If a cubic number multiplied by any number makes a cubic number, then the multiplied number is also cubic.

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

I say that *B* is cubic.

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

Now, 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 *D* and *C* are cubes, therefore they are similar solid numbers. Therefore two mean proportional numbers fall between *D* and *C.* And *D* is to *C* as *A* is to *B,* therefore two mean proportional numbers fall between *A* and *B,* too.

And *A* is a cube, therefore *B* is also a cube.

Therefore, *if a cubic number multiplied by any number makes a cubic number, then the multiplied number is also cubic.*

Q.E.D.