G.274 least numbers ratio

VII.20

The least numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater and the less the less.
least_numbers_ratio()
prove / C D / measures A the same number of times that / E F / measures B
A B C G D E H F
assert Let / C D /, / E F / be the least numbers of those which have the same ratio with A, B
assert Now / C D / is not parts of A
assert If possible, let it be so
∵ numbers_proportional_proportional, numbers_proportional_multiple_defassert / E F / is also the same parts of B that / C D / is of A
assert as many parts of A as there are in / C D /, so many parts of B are there also in / E F /
assert Let / C D / be divided into the parts of A, namely / C G /, / G D /, and / E F / into the parts of B, namely / E H /, / H F /
assert the multitude of / C G /, / G D / will be equal to the multitude of / E H /, / H F /
assert the numbers / C G /, / G D / =, and the numbers / E H /, / H F / =, while the multitude of / C G /, / G D / is equal to the multitude of / E H /, / H F /,
assert as / C G / is to / E H /, so is / G D / to / H F /
∵ there_many_numbersassert also, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents
assert as / C G / is to / E H /, so is / C D / to / E F /
assert / C G /, / E H / are in the same ratio with / C D /, / E F /, being less than they ⇒⇐, for by hypothesis / C D /, / E F / are the least numbers of those which have the same ratio with them
∵ number_part_partsassert / C D / is not parts of A ; therefore it is a part of it
∵ numbers_proportional_proportional, numbers_proportional_multiple_defassert / E F / is the same part of B that / C D / is of A
∴ / C D / measures A the same number of times that / E F / measures B

Dependency Graph

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Required for

G.275, G.278, G.279, G.280, G.281, G.285, G.289, G.290, G.292, G.295, G.296, G.297, G.298, G.299, G.301, G.303, G.304, G.305, G.306, G.314, G.315, G.316, G.317, G.322, G.323, G.324, G.325, G.326, G.327, G.328, G.329, G.330, G.331, G.332, G.333, G.334, G.335, G.337, G.338, G.339, G.342, G.343, G.344, G.345, G.346, G.347, G.348, G.349, G.364, G.370, G.381, G.383, G.384, G.385, G.389, G.390, G.396, G.397, G.398, G.399, G.400, G.401, G.402, G.403, G.404, G.405, G.406, G.407, G.408, G.409, G.410, G.411, G.412, G.413, G.414, G.415, G.416, G.417, G.418, G.419, G.420, G.421, G.423, G.425, G.427, G.429, G.431, G.433, G.434, G.435, G.436, G.437, G.438, G.439, G.440, G.441, G.442, G.443, G.444, G.445, G.446, G.447, G.448, G.449, G.450, G.451, G.452, G.453, G.457, G.458, G.459, G.460, G.461, G.462, G.463, G.464, G.465, G.466, G.467, G.468, G.470, G.472, G.474, G.476, G.477, G.478, G.479, G.480, G.481, G.482, G.483, G.484, G.485, G.486, G.487, G.488, G.489, G.490, G.491, G.492, G.493, G.494, G.495, G.496, G.497, G.498, G.499, G.500, G.584, G.589, G.597, G.600, G.601