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.
Exit book to another website.Branko Grunbaum (1929–?) and Exit book to another website.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)
      Saccheri Quadrilateral   (Exercises 2.62–2.64)

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 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 Exit book to another website.Euclid's Elements (1997) or Exit book to another website.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".
Exit book to another website.Kurt Gφdel (1906–1978)

Chapter One Axiomatic SystemsBack to Chapter One IndexNext to Chapter Three Index Chapter Three Transformational Geometry

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  Timothy Peil  Mathematics Dept.  MSU Moorhead

© Copyright 2005, 2006 - Timothy Peil