Similar solid numbers have to one another the ratio which a cubic number has to a cubic number.

Let *A* and *B* be similar solid numbers.

I say that *A* has to *B* the ratio which cubic number has to cubic number.

Since *A* and *B* are similar solid numbers, therefore two mean proportional numbers *C* and *D* fall between *A* and *B.*

Take *E, F, G,* and *H,* the least numbers of those which have the same ratio with *A, C, D,* and *B,* and equal with them in multitude.

Therefore the extremes of them, *E* and *H,* are cubes. And *E* is to *H* as *A* is to *B,* therefore *A* also has to *B* the ratio which a cubic number has to a cubic number.

Therefore, *similar solid numbers have to one another the ratio which a cubic number has to a cubic number.*

Q.E.D.