Magnitudes commensurable with the same magnitude are also commensurable with one another.

Let each of the magnitudes *A* and *B* be commensurable with *C*.

I say that *A* is also commensurable with *B*.

Since *A* is commensurable with *C*, therefore *A* has to *C* the ratio which a number has to a number. Let it have the ratio which *D* has to *E*. Again, since *C* is commensurable with *B*, therefore *C* has to *B* the ratio which a number has to a number. Let it have the ratio which *F* has to *G*.

And, given any number of ratios we please, namely the ratio which *D* has to *E* and that which *F* has to *G*, take the numbers *H*, *K*, and *L* continuously in the given ratios, so that *D* is to *E* as *H* is to *K*, and *F* is to *G* as *K* is to *L*.

Since *A* is to *C* as *D* is to *E*, while *D* is to *E* as *H* is to *K*, therefore *A* is to *C* as *H* is to *K*. Again, since *C* is to *B* as *F* is to *G*, while *F* is to *G* as *K* is to *L*, therefore *C* is to *B* as *K* is to *L*.

But *A* is to *C* as *H* is to *K*, therefore, *ex aequali*, *A* is to *B* as *H* is to *L*.

Therefore *A* has to *B* the ratio which a number has to a number. Therefore *A* is commensurable with *B*.

Therefore, *magnitudes commensurable with the same magnitude are also commensurable with one another.*

Q.E.D.

This proposition is used in frequently in Book X starting with the next proposition. It is also used in XIII.11.