Rent 9780131437005 | Foundations of Geometry | Venema, Gerard A. @knetbooks.com

Summary Table of Contents

For sophomore/junior-level courses in Geometry; especially appropriate for students that will go on to teach high-school mathematics.This text comfortably serves as a bridge between lower-level mathematics courses (calculus and linear algebra) and upper-level courses (real analysis and abstract algebra). It fully implements the latest national standards and recommendations regarding geometry for the preparation of high school mathematics teachers. Foundations of Geometry particularly teaches good proof-writing skills, emphasizes the historical development of geometry, and addresses certain issues concerning the place of geometry in human culture.

Preface

1 Euclid's Elements

(16)

1.1 Geometry before Euclid

(1)

1.2 The logical structure of Euclid's Elements

... MORE

(1)

1.3 The historical importance of Euclid's Elements

(2)

1.4 A look at Book I of the Elements

(3)

1.5 A critique of Euclid's Elements

(3)

1.6 Final observations about the Elements

(6)

2 Axiomatic Systems and Incidence Geometry

(14)

2.1 Undefined and defined terms

(1)

2.2 Axioms

(1)

2.3 Theorems

(1)

2.4 Models

(1)

2.5 An example of an axiomatic system

(6)

2.6 The parallel postulates

(2)

2.7 Axiomatic systems and the real world

(4)

3 Theorems, Proofs, and Logic

(12)

3.1 The place of proof in mathematics

(1)

3.2 Mathematical language

(2)

3.3 Stating theorems

(3)

3.4 Writing proofs

(2)

3.5 Indirect proof

(1)

3.6 The theorems of incidence geometry

(3)

4 Set Notation and the Real Numbers

(9)

4.1 Some elementary set theory

(2)

4.2 Properties of the real numbers

(3)

4.3 Functions

(1)

4.4 The foundations of mathematics

(3)

5 The Axioms of Plane Geometry

(42)

5.1 Systems of axioms for geometry

(3)

5.2 The undefined terms

(1)

5.3 Existence and incidence

(1)

5.4 Distance

(6)

5.5 Plane separation

(4)

5.6 Angle measure

(3)

5.7 Betweenness and the Crossbar Theorem

(14)

5.8 Side-Angle-Side

(5)

5.9 The parallel postulates

(1)

5.10 Models

(4)

6 Neutral Geometry

(41)

6.1 Geometry without the parallel postulate

(1)

6.2 Angle-Side-Angle and its consequences

(2)

6.3 The Exterior Angle Theorem

(5)

6.4 Three inequalities for triangles

102

(5)

6.5 The Alternate Interior Angles Theorem

107

(3)

6.6 The Saccheri-Legendre Theorem

110

(3)

6.7 Quadrilaterals

113

(3)

6.8 Statements equivalent to the Euclidean Parallel Postulate

116

(7)

6.9 Rectangles and defect

123

(8)

6.10 The Universal Hyperbolic Theorem

131

(4)

7 Euclidean Geometry

135

(26)

7.1 Geometry with the parallel postulate

135

(2)

7.2 Basic theorems of Euclidean geometry

137

(2)

7.3 The Parallel Projection Theorem

139

(2)

7.4 Similar triangles

141

(2)

7.5 The Pythagorean Theorem

143

(2)

7.6 Trigonometry

145

(2)

7.7 Exploring the Euclidean geometry of the triangle

147

(14)

8 Hyperbolic Geometry

161

(33)

8.1 The discovery of hyperbolic geometry

161

(2)

8.2 Basic theorems of hyperbolic geometry

163

(5)

8.3 Common perpendiculars

168

(3)

8.4 Limiting parallel rays and asymptotically parallel lines

171

(10)

8.5 Properties of the critical function

181

(4)

8.6 The defect of a triangle

185

(4)

8.7 Is the real world hyperbolic?

189

(5)

9 Area

194

(31)

9.1 The Neutral Area Postulate

195

(3)

9.2 Area in Euclidean geometry

198

(8)

9.3 Dissection theory in neutral geometry

206

(7)

9.4 Dissection theory in Euclidean geometry

213

(3)

9.5 Area and defect in hyperbolic geometry

216

(9)

10 Circles

225

(39)

10.1 Basic definitions

226

(1)

10.2 Circles and lines

227

(4)

10.3 Circles and triangles

231

(7)

10.4 Circles in Euclidean geometry

238

(6)

10.5 Circular continuity

244

(3)

10.6 Circumference and area of Euclidean circles

247

(8)

10.7 Exploring Euclidean circles

255

(9)

11 Constructions

264

(20)

11.1 Compass and straightedge constructions

265

(2)

11.2 Neutral constructions

267

(3)

11.3 Euclidean constructions

270

(2)

11.4 Construction of regular polygons

272

(4)

11.5 Area constructions

276

(3)

11.6 Three impossible constructions

279

(5)

12 Transformations

284

(43)

12.1 The transformational perspective

285

(1)

12.2 Properties of isometrics

286

(6)

12.3 Rotations, translations, and glide reflections

292

(8)

12.4 Classification of Euclidean motions

300

(3)

12.5 Classification of hyperbolic motions

303

(1)

12.6 A transformational approach to the foundations

304

(6)

12.7 Euclidean inversions in circles

310

(17)

13 Models

327

(18)

13.1 The significance of models for hyperbolic geometry

327

(2)

13.2 The Cartesian model for Euclidean geometry

329

(2)

13.3 The Poincar� disk model for hyperbolic geometry

331

(5)

13.4 Other models for hyperbolic geometry

336

(5)

13.5 Models for elliptic geometry

341

(4)

14 Polygonal Models and the Geometry of Space

345

(47)

14.1 Curved surfaces

346

(11)

14.2 Approximate models for the hyperbolic plane

357

(6)

14.3 Geometric surfaces

363

(6)

14.4 The geometry of the universe

369

(6)

14.5 Conclusion

375

(1)

14.6 Further study

375

(9)

14.7 Templates

384

(8)

APPENDICES

A Euclid's Book I

392

(6)

A.1 Definitions

392

(2)

A.2 Postulates

394

(1)

A.3 Common Notions

394

(1)

A.4 Propositions

394

(4)

B Other Axiom Systems

398

(10)

B.1 Hilbert's axioms

398

(2)

B.2 Birkhoff's axioms

400

(1)

B.3 MacLane's axioms

401

(1)

B.4 SMSG axioms

402

(3)

B.5 UCSMP axioms

405

(3)

C The Postulates Used in this Book

408

(3)

C.1 The undefined terms

408

(1)

C.2 The postulates of neutral geometry

408

(1)

C.3 The parallel postulates

409

(1)

C.4 The area postulates

409

(1)

C.5 The reflection postulate

410

(1)

C.6 Logical relationships

410

(1)

D The van Hiele Model

411

(1)

E Hints for Selected Exercises

412

(9)

Bibliography

421

(4)

Index

425

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