Collapsible Compass

Date: 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/