# Ratios of Numbers is Distributive over Addition

## Theorem

In the words of Euclid:

If there be as many 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 $A : B = A + C : B + D$. We have that $A : B = C : D$.

So from Book $\text{VII}$ Definition $20$: Proportional, whatever aliquot part or aliquant part $A$ is of $B$, the same aliquot part or aliquant part is $C$ of $D$ also.

Therefore from:

Proposition $5$ of Book $\text{VII}$: Divisors obey Distributive Law

and:

Proposition $6$ of Book $\text{VII}$: Multiples of Divisors obey Distributive Law

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

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

$\blacksquare$

## Historical Note

This proof is Proposition $12$ of Book $\text{VII}$ of Euclid's The Elements.