"Kepler imagined a given geometrical figure to be decomposed into infinitesimal figures, whose areas or volumes he added up in some ad hoc way to obtain the area or volume... Cavalieri proceeded by setting up a one-to-one correspondence between the indivisible elements of two geometrical figures. If corresponding indivisibles of the two figures had a certain (constant) ratio, he concluded that the areas of volumes of one of the figures had the same ratio. Typically, the area or volume of one of the figures was known in advance, so this gave the other. ... Kepler thought of a geometrical figure as being composed of indivisibles of the same dimension [as the original figure]... from some process of successive subdivision... However, Cavalieri generally considered a geometrical figure to be composed of an indefinitely large number of indivisibles of lower dimension. ...an area as consisting of ...line segments, and a volume as consisting of... plane sections... Rigor, he wrote in the Exercitationes, is the affair of philosophy rather than mathematics."