Birkhoff's axioms

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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]


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 {ab, ...} 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 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 2π) so that if A and B are points (not equal to O) of and m, respectively, the difference am − a (mod 2π) of the numbers associated with the lines and m 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: 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.

See also[edit]


  1. ^ 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
  2. ^ Birkhoff, George David; Beatley, Ralph (2000) [first edition, 1940], Basic Geometry (3rd ed.), American Mathematical Society, ISBN 978-0-8218-2101-5
  3. ^ Kelly, Paul Joseph; Matthews, Gordon (1981), The non-Euclidean, hyperbolic plane: its structure and consistency, Springer-Verlag, ISBN 0-387-90552-9