Multiples of Ratios of Numbers

Theorem

As Euclid defined it:

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

(The Elements: Book VII: Proposition $17$)

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 $D$.

So from Book VII Definition 20: Proportional, $F : A = B : D$.

For the same reason, $F : A = C : E$.

Therefore also $B : D = C : E$.

So from Book VII Proposition 13: Proportional Numbers are Proportional Alternately, $B : C = D : E$.

$\blacksquare$

Historical Note

This is Proposition 17 of Book VII of Euclid's The Elements.