# Definition:Euclid's Definitions - Book V/5 - Equality of Ratios

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## Definition

In the words of Euclid:

*Magnitudes are said to***be in the same ratio**, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

(*The Elements*: Book $\text{V}$: Definition $5$)

## Historical Note

The definition of equality of ratios was first put together by Eudoxus of Cnidus.

Some sources are of the opinion that this definition is "epochal".

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 2*(2nd ed.) ... (previous) ... (next): Book $\text{V}$. Definitions