Ratios of Multiples of Numbers

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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 $18$)


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:

$A : B = D : E$


Historical Note

This proof is Proposition $18$ of Book $\text{VII}$ of Euclid's The Elements.