 (diagram 1) Let there be equal magnitudes, A, B, and some other magnitude, whatever happens, G. I say that each of A, B has the same ratio to G, and G to each of A, B. (diagram 2 = gen. diag) For let equal-times multiples of A, B be taken, D, E and of G another multiple, whatever happens, Z. And so, since D is equally-times a multiple of A as E of B, but A is equal to B, therefore D is also equal to E. But Z is another, whatever happens. Therefore, if D exceeds Z, E also exceeds Z, and if equal, equal, and if smaller, smaller. And D, E, are equal-times multiples of A, B, while Z is another multiple, whatever happens, of G. Therefore, it is: as A to G, so B to G. I say, in fact, that G also has the same ratio to each of A, B. For when the same things are constructed, we will show similarly that D is equal to E And Z is some other. Therefore, if Z exceeds D, it also exceeds E, and if equal, equal, and if smaller, smaller. And S is a multiple of G while D, E are other, whatever happens, equal-times multiples, of A, B. Therefore, it is: as G to A, so G to B. Therefore, equals have the same ratio to the same and the same to equals, just what it was required to show.