Book 5
20
For, since A is greater than C, and B is some other magnitude, and the greater has to the same a greater ratio than the less has, [V. 8] therefore A has to B a greater ratio than C has to B.
20
But, as A is to B, so is D to E, and, as C is to B, inversely, so is F to E; therefore D has also to E a greater ratio than F has to E. [V. 13]
20
But, of magnitudes which have a ratio to the same, that which has a greater ratio is greater; [V. 10] therefore D is greater than F.
20
Similarly we can prove that, if A be equal to C, D will also be equal to F; and if less, less.
20
Therefore etc. Q. E. D.
PROPOSITION 21.
21
If there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then, if ex aequali the first magnitude is greater than the third, the fourth will also be greater than the sixth; if equal, equal; and if less, less.
21
Let there be three magnitudes A, B, C, and others D, E, F equal to them in multitude, which taken two and two are in the same ratio, and let the proportion of them be perturbed, so that, as A is to B, so is E to F, and, as B is to C, so is D to E, and let A be greater than C ex aequali; I say that D will also be greater than F; if A is equal to C, equal; and if less, less.
21
For, since A is greater than C, and B is some other magnitude, therefore A has to B a greater ratio than C has to B. [V. 8]
