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. q.e.d.
