Cor. 3.—In right prisms of equal altitudes the volumes are to one another as the areas of their bases. This may be proved by dividing the bases into parts so that the subdivisions will be equal, and then the volumes proportional to the number of subdivisions in their respective bases, that is, to their areas.
Cor. 4.—The volume of a rectangular parallelopiped is measured by the continued product of its three dimensions.
PROP. II.—Theorem.
Parallelopipeds (ABCD–EFGH, ABCD–MNOP), having a common base (ABCD) and equal altitudes, are equal.
1∘. Let the edges MN, EF be in one right line; then GH, OP must be in one right line. Now EF = MN, because each equal AB; therefore ME = NF; therefore the prisms AEM–DHP, and BFN–CGO, have their triangular bases AEM, BFN identically equal, and they have equal altitudes; hence they are equal; and supposing them taken away from the entire solid, the remaining parallelopipeds ABCD–EFGH, ABCD–MNOP are equal.
2∘. Let the edges EF, MN be in different lines; then produce ON, PM to meet the lines EF and GH produced in the points J, K, L, I. Then by 1∘ the parallelopipeds ABCD–EFGH, ABCD–MNOP are each equal to the parallelopiped ABCD–IJKL. Hence they an equal to one another.
Cor.—The volume of any parallelopiped is equal to the product of its base and altitude.
PROP. III.—Theorem.
A diagonal plane of a parallelopiped divides it into two prisms of equal volume.
1∘. When the parallelopiped is rectangular the proposition is evident.