If two incommensurable magnitudes are added together, the sum is also incommensurable with each of them; but, if the sum is incommensurable with one of them, then the original magnitudes are also incommensurable.

Let the two incommensurable magnitudes *AB* and *BC* be added together.

I say that the whole *AC* is also incommensurable with each of the magnitudes *AB* and *BC*.

For, if *CA* and *AB* are not incommensurable, then some magnitude *D* measures them.

Since then *D* measures *CA* and *AB*, therefore it also measures the remainder *BC*. But it also measures *AB*, therefore *D* measures *AB* and *BC*. Therefore *AB* and *BC* are commensurable, but they were also, by hypothesis, incommensurable, which is impossible.

Therefore no magnitude measures *CA* and *AB*. Therefore *CA* and *AB* are incommensurable.

Similarly we can prove that *AC* and *CB* are also incommensurable. Therefore *AC* is incommensurable with each of the magnitudes *AB* and *BC*.

Next, let *AC* be incommensurable with one of the magnitudes *AB* or *BC*.

First, let it be incommensurable with *AB*.

I say that *AB* and *BC* are also incommensurable.

For, if they are commensurable, then some magnitude *D* measures them.

Since, then, *D* measures *AB* and *BC*, therefore it also measures the whole *AC*. But it also measures *AB*, therefore *D* measures *CA* and *AB*. Therefore *CA* and *AB* are commensurable, but they were also, by hypothesis, incommensurable, which is impossible.

Therefore no magnitude measures *AB* and *BC*. Therefore *AB* and *BC* are incommensurable.

Therefore, *if two incommensurable magnitudes are added together, the sum is also incommensurable with each of them; but, if the sum is incommensurable with one of them, then the original magnitudes are also incommensurable.*

Q.E.D.