Euclid,

ElementsI 3©

translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 3: With two given, unequal straight-lines to take away from the larger a straight-line equal to the smaller.

(diagram 1). Let the given two unequal straight-lines be AB, G, for which let AB be the larger. It is, in fact, required to take away from the larger, AB, a straight-line equal to G.

(diagram 2) Let an equal to straight-line G, AD, be positioned at point A (prop. 2), (diagram 3) and with a center, A, and an extension, AD, let a circle be described, DEZ (diagram 4) And since point A is center of circle DEZ, AE is equal to AD. But G is also equal to AD. Therefore, each of AE, G is equal to AD. Thus AD is equal to G.

Therefore, with two given, unequal straight-lines, AB, G, from the larger, AB, an equal, AE, to the smaller, G, has been taken away, just what it was required to make.

Note: the construction of circle DEZ is strictly speaking unnecessary, since it was already constructed in the construction of AD. In effect, one hides how estrablished constructions got established.

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