G.387 commensurable magnitudes added

X.15

If two commensurable magnitudes be added together, the whole is commensurable with each of them; and, if the whole be commensurable with one of them, the original magnitudes is commensurable.
commensurable_magnitudes_added()
prove / A C / is also commensurable with each of the magnitudes / A B /, / B C /; / A B /, / B C / are also commensurable
A B C D
construct Let the two commensurable magnitudes / A B /, / B C / be added together
assert / A B /, / B C / are commensurable, some magnitude will measure them
assert Let it measure them, and let it be D
assert then D measures / A B /, / B C /, it will also measure / A C /
assert It measures / A B /, / B C / also
∵ magnitudes_said_commensurable_defassert D measures / A B /, / B C /, / A C /; therefore / A C / is commensurable with each of the magnitudes / A B /, / B C /
assert Next, let / A C / be commensurable with / A B /
assert / A C /, / A B / are commensurable, some magnitude will measure them
assert Let it measure them, and let it be D
assert then D measures / A C /, / A B /, it will also measure / B C /
assert It measures / A B / also
∵ magnitudes_said_commensurable_def∴ D will measure / A B /, / B C /; therefore / A B /, / B C / are commensurable

Dependency Graph

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Required for

G.389, G.390, 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.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