From: cet1@cus.cam.ac.uk (Chris Thompson)
Newsgroups: sci.math
Subject: Re: bisecting line segment w/ compass
Date: 12 Dec 1997 11:14:28 GMT
In article <3490E328.670F@descartes.uwaterloo.ca>,
Jim Nastos wrote:
[...]
> Similarly, Jakob Steiner showed that every compass & straightedge
>construction in existence can be carried out with straightedge alone,
>provided there exists one circle on the plane to work on.
One circle *and* its centre.
A proof that the straightedge is insufficient to construct the
centre of a given circle (due to one of Hilbert's students, I think)
can be found in Rademacher & Toeplitz "The Enjoyment of Mathematics"
(originally "Von Zahlen und Figuren").
Chris Thompson
Email: cet1@cam.ac.uk
==============================================================================
From: John McGowan
Subject: Re: bisecting line segment w/ compass
Newsgroups: sci.math
Date: 12 Dec 97 15:14:07 GMT
In sci.math Chris Thompson wrote:
> One circle *and* its centre.
> A proof that the straightedge is insufficient to construct the
> centre of a given circle (due to one of Hilbert's students, I think)
> can be found in Rademacher & Toeplitz "The Enjoyment of Mathematics"
> (originally "Von Zahlen und Figuren").
Actually, it is quite simple. A point projection (as one gets in taking a
photograph through a pinhole camera) is not linear but maps lines to
lines. It does not preserve midpoints of segments. Any construction on
points using just a straight-edge maps to the same construction on the
image points. A construction purporting to find the center of a segment
(and which happens to find it on the original set of points) would map to
the same construction purporting to find the center but finding a point
OTHER than the center on the image points.
[I believe that if one uses the composition of two point projections one
can map a circle in the original set to a circle... however the center
is not preserved -- nor are midpoints -- indicating that it is not
sufficient to have a circle to manage constructions with a
straightedge... though a circle and center does suffice -- I believe even
a small arc of a circle along with the circle's center suffices]