# Multiples of Ratios of Numbers

## Theorem

In the words of Euclid:

If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.

## Proof

Let the number $A$ by multiplying the two numbers $B, C$ to make $D, E$.

We need to show that:

$B : C = D : E$

We have that:

$A \times B = D$

Therefore $B$ measures $D$ according to the units in $A$.

But the unit $F$ also measures $A$ according to the units in it.

Therefore $F$ measures $A$ the same number of times that $B$ measures $D$.

$F : A = B : D$

For the same reason:

$F : A = C : E$

Therefore also:

$B : D = C : E$
$B : C = D : E$

$\blacksquare$

## Historical Note

This theorem is Proposition $17$ of Book $\text{VII}$ of Euclid's The Elements.