parallel postulate
nounEtymology 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.
1
Geometry, Mathematics, Sciences 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.
- Before we examine the particulars of the parallel postulates of Hyperbolic and Elliptical geometries, we must see their logic.1962, Mary Irene Solon, The Parallel Postulates of Non-Euclidean Geometry: The Pentagon: A Mathematics Magazine for Students, Volumes 22-25, page 28:
- The earliest known attempt to prove the parallel postulates from the other axioms was by Ptolemy in about 150 A.D.1969, John B. Fraleigh, Mainstreams of Mathematics, Addison-Wesley, page 95:
3 more examples
- 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.1989, Walter Prenowitz, Meyer Jordan, Basic Concepts of Geometry, Ardsley House Publishers, published 1965, page 28:
- 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.1998, David A. Singer, Geometry: Plane and Fancy, Springer, page 16:
- 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.2006, John Ratcliffe, Foundations of Hyperbolic Manifolds, Springer, page 7:
- Synonyms
- Euclid's fifth postulate, triangle postulate
- Related terms
- Euclidean geometry, non-Euclidean geometry