Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/180

156. And if the first have a greater ratio to the second than the third has to the fourtli, but the third the same ratio to the fourth that the fifth has to the sixth, it may be shewn, in the same manner, that the first has a greater ratio to the second than the fifth has to the sixth.

PROPOSITION 14. THEOREM.

If the first have the same ratio to the second that the third has to the fourth, then if the first he greater than the third the second shall he greater than the fourth; and if equal, equal; and if less, less.

Let A the first have the same ratio to B the second that C the third has to D the fourth: if A be greater than C, B shall be greater than D; if equal, equal; and if less,

First, let A be greater than C: B shall be greater than D. For, because A is greater than C, [Hypothesis. 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 C is to D. [Hypothesis. Therefore C has to D a greater ratio than C has to B. [V. 13.

But of two magnitudes, that to which the same has the greater ratio is the less. [V. 10. Therefore D is less than B, that is. B is greater than D. Secondly, let A be equal to C: B shall be equal to D. For, ^ is to ^ as C, that is A, is to D. [Hypothesis. Therefore B is equal to D. [V. 9.