Let a first magnitude A have to a second B the same ratio as a third C to a fourth D, and let equimultiples E and F be taken of A and C, and G and H other, arbitrary, equimultiples of B and D.
I say that E is to G as F is to H.
Take equimultiples K and L of E and F, and other, arbitrary, equimultiples M and N of G and H.
Since E is the same multiple of A that F is of C, and equimultiples K and L of E and F have been taken, therefore K is the same multiple of A that L is of C. For the same reason M is the same multiple of B that N is of D.
And, since A is to B as C is to D, and equimultiples K and L have been taken of A and C, and other, arbitrary, equimultiples M and N of B and D, therefore, if K is in excess of M, then L is in excess of N; if it is equal, equal; and if less, less.
And K and L are equimultiples of E and F, and M and N are other, arbitrary, equimultiples of G and H, therefore E is to G as F is to H.
Therefore, if a first magnitude has to a second the same ratio as a third to a fourth, then any equimultiples whatever of the first and third also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order.
The statement of the proposition says that
Note how Euclid uses the definition to prove that the two ratios pa : qb and pc : qd are the same. (Here, a and b are magnitudes of one kind, and c and d are magnitudes of another kind, but p and q are numbers.) We are given a : b = c : d. That means for any numbers m and n that
We have to prove that pa : qb = pc : qd for any numbers p and q. That means, we have to prove that for any m and n,
But that’s just a special case of the given relation