Euclid, Elements X 5©
translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 5: Commensurable magnitudes have to each other a ratio that an arithmos has to an arithmos.

(general diagram)

(diagram 1) Let there be commensurable magnitudes, A, B. I say that A has a ratio to B that an arithmos has to an arithmos.

(diagram 2) For since A, B are commensurable, some magnitude measures them, and let it be G. (diagram 3) And as many-times as G measures A, let there be so-many units in D, and as many-times as G measures B, let there be so-many units in E. And so, since G measures A according to the units in D, but the unit also measures D according to the units in it, therefore the unit measures arithmos D equal-times as magnitude G measures A. (general diagram) Therefore, as G to A, so the unit to D. [VII def. 20] Therefore, inversely, it is: as A to G so D to the unit. [V 7 cor.] Again, since G measures B according to the units in E, but the unit also measures E according to the units in itself, therefore, the unit measures E equal-times as G measures B. Therefore, it is: as G to B so the unit to E. But it was shown also that as A to G, also D to the unit. Ex aequali, therefore, it is: as A to B, so arithmos D to E. [V 22] Therefore, commensurable magnitudes, A, B have to each other a ratio that an arithmos, D, has to an arithmos, E, just what it was required to show.

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