Appendix B - Birkhoff's Axioms for Euclidean Geometry  Acrobat Reader IconPrintout

Exit book to another website.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
Exit book to another website.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.

(Some web browsers display some characters incorrectly, an angle shows as and not equal to shows as  .)

Undefined Elements and Relations.

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

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

Next to Appendix A - Hilbert's Axioms for Euclidean GeometryAppendix A - Hilbert's Axioms      Next to Appendix C - SMSG Axioms for Euclidean Geometry.Appendix C - SMSG Axioms

Ch. 2 Euclidean/NonEuclidean TOC  Table of Contents

  Timothy Peil  Mathematics Dept.  MSU Moorhead

Copyright 2005, 2006 - Timothy Peil