Preface.

Acknowledgments.

**PART I FOUNDATIONS.**

**1 The First Geometers.**

1.1 Egypt.

1.2 Babylon.

1.3 China.

**2 Thales.**

2.1 The Axiomatic System.

2.2 Deductive Logic.

2.3 Proof Writing.

**3 Plato and Aristotle.**

3.1 Form.

3.2 Categorical Propositions..

3.3 Categorical Syllogisms.

3.4 Figures.

**PART II THE GOLDEN AGE.**

**4 Pythagoras.**

4.1 Number Theory.

4.2 The Pythagorean Theorem.

4.3 Archytas.

4.4 The Golden Ratio.

**5 Euclid.**

5.1 The Elements.

5.2 Constructions.

5.3 Triangles.

5.4 Parallel Lines.

5.5 Circles.

5.6 The Pythagorean Theorem Revisited.

**6 Archimedes.**

6.1 The Archimedean Library.

6.2 The Method of Exhaustion.

6.3 The Method.

6.4 Preliminaries to the Proof.

6.5 The Volume of a Sphere.

**PART III ENLIGHTENMENT.**

**7 François Viète.**

7.1 The Analytic Art.

7.2 Three Problems.

7.3 Conic Sections.

7.4 The Analytic Art in Two Variables.

**8 René Descartes.**

8.1 Compasses.

8.2 Method.

8.3 Analytic Geometry.

**9 Gérard Desargues.**

9.1 Projections.

9.2 Points at Infinity.

9.3 Theorems of Desargues and Menelaus.

9.4 Involutions.

**PART IV A STRANGE NEW WORLD.**

**10 Giovanni Saccheri.**

10.1 The Question of Parallels.

10.2 The Three Hypotheses.

10.3 Conclusions for Two Hypotheses.

10.4 Properties of Parallel Lines.

10.5 Parallelism Redefined.

**11 Johann Lambert.**

11.1 The Three Hypotheses Revisited.

11.2 Polygons.

11.3 Omega Triangles.

11.4 Pure Reason.

**12 Nicolai Lobachevski and János Bolyai.**

12.1 Parallel Fundamentals.

12.2 Horocycles.

12.3 The Surface of a Sphere.

12.4 Horospheres.

12.5 Evaluating the Pi Function.

**PART V NEW DIRECTIONS.**

**13 Bernhard Riemann.**

13.1 Metric Spaces.

13.2 Topological Spaces.

13.3 Stereographic Projection.

13.4 Consistency of Non-Euclidean Geometry.

**14 Jean-Victor Poncelet.**

14.1 The Projective Plane.

14.2 Duality.

14.3 Perspectivity.

14.4 Homogeneous Coordinates.

**15 Felix Klein.**

15.1 Group Theory.

15.2 Transformation Groups.

15.3 The Principal Group.

15.4 Isometries of the Plane.

15.5 Consistency of Euclidean Geometry.

References.

Index.