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History of non-Euclidean geometry

2016-06-05

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[[File:noneuclid.svg|right|thumb|400px|<center>Lines with a common perpendicular in 3 types of geometry</center>]]

The '''history of {{w|non-Euclidean geometry}}''' is the history of two geometries based on axioms closely related to those specifying [[Euclid|Euclidean]] [[geometry]].

== Quotes ==

* [The] empirical origin of Euclid's geometrical axioms and postulates was lost sight of, indeed was never even realised. As a result, Euclidean geometry was thought to derive its validity from certain self-evident universal truths; it appeared as the only type of consistent geometry of which the mind could conceive. [[Carl Friedrich Gauss|Gauss]] had certain misgivings on the matter, but... the honor of discovering non-Euclidean geometry fell to [[Nikolai Lobachevsky|Lobatchewski]] and [[w:János Bolyai|Bolyai]]. ... From the difference in geometric premises important variations followed. Thus, whereas in Euclidean geometry the sum of the angles of any triangles is always equal to two right angles, in non-Eudlidean geometry the value of this sum varies with the size of the triangles. It is always less than two right angles in Lobatchewski's, and always greater in [[Bernhard Riemann|Riemann]]'s. Again, in Euclidean geometry, similar figures of various sizes can exist; in non-Euclidean geometry, this is impossible. It appeared then, that the universal truth formerly credited to Euclidean geometry would have to be shared by these two other geometrical doctrines. But truth, when divested of its absoluteness, loses much of its significance, so this co-presence of conflicting universal truths brought the realisation that a geometry was true only in relation to our more or less arbitrary choice of a system of geometrical postulates. ...The character of self-evidence which had been formerly credited to the Euclidean axioms was seen to be illusory.

** A. D'Abro, ''The Evolution of Scientific Thought from Newton to Einstein'' (1927) pp. 35-36

* In the field of [[w:Non-Euclidean geometry|non-Euclidean geometry]], [[Bernhard Riemann|Riemann]]... began by calling attention to a distinction that seems obvious once it is pointed out: the distinction between an unbounded straight line and an infinite line. The distinction between unboundedness and infiniteness is readily illustrated. A circle is an unbounded figure in that it never comes to an end, and yet it is of finite length. On the other hand, the usual [[Euclid|Euclidean]] concept of a straight line is also unbounded in that it never reaches an end but is of infinite length. ...he proposed to replace the infiniteness of the Euclidean straight line by the condition that it is merely unbounded. He also proposed to adopt a new [[w:Parallel postulate|parallel axiom]]... In brief, there are no parallel lines. This ... had been tried... in conjunction with the infiniteness of the straight line and had led to contradictions. However... Riemann found that he could construct another consistent non-Euclidean geometry.

** [[w:Morris Kline|Morris Kline]], ''Mathematics and the Physical World'' (1959) Ch. 26: Non-Euclidean Geometries.

=== ''History of Modern Mathematics'' (1896) ===

:[[w:David Eugene Smith|David Eugene Smith]], ''[https://books.google.com/books?id=yUJBAAAAYAAJ source]

* The Non-Euclidean Geometry is a natural result of the futile attempts which had been made from the time of [[w:Proclus|Proklos]] to the opening of the nineteenth century to prove the [[w:Parallel postulate|fifth postulate]], (also called the twelfth axiom, and sometimes the eleventh or thirteenth) of Euclid. The first scientific investigation of this part of the foundation of geometry was made by [[w:Giovanni Girolamo Saccheri|Saccheri]] (1733), a work which was not looked upon as a precursor of [[Nikolai Lobachevsky|Lobachevsky]], however, until [[w:Eugenio Beltrami|Beltrami]] (1889) called attention to the fact. [[w:Johann Heinrich Lambert|Lambert]] was the next to question the validity of Euclid's postulate in his Theorie der Parallellinien (posthumous, 1786), the most important of many treatises on the subject between the publication of Saccheri's work and those of Lobachevsky and [[w:János Bolyai|Bolyai]]. [[w:Adrien-Marie Legendre|Legendre]] also worked in the field, but failed to bring himself to view the matter outside the Euclidean limitations.

** pp. 565-566.

* During the closing years of the eighteenth century [[Immanuel Kant|Kant]]'s doctrine of absolute space, and his assertion of the necessary postulates of geometry, were the object of much scrutiny and attack. At the same time [[Carl Friedrich Gauss|Gauss]] was giving attention to the fifth postulate, though on the side of proving it. It was at one time surmised that Gauss was the real founder of the non-Euclidean geometry, his influence being exerted on [[Nikolai Lobachevsky|Lobachevsky]] through his friend [[w:Johann Christian Martin Bartels|Bartels]], and on [[w:János Bolyai|Johann Bolyai]] through the father [[Farkas Bolyai|Wolfgang]], who was a fellow student of Gauss's. But it is now certain that Gauss can lay no claim to priority of discovery, although the influence of himself and of Kant, in a general way, must have had its effect.

** p. 566.

* Bartels went to Kasan in 1807, and [[Nikolai Lobachevsky|Lobachevsky]] was his pupil. The latter's lecture notes show that Bartels never mentioned the subject of the fifth postulate to him, so that his investigations, begun even before 1823, were made on his own motion and his results were wholly original. Early in 1826 he sent forth the principles of his famous doctrine of parallels, based on the assumption that through a given point more than one line can be drawn which shall never meet a given line coplanar with it. The theory was published in full in 1829-30, and he contributed to the subject... until his death.

** p. 566.

* Johann Bolyai received through his father, Wolfgang, some of the inspiration to original research which the latter had received from Gauss. When only twenty-one he discovered, at about the same time as Lobachevsky, the principles of non-Euclidean geometry, and refers to them in a letter of November, 1823. They were committed to writing in 1825 and published in 1832. Gauss asserts in his correspondence with Schumacher (1831-32) that he had brought out a theory along the same lines as Lobachevsky and Bolyai, but the publication of their works seems to have put an end to his investigations. [[w:de:Ferdinand_Karl_Schweikart|Schweikart]] was also an independent discoverer of the non-Euclidean geometry, as his recently recovered letters show, but he never published anything on the subject, his work on the theory of parallels (1807), like that of his nephew [[w:Franz Taurinus|Taurinus]] (1825), showing no trace of the Lobachevsky-Bolyai idea.

** p. 567.

* The hypothesis was slowly accepted by the mathematical world. Indeed, it was about forty years after its publication that it began to attract any considerable attention. ... Of all these contributions the most noteworthy from the scientific standpoint is that of [[Bernhard Riemann|Riemann]]. In his [[wiktionary:habilitation#Noun|Habilitationsschrift]] (1854) he applied the methods of analytic geometry to the theory, and suggested a surface of negative curvature, which Beltrami calls "pseudo-spherical," thus leaving Euclid's geometry on a surface of zero curvature midway between his own and Lobachevsky's. He thus set forth three kinds of geometry, Bolyai having noted only two. These Klein (1871) has called the elliptic (Riemann's), parabolic (Euclid's), and hyperbolic (Lobachevsky's).

** pp. 567-568.

* There have contributed to the subject many of the leading mathematicians of the last quarter of a century, including... [[w:Arthur Cayley|Cayley]], [[w:Sophus Lie|Lie]], [[w:Felix Klein|Klein]], Newcomb, Pasch, [[Charles Sanders Peirce|C. S. Peirce]], Killing, Fiedler, Mansion, and McClintock. Cayley's contribution of his "metrical geometry" was not at once seen to be identical with that of Lobachevsky and Bolyai. It remained for Klein (1871) to show, this thus simplifying Cayley's treatment and adding one of the most important results of the entire theory. Cayley's metrical formulas are, when the Absolute is real, identical with those of the hyperbolic geometry; when it is imaginary, with the elliptic; the limiting case between the two gives the parabolic (Euclidean) geometry. The question raised by Cayley's memoir as to how far [[w:Projective geometry|projective geometry]] can be defined in terms of space without the introduction of distance had already been discussed by [[w:Karl Georg Christian von Staudt|von Staudt]] (1857) and has since been treated by Klein (1873) and by [[w:Ferdinand von Lindemann|Lindemann]] (1876).

=== ''Elements of Non-Euclidean Geometry'' (1919) ===

:Duncan M'Laren Young Sommerville, [https://books.google.com/books?id=6eASAQAAMAAJ source]

* The common notions of [[Euclid]] are five in number, and deal exclusively with equalities and inequalities of magnitudes. The postulates are also five in number and are exclusively geometrical. The first three refer to the construction of straight lines and circles. The fourth asserts the equality of all right angles, and the fifth is the famous [[w:Parallel postulate|Parallel Postulate]]... It seems impossible to suppose that Euclid ever imagined this to be self-evident, yet the history of the theory of parallels is full of reproaches against the lack of self-evidence of this "axiom." [[w:Henry Savile (Bible translator)|Sir Henry Savile]] referred to it as one of the great blemishes in the beautiful body of geometry; [[Jean le Rond d'Alembert|D'Alembert]] called it "l'écueil et le scandale des élémens de Géométrie." Such considerations induced geometers (and others), even up to the present day, to attempt its demonstration. From the invention of printing onwards a host of parallel-postulate demonstrators existed, rivalled only by the "circle-squarers," the "flat-earthers," and the candidates for the Wolfskehl "Fermat" prize. ...Modern research has vindicated Euclid, and justified his decision in putting this great proposition among the independent assumptions which are necessary for the development of euclidean geometry as a logical system. All this labour has not been fruitless, for it has led in modern times to a rigorous examination of the principles not only of geometry, but of the whole of mathematics, and even logic itself, the basis of mathematics. It has had a marked effect upon philosophy, and has given us a freedom of thought which in former times would have received the award meted out to the most deadly heresies.

** Chapter 1. Historical, pp. 2-4.

* One of the commonest of the equivalents used for [[w:Parallel postulate|Euclid's axiom]] in school text-books is [[w:Playfair's axiom|Playfair's axiom]] (really due to Ludlam).

** Chapter 1. Historical, p. 4.

* A... fallacy is contained in all proofs [of the Parallel Postulate] based upon the idea of ''direction''. ... Another class of demonstrations is based upon considerations of ''infinite areas''. [In] [[w:Joseph Bertrand|Bertrand]]'s Proof... The fallacy... consists in applying the principle of superposition to infinite areas as if they were finite magnitudes.

** Chapter 1. Historical, pp. 6-8.

* Non-euclidean geometry has made it clear that the ideas of parallelism and equidistance are quite distinct. The term parallel (Greek... running alongside) originally connoted equidistance, but the term is used by Euclid rather in the sense "asymptotic" (Greek... non-intersecting), and this term has come to be used in the limiting case of curves which tend to coincidence, or the limiting case between intersection and non-intersection. In non-euclidean geometry parallel straight lines are asymptotic in this sense, and equidistant straight lines in a plane do not exist. This is just one instance of two distinct ideas which are confused in euclidean geometry, but are quite distinct in non-euclidean.

** Chapter 1. Historical, pp. 10-11.

* Among the early postulate demonstrators there stands a unique figure that of a Jesuit [[w:Giovanni Girolamo Saccheri|Gerolamo Saccheri]], a contemporary and friend of [[w:Tommaso Ceva|Ceva]]. This man devised an entirely different mode of attacking the problem, in an attempt to institute a ''reductio ad absurdum''. At that time the favourite starting-point was the conception of parallels as equidistant straight lines, but Saccheri, like some of his predecessors, saw that it would not do to assume this in the definition. ...Saccheri keeps an open mind, and proposes three hypotheses: (1) The Hypothesis of the Right Angle. (2) The Hypothesis of the Obtuse Angle. (3) The Hypothesis of the Acute Angle. The object of his work is to demolish the last two hypotheses and leave the first, the Euclidean hypothesis, supreme; but the task turns out to be more arduous than he expected. He establishes a number of theorems, of which the most important are the following: If one of the three hypotheses is true in any one case, the same hypothesis is true in every case. On the hypothesis of the right angle, the obtuse angle, or the acute angle, the sum of the angles of a triangle is equal to, greater than, or less than two right angles. ... Saccheri demolishes the hypothesis of the obtuse angle in his Theorem 14 by showing that it contradicts Euclid I. 17 (that the sum of any two angles of a triangle is less than two right angles); but he requires nearly twenty more theorems before he can demolish the hypothesis of the acute angle, which he does by showing that two lines which meet in a point at infinity can be perpendicular at that point to the same straight line. In spite of all his efforts, however, he does not seem to be quite satisfied with the validity of his proof, and he offers another proof in which he loses himself, like many another, in the quicksands of the infinitesimal. If Saccheri had had a little more imagination and been less bound down by tradition, and a firmly implanted belief that Euclid's hypothesis was the only true one, he would have anticipated by a century the discovery of the two non-euclidean geometries which follow from his hypotheses of the obtuse and the acute angle.

** Chapter 1. Historical, pp. 11-13.

* [[w:Johann Heinrich Lambert|J. H. Lambert]], fifty years after Saccheri, also fell just short... His starting point is very similar to Saccheri's, and he distinguishes the same three hypotheses; but he went further than Saccheri. He actually showed that on the hypothesis of the obtuse angle the area of a triangle is proportional to the excess of the sum of its angles over two right angles, which is the case for the geometry on the sphere, and he concluded that the hypothesis of the acute angle would be verified on a sphere of [[w:Imaginary number|imaginary]] radius. ... He dismisses the hypothesis of the obtuse angle, since it requires that two straight lines should enclose a space, but his argument against the hypothesis of the acute angle, such as the non-existence of similar figures, he characterises as arguments ''ab amore et invidia ducta'' [guided by love and jealousy]. Thus he arrived at no definite conclusion, and his researches were only published some years after his death.

** Chapter 1. Historical, pp. 13-14.

* About... 1799 the genius of [[Carl Friedrich Gauss|Gauss]] was being attracted to the question, and, although he published nothing on the subject except a few reviews, it is clear from his correspondence and fragments of his notes that he was deeply interested in it. He was a keen critic of the attempts made by his contemporaries to establish the theory of parallels; and while at first he inclined to the orthodox belief, encouraged by [[Immanuel Kant|Kant]], that Euclidean geometry was an example of a necessary truth, he gradually came to see that it was impossible to demonstrate it. He declares that he refrained from publishing anything because he feared the clamour of the Boeotians, or, as we should say, the Wise Men of Gotham; indeed at this time the problem of parallel lines was greatly discredited, and anyone who occupied himself with it was liable to be considered as a crank.

** Chapter 1. Historical, p. 14.

* [[Carl Friedrich Gauss|Gauss]] was probably the first to obtain a clear idea of the possibility of a geometry other than that of Euclid, and we owe the very name Non-Euclidean Geometry to him. It is clear that about the year 1820 he was in possession of many theorems of non-euclidean geometry, and though he meditated publishing his researches when he had sufficient leisure to work them out in detail with his characteristic elegance, he was finally forestalled by receiving in 1832, from his friend [[Farkas Bolyai|W. Bolyai]], a copy of the now famous Appendix by his son, [[w:János Bolyai|John Bolyai]].

** Chapter 1. Historical, p. 14.

* Among the contemporaries and pupils of Gauss... [[w:de:Ferdinand_Karl_Schweikart|F. K. Schweikart]], Professor of Law in {{w|Marburg}}, sent to Gauss in 1818 a page of [[wiktionary:manuscript#Noun|MS.]] explaining a system of geometry which he calls "Astral Geometry," in which the sum of the angles of a triangle is always less than two right angles, and in which there is an absolute unit of length. He did not publish any account of his researches, but he induced his nephew, [[w:Franz Taurinus|F.A. Taurinus]], to take up the question. ...a few years later he attempted a treatment of the theory of parallels and having received some encouragement from Gauss he [Taurinus] published a small book, ''Theorie der Parallellinien'', in 1825. After its publication he came across [J. W.] Camerer's new edition of Euclid in Greek and Latin, which in an [[wiktionary:excursus#Noun|Excursus]] to Euclid I. 29, contains a very valuable history of the theory of parallels, and there he found that his methods had been anticipated by [[w:Giovanni Girolamo Saccheri|Saccheri]] and [[w:Johann Heinrich Lambert|Lambert]]. Next year, accordingly, he published another work, ''Oeometriae prima elementa'' and in the Appendix... works out some of the most important trigonometrical formulae for non-euclidean geometry by using the fundamental formulae of [[w:Spherical geometry|spherical geometry]] with an [[w:Imaginary number|imaginary]] radius. Instead of the notation of hyperbolic functions, which was then scarcely in use, he expresses his results in terms of logarithms and exponentials, and calls his geometry the "Logarithmic Spherical Geometry." Though Taurinus must be regarded as an independent discoverer of non-euclidean trigonometry, he always retained the belief, unlike Gauss and Schweikart, that Euclidean geometry was necessarily the true one. Taurinus himself was aware, however, of the importance of his contribution... and it was a bitter disappointment to him when he found that his work attracted no attention. In disgust he burned the remainder of the edition of his ''Elementa'', which is now one of the rarest of books.

** Chapter 1. Historical, pp. 14-15.

* The third... having arrived at the notion of a geometry in which Euclid's postulate is denied is [[w:de:Friedrich Ludwig Wachter|F. L. Wachter]], a student under Gauss. It is remarkable that he affirms that even if the postulate be denied, the geometry on a sphere becomes identical with the geometry of Euclid when the radius is indefinitely increased, though it is distinctly shown that the limiting surface is not a plane. This was one of the greatest discoveries of Lobachevsky and Bolyai. If Wachter had lived he might have been the discoverer of non-euclidean geometry, for his insight into the question was far beyond that of the ordinary parallel-postulate demonstrator.

** Chapter 1. Historical, p. 15.

* While Gauss, Schweikart, Taurinus and others were working in Germany,... just on the threshold of... discovery, in France and Britain... there was a considerable interest in the subject inspired chiefly by [[w:Adrien-Marie Legendre|A. M. Legendre]]. Legendre's researches were published in the various editions of his ''Éléments'', from 1794 to 1823. and collected in an extensive article in the Memoirs of the Paris Academy in 1833. Assuming all Euclid's definitions, axioms and postulates, except the parallel-postulate and all that follows from it, he proves some important theorems, two of which, Propositions A and B, are frequently referred to in later work as Legendre's First and Second Theorems. Prop. A. ''The sum of the three angles of a rectilinear triangle cannot be greater than two right angles (π)''. ... Prop. B. ''If there exists a single triangle in which the sum of the angles is equal to two right angles, then in every triangle the sum of the angles must likewise be equal to two right angles''. This proposition was already proved by [[w:Giovanni Girolamo Saccheri|Saccheri]], along with the corresponding theorem for the case in which the sum of the angles is less than two right angles... Legendre's proof... proceeds by constructing successively larger and larger triangles in each of which the sum of the angles = ''π''. ... In this proof there is a latent assumption and also a fallacy. ...Legendre's other attempts make use of infinite areas. He makes reference to [[w:Joseph Bertrand|Bertrand]]'s proof, and attempts to prove the necessity of [[w:Playfair's axiom|Playfair's axiom]]...

** Chapter 1. Historical, pp. 16-19.

* [[Nikolai Lobachevsky|Nikolai Ivanovich Lobachevsky]], Professor of Mathematics at [[w:Kazan|Kazan]], was interested in the theory of parallels from at least 1815. Lecture notes of the period 1815-17 are extant, in which Lobachevsky attempts in various ways to establish the Euclidean theory. He proves [[w:Adrien-Marie Legendre|Legendre]]'s two propositions, and employs also the ideas of direction and infinite areas. In 1823 he prepared a treatise on geometry for use in the University, but it obtained so unfavourable a report that it was not printed. The [[wiktionary:manuscript#Noun|MS]]. remained buried in the University Archives until it was discovered and printed in 1909. In this book he states that "a rigorous proof of [[w:Parallel postulate|the postulate of Euclid]] has not hitherto been discovered; those which have been given may be called explanations, and do not deserve to be considered as mathematical proofs in the full sense." Just three years afterwards, he read to the physical and mathematical section of the University of Kazan a paper entitled "Exposition succinte des principes de la géométrie avec une démonstration rigoureuse du théorème des parallèles." In this paper... Lobachevsky explains the principles of his "Imaginary Geometry," which is more general than Euclid's, and in which two parallels can be drawn to a given line through a given point, and in which the sum of the angles of a triangle is always less than two right angles.

** Chapter 1. Historical, p. 21.

* [[w:János Bolyai|Bolyai János]] (John) was the son of [[Farkas Bolyai|Bolyai Farkas]] (Wolfgang), a fellow-student and friend of [[Carl Friedrich Gauss|Gauss]] at [[w:University of Göttingen|Göttingen]]. The father was early interested in the theory of parallels, and without doubt discussed the subject with Gauss while at Göttingen. The professor of mathematics at that time, A. G. Kaestner, had himself attacked the problem and with his help G. S. Klügel, one of his pupils, compiled in 1763 the earliest history of the theory of parallels.

** Chapter 1. Historical, pp. 21-22.

* In 1804, [[Farkas Bolyai|Wolfgang Bolyai]]... sent to Gauss a "Theory of Parallels," the elaboration of his Göttingen studies. In this he gives a demonstration very similar to that of [Henry] Meikle and some of Perronet Thompson's, in which he tries to prove that a series of equal segments placed end to end at equal angles, like the sides of a regular polygon, must make a complete circuit. Though Gauss clearly revealed the fallacy, Bolyai persevered and sent Gauss, in 1808, a further elaboration of his proof. To this Gauss did not reply, and Bolyai, wearied with his ineffectual endeavours to solve the riddle of parallel lines, took refuge in poetry and composed dramas. During the next twenty years, amid various interruptions, he put together his system of mathematics, and at length in 1832-3, published in two volumes an elementary treatise on mathematical discipline which contains all his ideas with regard to the first principles of geometry. Meanwhile, John Bolyai... had been giving serious attention to the theory of parallels, in spite of his father's solemn adjuration to let the loathsome subject alone. At first, like his predecessors, he attempted to find a proof for the parallel-postulate, but gradually, as he focussed his attention more and more upon the results which would follow from a denial of the axiom, there developed in his mind the idea of a general or "Absolute Geometry" which would contain ordinary or euclidean geometry as a special or limiting case. Already, in 1823, he had worked out the main ideas of the non-euclidean geometry, and in a letter of 3rd November he announces to his father his intention of publishing a work on the theory of parallels, "for," he says, "I have made such wonderful discoveries that I am myself lost in astonishment, and it would be an irreparable loss if they remained unknown. When you read them, dear Father, you too will acknowledge it. I cannot say more now except that out of nothing I have created a new and another world. All that I have sent you hitherto is as a house of cards compared to a tower." Wolfgang advised his son, if his researches had really reached the desired goal, to get them published as soon as possible, for new ideas are apt to leak out, and further, it often happens that a new discovery springs up spontaneously in many places at once, "like the violets in springtime." Bolyai's presentment was truer than he suspected, for at this very moment Lobachevsky at Kazan, Gauss at Gottingen, Taurinus at Cologne, were all on the verge of this great discovery. It was not, however, till 1832 that... the work was published. It appeared in Vol. I of his father's ''Tentamen'', under the title "Appendix, scientiam absolute veram exhibens." ...the son, although he continued to work at his theory of space, published nothing further. Lobachevsky's ''Geometrische Untersuchungen'' came to his knowledge in 1848, and this spurred him on to complete the great work on "Raumlehre," which he had already planned at the time of the publication of his "Appendix," but he left this in large part as a ''rudis indigestaque moles'', and he never realised his hope of triumphing over his great Russian rival.

** Chapter 1. Historical, pp. 22-23.

* [[Nikolai Lobachevsky|Lobachevsky]] never seems to have heard of [[w:János Bolyai|Bolyai]], though both were directly or indirectly in communication with [[Carl Friedrich Gauss|Gauss]]. Much has been written on the relationship of these three discoverers, but it is now generally recognised that John Bolyai and Lobachevsky each arrived at their ideas independently of Gauss and of each other; and, since they possessed the convictions and the courage to publish them which Gauss lacked, to them alone is due the honour of the discovery.

** Chapter 1. Historical, p. 24.

* The ideas inaugurated by [[Nikolai Lobachevsky|Lobachevsky]] and [[w:János Bolyai|Bolyai]] did not for many years attain any wide recognition, and it was only after Baltzer had called attention to them in 1867, and at his request Hoüel had published French translations of the epoch making works, that the subject of non-euclidean geometry began to be seriously studied. It is remarkable that while [[w:Giovanni Girolamo Saccheri|Saccheri]] and [[w:Johann Heinrich Lambert|Lambert]] both considered the two hypotheses, it never occurred to Lobachevsky or Bolyai or their predecessors, [[Carl Friedrich Gauss|Gauss]], [F. K.] Schweikart, [F. A.] Taurinus, and [F. L.] Wachter, to admit the hypothesis that the sum of the angles of a triangle may be greater than two right angles. This involves the conception of a straight line as being unbounded but yet of finite length. Somewhere "at the back of beyond" the two ends of the line meet and close it. We owe this conception first to [[Bernhard Riemann]] in his Dissertation of 1854 (published only in 1866 after the author's death), but in his [[w:Spherical geometry|Spherical Geometry]] two straight lines intersect twice like two great circles on a sphere. The conception of a geometry in which the straight line is finite, and is, without exception, uniquely determined by two distinct points, is due to [[w:Felix Klein|Felix Klein]]. Klein attached the now usual nomenclature to the three geometries; the geometry of Lobachevsky he called ''Hyperbolic'', that of Riemann ''Elliptic'', and that of Euclid ''Parabolic''.

** Chapter 1. Historical, pp. 24-25.

== Also see ==

* [[Mathematics]]

* [[History of mathematics]]

== External links ==

{{Wikipedia}}

{{Wiktionary|non-Euclidean geometry}}

* [https://archive.org/details/noneuclideangeom00bonorich Non-Euclidean Geometry] by Roberto Bonola (1912).

[[Category:History of science]]

[[Category:Mathematics]]