Collapsible CompassDate: 11/21/2003 at 00:05:12 From: HEF Subject: Collapsible Compass I need to know what a collapsible compass is and what it is used for. All I know is that when you pick it up from the paper, you lose your place. Date: 11/21/2003 at 08:43:34 From: Doctor Peterson Subject: Re: Collapsible Compass Hi, HEF. The collapsible is not something that is "used"; rather, it represents the fact that Euclid wanted to make as few assumptions (postulates, or axioms) at the base of his proofs as possible. So rather than assume that it was possible to move a line around, keeping the same length (as you could do with a real, fixed compass), or equivalently that you can draw a circle with a given center and length, he assumed only that you can draw a circle with a given center and through a given point. Then he went on to prove that if you could do that, you COULD then construct a circle with a given radius, or move a line to a given place: Collapsible Compass http://mathforum.org/library/drmath/view/52601.html Once that was proved, you didn't have to use a collapsible compass, but could use a regular one, knowing that any construction you could do this way, you could do with a collapsible compass. The link in the page above explains this a little more fully. Note that a real "collapsible compass" would not work, because the radius would change as you drew the supposed circle! The idea is that it holds together and keeps its radius as long as the center point is in place, but loses it when you pick it up. I don't know of any design for a real-world compass that would work that way; this is a tool that exists only in the mind of a geometer, as a description of Euclid's postulate 3. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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