e, f, g, a b, c d / e, a b, c d

To one who is given back what was taken from him, there is no lack of the thing taken. / That quantity lacks nothing to which is restored what was taken from it. / That part is made whole again in its whole, to which its remainder is restored. / Therefore I shall remove a c f g e from the pyramid a c b d e and give it back correspondingly in the posterior part, that is in b d e. And of the rectilinear pyramidal body I shall have made a pyramid of curved lines and sides, which is below in a c b d e. And by the conception proposed above, this curved pyramid is exactly equal to the rectilinear pyramid proposed above. Pyramids of equal curvature placed on the same or similar bases as the rectilinear pyramids, between equidistant surfaces, will always be of equal quantity among themselves.

a b e f, c d g

The space a b c d is of quantity equal to the space c d e f. It is proved by the second [proposition] of this [book], which said that the thing moved acquired as much space as it left behind it. Therefore, by removing the part of the surface b d f from its remainder d f g through the whole space a b c d, necessity makes the space c d e f be equal to the space a b c d. This manner of proceeding is not simply geometric, but is subalternate and partakes both of philosophy and of geometry, because it is proved by motion, although in the end all the mathematics are philosophical speculations.