Birkhoff's axioms
Encyclopedia
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 number
s, 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).
A set of points {A, B, ...} on any line can be put into a 1:1 correspondence with the real number
s {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, ℓ, 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 ℓ and m, respectively, the difference am − aℓ (mod 2π) of the numbers associated with the lines ℓ and m is
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
B'A'C' = ±
BAC, then d(B', C' ) = kd(B, C), 
C'B'A' = ±
CBA, and 
A'C'B' = ±
ACB.
George David Birkhoff
-External links:* − from National Academies Press, by Oswald Veblen....
created a set of four postulates of Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
sometimes referred to as Birkhoff's axioms. These postulates are all based on basic geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
that can be confirmed experimentally with a scale
Vernier scale
A vernier scale is an additional scale which allows a distance or angle measurement to be read more precisely than directly reading a uniformly-divided straight or circular measurement scale...
and protractor
Protractor
In geometry, a protractor is a circular or semicircular tool for measuring an angle or a circle. The units of measurement utilized are usually degrees.Some protractors are simple half-discs; these have existed since ancient times...
. Since the postulates build upon the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, the approach is similar to a model
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
-based introduction to Euclidean geometry. Other often-used axiomizations of plane geometry are Hilbert's axioms
Hilbert's axioms
Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie , as the foundation for a modern treatment of Euclidean geometry...
and Tarski's axioms
Tarski's axioms
Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called "elementary," that is formulable in first-order logic with identity, and requiring no set theory . Other modern axiomizations of Euclidean geometry are those by Hilbert and George...
.
Birkhoff's axiom system was utilized in the secondary-school text Basic Geometry (first edition, 1940; see References).
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 number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s {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, ℓ, 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 ℓ and m, respectively, the difference am − aℓ (mod 2π) of the numbers associated with the lines ℓ and m is
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
B'A'C' = ±BAC, then d(B', C' ) = kd(B, C), C'B'A' = ±CBA, and A'C'B' = ±ACB.