| If two numbers have to one another the ratio which a cubic number has to a cubic number, and the first is a cube, then the second is also a cube. | ||
| Let the two numbers A and B have to one another the ratio which the cubic number C has to the cubic number D, and let A be a cube. | ||
| I say that B is also a cube. | ||
| Since C and D are cubes, C and D are similar solid numbers, therefore two mean proportional numbers fall between C and D. | VIII.19 | |
| Since as many numbers fall in continued proportion between those which have the same ratio with C and D as fall between C and D, therefore two mean proportional numbers E and F fall between A and B. | VIII.18 | |
| Since, then, the four numbers A, E, F, and B are in continued proportion, and A is a cube, therefore B is also a cube. | VIII.23 | |
| Therefore, if two numbers have to one another the ratio which a cubic number has to a cubic number, and the first is a cube, then the second is also a cube. | ||
| Q.E.D. | ||
Book VIII Introduction - Proposition VIII.24 - Proposition VIII.26.