Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/179

Rh ratio than the fifth to the sixth, the first shall have to the second a greater ratio than the fifth has to the sixth.

Let A the first have the same ratio to B the second that C the third has to D the fourth, but C the third a greater ratio to D the fourth than E the fifth to F the sixth: A the first shall have to B the second a greater ratio than E the fifth has to F the sixth.

For, because C has a greater ratio to D than E has to F, there are some equimultiples of C and E, and some equi- multiples of D and F, such that the multiple of C is greater than the multiple of D, but the multiple of E is not greater than the multiple of F. [V. Definition 7. Let such multiples be taken, and let G and H be the equi- multiples of C and E, and K and L the equimultiples of D and E; so that G is greater than K, but H is not greater than L. And whatever multiple G is of C, take M the same mul- tiple of A ; and whatever multiple K is of D, take N the same multiple of B.

Then, because A is to B as C is to D, [Hypothesis. and M and G are equimultiples of A and C, and N and K are equimultiples of B and D ; ['Construction. therefore if M be greater than 'N, G is greater than K; and if equal, equal ; and if less, less. [V. Definition 5. But G is greater than K ; [Construction. therefore M is greater than N. But H is not greater than L ; [Construction. and M and H are equimultiples of A and E, and N and L are equimultiples of B and F ; [Construction. therefore A has a greater ratio to B than E has to F,

Wherefore, if the first &c.