Birkhoff's axioms

Template:Short description In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry in the plane, sometimes referred to as Birkhoff's axioms.<ref>Template:Citation</ref> 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.<ref>Template:Citation</ref> 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.<ref>Template:Citation</ref>

Birkhoff's Four Postulates

The distance between two points Template:Math and Template:Math is denoted by Template:Math, and the angle formed by three points Template:Math is denoted by Template:Math.

Postulate I: Postulate of line measure. The set of points Template:Math on any line can be put into a 1:1 correspondence with the real numbers Template:Math so that Template:Math for all points Template:Math and Template:Math.

Postulate II: Point-line postulate. There is one and only one line Template:Math that contains any two given distinct points Template:Math and Template:Math.

Postulate III: Postulate of angle measure. The set of rays Template:Math through any point Template:Math can be put into 1:1 correspondence with the real numbers Template:Math so that if Template:Math and Template:Math are points (not equal to Template:Math) of Template:Math and Template:Math, respectively, the difference Template:Math of the numbers associated with the lines Template:Math and Template:Math is Template:Math. Furthermore, if the point Template:Math on Template:Math varies continuously in a line Template:Math not containing the vertex Template:Math, the number Template:Math varies continuously also.

Postulate IV: Postulate of similarity. Given two triangles Template:Math and Template:Math and some constant Template:Math such that Template:Math and Template:Math, then Template:Math, and Template:Math.

See also

• Euclidean geometry
• Euclidean space
• Foundations of geometry
• Hilbert's axioms
• Tarski's axioms

References

<references />