The Foundations of Geometry

Gerard A. Venema Department of Mathematics and Statistics Calvin College

SUB Gottingen 7 219 059 926

2006 A 7409

PEARSON Prentice Hall Upper Saddle River, New Jersey 07458 Contents

Preface ix Euclid's Elements 1 1.1 Geometry before Euclid 1 1.2 The logical structure of Euclid's Elements 2 1.3 The historical importance of Euclid's Elements 3 1.4 A look at Book I of the Elements 5 1.5 A critique of Euclid's Elements 8 1.6 Final observations about the Elements 11

Axiomatic Systems and Incidence Geometry 17 2.1 Undefined and defined terms 17 2.2 Axioms 18° 2.3 Theorems 18 2.4 Models 19 2.5 An example of an axiomatic system 19 2.6 The parallel postulates 25 2.7 Axiomatic systems and the real world 27

Theorems, Proofs, and Logic 31 3.1 The place of proof in mathematics 31 3.2 Mathematical language 32 3.3 Stating theorems 34 3.4 Writing proofs 37 3.5 Indirect proof \. . 39 3.6 The theorems of incidence geometry ';, . 40

Set Notation and the Real Numbers 43 4.1 Some elementary set theory 43 4.2 Properties of the real numbers 45 4.3 Functions ,48 4.4 The foundations of mathematics 49

The Axioms of Plane Geometry 52 5.1 Systems of axioms for geometry 53 5.2 The undefined terms 56 5.3 Existence and incidence 56 5.4 Distance 57 5.5 Plane separation 63 5.6 Angle measure 67 vi Contents

5.7 Betweenness and the Crossbar Theorem 70 5.8 Side-Angle-Side 84 5.9 The parallel postulates 89 5.10 Models 90

6 Neutral Geometry 94 6.1 Geometry without the parallel postulate 94 6.2 Angle-Side-Angle and its consequences 95 6.3 The Exterior Angle Theorem 97 6.4 Three inequalities for triangles 102 6.5 The Alternate Interior Angles Theorem 107 6.6 The Saccheri-Legendre Theorem 110 6.7 Quadrilaterals 113 6.8 Statements equivalent to the Euclidean Parallel Postulate 116 6.9 Rectangles and defect 123 6.10 The Universal Hyperbolic Theorem 131

7 Euclidean Geometry 135 7.1 Geometry with the parallel postulate 135 7.2 Basic theorems of Euclidean geometry 137 7.3 The Parallel Projection Theorem 139 7.4 Similar triangles 141 7.5 The Pythagorean Theorem 143 7.6 Trigonometry 145 7.7 Exploring the Euclidean geometry of the triangle 147

8 Hyperbolic Geometry 161 8.1 The discovery of hyperbolic geometry 161 8.2 Basic theorems of hyperbolic geometry 163 8.3 Common perpendiculars . 168 8.4 Limiting parallel rays and asymptotically parallel lines ,. . 171 8.5 Properties of the critical function . 181 8.6 The defect of a triangle 185 8.7 Is the real world hyperbolic? 189

9 Area 194 9.1 The Neutral Area Postulate 195 9.2 Area in Euclidean geometry 198 9.3 Dissection theory in neutral geometry 206 9.4 Dissection theory in Euclidean geometry 213 9.5 Area and defect in hyperbolic geometry . . • 216

10 Circles 225 10.1 Basic definitions 226 10.2 Circles and lines 227 Contents vii

10.3 Circles and triangles 231 10.4 Circles in Euclidean geometry 238 10.5 Circular continuity 244 10.6 Circumference and area of Euclidean circles 247 10.7 Exploring Euclidean circles 255

11 Constructions 264 11.1 Compass and straightedge constructions 265 11.2 Neutral constructions 267 11.3 Euclidean constructions 270 11.4 Construction of regular polygons 272 11.5 Area constructions 276 11.6 Three impossible constructions 279

12 Transformations 284 12.1 The transformational perspective 285 12.2 Properties of isometries 286 12.3 Rotations, translations, and glide reflections 292 12.4 Classification of Euclidean motions 300 12.5 Classification of hyperbolic motions 303 12.6 A transformational approach to the foundations 304 12.7 Euclidean inversions in circles 310

13 Models 327 13.1 The significance of models for hyperbolic geometry 327 13.2 The Cartesian model for Euclidean geometry 329 13.3 The Poincare disk model for hyperbolic geometry 331 13.4 Other models for hyperbolic geometry 336 13.5 Models for elliptic geometry 341

14 Polygonal Models and the Geometry of Space \ 345 14.1 Curved surfaces 346 14.2 Approximate models for the hyperbolic plane 357 14.3 Geometric surfaces 363 14.4 The geometry of the universe 369 14.5 Conclusion ,. 375 14.6 Further study 375 14.7 Templates 384

APPENDICES

A Euclid's Book I 392 A.I Definitions 392 A.2 Postulates 394 viii Contents

A.3 Common Notions 394 A.4 Propositions 394

B Other Axiom Systems 398 B.I Hilbert's axioms 398 B.2 Birkhoff's axioms 400 B.3 MacLane's axioms 401 B.4 SMSG axioms 402 B.5 UCSMP axioms 405

C The Postulates Used in this Book 408 C.I The undefined terms 408 C.2 The postulates of neutral geometry 408 C.3 The parallel postulates 409 C.4 The area postulates 409 C.5 The reflection postulate 410 C.6 Logical relationships 410

D The van Hiele Model 411

E Hints for Selected Exercises 412

Bibliography 421

Index 425