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G.188 magnitudes ratio greater¶

Of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less.

magnitudes_ratio_greater()

prove A > B; B < A

assert Let A have to C a greater ratio than B has to C

assert If not, A is either = to B or less

assert Now A is not = to B

∵ equal_magnitudes_ratioassert For in that case each of the magnitudes A, B would have had the same ratio to C

assert But they have not

assert A is not = to B

assert Nor again is A less than B

∵ unequal_magnitudes_greaterassert For in that case A would have had to C a less ratio than B has to C

assert But it has not

assert A is not less than B

assert It was proved not to be = either

assert A > B

assert Let C have to B a greater ratio than C has to A

assert If not, it is either = or greater

assert Now B is not = to A

∵ equal_magnitudes_ratioassert For in that case C would have had the same ratio to each of the magnitudes A, B

assert But it has not

assert A is not = to B

assert Nor again is B greater than A

∵ unequal_magnitudes_greaterassert For in that case C would have had to B a less ratio than it has to A

assert But it has not

assert B is not greater than A

assert It was proved that it is not = either

∴ B < A

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