Page "Axiom" Paragraph 14

Probably the oldest, and most famous, list of axioms are the 4 + 1 Euclid's postulates of plane geometry.

The axioms are referred to as " 4 + 1 " because for nearly two millennia the fifth ( parallel ) postulate (" through a point outside a line there is exactly one parallel ") was suspected of being derivable from the first four.

Ultimately, the fifth postulate was found to be independent of the first four.

Indeed, one can assume that exactly one parallel through a point outside a line exists, or that infinitely many exist.

This choice gives us two alternative forms of geometry in which the interior angles of a triangle add up to exactly 180 degrees or less, respectively, and are known as Euclidean and hyperbolic geometries.

If one also removes the second postulate (" a line can be extended indefinitely ") then elliptic geometry arises, where there is no parallel through a point outside a line, and in which the interior angles of a triangle add up to more than 180 degrees.