From the time of Euclid in 300 BCE until the 19th Century, Geometry was based on 5 axioms, considered to be clearly true, including the Parallel Postulate:
"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, meet on that side on which are the angles less than the two right angles."
Mathematicians were not satisfied by the obviousness of this statement and attempted to prove it from the other four axioms for 2000 years.
In 1733 Giovanni Girolamo Saccheri attempted to prove the validity of the 5th postulate by contradiction. In doing so he inadvertently discovered elliptical and hyperbolic geometry. While he considered these to be absurd, later geometers found them to have unexpected depth and applications.
lyrics
beyond the contradictions of the mind lie
the oceans of infinite space
beneath the foundations of arbitrary
and the postulate’s cracked facade
the churning of the plain unbounded
lines diverge and are drawn together
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