Let / A B /, / C D / be the two given numbers not prime to one another.
Thus it is required to find the greatest common measure of / A B /, / C D /.
If now / C D / measures / A B /—and it also measures itself—/ C D / is a common measure of / C D /, / A B /.
And it is manifest that it is also the greatest; for no greater number than / C D / will measure / C D /.
But, if / C D / does not measure / A B /, then, the less of the numbers / A B /, / C D / being continually subtracted from the greater, some number will be left which will measure the one before it.
For an unit will not be left; otherwise / A B /, / C D / will be prime to one another [G.250], which is contrary to the hypothesis.
Therefore some number will be left which will measure the one before it.
Now let / C D /, measuring / E B /, leave / A E / less than itself, let / A E /, measuring / F D /, leave / C F / less than itself, and let / C F / measure / A E /.
Since then, / C F / measures / A E /, and / A E / measures / F D /,
therefore / C F / will also measure / F D /.
But it also measures itself;
therefore it will also measure the whole / C D /.
But / C D / measures / E B /;
therefore / C F / also measures / E B /.
But it also measures / A E /;
therefore it will also measure the whole / A B /.
But it also measures / C D /;
therefore / C F / measures / A B /, / C D /.
Therefore / C F / is a common measure of / A B /, / C D /.
I say next that it is also the greatest.
For, if / C F / is not the greatest common measure of / A B /, / C D /, some number which is greater than / C F / will measure the numbers / A B /, / C D /.
Let such a number measure them, and let it be G.
Now, since G measures / C D /, while / C D / measures / E B /,
G also measures / E B /.
But it also measures the whole / A B /;
therefore it will also measure the remainder / A E /.
But / A E / measures / F D /;
therefore G will also measure / F D /.
But it also measures the whole / C D /;
therefore it will also measure the remainder / C F /, that is, the greater will measure the less: which is impossible.
Therefore no number which is greater than / C F / will measure the numbers / A B /, / C D /; therefore / C F / is the greatest common measure of / A B /, / C D /.
PORISM. From this it is manifest that, if a number measure two numbers, it will also measure their greatest common measure.
