Kitap 5
21
But, as A is to B, so is E to F, and, as C is to B, inversely, so is E to D. Therefore also E has to F a greater ratio than E has to D. [V. 13]
21
But that to which the same has a greater ratio is less; [V. 10] therefore F is less than D; therefore D is greater than F.
21
Similarly we can prove that, if A be equal to C, D will also be equal to F; and if less, less.
21
Therefore etc. Q. E. D.
PROPOSITION 22.
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If there be any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, they will also be in the same ratio ex aequali.
22
Let there be any number of magnitudes A, B, C, and others D, E, F equal to them in multitude, which taken two and two together are in the same ratio, so that, as A is to B, so is D to E, and, as B is to C, so is E to F; I say that they will also be in the same ratio ex aequali, ltthat is, as A is to C, so is D to Fgt.
22
For of A, D let equimultiples G, H be taken, and of B, E other, chance, equimultiples K, L; and, further, of C, F other, chance, equimultiples M, N.
22
Then, since, as A is to B, so is D to E, and of A, D equimultiples G, H have been taken, and of B, E other, chance, equimultiples K, L, therefore, as G is to K, so is H to L. [V. 4]
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For the same reason also, as K is to M, so is L to N.
