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 axiomatic 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]

Birkhoff's Four Postulates

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 $a (mod 2 π)$ so that if $A$ and $B$ are points (not equal to $O$ ) of $ℓ$ and $m$ , respectively, the difference $a m - a ℓ (mod 2π)$ 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 $∠ B'A'C' = \pm∠ BAC$ , then $d (B', C') = kd (B, C), ∠ C'B'A' = \pm∠ CBA$ , and $∠ A'C'B' = \pm∠ ACB$ .

References

^ 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

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