- Parallel postulate
In

geometry , the**parallel postulate**, also calledsince it is the fifth postulate in Euclid's "Elements", is a distinctiveEuclid 's fifth postulateaxiom in what is now calledEuclidean geometry . It states that:"If a

line segment intersects two straight lines forming two interior angles on the same side that sum to less than tworight angle s, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.""Euclidean geometry" is the study of geometry that satisfies all of Euclid's axioms, "including" the parallel postulate. A geometry where the parallel postulate cannot hold is known as a

non-euclidean geometry . Geometry that is "independent" of Euclid's fifth postulate (i.e., only assumes the first four postulates) is known asabsolute geometry (or, in other places known asneutral geometry ).**Converse of Euclid's parallel postulate**Euclid did not postulate the converse of his fifth postulate, which is one way to distinguish Euclidean geometry from

elliptic geometry . The Elements contains the proof of an equivalent statement (Book I, Proposition 17): "Any twoangle s of atriangle are together less than tworight angle s." The proof depends on an earlier proposition: "In a triangle ABC, theexterior angle at C is greater than either of theinterior angle s A or B". This in turn depends on Euclid's unstated assumption that two straight lines meet in only one point, a statement not true of elliptic geometry.In other words, the converse of the fifth postulate follows from Euclid's axioms minus the fifth postulate, plus an axiom stating that two distinct non-parallel straight lines meet in only one point.

**Logically equivalent properties**Euclid's parallel postulate is often stated in a similar form known as

**Playfair's axiom**, named after the Scottishmathematician John Playfair . It states:"Exactly one line can be drawn through any point not on a given line parallel to the given line in a plane." [

*[*]*http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI30.html Euclid's Parallel Postulate and Playfair's Axiom*]Many other equivalent statements have been suggested, some of them appearing at first to be unrelated to parallelism, and someseeming so self-evident that they were unconsciously assumed by people who claimed to have proven the parallel postulate from Euclid's other postulates.

# The sum of the

angle s in everytriangle is 180°.

# There exists a triangle whose angles add up to 180°.

# The sum of the angles is the same for every triangle.

# There exists a pair of similar, but not congruent, triangles.

# Every triangle can becircumscribe d.

# If three angles of aquadrilateral areright angle s, then the fourth angle is also a right angle.

# There exists a quadrilateral of which all angles are right angles.

# There exists a pair of straight lines that are at constantdistance from each other.

# Two lines that are parallel to the same line are also parallel to each other.

# Given two parallel lines, any line that intersects one of them also intersects the other.

# In aright-angled triangle , the square of the hypotenuse equals the sum of the squares of the other two sides (Pythagoras' Theorem ).

# There is no upper limit to the area of a triangle. [*http://www.cut-the-knot.com/triangle/pythpar/Fifth.shtml*]However, the alternatives which employ the word "parallel" cease appearing so simple when one is obliged to explain which of the three common definitions of "parallel" is meant - constant separation, never meeting or same angles where crossed by a third line - since the equivalence of these three is itself one of the unconsciously obvious assumptions equivalent to Euclid's fifth postulate.

**History**For two thousand years, many attempts were made to prove the parallel postulate using Euclid's first four postulates. The main reason that such a proof was so highly sought after was that the fifth postulate isn't self-evident unlike the other postulates. If the order the postulates were listed in the Elements is significant, it indicates that Euclid included this postulate only when he realised he could not prove it or proceed without it [

*cite journal|title=History of the Parallel Postulate|journal=The American Mathematical Monthly|volume=27|issue=1|pages=16–23|year=Jan 1920|author=Florence P. Lewis|doi=10.2307/2973238*] .Omar Khayyám (1050-1123) recognized that three possibilities arose from omitting Euclid's Fifth; if two perpendiculars to one line cross another line, judicious choice of the last can make the internal angles where it meets the two perpendiculars equal (it is then parallel to the first line). If those equal internal angles are right angles, we get Euclid's Fifth; otherwise, they must be either acute or obtuse. He persuaded himself that the acute and obtuse cases lead to contradiction, but had made a tacit assumption equivalent to the fifth to get there.Girolamo Saccheri (1667-1733) pursued the same line of reasoning more thoroughly, correctly obtaining absurdity from the obtuse case (proceeding, like Euclid, from the implicit assumption that lines can be extended indefinitely and have infinite length), but failing to debunk the acute case (although he managed to wrongly persuade himself that he had).Where Saccheri and Khayyám had attempted to prove Euclid's fifth by disproving the only possible alternatives, the nineteenth century finally saw mathematicians exploring those alternatives and discovering the

logically consistent geometries which result.In 1829, Nikolai Ivanovich Lobachevsky published an account of acute geometry in an obscure Russian journal (later re-published in 1840 in German).

In 1831, János Bolyai included, in a book by his father, an appendix describing acute geometry, which, doubtlessly, he had developed independently of Lobachevsky.

Carl Friedrich Gauss had actually studied the problem before that, but he didn't have a courage to publish any of his results. [*Isaac Asimov, The left hand of the electron, pp 159.*]The resulting geometries were later developed by Lobachevsky, Riemann and Poincaré into

hyperbolic geometry (the acute case) andspherical geometry (the obtuse case).The independence of the parallel postulate from Euclid's other axioms was finally demonstrated by

Eugenio Beltrami in1868 .**Criticism**Attempts to logically prove this postulate, rather than the eighth axiom, were criticized by

Schopenhauer , as described inSchopenhauer's criticism of the proofs of the Parallel Postulate .**ee also***For more information, see the history of non-Euclidean geometry.

**References****Further reading***

Lewis Carroll , "Euclid and His Modern Rivals", Dover, ISBN 0-486-22968-8

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