S. Brodie has shown that the parallel postulate is equivalent to the Pythagorean Parallel axiom in Hilbert's axioms which is equivalent The parallel postulate is equivalent to the equidistance postulate, Playfair's axiom, Proclus' Postulate describes equally valid (though less intuitive) types of geometries known No line which passes," or "exist at least two lines which pass," the "exists one and only one straight line which passes" is replaced by "exists
(Gauss had also discoveredīut suppressed the existence of non-Euclidean geometries.)Īs stated above, the parallel postulate describes the type of geometry now known as Euclidean geometry. In 1823, Janos Bolyai and Lobachevsky independently realized that entirely self-consistent " non-Euclidean geometries" couldīe created in which the parallel postulate did not hold. His "axiom" states that any triangle can be made bigger or smaller without distorting its proportions or angles (Greenberg 1994, pp. 152-153). John Wallis proposed a new axiom that implied the parallel postulate and was also intuitively appealing. 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. However, none were correct, including the 28 "proofs" G. S. Klügel analyzed in his dissertation of 1763 (Hofstadter 1989). Over the years, many purported proofs of the parallel postulate were published. (That part of geometry which could be derived using only postulatesġ-4 came to be known as absolute geometry.) To the fifth of Euclid's postulates, whichĮuclid himself avoided using until proposition 29 in the Elements.įor centuries, many mathematicians believed that this statement was not a true postulate,īut rather a theorem which could be derived from the first four of Euclid's The first line, no matter how far they are extended.
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
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