## Proposition 15

 If two commensurable magnitudes are added together, then the whole is also commensurable with each of them; and, if the whole is commensurable with one of them, then the original magnitudes are also commensurable. Let the two commensurable magnitudes AB and BC be added together. I say that the whole AC is also commensurable with each of the magnitudes AB and BC. Since AB and BC are commensurable, some magnitude D measures them. Since then D measures AB and BC, therefore it also measures the whole AC. But it measures AB and BC also, therefore D measures AB, BC, and AC. Therefore AC is commensurable with each of the magnitudes AB and BC. X.Def.1 Next, let AC be commensurable with AB. I say that AB and BC are also commensurable. Since AC and AB are commensurable, some magnitude D measures them. Since then D measures CA and AB, therefore it also measures the remainder BC. But it measures AB also, therefore D measures AB and BC. Therefore AB and BC are commensurable. X.Def.1 Therefore, if two commensurable magnitudes are added together, then the whole is also commensurable with each of them; and, if the whole is commensurable with one of them, then the original magnitudes are also commensurable. Q.E.D.
This fundamental proposition on commensurability of sums and differences is used in very frequently in Book X starting with X.17. It is also used in XIII.11.