parallel postulate
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.
Noun
- 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...
- 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:
- 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...
Origin
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.