In geometry, the parallel postulate is the fifth postulate in Euclid's Elements and a distinctive axiom in Euclidean geometry. It states that, in two-dimensional...

evident in his criticism of contemporaneous attempts to prove the parallel postulate in Euclidean geometry. Writing shortly before the discovery of hyperbolic...

axioms (postulates) and deducing many other propositions (theorems) from these. One of those is the parallel postulate which relates to parallel lines on...

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...

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 to the given line...

affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former...

triangle postulate states that the sum of the angles of a triangle is two right angles. This postulate is equivalent to the parallel postulate. In the...

seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute...

direction (not necessarily the same length). Parallel lines are the subject of Euclid's parallel postulate. Parallelism is primarily a property of affine...

or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: For any given line R and point...

without the parallel postulate or any of its alternatives. Traditionally, this has meant using only the first four of Euclid's postulates. The term was...

comparison of magnitudes. While postulates 1 through 4 are relatively straightforward, the 5th is known as the parallel postulate and particularly famous. Book...

consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent...

geometry sufficient to resolve the ancient problem of whether the parallel postulate is a logical consequence of the rest of Euclid's axioms of plane geometry...

Later in the 19th century, it appeared that geometries without the parallel postulate (non-Euclidean geometries) can be developed without introducing any...

the Parallel Postulate, as shown by Max Dehn. In the presence of the Archimedean axiom, the Lotschnittaxiom is equivalent with the Parallel Postulate. As...

angles, and linear pairs. As a consequence of Euclid's parallel postulate, if the two lines are parallel, consecutive angles and linear pairs are supplementary...

ISBN 1-58488-347-2. The parallel postulate is equivalent to the Equidistance postulate, Playfair axiom, Proclus axiom, the Triangle postulate and the Pythagorean...

impossible to prove (parallel postulate). After the introduction at the end of the 19th century of non-Euclidean geometries, the old postulates were re-formalized...

a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must...

unprovability of the parallel postulate lead to several philosophical problems, the main one being that before this discovery, the parallel postulate and all its...

that does not include the parallel postulate, called hyperbolic geometry. In this geometry, an infinite number of parallel lines pass through the point...

comparison of magnitudes. While postulates 1 through 4 are relatively straightforward, the 5th is known as the parallel postulate and particularly famous. Book...

parallel postulate from Euclid's Elements is equivalent to the statement that given a straight line and a point not on that line, only one parallel to...

vindicatus (Euclid freed of every flaw), an attempt to prove the parallel postulate using the method reductio ad absurdum. Such a quadrilateral is sometimes...

particular the parallel postulate. A projective plane may be constructed by adding "points at infinity" where two otherwise parallel lines would intersect...

constructibility Rank-into-rank Kripke–Platek axioms Diamond principle Parallel postulate Birkhoff's axioms (4 axioms) Hilbert's axioms (20 axioms) Tarski's...

very old problem of proving Euclid's Fifth Postulate, the "Parallel Postulate", from his first four postulates had never been forgotten. Beginning not long...

self-contradictory.: 270 : 133  His elaborate attempt to prove the parallel postulate was significant for the further development of geometry, as it clearly...

space is always 180 degrees. This fact is equivalent to Euclid's parallel postulate. This allows the determination of the measure of the third angle of...