If a cubic number multiplied by a cubic number makes some number, then the product is a cube.

Let the cubic number *A* multiplied by the cubic number *B* make *C.*

I say that *C* is cubic.

Multiply *A* by itself to make *D.* Then *D* is a cube.

Since *A* multiplied by itself makes *D,* and multiplied by *B* makes *C,* therefore *A* is to *B* as *D* is to *C.* And, since *A* and *B* are cubic numbers, therefore *A* and *B* are similar solid numbers. Therefore two mean proportional numbers fall between *A* and *B,* so that two mean proportional numbers fall between *D* and *C* also.

And *D* is a cube, therefore *C* is also a cube.

Therefore, *if a cubic number multiplied by a cubic number makes some number, then the product is cubic.*

Q.E.D.