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