The review traces the progress from the early application of the Hollerith Tabulating Machine to the construction of astronomical tables by interpolation and to the evaluation of Brown’s Tables of the Moon by L. J. Comrie in the late 1920’s and early 1930’s to the introduction of large programmable electronic calculators in the post World War II era. The early application of electromechanical punched-card machines to astronomical problems stemmed from the needs of the national ephemeris offices to evaluate theories of the Sun, Moon and planets and to subtabulate these positions to form the tables of daily values given in the national almanacs. These techniques were quickly applied to other astronomical problems such as the construction of star catalogs, the reduction of astrometric observations and the numerical integration of the equations of motion of objects in the solar system.