## Table of Contents

**1. Prologue: Euclid’s ***Elements*

1.1 Geometry before Euclid

1.2 The logical structure of Euclid’s *Elements*

1.3 The historical significance of Euclid’s *Elements*

1.4 A look at Book I of the *Elements*

1.5 A critique of Euclid’s *Elements*

1.6 Final observations about the *Elements*

**2. Axiomatic Systems and Incidence Geometry**

2.1 The structure of an axiomatic system

2.2 An example: Incidence geometry

2.3 The 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 The 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 The Crossbar Theorem and the Linear Pair Theorem

3.6 The Side-Angle-Side Postulate

3.7 The parallel postulates and models

**4. Neutral Geometry**

4.1 The Exterior Angle Theorem and perpendiculars

4.2 Triangle congruence conditions

4.3 Three inequalities for triangles

4.4 The Alternate Interior Angles Theorem

4.5 The Saccheri-Legendre Theorem

4.6 Quadrilaterals

4.7 Statements equivalent to the Euclidean Parallel Postulate

4.8 Rectangles and defect

4.9 The Universal Hyperbolic Theorem

**5. Euclidean Geometry**

5.1 Basic theorems of Euclidean geometry

5.2 The Parallel Projection Theorem

5.3 Similar triangles

5.4 The Pythagorean Theorem

5.5 Trigonometry

5.6 Exploring the Euclidean geometry of the triangle

**6. Hyperbolic Geometry**

6.1 The discovery of hyperbolic geometry

6.2 Basic theorems of hyperbolic geometry

6.3 Common perpendiculars

6.4 Limiting parallel rays and asymptotically parallel lines

6.5 Properties of the critical function

6.6 The defect of a triangle

6.7 Is the real world hyperbolic?

**7. Area**

7.1 The Neutral Area Postulate

7.2 Area in Euclidean geometry

7.3 Dissection theory in neutral geometry

7.4 Dissection theory in Euclidean geometry

7.5 Area and defect in hyperbolic geometry

**8. Circles**

8.1 Basic definitions

8.2 Circles and lines

8.3 Circles and triangles

8.4 Circles in Euclidean geometry

8.5 Circular continuity

8.6 Circumference and area of Euclidean circles

8.7 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 The transformational perspective

10.2 Properties of isometries

10.3 Rotations, translations, and glide reflections

10.4 Classification of Euclidean motions

10.5 Classification of hyperbolic motions

10.6 Similarity transformations in Euclidean geometry

10.7 A transformational approach to the foundations

10.8 Euclidean inversions in circles

**11. Models**

11.1 The significance of models for hyperbolic geometry

11.2 The Cartesian model for Euclidean geometry

11.3 The Poincaré disk model for hyperbolic geometry

11.4 Other models for hyperbolic geometry

11.5 Models for elliptic geometry

11.6 Regular Tessellations

**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 The 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 Filling in Euclid’s gaps

B.2 Hilbert’s axioms

B.3 Birkhoff’s axioms

B.4 MacLane’s axioms

B.5 SMSG axioms

B.6 UCSMP axioms

**C. The Postulates Used in this Book**

C.1 The undefined terms

C.2 Neutral postulates

C.3 Parallel postulates

C.4 Area postulates

C.5 The reflection postulate

C.6 Logical relationships

**D. Set Notation and the Real Numbers**

D.1 Some elementary set theory

D.2 Properties of the real numbers

D.3 Functions

**E. The van Hiele Model**

**F. Hints for Selected Exercises**

**Bibliography**

**Index**