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A '''postulate''' is a statement that is assumed to be true without proof. [[Euclid]], the father of geometry, based ''The Elements'' on ten such statements, divided into five "[[axiom]]s" and five "postulates."

A '''postulate''' is a statement that is assumed to be true without proof. [[Euclid]], the father of geometry, based ''The Elements'' on ten such statements, divided into five "[[axiom]]s" and five "postulates."<ref>Euclid.  ''The Elements''.</ref>  These are used as foundation for geometry<ref>"1.1:  Definitions, Theorems, and Postulates."  ''Beauty, Rigor, Surprise''.</ref> and occasionally applied to other sciences, such as [[special relativity]].<ref>http://casa.colorado.edu/~ajsh/sr/postulate.html</ref><ref>https://www.nobelprize.org/educational/physics/relativity/postulates-1.html</ref>

==Euclid's fifth postulate==

==Euclid's fifth postulate==

:Given a line and a point not on that line exactly one line can be drawn through that point which does not intersect the original line.

:Given a line and a point not on that line exactly one line can be drawn through that point which does not intersect the original line.

That has never seemed as "self-evident" as the others, and in fact for centuries mathematicians thought it could be proved and tried to produce proofs. Eventually, mathematicians realized that this was impossible. If you negated the parallel postulate, you ended up with a perfectly logical, consistent system, a ''non-Euclidean geometry'' that simply happened to describe a kind of geometry ''different'' from plane geometry.

That has never seemed as "self-evident" as the others, and in fact for centuries mathematicians thought it could be proved and tried to produce proofs. Eventually, mathematicians realized that this was impossible. If you negated the parallel postulate, you ended up with a perfectly logical, consistent system, a ''non-Euclidean geometry'' that simply happened to describe a kind of geometry ''different'' from plane geometry.<ref>"Geometry."  ''The New Encyclopedia Britannica'', 15th ed.</ref><ref>"4.4:  The Playfair Postulate and its Consequences."  Beauty, Rigor, Surprise.</ref>

===Equivalent statements of the fifth postulate===

===Equivalent statements of the fifth postulate===