Ratios of Multiples of Numbers

Theorem

As Euclid defined it:

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

(The Elements: Book VII: Proposition $18$)

Proof

Let two (natural) numbers $A, B$ by multiplying any number $C$ make $D, E$.

Then we need to show that $A : B = D : E$.

We have that $A \times C = D$.

So from Natural Number Multiplication is Commutative, also $C \times A = D$.

For the same reason, $C \times B = E$.

Therefore from Book VII Proposition 17: Multiples of Ratios of Numbers, $A : B = D : E$.

$\blacksquare$

Historical Note

This is Proposition 18 of Book VII of Euclid's The Elements.