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

Let the two given commensurable magnitudes be *AB* and *CD* with *AB* the less.

It is required to find the greatest common measure of *AB* and *CD*.

Now the magnitude *AB* either measures *CD* or it does not.

If it measures it, and it does measures itself, then *AB* is a common measure of *AB* and *CD*. And it is manifest that it is also the greatest, for a greater magnitude than the magnitude *AB* does not measure *AB*.

Next, let *AB* not measure *CD*.

Then, if the less is continually subtracted in turn from the greater, then that which is left over will sometime measure the one before it, because *AB* and *CD* are not incommensurable.

Let *AB*, measuring *ED*, leave *EC* less than itself, let *EC*, measuring *FB*, leave *AF* less than itself, and let *AF* measure *CE*.

Since, then, *AF* measures *CE*, while *CE* measures *FB*, therefore *AF* also measures *FB*. But it measures itself also, therefore *AF* also measures the whole *AB*. But *AB* measures *DE*, therefore *AF* also measures *ED*. But it measures *CE* also, therefore it also measures the whole *CD*.

Therefore *AF* is a common measure of *AB* and *CD*.

I say next that it is also the greatest.

If not, then there there is some magnitude *G* greater than *AF* which measures *AB* and *CD*. Since then *G* measures *AB*, while *AB* measures *ED*, therefore *G* also measures *ED*.

But it measures the whole *CD* also, therefore *G* measures the remainder *CE*. But *CE* measures *FB*, therefore *G* also measures *FB*.

But it measures the whole *AB* also, and it therefore measures the remainder *AF*, the greater the less, which is impossible.

Therefore no magnitude greater than *AF* measures *AB* and *CD*. Therefore *AF* is the greatest common measure of *AB* and *CD*.

Therefore the greatest common measure of the two given commensurable magnitudes *AB* and *CD* has been found.

Q.E.D.

From this it is clear that, *if a magnitude measures two magnitudes, then it also measures their greatest common measure*.

This proposition and its corollary are used in the next proposition.