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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 states that, in two-dimensional geometry:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, 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 does not 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 as absolute geometry (or, in other places known as neutral geometry).

Converse of Euclid's parallel postulate
If the sum of the two interior angles equals 180°, the lines are parallel and will never intersect.

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 27): If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another. As De Morgan[1] pointed out, this is logically equivalent to (Book I, Proposition 16). These results do not depend upon the fifth postulate, but they do require the second postulate[2] which is violated in elliptic geometry.
Logically equivalent properties

Probably the best known equivalent of Euclid's parallel postulate is Playfair's axiom, named after the Scottish mathematician John Playfair, which states:

At most one line can be drawn through any point not on a given line parallel to the given line in a plane.[3]

Many other statements equivalent to the parallel postulate have been suggested, some of them appearing at first to be unrelated to parallelism, and some seeming so self-evident that they were unconsciously assumed by people who claimed to have proven the parallel postulate from Euclid's other postulates. This is a summary

There is at most one line that can be drawn parallel to another given one by an external point. (Playfair's axiom)
The sum of the angles in every triangle is 180° (triangle postulate).
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 be circumscribed.
If three angles of a quadrilateral are right angles, 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 constant distance from each other.
Two lines that are parallel to the same line are also parallel to each other.
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (Pythagoras' Theorem).[4][5]
There is no upper limit to the area of a triangle. (Wallis axiom)[6]
The summit angles of the Saccheri quadrilateral are 90°.
If a line intersects one of two parallel lines, both of which are coplanar with the original line, then it also intersects the other. (Proclus' axiom)[7]

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. For example, if the word "parallel" in Playfair's axiom is taken to mean 'constant separation', then it is no longer equivalent to Euclid's fifth postulate, and is provable from the first four (the axiom says 'There is at most one line...', which is consistent with there being no such lines). However, if the definition is taken so that parallel lines are lines that do not intersect, Playfair's axiom is equivalent to Euclid's fifth postulate and is thus independent of the first four postulates.
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, unlike the first four postulates, the parallel postulate isn't self-evident. 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.[8] Many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods of time until the mistake was found. Invariably the mistake was assuming some 'obvious' property which turned out to be equivalent to the fifth postulate (Playfair's axiom). Although known from the time of Proclus, this became known as Playfair's Axiom after John Playfair wrote a famous commentary on Euclid in 1795 in which he proposed replacing Euclid's fifth postulate by his own axiom.

Proclus (410-485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four, in particular he notes that Ptolemy had produced a false 'proof'. Proclus then goes on to give a false proof of his own. However he did give a postulate which is equivalent to the fifth postulate.

Ibn al-Haytham (Alhazen) (965-1039), an Arab mathematician, made the first attempt at proving the parallel postulate using a proof by contradiction,[9][not in citation given] where he introduced the concept of motion and transformation into geometry.[10][not in citation given] He formulated the Lambert quadrilateral, which Boris Abramovich Rozenfeld names the "Ibn al-Haytham–Lambert quadrilateral",[11] and his attempted proof also shows similarities to Playfair's axiom.[12]

Omar Khayyám (1050–1123), a Persian, made the first attempt at formulating a non-Euclidean postulate as an alternative to the parallel postulate,[13] and he was the first to consider the cases of elliptical geometry and hyperbolic geometry, though he excluded the latter.[14] The Khayyam-Saccheri quadrilateral was also first considered by Omar Khayyam in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid.[11] Unlike many commentators on Euclid before and after him (including Giovanni Girolamo Saccheri), Khayyam was not trying to prove the parallel postulate as such but to derive it from an equivalent postulate: "Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge."[15] He 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.[citation needed]

Nasir al-Din al-Tusi (1201–1274), in his Al-risala al-shafiya'an al-shakk fi'l-khutut al-mutawaziya (Discussion Which Removes Doubt about Parallel Lines) (1250), wrote detailed critiques of the parallel postulate and on Khayyám's attempted proof a century earlier. Nasir al-Din attempted to derive a proof by contradiction of the parallel postulate.[16] He was also one of the first to consider the cases of elliptical geometry and hyperbolic geometry, though he ruled out both of them.[14]
Euclidean, elliptical and hyperbolic geometry. The Parallel Postulate is satisfied only for models of Euclidean geometry.

Nasir al-Din's son, Sadr al-Din (sometimes known as "Pseudo-Tusi"), wrote a book on the subject in 1298, based on his father's later thoughts, which presented one of the earliest arguments for a non-Euclidean hypothesis equivalent to the parallel postulate. "He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements."[16][17] His work was published in Rome in 1594 and was studied by European geometers. This work marked the starting point for Saccheri's work on the subject [16] wherein he criticised this work as well as the work of Wallis.[18]

Giordano Vitale (1633-1711), in his book Euclide restituo (1680, 1686), used the Khayyam-Saccheri quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant. 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 Khayyám and Saccheri 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 also studied the problem, but he did not publish any of his results. Upon hearing of Bolyai's results in a letter from Bolyai's father, Farkas Bolyai, he stated:

"If I commenced by saying that I am unable to praise this work, you would certainly be surprised for a moment. But I cannot say otherwise. To praise it would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the last thirty or thirty-five years."[19]

The resulting geometries were later developed by Lobachevsky, Riemann and Poincaré into hyperbolic geometry (the acute case) and spherical geometry (the obtuse case). The independence of the parallel postulate from Euclid's other axioms was finally demonstrated by Eugenio Beltrami in 1868.
Criticism

Attempts to logically prove this postulate, rather than the eighth axiom, were criticized by Schopenhauer, as described in Schopenhauer's criticism of the proofs of the Parallel Postulate. However, the argument used by Schopenhauer was that the postulate is evident by perception, not that it was not a logical consequence of the other axioms.
See also

For more information, see the history of non-Euclidean geometry.

Notes and references

^ Heath, T.L., The thirteen books of Euclid's Elements, Vol.1, Dover, 1956, pg.309.
^ Coxeter, H.S.M., Non-Euclidean Geometry, 6th Ed., MAA 1998, pg.3
^ Euclid's Parallel Postulate and Playfair's Axiom
^ Eric W. Weisstein (2003), CRC concise encyclopedia of mathematics (2nd ed.), p. 2147, ISBN 1-58488-347-2, "The parallel postulate is equivalent to the Equidistance postulate, Playfair axiom, Proclus axiom, the Triangle postulate and the Pythagorean theorem."
^ Alexander R. Pruss (2006), The principle of sufficient reason: a reassessment, Cambridge University Press, p. 11, ISBN 0-521-85959-X, "We could include...the parallel postulate and derive the Pythagorean theorem. Or we could instead make the Pythagorean theorem among the other axioms and derive the parallel postulate."
^ Bogomolny, Alexander. "Euclid's Fifth Postulate". Cut The Knot. Retrieved 30 September 2011.
^ Weisstein, Eric W.. "Proclus' Axiom – MathWorld". Retrieved 2009-09-05.
^ Florence P. Lewis (Jan 1920), "History of the Parallel Postulate", The American Mathematical Monthly (The American Mathematical Monthly, Vol. 27, No. 1) 27 (1): 16–23, doi:10.2307/2973238, JSTOR 2973238.
^ Eder (2000)
^ (Katz 1998, p. 269):

In effect, this method characterized parallel lines as lines always equidistant from one another and also introduced the concept of motion into geometry.

^ a b (Rozenfeld 1988, p. 65)
^ (Smith 1992)
^ Victor J. Katz (1998), History of Mathematics: An Introduction, p. 270, Addison-Wesley, ISBN 0-321-01618-1:

"In some sense, his treatment was better than ibn al-Haytham's because he explicitly formulated a new postulate to replace Euclid's rather than have the latter hidden in a new definition."

^ a b Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447-494 [469], Routledge, London and New York:

"Khayyam's postulate had excluded the case of the hyperbolic geometry whereas al-Tusi's postulate ruled out both the hyperbolic and elliptic geometries."

^ Boris A Rosenfeld and Adolf P Youschkevitch (1996), Geometry, p.467 in Roshdi Rashed, Régis Morelon (1996), Encyclopedia of the history of Arabic science, Routledge, ISBN 0-415-12411-5.
^ a b c Victor J. Katz (1998), History of Mathematics: An Introduction, p. 270-271, Addison-Wesley, ISBN 0-321-01618-1:

"But in a manuscript probably written by his son Sadr al-Din in 1298, based on Nasir al-Din's later thoughts on the subject, there is a new argument based on another hypothesis, also equivalent to Euclid's, [...] The importance of this latter work is that it was published in Rome in 1594 and was studied by European geometers. In particular, it became the starting point for the work of Saccheri and ultimately for the discovery of non-Euclidean geometry."

^ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447-494 [469], Routledge, London and New York:

"In Pseudo-Tusi's Exposition of Euclid, [...] another statement is used instead of a postulate. It was independent of the Euclidean postulate V and easy to prove. [...] He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements."

^ MacTutor's Giovanni Girolamo Saccheri
^ On Gauss' Mountains. http://www.mathpages.com/rr/s8-06/8-06.htm

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Parallel postulate