Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/194

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But A is to B as E is to F, [Hypothesis. Therefore G is to H as M is to N. [V. 11.

And because B is to C as D is to E, [Hypothesis. and that H and K are equimultiples of B and D, and L and M are equimultiples of C and E; [Constr. therefore H is to L as K is to M. [V. 4.

And it has been shewn that G is to H as M is to N.

Then since there are three magnitudes G, H, L, and other three K, M,N, which have the same ratio, taken two and two in a cross order; therefore if G be greater than L, K is greater than N; and if equal, equal; and if less, less. [V. 21. But G and K are any equimultiples whatever of A and D, and L and 'N are any equimultiples whatever of C 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 other four E, F, G, H, which have the same ratio, taken two and two in a cross order; namely, let A be to B as G is to H, and B to C as F is to G, and C to D as E is to F: A shall be to D as E is to H.

For, because A, B, C are three magnitudes, and F, G, H other three, which have the same ratio, taken two and two in a cross order; [Hypothesis. therefore, by the first case, A is to C as F is to H. But C is to D as E is to F; [Hypothesis. therefore also, by the first case, A is to D as E is to H.

And so on, whatever be the number of magnitudes.

Wherefore, if there he any number &c.