# Proposition: 5.15: Ratio Equals its Multiples

### (Proposition 15 from Book 5 of Euclid's “Elements”)

Parts have the same ratio as similar multiples, taken in corresponding order.

• For let $AB$ and $DE$ be equal multiples of $C$ and $F$ (respectively).
• I say that as $C$ is to $F$, so $AB$ (is) to $DE$.

### Modern Formulation

In modern notation, this proposition reads that $\frac\alpha\beta=\frac{m\,\alpha}{m\,\beta},$

for all positive real numbers $$\alpha$$, $$\beta$$, and all multiples of aliquot parts $$m > 1$$.

### Generalized Modern Formulation

since $\frac mm=1$, this follows immediately from the existence and uniqueness of real 1.

Proofs: 1

Proofs: 1 2 3 4 5
Sections: 6

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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