Circles: A Mathematical View

Cambridge University Press, 1995 - Mathematics - 102 pages
This revised edition of a mathematical classic originally published in 1957 will bring to a new generation of students the enjoyment of investigating that simplest of mathematical figures, the circle. As a concession to the general neglect of geometry in school and college curricula, however, the author has supplemented this new edition with a chapter 0 designed to introduce readers to the special vocabulary of circle concepts with which the author could assume his readers of two generations ago were familiar. For example, Pedoe carefully explains what is meant by the circumcircle, incircle, and excircles of a triangle as well as the circumcentre, incentre, and otrthocentre. The reader can then understand his discussion in Chapter 1 of the nine-point circle, and of Feuerbach's theorem. As an appendix, Pedoe includes a biographical article by Laura Guggenbuhl on Karl Wilhelm Feuerbach, a little-known mathematician with a tragically short life, who published his theorem in a slender geometric treatise in 1822. Readers of Circles need only be armed with paper, pencil, compass and straightedge to find great pleasure in following the constructions and theorems. Those who think that geometry using Euclidean tools died out with the ancient Greeks will be pleasantly surprised to learn many interesting results which were only discovered in modern time. And those who think that they learned all they needed to know about circles in high school will find much to enlighten them in chapters dealing with the representation of a circle by a point in three-space, a model for non-Euclidean geometry, and isoperimetric property of the circle. -- from back cover.

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Contents

 CHAPTER I 1 Inversion 4 Feuerbachs theorem 9 Extension of Ptolemys theorem 10 Fermats problem 11 The centres of similitude of two circles 12 Coaxal systems of circles 14 Canonical form for coaxal system 16
 Modulus and argument 45 Circles as level curves 46 The crossratio of four complex numbers 47 Mobius transformations of the flplane 50 A Mobius transformation dissected 51 The group property 53 Special transformations 55 The Poincar6 model 58

 Further properties 19 Problem of Apollonius 22 Compass geometry 23 Representation of a circle 26 First properties of the representation 28 Coaxal systems 29 Deductions from the representation 30 Conjugacy relations 33 Circles cutting at a given angle 35 Representation of inversion 36 The envelope of a system 37 Some further applications 39 Some anallagmatic curves 43 CHAPTER III 44
 The parallel axiom 61 CHAPTER IV 64 Existence of a solution 65 Method of solution 66 Area of a polygon 67 Regular polygons 69 Rectifiable curves 71 Approximation by polygons 73 Area enclosed by a curve 76 Exercises 79 Solutions 84 Karl Wilhelm Feuerbach Mathematician by Laura Guggenbuhl 89 Index 101 Copyright