Foundations of geometry /
Synopsis: Foundations of Geometry, Second Edition is written to help enrich the education of all mathematics majors and facilitate a smooth transition into more advanced mathematics courses. The text also implements the latest national standards and recommendations regarding geometry for the prepara...
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| Format: | Book |
| Language: | English |
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Boston :
Pearson,
[2012]
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| Edition: | 2nd ed. |
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Table of Contents:
- Preface
- 1: Prologue: Euclid's Elements
- 1-1: Geometry before Euclid
- 1-2: Logical structure of Euclid's Elements
- 1-2: Logical structure of Euclid's Elements
- 1-3: Historical significance of Euclid's Elements
- 1-4: Look at Book I of the Elements
- 1-5: Critique of Euclid's Elements
- 1-6: Final observations about the Elements
- 2: Axiomatic Systems And Incidence Geometry
- 2-1: Structure of an axiomatic system
- 2-2: Example: incidence geometry
- 2-3: Parallel postulates in incidence geometry
- 2-4: Axiomatic systems and the real world
- 2-5: Theorems, proofs, and logic
- 2-6: Some theorems from incidence geometry
- 3: Axioms For Plane Geometry
- 3-1: Undefined terms and two fundamental axioms
- 3-2: Distance and the ruler postulate
- 3-3: Plane separation
- 3-4: Angle measure and the protractor postulate
- 3-5: Crossbar theorem and the Linear Pair theorem
- 3-6: Side-angle-side postulate
- 3-7: Parallel postulates and models
- 4: Neutral Geometry
- 4-1: Exterior Angle theorem and existence of perpendiculars
- 4-2: Triangle congruence conditions
- 4-3: Three inequalities for triangles
- 4-4: Alternate interior angles theorem
- 4-5: Saccheri-Legendre theorem
- 4-6: Quadrilaterals
- 4-7: Statements equivalent to the Euclidean parallel postulate
- 4-8: Rectangles and defect
- 4-9: Universal hyperbolic theorem
- 5: Euclidean Geometry
- 5-1: Basic theorems of Euclidean geometry
- 5-2: Parallel projection theorem
- 5-3: Similar triangles
- 5-4: Pythagorean theorem
- 5-5: Trigonometry
- 5-6: Exploring the Euclidean geometry of the triangle
- 6: Hyperbolic Geometry
- 6-1: Basic theorems of hyperbolic geometry
- 6-2: Common perpendiculars
- 6-3: Angle of parallelism
- 6-4: Limiting parallel rays
- 6-5: Asymptotic triangles
- 6-6: Classification of parallels
- 6-7: Properties of the critical function
- 6-8: Defect of a triangle
- 6-9: Is the real world hyperbolic?
- 7: Area
- 7-1: Neutral area postulate
- 7-2: Area in Euclidean geometry
- 7-3: Dissection theory
- 7-4: Proof of the dissection theorem in Euclidean geometry
- 7-5: Associated Saccheri quadrilateral
- 7-6: Area and defect in hyperbolic geometry
- 8: Circles
- 8-1: Circles and lines in neutral geometry
- 8-2: Circles and triangles in neutral geometry
- 8-3: Circles in Euclidean geometry
- 8-4: Circular continuity
- 8-5: Circumference and area of Euclidean circles
- 8-6: Exploring Euclidean circles.
- 9: Constructions
- 9-1: Compass and straightedge constructions
- 9-2: Neutral constructions
- 9-3: Euclidean constructions
- 9-4: Construction of regular polygons
- 9-5: Area constructions
- 9-6: Three impossible constructions
- 10: Transformations
- 10-1: Properties of isometries
- 10-2: Rotations, translations, and glide reflections
- 10-3: Classification of Euclidean motions
- 10-4: Classification of hyperbolic motions
- 10-5: Transformational approach to the foundations
- 10-6: Similarity transformations in Euclidean geometry
- 10-7: Euclidean inversions in circles
- 11: Models
- 11-1: Cartesian model for Euclidean geometry
- 11-2: Poincare disk model for hyperbolic geometry
- 11-3: Other models for hyperbolic geometry
- 11-4: Models for elliptic geometry
- 12: Polygonal Models And The Geometry Of Space
- 12-1: Curved surfaces
- 12-2: Approximate models for the hyperbolic plane
- 12-3: Geometric surfaces
- 12-4: Geometry of the universe
- 12-5: Conclusion
- 12-6: Further study
- 12-7: Templates
- Appendices:
- A: Euclid's Book I
- A-1: Definitions
- A-2: Postulates
- A-3: Common notions
- A-4: Propositions
- B: Systems of axioms for geometry
- B-1: Hilbert's axioms
- B-2: Birkoff's axioms
- B-3: MacLane's axioms
- B-4: SMSG axioms
- B-5: UCSMP axioms
- C: Postulates used in this book
- C-1: Criteria used in selecting the postulates
- C-2: Statements of the postulates
- C-3: Logical relationships
- D: Van Hiele model
- E: Set notation and the real numbers
- E-1: Some elementary set theory
- E-2: Axioms for the real numbers
- E-3: Properties of the real numbers
- E-4: One-to-one and onto functions
- E-5: Continuous functions
- F: Hints for selected exercises
- Bibliography
- Index.