IX.4

If a cube number by multiplying a cube number make some number, the product will be cube.

Proof

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

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.