Euclid, Elements V 15©
translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 15: The parts have the same ratio as the multiples in the same way, taken respectively.

(general diagram)

Diagrams for book 5 proposition 15, following the display, construction, and demonstration(diagram 1) For let there be an equal-times multiple, AB of G as DE of Z. I say that it is: as G to Z, so is AB to DE. For since AB is an equal-times multiple of G as DE of Z, therefore as many magnitudes are in AB equal to G, so-many are also in DE equal to Z. (diagram 2=gen. diag.) Let AB be divided into the equals to G, i.e., AH, HQ, QB and DE into the equals to Z, i.e., DK, KL, LE. In fact, the plĂȘthos of AH, HQ, QB will be equal to the plĂȘthos of DK, KL, LE. And since AH, HQ, QB are equal to one another, and DK, KL, LE are also equal to one another, therefore, it is: as AH to DK, so HQ to KL, and QB to LE. (v 8) Therefore, as the one of the leaders will be to one of the followers, so too all the leaders to all the followers. (v 12) Therefore, it is: as AH to DK, so AB to DE. But AH is equal to G and DK to Z. Therefore, it is: as G to Z, so AB to DE. Therefore, the parts have the same ratio as the multiples in the same way, taken respectively, just what it was required to show.

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