If four magnitudes are proportional, and the first is commensurable with the second, then the third also is commensurable with the fourth; but, if the first is incommensurable with the second, then the third also is incommensurable with the fourth.

Let *A*, *B*, *C*, and *D* be four magnitudes in proportion, so that *A* is to *B* as *C* is to *D*, and let *A* be commensurable with *B*.

I say that *C* is also commensurable with *D*.

Since *A* is commensurable with *B*, therefore *A* has to *B* the ratio which a number has to a number.

And *A* is to *B* as *C* is to *D*, therefore *C* also has to *D* the ratio which a number has to a number. Therefore *C* is commensurable with *D*.

Next, let *A* be incommensurable with *B*.

I say that *C* is also incommensurable with *D*.

Since *A* is incommensurable with *B*, therefore *A* does not have to *B* the ratio which a number has to a number.

And *A* is to *B* as *C* is to *D*, therefore neither has *C* to *D* the ratio which a number has to a number. Therefore *C* is incommensurable with *D*.

Therefore, *if four magnitudes are proportional, and the first is commensurable with the second, then the third also is commensurable with the fourth; but, if the first is incommensurable with the second, then the third also is incommensurable with the fourth.*

Q.E.D.

This proposition is used in repeatedly in Book X starting with X.14. It is also used in the previous proposition which was, no doubt, not in the original *Elements*.