If two magnitudes are commensurable, and one of them is incommensurable with any magnitude, then the remaining one is also incommensurable with the same.

Let *A* and *B* be two commensurable magnitudes, and let one of them, *A*, be incommensurable with some other magnitude *C*.

I say that the remaining one, *B*, is also incommensurable with *C*.

If *B* is commensurable with *C*, while *A* is also commensurable with *B*, then *A* is also commensurable with *C*.

But it is also incommensurable with it, which is impossible. Therefore *B* is not commensurable with *C*. Therefore it is incommensurable with it.

Therefore, *if two magnitudes are commensurable, and one of them is incommensurable with any magnitude, then the remaining one is also incommensurable with the same.*

Q.E.D.