eBook ISBN:  9781614445166 
Product Code:  SPEC/23.E 
List Price:  $30.00 
MAA Member Price:  $22.50 
AMS Member Price:  $22.50 
eBook ISBN:  9781614445166 
Product Code:  SPEC/23.E 
List Price:  $30.00 
MAA Member Price:  $22.50 
AMS Member Price:  $22.50 

Book DetailsSpectrumVolume: 23; 1998; 336 pp
Throughout most of this book, nonEuclidean geometries in spaces of two or three dimensions are treated as specializations of real projective geometry in terms of a simple set of axioms concerning points, lines, planes, incidence, order and continuity, with no mention of the measurement of distances or angles. This synthetic development is followed by the introduction of homogeneous coordinates, beginning with Von Staudt's idea of regarding points as entities that can be added or multiplied. Transformations that preserve incidence are called collineations. They lead in a natural way to isometries or 'congruent transformations.' Following a recommendation by Bertrand Russell, continuity is described in terms of order. Elliptic and hyperbolic geometries are derived from real projective geometry by specializing in elliptic or hyperbolic polarity which transforms points into lines (in two dimensions), planes (in three dimensions), and vice versa.
An unusual feature of the book is its use of the general linear transformation of coordinates to derive the formulas of elliptic and hyperbolic trigonometry. The area of a triangle is related to the sum of its angles by means of an ingenious idea of Gauss.
This treatment can be enjoyed by anyone who is familiar with algebra up to the elements of group theory. The present (sixth) edition clarifies some obscurities in the fifth, and includes a new section 15.9 on the author's useful concept of inversive distance.

Table of Contents

Chapters

I. THE HISTORICAL DEVELOPMENT OF NONEUCLIDEAN GEOMETRY

II. REAL PROJECTIVE GEOMETRY: FOUNDATIONS

III. REAL PROJECTIVE GEOMETRY: POLARITIES, CONICS AND QUADRICS

IV. HOMOGENEOUS COORDINATES

V. ELLIPTIC GEOMETRY IN ONE DIMENSION

VI. ELLIPTIC GEOMETRY IN TWO DIMENSIONS

VII. ELLIPTIC GEOMETRY IN THREE DIMENSIONS

VIII. DESCRIPTIVE GEOMETRY

IX. EUCLIDEAN AND HYPERBOLIC GEOMETRY

X. HYPERBOLIC GEOMETRY IN TWO DIMENSIONS

XI. CIRCLES AND TRIANGLES

XII. THE USE OF A GENERAL TRIANGLE OF REFERENCE

XIII. AREA

XIV. EUCLIDEAN MODELS

XV. CONCLUDING REMARKS

APPENDIX: ANGLES AND ARCS IN THE HYPERBOLIC PLANE


Reviews

This book presents a very readable account of the fundamental principles of hyperbolic and elliptic geometries. The approach is by way of projective geometry, the three chapters immediately following a brief historical sketch being devoted to an excellent survey of the foundations of real projective geometry.
L. M. Blumenthal, Mathematical Reviews 
No living geometer writes more clearly and beautifully about difficult topics than world famous professor H. S. M. Coxeter. When nonEuclidean geometry was first developed, it seemed little more than a curiosity with no relevance to the real world. Then to everyone's amazement, it turned out to be essential to Einstein's general theory of relativity! Coxeter's book has remained out of print for too long. Hats off to the MAA for making this classic available once more.
Martin Gardner 
Coxeter's geometry books are a treasure that should not be lost. I am delighted to see “NonEuclidean Geometry” back in print.
Doris Schattschneider


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Throughout most of this book, nonEuclidean geometries in spaces of two or three dimensions are treated as specializations of real projective geometry in terms of a simple set of axioms concerning points, lines, planes, incidence, order and continuity, with no mention of the measurement of distances or angles. This synthetic development is followed by the introduction of homogeneous coordinates, beginning with Von Staudt's idea of regarding points as entities that can be added or multiplied. Transformations that preserve incidence are called collineations. They lead in a natural way to isometries or 'congruent transformations.' Following a recommendation by Bertrand Russell, continuity is described in terms of order. Elliptic and hyperbolic geometries are derived from real projective geometry by specializing in elliptic or hyperbolic polarity which transforms points into lines (in two dimensions), planes (in three dimensions), and vice versa.
An unusual feature of the book is its use of the general linear transformation of coordinates to derive the formulas of elliptic and hyperbolic trigonometry. The area of a triangle is related to the sum of its angles by means of an ingenious idea of Gauss.
This treatment can be enjoyed by anyone who is familiar with algebra up to the elements of group theory. The present (sixth) edition clarifies some obscurities in the fifth, and includes a new section 15.9 on the author's useful concept of inversive distance.

Chapters

I. THE HISTORICAL DEVELOPMENT OF NONEUCLIDEAN GEOMETRY

II. REAL PROJECTIVE GEOMETRY: FOUNDATIONS

III. REAL PROJECTIVE GEOMETRY: POLARITIES, CONICS AND QUADRICS

IV. HOMOGENEOUS COORDINATES

V. ELLIPTIC GEOMETRY IN ONE DIMENSION

VI. ELLIPTIC GEOMETRY IN TWO DIMENSIONS

VII. ELLIPTIC GEOMETRY IN THREE DIMENSIONS

VIII. DESCRIPTIVE GEOMETRY

IX. EUCLIDEAN AND HYPERBOLIC GEOMETRY

X. HYPERBOLIC GEOMETRY IN TWO DIMENSIONS

XI. CIRCLES AND TRIANGLES

XII. THE USE OF A GENERAL TRIANGLE OF REFERENCE

XIII. AREA

XIV. EUCLIDEAN MODELS

XV. CONCLUDING REMARKS

APPENDIX: ANGLES AND ARCS IN THE HYPERBOLIC PLANE

This book presents a very readable account of the fundamental principles of hyperbolic and elliptic geometries. The approach is by way of projective geometry, the three chapters immediately following a brief historical sketch being devoted to an excellent survey of the foundations of real projective geometry.
L. M. Blumenthal, Mathematical Reviews 
No living geometer writes more clearly and beautifully about difficult topics than world famous professor H. S. M. Coxeter. When nonEuclidean geometry was first developed, it seemed little more than a curiosity with no relevance to the real world. Then to everyone's amazement, it turned out to be essential to Einstein's general theory of relativity! Coxeter's book has remained out of print for too long. Hats off to the MAA for making this classic available once more.
Martin Gardner 
Coxeter's geometry books are a treasure that should not be lost. I am delighted to see “NonEuclidean Geometry” back in print.
Doris Schattschneider