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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]

  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

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