G.266 numbers proportional proportional — Heath

VII.13

If four numbers be proportional, they will also be proportional alternately.

Let the four numbers A, B, C, D be proportional, so that, as A is to B, so is C to D;

I say that they will also be proportional alternately, so that, as A is to C, so will B be to D.

A B C D
For since, as A is to B, so is C to D,
therefore, whatever part or parts A is of B, the same part or the same parts is C of D also. [G.263]
Therefore, alternately, whatever part or parts A is of C, the same part or the same parts is B of D also. [G.262]
[G.263]Therefore, as A is to C, so is B to D.
Q. E. D.

Dependency Graph

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Required for

G.267, G.271, G.272, G.273, G.274, G.275, G.276, 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.307, G.309, G.312, G.313, G.314, G.315, G.316, G.317, G.319, G.321, 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.368, 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