For let A by multiplying itself make D; therefore D is cube. [[IX. 3](elem.9.3)]

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](elem.7.17)]

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

Therefore two mean proportional numbers fall between D, C. [[VIII. 19](elem.8.19)]

And, as D is to C, so is A to B; therefore two mean proportional numbers fall between A, B also. [[VIII. 8](elem.8.8)]

And A is cube; therefore B is also cube. [[VIII. 23](elem.8.23)]