Ratios of Numbers is Distributive over Addition

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