Euclid, Elements X 2©
translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 2: If upon two unequal magnitudes being displayed, when the smaller is repeatedly taken away in turn from the larger what’s left over never measures out that before it, the magnitudes will be incommensurable.

(general diagram)

(diagram 1) For upon two magnitudes being unequal, AB, GD, and a smaller, AB, being repeatedly taken way, the smaller form the larger, let what remains never measure out what’s before it. I say that magnitudes AB, GD, are incommensurable. (diagram 2) For if they are commensurable, let some magnitude measure them, if possible, and let it be E. (diagram 3) And let AB in measuring out ZD leave a smaller than it, AH, and let this repeatedly come about, until there is left some magnitude which is smaller than E. (diagram 4) Let it come about, and let AH, smaller than E, be left. And so, since E measures AB, but AB measures DZ, therefore, E also will measure ZD. But it also measures a whole GD. Therefore, it will also measure a remainder, GZ. But GZ measures BG. Therefore, E also measures BG. But it also measures a whole, AB. Threfore it will also measure a remainder AH, the larger the smaller, which is impossible. Therefore, some magnitude will not measure magnitudes AB, GD. Therefore, magnitudes AB, GD are incommensurable. [X def. 1] Therefore, if upon two unequal magnitudes, etc.

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