G.189 ratios ratio — Heath

V.11

Ratios which are the same with the same ratio are also the same with one another.

For, as A is to B, so let C be to D, and, as C is to D, so let E be to F;

I say that, as A is to B, so is E to F.

A B G L C D H M E F K N

For of A, C, E let equimultiples G, H, K be taken, and of B, D, F other, chance, equimultiples L, M, N.

Then since, as A is to B, so is C to D, and of A, C equimultiples G, H have been taken, and of B, D other, chance, equimultiples L, M,

therefore, if G is in excess of L, H is also in excess of M, if equal, equal, and if less, less.

Again, since, as C is to D, so is E to F, and of C, E equimultiples H, K have been taken, and of D, F other, chance, equimultiples M, N,

therefore, if H is in excess of M, K is also in excess of N, if equal, equal, and if less, less.

But we saw that, if H was in excess of M, G was also in excess of L; if equal, equal; and if less, less;

so that, in addition, if G is in excess of L, K is also in excess of N, if equal, equal, and if less, less.

And G, K are equimultiples of A, E, while L, N are other, chance, equimultiples of B, F;

therefore, as A is to B, so is E to F.
Therefore etc.

Dependency Graph

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Required for

G.195, G.199, G.201, G.206, G.207, G.208, G.210, G.211, G.212, G.213, G.214, G.215, G.216, G.218, G.219, G.220, G.221, G.222, G.223, G.225, G.226, G.227, G.228, G.229, G.231, G.232, G.234, G.235, G.236, G.237, G.238, G.239, G.240, G.241, G.243, G.244, G.245, G.377, G.378, G.381, G.382, G.383, G.384, G.385, G.386, G.389, G.390, G.393, G.394, G.395, 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.501, G.520, G.528, G.536, G.541, G.542, G.543, G.544, G.546, G.547, G.550, G.551, G.552, G.554, G.555, G.556, G.557, G.558, G.559, G.560, G.563, G.565, G.570, G.571, G.572, G.573, G.577, G.578, G.579, G.580, G.582, G.583, G.584, G.586, G.587, G.588, G.589, G.592, G.597, G.600, G.601