Birkhoff's Axioms

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## Postulates

## See also

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Birkhoff's Axioms

In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry in the plane, sometimes referred to as **Birkhoff's axioms**.^{[1]} These postulates are all based on basic geometry that can be confirmed experimentally with a scale and protractor. Since the postulates build upon the real numbers, the approach is similar to a model-based introduction to Euclidean geometry.

Birkhoff's axiom system was utilized in the secondary-school textbook by Birkhoff and Beatley.^{[2]}
These axioms were also modified by the School Mathematics Study Group to provide a new standard for teaching high school geometry, known as SMSG axioms.
A few other textbooks in the foundations of geometry use variants of Birkhoff's axioms.^{[3]}

The distance between two points *A* and *B* is denoted by *d*(*A, B*), and the angle formed by three points *A*, *B*, *C* is denoted by ? *ABC*.

**Postulate I: Postulate of line measure**.
The set of points {*A*, *B*, ...} on any line can be put into a 1:1 correspondence with the real numbers {*a*, *b*, ...} so that |*b* − *a*| = *d*(*A, B*) for all points *A* and *B*.

**Postulate II: Point-line postulate**.
There is one and only one line *l* that contains any two given distinct points *P* and *Q*.

**Postulate III: Postulate of angle measure**.
The set of rays {*l, m, n*, ...} through any point *O* can be put into 1:1 correspondence with the real numbers *a* (mod 2*?*) so that if *A* and *B* are points (not equal to *O*) of *l* and *m*, respectively, the difference *a*_{m} − *a*_{l} (mod 2?) of the numbers associated with the lines *l* and *m* is ? *AOB*. Furthermore, if the point *B* on *m* varies continuously in a line *r* not containing the vertex *O*, the number *a*_{m} varies continuously also.

**Postulate IV: Postulate of similarity**.
Given two triangles *ABC* and *A'B'C' * and some constant *k* > 0 such that *d*(*A', B' *) = *kd*(*A, B*), *d*(*A', C' *) = *kd*(*A, C*) and ? *B'A'C' * = ±? *BAC*, then *d*(*B', C' *) = *kd*(*B, C*), ? *C'B'A' * = ±? *CBA*, and ? *A'C'B' * = ±? *ACB*.

**^**Birkhoff, George David (1932), "A Set of Postulates for Plane Geometry (Based on Scale and Protractors)",*Annals of Mathematics*,**33**(2): 329-345, doi:10.2307/1968336, hdl:10338.dmlcz/147209, JSTOR 1968336**^**Birkhoff, George David; Beatley, Ralph (2000) [first edition, 1940],*Basic Geometry*(3rd ed.), American Mathematical Society, ISBN 978-0-8218-2101-5**^**Kelly, Paul Joseph; Matthews, Gordon (1981),*The non-Euclidean, hyperbolic plane: its structure and consistency*, Springer-Verlag, ISBN 0-387-90552-9

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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