G.271 number multiplying numbers — Heath

VII.17

If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.

For let the number A by multiplying the two numbers B, C make D, E;

I say that, as B is to C, so is D to E.

For, since A by multiplying B has made D,
therefore B measures D according to the units in A.
A B C D E F
But the unit F also measures the number A according to the units in it;
therefore the unit F measures the number A the same number of times that B measures D.
[G.263]Therefore, as the unit F is to the number A, so is B to D.
For the same reason,
as the unit F is to the number A, so also is C to E;
therefore also, as B is to D, so is C to E.
[G.266]Therefore, alternately, as B is to C, so is D to E.
Therefore etc. Q. E. D.

Dependency Graph

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"G.254" [label="G.254", tooltip="G.254 (VII.def.4): Butparts when it does notmeasure it.", fillcolor="#224422", URL="../G.254/index.html", target="_top"]; "G.256" [label="G.256", tooltip="G.256 (VII.def.5): The greaternumber is amultiple of the less when it ismeasured by the less.", fillcolor="#224422", URL="../G.256/index.html", target="_top"]; "G.257" [label="G.257", tooltip="G.257 (VII.5): If a number be a part of a number, and another be the same part of another, the sum is the same part of the sum that the one is of the one.", fillcolor="#222244", URL="../G.257/index.html", target="_top"]; "G.258" [label="G.258", tooltip="G.258 (VII.6): If a number be parts of a number, and another be the same parts of another, the sum is the same parts of the sum that the one is of the one.", fillcolor="#222244", URL="../G.258/index.html", target="_top"]; "G.261" [label="G.261", tooltip="G.261 (VII.9): If a number be a part of a number, and another be the same part of another, alternately also, whatever part or parts the first is of the third, the same part, or the same parts, wi...", fillcolor="#222244", URL="../G.261/index.html", target="_top"]; "G.262" [label="G.262", tooltip="G.262 (VII.10): If a number be parts of a number, and another be the same parts of another, alternately also, whatever parts or part the first is of the third, the same parts or the same part will...", fillcolor="#222244", URL="../G.262/index.html", target="_top"]; "G.263" [label="G.263", tooltip="G.263 (VII.def.20): Numbers areproportional when the first is thesame multiple, or thesame part, or thesame parts, of the second that the third is of the fourth.", fillcolor="#224422", URL="../G.263/index.html", target="_top"]; "G.266" [label="G.266", tooltip="G.266 (VII.13): If four numbers be proportional, they is proportional alternately.", fillcolor="#222244", URL="../G.266/index.html", target="_top"]; "G.269" [label="G.269", tooltip="G.269 (VII.def.15): Anumber is said tomultiply a number when that which is multiplied is added to itself as many times as there areunits in the other, and thus some number is produced.", fillcolor="#224422", URL="../G.269/index.html", target="_top"]; 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Required for

G.272, G.273, G.276, G.278, G.279, G.280, G.281, G.285, G.289, G.290, G.292, G.295, G.297, G.298, G.299, G.301, G.303, G.304, G.305, G.306, G.307, G.309, G.312, 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.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