Meaning of parallel postulate | Babel Free

Definitions

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.

Equivalents

Examples

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

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

“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.”

“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.”

“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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