien's suggestion are discussed.