# Commensurability of Elements of Proportional Magnitudes

## Theorem

In the words of Euclid:

If four magnitudes be proportional, and the first be commensurable with the second, the third will also be commensurable with the fourth; and, if the first be incommensurable with the second, the third will also be incommensurable with the fourth.

## Proof

Let $A$, $B$, $C$ and $D$ be four magnitudes in proportion:

$A : B = C : D$

Let $A$ be commensurable with $B$.

Then from Ratio of Commensurable Magnitudes:

$A$ has to $B$ the ratio which a number has to a number.

As $A : B = C : D$ it follows that:

$C$ has to $D$ the ratio which a number has to a number.

From Magnitudes with Rational Ratio are Commensurable it follows that:

$C$ is commensurable with $D$.

$\Box$

Let $A$ be incommensurable with $B$.

$A$ does not have to $B$ the ratio which a number has to a number.

As $A : B = C : D$ it follows that:

$C$ does not have to $D$ the ratio which a number has to a number.

From Magnitudes with Irrational Ratio are Incommensurable it follows that:

$C$ is incommensurable with $D$.

$\blacksquare$

## Historical Note

This proof is Proposition $11$ of Book $\text{X}$ of Euclid's The Elements.