G.384 magnitudes commensurable magnitude

X.12

Magnitudes commensurable with the same magnitude are commensurable with one another also.

magnitudes_commensurable_magnitude()

prove A is also commensurable with B

assert Let each of the magnitudes A, B be commensurable with C

assert A is commensurable with C,

∵ commensurable_magnitudes_ratioassert A has to C the ratio which a number has to a number

assert Let it have the ratio which D has to E

assert C is commensurable with B,

∵ commensurable_magnitudes_ratioassert C has to B the ratio which a number has to a number

assert Let it have the ratio which F has to G

construct many_ratios_please: And, given any number of ratios we please, namely the ratio which D has to E and that which F has to G, let the numbers H, K, L be taken continuously in the given ratios

assert So that, as D is to E, so is H to K, and, as F is to G, so is K to L

assert Since, then, as A is to C, so is D to E, while, as D is to E, so is H to K,

∵ ratios_ratioassert also, as A is to C, so is H to K

assert Since, as C is to B, so is F to G, while, as F is to G, so is K to L,

∵ ratios_ratioassert also, as C is to B, so is K to L

assert Also, as A is to C, so is H to K

∵ there_number_magnitudes_v_22assert :term:`ex aequali <G.174>`, as A is to B, so is H to L

∵ magnitudes_ratio_number∴ A has to B the ratio which a number has to a number; therefore A is commensurable with B

Dependency Graph

Required for

G.385, G.389, G.390, G.396, G.397, G.398, G.399, G.400, G.401, G.402, 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.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