Chapter Two
Euclidean and Non-Euclidean Geometry
Mathematicians have long since regarded it as demeaning to work on problems related to elementary geometry in two or three dimensions, in spite of the fact that it is precisely this sort of mathematics which is of practical value.

Branko Grunbaum (1929?) and G. C. Shephard (1926?)

Section 2.1 -
Introduction   (Exercises 2.12.2)
Historical Overview
Analytic Models for Plane Geometry   (Exercises 2.32.10)

Section 2.2 - Incidence Axioms   (Exercises 2.112.16)

Section 2.3 - Distance and Ruler Axioms   (Exercises 2.172.27)

Section 2.4 -
Plane Separation Postulate   (Exercises 2.282.31)
Angles and Angle Measure   (Exercises 2.322.40)

Section 2.5 -
Supplement Postulate   (Exercises 2.412.47)
SAS Postulate   (Exercises 2.482.54)

Section 2.6 -
Parallel Lines without a Parallel Postulate   (Exercises 2.552.61)

Section 2.7 -
Euclidean Parallel Postulate   (Exercises 2.652.67)
Hyperbolic Parallel Postulate   (Exercises 2.682.74)
Elliptic Parallel Postulate   (Exercises 2.752.79)

Section 2.8 - Euclidean, Hyperbolic, and Elliptic Geometries   (Exercise 2.80)

Chapter Two Exercises. Exercises from all the sections.
Solutions.
Solutions to selected exercises.
Self-Assessment Quizzes. Quizzes for all sections.

Appendix A - Hilbert's Axioms for Euclidean Geometry
Appendix B - Birkhoff's Axioms for Euclidean Geometry
Appendix C - SMSG Axioms for Euclidean Geometry
Appendix D - Links to two on-line editions of Euclid's Elements: David E. Joyce's Java edition of Euclid's Elements (1997) or Oliver Byrne's edition of Euclid published in 1847.

Geometrical intuition, strictly speaking, is not mathematical, but rather a priori physical intuition. In its purely mathematical aspect our Euclidean space intuition is perfectly correct, namely, it represents correctly a certain structure existing in the realm of mathematical objects. Even physically it is correct "in the small".

Kurt Gφdel (19061978)

 Chapter One Axiomatic Systems Chapter Three Transformational Geometry © Copyright 2005, 2006 - Timothy Peil