given/ A B /, / C D / be the two given numbers not prime to one another
construct PORISM. From this it is manifest that, if a number measure two numbers, it will also measure their greatest common measure
given/ A B /, / C D / be the two given numbers not prime to one another
assert If now / C D / measures / A B /—and it also measures itself—/ C D / is a common measure of / C D /, / A B /
assert It is manifest that it is also the greatest
assert For no greater number than / C D / will measure / C D /
assert 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
∵unequal_numbers_setassert For an unit will not be left; otherwise / A B /, / C D / will be prime to one another, which is contrary to the hypothesis
assert some number will be left which will measure the one before it
assert 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 /
assert then, / C F / measures / A E /, and / A E / measures / F D /,
assert/ C F / will also measure / F D /
assert It also measures itself
assert it will also measure / C D /
assert/ C D / measures / E B /
assert/ C F / also measures / E B /
assert It also measures / A E /
assert it will also measure / A B /
assert It also measures / C D /
assert/ C F / measures / A B /, / C D /
assert/ C F / is a common measure of / A B /, / C D /
assert I say next that it is also the greatest
assert If / C F / is not the greatest common measure of / A B /, / C D /, some number which > / C F / will measure the numbers / A B /, / C D /
assert Let such a number measure them, and let it be G
assertG measures / C D /, while / C D / measures / E B /,
assertG also measures / E B /
assert It also measures / A B /
assert it will also measure / A E /
assert/ A E / measures / F D /
assertG will also measure / F D /
assert It also measures / C D /
assert it will also measure / C F /, that is, the greater will measure the less ⇒⇐
assert no number which > / 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
