| |

| |

| |

Prologue: Euclid's Elements | |

| |

| |

| |

Geometry before Euclid | |

| |

| |

| |

The logical structure of Euclid's Elements | |

| |

| |

| |

The historical significance of Euclid's Elements | |

| |

| |

| |

A look at Book I of the Elements | |

| |

| |

| |

A critique of Euclid's Elements | |

| |

| |

| |

Final observations about the Elements | |

| |

| |

| |

Axiomatic Systems and Incidence Geometry | |

| |

| |

| |

The structure of an axiomatic system | |

| |

| |

| |

An example: Incidence geometry | |

| |

| |

| |

The parallel postulates in incidence geometry | |

| |

| |

| |

Axiomatic systems and the real world | |

| |

| |

| |

Theorems, proofs, and logic | |

| |

| |

| |

Some theorems from incidence geometry | |

| |

| |

| |

Axioms for Plane Geometry | |

| |

| |

| |

The undefined terms and two fundamental axioms | |

| |

| |

| |

Distance and the Ruler Postulate | |

| |

| |

| |

Plane separation | |

| |

| |

| |

Angle measure and the Protractor Postulate | |

| |

| |

| |

The Crossbar Theorem and the Linear Pair Theorem | |

| |

| |

| |

The Side-Angle-Side Postulate | |

| |

| |

| |

The parallel postulates and models | |

| |

| |

| |

Neutral Geometry | |

| |

| |

| |

The Exterior Angle Theorem and perpendiculars | |

| |

| |

| |

Triangle congruence conditions | |

| |

| |

| |

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 | |

| |

| |

| |

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 | |

| |

| |

| |

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 | |

| |

| |

| |

Similarity transformations in Euclidean geometry | |

| |

| |

| |

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 Poincarï¿½ disk model for hyperbolic geometry | |

| |

| |

| |

Other models for hyperbolic geometry | |

| |

| |

| |

Models for elliptic geometry | |

| |

| |

| |

Regular Tessellations | |

| |

| |

| |

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 | |

| |

| |

Appendices | |

| |

| |

| |

Euclid's Book I | |

| |

| |

| |

Definitions | |

| |

| |

| |

Postulates | |

| |

| |

| |

Common Notions | |

| |

| |

| |

Propositions | |

| |

| |

| |

Systems of Axioms for Geometry | |

| |

| |

| |

Filling in Euclid's gaps | |

| |

| |

| |

Hilbert's axioms | |

| |

| |

| |

Birkhoff's axioms | |

| |

| |

| |

MacLane's axioms | |

| |

| |

| |

SMSG axioms | |

| |

| |

| |

UCSMP axioms | |

| |

| |

| |

The Postulates Used in this Book | |

| |

| |

| |

The undefined terms | |

| |

| |

| |

Neutral postulates | |

| |

| |

| |

Parallel postulates | |

| |

| |

| |

Area postulates | |

| |

| |

| |

The reflection postulate | |

| |

| |

| |

Logical relationships | |

| |

| |

| |

Set Notation and the Real Numbers | |

| |

| |

| |

Some elementary set theory | |

| |

| |

| |

Properties of the real numbers | |

| |

| |

| |

Functions | |

| |

| |

| |

The van Hiele Model | |

| |

| |

| |

Hints for Selected Exercises | |

| |

| |

Bibliography | |

| |

| |

Index | |