In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry in the plane, 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. Birkhoff's axiom system was utilized in the secondary-school textbook by Birkhoff and Beatley. 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. Postulates The distance between two points and is denoted by , and the angle formed by three points is denoted by . Postulate I: Postulate of line measure. The set of points on any line can be put into a 1:1 correspondence with the real numbers so that for all points and . Postulate II: Point-line postulate. There is one and only one line that contains any two given distinct points and . Postulate III: Postulate of angle measure. The set of rays through any point can be put into 1:1 correspondence with the real numbers so that if and are points (not equal to ) of and , respectively, the difference of the numbers associated with the lines and is . Furthermore, if the point on varies continuously in a line not containing the vertex , the number varies continuously also. Postulate IV: Postulate of similarity. Given two triangles and and some constant such that and , then , and . See also Euclidean geometry Euclidean space Foundations of geometry Hilbert's axioms Tarski's axioms References Foundations of geometry Elementary geometry