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.
(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.
Birkhoff, George D., A set of postulates for plane geometry, Annals of Mathematics 33: pp. 329-345, 1932.
© Copyright 2005, 2006 - Timothy Peil