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