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

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

I say that *B* is cubic.

Take *C,* the side of *A.* Multiply *C* by itself make *D.* It is then manifest that *C* multiplied by *D* makes *A.*

Now, since *C* multiplied by itself makes *D,* therefore *C* measures *D* according to the units in itself. But further the unit also measures *C* according to the units in it, therefore the unit is to *C* as *C* is to *D.*

Again, since *C* multiplied by *D* makes *A,* therefore *D* measures *A* according to the units in *C.* But the unit also measures *C* according to the units in it, therefore the unit is to *C* as *D* is to *A.* But the unit is to *C* as *C* is to *D,* therefore the unit is to *C* as *C* is to *D,* and as *D* is to *A.*

Therefore between the unit and the number *A* two mean proportional numbers *C* and *D* have fallen in continued proportion.

Again, 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.*

But between the unit and *A* two mean proportional numbers have fallen, therefore two mean proportional numbers also fall between *A* and *B.*

But, if two mean proportional numbers fall between two numbers, and the first is a cube, then the second is also a cube. And *A* is a cube, therefore *B* is also a cube.

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

Q.E.D.