Euclid's fifth axiom | Euclid's fifth postulate | parallel axiom

Given any straight line and a point not on it, there "exists one and only one straight line which passes" through that point and never intersects the first line, no matter how far they are extended. This statement is equivalent to the fifth of Euclid's postulates, which Euclid himself avoided using until proposition 29 in the Elements. For centuries, many mathematicians believed that this statement was not a true postulate, but rather a theorem which could be derived from the first four of Euclid's postulates. (That part of geometry which could be derived using only postulates 1-4 came to be known as absolute geometry.)

absolute geometry | Euclidean geometry | Euclid's postulates | Hilbert's axioms | non-Euclidean geometry | Playfair's axiom | Proclus' axiom | Pythagorean theorem | triangle postulate