Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/191

Rh For, because A is greater than C, and B is any other magnitude, therefore A has to B a greater ratio than C has to B. [V. 8. But A is to B as E is to F; [Hypothesis. therefore E has to F a greater ratio than C has to B. [V. 13. And because B is to C as D is to E, [Hypothesis. therefore, by inversion, C is to B as E is to D. [V. B. And it was shewn that E has to F a greater ratio than C has to B; therefore E has to F a greater ratio than E has to D; [V. 13, Cor. therefore F is less than D; [V. 10. that is, D is greater than F.

Secondly, let A be equal to C: D shall be equal to F. For, because A is equal to C, and B is any other magnitude, therefore A is to B as C is to B. [V. 7. But A is to B as E is to F; [Hyp. and C is to B as E is to D; [Hyp. V. B. therefore E is to F as E is to D; [V. 11. and therefore D is equal to F. [V. 9.

Lastly, let A be less than C: D shall be less than F. For C is greater than A; and, as was shewn in the first case, C is to B as E is to D; and in the same manner, B is to A as F is to E; therefore, by the first case, F is greater than D; that is, D is less than F.

Wherefore, if there be three &c.