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A geometry where the parallel postulate does not hold is known as a non-Euclidean geometry. Geometry that is independent of Euclid's fifth postulate (i.e., only assumes the modern equivalent of the first four postulates) is known as absolute geometry (or sometimes "neutral geometry").
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements.Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions from these.
Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line l and a point A, which is not on l, there is exactly one line through A that does not intersect l.
Although Euclid was known to Cicero, for instance, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century. [2] The Arabs received the Elements from the Byzantines around 760; this version was translated into Arabic under Harun al-Rashid ( c. 800). [ 2 ]
1.2 Proof that Euclid’s fifth postulate cannot be proven. 3 comments. 1.3 2.8651496649... 3 comments. Toggle the table of contents.
Playfair's axiom, an alternative to Euclid's fifth postulate on parallel lines, first stated by Proclus in the 5th century AD but named after John Playfair after he included it in his 1795 book Elements of Geometry and credited it to William Ludlam.
Euclid's 5th postulate also was not written as a biconditional statement. So, its claim is about if the angles are less than 2 right angles on one side, then they meet on that side does NOT imply that if they do NOT equal 2 right angles then they do NOT meet on that side.
The classical equivalence between Playfair's axiom and Euclid's fifth postulate collapses in the absence of triangle congruence. [18] This is shown by constructing a geometry that redefines angles in a way that respects Hilbert's axioms of incidence, order, and congruence, except for the Side-Angle-Side (SAS) congruence.