# Ratio of Commensurable Magnitudes

## Theorem

In the words of Euclid:

Commensurable magnitudes have to one another the ratio which a number has to a number.

## Proof

Let $A$ and $B$ be commensurable magnitudes.

By definition of commensurable, some magnitude will measure them both.

Let $C$ be a common measure of $A$ and $B$.

Let $D$ be the number of times $C$ is the measure of $A$.

Let $E$ be the number of times $C$ is the measure of $B$.

Since:

$C$ measures $A$ according to the number of units of $D$

and

the unit measures $D$ according to the number of units of $D$

it follows that the unit measures $D$ the same number of times $C$ measures $A$.

$\dfrac C A = \dfrac 1 D$

and:

$\dfrac A C = \dfrac D 1 = D$

Similarly, since:

$C$ measures $B$ according to the number of units of $E$

and

the unit measures $E$ according to the number of units of $E$

it follows that the unit measures $E$ the same number of times $C$ measures $B$.

$\dfrac C B = \dfrac 1 E$
$\dfrac A B = \dfrac D E$

Hence the result.

$\blacksquare$

## Historical Note

This proof is Proposition $5$ of Book $\text{X}$ of Euclid's The Elements.