Ratios of Multiples of Numbers

Theorem

In the words of Euclid:

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

(The Elements: Book $\text{VII}$: 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 Proposition $17$ of Book $\text{VII} $: Multiples of Ratios of Numbers:

$A : B = D : E$

$\blacksquare$

Historical Note

This proof is Proposition $18$ of Book $\text{VII}$ 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 $\text{VII}$. Propositions