Euclid, Elements V 7©
translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 7: Equals have the same ratio to the same and the same to equals.

(general diagram)

Diagrams for book 5 proposition 7, following the display, construction, and demonstration(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.

Provided (Corollary) From this, in fact, it is obvious that if some magnitudes are proportional, they will be inversely proportional, just what it was required to show.

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