X.8

If two magnitudes have not to one another the ratio which a number has to a number, the magnitudes will be incommensurable.

Proof

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.

Therefore etc.