In 1932,
G. D. Birkhoff created a set of four
postulates of
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
in the plane, sometimes referred to as Birkhoff's axioms. These postulates are all based on basic
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
that can be confirmed experimentally with a
scale and
protractor. 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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 , .
Models can be divided in ...
-based introduction to Euclidean geometry.
Birkhoff's
axiomatic system was utilized in the secondary-school textbook by Birkhoff and Beatley.
These axioms were also modified by the
School Mathematics Study Group
The School Mathematics Study Group (SMSG) was an American academic think tank focused on the subject of reform in mathematics education. Directed by Edward G. Begle and financed by the National Science Foundation, the group was created in 1958 i ...
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 geometry, non-Euclidean geometries. These are fundamental to the study and of hist ...
use variants of Birkhoff's axioms.
Birkhoff's Four 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 number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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''.
See also
*
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
*
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
*
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 geometry, non-Euclidean geometries. These are fundamental to the study and of hist ...
*
Hilbert's axioms
Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book ''Grundlagen der Geometrie'' (tr. ''The Foundations of Geometry'') as the foundation for a modern treatment of Euclidean geometry. Other well-known modern ax ...
*
Tarski's axioms
References
Foundations of geometry
Elementary geometry