Euclid, Elements of Geometry, Book I, Proposition 3 (Edited by Sir Thomas L. Heath, 1908)

Proposition 3[Euclid, ed. Heath, 1908, on archive.org]

Given two unequal straight lines, to cut off from the greater a straight line equal to the less.

Let AB, C be the two given unequal straight lines, and let AB be the greater of them.

Thus it is required to cut off from AB the greater a straight line equal to C the less.

At the point A let AD be placed equal to the straight line C; [I. 2]
and with centre A and distance AD let the circle DEF be described. [Post. 3]

Now, since the point A is the centre of the circle DEF,
AE is equal to AD. [Def. 15]

But C is also equal to AD.

Therefore each of the straight lines AE, C is equal to AD; so that AE is also equal to C. [C.N. 1]

Therefore, given the two straight lines AB, C, from AB the greater AE has been cut off equal to C the less.
(Being) what it was required to do.

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