Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides/Porism

Theorem

In the words of Euclid:

From this it is manifest that, if four straight lines be < continuously > proportional, as the first is to the fourth, so will a parallelepidedal solid on the first be to the similar and similarly described parallelepidedal solid on the second, inasmuch as the first has to the fourth and the ratio triplicate of that which it has to the second.

(The Elements: Book $\text{XI}$: Proposition $33$ : Porism)

Proof

Apparent from the construction.

$\blacksquare$

Historical Note

This proof is Proposition $33$ of Book $\text{XI}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{XI}$. Propositions