# 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.*

(*The Elements*: Book $\text{X}$: Proposition $5$)

## 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:

and

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:

and

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.

$\blacksquare$

## Historical Note

This proof is Proposition $5$ of Book $\text{X}$ 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 $\text{X}$. Propositions