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ElementsX 8©

translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 8: If two magnitudes do not have a ratio to one another that an arithmos has to an arithmos, the magnitudes will be incommensurable.

For let two magnitudes, A, B, not have a ratio to one another that an arithmos has to an arithmos. I say that magnitudes A, B are incommensurable.

For if they will be commensurable, A will have a ratio to B that an arithmos has to an arithmos. Therefore, magnitudes A, B are incommensurable.
Therefore, if two magnitudes to one another, etc.

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