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.
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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.

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