TOC & Ch. 0 & Ch. 1 Axiom Table of Contents Ch. 0 Introduction Ch. 1 Axiomatic Systems 1.1.1 Introduction 1.1.2 Examples 1.1.3 History 1.2 A Finite Geometry 1.3 Finite Projective 1.4 Applications Ch. 2 Neutral Geometry Ch. 2 Table of Contents 2.1.1 Introduction 2.1.2 History 2.1.3 Analytic Models 2.2 Incidence Axioms 2.3 Distance/Ruler Axioms 2.4.1 Plane Separation Axiom 2.4.2 Angle & Measure 2.5.1 Supplement Postulate 2.5.2 SAS Postulate 2.6.1 Parallel Lines 2.6.2 Saccheri Quadrilateral 2.7.1 Euclid Parallel Postulate 2.7.2 Hyperbolic Parallel Postulate 2.7.3 Elliptic Parallel Postulate 2.8 Euclid/Hyperbolic/Elliptic Birkhoff's Axioms Hilbert's Axioms SMSG Axioms Ch. 3 Transformational Ch. 3 Table of Contents 3.1.1 Introduction 3.1.2 History 3.2.1 Definitions 3.2.2 Analytic Model 3.2.3 Affine Transformation 3.3.1 Isometry 3.3.2 Model/Collinearity 3.3.3 Model/Isometry 3.4.1 Direct Isometry 3.4.2 Model/Direct 3.5.1 Indirect Isometry 3.5.2 Model/Indirect 3.6.1 Similarity Transformation 3.6.2 Model/Similarity 3.7 Other Affine Transformations Ch. 4 Projective Geometry Ch. 4 Table of Contents 4.1.1 Introduction 4.1.2 Historical 4.2.1 Axioms 4.2.2 Basic Theorems 4.3 Duality 4.4 Desargue's Theorem 4.5.1 Harmonic Sets 4.5.2 Music & Harmonic Sets 4.6.1 Definitions for Projectivity 4.6.2 Fundamental Theorem 4.6.3 Projectivity/Harmonic Sets 4.6.4 Alternate Construction 4.7.1 Conics 4.7.2 Pascal's Theorem 4.7.3 Tangents to Conics Other Topics Ch. 5 Spherical Geometry Ch. 6 Fractal Geometry Ch. 7 Topology Appendices Internet Resources Index Geometer's Sketchpad/GeoGebra JavaSketchpad/GeoGebraHTML Video Lectures Logic Review References Acknowledgements

Appendix B - Birkhoff's Axioms for Euclidean Geometry  Printout

George D. Birkhoff would be referred to by mathematicians as the author of this web course's "great-great-grandfather" as a direct line of thesis advisors, since Timothy Peil's Ph.D. thesis advisor was Allan Peterson whose advisor was John Barrett whose advisor was Hyman Ettlinger whose advisor was George Birkhoff.
- from the
Mathematics Genealogy Project
Timothy Peil authored this web course.

Introductory Note. Birkhoff's Axiom set is an example of what is called a metric geometry. A metric geometry has axioms for distance and angle measure, then betweenness and congruence are defined from distance and angle measures and properties of congruence are developed in theorems.

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Undefined Elements and Relations.

• points A, B, ...
• sets of points called lines l, m, ...
• distance between any two points A and B: a nonnegative real number d(A,B) such that d(A,B) = d(B,A)
• angle formed by three ordered points A, O, B (A O, B O) : AOB a real number (mod 2π). The point O is called the vertex of the angle.

Postulate I. (Postulate of Line Measure) The points A,B, ...  of any line l can be placed into one-to-one correspondence with the real numbers x so that |xA - xB| = d(A,B) for all points A, B.

Postulate II. (Point-Line Postulate) One and only one line l contains two given points P, Q (P Q).

Postulate III. (Postulate of Angle Measure) The half-lines l, m, ... through any point O can be put into one-to-one correspondence with the real numbers a(mod 2π), so that, if A O and B O are points of l and m, respectively, the difference am - al(mod 2π) is AOB. Furthermore if the point B on m varies continuously in a line r not containing the vertex O, the number am varies continuously also.

Postulate IV. (Similarity Postulate) If in two triangles ABC, A'B'C' and for some constant k > 0, d(A',B') = kd(A,B), d(A',C') = kd(A,C), and B'A'C' = ±BAC, then also d(B',C') = kd(B,C), A'B'C' = ±ABC, and A'C'B' = ±ACB.

Defined Terms

• A point B is between A and C (A C), if d(A,B) + d(B,C) = d(A,C).
• The half-line l' with endpoint O is defined by two points O, A in line l (A O) as the set of all points A' of l such that O is not between A and A'
• The points A and C, together with all point B between A and C, for segment AC.
• If A, B, C are three distinct points, the segments AB, BC, CA are said to form a triangle ABC with sides AB, BC, CA and vertices A, B, C.

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Birkhoff, George D., A set of postulates for plane geometry, Annals of Mathematics 33: pp. 329-345, 1932.

 Appendix A - Hilbert's Axioms      Appendix C - SMSG Axioms © Copyright 2005, 2006 - Timothy Peil