In 1932, G. D. Birkhoff created a set of four

postulate An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...

s of

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

in the plane, sometimes referred to as Birkhoff's axioms. These postulates are all based on basic

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

that can be confirmed experimentally with a scale and

protractor A protractor is a measuring instrument, typically made of transparent plastic or glass, for measuring angles. Some protractors are simple half-discs or full circles. More advanced protractors, such as the bevel protractor, have one or two sw ...

. Since the postulates build upon the

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s, the approach is similar to a

model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...

-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 a
SMSG axioms
A few other textbooks in the

foundations of geometry Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but ...

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

s 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 {{math, 1=∠ ''A'C'B' '' = ±∠ ''ACB''.

References

Foundations of geometry Elementary geometry

Postulates

See also

References