parallel postulate
Noun
Meaning
parallel postulate (plural parallel postulates)
(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.
Synonyms
Examples
Closely related to the problem of the parallel postulate is the problem of whether physical space is infinite. Euclid assumes.
Although Schopenhauer could see no justification for trying to prove Euclid's parallel postulate, he did see a reason for examining another of Euclid's axioms.
As the first 28 propositions of Euclid (in The Elements) do not require the use of the parallel postulate or anything equivalent to it, they are all true statements in absolute geometry.
Ball, p. 485 Since non-Euclidean geometry is provably relatively consistent with Euclidean geometry, the parallel postulate cannot be proved from the other postulates.
Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate ) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense.
Consequently, rectangles exist (a statement equivalent to the parallel postulate) only in Euclidean geometry.