X.12

Magnitudes commensurable with the same magnitude are commensurable with one another also.

Proof

For let each of the magnitudes A, B be commensurable with C; I say that A is also commensurable with B.

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

Let it have the ratio which D has to E.

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

Let it have the ratio which F has to G.

And, given any number of ratios we please, namely the ratio which D has to E and that which F has to G, let the numbers H, K, L be taken continuously in the given ratios; cf. VIII. 4 so that, as D is to E, so is H to K, and, as F is to G, so is K to L.

Since, then, as A is to C, so is D to E, while, as D is to E, so is H to K, therefore also, as A is to C, so is H to K. V. 11

Again, since, as C is to B, so is F to G, while, as F is to G, so is K to L, therefore also, as C is to B, so is K to L. V. 11

But also, as A is to C, so is H to K; therefore, ex aequali, as A is to B, so is H to L. V. 22

Therefore A has to B the ratio which a number has to a number; therefore A is commensurable with B. X. 6

Therefore etc. Q. E. D.