volume V and the edges a, b, c, and the second, the volume V' and the

edges a', b', c', we may compare them by aid of two new ones which have respectively the edges a', b, c and a', b', c, and the volumes V1 and V2. We then have

V : V1 = a : a'; V1 : V2 = b : b', V2 : V' = c : c'.

Compounding these, we have

V : V' = (a : a')(b : b')(c : c'),

or

V a b c -- = -- . -- . --. V' a' b' c'

Hence, as a special case, making V' equal to the unit cube U on u we get

V a b c -- = -- . -- . -- = [alpha].[beta].[gamma], U u u u

where [alpha], [beta], [gamma] are the numerical values of a, b, c; that is, _The number of unit cubes in a rectangular parallelepiped_ is equal to the product of the numerical values of its three edges. This is generally expressed by saying the volume of a rectangular parallelepiped is measured by the product of its sides, or by the product of its base into its altitude, which in this case is the same.

Prop. 31 allows us to extend this to any parallelepipeds, and Props. 28 or 40, to triangular prisms.

_The volume of any parallelepiped, or of any triangular prism, is measured by the product of base and altitude._

The consideration that any polygonal prism may be divided into a number of triangular prisms, which have the same altitude and the sum of their bases equal to the base of the polygonal prism, shows further that the same holds for any prism whatever.