## Proposition 12

 Magnitudes commensurable with the same magnitude are also commensurable with one another. Let each of the magnitudes A and B be commensurable with C. I say that A is also commensurable with B. Since A is commensurable with C, therefore A has to C the ratio which a number has to a number. 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. Let it have the ratio which F has to G. X.5 And, given any number of ratios we please, namely the ratio which D has to E and that which F has to G, take the numbers H, K, and L continuously in the given ratios, so that D is to E as H is to K, and F is to G as K is to L. VIII.4 Since A is to C as D is to E, while D is to E as H is to K, therefore A is to C as H is to K. Again, since C is to B as F is to G, while F is to G as K is to L, therefore C is to B as K is to L. V.11 But A is to C as H is to K, therefore, ex aequali, A is to B as H is 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, magnitudes commensurable with the same magnitude are also commensurable with one another. Q.E.D.
The proof is primarily an application of VIII.4.

This proposition is used in frequently in Book X starting with the next proposition. It is also used in XIII.11.