**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 *

Timothy Peil authored this web course.

** Introductory Note.** Birkhoff's Axiom set is an example of what
is called a

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

**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 2*π*). 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 |*x _{A} - x*

**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 2*π*),
so that, if *A *
≠* *
*O* and *B *≠*
O*
are points of* l* and *m*, respectively, the difference *a _{m}
- a_{l}*(mod 2

**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*)*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.

© Copyright 2005, 2006 - Timothy Peil |