Euclid, Elements V 21©
translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 21: If three magnitudes and others equal to them in plêthos, taken by two's, are also in the same ratio, and their proportion is perturbed (perturbando), and through an equal (ex aequali) the first is larger than the third, the fourth will also be larger than the sixth, and if equal, equal, and if smaller, smaller.

(general diagram = diagram 1)

Diagrams for book 5 proposition 21, 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 and in the same ratio, and let the proportion of them be perturbed, as A to B so E to Z and as B to G so D to E, but through an equal (ex aequali) let A 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 other, therefore A to B has a larger ratio than G to B. But as A to B so E to Z, and as G to B, inversely, so E to D. (v 7 coroll.) Therefore E to Z also has a larger ratio than E to D. (v 13) But that to the same has a larger ratio is smaller. (v 10) Therefore, Z is smaller than D. Therefore, D is larger than Z. In fact, we will similarly show that (diagram 2) also if A is equal to G, D will also be equal to Z (v 7 and coroll., 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, are also in the same ratio, and their proportion is perturbed (perturbando), and through an equal (ex aequali) the first is larger than the third, the fourth will also be larger than the sixth, and if equal, equal, and if smaller, smaller, just what it was required to show.

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