The Foundations of Geometry

Transcription

1 The Foundations of Geometry Gerard A. Venema Department of Mathematics and Statistics Calvin College SUB Gottingen A 7409 PEARSON Prentice Hall Upper Saddle River, New Jersey 07458

2 Contents Preface Euclid's Elements Geometry before Euclid The logical structure of Euclid's Elements The historical importance of Euclid's Elements A look at Book I of the Elements A critique of Euclid's Elements Final observations about the Elements 11 Axiomatic Systems and Incidence Geometry Undefined and defined terms Axioms Theorems Models An example of an axiomatic system The parallel postulates Axiomatic systems and the real world 27 Theorems, Proofs, and Logic The place of proof in mathematics Mathematical language Stating theorems Writing proofs Indirect proof \ The theorems of incidence geometry ';,. 40 Set Notation and the Real Numbers Some elementary set theory Properties of the real numbers Functions, The foundations of mathematics 49 The Axioms of Plane Geometry Systems of axioms for geometry The undefined terms Existence and incidence Distance Plane separation Angle measure 67 ix

3 vi Contents 5.7 Betweenness and the Crossbar Theorem Side-Angle-Side The parallel postulates Models 90 6 Neutral Geometry Geometry without the parallel postulate Angle-Side-Angle and its consequences The Exterior Angle Theorem Three inequalities for triangles The Alternate Interior Angles Theorem The Saccheri-Legendre Theorem Quadrilaterals Statements equivalent to the Euclidean Parallel Postulate Rectangles and defect The Universal Hyperbolic Theorem Euclidean Geometry Geometry with the parallel postulate Basic theorems of Euclidean geometry The Parallel Projection Theorem Similar triangles The Pythagorean Theorem Trigonometry Exploring the Euclidean geometry of the triangle Hyperbolic Geometry The discovery of hyperbolic geometry Basic theorems of hyperbolic geometry Common perpendiculars Limiting parallel rays and asymptotically parallel lines, Properties of the critical function The defect of a triangle Is the real world hyperbolic? Area The Neutral Area Postulate Area in Euclidean geometry Dissection theory in neutral geometry Dissection theory in Euclidean geometry Area and defect in hyperbolic geometry Circles Basic definitions Circles and lines 227

4 Contents vii 10.3 Circles and triangles Circles in Euclidean geometry Circular continuity Circumference and area of Euclidean circles Exploring Euclidean circles Constructions Compass and straightedge constructions Neutral constructions Euclidean constructions Construction of regular polygons Area constructions Three impossible constructions Transformations The transformational perspective Properties of isometries Rotations, translations, and glide reflections Classification of Euclidean motions Classification of hyperbolic motions A transformational approach to the foundations Euclidean inversions in circles Models The significance of models for hyperbolic geometry The Cartesian model for Euclidean geometry The Poincare disk model for hyperbolic geometry Other models for hyperbolic geometry Models for elliptic geometry Polygonal Models and the Geometry of Space \ Curved surfaces Approximate models for the hyperbolic plane Geometric surfaces The geometry of the universe Conclusion, Further study Templates 384 APPENDICES A Euclid's Book I 392 A.I Definitions 392 A.2 Postulates 394

5 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