Birkhoff's axioms

In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry sometimes referred to as Birkhoff's axioms. 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. Other often-used axiomizations of plane geometry are Hilbert's axioms and Tarski's axioms.

Birkhoff's axiom system was utilized in the secondary-school text Basic Geometry (first edition, 1940; see References). Birkhoff's axioms were also modified by the School Mathematics Study Group to provide a new standard for teaching high school geometry, known as SMSG axioms.

Postulates

Postulate I: Postulate of Line Measure. A 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, , that contains any two given distinct points P and Q.

Postulate III: Postulate of Angle Measure. A 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 $\angle$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, d(A', B' ) = kd(A, B), d(A', C' ) = kd(A, C) and $\angle$B'A'C'  = ±$\angle$BAC, then d(B', C' ) = kd(B, C), $\angle$C'B'A'  = ±$\angle$CBA, and $\angle$A'C'B'  = ±$\angle$ACB.