Ratios of Numbers is Distributive over Addition

Theorem

As Euclid defined it:

If there be as many (natural) numbers as we please in proportion, then, as one of the antecedents is to one of the consequents, so are all the antecedents to all of the consequents.

(The Elements: Book VII: Proposition $12$)

Proof

Let $A, B, C, D$ be as many numbers as we please in proportion, so that $A : B = C : D$.

We need to show that $A : B = A + C : B + D$.

We have that $A : B = C : D$.

So from Book VII Definition 20: Proportional, whatever part or parts $A$ is of $B$, the same part or parts is $C$ of $D$ also.

Therefore from Book VII Proposition 5: Divisors Obey Distributive Law and Book VII Proposition 6: Multiples of Divisors Obey Distributive Law, $A + C$ is the same part or parts of $C + D$ that $A$ is of $B$.

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

$\blacksquare$

Historical Note

This is Proposition 12 of Book VII of Euclid's The Elements.