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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 "axioms" and five "postulates."

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 "axioms" and five "postulates."

==The fifth postulate==

==The fifth postulate==

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 removed 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 removed 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.