Cover: Euclidean and Non-Euclidean Geometries, 4th Edition by Marvin J. Greenberg

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Euclidean and Non-Euclidean Geometries

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Fourth  Edition©2008  Marvin J. Greenberg

ISBN:9781429281331

ISBN:9780716799481

About

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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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

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 Murphys 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.

Marvin Jay Greenberg