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 | = α·β·γ, |
| U | u | u | u |
where α, β, γ 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.
Book XII.