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.

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VIII.15

For if C measures D, then A also measures B. But A does not measure B, therefore neither does C measure D.

Next, let C not measure D.

I say that neither does A measure B.

VIII.15

If A measures B, then C also measures D. But C does not measure D, therefore neither does A measure B.

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.

Guide

This proposition is simply the contrapositive of VIII.15.

“Contrariwise,” continued Tweedledee, “if it was so, it would be; but as it isn’t, it ain’t. That’s logic.”