From: cet1@cus.cam.ac.uk (Chris Thompson) Newsgroups: sci.math Subject: Re: The math of compasses Date: 5 Feb 1995 15:59:21 GMT In article , mig@bnr.ca (Michel Gallant) writes: |> |> The construction with straight-edge alone is the Macheroni construction and |> the one with compass alone is the Steiner construction (or is it the other |> way around) ? Both are described in "100 Great Problems in Elementary |> Mathematics" by Dorrie (Dover reprint). That's "Mascheroni", and yes, you have him and Steiner backwards. Mascheroni showed in [1] (although he was anticipated by Georg Mohr in 1672) that all Euclidean constructions can be performed by use of the compass alone [taking a line to have been "constructed" if two distinct points on it have been]. Steiner showed in [2] that all Euclidean contructions can be performed by use of the straightedge alone, provided one fixed circle and its centre are provided [again, taking a circle to have been "contructed" if its centre and a point on its circumference have been]. Hilbert raised the question of whether Steiner's prerequsites were minimal, and the results of the subsequent investigations were published by Cauer (one of Hilbert's students) in [3]. A fixed circle without its centre, or two such circles if non-intersecting, are insufficient: in particular one cannot construct the centre(s). [The method of proof involves finding a real projective transformation that preserves the circle(s) but not the centre(s).] On the other hand, two intersecting circles, or three non- intersecting circles, without their centres, are sufficient. There is a splendid exposition of Cauer's work in [4], from which the other references have shamelessly been taken. [1] Lorenzo Mascheroni (1750-1800) "La geometria del compasso" Pavia, 1797 [2] Jacob Steiner (1796-1863) "Die geometrischen Konstructionen, ausgef\"uhrt mittelst der geraden Linie und eines festen Kreises, als Lehrgegenstand auf h\"oheren Unterrichts-Anstalten und zur praktischen Benutzung" Berlin, 1833 [3] Detlef Cauer Math. Ann. 73 (1912) 90-94 & 74 (1913) 462-464 [4] Hans Rademacher & Otto Topelitz "Von Zahlen und Figuren" (Springer, 1933) tranlated into English by Herbert Zuckerman as "The Enjoyment of Mathematics" (Princeton, 1957, 1966) republished by Dover, 1990 Chris Thompson Internet: cet1@cus.cam.ac.uk JANET: cet1@uk.ac.cam.cus