Props. 29 and 30 contain special cases of this theorem leading up to the proof of the general theorem.
As consequences of this fundamental theorem we get
Prop. 32. Solid parallelepipeds, which have the same altitude, are to one another as their bases; and
Prop. 33. Similar solid parallelepipeds are to one another in the triplicate ratio of their homologous sides.
If we consider, as in § 67, the ratios of lines as numbers, we may also say—
The ratio of the volumes of similar parallelepipeds is equal to the ratio of the third powers of homologous sides.
Parallelepipeds which are not similar but equal are compared by aid of the theorem
Prop. 34. The bases and altitudes of equal solid parallelepipeds are reciprocally proportional; and if the bases and altitudes be reciprocally proportional, the solid parallelepipeds are equal.
§ 83. Of the following propositions the 37th and 40th are of special interest.
Prop. 37. If four straight lines be proportionals, the similar solid parallelepipeds, similarly described from them, shall also be proportionals; and if the similar parallelepipeds similarly described from four straight lines be proportionals, the straight lines shall be proportionals.