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.

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VIII.14

If C measures D, 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.14

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

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.

Guide

This is simply the contrapositive of VIII.14. It is unclear why papyrus was wasted to state and prove it.