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Proposition 3

Thu, 2006-10-12 17:30 — ashley

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 $; [Prop. 1.2]

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 $.

Therefore, given the two straight lines $ AB $,$ C $ from $ AB $ the greater $ AE $ has been cut of equal to $ C $ the less.

(Being) what it was required to do.