Incommensurable magnitudes do not have to one another the ratio which a number has to a number.

Let *A* and *B* be incommensurable magnitudes.

I say that *A* does not have to *B* the ratio which a number has to a number.

If *A* does have to *B* the ratio which a number has to a number, then *A* is commensurable with *B*.

But it is not, therefore *A* does not have to *B* the ratio which a number has to a number.

Therefore, *incommensurable magnitudes do not have to one another the ratio which a number has to a number.*

Q.E.D.