Table of contents

1. Euclidean constructions
2. Reduction to circular constructions
3. Summary of inversion
4. Stereographic projection
5. Construction PC to invert a point in a circle
6. Construction C to find the center of a circle
7. Inversive geometry and involutory quandles

   Introduction

   With compasses alone, all the points that can be constructed with straightedges and compasses can be constructed. That means that straightedges are only necessary for the actual drawing of lines. One would not want to dispense with straightedges, however, since the constructions with compasses alone are much more complicated.

   The geometry of compasses was developed independently by G. Mohr in Denmark in 1672, and by L. Mascheroni in Italy in 1797. The easiest way, however, to show that compasses are sufficient depends on circle inversion which wasn't invented until 1828 by Jacob Steiner.

   References

   1. R. Courant and H.E. Robbins, What is Mathematics? Oxford Univ. Pr., New York, 1953.
   2. H.S.M. Coxeter, Introduction to Geometry, Wiley, New York, 1961.
   3. Euclid, Elements, http://aleph0.clarku.edu/~djoyce/java/elements/elements.html.
   4. D. Pedoe, Circles, Dover, New York, 1957.

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   Table of contents Introduction 1. Euclidean constructions 2. Reduction to circular constructions 3. Summary of inversion 4. Stereographic projection 5. Invert a point in a circle 6. Find the center of a circle 7. Involutory quandles

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   April, 1998; March, 2002.
   David E. Joyce
   Department of Mathematics and Computer Science
   Clark University
   Worcester, MA 01610

   Email: djoyce@clarku.edu
   My Homepage