G.375 commensurable magnitudes find

X.4

Given three commensurable magnitudes, to find their greatest common measure.

commensurable_magnitudes_find_x_4(A, B, C)

given A, B, C be the three given commensurable magnitudes

construct PORISM. From this it is manifest that, if a magnitude measure three magnitudes, it will also measure their greatest common measure

given A, B, C be the three given commensurable magnitudes

construct Let the greatest common measure of the two magnitudes A, B be taken

∵ commensurable_magnitudes_findassert Let it be D

assert Then D either measures C, or does not measure it

assert First, let it measure it

assert then D measures C, while it also measures A, B,

assert D is a common measure of A, B, C

assert It is manifest that it is also the greatest ; for a greater magnitude than the magnitude D does not measure A, B

assert Next, let D not measure C

assert I say first that C, D are commensurable

assert A, B, C are commensurable, some magnitude will measure them, and this will of course measure A, B also

∵ commensurable_magnitudes_findassert So that it will also measure the greatest common measure of A, B, namely D

assert It also measures C

assert So that the said magnitude will measure C, D ; therefore C, D are commensurable

construct Now let their greatest common measure be taken

∵ commensurable_magnitudes_findassert Let it be E

assert then E measures D, while D measures A, B,

assert E will also measure A, B

assert It measures C also

assert E measures A, B, C ; therefore E is a common measure of A, B, C

assert I say next that it is also the greatest

assert If possible, let there be some magnitude F greater than E, and let it measure A, B, C

∵ commensurable_magnitudes_findassert F measures A, B, C, it will also measure A, B, and will measure the greatest common measure of A, B

assert The greatest common measure of A, B is D

assert F measures D

assert It measures C also

∵ commensurable_magnitudes_findassert F measures C, D ; therefore F will also measure the greatest common measure of C, D

assert That is E

assert F will measure E, the greater the less ⇒⇐

assert no magnitude greater than the magnitude E will measure A, B, C ; therefore E is the greatest common measure of A, B, C if D do not measure C, and, if it measure it, D is itself the greatest common measure

assert the greatest common measure of the three given commensurable magnitudes has been found

∴ PORISM. From this it is manifest that, if a magnitude measure three magnitudes, it will also measure their greatest common measure

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