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Proposition 17

If a cubic number does not measure a cubic number, then neither does the side measure the side; and, if the side does not measure the side, then neither does the cube measure the cube.
Let the cubic number A not measure the cubic number B, and let C be the side of A, and D of B. I say that C does not measure D.
| For if C measures D, then A also measures B. But A does not measure B, therefore neither does C measure D. | VIII.15
Next, let C not measure D. I say that neither does A measure B.
If A measures B, then C also measures D. But C does not measure D, therefore neither does A measure B. | VIII.15
Therefore, if a cubic number does not measure a cubic number, then neither does the side measure the side; and, if the side does not measure the side, then neither does the cube measure the cube.
Q.E.D.

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(Forthcoming)

Book VIII Introduction - Proposition VIII.16 - Proposition VIII.18.

© 1996
D.E.Joyce
Clark University