Euclid, Elements V 20©
translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 20: If three magnitudes and others equal to them in plêthos, taken by two's and in the same ratio, and through an equal (ex aequali) the first is larger than the third, the fourth will be larger than the sixth, and if equal, equal, and if smaller, smaller.

(general diagram = diagram 1)

Diagrams for book 5 proposition 20, following the display, construction, and demonstrationLet there be three magnitudes A, B, G and others equal to them in plĂȘthos, D, E, Z, taken by two's in the same ratio, as A to B so D to E, and as B to G so E to Z, but let A through an equal (ex aequali) be larger than G. I say that D is also larger than Z, and if equal equal, and if smaller smaller.

For since A is larger than G, but B is some something else, but the larger to the same has a larger ratio than does the smaller (v 7), therefore A to B has a larger ratio than G to B. But as A to B so D to E and as G to B, inversely, so Z to E. (v 7 coroll.) Therefore, D to E also has a larger ratio than Z to E. (v 13) Of those having a ratio to the same that having the larger ratio is larger. (v 10) Therefore D is larger than Z. In fact, we will show similarly that (diagram 2) also if A is equal to G, D will also be equal to Z (v 7, v 9, v 11), (diagram 3) and if smaller smaller. Therefore, if three magnitudes and others equal to them in plêthos, taken by two's and in the same ratio, and through an equal (ex aequali) the first is larger than the third, the fourth will be larger than the sixth, and if equal, equal, and if smaller, smaller, just what it was required to show.

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