To find the greatest common measure of three given commensurable magnitudes.

Let *A*, *B*, and *C* be the three given commensurable magnitudes.

It is required to find the greatest common measure of *A*, *B*, and *C*.

Take the greatest common measure *D* of the two magnitudes *A* and *B*.

Either *D* measures *C*, or it does not measure it.

First, let it measure it.

Since then *D* measures *C*, while it also measures *A* and *B*, therefore *D* is a common measure of *A*, *B*, and *C*. And it is manifest that it is also the greatest, for a greater magnitude than the magnitude *D* does not measure *A* and *B*.

Next, let *D* not measure *C*.

I say first that *C* and *D* are commensurable.

Since *A*, *B*, and *C* are commensurable, some magnitude measures them, and this of course measures *A* and *B* also, so that it also measures the greatest common measure of *A* and *B*, namely *D*.

But it also measures *C*, so that the said magnitude measures *C* and *D*, therefore *C* and *D* are commensurable.

Now take their greatest common measure *E*.

Since *E* measures *D*, while *D* measures *A* and *B*, therefore *E* also measures *A* and *B*. But it measures *C* also, therefore *E* measures *A*, *B*, and *C*. Therefore *E* is a common measure of *A*, *B*, and *C*.

I say next that it is also the greatest.

For, if possible, let there be some magnitude *F* greater than *E*, and let it measure *A*, *B*, and *C*.

Now, since *F* measures *A*, *B*, and *C*, it also measures *A* and *B*, and therefore measures the greatest common measure of *A* and *B*.

But the greatest common measure of *A* and *B* is *D*, therefore *F* measures *D*.

But it measures *C* also, therefore *F* measures *C* and *D*. Therefore *F* also measures the greatest common measure of *C* and *D*. But that is *E*, therefore *F* measures *E*, the greater the less, which is impossible.

Therefore no magnitude greater than the magnitude *E* measures *A*, *B*, and *C*. Therefore *E* is the greatest common measure of *A*, *B*, and *C* if *D* does not measure *C*, but if it measures it, then *D* is itself the greatest common measure.

Therefore the greatest common measure of the three given commensurable magnitudes has been found.

Q.E.D.

From this it is clear that, *if a magnitude measures three magnitudes, then it also measures their greatest common measure. The greatest common measure can be found similarly for more magnitudes, and the corollary extended.*