G.191 magnitude ratio greater — Heath

V.13

If a first magnitude have to a second the same ratio as a third to a fourth, and the third have to the fourth a greater ratio than a fifth has to a sixth, the first will also have to the second a greater ratio than the fifth to the sixth.

For let a first magnitude A have to a second B the same ratio as a third C has to a fourth D, and let the third C have to the fourth D a greater ratio than a fifth E has to a sixth F;

I say that the first A will also have to the second B a greater ratio than the fifth E to the sixth F.

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

For, since there are some equimultiples of C, E, and of D, F other, chance, equimultiples, such that the multiple of C is in excess of the multiple of D, while the multiple of E is not in excess of the multiple of F, [G.185]

let them be taken, and let G, H be equimultiples of C, E, and K, L other, chance, equimultiples of D, F, so that G is in excess of K, but H is not in excess of L; and, whatever multiple G is of C, let M be also that multiple of A, and, whatever multiple K is of D, let N be also that multiple of B.

Now, since, as A is to B, so is C to D, and of A, C equimultiples M, G have been taken, and of B, D other, chance, equimultiples N, K,

therefore, if M is in excess of N, G is also in excess of K, if equal, equal, and if less, less. [G.176]
But G is in excess of K;
therefore M is also in excess of N.
But H is not in excess of L;

and M, H are equimultiples of A, E, and N, L other, chance, equimultiples of B, F;

[G.185]therefore A has to B a greater ratio than E has to F.
Therefore etc. Q. E. D.

Dependency Graph

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Required for

G.192, G.195, G.199, G.201, G.202, G.204, G.205, G.206, G.207, G.208, G.213, G.214, G.215, G.216, G.218, G.223, G.229, G.231, G.232, G.234, G.236, G.237, G.238, G.239, G.240, G.241, G.243, G.244, G.245, G.376, G.377, G.378, G.379, 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.528, G.536, G.551, G.552, G.555, G.556, G.557, G.558, G.559, G.563, G.565, G.570, G.571, G.572, G.573, G.577, G.578, G.583, G.584, G.586, G.587, G.588, G.589, G.592, G.597, G.600, G.601