X.11

If four magnitudes be proportional, and the first be commensurable with the second, the third will also be commensurable with the fourth; and, if the first be incommensurable with the second, the third will also be incommensurable with the fourth.

Proof

Let A, B, C, D be four magnitudes in proportion, so that, as A is to B, so is C to D, and let A be commensurable with B; I say that C will also be commensurable with D.

For, since A is commensurable with B, therefore A has to B the ratio which a number has to a number. X. 5

And, as A is to B, so is C to D; therefore C also has to D the ratio which a number has to a number; therefore C is commensurable with D. X. 6

Next, let A be incommensurable with B; I say that C will also be incommensurable with D.

For, since A is incommensurable with B, therefore A has not to B the ratio which a number has to a number. X. 7

And, as A is to B, so is C to D; therefore neither has C to D the ratio which a number has to a number; therefore C is incommensurable with D. X. 8

Therefore etc.