Euclid, Elements X 7©
translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 7: Incommensurable magnitudes do not have a ratio to one another that an arithmos has to another.

Let there be incommensurable magnitudes A, B. I say that A does not have a ratio to B that an arithmos has to an arithmos.

For if A has to B a ratio that an arithmos has to an arithmos, A will be commensurable with B. But it isn’t. Therefore, A does not have to B a ratio that an arithmos has to an arithmos.

Therefore, incommensurable magnitudes do not have a ratio to one another, etc.

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