If two magnitudes do not have to one another the ratio which a number has to a number, then the magnitudes are incommensurable.

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.*

Q.E.D.