Ratios of Numbers is Distributive over Addition

Theorem

 * ''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.

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 $A : B = A + C : B + D$.


 * Euclid-VII-12.png

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

So from, whatever part or parts $A$ is of $B$, the same part or parts is $C$ of $D$ also.

Therefore from:

and:

$A + C$ is the same part or parts of $C + D$ that $A$ is of $B$.

So from, $A : B = A + C : B + D$.