If a cubic number measures a cubic number, then the side also measures the side; and, if the side measures the side, then the cube also measures the cube.

Let the cubic number *A* measure the cube *B,* and let *C* be the side of *A* and *D* the side of *B.*

I say that *C* measures *D.*

Multiply *C* by itself to make *E,* multiply *D* by itself to make *G,* multiply *C* by *D* to make *F,* and multiply *C* and *D* by *F* to make *H* and *K* respectively.

Now it is clear that *E, F,* and *G* and *A, H, K,* and *B* are continuously proportional in the ratio of *C* to *D.* And, since *A, H, K,* and *B* are continuously proportional, and *A* measures *B* and *G* therefore it also measures *H.*

And *A* is to *H* as *C* is to *D,* therefore *C* also measures *D.*

Next, let *C* measure *D.*

I say that *A* also measures *B.*

With the same construction, we can prove in a similar manner that *A, H, K,* and *B* are continuously proportional in the ratio of *C* to *D.* And, since *C* measures *D,* and *C* is to *D* as *A* is to *H,* therefore *A* also measures *H,* so that *A* measures *B* also.

Therefore, *if a cubic number measures a cubic number, then the side also measures the side; and, if the side measures the side, then the cube also measures the cube.*

Q.E.D.

This proposition is used to prove its contrapositive, VIII.17.