Preface. Chapter 0. Chapter I. 1. The nine-point circle. 2. Inversion. 3. Feuerbach's theorem. 4. Extension of Ptolemy's theorem. 5. Fermat's problem. 6. The centres of similitude of two circles. 7. Coaxal systems of circles. 8. Canonical form of coaxal system. 9. Further properties. 10. Problem of Apollonius. 11. Compass geometry. Chapter II. 1. Representation of a circle. 2. Euclidean three-space, E3. 4. First properties of the representation. 4. Coaxal systems. 5. Deductions from the representation. 6. Conjugacy relations. 7. Circles cutting at a given angle. 8. Representation of inversion. 9. The envelope of a system. 10. Some further applications. 11. Some anallagmatic curves. Chapter III. 1. Complex numbers. 2. The Argand diagram. 3. Modulus and argument. 4. Circles as level curves. 5. The cross-ratio of four complex numbers. 6. MSbius transformation of the z-plane. 7. A M6bius transformation dissected. 8, The group property. 9. Special transformations. 10. The fundamental theorem. 11. The Poincar~ model. 12. The parallel axiom. 13. Non-Euclidean distance. Chapter IV. 1. Steiner's enlarging process. 2. Existence of a solution. 3. Method of solution. 4. Area of a polygon. 5. Regular polygons. 6. Rectifiable curves. 7. Approximation by polygons. 8. Area enclosed by a curve. Exercises. Solutions. Appendix: Karl Wilhelm Feuerbach, Mathematician (Laura Guggenbuhl), Index.