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# Euclidean and Non-Euclidean Geometries

## Fourth EditionMarvin J. Greenberg

©2008ISBN:9781429281331

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ISBN:9780716799481

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This is the definitive presentation of the history, development and philosophical significance of non-Euclidean geometry as well as of the rigorous foundations for it and for elementary Euclidean geometry, essentially according to Hilbert. Appropriate for liberal arts students, prospective high school teachers, math. majors, and even bright high school students. The first eight chapters are mostly accessible to any educated reader; the last two chapters and the two appendices contain more advanced material, such as the classification of motions, hyperbolic trigonometry, hyperbolic constructions, classification of Hilbert planes and an introduction to Riemannian geometry.

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Learn More## Table of Contents

**Chapter 1 Euclid’s Geometry**

Very Brief Survey of the Beginnings of Geometry

The Pythagoreans

Plato

Euclid of Alexandria

The Axiomatic Method

Undefined Terms

Euclid’s First Four Postulates

The Parallel Postulate

Attempts to Prove the Parallel Postulate

The Danger in Diagrams

The Power of Diagrams

Straightedge-and-Compass Constructions, Briefly

Descartes’ Analytic Geometry and Broader Idea of Constructions

Briefly on the Number ð

Conclusion

**Chapter 2 Logic and Incidence Geometry**

Elementary Logic

Theorems and Proofs

RAA Proofs

Negation

Quantifiers

Implication

Law of Excluded Middle and Proof by Cases

Brief Historical Remarks

Incidence Geometry

Models

Consistency

Isomorphism of Models

Projective and Affine Planes

Brief History of Real Projective Geometry

Conclusion

**Chapter 3 Hilbert’s Axioms**Flaws in Euclid

Axioms of Betweenness

Axioms of Congruence

Axioms of Continuity

Hilbert’s Euclidean Axiom of Parallelism

Conclusion

**Chapter 4 Neutral Geometry**

Geometry without a Parallel Axiom

Alternate Interior Angle Theorem

Exterior Angle Theorem

Measure of Angles and Segments

Equivalence of Euclidean Parallel Postulates

Saccheri and Lambert Quadrilaterals

Angle Sum of a Triangle

Conclusion

**Chapter 5 History of the Parallel Postulate**

Review

Proclus

Equidistance

Wallis

Saccheri

Clairaut’s Axiom and Proclus’ Theorem

Legendre

Lambert and Taurinus

Farkas Bolyai

**Chapter 6 The Discovery of Non-Euclidean Geometry**

János Bolyai

Gauss

Lobachevsky

Subsequent Developments

Non-Euclidean Hilbert Planes

The Defect

Similar Triangles

Parallels Which Admit a Common Perpendicular

Limiting Parallel Rays, Hyperbolic Planes

Classification of Parallels

Strange New Universe?

**Chapter 7 Independence of the Parallel Postulate**

Consistency of Hyperbolic Geometry

Beltrami’s Interpretation

The Beltrami–Klein Model

The Poincaré Models

Perpendicularity in the Beltrami–Klein Model

A Model of the Hyperbolic Plane from Physics

Inversion in Circles, Poincaré Congruence

The Projective Nature of the Beltrami–Klein Model

Conclusion

**Chapter 8 Philosophical Implications, Fruitful Applications**What Is the Geometry of Physical Space?

What Is Mathematics About?

The Controversy about the Foundations of Mathematics

The Meaning

The Fruitfulness of Hyperbolic Geometry for Other Branches of Mathematics, Cosmology, and Art

**Chapter 9 Geometric Transformations**

Klein’s Erlanger Programme

Groups

Applications to Geometric Problems

Motions and Similarities

Reflections

Rotations

Translations

Half-Turns

Ideal Points in the Hyperbolic Plane

Parallel Displacements

Glides

Classification of Motions

Automorphisms of the Cartesian Model

Motions in the Poincaré Model

Congruence Described by Motions

Symmetry

**Chapter 10 Further Results in Real Hyperbolic Geometry**

Area and Defect

The Angle of Parallelism

Cycles

The Curvature of the Hyperbolic Plane

Hyperbolic Trigonometry

Circumference and Area of a Circle

Saccheri and Lambert Quadrilaterals

Coordinates in the Real Hyperbolic Plane

The Circumscribed Cycle of a Triangle

Bolyai’s Constructions in the Hyperbolic Plane

Appendix A

Appendix B

Axioms

Bibliography

Symbols

Name Index

Subject Index