# Proposition: 7.12: Ratios of Numbers is Distributive over Addition

### (Proposition 12 from Book 7 of Euclid's “Elements”)

If any multitude whatsoever of numbers are proportional then as one of the leading (numbers is) to one of the following so (the sum of) all of the leading (numbers) will be to (the sum of) all of the following.

• Let any multitude whatsoever of numbers, $A$, $B$, $C$, $D$, be proportional, (such that) as $A$ (is) to $B$, so $C$ (is) to $D$.
• I say that as $A$ is to $B$, so $A$, $C$ (is) to $B$, $D$.

### Modern Formulation

In modern notation, this proposition states that if $\frac AB=\frac CD,$ then $\frac AB=\frac{A+C}{B+D}.$

Proofs: 1

Proofs: 1 2 3

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### References

#### Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

#### Adapted from CC BY-SA 3.0 Sources:

1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016