Four Straight Lines are Proportional iff Similar Parallelepipeds formed on them are Proportional

Theorem

In the words of Euclid:

If four straight lines be proportional, the parallelepipedal solids on them which are similar and similarly described will also be proportional; and if the parallelepipedal solids on them which are similar and similarly described be proportional, the straight lines will themselves also be proportional.

(The Elements: Book $\text{XI}$: Proposition $37$)

Proof

Let $AB$, $CD$, $EF$ and $GH$ be four straight lines in proportion, so that:

$AB : CD = EF : GH$

Let similar and similarly situated parallelepipeds $KA$, $LC$, $ME$ and $NG$ be described on $AB$, $CD$, $EF$ and $GH$ respectively.

It is to be demonstrated that:

$KA : LC = ME : NG$

From Proposition $33$ of Book $\text{XI} $: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides:

$KA : LC = AB^3 : CD^3$

For the same reason:

$ME : NG = EF^3 : GH^3$

We have that:

$AB : CD = EF : GH$

Therefore:

$KA : LC = ME : NG$

$\Box$

Suppose that:

$KA : LC = ME : NG$

Then from Proposition $33$ of Book $\text{XI} $: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides:

$KA : LC = AB^3 : CD^3$

For the same reason:

$ME : NG = EF^3 : GH^3$

We have that:

$KA : LC = ME : NG$

Therefore:

$AB : CD = EF : GH$

$\blacksquare$

Historical Note

This proof is Proposition $37$ 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