Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/192

168 PROPOSITION 22. THEOREM.

If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in order, the first shall ham to the last of the first magnitudes the same ratio which the first of the others has to the last.

[This proposition is usually cited by the words ex aequali.]

First, let there be three magnitudes A, B, C, and other three D, E, F, which have the same ratio, taken two and two in order; that is, let A be to B as D is to E, and let B be to C as E is to F: A shall be to C as D is to F.

Take of A and D any equimultiples whatever G and H; and of B and E any equimultiples whatever K and L; and of C and F any equimultiples whatever M and N. Then, because A is to B as D is to E; [Hypothesis. and that G and H are equimultiples of A and D, and K and L equimultiples of B and E; [Construction. therefore G is to K as H is to L. [V. 4.

For the same reason, K is to M as L is to N. And because there are three magnitudes G, K, M, and other three H, L, N, which have the same ratio taken two and two, therefore if G be greater than M, H is greater than N, and if equal, equal; and if less, less. [V. 20. But G and H are any equimultiples whatever of A and D, and M and N are any equimultiples whatever of (7 and F.

Therefore A is to C as D is to F. [V. Definition 5. Next, let there be four magnitudes, A, B, C, D, and