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]