For let A by multiplying itself make D;
therefore D is cube. [IX. 3]
And, 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 A, B are cube numbers, A, B are similar solid numbers.
Therefore two mean proportional numbers fall between A, B; [VIII. 19]
so that two mean proportional numbers will fall between D, C also. [VIII. 8]
And D is cube;
therefore C is also cube [VIII. 23]
Q. E. D.