IX.5

If a cube number by multiplying any number make a cube number, the multiplied number will also be cube.

Proof

For let the cube number A by multiplying any number B make the cube number C; I say that B is cube.

For let A by multiplying itself make D; therefore D is cube. IX. 3

Now, since A by multiplying itself has made D, and by multiplying B has made C, therefore, as A is to B, so is D to C. VII. 17

And since D, C are cube, they are similar solid numbers.

Therefore two mean proportional numbers fall between D, C. VIII. 19

And, as D is to C, so is A to B; therefore two mean proportional numbers fall between A, B also. VIII. 8

And A is cube; therefore B is also cube. VIII. 23