X.7

Incommensurable magnitudes have not to one another the ratio which a number has to a number.

Proof

Let A, B be incommensurable magnitudes; I say that A has not to B the ratio which a number has to a number.

For, if A has to B the ratio which a number has to a number, A will be commensurable with B. X. 6

But it is not; therefore A has not to B the ratio which a number has to a number.

Therefore etc.