In symbols it says—
If a : b = c : d, then a³ : b³ = c³: d³.
Prop. 40 teaches how to compare the volumes of triangular prisms with those of parallelepipeds, by proving that a triangular prism is equal in volume to a parallelepiped, which has its altitude and its base equal to the altitude and the base of the triangular prism.
§ 84. From these propositions follow all results relating to the mensuration of volumes. We shall state these as we did in the case of areas. The starting-point is the “rectangular” parallelepiped, which has every edge perpendicular to the planes it meets, and which takes the place of the rectangle in the plane. If this has all its edges equal we obtain the “cube.”
If we take a certain line u as unit length, then the square on u is the unit of area, and the cube on u the unit of volume, that is to say, if we wish to measure a volume we have to determine how many unit cubes it contains.
A rectangular parallelepiped has, as a rule, the three edges unequal, which meet at a point. Every other edge is equal to one of them. If a, b, c be the three edges meeting at a point, then we may take the rectangle contained by two of them, say by b and c, as base and the third as altitude. Let V be its volume, V′ that of another rectangular parallelepiped which has the edges a′, b, c, hence the same base as the first. It follows then easily, from Prop. 25 or 32, that V : V′ = a : a′; or in words,
Rectangular parallelepipeds on equal bases are proportional to their altitudes.
If we have two rectangular parallelepipeds, of which the first has the 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