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...
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Arthur Schopenhauer (redirect from Arthur Schopenhauer's criticism of the proofs of the parallel postulate)
evident in his criticism of contemporaneous attempts to prove the parallel postulate in Euclidean geometry. Writing shortly before the discovery of hyperbolic...
153 KB (19,999 words) - 09:48, 5 September 2024
Euclidean geometry (redirect from Euclid's second postulate)
seemed so intuitively obvious (with the possible exception of the parallel postulate) that theorems proved from them were deemed absolutely true, and thus...
58 KB (7,005 words) - 02:24, 31 August 2024
Sum of angles of a triangle (redirect from Triangle postulate)
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...
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direction (not necessarily the same length). Parallel lines are the subject of Euclid's parallel postulate. Parallelism is primarily a property of affine...
23 KB (2,818 words) - 14:57, 7 September 2024
Playfair's axiom (redirect from Playfair's Postulate)
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...
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affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former...
44 KB (6,023 words) - 21:22, 27 August 2024
seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute...
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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...
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consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent...
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and alternate angles. As a consequence of Euclid's parallel postulate, if the two lines are parallel, consecutive interior angles are supplementary, corresponding...
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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...
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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...
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Hyperbolic geometry (redirect from Ultra-parallel)
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...
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assumed a different form of the parallel postulate. It is in fact possible to create a valid geometry without the fifth postulate entirely, or with different...
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ISBN 1-58488-347-2. The parallel postulate is equivalent to the Equidistance postulate, Playfair axiom, Proclus axiom, the Triangle postulate and the Pythagorean...
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Later in the 19th century, it appeared that geometries without the parallel postulate (non-Euclidean geometries) can be developed without introducing any...
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particular the parallel postulate. A projective plane may be constructed by adding "points at infinity" where two otherwise parallel lines would intersect...
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comparison of magnitudes. While postulates 1 through 4 are relatively straightforward, the 5th is known as the parallel postulate and particularly famous. Book...
44 KB (4,312 words) - 20:35, 12 September 2024
vindicatus (Euclid freed of every flaw), an attempt to prove the parallel postulate using the method reductio ad absurdum. Such a quadrilateral is sometimes...
8 KB (1,126 words) - 22:17, 11 May 2024
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...
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that does not include the parallel postulate, called hyperbolic geometry. In this geometry, an infinite number of parallel lines pass through the point...
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List of axioms (redirect from List of postulates)
constructibility Rank-into-rank Kripke–Platek axioms Diamond principle Parallel postulate Birkhoff's axioms (4 axioms) Hilbert's axioms (20 axioms) Tarski's...
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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...
49 KB (5,964 words) - 09:57, 12 September 2024
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...
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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...
47 KB (6,964 words) - 03:20, 8 September 2024
the Parallel Postulate, as shown by Max Dehn. In the presence of the Archimedean axiom, the Lotschnittaxiom is equivalent with the Parallel Postulate. As...
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result in absolute geometry because its proof does not depend upon the parallel postulate. In several high school treatments of geometry, the term "exterior...
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no need to break into different cases, such as parallel to the x {\displaystyle x} -axis and non-parallel. The other advantage is that these homogeneous...
22 KB (3,424 words) - 23:07, 25 July 2023