# Ratio Equals its Multiples

## Theorem

In the words of Euclid:

(*The Elements*: Book $\text{V}$: Proposition $15$)

That is:

- $a : b \implies ma = mb$

## Proof

Let $AB$ be the same multiple of $C$ that $DE$ is of $F$.

So as many magnitudes as there are in $AB$ equal to $C$, so many are there also in $DE$ equal to $F$.

Let $AB$ be divided into the magnitudes $AG, GH, HB$ equal to $C$.

Let $DE$ be divided into the magnitudes $DK, KL, LE$ equal to $F$.

Then the number of magnitudes $AG, GH, GB$ is the same as the number of magnitudes in $DK, KL, LE$.

We have that $AG = GH = HB$ and $DK = KL = LE$.

So from Ratios of Equal Magnitudes it follows that $AG : DK = GH : KL = HB : LE$.

Then from Sum of Components of Equal Ratios $AG : DK = AB : DE$.

But $AG = C$ and $DK = F$.

$\blacksquare$

## Historical Note

This theorem is Proposition $15$ of Book $\text{V}$ of Euclid's *The Elements*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 2*(2nd ed.) ... (previous) ... (next): Book $\text{V}$. Propositions