Ratio of Commensurable Magnitudes

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 $A$.

From Ratios of Fractions in Lowest Terms:
 * $\dfrac C A = \dfrac 1 D$

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 $B$.

From Ratios of Fractions in Lowest Terms:
 * $\dfrac C B = \dfrac 1 E$

So by Equality of Ratios Ex Aequali:
 * $\dfrac A B = \dfrac D E$

Hence the result.