Marvin J. Greenberg

Euclidean and Non-Euclidean Geometries

Development and History

• Gebundenes Buch

Produktbeschreibung

The fourth edition provides students with an overview of both classic and hyperbolic geometries, whilst also placing the work of key mathematicians and philosophers in a historical context. Including coverage on geometric transformations, and augmented by review notes and essay topics, this text is an excellent accompaniment to any geometry course.

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Produktdetails

• Verlag: Macmillan Education
• 4. Aufl.
• Seitenzahl: 512
• Erscheinungstermin: 1. September 2007
• Englisch, Unbestimmt
• Abmessung: 246mm x 164mm x 37mm
• Gewicht: 1082g
• ISBN-13: 9780716799481
• ISBN-10: 0716799480
• Artikelnr.: 21410642

Autorenporträt

MARVIN JAY GREENBERG, University of California, Santa Cruz, USA.

Inhaltsangabe

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 DIV>.
From the contents:
Euclid's Geometry - Logic and Incidence Geometry - Hilbert's Axioms - Neutral Geometries - History of the Parallel Postulate - The Discovery of Non-Euclidean Geometry - Independence of the Parallel Postulate - Philosophical Implications - Review Exercises - Some Topics for Essays - Index -