Editing Non-Euclidean geometry (section)

==History==<!-- This section is linked from [[Parallel postulate]] -->
{{see also|Euclidean geometry#History|History of geometry|Hyperbolic geometry#History}}

===Background===

[[Euclidean geometry]], named after the [[Greek mathematics|Greek mathematician]] [[Euclid]], includes some of the oldest known mathematics, and geometries that deviated from this were not widely accepted as legitimate until the 19th century.

The debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclid wrote ''[[Euclid's Elements|Elements]]''. In the ''Elements'', Euclid begins with a limited number of assumptions (23 definitions, five common notions, and five postulates) and seeks to prove all the other results ([[proposition]]s) in the work. The most notorious of the postulates is often referred to as "Euclid's Fifth Postulate", or simply the ''[[parallel postulate]]'', which in Euclid's original formulation is:

<blockquote>
If a straight line falls on two straight lines in such a manner that the interior angles on the same side are together less than two right angles, then the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
</blockquote>

Other mathematicians have devised simpler forms of this property. Regardless of the form of the postulate, however, it consistently appears more complicated than [[Euclidean geometry#Axioms|Euclid's other postulates]]:

# To draw a straight line from any point to any point.
# To produce [extend] a finite straight line continuously in a straight line.
# To describe a circle with any centre and distance [radius].
# That all right angles are equal to one another.

For at least a thousand years, [[geometer]]s were troubled by the disparate complexity of the fifth postulate, and believed it could be proved as a theorem from the other four. Many attempted to find a [[proof by contradiction]], including [[Ibn al-Haytham]] (Alhazen, 11th century),<ref>{{Citation |last=Eder |first=Michelle |year=2000 |title=Views of Euclid's Parallel Postulate in Ancient Greece and in Medieval Islam |url=http://www.math.rutgers.edu/~cherlin/History/Papers2000/eder.html |publisher=[[Rutgers University]] |access-date=2008-01-23 }}</ref> [[Omar Khayyám]] (12th century), [[Nasīr al-Dīn al-Tūsī]] (13th century), and [[Giovanni Girolamo Saccheri]] (18th century).

The theorems of Ibn al-Haytham, Khayyam and al-Tusi on [[quadrilateral]]s, including the [[Lambert quadrilateral]] and [[Saccheri quadrilateral]], were "the first few theorems of the [[Hyperbolic geometry|hyperbolic]] and the [[Elliptical geometry|elliptic geometries]]". These theorems along with their alternative postulates, such as [[Playfair's axiom]], played an important role in the later development of non-Euclidean geometry.  These early attempts at challenging the fifth postulate had a considerable influence on its development among later European geometers, including [[Witelo]], [[Levi ben Gerson]], [[Abner of Burgos|Alfonso]], [[John Wallis]] and Saccheri.<ref>Boris A. Rosenfeld & Adolf P. Youschkevitch, "Geometry", p. 470, in Roshdi Rashed & Régis Morelon (1996), ''[[Encyclopedia of the History of Arabic Science]]'', vol. 2, pp. 447–494, [[Routledge]], London and New York: {{blockquote|"Three scientists, Ibn al-Haytham, Khayyam, and al-Tusi, had made the most considerable contribution to this branch of geometry, whose importance was completely recognized only in the nineteenth century. In essence, their propositions concerning the properties of quadrangle—which they considered assuming that some of the angles of these figures were acute of obtuse—embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between this postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investigations of their European counterparts. The first European attempt to prove the postulate on parallel lines&nbsp;– made by [[Witelo]], the Polish scientists of the thirteenth century, while revising [[Ibn al-Haytham]]'s ''[[Book of Optics]]'' (''Kitab al-Manazir'')&nbsp;– was undoubtedly prompted by Arabic sources. The proofs put forward in the fourteenth century by the Jewish scholar [[Levi ben Gerson]], who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that ''Pseudo-Tusi's Exposition of Euclid'' had stimulated both J. Wallis's and G. [[Saccheri]]'s studies of the theory of parallel lines."}}</ref>  All of these early attempts made at trying to formulate non-Euclidean geometry, however, provided flawed proofs of the parallel postulate, depending on assumptions that are now recognized as essentially equivalent to the parallel postulate.  These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries.

Khayyam, for example, tried to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" ([[Aristotle]]): "Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge."<ref>Boris A. Rosenfeld & Adolf P. Youschkevitch (1996), "Geometry", p. 467, in Roshdi Rashed & Régis Morelon (1996), ''[[Encyclopedia of the History of Arabic Science]]'', vol. 2, pp. 447–494, [[Routledge]], {{isbn|0-415-12411-5}}</ref> Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he correctly refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid, which he didn't realize was equivalent to his own postulate. Another example is al-Tusi's son, Sadr al-Din (sometimes known as "Pseudo-Tusi"), who wrote a book on the subject in 1298, based on al-Tusi's later thoughts, which presented another 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''."<ref name=Katz/><ref>Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., ''[[Encyclopedia of the History of Arabic Science]]'', vol. 2, pp. 447–494 [469], [[Routledge]], London and New York: {{blockquote|"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''."}}</ref> His work was published in [[Rome]] in 1594 and was studied by European geometers, including Saccheri<ref name=Katz>Victor J. Katz (1998), ''History of Mathematics: An Introduction'', pp. 270–271, [[Addison–Wesley]], {{isbn|0-321-01618-1}}: <blockquote>"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."</blockquote></ref> who criticised this work as well as that of Wallis.<ref>{{MacTutor |title= Giovanni Girolamo Saccheri |id=Saccheri }}</ref>

[[Giordano Vitale]], in his book ''Euclide restituo'' (1680, 1686), used the 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.

In a work titled ''Euclides ab Omni Naevo Vindicatus'' (''Euclid Freed from All Flaws''), published in 1733, Saccheri quickly discarded elliptic geometry as a possibility (some others of Euclid's axioms must be modified for elliptic geometry to work) and set to work proving a great number of results in hyperbolic geometry.

He finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry. His claim seems to have been based on Euclidean presuppositions, because no ''logical'' contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, but did not realize it.

In 1766 [[Johann Heinrich Lambert|Johann Lambert]] wrote, but did not publish, ''Theorie der Parallellinien'' in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure now known as a ''Lambert quadrilateral'', a quadrilateral with three right angles (can be considered half of a Saccheri quadrilateral). He quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle. Unlike Saccheri, he never felt that he had reached a contradiction with this assumption. He had proved the non-Euclidean result that the sum of the angles in a triangle increases as the area of the triangle decreases, and this led him to speculate on the possibility of a model of the acute case on a sphere of imaginary radius. He did not carry this idea any further.<ref>{{MacTutor |title=Johann Heinrich Lambert|id=Lambert }}</ref>

At this time it was widely believed that the universe worked according to the principles of Euclidean geometry.<ref>A notable exception is David Hume, who as early as 1739 seriously entertained the possibility that our universe was non-Euclidean; see David Hume (1739/1978) ''A Treatise of Human Nature'', L.A. Selby-Bigge, ed. (Oxford: Oxford University Press), pp. 51–52.</ref>

===Development of non-Euclidean geometry===
The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry.
Circa 1813, [[Carl Friedrich Gauss]] and independently around 1818, the German professor of law [[Ferdinand Karl Schweikart]]<ref>In a letter of December 1818, Ferdinand Karl Schweikart (1780–1859) sketched a few insights into non-Euclidean geometry.  The letter was forwarded to Gauss in 1819 by Gauss's former student Gerling.  In his reply to Gerling, Gauss praised Schweikart and mentioned his own, earlier research into non-Euclidean geometry.  See:
* Carl Friedrich Gauss, ''Werke'' (Leipzig, Germany:  B. G. Teubner, 1900), vol. 8, [http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN236010751&DMDID=DMDLOG_0058&LOGID=LOG_0058&PHYSID=PHYS_0187 pp. 180–182.]
* English translations of Schweikart's letter and Gauss's reply to Gerling appear in:[http://www.math.uwaterloo.ca/~snburris/htdocs/noneucl.pdf Course notes:  "Gauss and non-Euclidean geometry", University of Waterloo, Ontario, Canada]; see especially pages 10 and 11.
* Letters by Schweikart and the writings of his nephew [[Franz Adolph Taurinus]], who also was interested in non-Euclidean geometry and who in 1825 published a brief book on the parallel axiom, appear in:  Paul Stäckel and Friedrich Engel, ''Die theorie der Parallellinien von Euklid bis auf Gauss, eine Urkundensammlung der nichteuklidischen Geometrie'' (The theory of parallel lines from Euclid to Gauss, an archive of non-Euclidean geometry), (Leipzig, Germany:  B. G. Teubner, 1895), [http://quod.lib.umich.edu/u/umhistmath/abq9565.0001.001/254?rgn=full+text;view=pdf pages 243 ff.]</ref> had the germinal ideas of non-Euclidean geometry worked out, but neither published any results. Schweikart's nephew [[Franz Taurinus]] did publish important results of hyperbolic trigonometry in two papers in 1825 and 1826, yet while admitting the internal consistency of hyperbolic geometry, he still believed in the special role of Euclidean geometry.<ref>{{citation|author=Bonola, R.|title=Non-Euclidean geometry: A critical and historical study of its development|year=1912|location=Chicago|publisher=Open Court|url=https://archive.org/details/noneuclideangeom00bono}}</ref>

Then, in 1829–1830 the [[Russia]]n mathematician [[Nikolai Ivanovich Lobachevsky]] and in 1832 the [[Hungary|Hungarian]] mathematician [[János Bolyai]] separately and independently published treatises on hyperbolic geometry.  Consequently, hyperbolic geometry is called Lobachevskian or Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors of non-Euclidean geometry. [[Carl Friedrich Gauss|Gauss]] mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before,<ref>In the letter to Wolfgang (Farkas) Bolyai of March 6, 1832 Gauss claims to have worked on the problem for thirty or thirty-five years {{Harvard citation|Faber|1983|loc=p. 162}}. In his 1824 letter to Taurinus {{Harvard citation|Faber|1983|loc=p. 158}} he claimed that he had been working on the problem for over 30 years and provided enough detail to show that he actually had worked out the details. According to {{harvtxt|Faber|1983|loc=p. 156}} it wasn't until around 1813 that Gauss had come to accept the existence of a new geometry.</ref> though he did not publish. While Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai worked out a geometry where both the Euclidean and the hyperbolic geometry are possible depending on a parameter&nbsp;''k''. Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences.

[[Bernhard Riemann]], in a famous lecture in 1854, founded the field of [[Riemannian geometry]], discussing in particular the ideas now called [[manifold]]s, [[Riemannian metric]], and [[curvature]].
He constructed an infinite family of non-Euclidean geometries by giving a formula for a family of Riemannian metrics on the unit ball in [[Euclidean space]]. The simplest of these is called [[elliptic geometry]] and it is considered a non-Euclidean geometry due to its lack of parallel lines.<ref>However, other axioms besides the parallel postulate must be changed to make this a feasible geometry.</ref>

By formulating the geometry in terms of a curvature [[tensor]], Riemann allowed non-Euclidean geometry to apply to higher dimensions. Beltrami (1868) was the first to apply Riemann's geometry to spaces of negative curvature.

===Terminology===
It was Gauss who coined the term "non-Euclidean geometry".<ref>Felix Klein, ''Elementary Mathematics from an Advanced Standpoint: Geometry'', Dover, 1948 (Reprint of English translation of 3rd Edition, 1940. First edition in German, 1908.) p. 176.</ref> He was referring to his own work, which today we call ''hyperbolic geometry'' or ''Lobachevskian geometry''. Several modern authors still use the generic term ''non-Euclidean geometry'' to mean ''hyperbolic geometry''.<ref>For example:
{{citation |last=Kulczycki |first=Stefan |year=1961 |title=Non-Euclidean Geometry |publisher=Pergamon |url=https://archive.org/details/noneuclideangeom0000stef/page/53 |url-access=limited |page=53 }}<br/ >
{{citation |last=Iwasawa |first=Kenkichi |year=1993 |title=Algebraic Functions |publisher=American Mathematical Society |page=140 |isbn=978-0-8218-4595-0 |url=https://archive.org/details/algebraicfunctio0000iwas/page/140/ |url-access=limited }}
</ref>

[[Arthur Cayley]] noted that distance between points inside a conic could be defined in terms of [[logarithm]] and the projective [[cross-ratio]] function. The method has become called the [[Cayley–Klein metric]] because [[Felix Klein]] exploited it to describe the non-Euclidean geometries in articles<ref>F. Klein, Über die sogenannte nichteuklidische Geometrie, ''Mathematische Annalen'', '''4'''(1871).</ref> in 1871 and 1873 and later in book form. The Cayley–Klein metrics provided working models of hyperbolic and elliptic metric geometries, as well as Euclidean geometry.

Klein is responsible for the terms "hyperbolic" and "elliptic" (in his system he called Euclidean geometry ''parabolic'', a term that generally fell out of use<ref>The Euclidean plane is still referred to as ''parabolic'' in the context of [[conformal geometry]]: see [[Uniformization theorem]].</ref>). His influence has led to the current usage of the term "non-Euclidean geometry" to mean either "hyperbolic" or "elliptic" geometry.

There are some mathematicians who would extend the list of geometries that should be called "non-Euclidean" in various ways.<ref>for instance, {{harvnb|Manning|1963}} and Yaglom 1968</ref>

There are many kinds of geometry that are quite different from Euclidean geometry but are also not necessarily included in the conventional meaning of "non-Euclidean geometry", such as more general instances of [[Riemannian geometry]].