Let the two given commensurable magnitudes be / A B /, / C D / of which / A B / is the less ; thus it is required to find the greatest common measure of / A B /, / C D /.
Now the magnitude / A B / either measures / C D / or it does not.
If then it measures it—and it measures itself also—/ A B / is a common measure of / A B /, / C D /.
And it is manifest that it is also the greatest ; for a greater magnitude than the magnitude / A B / will not measure / A B /.
Next, let / A B / not measure / C D /. Then, if the less be continually subtracted in turn from the greater, that which is left over will sometime measure the one before it, because / A B /, / C D / are not incommensurable ; [G.373] let / A B /, measuring / E D /, leave / E C / less than itself, let / E C /, measuring / F B /, leave / A F / less than itself, and let / A F / measure / C E /.
Since, then, / A F / measures / C E /, while / C E / measures / F B /,
therefore / A F / will also measure / F B /.
But it measures itself also ;
therefore / A F / will also measure the whole / A B /.
But / A B / measures / D E / ;
therefore / A F / will also measure / E D /.
But it measures / C E / also ;
therefore it also measures the whole / C D /.
Therefore / A F / is a common measure of / A B /, / C D /.
I say next that it is also the greatest.
For, if not, there will be some magnitude greater than / A F / which will measure / A B /, / C D /. Let it be G.
Since then G measures / A B /, while / A B / measures / E D /,
therefore G will also measure / E D /.
But it measures the whole / C D / also ;
therefore G will also measure the remainder / C E /.
But / C E / measures / F B / ;
therefore G will also measure / F B /.
But it measures the whole / A B / also,
and it will therefore measure the remainder / A F /, the greater the less : which is impossible.
Therefore no magnitude greater than / A F / will measure / A B /, / C D / ; therefore / A F / is the greatest common measure of / A B /, / C D /.
Therefore the greatest common measure of the two given commensurable magnitudes / A B /, / C D / has been found. Q. E. D.
PORISM. From this it is manifest that, if a magnitude measure two magnitudes, it will also measure their greatest common measure.
