In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, but did not realize it. In Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure that today we call a Lambert quadrilateral , a quadrilateral with three right angles can be considered half of a Saccheri quadrilateral. He quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle.

Unlike Saccheri, he never felt that he had reached a contradiction with this assumption. He had proved the non-Euclidean result that the sum of the angles in a triangle increases as the area of the triangle decreases, and this led him to speculate on the possibility of a model of the acute case on a sphere of imaginary radius.

He did not carry this idea any further. At this time it was widely believed that the universe worked according to the principles of Euclidean geometry. The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry. Circa , Carl Friedrich Gauss and independently around , the German professor of law Ferdinand Karl Schweikart [9] had the germinal ideas of non-Euclidean geometry worked out, but neither published any results. Schweikart's nephew Franz Taurinus did publish important results of hyperbolic trigonometry in two papers in and , yet while admitting the internal consistency of hyperbolic geometry, he still believed in the special role of Euclidean geometry.

Consequently, hyperbolic geometry is called Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors of non-Euclidean geometry. Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before, [11] though he did not publish. Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences.

Bernhard Riemann , in a famous lecture in , founded the field of Riemannian geometry , discussing in particular the ideas now called manifolds , Riemannian metric , and curvature. He constructed an infinite family of geometries which are not Euclidean by giving a formula for a family of Riemannian metrics on the unit ball in Euclidean space. The simplest of these is called elliptic geometry and it is considered to be a non-Euclidean geometry due to its lack of parallel lines. By formulating the geometry in terms of a curvature tensor , Riemann allowed non-Euclidean geometry to be applied to higher dimensions.

Beltrami was the first to apply Riemann's geometry to spaces of negative curvature. It was Gauss who coined the term "non-Euclidean geometry". Several modern authors still consider "non-Euclidean geometry" and "hyperbolic geometry" to be synonyms. Arthur Cayley noted that distance between points inside a conic could be defined in terms of logarithm and the projective cross-ratio function. The method has become called the Cayley-Klein metric because Felix Klein exploited it to describe the non-euclidean geometries in articles [14] in and 73 and later in book form.

The Cayley-Klein metrics provided working models of hyperbolic and elliptic metric geometries, as well as Euclidean geometry. Klein is responsible for the terms "hyperbolic" and "elliptic" in his system he called Euclidean geometry "parabolic", a term which generally fell out of use [15]. His influence has led to the current usage of the term "non-Euclidean geometry" to mean either "hyperbolic" or "elliptic" geometry.

There are some mathematicians who would extend the list of geometries that should be called "non-Euclidean" in various ways. Euclidean geometry can be axiomatically described in several ways. Unfortunately, Euclid's original system of five postulates axioms is not one of these as his proofs relied on several unstated assumptions which should also have been taken as axioms.

Hilbert's system consisting of 20 axioms [17] most closely follows the approach of Euclid and provides the justification for all of Euclid's proofs. Other systems, using different sets of undefined terms obtain the same geometry by different paths. In all approaches, however, there is an axiom which is logically equivalent to Euclid's fifth postulate, the parallel postulate. Hilbert uses the Playfair axiom form, while Birkhoff , for instance, uses the axiom which says that "there exists a pair of similar but not congruent triangles.

As the first 28 propositions of Euclid in The Elements do not require the use of the parallel postulate or anything equivalent to it, they are all true statements in absolute geometry.

## Non-Euclidean Geometry -- from Wolfram MathWorld

To obtain a non-Euclidean geometry, the parallel postulate or its equivalent must be replaced by its negation. Negating the Playfair's axiom form, since it is a compound statement Two dimensional Euclidean geometry is modelled by our notion of a "flat plane. The simplest model for elliptic geometry is a sphere, where lines are " great circles " such as the equator or the meridians on a globe , and points opposite each other called antipodal points are identified considered to be the same. This is also one of the standard models of the real projective plane.

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. Even after the work of Lobachevsky, Gauss, and Bolyai, the question remained: "Does such a model exist for hyperbolic geometry? The model for hyperbolic geometry was answered by Eugenio Beltrami , in , who first showed that a surface called the pseudosphere has the appropriate curvature to model a portion of hyperbolic space and in a second paper in the same year, defined the Klein model which models the entirety of hyperbolic space, and used this to show that Euclidean geometry and hyperbolic geometry were equiconsistent so that hyperbolic geometry was logically consistent if and only if Euclidean geometry was.

The reverse implication follows from the horosphere model of Euclidean geometry. In these models the concepts of non-Euclidean geometries are being represented by Euclidean objects in a Euclidean setting. This introduces a perceptual distortion wherein the straight lines of the non-Euclidean geometry are being represented by Euclidean curves which visually bend.

This "bending" is not a property of the non-Euclidean lines, only an artifice of the way they are being represented. In three dimensions, there are eight models of geometries. Euclidean and non-Euclidean geometries naturally have many similar properties, namely those which do not depend upon the nature of parallelism.

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This commonality is the subject of absolute geometry also called neutral geometry. However, the properties which distinguish one geometry from the others are the ones which have historically received the most attention. Besides the behavior of lines with respect to a common perpendicular, mentioned in the introduction, we also have the following:. Furthermore, since the substance of the subject in synthetic geometry was a chief exhibit of rationality, the Euclidean point of view represented absolute authority.

The discovery of the non-Euclidean geometries had a ripple effect which went far beyond the boundaries of mathematics and science. The philosopher Immanuel Kant 's treatment of human knowledge had a special role for geometry. Unfortunately for Kant, his concept of this unalterably true geometry was Euclidean.

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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. Non-Euclidean geometry is an example of a scientific revolution in the history of science , in which mathematicians and scientists changed the way they viewed their subjects.

The existence of non-Euclidean geometries impacted the intellectual life of Victorian England in many ways [26] and in particular was one of the leading factors that caused a re-examination of the teaching of geometry based on Euclid's Elements. This curriculum issue was hotly debated at the time and was even the subject of a book, Euclid and his Modern Rivals , written by Charles Lutwidge Dodgson — better known as Lewis Carroll , the author of Alice in Wonderland.

For planar algebra, non-Euclidean geometry arises in the other cases. This approach to non-Euclidean geometry explains the non-Euclidean angles: the parameters of slope in the dual number plane and hyperbolic angle in the split-complex plane correspond to angle in Euclidean geometry. Indeed, they each arise in polar decomposition of a complex number z. Hyperbolic geometry found an application in kinematics with the physical cosmology introduced by Hermann Minkowski in Minkowski introduced terms like worldline and proper time into mathematical physics.

He realized that the submanifold , of events one moment of proper time into the future, could be considered a hyperbolic space of three dimensions. The relevant structure is now called the hyperboloid model of hyperbolic geometry. The non-Euclidean planar algebras support kinematic geometries in the plane. Furthermore, multiplication by z amounts to a Lorentz boost mapping the frame with rapidity zero to that with rapidity a.

Another view of special relativity as a non-Euclidean geometry was advanced by E.

They revamped the analytic geometry implicit in the split-complex number algebra into synthetic geometry of premises and deductions. Non-Euclidean geometry often makes appearances in works of science fiction and fantasy. Three scientists, Ibn al-Haytham, Khayyam and al-Tusi, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the nineteenth century.

In essence their propositions concerning the properties of quadrangles 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 proofs put forward in the fourteenth century by the Jewish scholar Levi ben Gerson , who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Wallis's and G. Free Shipping Free global shipping No minimum order. This book will be of great value to mathematics, liberal arts, and philosophy major students.

## Introduction to Non-Euclidean Geometry by Wolfe

Introduction 2. Euclid's Stated Assumptions 3. The Extent of a Straight Line 4. Euclid's Theory of Parallels 5. Further Consequences of Postulate 5 6. Euclid's Choices 3. Posidonius and His Followers 4. Ptolemy and Proclus 5.

Saccheri 6. Lambert 7.

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Legendre 8. The Basis E 3. The Initial Theorems of Hyperbolic Geometry 4. The Hyperbolic Parallel Postulate 5. Immediate Consequences of the Postulate 6. Further Properties of Quadrilaterals 7. Parallels With a Common Perpendicular 8. The Angle-Sum of a Triangle 9. The Defect of a Triangle Quadrilaterals Associated with a Triangle The Equivalence of Triangles Area of a Triangle Implications of the Area Formula Parallels Without a Common Perpendicular 3. Properties of Boundary Parallels 4.

## Advanced Euclidean Geometry

Trilateral 5. Angles of Parallelism 6. Distance Between Two Lines 7. Corresponding Points 3. Definition of a Horocycle 4. Arcs and Chords of a Horocycle 5. Codirectional Horocycles 6. Arc Length on a Horocycle 7. Associated Right Triangles 3. Improved Angle of Parallelism Formulas 4. Remarks on the Trigonometric Functions 5. Right Triangle Formulas 6. Comparison with Euclidean Formulas 7.