See parallel postulate in All languages combined, or Wiktionary
{
"etymology_text": "From the reference to parallel lines in the definition as formulated below, following Scottish mathematician John Playfair; this wording leads to a convenient basic categorization of Euclidean and non-Euclidean geometries. The original formulation in Euclid's Elements makes no mention of parallels.",
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"ref": "1962, Mary Irene Solon, The Parallel Postulates of Non-Euclidean Geometry: The Pentagon: A Mathematics Magazine for Students, Volumes 22-25, page 28:",
"text": "Before we examine the particulars of the parallel postulates of Hyperbolic and Elliptical geometries, we must see their logic.",
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"ref": "1969, John B. Fraleigh, Mainstreams of Mathematics, Addison-Wesley, page 95:",
"text": "The earliest known attempt to prove the parallel postulates from the other axioms was by Ptolemy in about 150 A.D.",
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"ref": "1989, Walter Prenowitz, Meyer Jordan, Basic Concepts of Geometry, Ardsley House Publishers, published 1965, page 28:",
"text": "Euclid's parallel postulate may not have seemed so important when you studied geometry in high school, since it is used only once in order to derive the basic result 1 on alternate interior angles, which is then constantly used to derive further results.",
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"ref": "1998, David A. Singer, Geometry: Plane and Fancy, Springer, page 16:",
"text": "In order to understand this section, it is vital to keep in mind an important fact: The \"proof\" of the parallel postulate presented here is not correct. In fact, there can be no proof of the parallel postulate that relies only on the other axioms and postulates of Euclid.",
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"ref": "2006, John Ratcliffe, Foundations of Hyperbolic Manifolds, Springer, page 7:",
"text": "After enduring twenty centuries of criticism, Euclid's theory of parallels was fully vindicated in 1868 when Eugenio Beltrami proved the independence of Euclid's parallel postulate by constructing a Euclidean model of the hyperbolic plane.",
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"An axiom in Euclidean geometry: given a straight line L and a point p not on L, there exists exactly one straight line parallel to L that passes through p; a variant of this axiom, such that the number of lines parallel to L that pass through p may be zero or more than one."
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"(geometry) An axiom in Euclidean geometry: given a straight line L and a point p not on L, there exists exactly one straight line parallel to L that passes through p; a variant of this axiom, such that the number of lines parallel to L that pass through p may be zero or more than one."
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"etymology_text": "From the reference to parallel lines in the definition as formulated below, following Scottish mathematician John Playfair; this wording leads to a convenient basic categorization of Euclidean and non-Euclidean geometries. The original formulation in Euclid's Elements makes no mention of parallels.",
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"text": "In order to understand this section, it is vital to keep in mind an important fact: The \"proof\" of the parallel postulate presented here is not correct. In fact, there can be no proof of the parallel postulate that relies only on the other axioms and postulates of Euclid.",
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This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-12-21 from the enwiktionary dump dated 2024-12-04 using wiktextract (d8cb2f3 and 4e554ae). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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