parallel postulate

From the reference to parallel lines in the definition as formulated below, following Scottish mathematician John Playfair; this wording leads to a convenient basic categorisation of Euclidean and non-Euclidean geometries. The original formulation in Euclid's Elements makes no mention of parallels.

parallel postulate (plural parallel postulates)

1. (geometry) An axiom of Euclidean geometry equivalent to the statement that, given a straight line L and a point P not on the line, 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.
   • 1962, Mary Irene Solon, The Parallel Postulates of Non-Euclidean Geometry, The Pentagon: A Mathematics Magazine for Students, Volumes 22-25, page 28,

     Before we examine the particulars of the parallel postulates of Hyperbolic and Elliptical geometries, we must see their logic.

   • 1969, John B. Fraleigh, Mainstreams of Mathematics, Addison-Wesley, page 95,

     The earliest known attempt to prove the parallel postulates from the other axioms was by Ptolemy in about 150 A.D.

   • 1989 [1965], Walter Prenowitz, Meyer Jordan, Basic Concepts of Geometry, Ardsley House Publishers, page 28,

     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.

   • 1998, David A. Singer, Geometry: Plane and Fancy, Springer, page 16,

     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.

   • 2006, John Ratcliffe, Foundations of Hyperbolic Manifolds, Springer, page 7,

     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.

Equivalent formulations include:

• (From Euclid's Elements): If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
• (The triangle postulate): The sum of the angles in any triangle equals a straight angle.
• Numerous others listed at Parallel postulate#Equivalent properties on Wikipedia.Wikipedia

Variations can be classified as follows:[1]

• (elliptic parallel postulate): No straight line exists that is parallel to L and passes through P;
• (Euclidean parallel postulate): Exactly one straight line exists that is parallel to L and passes through P;
• (hyperbolic parallel postulate): At least two straight lines exist that are parallel to L and pass through P.

It is also possible to forego the postulate entirely, as is the case in absolute geometry.

• (axiom of Euclidean geometry): Euclid's fifth postulate, triangle postulate

• Euclidean geometry
• non-Euclidean geometry

• Absolute geometry on Wikipedia.Wikipedia
• Non-Euclidean geometry on Wikipedia.Wikipedia