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). The main motivation for all of this effort was that Euclid's parallel postulate did not seem as "intuitive" as the other axioms, but it was needed to prove important results. John Wallis proposed a new axiom that implied the parallel postulate and was also intuitively appealing. His "axiom" states that any triangle can be made bigger or smaller without distorting its proportions or angles (Greenberg 1994, pp. 152-153). However, Wallis's axiom never caught on.

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 Euclidean geometry. If, however, the phrase "exists one and only one straight line which passes" is replaced by "exists 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, the triangle postulate, and the Pythagorean theorem. There is also a single parallel axiom in Hilbert's axioms which is equivalent to Euclid's parallel postulate.

S. Brodie has shown that the parallel postulate is equivalent to the Pythagorean theorem.

REFERENCES

Brodie, S. E. "The Pythagorean Theorem Is Equivalent to the Parallel Postulate." http://www.cut-the-knot.org/triangle/pythpar/PTimpliesPP.shtml.Dixon, R. Mathographics. New York: Dover, p. 27, 1991.

Greenberg, M. J. Euclidean and Non-Euclidean Geometries: Development and History, 3rd ed. San Francisco, CA: W. H. Freeman, 1994.

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.