G.258 number parts number — Heath

VII.6

If a number be parts of a number, and another be the same parts of another, the sum will also be the same parts of the sum that the one is of the one.

For let the number / A B / be parts of the number C, and another, / D E /, the same parts of another, F, that / A B / is of C;

I say that the sum of / A B /, / D E / is also the same parts of the sum of C, F that / A B / is of C.

A G B C D H E F
For since, whatever parts / A B / is of C, / D E / is also the same parts of F,
therefore, as many parts of C as there are in / A B /, so many parts of F are there also in / D E /.

Let / A B / be divided into the parts of C, namely / A G /, / G B /, and / D E / into the parts of F, namely / D H /, / H E /;

thus the multitude of / A G /, / G B / will be equal to the multitude of / D H /, / H E /.
And since, whatever part / A G / is of C, the same part is / D H / of F also,
therefore, whatever part / A G / is of C, the same part also is the sum of / A G /, / D H / of the sum of C, F. [G.257]
For the same reason,
whatever part / G B / is of C, the same part also is the sum of / G B /, / H E / of the sum of C, F.
Therefore, whatever parts / A B / is of C, the same parts also is the sum of / A B /, / D E / of the sum of C, F.
Q. E. D.

Dependency Graph

digraph { bgcolor="transparent"; node [shape=box, style="rounded,filled", fontname="Helvetica", fontsize=11, color="#d6d3cd", fontcolor="#ffffff"]; edge [color="#8c857b", fontcolor="#8c857b", arrowsize=0.75]; rankdir="TB"; "G.169" [label="G.169", tooltip="G.169 (V.def.1): A magnitude is apart of a magnitude, the less of the greater, when it measures the greater.", fillcolor="#224422", URL="../G.169/index.html", target="_top"]; "G.170" [label="G.170", tooltip="G.170 (V.def.2): The greater is amultiple of the less when it ismeasured by the less.", fillcolor="#224422", URL="../G.170/index.html", target="_top"]; "G.247" [label="G.247", tooltip="G.247 (VII.def.1): Anunit is that by virtue of which each of the things that exist is called one.", fillcolor="#224422", URL="../G.247/index.html", target="_top"]; "G.248" [label="G.248", tooltip="G.248 (VII.def.2): Anumber is a multitude composed ofunits.", fillcolor="#224422", URL="../G.248/index.html", target="_top"]; "G.253" [label="G.253", tooltip="G.253 (VII.def.3): Anumber is apart of a number, the less of the greater, when itmeasures the greater;", fillcolor="#224422", URL="../G.253/index.html", target="_top"]; "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.", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", URL="index.html", target="_top"]; "G.170" -> "G.169"; "G.253" -> "G.169"; "G.254" -> "G.169"; "G.256" -> "G.169"; "G.257" -> "G.170"; "G.248" -> "G.247"; "G.253" -> "G.248"; "G.256" -> "G.248"; "G.257" -> "G.248"; "G.258" -> "G.248"; "G.254" -> "G.253"; "G.256" -> "G.253"; "G.257" -> "G.253"; "G.258" -> "G.253"; "G.258" -> "G.254"; "G.257" -> "G.256"; "G.258" -> "G.257"; }

Required for

G.261, G.262, G.265, G.266, G.267, G.268, G.270, 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.293, G.294, 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.340, 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