Let the two magnitudes A and B not have to one another the ratio which a number has to a number.
I say that the magnitudes A and B are incommensurable.
For, if they are commensurable, then A has to B the ratio which a number has to a number.
But it does not, therefore the magnitudes A and B are incommensurable.
Therefore, if two magnitudes do not have to one another the ratio which a number has to a number, then the magnitudes are incommensurable.