Changes

:It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.

:It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.

This is sometimes stated in an equivalent form:

This is sometimes stated in an equivalent form:

:Exactly one line can be drawn through any point not on a given line parallel to the given line.

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