Parallel Postulate

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.)

Over the years, many purported proofs of the parallel postulate were published. However, none were correct, including the 28 ``proofs'' G. S. Klügel analyzed in his dissertation of 1763 (Hofstadter 1989). In 1823, Janos Bolyai and Lobachevsky independently realized that entirely self-consistent ``Non-Euclidean Geometries'' could be created in which the parallel postulate did not hold. (Gauß had also discovered but suppressed the existence of non-Euclidean geometries.)

As stated above, the parallel postulate describes the type of geometry now known as Parabolic Geometry. If, however, the phrase ``exists one and only one straight line which passes'' is replace by ``exist no line which passes,'' or ``exist at least two lines which pass,'' the postulate describes equally valid (though less intuitive) types of geometries known as Elliptic and Hyperbolic Geometries, respectively.

The parallel postulate is equivalent to the Equidistance Postulate, Playfair's Axiom, Proclus' Axiom, Triangle Postulate. There is also a single parallel axiom in Hilbert's Axioms which is equivalent to Euclid's parallel postulate.

See also Absolute Geometry, Euclid's Axioms, Euclidean Geometry, Hilbert's Axioms, Non-Euclidean Geometry, Playfair's Axiom, Triangle Postulate

References

Dixon, R. Mathographics. New York: Dover, p. 27, 1991.

Hilbert, D. The Foundations of Geometry, 2nd ed. Chicago, IL: Open Court, 1980.

Hofstadter, D. R. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, pp. 88-92, 1989.

Iyanaga, S. and Kawada, Y. (Eds.). ``Hilbert's System of Axioms.'' §163B in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 544-545, 1980.