|If a cubic number multiplied by a cubic number makes some number, then the product is a cube.|
|Let the cubic number A multiplied by the cubic number B make C.
I say that C is cubic.
|Multiply A by itself to make D. Then D is a cube.||IX.3|
|Since A multiplied by itself makes D, and multiplied by B makes C, therefore A is to B as D is to C. And, since A and B are cubic numbers, therefore A and B are similar solid numbers. Therefore two mean proportional numbers fall between A and B, so that two mean proportional numbers fall between D and C also.||VII.17
|And D is a cube, therefore C is also a cube.||VIII.23|
|Therefore, if a cubic number multiplied by a cubic number makes some number, then the product is cubic.|
Next proposition: IX.5