The extended Kantorovich method is used to obtain solutions for a large number of previously unsolved elastic buckling problems of thin rectangular plates. In the present work, this method is specially adapted to a numerical method of solution. Verification of the solution method is made by solving a large number of plate buckling problems with known classical solutions. Included among the new problems solved are plates with a variety of boundary conditions, plates on elastic foundations, and plates with a variable inplane compressive load applied to only one edge, together with several different in-plane prestress configurations to increase the magnitude of the critical stress.