eBook ISBN:  9781470457327 
Product Code:  SPEC/12.E 
List Price:  $50.00 
MAA Member Price:  $37.50 
AMS Member Price:  $37.50 
eBook ISBN:  9781470457327 
Product Code:  SPEC/12.E 
List Price:  $50.00 
MAA Member Price:  $37.50 
AMS Member Price:  $37.50 

Book DetailsSpectrumVolume: 12; 1997; 102 pp
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. The author has supplemented this new edition with a special chapter designed to introduce readers to the vocabulary of circle concepts with which the readers of two generations ago were familiar. Readers of Circles need only be armed with paper, pencil, compass, and straight edge 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 times. Novices and experts alike will find much to enlighten them in chapters dealing with the representation of a circle by a point in threespace, a model for nonEuclidean geometry, and the isoperimetric property of the circle.

Table of Contents

Articles

The ninepoint circle

Inversion

Feuerbach’s theorem

Extension of Ptolemy’s theorem

Fermat’s problem

The centres of similitude of two circles

Coaxal systems of circles

Canonical form for coaxal system

Further properties

Problem of Apollonius

Compass geometry

Representation of a circle

Euclidean threespace $E_3$

First properties of the representation

Coaxal systems

Deductions from the representation

Conjugacy relations

Circles cutting at a given angle

Representation of inversion

The envelope of a system

Some further applications

Some anallagmatic curves

Complex numbers

The Argand diagram

Modulus and argument

Circles as level curves

The crossratio of four complex numbers

Möbius transformations of the ${\frak s}$plane

A Möbius transformation dissected

The group property

Special transformations

The fundamental theorem

The Poincaré model

The parallel axiom

NonEuclidean distance

Steiner’s enlarging process

Existence of a solution

Method of solution

Area of a polygon

Regular polygons

Rectifiable curves

Approximation by polygons

Area enclosed by a curve


Reviews

This excellent little book is an invaluable addition to the library of any teacher of geometry at any level...Discussions are straightforward; figures are well done; and the tone of the discourse is 'friendly.'
The Mathematics Teacher, May 1996


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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. The author has supplemented this new edition with a special chapter designed to introduce readers to the vocabulary of circle concepts with which the readers of two generations ago were familiar. Readers of Circles need only be armed with paper, pencil, compass, and straight edge 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 times. Novices and experts alike will find much to enlighten them in chapters dealing with the representation of a circle by a point in threespace, a model for nonEuclidean geometry, and the isoperimetric property of the circle.

Articles

The ninepoint circle

Inversion

Feuerbach’s theorem

Extension of Ptolemy’s theorem

Fermat’s problem

The centres of similitude of two circles

Coaxal systems of circles

Canonical form for coaxal system

Further properties

Problem of Apollonius

Compass geometry

Representation of a circle

Euclidean threespace $E_3$

First properties of the representation

Coaxal systems

Deductions from the representation

Conjugacy relations

Circles cutting at a given angle

Representation of inversion

The envelope of a system

Some further applications

Some anallagmatic curves

Complex numbers

The Argand diagram

Modulus and argument

Circles as level curves

The crossratio of four complex numbers

Möbius transformations of the ${\frak s}$plane

A Möbius transformation dissected

The group property

Special transformations

The fundamental theorem

The Poincaré model

The parallel axiom

NonEuclidean distance

Steiner’s enlarging process

Existence of a solution

Method of solution

Area of a polygon

Regular polygons

Rectifiable curves

Approximation by polygons

Area enclosed by a curve

This excellent little book is an invaluable addition to the library of any teacher of geometry at any level...Discussions are straightforward; figures are well done; and the tone of the discourse is 'friendly.'
The Mathematics Teacher, May 1996