Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/177

Rh PROPOSITION 11. THEOREM.

Ratios that are the same to the same ratio, are the same to one another.

Let A be to B as C is to D, and let C be to D as E is to F: A shall be to B as E is to F.

Take of A, C, E any equimultiples whatever G, H, K; and of B, D, F any equimultiples whatever L, M, N.

Then, because A is to B as C is to D, [Hypothesis. and that G and H are equimultiples of A and C, and L and M are equimultiples of B and D; [Construction. therefore if G be greater than L, H is greater than 'N; and if equal, equal; and if less, less. [V. Definition 5.

Again, because C is to D as E is to F, [Hypothesis. and that H and K are equimultiples of C and E, and M and N are equimultiples of D and F; [Construction. therefore if H be greater than M, K is greater than N; and if equal, equal; and if less, less. [V. Definition 5.

But it has been shewn that if G be greater than L, H is greater than M; and if equal, equal; and if less, less. Therefore if G be greater than L, K is greater than N; and if equal, equal; and if less, less. And G and K are any equimultiples whatever of A and E, and L and N are any equimultiples whatever of B and F. Therefore A is to B as E is to F. [V. Definition 6.

Wherefore, ratios that are the same &c.