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

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](elem.7.17)]

And, since A, B are cube numbers, A, B are similar solid numbers.

Therefore two mean proportional numbers fall between A, B; [[VIII. 19](elem.8.19)] so that two mean proportional numbers will fall between D, C also. [[VIII. 8](elem.8.8)]

And D is cube; therefore C is also cube [[VIII. 23](elem.8.23)] Q. E. D.