Between two cubic numbers there are two mean proportional numbers, and the cube has to the cube the triplicate ratio of that which the side has to the side.

Let *A* and *B* be cubic numbers, and let *C* be the side of *A,* and *D* of *B.*

I say that between *A* and *B* there are two mean proportional numbers, and *A* has to *K* the ratio triplicate of that which *C* has to *D.*

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

Now, since *A* is a cube, and *C* its side, and *C* multiplied by itself makes *E,* therefore *C* multiplied by itself makes *E* and multiplied by *E* makes *A.* For the same reason also *D* multiplied by itself makes *G* and multiplied by *G* makes *B.*

And, since *C* multiplied by the numbers *C* and *D* makes *E* and *F* respectively, therefore *C* is to *D* as *E* is to *F.* For the same reason also *C* is to *D* as *F* is to *G.* Again, since *C* multiplied by the numbers *E* and *F* makes *A* and *H* respectively, therefore *E* is to *F* as *A* is to *H.* But *E* is to *F* as *C* is to *D.* Therefore *C* is to *D* as *A* is to *H.*

Again, since the numbers *C* and *D* multiplied by *F* make *H* and *K* respectively, therefore *C* is to *D* as *H* is to *K.* Again, since *D* multiplied by each of the numbers *F* and *G* makes *K* and *B* respectively, therefore *F* is to *G* as *K* is to *B.*

But *F* is to *G* as *C* is to *D,* therefore *C* is to *D* as *A* is to *H,* as *H* is to *K,* and as *K* is to *B.*

Therefore *H* and *K* are two mean proportionals between *A* and *B.*

I say next that *A* also has to *B* the ratio triplicate of that which *C* has to *D.*

Since *A, H, K,* and *B* are four numbers in proportion, therefore *A* has to *B* the ratio triplicate of that which *A* has to *H.*

But *A* is to *H* as *C* is to *D,* therefore *A* also has to *B* the ratio triplicate of that which *C* has to *D.*

Therefore, *Between two cubic numbers there are two mean proportional numbers, and the cube has to the cube the triplicate ratio of that which the side has to the side.*

Q.E.D.

This proposition is used in VIII.15.