G.374 commensurable magnitudes find¶
Given two commensurable magnitudes, to find their greatest common measure.
commensurable_magnitudes_find(/ A B /, / C D /)
given the two given commensurable magnitudes be / A B /, / C D / of which / A B / is the less
construct PORISM. From this it is manifest that, if a magnitude measure two magnitudes, it will also measure their greatest common measure
given the two given commensurable magnitudes be / A B /, / C D / of which / A B / is the less
assert Now the magnitude / A B / either measures / C D / or it does not
assert If then it measures it—and it measures itself also—/ A B / is a common measure of / A B /, / C D /
assert It is manifest that it is also the greatest ; for a greater magnitude than the magnitude / A B / will not measure / A B /
assert Next, let / A B / not measure / C D /
∵ less_unequal_magnitudesassert 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
assert 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 /
assert Since, then, / A F / measures / C E /, while / C E / measures / F B /,
assert / A F / will also measure / F B /
assert It measures itself also
assert / A F / will also measure / A B /
assert / A B / measures / D E /
assert / A F / will also measure / E D /
assert It measures / C E / also
assert it also measures / C D /
assert / A F / is a common measure of / A B /, / C D /
assert I say next that it is also the greatest
assert If not, there will be some magnitude greater than / A F / which will measure / A B /, / C D /
assert Let it be G
assert then G measures / A B /, while / A B / measures / E D /,
assert G will also measure / E D /
assert It measures / C D / also
assert G will also measure / C E /
assert / C E / measures / F B /
assert G will also measure / F B /
assert It measures / A B / also,
assert it will therefore measure / A F /, the greater the less ⇒⇐
assert 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 /
∴ PORISM. From this it is manifest that, if a magnitude measure two magnitudes, it will also measure their greatest common measure