# Category: Foundations of geometry

Geometric Algebra (book)
Geometric Algebra is a book written by Emil Artin and published by Interscience Publishers, New York, in 1957. It was republished in 1988 in the Wiley Classics series (ISBN 0-471-60839-4). In 1962 Alg
Aristotle's axiom
Aristotle's axiom is an axiom in the foundations of geometry, proposed by Aristotle in On the Heavens. It states: If is an acute angle and AB is any segment, then there exists a point P on the ray and
Hilbert's fourth problem
In mathematics, Hilbert's fourth problem in the 1900 list of Hilbert's problems is a foundational question in geometry. In one statement derived from the original, it was to find — up to an isomorphis
Parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It states that, in two-d
Playfair's axiom
In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate): In a plane, given a line and a point not on it, at most one line parallel
Euclid's Elements
The Elements (Ancient Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC. It is
Tarski's axioms
Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry that is formulable in first-order logic with identity, and requiring no set theory (i.e., tha
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
Crossbar theorem
In geometry, the crossbar theorem states that if ray AD is between ray AC and ray AB, then ray AD intersects line segment BC. This result is one of the deeper results in axiomatic plane geometry. It i
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 Euclid
Birkhoff's axioms
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 co
Pasch's theorem
In geometry, Pasch's theorem, stated in 1882 by the German mathematician Moritz Pasch, is a result in plane geometry which cannot be derived from Euclid's postulates.
Point–line–plane postulate
In geometry, the point–line–plane postulate is a collection of assumptions (axioms) that can be used in a set of postulates for Euclidean geometry in two (plane geometry), three (solid geometry) or mo
Pasch's axiom
In geometry, Pasch's axiom is a statement in plane geometry, used implicitly by Euclid, which cannot be derived from the postulates as Euclid gave them. Its essential role was discovered by Moritz Pas
Lotschnittaxiom
The Lotschnittaxiom (German for "axiom of the intersecting perpendiculars") is an axiom in the foundations of geometry, introduced and studied by Friedrich Bachmann. It states: Perpendiculars raised o