Non-Euclidean geometry - Knowledge (XXG)

3428:"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 – made by 1532:): "Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge." 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 1814:", does not give a consistent set of axioms. This follows since parallel lines exist in absolute geometry, but this statement says that there are no parallel lines. This problem was known (in a different guise) to Khayyam, Saccheri and Lambert and was the basis for their rejecting what was known as the "obtuse angle case". To obtain a consistent set of axioms that includes this axiom about having no parallel lines, some other axioms must be tweaked. These adjustments depend upon the axiom system used. Among others, these tweaks have the effect of modifying Euclid's second postulate from the statement that line segments can be extended indefinitely to the statement that lines are unbounded. 1583:, 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. 91: 2202:'s treatment of human knowledge had a special role for geometry. It was his prime example of synthetic a priori knowledge; not derived from the senses nor deduced through logic — our knowledge of space was a truth that we were born with. Unfortunately for Kant, his concept of this unalterably true geometry was Euclidean. Theology was also affected by the change from absolute truth to relative truth in the way that mathematics is related to the world around it, that was a result of this paradigm shift. 1840: 3517:"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." 1832: 2096: 4766: 2106: 4470: 36: 70: 3223:

is characterized by its non-Euclidean geometry. It is heavily implied this is achieved as a side effect of not following the natural laws of this universe rather than simply using an alternate geometric model, as the sheer innate wrongness of it is said to be capable of driving those who look upon it

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of a triangle is the numerical value (180° − sum of the measures of the angles of the triangle). This result may also be stated as: the defect of triangles in hyperbolic geometry is positive, the defect of triangles in Euclidean geometry is zero, and the defect of triangles in elliptic geometry

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on a sphere are identified in a model of the elliptic plane. In the story, in the midst of a thunderstorm, Sidney Davidson sees "Waves and a remarkably neat schooner" while working in an electrical laboratory at Harlow Technical College. At the story's close, Davidson proves to have witnessed H.M.S.

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In three dimensions, there are eight models of geometries. There are Euclidean, elliptic, and hyperbolic geometries, as in the two-dimensional case; mixed geometries that are partially Euclidean and partially hyperbolic or spherical; twisted versions of the mixed geometries; and one unusual geometry

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mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before, 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

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and Saccheri. 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

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In these models, the concepts of non-Euclidean geometries are represented by Euclidean objects in a Euclidean setting. This introduces a perceptual distortion wherein the straight lines of the non-Euclidean geometry are represented by Euclidean curves that visually bend. This "bending" is not a

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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

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Either there will exist more than one line through the point parallel to the given line or there will exist no lines through the point parallel to the given line. In the first case, replacing the parallel postulate (or its equivalent) with the statement "In a plane, given a point P and a line

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On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly

1763:, for instance, uses the axiom that says that, "There exists a pair of similar but not congruent triangles." In any of these systems, removal of the one axiom equivalent to the parallel postulate, in whatever form it takes, and leaving all the other axioms intact, produces 1431:

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.

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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.

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Euclidean geometry can be axiomatically described in several ways. However, Euclid's original system of five postulates (axioms) is not one of these, as his proofs relied on several unstated assumptions that should also have been taken as axioms.

3545:, 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 3140: 1561:), 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. 1719:

exploited it to describe the non-Euclidean geometries in articles 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.

1508:, 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 1636:. 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. 1606:

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.

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from each other (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) and are known as

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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

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and they have equal measure. The summit angles of a Saccheri quadrilateral are acute if the geometry is hyperbolic, right angles if the geometry is Euclidean and obtuse angles if the geometry is elliptic.

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The sum of the measures of the angles of any triangle is less than 180° if the geometry is hyperbolic, equal to 180° if the geometry is Euclidean, and greater than 180° if the geometry is elliptic. The

2828: 1960:. The difference is that as a model of elliptic geometry a metric is introduced permitting the measurement of lengths and angles, while as a model of the projective plane there is no such metric. 2894: 1359:

In hyperbolic geometry, they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called

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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

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Euclidean and non-Euclidean geometries naturally have many similar properties, namely those that do not depend upon the nature of parallelism. This commonality is the subject of

1727:, a term that generally fell out of use). His influence has led to the current usage of the term "non-Euclidean geometry" to mean either "hyperbolic" or "elliptic" geometry. 3674:, 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 3289:, allowing the player to experience many properties of this geometry. Many mechanics, quests, and locations are strongly dependent on the features of hyperbolic geometry. 1550:(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. 3030: 1755:

obtain the same geometry by different paths. All approaches, however, have an axiom that is logically equivalent to Euclid's fifth postulate, the parallel postulate.

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consisting of 20 axioms most closely follows the approach of Euclid and provides the justification for all of Euclid's proofs. Other systems, using different sets of

4043:(1979) A simple non-Euclidean geometry and its physical basis : an elementary account of Galilean geometry and the Galilean principle of relativity, Springer 1673:, 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. 3450:, 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 1436:

Other mathematicians have devised simpler forms of this property. Regardless of the form of the postulate, however, it consistently appears more complicated than

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Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both

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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)

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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

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contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, but did not realize it.

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Richard C. Tolman (2004) Theory of Relativity of Motion, page 194, §180 Non-Euclidean angle, §181 Kinematical interpretation of angle in terms of velocity

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and communications is possible through the use of Hsieh Ho's Polydimensional Non-Euclidean Geometry, published sometime in the middle of the 22nd century.

1417:, Euclid begins with a limited number of assumptions (23 definitions, five common notions, and five postulates) and seeks to prove all the other results ( 1219: 1404:, includes some of the oldest known mathematics, and geometries that deviated from this were not widely accepted as legitimate until the 19th century. 5209: 3233: 2198:

The discovery of the non-Euclidean geometries had a ripple effect which went far beyond the boundaries of mathematics and science. The philosopher

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The second case is not dealt with as easily. Simply replacing the parallel postulate with the statement, "In a plane, given a point P and a line

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Before the models of a non-Euclidean plane were presented by Beltrami, Klein, and Poincaré, Euclidean geometry stood unchallenged as the

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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 (

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utilizes non-Euclidean geometry to explain instantaneous transport through space and time and between parallel and fictional universes.

4553: 3017:{\displaystyle {\begin{pmatrix}x'\\t'\end{pmatrix}}={\begin{pmatrix}1&v\\0&1\end{pmatrix}}{\begin{pmatrix}x\\t\end{pmatrix}}.} 3635:(The theory of parallel lines from Euclid to Gauss, an archive of non-Euclidean geometry), (Leipzig, Germany: B. G. Teubner, 1895), 1268:, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the 5490: 5474: 5330: 4433: 4413: 4392: 4370: 4352: 4292: 4261: 4231: 4215: 4171: 4048: 3866: 3736: 3511: 3484: 1771:) do not require the use of the parallel postulate or anything equivalent to it, they are all true statements in absolute geometry. 285: 54: 4516: 4474: 2126:

Besides the behavior of lines with respect to a common perpendicular, mentioned in the introduction, we also have the following:

4066:(1912) "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics" Proceedings of the 2228:

in many ways and in particular was one of the leading factors that caused a re-examination of the teaching of geometry based on

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The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry. Circa 1813,

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Khayyam, for example, tried to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" (

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There are some mathematicians who would extend the list of geometries that should be called "non-Euclidean" in various ways.

1212: 1166: 772: 231: 4521: 4079: 3861:. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. 3446:) – was undoubtedly prompted by Arabic sources. The proofs put forward in the fourteenth century by the Jewish scholar 2025:, which models the entirety of hyperbolic space, and used this to show that Euclidean geometry and hyperbolic geometry were 2775: 5554: 4890: 4512: 4448:, Critical edition of Lambert's memoir with a French translation, with historical and mathematical notes and commentaries 4406: 4306: 2837: 1615: 4441: 3469:

Boris A. Rosenfeld & Adolf P. Youschkevitch (1996), "Geometry", p. 467, in Roshdi Rashed & Régis Morelon (1996),

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in 1594 and was studied by European geometers, including Saccheri who criticised this work as well as that of Wallis.

1421:) in the work. The most notorious of the postulates is often referred to as "Euclid's Fifth Postulate", or simply the 4489: 3636: 2274: 1407:

The debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclid wrote

2123:). However, the properties that distinguish one geometry from others have historically received the most attention. 1602:

had the germinal ideas of non-Euclidean geometry worked out, but neither published any results. Schweikart's nephew

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Boris A. Rosenfeld & Adolf P. Youschkevitch, "Geometry", p. 470, in Roshdi Rashed & Régis Morelon (1996),

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published the short story "The Remarkable Case of Davidson's Eyes". To appreciate this story one should know how

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It was Gauss who coined the term "non-Euclidean geometry". He was referring to his own work, which today we call

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At this time it was widely believed that the universe worked according to the principles of Euclidean geometry.

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Die theorie der Parallellinien von Euklid bis auf Gauss, eine Urkundensammlung der nichteuklidischen Geometrie

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of geometries which are non-Euclidean in the sense that it is not the case that exactly one line can be drawn

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in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure now known as a

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while only two lines are postulated, it is easily shown that there must be an infinite number of such lines.

2831: 1572: 1297: 600: 280: 137: 3700:, Dover, 1948 (Reprint of English translation of 3rd Edition, 1940. First edition in German, 1908.) p. 176. 2213:, in which mathematicians and scientists changed the way they viewed their subjects. Some geometers called 1997:

Even after the work of Lobachevsky, Gauss, and Bolyai, the question remained: "Does such a model exist for

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Klein is responsible for the terms "hyperbolic" and "elliptic" (in his system he called Euclidean geometry

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form, since it is a compound statement (... there exists one and only one ...), can be done in two ways:

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However, other axioms besides the parallel postulate must be changed to make this a feasible geometry.

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was a chief exhibit of rationality, the Euclidean point of view represented absolute authority.

3713: 3215:. In his works, many unnatural things follow their own unique laws of geometry: in Lovecraft's 3163:

in 1912. They revamped the analytic geometry implicit in the split-complex number algebra into

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The non-Euclidean planar algebras support kinematic geometries in the plane. For instance, the

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This approach to non-Euclidean geometry explains the non-Euclidean angles: the parameters of

5601: 5522: 5506: 5432: 5386: 4951: 4860: 4620: 4340: 4305:, Translator and Editor: A. Papadopoulos, Heritage of European Mathematics Series, vol. 4, 4284: 4195: 4095: 4063: 3882:, Dieter Henrich, ed. (Schriftenreihe der Universität Regensburg, band 7, 1982) pp. 141–204. 3854: 3724: 3712: 3621: 3346: 3286: 3201: 3192: 3153: 2714: 2662: 2339: 2261: 2018: 2006: 1953: 1760: 1752: 1639: 1131: 872: 582: 460: 395: 253: 238: 103: 4283:, (2012) " A New Perspective on Relativity : An Odyssey In Non-Euclidean Geometries", 4029: 2157:

is a quadrilateral with two sides of equal length, both perpendicular to a side called the

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is a quadrilateral with three right angles. The fourth angle of a Lambert quadrilateral is

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Course notes: "Gauss and non-Euclidean geometry", University of Waterloo, Ontario, Canada

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Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed.,

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with an alternative, or relaxing the metric requirement. In the former case, one obtains

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property of the non-Euclidean lines, only an artifice of the way they are represented.

1956:) are identified (considered the same). This is also one of the standard models of the 1856: 1603: 1466: 1182: 1156: 1090: 1034: 907: 787: 767: 747: 652: 3869:(in depth explanation of the eight geometries and the proof that there are only eight) 2095: 17: 5722: 5457: 5452: 5427: 5236: 5177: 5088: 5016: 4946: 4941: 4524:, a digest of the axioms used, and theorems proved, by Wilson and Lewis. Archived by 4179: 4082:, a digest of the axioms used, and theorems proved, by Wilson and Lewis. Archived by 4016: 3937:

This is a quote from G. B. Halsted's translator's preface to his 1914 translation of

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Non-Euclidean geometry is sometimes connected with the influence of the 20th-century

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Two geometries based on axioms closely related to those specifying Euclidean geometry

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exist (a statement equivalent to the parallel postulate) only in Euclidean geometry.

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Behavior of lines with a common perpendicular in each of the three types of geometry

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English translations of Schweikart's letter and Gauss's reply to Gerling appear in:

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if and only if Euclidean geometry was. (The reverse implication follows from the

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and it is considered a non-Euclidean geometry due to its lack of parallel lines.

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To obtain a non-Euclidean geometry, the parallel postulate (or its equivalent)

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noted that distance between points inside a conic could be defined in terms of

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Comparison of elliptic, Euclidean and hyperbolic geometries in two dimensions

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Views of Euclid's Parallel Postulate in Ancient Greece and in Medieval Islam

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The existence of non-Euclidean geometries impacted the intellectual life of

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In the hyperbolic model, within a two-dimensional plane, for any given line

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For planar algebra, non-Euclidean geometry arises in the other cases. When

1894:." The simplest model for elliptic geometry is a sphere, where lines are " 4469: 3653:

Non-Euclidean geometry: A critical and historical study of its development

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not passing through P, there exist two lines through P, which do not meet

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describes the struggle of living on a planet with the form of a rotating

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In elliptic geometry, the lines "curve toward" each other and intersect.

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The essential difference between the metric geometries is the nature of

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the hyperbolic geometry are possible depending on a parameter

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To produce a finite straight line continuously in a straight line.

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Mathematical Visions: The Pursuit of Geometry in Victorian England

4336:, Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 1, pp. 9–24. 3946: 3878:

Imre Toth, "Gott und Geometrie: Eine viktorianische Kontroverse,"

3432:, the Polish scientists of the thirteenth century, while revising 2666: 2658: 2238:, written by Charles Lutwidge Dodgson (1832–1898) better known as 2188: 2161:. The other two angles of a Saccheri quadrilateral are called the 2094: 1949: 1907: 1241: 1024: 948: 882: 727: 331: 326: 68: 4141: 1525:

some early properties of the hyperbolic and elliptic geometries.

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Euclidean and Non-Euclidean Geometries: Development and History

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a 21st axiom appeared in the French translation of Hilbert's

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of Geometry" due to the revolutionary character of his work.

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Elementary Mathematics from an Advanced Standpoint: Geometry

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Non-Euclidean geometry often makes appearances in works of

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mapping the frame with rapidity zero to that with rapidity

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emerges as the most natural geometry satisfying this axiom.

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and independently around 1818, the German professor of law

4241:

Ideas of Space: Euclidean, Non-Euclidean, and Relativistic

3751:

F. Klein, Über die sogenannte nichteuklidische Geometrie,

1867:. In hyperbolic geometric models, by contrast, there are 1890:

Euclidean geometry is modelled by our notion of a "flat

1256:, non-Euclidean geometry arises by either replacing the 3924:

Author attributes this quote to another mathematician,

3331:

Victoria Line visits all kinds of non-Euclidean worlds.

1484:

The theorems of Ibn al-Haytham, Khayyam and al-Tusi on

2990: 2954: 2915: 2009:, in 1868, who first showed that a surface called the 1351:

In Euclidean geometry, the lines remain at a constant

4156:

Teoria fondamentale degli spazî di curvatura costante

3627:

Letters by Schweikart and the writings of his nephew

3033: 2909: 2840: 2823:{\displaystyle z=x+y\epsilon ,\quad \epsilon ^{2}=0,} 2778: 2526: 2380: 2277: 2191:. Furthermore, since the substance of the subject in 4450:éd. Blanchard, coll. Sciences dans l'Histoire, Paris 4446:

La théorie des parallèles de Johann Heinrich Lambert

2889:{\displaystyle x^{\prime }=x+vt,\quad t^{\prime }=t} 2830:

to represent the classical description of motion in

2021:

and in a second paper in the same year, defined the

1810:

not passing through P, all the lines through P meet

1689:. Several modern authors still use the generic term 1669:

By formulating the geometry in terms of a curvature

1642:, in a famous lecture in 1854, founded the field of 1444:

To draw a straight line from any point to any point.

1248:. As Euclidean geometry lies at the intersection of 5655: 5589: 5466: 5405: 5322: 5269: 5081: 5050: 4965: 4909: 4853: 4822: 4727: 4696: 4646: 4594: 4224:

Foundations of Euclidean and Non-Euclidean Geometry

1450:

To describe a circle with any centre and distance .

1276:that have also been called non-Euclidean geometry. 4490:MacTutor Archive article on non-Euclidean geometry 4421: 3614:(Leipzig, Germany: B. G. Teubner, 1900), vol. 8, 3134: 3016: 2888: 2822: 2669:in Euclidean geometry. Indeed, they each arise in 2612: 2461: 2327: 4271:Mathematical Thought from Ancient to Modern Times 3372:Spherical geometry#Relation to similar geometries 2609: 1320:. In hyperbolic geometry, by contrast, there are 4273:, Chapter 36 Non-Euclidean Geometry, pp 861–81, 1646:, discussing in particular the ideas now called 3859:Three-dimensional geometry and topology. Vol. 1 3539: 3426: 2328:{\displaystyle C=\{(x,y):x,y\in \mathbb {R} \}} 2110:Saccheri quadrilaterals in the three geometries 1453:That all right angles are equal to one another. 1332:, while in elliptic geometry, any line through 5307:Fourth Great Debate in international relations 3731:. American Mathematical Society. p. 140. 4796: 4547: 4185:, New York: Barnes and Noble, 2009 (reprint) 3531:Encyclopedia of the History of Arabic Science 3472:Encyclopedia of the History of Arabic Science 3418:Encyclopedia of the History of Arabic Science 3367:Parallel (geometry)#In non-Euclidean geometry 1767:. As the first 28 propositions of Euclid (in 1427:, which in Euclid's original formulation is: 1213: 8: 5686: 5296: 5286: 5276: 4992: 3768:The Euclidean plane is still referred to as 3458:'s studies of the theory of parallel lines." 3148:as a non-Euclidean geometry was advanced by 2685:Hyperbolic geometry found an application in 2363:The Euclidean plane corresponds to the case 2322: 2284: 2100:Lambert quadrilateral in hyperbolic geometry 2087:(i.e. every direction behaves differently). 4158:, Annali. di Mat., ser II 2 (1868), 232–255 4126:A'Campo, Norbert and Papadopoulos, Athanase 3670:, p. 162). In his 1824 letter to Taurinus ( 2146:if the geometry is elliptic. Consequently, 1799:" and keeping all the other axioms, yields 1711:function. The method has become called the 5266: 5255: 4819: 4803: 4789: 4781: 4554: 4540: 4532: 4256:, 4th ed., New York: W. H. Freeman, 2007. 2721:was charting this submanifold through his 2717:of three dimensions. Already in the 1890s 2205:Non-Euclidean geometry is an example of a 1963:In the elliptic model, for any given line 1952:), and points opposite each other (called 1220: 1206: 935: 454: 89: 78: 5210:Relationship between religion and science 4030:Non-Euclidean Style of Special Relativity 3234:Zen and the Art of Motorcycle Maintenance 3051: 3038: 3032: 2985: 2949: 2910: 2908: 2874: 2845: 2839: 2805: 2777: 2697:in 1908. Minkowski introduced terms like 2665:in the split-complex plane correspond to 2603: 2590: 2575: 2555: 2534: 2525: 2453: 2440: 2388: 2379: 2318: 2317: 2276: 1742:Axiomatic basis of non-Euclidean geometry 1273: 4333:Hyperbolic geometry: The first 150 years 3974: 2104: 2072:Three-dimensional non-Euclidean geometry 1838: 1830: 5531:The Structure of Scientific Revolutions 3892: 3790: 3500:History of Mathematics: An Introduction 3454:had stimulated both J. Wallis's and G. 3382: 2472:and this quantity is the square of the 1174: 1108: 1057: 986: 938: 700: 562: 539: 506: 478: 81: 4886:Machian positivism (empirio-criticism) 3494: 3492: 3342:Euclidean space#Other geometric spaces 1496:, were "the first few theorems of the 1244:closely related to those that specify 320:Straightedge and compass constructions 4068:American Academy of Arts and Sciences 3820: 3807: 3675: 3671: 3667: 3561:MacTutor's Giovanni Girolamo Saccheri 3160:American Academy of Arts and Sciences 1347:to a third line (in the same plane): 7: 4752:List of differential geometry topics 3880:Evolutionstheorie und ihre Evolution 1759:uses the Playfair axiom form, while 1312:, there is exactly one line through 1269: 1240:consists of two geometries based on 4442:A. Papadopoulos et Guillaume Théret 4315:Introductory Non-Euclidean Geometry 4208:Mathematical Association of America 1590:Discovery of non-Euclidean geometry 5163:Nomothetic–idiographic distinction 3841:Book I Proposition 27 of Euclid's 3543:Pseudo-Tusi's Exposition of Euclid 3452:Pseudo-Tusi's Exposition of Euclid 1662:. The simplest of these is called 1385:Hyperbolic geometry § History 25: 5491:The Logic of Scientific Discovery 5475:Materialism and Empirio-criticism 5331:The Course in Positive Philosophy 3912:. Touchstone Books. p. 294. 3624:; see especially pages 10 and 11. 3027:With dual numbers the mapping is 2769:Kinematic study makes use of the 2754:. Furthermore, multiplication by 2138:if the geometry is hyperbolic, a 1555:Euclides ab Omni Naevo Vindicatus 1377:Euclidean geometry § History 286:Noncommutative algebraic geometry 4765: 4764: 4468: 4151:, second edition, Springer, 2005 3572:O'Connor, J.J.; Robertson, E.F. 2576: 2556: 2142:if the geometry is Euclidean or 2029:so that hyperbolic geometry was 1849:Models of non-Euclidean geometry 34: 5483:History and Class Consciousness 4517:Springer Science+Business Media 4387:, New York: Perseus Publishing 4162:Blumenthal, Leonard M. (1980), 2869: 2800: 1863:through a point that is not on 1457:For at least a thousand years, 5347:Critical History of Philosophy 4403:Sources of Hyperbolic Geometry 4365:, Pacific Grove: Brooks/Cole, 4313:Manning, Henry Parker (1963), 3607:non-Euclidean geometry. See: 3123: 3108: 3096: 3081: 3078: 3063: 2580: 2563: 2560: 2543: 2480:and the origin. For instance, 2430: 2415: 2412: 2397: 2342:are sometimes identified with 2299: 2287: 2037:model of Euclidean geometry.) 1936:is a sphere, where lines are " 1918:to model hyperbolic geometry. 679:- / other-dimensional 1: 5555:Knowledge and Human Interests 4891:Rankean historical positivism 4513:European Mathematical Society 4407:American Mathematical Society 4322:Meschkowski, Herbert (1964), 4307:European Mathematical Society 2661:in the dual number plane and 1616:Nikolai Ivanovich Lobachevsky 1536:." His work was published in 5673: 5339:A General View of Positivism 4424:The Non-Euclidean Revolution 4420:Trudeau, Richard J. (1987), 4183:Euclid and His Modern Rivals 4130:Notes on hyperbolic geometry 4015:Hermann Minkowski (1908–9). 3362:Non-Euclidean surface growth 3167:of premises and deductions. 2235:Euclid and his Modern Rivals 1575:wrote, but did not publish, 46:Non-Euclidean surface growth 5539:Conjectures and Refutations 5371:The Logic of Modern Physics 5188:Deductive-nomological model 4363:Modern Geometries (5th Ed.) 4226:, New York: Marcel Dekker, 4204:University of Toronto Press 3991:Complex Numbers in Geometry 3253:Christopher Priest's novel 1559:Euclid Freed from All Flaws 43:It has been suggested that 5745: 5499:The Poverty of Historicism 5395:The Universe in a Nutshell 5379:Language, Truth, and Logic 5363:The Analysis of Sensations 4347:, Boston: Academic Press, 4326:, New York: Academic Press 4222:Faber, Richard L. (1983), 3723:Iwasawa, Kenkichi (1993). 3711:Kulczycki, Stefan (1961). 3595:A Treatise of Human Nature 2075: 1990: 1925: 1577:Theorie der Parallellinien 1479:Giovanni Girolamo Saccheri 1374: 5707: 5579:The Rhetoric of Economics 5265: 5260:Positivist-related debate 5254: 4818: 4760: 4569: 4164:A Modern View of Geometry 3574:"Johann Heinrich Lambert" 1600:Ferdinand Karl Schweikart 1469:(Alhazen, 11th century), 1438:Euclid's other postulates 60:Proposed since July 2024. 5515:Two Dogmas of Empiricism 5232:Structural functionalism 5158:Naturalism in literature 4505:Non-Euclidean geometries 4361:Smart, James R. (1997), 3957:that was Lobachevsky to 3926:William Kingdon Clifford 3804:Grundlagen der Geometrie 3534:, vol. 2, pp. 447–494 , 3313:faster-than-light travel 2733:of hyperbolic geometry. 2709:. He realized that the 1316:that does not intersect 1292:'s fifth postulate, the 175:Non-Archimedean geometry 5642:Willard Van Orman Quine 5355:Idealism and Positivism 4947:Critique of metaphysics 4881:Sociological positivism 4747:List of geometry topics 4330:Milnor, John W. (1982) 4275:Oxford University Press 3939:The Theory of Parallels 3719:. Pergamon. p. 53. 3538:, London and New York: 3498:Victor J. Katz (1998), 3475:, vol. 2, pp. 447–494, 3425:, London and New York: 3421:, vol. 2, pp. 447–494, 3390:Eder, Michelle (2000), 3269:The Number of the Beast 2832:absolute time and space 2517:replaces epsilon. Then 2360:where ε ∈ { –1, 0, 1}. 1932:The simplest model for 1887:for more information.) 1610:Then, in 1829–1830 the 281:Noncommutative geometry 5729:Non-Euclidean geometry 5687: 5656:Concepts in contention 5297: 5287: 5277: 5168:Objectivity in science 5066:Non-Euclidean geometry 5032:Methodological dualism 4993: 4495:Non-euclidean geometry 4486:, Open Court, Chicago. 4484:Non-Euclidean Geometry 4482:Roberto Bonola (1912) 4475:Non-Euclidean geometry 4428:, Boston: Birkhauser, 4200:Non-Euclidean Geometry 3778:Uniformization theorem 3715:Non-Euclidean Geometry 3656:. Chicago: Open Court. 3610:Carl Friedrich Gauss, 3551: 3519: 3460: 3247:The Brothers Karamazov 3241:on multiple occasions. 3227:The main character in 3136: 3018: 2890: 2824: 2727:hyperbolic quaternions 2614: 2463: 2329: 2155:Saccheri quadrilateral 2112: 2102: 2060:that do not intersect 2017:to model a portion of 1845: 1836: 1691:non-Euclidean geometry 1687:Lobachevskian geometry 1494:Saccheri quadrilateral 1463:proof by contradiction 1434: 1238:non-Euclidean geometry 249:Discrete/Combinatorial 76: 18:Non-Euclidean Geometry 5563:The Poverty of Theory 5183:Philosophy of science 5072:Uncertainty principle 4719:Differential geometry 4324:Noneuclidean Geometry 4251:Greenberg, Marvin Jay 4096:"The Call of Cthulhu" 3753:Mathematische Annalen 3629:Franz Adolph Taurinus 3515: 3219:, the sunken city of 3137: 3019: 2891: 2825: 2615: 2464: 2367:since the modulus of 2330: 2266:Cartesian coordinates 2207:scientific revolution 2132:Lambert quadrilateral 2108: 2098: 1958:real projective plane 1842: 1834: 1581:Lambert quadrilateral 1490:Lambert quadrilateral 1429: 1272:, which give rise to 232:Discrete differential 72: 5571:The Scientific Image 5242:Structuration theory 5205:Qualitative research 5106:Criticism of science 5101:Critical rationalism 5037:Problem of induction 4509:Encyclopedia of Math 4477:at Wikimedia Commons 4110:"HyperRogue website" 4028:Scott Walter (1999) 3031: 2907: 2896:are equivalent to a 2838: 2776: 2738:split-complex number 2719:Alexander Macfarlane 2707:mathematical physics 2681:Kinematic geometries 2673:of a complex number 2524: 2509:split-complex number 2378: 2275: 2031:logically consistent 2013:has the appropriate 1975:, all lines through 1914:has the appropriate 1596:Carl Friedrich Gauss 1477:(13th century), and 1475:Nasīr al-Dīn al-Tūsī 1298:Playfair's postulate 1274:kinematic geometries 53:into this article. ( 5547:One-Dimensional Man 4995:Geisteswissenschaft 4978:Confirmation holism 4522:Synthetic Spacetime 4299:Nikolai Lobachevsky 4239:Jeremy Gray (1989) 4206:, reissued 1998 by 4166:, New York: Dover, 4149:Hyperbolic Geometry 4147:Anderson, James W. 4080:Synthetic Spacetime 3727:Algebraic Functions 3650:Bonola, R. (1912). 3357:Projective geometry 3239:Riemannian geometry 3158:Proceedings of the 2900:in linear algebra: 2671:polar decomposition 2511:and conventionally 2245:Alice in Wonderland 2091:Uncommon properties 2083:that is completely 2056:many lines through 2003:hyperbolic geometry 1999:hyperbolic geometry 1993:Hyperbolic geometry 1987:Hyperbolic geometry 1881:hyperbolic geometry 1871:many lines through 1853:mathematical models 1801:hyperbolic geometry 1778:be replaced by its 1736:Riemannian geometry 1713:Cayley–Klein metric 1707:and the projective 1695:hyperbolic geometry 1683:hyperbolic geometry 1644:Riemannian geometry 1502:elliptic geometries 1399:Greek mathematician 1381:History of geometry 1324:many lines through 1296:, is equivalent to 1262:hyperbolic geometry 499:Pythagorean theorem 5622:Hans-Georg Gadamer 5423:Alexander Bogdanov 5299:Positivismusstreit 5094:Post-behavioralism 5058:history of science 4910:Principal concepts 4866:Logical positivism 4714:Algebraic geometry 4281:Bernard H. Lavenda 4154:Beltrami, Eugenio 3910:Men of Mathematics 3774:conformal geometry 3772:in the context of 3398:Rutgers University 3266:Robert Heinlein's 3165:synthetic geometry 3146:special relativity 3132: 3014: 3005: 2979: 2940: 2886: 2820: 2745:frame of reference 2723:Algebra of Physics 2691:physical cosmology 2610: 2474:Euclidean distance 2459: 2325: 2264:is described with 2211:history of science 2193:synthetic geometry 2185:mathematical model 2113: 2103: 2048:, which is not on 2001:?". The model for 1971:, which is not on 1846: 1837: 1424:parallel postulate 1397:, named after the 1395:Euclidean geometry 1308:, which is not on 1294:parallel postulate 1258:parallel postulate 1246:Euclidean geometry 77: 5716: 5715: 5703: 5702: 5699: 5698: 5597:Theodor W. Adorno 5413:Richard Avenarius 5289:Werturteilsstreit 5250: 5249: 5198:Sense-data theory 4896:Polish positivism 4871:Positivist school 4778: 4777: 4473:Media related to 4457:978-2-85367-266-5 4341:Richards, Joan L. 4317:, New York: Dover 4191:978-1-4351-2348-9 4138:978-3-03719-105-7 3919:978-0-671-62818-5 3309:role-playing-game 2731:hyperboloid model 2695:Hermann Minkowski 2258:analytic geometry 2230:Euclid's Elements 2226:Victorian England 2117:absolute geometry 2078:Thurston geometry 1934:elliptic geometry 1928:Elliptic geometry 1922:Elliptic geometry 1885:elliptic geometry 1820:elliptic geometry 1765:absolute geometry 1664:elliptic geometry 1652:Riemannian metric 1553:In a work titled 1328:not intersecting 1266:elliptic geometry 1230: 1229: 1195: 1194: 918:List of geometers 601:Three-dimensional 590: 589: 67: 66: 62: 16:(Redirected from 5736: 5692: 5678: 5602:Gaston Bachelard 5523:Truth and Method 5507:World Hypotheses 5387:The Two Cultures 5302: 5292: 5282: 5267: 5256: 4998: 4952:Unity of science 4861:Legal positivism 4820: 4805: 4798: 4791: 4782: 4768: 4767: 4556: 4549: 4542: 4533: 4472: 4438: 4427: 4375: 4357: 4327: 4318: 4287:, pp. 696, 4285:World Scientific 4236: 4196:H. S. M. Coxeter 4176: 4127: 4114: 4113: 4106: 4100: 4099: 4092: 4086: 4077: 4071: 4064:Gilbert N. Lewis 4057: 4051: 4038: 4032: 4026: 4020: 4017:"Space and Time" 4013: 4007: 4004: 3998: 3984: 3978: 3971: 3965: 3935: 3929: 3923: 3902: 3896: 3889: 3883: 3876: 3870: 3855:William Thurston 3851: 3845: 3839: 3833: 3830: 3824: 3817: 3811: 3800: 3794: 3787: 3781: 3766: 3760: 3749: 3743: 3742: 3730: 3720: 3718: 3707: 3701: 3694: 3688: 3685: 3679: 3664: 3658: 3657: 3647: 3641: 3604: 3598: 3591: 3585: 3584: 3582: 3580: 3569: 3563: 3558: 3552: 3526: 3520: 3496: 3487: 3467: 3461: 3444:Kitab al-Manazir 3413: 3407: 3406: 3405: 3404: 3387: 3347:Hyperbolic space 3287:hyperbolic plane 3285:game set on the 3202:Antipodes Island 3193:antipodal points 3144:Another view of 3141: 3139: 3138: 3133: 3056: 3055: 3043: 3042: 3023: 3021: 3020: 3015: 3010: 3009: 2984: 2983: 2945: 2944: 2937: 2925: 2895: 2893: 2892: 2887: 2879: 2878: 2850: 2849: 2834:: The equations 2829: 2827: 2826: 2821: 2810: 2809: 2715:hyperbolic space 2676: 2663:hyperbolic angle 2649: 2645: 2634: 2619: 2617: 2616: 2611: 2608: 2607: 2595: 2594: 2579: 2559: 2539: 2538: 2516: 2506: 2502: 2491: 2479: 2468: 2466: 2465: 2460: 2458: 2457: 2445: 2444: 2393: 2392: 2370: 2366: 2359: 2334: 2332: 2331: 2326: 2321: 2242:, the author of 2121:neutral geometry 2063: 2051: 2043: 2019:hyperbolic space 2007:Eugenio Beltrami 2005:was answered by 1982: 1974: 1966: 1954:antipodal points 1859:to a given line 1813: 1809: 1798: 1794: 1784:Playfair's axiom 1749:Hilbert's system 1640:Bernhard Riemann 1618:and in 1832 the 1548:Euclide restituo 1506:Playfair's axiom 1488:, including the 1481:(18th century). 1473:(12th century), 1339: 1331: 1319: 1311: 1303: 1222: 1215: 1208: 936: 455: 388:Zero-dimensional 93: 79: 58: 38: 37: 30: 21: 5744: 5743: 5739: 5738: 5737: 5735: 5734: 5733: 5719: 5718: 5717: 5712: 5695: 5651: 5617:Paul Feyerabend 5612:Wilhelm Dilthey 5585: 5462: 5401: 5318: 5261: 5246: 5193:Ramsey sentence 5148:Instrumentalism 5077: 5055: 5053:paradigm shifts 5046: 4983:Critical theory 4961: 4957:Verificationism 4905: 4901:Russian Machism 4849: 4814: 4809: 4779: 4774: 4756: 4723: 4692: 4649: 4642: 4597: 4590: 4565: 4560: 4465: 4436: 4419: 4373: 4360: 4355: 4339: 4321: 4312: 4245:Clarendon Press 4243:, 2nd edition, 4234: 4221: 4174: 4161: 4142:DOI:10.4171/105 4125: 4122: 4117: 4108: 4107: 4103: 4094: 4093: 4089: 4078: 4074: 4060:Edwin B. Wilson 4058: 4054: 4039: 4035: 4027: 4023: 4014: 4010: 4005: 4001: 3985: 3981: 3972: 3968: 3936: 3932: 3920: 3904: 3903: 3899: 3890: 3886: 3877: 3873: 3852: 3848: 3840: 3836: 3831: 3827: 3818: 3814: 3801: 3797: 3793:and Yaglom 1968 3788: 3784: 3767: 3763: 3750: 3746: 3739: 3722: 3721: 3710: 3708: 3704: 3695: 3691: 3686: 3682: 3665: 3661: 3649: 3648: 3644: 3605: 3601: 3592: 3588: 3578: 3576: 3571: 3570: 3566: 3559: 3555: 3527: 3523: 3502:, pp. 270–271, 3497: 3490: 3468: 3464: 3448:Levi ben Gerson 3414: 3410: 3402: 3400: 3389: 3388: 3384: 3380: 3338: 3297:science fiction 3294:Renegade Legion 3213:H. P. Lovecraft 3177:science fiction 3173: 3047: 3034: 3029: 3028: 3004: 3003: 2997: 2996: 2986: 2978: 2977: 2972: 2966: 2965: 2960: 2950: 2939: 2938: 2930: 2927: 2926: 2918: 2911: 2905: 2904: 2870: 2841: 2836: 2835: 2801: 2774: 2773: 2683: 2674: 2647: 2643: 2624: 2599: 2586: 2530: 2522: 2521: 2512: 2504: 2500: 2481: 2477: 2449: 2436: 2384: 2376: 2375: 2368: 2364: 2346: 2344:complex numbers 2273: 2272: 2254: 2252:Planar algebras 2181: 2111: 2101: 2093: 2080: 2074: 2061: 2049: 2041: 1995: 1989: 1980: 1979:will intersect 1972: 1964: 1940:" (such as the 1930: 1924: 1898:" (such as the 1829: 1811: 1807: 1796: 1792: 1782:. Negating the 1753:undefined terms 1744: 1679: 1660:Euclidean space 1592: 1544:Giordano Vitale 1514:Levi ben Gerson 1392: 1387: 1373: 1337: 1329: 1317: 1309: 1301: 1282: 1270:planar algebras 1254:affine geometry 1250:metric geometry 1226: 1197: 1196: 933: 932: 923: 922: 713: 712: 696: 695: 681: 680: 668: 667: 604: 603: 592: 591: 452: 451: 449:Two-dimensional 440: 439: 413: 412: 410:One-dimensional 401: 400: 391: 390: 379: 378: 312: 311: 310: 293: 292: 141: 140: 129: 106: 75: 63: 39: 35: 28: 23: 22: 15: 12: 11: 5: 5742: 5740: 5732: 5731: 5721: 5720: 5714: 5713: 5708: 5705: 5704: 5701: 5700: 5697: 5696: 5694: 5693: 5684: 5679: 5670: 5665: 5659: 5657: 5653: 5652: 5650: 5649: 5644: 5639: 5634: 5629: 5624: 5619: 5614: 5609: 5604: 5599: 5593: 5591: 5587: 5586: 5584: 5583: 5575: 5567: 5559: 5551: 5543: 5535: 5527: 5519: 5511: 5503: 5495: 5487: 5479: 5470: 5468: 5464: 5463: 5461: 5460: 5455: 5450: 5445: 5440: 5438:Émile Durkheim 5435: 5430: 5425: 5420: 5415: 5409: 5407: 5403: 5402: 5400: 5399: 5391: 5383: 5375: 5367: 5359: 5351: 5343: 5335: 5326: 5324: 5320: 5319: 5317: 5316: 5310: 5304: 5294: 5284: 5279:Methodenstreit 5273: 5271: 5263: 5262: 5259: 5252: 5251: 5248: 5247: 5245: 5244: 5239: 5234: 5229: 5228: 5227: 5220:Social science 5217: 5212: 5207: 5202: 5201: 5200: 5195: 5190: 5180: 5175: 5173:Operationalism 5170: 5165: 5160: 5155: 5150: 5145: 5140: 5139: 5138: 5133: 5128: 5123: 5118: 5108: 5103: 5098: 5097: 5096: 5085: 5083: 5082:Related topics 5079: 5078: 5076: 5075: 5069: 5062: 5060: 5048: 5047: 5045: 5044: 5039: 5034: 5029: 5024: 5019: 5014: 5009: 5004: 4999: 4990: 4988:Falsifiability 4985: 4980: 4975: 4973:Antipositivism 4969: 4967: 4963: 4962: 4960: 4959: 4954: 4949: 4944: 4939: 4934: 4929: 4924: 4919: 4913: 4911: 4907: 4906: 4904: 4903: 4898: 4893: 4888: 4883: 4878: 4876:Postpositivism 4873: 4868: 4863: 4857: 4855: 4851: 4850: 4848: 4847: 4842: 4837: 4832: 4826: 4824: 4816: 4815: 4810: 4808: 4807: 4800: 4793: 4785: 4776: 4775: 4773: 4772: 4761: 4758: 4757: 4755: 4754: 4749: 4744: 4743: 4742: 4731: 4729: 4725: 4724: 4722: 4721: 4716: 4711: 4706: 4700: 4698: 4694: 4693: 4691: 4690: 4685: 4680: 4675: 4670: 4665: 4660: 4654: 4652: 4648:Non-Euclidean 4644: 4643: 4641: 4640: 4638:Solid geometry 4635: 4634: 4633: 4628: 4621:Plane geometry 4618: 4613: 4608: 4602: 4600: 4592: 4591: 4589: 4588: 4583: 4582: 4581: 4570: 4567: 4566: 4561: 4559: 4558: 4551: 4544: 4536: 4530: 4529: 4519: 4502: 4492: 4487: 4479: 4478: 4464: 4463:External links 4461: 4460: 4459: 4439: 4434: 4417: 4399:John Stillwell 4396: 4376: 4371: 4358: 4353: 4337: 4328: 4319: 4310: 4296: 4278: 4264: 4248: 4237: 4232: 4219: 4193: 4180:Carroll, Lewis 4177: 4172: 4159: 4152: 4145: 4121: 4118: 4116: 4115: 4101: 4087: 4072: 4052: 4033: 4021: 4008: 3999: 3995:Academic Press 3979: 3966: 3963:W. K. Clifford 3930: 3918: 3897: 3895:, pp. vii–viii 3884: 3871: 3846: 3834: 3825: 3812: 3795: 3789:for instance, 3782: 3761: 3744: 3737: 3702: 3689: 3680: 3659: 3642: 3640: 3639: 3625: 3618: 3599: 3586: 3564: 3553: 3521: 3504:Addison–Wesley 3488: 3462: 3439:Book of Optics 3434:Ibn al-Haytham 3408: 3381: 3379: 3376: 3375: 3374: 3369: 3364: 3359: 3354: 3349: 3344: 3337: 3334: 3333: 3332: 3316: 3290: 3273: 3264: 3256:Inverted World 3251: 3242: 3225: 3217:Cthulhu Mythos 3209:horror fiction 3205: 3172: 3169: 3131: 3128: 3125: 3122: 3119: 3116: 3113: 3110: 3107: 3104: 3101: 3098: 3095: 3092: 3089: 3086: 3083: 3080: 3077: 3074: 3071: 3068: 3065: 3062: 3059: 3054: 3050: 3046: 3041: 3037: 3025: 3024: 3013: 3008: 3002: 2999: 2998: 2995: 2992: 2991: 2989: 2982: 2976: 2973: 2971: 2968: 2967: 2964: 2961: 2959: 2956: 2955: 2953: 2948: 2943: 2936: 2933: 2929: 2928: 2924: 2921: 2917: 2916: 2914: 2885: 2882: 2877: 2873: 2868: 2865: 2862: 2859: 2856: 2853: 2848: 2844: 2819: 2816: 2813: 2808: 2804: 2799: 2796: 2793: 2790: 2787: 2784: 2781: 2693:introduced by 2682: 2679: 2637:unit hyperbola 2621: 2620: 2606: 2602: 2598: 2593: 2589: 2585: 2582: 2578: 2574: 2571: 2568: 2565: 2562: 2558: 2554: 2551: 2548: 2545: 2542: 2537: 2533: 2529: 2470: 2469: 2456: 2452: 2448: 2443: 2439: 2435: 2432: 2429: 2426: 2423: 2420: 2417: 2414: 2411: 2408: 2405: 2402: 2399: 2396: 2391: 2387: 2383: 2336: 2335: 2324: 2320: 2316: 2313: 2310: 2307: 2304: 2301: 2298: 2295: 2292: 2289: 2286: 2283: 2280: 2253: 2250: 2180: 2177: 2176: 2175: 2166: 2151: 2109: 2099: 2092: 2089: 2076:Main article: 2073: 2070: 2027:equiconsistent 1991:Main article: 1988: 1985: 1926:Main article: 1923: 1920: 1828: 1825: 1824: 1823: 1804: 1743: 1740: 1678: 1675: 1622:mathematician 1614:mathematician 1604:Franz Taurinus 1591: 1588: 1573:Johann Lambert 1546:, in his book 1486:quadrilaterals 1467:Ibn al-Haytham 1455: 1454: 1451: 1448: 1445: 1391: 1388: 1372: 1369: 1368: 1367: 1364: 1361:ultraparallels 1357: 1281: 1278: 1228: 1227: 1225: 1224: 1217: 1210: 1202: 1199: 1198: 1193: 1192: 1191: 1190: 1185: 1177: 1176: 1172: 1171: 1170: 1169: 1164: 1159: 1154: 1149: 1144: 1139: 1134: 1129: 1124: 1119: 1111: 1110: 1106: 1105: 1104: 1103: 1098: 1093: 1088: 1083: 1078: 1073: 1068: 1060: 1059: 1055: 1054: 1053: 1052: 1047: 1042: 1037: 1032: 1027: 1022: 1017: 1012: 1007: 1002: 997: 989: 988: 984: 983: 982: 981: 976: 971: 966: 961: 956: 951: 943: 942: 934: 930: 929: 928: 925: 924: 921: 920: 915: 910: 905: 900: 895: 890: 885: 880: 875: 870: 865: 860: 855: 850: 845: 840: 835: 830: 825: 820: 815: 810: 805: 800: 795: 790: 785: 780: 775: 770: 765: 760: 755: 750: 745: 740: 735: 730: 725: 720: 714: 710: 709: 708: 705: 704: 698: 697: 694: 693: 688: 682: 675: 674: 673: 670: 669: 666: 665: 660: 655: 653:Platonic Solid 650: 645: 640: 635: 630: 625: 624: 623: 612: 611: 605: 599: 598: 597: 594: 593: 588: 587: 586: 585: 580: 575: 567: 566: 560: 559: 558: 557: 552: 544: 543: 537: 536: 535: 534: 529: 524: 519: 511: 510: 504: 503: 502: 501: 496: 491: 483: 482: 476: 475: 474: 473: 468: 463: 453: 447: 446: 445: 442: 441: 438: 437: 432: 431: 430: 425: 414: 408: 407: 406: 403: 402: 399: 398: 392: 386: 385: 384: 381: 380: 377: 376: 371: 366: 360: 359: 354: 349: 339: 334: 329: 323: 322: 313: 309: 308: 305: 301: 300: 299: 298: 295: 294: 291: 290: 289: 288: 278: 273: 268: 263: 258: 257: 256: 246: 241: 236: 235: 234: 229: 224: 214: 213: 212: 207: 197: 192: 187: 182: 177: 172: 171: 170: 165: 164: 163: 148: 142: 136: 135: 134: 131: 130: 128: 127: 117: 111: 108: 107: 94: 86: 85: 73: 65: 64: 42: 40: 33: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 5741: 5730: 5727: 5726: 5724: 5711: 5706: 5691: 5690: 5685: 5683: 5680: 5677: 5676: 5671: 5669: 5666: 5664: 5661: 5660: 5658: 5654: 5648: 5645: 5643: 5640: 5638: 5635: 5633: 5632:György Lukács 5630: 5628: 5625: 5623: 5620: 5618: 5615: 5613: 5610: 5608: 5605: 5603: 5600: 5598: 5595: 5594: 5592: 5588: 5581: 5580: 5576: 5573: 5572: 5568: 5565: 5564: 5560: 5557: 5556: 5552: 5549: 5548: 5544: 5541: 5540: 5536: 5533: 5532: 5528: 5525: 5524: 5520: 5517: 5516: 5512: 5509: 5508: 5504: 5501: 5500: 5496: 5493: 5492: 5488: 5485: 5484: 5480: 5477: 5476: 5472: 5471: 5469: 5465: 5459: 5458:Vienna Circle 5456: 5454: 5453:Berlin Circle 5451: 5449: 5446: 5444: 5441: 5439: 5436: 5434: 5433:Eugen Dühring 5431: 5429: 5428:Auguste Comte 5426: 5424: 5421: 5419: 5416: 5414: 5411: 5410: 5408: 5404: 5397: 5396: 5392: 5389: 5388: 5384: 5381: 5380: 5376: 5373: 5372: 5368: 5365: 5364: 5360: 5357: 5356: 5352: 5349: 5348: 5344: 5341: 5340: 5336: 5333: 5332: 5328: 5327: 5325: 5323:Contributions 5321: 5314: 5311: 5308: 5305: 5301: 5300: 5295: 5291: 5290: 5285: 5281: 5280: 5275: 5274: 5272: 5268: 5264: 5257: 5253: 5243: 5240: 5238: 5237:Structuralism 5235: 5233: 5230: 5226: 5223: 5222: 5221: 5218: 5216: 5213: 5211: 5208: 5206: 5203: 5199: 5196: 5194: 5191: 5189: 5186: 5185: 5184: 5181: 5179: 5178:Phenomenalism 5176: 5174: 5171: 5169: 5166: 5164: 5161: 5159: 5156: 5154: 5151: 5149: 5146: 5144: 5141: 5137: 5134: 5132: 5129: 5127: 5124: 5122: 5119: 5117: 5114: 5113: 5112: 5109: 5107: 5104: 5102: 5099: 5095: 5092: 5091: 5090: 5089:Behavioralism 5087: 5086: 5084: 5080: 5073: 5070: 5067: 5064: 5063: 5061: 5059: 5054: 5049: 5043: 5040: 5038: 5035: 5033: 5030: 5028: 5025: 5023: 5020: 5018: 5017:Human science 5015: 5013: 5010: 5008: 5005: 5003: 5000: 4997: 4996: 4991: 4989: 4986: 4984: 4981: 4979: 4976: 4974: 4971: 4970: 4968: 4964: 4958: 4955: 4953: 4950: 4948: 4945: 4943: 4942:Pseudoscience 4940: 4938: 4937:Justification 4935: 4933: 4930: 4928: 4925: 4923: 4920: 4918: 4915: 4914: 4912: 4908: 4902: 4899: 4897: 4894: 4892: 4889: 4887: 4884: 4882: 4879: 4877: 4874: 4872: 4869: 4867: 4864: 4862: 4859: 4858: 4856: 4852: 4846: 4843: 4841: 4838: 4836: 4833: 4831: 4828: 4827: 4825: 4821: 4817: 4813: 4806: 4801: 4799: 4794: 4792: 4787: 4786: 4783: 4771: 4763: 4762: 4759: 4753: 4750: 4748: 4745: 4741: 4738: 4737: 4736: 4733: 4732: 4730: 4726: 4720: 4717: 4715: 4712: 4710: 4707: 4705: 4702: 4701: 4699: 4695: 4689: 4686: 4684: 4681: 4679: 4676: 4674: 4671: 4669: 4666: 4664: 4661: 4659: 4656: 4655: 4653: 4651: 4645: 4639: 4636: 4632: 4629: 4627: 4624: 4623: 4622: 4619: 4617: 4614: 4612: 4609: 4607: 4606:Combinatorial 4604: 4603: 4601: 4599: 4593: 4587: 4584: 4580: 4577: 4576: 4575: 4572: 4571: 4568: 4564: 4557: 4552: 4550: 4545: 4543: 4538: 4537: 4534: 4527: 4523: 4520: 4518: 4514: 4510: 4506: 4503: 4500: 4496: 4493: 4491: 4488: 4485: 4481: 4480: 4476: 4471: 4467: 4466: 4462: 4458: 4454: 4451: 4447: 4443: 4440: 4437: 4435:0-8176-3311-1 4431: 4426: 4425: 4418: 4415: 4414:0-8218-0529-0 4411: 4408: 4404: 4400: 4397: 4394: 4393:0-7382-0675-X 4390: 4386: 4385: 4380: 4377: 4374: 4372:0-534-35188-3 4368: 4364: 4359: 4356: 4354:0-12-587445-6 4350: 4346: 4342: 4338: 4335: 4334: 4329: 4325: 4320: 4316: 4311: 4308: 4304: 4300: 4297: 4294: 4293:9789814340489 4290: 4286: 4282: 4279: 4276: 4272: 4268: 4265: 4263: 4262:0-7167-9948-0 4259: 4255: 4252: 4249: 4246: 4242: 4238: 4235: 4233:0-8247-1748-1 4229: 4225: 4220: 4217: 4216:0-88385-522-4 4213: 4209: 4205: 4201: 4197: 4194: 4192: 4188: 4184: 4181: 4178: 4175: 4173:0-486-63962-2 4169: 4165: 4160: 4157: 4153: 4150: 4146: 4143: 4139: 4135: 4131: 4124: 4123: 4119: 4111: 4105: 4102: 4097: 4091: 4088: 4085: 4081: 4076: 4073: 4069: 4065: 4061: 4056: 4053: 4050: 4049:0-387-90332-1 4046: 4042: 4037: 4034: 4031: 4025: 4022: 4019:(Wikisource). 4018: 4012: 4009: 4003: 4000: 3996: 3992: 3988: 3983: 3980: 3976: 3975:Richards 1988 3970: 3967: 3964: 3960: 3956: 3952: 3948: 3944: 3940: 3934: 3931: 3927: 3921: 3915: 3911: 3907: 3901: 3898: 3894: 3888: 3885: 3881: 3875: 3872: 3868: 3867:0-691-08304-5 3864: 3860: 3856: 3850: 3847: 3844: 3838: 3835: 3829: 3826: 3822: 3816: 3813: 3809: 3806:according to 3805: 3799: 3796: 3792: 3786: 3783: 3779: 3775: 3771: 3765: 3762: 3758: 3754: 3748: 3745: 3740: 3738:9780821845950 3734: 3729: 3728: 3717: 3716: 3709:For example: 3706: 3703: 3699: 3696:Felix Klein, 3693: 3690: 3684: 3681: 3677: 3673: 3669: 3663: 3660: 3655: 3654: 3646: 3643: 3638: 3637:pages 243 ff. 3634: 3630: 3626: 3623: 3619: 3617: 3613: 3609: 3608: 3603: 3600: 3596: 3590: 3587: 3575: 3568: 3565: 3562: 3557: 3554: 3550: 3548: 3544: 3537: 3533: 3532: 3525: 3522: 3518: 3513: 3512:0-321-01618-1 3509: 3505: 3501: 3495: 3493: 3489: 3486: 3485:0-415-12411-5 3482: 3478: 3474: 3473: 3466: 3463: 3459: 3457: 3453: 3449: 3445: 3441: 3440: 3435: 3431: 3424: 3420: 3419: 3412: 3409: 3399: 3395: 3394: 3386: 3383: 3377: 3373: 3370: 3368: 3365: 3363: 3360: 3358: 3355: 3353: 3352:Lénárt sphere 3350: 3348: 3345: 3343: 3340: 3339: 3335: 3330: 3326: 3325: 3321: 3320:Ian Stewart's 3317: 3314: 3311:and fiction, 3310: 3306: 3302: 3298: 3295: 3291: 3288: 3284: 3280: 3279: 3275:Zeno Rogue's 3274: 3271: 3270: 3265: 3262: 3258: 3257: 3252: 3249: 3248: 3243: 3240: 3236: 3235: 3230: 3229:Robert Pirsig 3226: 3222: 3218: 3214: 3210: 3206: 3203: 3199: 3194: 3190: 3186: 3185: 3184: 3182: 3178: 3170: 3168: 3166: 3162: 3161: 3155: 3154:Gilbert Lewis 3151: 3147: 3142: 3129: 3126: 3120: 3117: 3114: 3111: 3105: 3102: 3099: 3093: 3090: 3087: 3084: 3075: 3072: 3069: 3066: 3060: 3057: 3048: 3044: 3035: 3011: 3006: 3000: 2993: 2987: 2980: 2974: 2969: 2962: 2957: 2951: 2946: 2941: 2934: 2931: 2922: 2919: 2912: 2903: 2902: 2901: 2899: 2898:shear mapping 2883: 2880: 2871: 2866: 2863: 2860: 2857: 2854: 2851: 2842: 2833: 2817: 2814: 2811: 2806: 2802: 2797: 2794: 2791: 2788: 2785: 2782: 2779: 2772: 2767: 2765: 2761: 2760:Lorentz boost 2758:amounts to a 2757: 2753: 2750: 2746: 2742: 2739: 2734: 2732: 2728: 2724: 2720: 2716: 2712: 2708: 2704: 2700: 2696: 2692: 2688: 2680: 2678: 2672: 2668: 2664: 2660: 2655: 2653: 2640: 2638: 2632: 2628: 2604: 2600: 2596: 2591: 2587: 2583: 2572: 2569: 2566: 2552: 2549: 2546: 2540: 2535: 2531: 2527: 2520: 2519: 2518: 2515: 2510: 2497: 2495: 2489: 2485: 2475: 2454: 2450: 2446: 2441: 2437: 2433: 2427: 2424: 2421: 2418: 2409: 2406: 2403: 2400: 2394: 2389: 2385: 2381: 2374: 2373: 2372: 2361: 2357: 2353: 2349: 2345: 2341: 2314: 2311: 2308: 2305: 2302: 2296: 2293: 2290: 2281: 2278: 2271: 2270: 2269: 2267: 2263: 2259: 2251: 2249: 2247: 2246: 2241: 2240:Lewis Carroll 2237: 2236: 2231: 2227: 2222: 2220: 2216: 2212: 2208: 2203: 2201: 2200:Immanuel Kant 2196: 2194: 2190: 2186: 2178: 2172: 2167: 2164: 2163:summit angles 2160: 2156: 2152: 2149: 2145: 2141: 2137: 2133: 2129: 2128: 2127: 2124: 2122: 2119:(also called 2118: 2107: 2097: 2090: 2088: 2086: 2079: 2071: 2069: 2065: 2059: 2055: 2047: 2038: 2036: 2032: 2028: 2024: 2020: 2016: 2012: 2008: 2004: 2000: 1994: 1986: 1984: 1978: 1970: 1961: 1959: 1955: 1951: 1947: 1943: 1939: 1938:great circles 1935: 1929: 1921: 1919: 1917: 1913: 1909: 1905: 1901: 1897: 1896:great circles 1893: 1888: 1886: 1882: 1878: 1874: 1870: 1866: 1862: 1858: 1854: 1850: 1841: 1833: 1826: 1821: 1817: 1805: 1802: 1789: 1788: 1787: 1785: 1781: 1777: 1772: 1770: 1766: 1762: 1758: 1754: 1750: 1741: 1739: 1737: 1731: 1728: 1726: 1721: 1718: 1714: 1710: 1706: 1702: 1701:Arthur Cayley 1698: 1696: 1692: 1688: 1684: 1676: 1674: 1672: 1667: 1665: 1661: 1657: 1653: 1649: 1645: 1641: 1637: 1635: 1630: 1625: 1621: 1617: 1613: 1608: 1605: 1601: 1597: 1589: 1587: 1584: 1582: 1578: 1574: 1569: 1567: 1562: 1560: 1556: 1551: 1549: 1545: 1541: 1539: 1535: 1531: 1526: 1523: 1519: 1515: 1511: 1507: 1503: 1499: 1495: 1491: 1487: 1482: 1480: 1476: 1472: 1468: 1464: 1460: 1452: 1449: 1446: 1443: 1442: 1441: 1439: 1433: 1428: 1426: 1425: 1420: 1416: 1412: 1411: 1405: 1403: 1400: 1396: 1389: 1386: 1382: 1378: 1370: 1365: 1362: 1358: 1354: 1350: 1349: 1348: 1346: 1345:perpendicular 1341: 1335: 1327: 1323: 1315: 1307: 1299: 1295: 1291: 1287: 1279: 1277: 1275: 1271: 1267: 1263: 1259: 1255: 1251: 1247: 1243: 1239: 1235: 1223: 1218: 1216: 1211: 1209: 1204: 1203: 1201: 1200: 1189: 1186: 1184: 1181: 1180: 1179: 1178: 1173: 1168: 1165: 1163: 1160: 1158: 1155: 1153: 1150: 1148: 1145: 1143: 1140: 1138: 1135: 1133: 1130: 1128: 1125: 1123: 1120: 1118: 1115: 1114: 1113: 1112: 1107: 1102: 1099: 1097: 1094: 1092: 1089: 1087: 1084: 1082: 1079: 1077: 1074: 1072: 1069: 1067: 1064: 1063: 1062: 1061: 1056: 1051: 1048: 1046: 1043: 1041: 1038: 1036: 1033: 1031: 1028: 1026: 1023: 1021: 1018: 1016: 1013: 1011: 1008: 1006: 1003: 1001: 998: 996: 993: 992: 991: 990: 985: 980: 977: 975: 972: 970: 967: 965: 962: 960: 957: 955: 952: 950: 947: 946: 945: 944: 941: 937: 927: 926: 919: 916: 914: 911: 909: 906: 904: 901: 899: 896: 894: 891: 889: 886: 884: 881: 879: 876: 874: 871: 869: 866: 864: 861: 859: 856: 854: 851: 849: 846: 844: 841: 839: 836: 834: 831: 829: 826: 824: 821: 819: 816: 814: 811: 809: 806: 804: 801: 799: 796: 794: 791: 789: 786: 784: 781: 779: 776: 774: 771: 769: 766: 764: 761: 759: 756: 754: 751: 749: 746: 744: 741: 739: 736: 734: 731: 729: 726: 724: 721: 719: 716: 715: 707: 706: 703: 699: 692: 689: 687: 684: 683: 678: 672: 671: 664: 661: 659: 656: 654: 651: 649: 646: 644: 641: 639: 636: 634: 631: 629: 626: 622: 619: 618: 617: 614: 613: 610: 607: 606: 602: 596: 595: 584: 581: 579: 578:Circumference 576: 574: 571: 570: 569: 568: 565: 561: 556: 553: 551: 548: 547: 546: 545: 542: 541:Quadrilateral 538: 533: 530: 528: 525: 523: 520: 518: 515: 514: 513: 512: 509: 508:Parallelogram 505: 500: 497: 495: 492: 490: 487: 486: 485: 484: 481: 477: 472: 469: 467: 464: 462: 459: 458: 457: 456: 450: 444: 443: 436: 433: 429: 426: 424: 421: 420: 419: 416: 415: 411: 405: 404: 397: 394: 393: 389: 383: 382: 375: 372: 370: 367: 365: 362: 361: 358: 355: 353: 350: 347: 346:Perpendicular 343: 342:Orthogonality 340: 338: 335: 333: 330: 328: 325: 324: 321: 318: 317: 316: 306: 303: 302: 297: 296: 287: 284: 283: 282: 279: 277: 274: 272: 269: 267: 266:Computational 264: 262: 259: 255: 252: 251: 250: 247: 245: 242: 240: 237: 233: 230: 228: 225: 223: 220: 219: 218: 215: 211: 208: 206: 203: 202: 201: 198: 196: 193: 191: 188: 186: 183: 181: 178: 176: 173: 169: 166: 162: 159: 158: 157: 154: 153: 152: 151:Non-Euclidean 149: 147: 144: 143: 139: 133: 132: 125: 121: 118: 116: 113: 112: 110: 109: 105: 101: 97: 92: 88: 87: 84: 80: 71: 61: 56: 52: 48: 47: 41: 32: 31: 19: 5577: 5569: 5561: 5553: 5545: 5537: 5529: 5521: 5513: 5505: 5497: 5489: 5481: 5473: 5393: 5385: 5377: 5369: 5361: 5353: 5345: 5337: 5329: 5313:Science wars 5111:Epistemology 5065: 5042:Reflectivism 5002:Hermeneutics 4854:Declinations 4830:Antihumanism 4823:Perspectives 4704:Trigonometry 4647: 4508: 4445: 4423: 4402: 4382: 4379:Stewart, Ian 4362: 4344: 4331: 4323: 4314: 4302: 4270: 4267:Morris Kline 4253: 4240: 4223: 4199: 4182: 4163: 4155: 4148: 4129: 4104: 4090: 4075: 4055: 4041:Isaak Yaglom 4036: 4024: 4011: 4002: 3990: 3987:Isaak Yaglom 3982: 3969: 3938: 3933: 3909: 3900: 3893:Trudeau 1987 3887: 3879: 3874: 3858: 3849: 3842: 3837: 3828: 3815: 3803: 3798: 3791:Manning 1963 3785: 3769: 3764: 3756: 3752: 3747: 3726: 3714: 3705: 3697: 3692: 3683: 3662: 3652: 3645: 3632: 3616:pp. 180–182. 3611: 3602: 3594: 3589: 3579:16 September 3577:. Retrieved 3567: 3556: 3546: 3542: 3540: 3529: 3524: 3516: 3499: 3470: 3465: 3451: 3443: 3437: 3427: 3416: 3411: 3401:, retrieved 3392: 3385: 3322: 3299:setting for 3276: 3267: 3261:pseudosphere 3254: 3245: 3232: 3197: 3174: 3157: 3150:E. B. Wilson 3143: 3026: 2771:dual numbers 2768: 2763: 2755: 2751: 2740: 2735: 2722: 2684: 2656: 2641: 2630: 2626: 2622: 2513: 2498: 2487: 2483: 2471: 2371:is given by 2365:ε = −1 2362: 2355: 2351: 2347: 2337: 2255: 2243: 2233: 2223: 2204: 2197: 2182: 2174:is negative. 2170: 2162: 2158: 2125: 2120: 2114: 2081: 2066: 2057: 2052:, there are 2045: 2044:and a point 2039: 2011:pseudosphere 1996: 1976: 1968: 1967:and a point 1962: 1931: 1912:pseudosphere 1889: 1876: 1875:parallel to 1872: 1864: 1860: 1848: 1847: 1775: 1773: 1769:The Elements 1768: 1745: 1732: 1729: 1724: 1722: 1699: 1694: 1690: 1686: 1682: 1680: 1668: 1638: 1633: 1624:János Bolyai 1609: 1593: 1585: 1580: 1576: 1570: 1565: 1563: 1558: 1554: 1552: 1547: 1542: 1533: 1527: 1483: 1471:Omar Khayyám 1465:, including 1456: 1435: 1430: 1422: 1419:propositions 1414: 1408: 1406: 1393: 1342: 1333: 1325: 1313: 1305: 1304:and a point 1283: 1237: 1231: 1050:Parameshvara 863:Parameshvara 633:Dodecahedron 217:Differential 150: 59: 44: 5668:Objectivity 5637:Karl Popper 5627:Thomas Kuhn 5607:Mario Bunge 5358:(1879–1884) 5293:(1909–1959) 5027:Metaphysics 5007:Historicism 4922:Demarcation 4917:Consilience 4840:Rationalism 4395:(softcover) 4384:Flatterland 4303:Pangeometry 3961:." — 3906:Bell, E. T. 3676:Faber (1983 3329:protagonist 3324:Flatterland 3189:H. G. Wells 2711:submanifold 2703:proper time 2652:dual number 2494:unit circle 2215:Lobachevsky 2140:right angle 2085:anisotropic 2023:Klein model 1717:Felix Klein 1709:cross-ratio 1677:Terminology 1522:John Wallis 1336:intersects 1234:mathematics 1175:Present day 1122:Lobachevsky 1109:1700s–1900s 1066:Jyeṣṭhadeva 1058:1400s–1700s 1010:Brahmagupta 833:Lobachevsky 813:Jyeṣṭhadeva 763:Brahmagupta 691:Hypersphere 663:Tetrahedron 638:Icosahedron 210:Diophantine 5448:Ernst Mach 5443:Ernst Laas 5418:A. J. Ayer 5406:Proponents 5225:Philosophy 5022:Humanities 4966:Antitheses 4835:Empiricism 4812:Positivism 4688:Riemannian 4683:Projective 4668:Symplectic 4663:Hyperbolic 4596:Euclidean 4499:PlanetMath 4120:References 4070:48:387–507 3951:Copernicus 3821:Smart 1997 3808:Smart 1997 3672:Faber 1983 3668:Faber 1983 3403:2008-01-23 3278:HyperRogue 3237:mentioned 2687:kinematics 2219:Copernicus 2179:Importance 2148:rectangles 2054:infinitely 2035:horosphere 1869:infinitely 1498:hyperbolic 1390:Background 1375:See also: 1356:parallels. 1322:infinitely 1280:Principles 1035:al-Yasamin 979:Apollonius 974:Archimedes 964:Pythagoras 954:Baudhayana 908:al-Yasamin 858:Pythagoras 753:Baudhayana 743:Archimedes 738:Apollonius 643:Octahedron 494:Hypotenuse 369:Similarity 364:Congruence 276:Incidence 227:Symplectic 222:Riemannian 205:Arithmetic 180:Projective 168:Hyperbolic 96:Projecting 5689:Verstehen 5675:Phronesis 5663:Knowledge 5647:Max Weber 5467:Criticism 5215:Sociology 5153:Modernism 5131:pluralism 5116:anarchism 5012:Historism 4932:Induction 4845:Scientism 4709:Lie group 4673:Spherical 4128:, (2012) 3823:, p. 366) 3770:parabolic 3536:Routledge 3477:Routledge 3423:Routledge 3283:roguelike 3187:In 1895, 3127:ϵ 3094:ϵ 3076:ϵ 3058:ϵ 3053:′ 3040:′ 2876:′ 2847:′ 2803:ϵ 2795:ϵ 2699:worldline 2689:with the 2597:− 2570:− 2536:∗ 2428:ϵ 2422:− 2410:ϵ 2390:∗ 2315:∈ 2015:curvature 1946:meridians 1916:curvature 1904:meridians 1725:parabolic 1705:logarithm 1656:curvature 1648:manifolds 1620:Hungarian 1530:Aristotle 1459:geometers 1413:. In the 1152:Minkowski 1071:Descartes 1005:Aryabhata 1000:Kātyāyana 931:by period 843:Minkowski 818:Kātyāyana 778:Descartes 723:Aryabhata 702:Geometers 686:Tesseract 550:Trapezoid 522:Rectangle 315:Dimension 200:Algebraic 190:Synthetic 161:Spherical 146:Euclidean 5723:Category 5710:Category 5126:nihilism 5121:idealism 5051:Related 4927:Evidence 4770:Category 4658:Elliptic 4650:geometry 4631:Polyform 4616:Discrete 4598:geometry 4579:Timeline 4563:Geometry 4343:(1988), 3943:Vesalius 3941:: "What 3908:(1986). 3843:Elements 3810:, p. 416 3547:Elements 3456:Saccheri 3336:See also 2935:′ 2923:′ 2749:rapidity 2476:between 1857:parallel 1780:negation 1761:Birkhoff 1715:because 1693:to mean 1571:In 1766 1534:Elements 1500:and the 1415:Elements 1410:Elements 1353:distance 1286:parallel 1142:Poincaré 1086:Minggatu 1045:Yang Hui 1015:Virasena 903:Yang Hui 898:Virasena 868:Poincaré 848:Minggatu 628:Cylinder 573:Diameter 532:Rhomboid 489:Altitude 480:Triangle 374:Symmetry 352:Parallel 337:Diagonal 307:Features 304:Concepts 195:Analytic 156:Elliptic 138:Branches 124:Timeline 83:Geometry 5590:Critics 5315:(1990s) 5309:(1980s) 5303:(1960s) 5283:(1890s) 5136:realism 5068:(1830s) 5056:in the 4626:Polygon 4574:History 4526:WebCite 4444:(2014) 4401:(1996) 4381:(2001) 4301:(2010) 4269:(1972) 4198:(1942) 4084:WebCite 3989:(1968) 3955:Ptolemy 3953:was to 3949:, what 3945:was to 3759:(1871). 3305:wargame 3292:In the 3224:insane. 3211:writer 3181:fantasy 3171:Fiction 2646:, then 2635:is the 2503:, then 2492:is the 2209:in the 1944:or the 1942:equator 1902:or the 1900:equator 1816:Riemann 1757:Hilbert 1612:Russian 1566:logical 1518:Alfonso 1371:History 1288:lines. 1167:Coxeter 1147:Hilbert 1132:Riemann 1081:Huygens 1040:al-Tusi 1030:Khayyám 1020:Alhazen 987:1–1400s 888:al-Tusi 873:Riemann 823:Khayyám 808:Huygens 803:Hilbert 773:Coxeter 733:Alhazen 711:by name 648:Pyramid 527:Rhombus 471:Polygon 423:segment 271:Fractal 254:Digital 239:Complex 120:History 115:Outline 55:Discuss 5582:(1986) 5574:(1980) 5566:(1978) 5558:(1968) 5550:(1964) 5542:(1963) 5534:(1962) 5526:(1960) 5518:(1951) 5510:(1942) 5502:(1936) 5494:(1934) 5486:(1923) 5478:(1909) 5398:(2001) 5390:(1959) 5382:(1936) 5374:(1927) 5366:(1886) 5350:(1869) 5342:(1848) 5334:(1830) 5270:Method 5143:Holism 5074:(1927) 4678:Affine 4611:Convex 4455: 4432: 4412: 4391: 4369: 4351: 4291: 4260: 4230: 4214: 4189: 4170: 4136: 4062:& 4047: 3997:, N.Y. 3959:Euclid 3916: 3865: 3776:: see 3735: 3510: 3483: 3430:Witelo 3221:R'lyeh 3198:Fulmar 2633:* = 1} 2501:ε = +1 2490:* = 1} 2340:points 2171:defect 2144:obtuse 1827:Models 1671:tensor 1654:, and 1510:Witelo 1402:Euclid 1383:, and 1290:Euclid 1242:axioms 1188:Gromov 1183:Atiyah 1162:Veblen 1157:Cartan 1127:Bolyai 1096:Sakabe 1076:Pascal 969:Euclid 959:Manava 893:Veblen 878:Sakabe 853:Pascal 838:Manava 798:Gromov 783:Euclid 768:Cartan 758:Bolyai 748:Atiyah 658:Sphere 621:cuboid 609:Volume 564:Circle 517:Square 435:Length 357:Vertex 261:Convex 244:Finite 185:Affine 100:sphere 51:merged 5682:Truth 4740:Lists 4735:Shape 4728:Lists 4697:Other 4586:Lists 4507:from 3947:Galen 3612:Werke 3378:Notes 3281:is a 2705:into 2667:angle 2659:slope 2650:is a 2644:ε = 0 2642:When 2507:is a 2262:plane 2217:the " 2189:space 2136:acute 1950:globe 1948:on a 1908:globe 1906:on a 1892:plane 1844:180°. 1629:Gauss 1137:Klein 1117:Gauss 1091:Euler 1025:Sijzi 995:Zhang 949:Ahmes 913:Zhang 883:Sijzi 828:Klein 793:Gauss 788:Euler 728:Ahmes 461:Plane 396:Point 332:Curve 327:Angle 104:plane 102:to a 4515:and 4453:ISBN 4430:ISBN 4410:ISBN 4389:ISBN 4367:ISBN 4349:ISBN 4289:ISBN 4258:ISBN 4228:ISBN 4212:ISBN 4187:ISBN 4168:ISBN 4134:ISBN 4045:ISBN 3914:ISBN 3891:see 3863:ISBN 3733:ISBN 3581:2011 3541:"In 3508:ISBN 3481:ISBN 3327:the 3301:FASA 3200:off 3179:and 3152:and 2725:and 2701:and 2623:and 2338:The 2159:base 1883:and 1851:are 1776:must 1538:Rome 1492:and 1264:and 1252:and 1101:Aida 718:Aida 677:Four 616:Cube 583:Area 555:Kite 466:Area 418:Line 4511:of 4497:at 3436:'s 3318:In 3303:'s 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Index