Changes

: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 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 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.

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

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