If two magnitudes have not to one another the ratio which a number has to a number, the magnitudes will be incommensurable.
For let the two magnitudes A, B not have to one another the ratio which a number has to a number; I say that the magnitudes A, B are incommensurable.
For, if they are commensurable, A will have to B the ratio which a number has to a number. [X. 5]
But it has not; therefore the magnitudes A, B are incommensurable.
Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.
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