multiples whatever of the first and the third, and also any equimultiples whatever of the second and the fourth then the multiple of the first shall have the same ratio to the multiple of the second, that the multiple of the third has to the multiple of the fourth.
Let A the first have to B the second, the same ratio that C the third has to D the fourth; and of A and C let there be taken any equimultiples whatever E and F, and B and D any equimultiples whatever G and H: E shall have the same ratio to G that F has to H.
Take of E and F any equimultiples whatever K and L, and of G and H any equimultiples whatever M and N.
Then, because E is the same multiple of A that F is of C, and of E and F have been taken equimultiples K and L;
therefore K is the same multiple of A that L is of C. [V. 3.
For the same reason, M is the same multiple of B that N is of D.
And because A is to B as C is to D, [Hypothesis.
and of A and C have been taken certain equimultiples K and L, and of B and D have been taken certain equimultiples M and N;
therefore if K be greater than M,L is greater than N-, and if equal, equal; and if less, less. [V. Definition 5.
But K and L are any equimultiples whatever of E and F, and M and N are any equimultiples whatever of G and H; therefore E is to G' as F is to H. [V. Definition 5.
Wherefore, if the first &c. q.e.d.
Corollary. Also if the first have the same ratio to the second that the third has to the fourth, then any equimultiples whatever of the first and third shall have the same ratio to the second and fourth: and the first and
