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- Euclidean and Non-Euclidean Geometries

# Euclidean and Non-Euclidean Geometries

## Development and HistoryFourth Edition| ©2008 Marvin J. Greenberg

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 h...

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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Features

New to This Edition

**Additional Coverage**This edition offers greater coverage of key events and contributors throughout history including additional information on the Pythagoreans, Plato, Euclid, Proclus, Saccheri and Bolyai.

**Revised Coverage**This edition offers clearer coverage of Models, Isomorphism of Models, Axioms of Betweenness, Axioms of Congruence, Geometry without a Parallel Axiom, Measures of Angles and Segments, Angle Sum of a Triangle, Similar Triangles, and Saccheri and Lambert Quadrilaterals.

**New Coverage**A new section has been added to discuss

*pi*(Chapter 1)

New section on Straightedge and Compass, Constructions (Chapter1)

New Section on "Brief Historical Remarks" (Chapter 2)

New Section on "Consistency" (Chapter 2)

New Section on "Brief History of Real Projective Geometry" (Chapter 2)

New section on "Equidistant Curves" (Chapter 5)

New section on the "Beltrami’s Interpretation" (Chapter 7)

New section on the "Fruitfulness of Hyperbolic Geometry for Other Branches of Mathematics, Cosmology, and Art" (Chapter 8)

New Section on “Bolyai's Constructions in the Hyperbolic Plane” (Chapter 10)

**More than 50% additional projects added to the text**, plus new exercises included in every chapter.

**
Euclidean and Non-Euclidean Geometries**

Fourth Edition| ©2008

Marvin J. Greenberg

# Digital Options

**Euclidean and Non-Euclidean Geometries**

Fourth Edition| 2008

Marvin J. Greenberg

## 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

## Authors

### Marvin J. Greenberg

**Marvin Jay Greenberg**is Emeritus Professor of Mathematics, University of California at Santa Cruz. He received his undergraduate degree from Columbia University, where he was a Ford Scholar. His PhD is from Princeton University, his thesis adviser having been the brilliant and fiery Serge Lang. He was subsequently an Assistant Professor at U.C. Berkeley for five years (two years of which he spent on NSF Postdoctoral Fellowships at Harvard and at the I.H.E.S. in Paris), an Associate Professor at Northeastern University for two years, and Full Professor at U.C. Santa Cruz for twenty five years. He took early retirement from that campus at age 57. His first published book was

*Lectures on Algebraic Topology*(Benjamin, 1967), which was later expanded into a joint work with John Harper,

*Algebraic Topology: A First Course*(Westview, 1981). His second book

*Lectures on Forms in Many Variables*(Benjamin, 1969) was about the subject started by Serge Lang in his thesis and subsequently developed by himself and others, culminating in the great theorem of Ax and Kochen showing that the conjecture of Emil Artin that p-adic fields are C2 is "almost true" (Terjanian found the first counter-example to the full conjecture). His Freeman text

*Euclidean and Non-Euclidean Geometries: Development and History*had its first edition appear in 1974, and is now in its vastly expanded fourth edition. His early journal publications are in the subject of algebraic geometry, where he discovered a functor J.-P. Serre named after him and an approximation theorem J. Nicaise and J. Sebag named after him. He is also the translator of Serre’s Corps Locaux. In later years, he published some articles on the foundations of geometry, most of whose results are included in his Freeman text. His latest publication appeared in the March 2010 issue of the

*American Mathematical Monthly*, entitled "Old and New Results in the Foundations of Elementary Euclidean and Non-Euclidean Geometries"; a copy of that paper is sent along with the Instructors' Manual to any instructor who requests it. Professor Greenberg lives alone in Berkeley, CA, and has an adult son who lives on the boat his son owns. His main interests outside of mathematics are (1) golf, where he is a founding member of the Shivas Irons Society based on Michael Murphy's classic book

*Golf in the Kingdom*(now made into a movie); (2) the economy and the stock market, where he is very concerned about the hard times that have befallen the U.S., due in large part to the fiat fractional reserve monetary system that enabled very dangerous levels of debt to be transacted; and (3) the quest for enlightenment, the topic of a course he taught at Crown College, UCSC, around 1970.

**Euclidean and Non-Euclidean Geometries**

Fourth Edition| 2008

Marvin J. Greenberg

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