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Euclidean and Non-Euclidean Geometries Development and History

ISBN-10: 0716799480
ISBN-13: 9780716799481
Edition: 4th 2008 (Revised)
List price: $190.99 Buy it from $74.48 Rent it from $26.11
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Description: Providing an overview of classic and hyperbolic geometries, and placing the work of key mathematicians and philosophers in a historical context, this title includes coverage on geometric transformations, and modes of the hyperbolic planes and  More...

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Book details

List price: $190.99
Edition: 4th
Copyright year: 2008
Publisher: W. H. Freeman & Company
Publication date: 9/28/2007
Binding: Hardcover
Pages: 512
Size: 7.00" wide x 7.75" long x 1.00" tall
Weight: 2.398

Providing an overview of classic and hyperbolic geometries, and placing the work of key mathematicians and philosophers in a historical context, this title includes coverage on geometric transformations, and modes of the hyperbolic planes and pseudospheres, with review notes and essay topics.

Preface
Introduction
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 [pi]
Conclusion
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
Hilbert's Axioms
Flaws in Euclid
Axioms of Betweenness
Axioms of Congruence
Axioms of Continuity
Hilbert's Euclidean Axiom of Parallelism
Conclusion
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
History of the Parallel Postulate
Review
Proclus
Equidistance
Wallis
Saccheri
Clairaut's Axiom and Proclus' Theorem
Legendre
Lambert and Taurinus
Farkas Bolyai
The Discovery of Non-Euclidean Geometry
Janos 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?
Independence of the Parallel Postulate
Consistency of Hyperbolic Geometry
Beltrami's Interpretation
The Beltrami-Klein Model
The Poincare Models
Perpendicularity in the Beltrami-Klein Model
A Model of the Hyperbolic Plane from Physics
Inversion in Circles, Poincare Congruence
The Projective Nature of the Beltrami-Klein Model
Conclusion
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
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 Poincare Model
Congruence Described by Motions
Symmetry
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
Elliptic and Other Riemannian Geometries
Hilbert's Geometry Without Real Numbers
Axioms
Bibliography
Symbols
Name Index
Subject Index

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