| If as many numbers as we please beginning from a unit are in continued proportion, and the number after the unit is square, then all the rest are also square; and if the number after the unit is cubic, then all the rest are also cubic. | ||
| Let there be as many numbers as we please, A, B, C, D, E, and F, beginning from a unit and in continued proportion, and let A, the number after the unit, be square. | ||
| I say that all the rest are also square.
Now it was proved that B, the third from the unit, is square as are all those which leave out one. I say that all the rest are also square. |
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| Since A, B, and C are in continued proportion, and A is square, therefore C is also square. Again, since B, C, and D are in continued proportion, and B is square, therefore D is also square. Similarly we can prove that all the rest are also square. | VIII.22 | |
| Next, let A be a cube.
I say that all the rest are also cubes. | ||
| Now it was proved that C, the fourth from the unit, is a cube as are all those which leave out two. | IX.8 | |
| I say that all the rest are also cubic.
Since the unit is to A as A is to B, therefore the unit measures A the same number of times as A measures B. But the unit measures A according to the units in it, therefore A also measures B according to the units in itself, therefore A multiplied by itself makes B. | ||
| And A is cubic. But, if cubic number multiplied by itself makes some number, then the product is also a cube, therefore B is also a cube. | IX.3 | |
| And, since the four numbers A, B, C, and D are in continued proportion, and A is a cube, therefore D also is a cube. | VIII.23 | |
| For the same reason E is also a cube, and similarly all the rest are cubes. | ||
| Therefore, if as many numbers as we please beginning from a unit are in continued proportion, and the number after the unit is square, then all the rest are also square; and if the number after the unit is cubic, then all the rest are also cubic. | ||
| Q.E.D. | ||
Book IX Introduction - Proposition IX.8 - Proposition IX.10.