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And the result is a theorem.
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>
In ordinary discourse about a subject in mathematics or science, the terms "axiom" and "postulate" are often not distinguished—they are the statements that one takes as true from the outset, that one does not need to prove. One then uses those to prove other statements, which are then called [[theorem]]s. The difference is that postulates are initial statements relating to the subject matter, such as the geometrical postulates (including Euclid's fifth postulate, below) about angles and lines. Axioms, on the other hand, tend to be more universal statements about logic, such as "if X=Y, then Y=X".
==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===
Without the fifth postulate many familiar geometric ideas are no longer true, these include,
Without the fifth postulate many familiar geometric ideas are no longer true, these include,
* The sum of the interior degrees of a triangle are two right angle (180 degrees)
* The sum of the interior degrees of a triangle are two right angle (180 degrees)
== References ==
{{Reflist}}
[[Category:Mathematics]]
[[Category:Mathematics]]
[[category:Geometry]]
[[Category:Geometry]]