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Proposition 16

If a square does not measure a square, then neither does the side measure the side; and, if the side does not measure the side, then neither does the square measure the square.
Let A and B be square numbers, and let C and D be their sides, and let A not measure B. I say that neither does C measure D.
| If C measures D, A also measures B. But A does not measure B, therefore neither does C measure D. | VIII.14
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.14
Therefore, if a square does not measure a square, then neither does the side measure the side; and, if the side does not measure the side, then neither does the square measure the square.
Q.E.D.

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(Forthcoming)

Book VIII Introduction - Proposition VIII.15 - Proposition VIII.17.

© 1996
D.E.Joyce
Clark University