If a number multiplied by itself makes a cubic number, then it itself is also cubic.

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

I say that *A* is also cubic.

Multiply *A* by *B* to make *C.* Since, then, *A* multiplied by itself makes *B,* and multiplied by *B* makes *C,* therefore *C* is a cube. And, since *A* multiplied by itself makes *B,* therefore *A* measures *B* according to the units in itself.

But the unit also measures *A* according to the units in it. Therefore the unit is to *A* as *A* is to *B.* And, since *A* multiplied by *B* makes *C,* therefore *B* measures *C* according to the units in *A.*

But the unit also measures *A* according to the units in it.

Therefore the unit is to *A* as *B* is to *C.* But the unit is to *A* as *A* is to *B,* therefore *A* is to *B* as *B* is to *C.*

And, since *B* and *C* are cubes, therefore they are similar solid numbers.

Therefore there are two mean proportional numbers between *B* and *C.* And *B* is to *C* as *A* is to *B.* Therefore there are two mean proportional numbers between *A* and *B* also.

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

Therefore, *if a number multiplied by itself makes a cubic number, then it itself is also cubic.*

Q.E.D.