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*.

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.

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.