Although these non-Euclidean geometries are interesting in their own right, they do not describe our physical universe. Non-Euclidean geometries in which the fifth postulate does not hold were developed as a result of efforts to prove Euclid's fifth postulate. However, it is possible to prove the fifth postulate using other axioms and postulates. The parallel postulate is not necessarily self-evident in fact, it is quite easy to come up with counterexamples in which the conclusion does not hold. The postulate states that if a line segment intersects two straight lines in such a way that the interior angles on one side of the line segment are less than two right angles, then the lines, if extended far enough, will meet on that side on which the angles are less than two right angles. In geometry, Euclid's fifth postulate, also known as the parallel postulate, is a statement that is equivalent to Playfair's axiom. Although Wallis' proof was ultimately unsuccessful, it did spark a great deal of interest in the topic and led to the development of non-Euclidean geometries in which the fifth postulate does not hold. Wallis' proof made use of a now-disproved theorem known as Playfair's axiom, which stated that through any given point not on a given line there is exactly one line parallel to the given line. One of the most famous attempts at proving the fifth postulate was made by mathematician John Wallis in 1655. The postulate states that if a line segment intersects two straight lines in such a way that the interior angles on one side of the line segment are less than two right angles, then the lines, if extended far enough, will meet on that side on which the angles are less than two right angles.Įuclid's fifth postulate is not necessarily self-evident in fact, it is quite easy to come up with counterexamples in which the conclusion does not hold. Euclid's Fifth Postulate: The Parallel Postulate
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