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If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side. In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry.
It is equivalent to Euclid's parallel postulate in the context of Euclidean geometry [2] and was named after the Scottish mathematician John Playfair. The "at most" clause is all that is needed since it can be proved from the first four axioms that at least one parallel line exists given a line L and a point P not on L, as follows:
Taken as a physical description of space, postulate 2 (extending a line) asserts that space does not have holes or boundaries; postulate 4 (equality of right angles) says that space is isotropic and figures may be moved to any location while maintaining congruence; and postulate 5 (the parallel postulate) that space is flat (has no intrinsic ...
Callahan was an academic who studied Euclidean geometry. The first volume of his book Euclid or Einstein? A Proof of the Parallel Theory and a Critique of Metageometry claimed to have proved Euclid's fifth "parallel" postulate, by re-ordering the logical structure of Euclid's Elements. Callahan proved that for any point not on a given line ...
In this work Klügel examined 28 attempts to prove Postulate 5 (including Saccheri's), found them all deficient, and offered the opinion that Postulate 5 is unprovable and is supported solely by the judgment of our senses. The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry.
Line segments and Euclidean vectors are parallel if they have the same direction or opposite direction (not necessarily the same length). [1] Parallel lines are the subject of Euclid's parallel postulate. [2] Parallelism is primarily a property of affine geometries and Euclidean geometry is a special instance of this type of geometry.
Removing five axioms mentioning "plane" in an essential way, namely I.4–8, and modifying III.4 and IV.1 to omit mention of planes, yields an axiomatization of Euclidean plane geometry. Hilbert's axioms, unlike Tarski's axioms , do not constitute a first-order theory because the axioms V.1–2 cannot be expressed in first-order logic .
[5] Sir John Playfair by Sir Francis Chantrey. In 1795 Playfair published an alternative, more stringent formulation of Euclid's parallel postulate, which is now called Playfair's axiom. Although the axiom bears Playfair's name, he did not create it, but credited others, in particular William Ludlam with its prior use. [6]