41. Non-Euclidean Geometry The use and assumption of these five axioms is what it means for something to be categorized as euclidean geometry, which is obviously named after Euclid, http://www.geocities.com/CapeCanaveral/7997/noneuclid.html

Non-Euclidean Geometry Introduction: Unlike other branches of math, geometry has been connected with two purposes since the ancient Greeks. Not only is it an intellectual discipline, but also, it has been considered an accurate description of our physical space. However in order to talk about the different types of geometries, we must not confuse the term geometry with how physical space really works. Geometry was devised for practical purposes such as constructions, and land surveying. Ancient Greeks, such as Pythagoras (around 500 BC) used geometry, but the various geometric rules that were being passed down and inherited were not well connected. So around 300 BC, Euclid was studying geometry in Alexandria and wrote a thirteen-volume book that compiled all the known and accepted rules of geometry called The Elements, and later referred to as Euclid’s Elements. Because math was a science where every theorem is based on accepted assumptions, Euclid first had to establish some axioms with which to use as the basis of other theorems. He used five axioms as the 5 assumptions, which he needed to prove all other geometric ideas. The use and assumption of these five axioms is what it means for something to be categorized as Euclidean geometry, which is obviously named after Euclid, who literally wrote the book on geometry. The first four of his axioms are fairly straightforward and easy to accept, and no mathematician has ever seriously doubted them. The first four of Euclid’s axioms are:

42. Non-Euclidean Geometry - The History Of Non-euclidean Geometry, The Founders Of Noneuclidean geometry refers to certain types of geometry which differ from plane and solid geometry which dominated the realm of mathematics for several http://science.jrank.org/pages/4705/Non-Euclidean-Geometry.html

Other Free Encyclopedias Science Encyclopedia Science Encyclopedia Vol 4 Non-Euclidean Geometry - The History Of Non-euclidean Geometry, The Founders Of Non-euclidean Geometry, Elliptic Non-euclidean Geometry Non-Euclidean geometry refers to certain types of geometry which differ from plane and solid geometry which dominated the realm of mathematics algebra A straight line can be drawn from any point to any point. A finite straight line can be produced continuously in a straight line. A circle may be described with any point as center and any distance as a radius. All right angles are equal to one another. If a transversal falls on two lines in such a way that the interior angles on one side of the transversal are less than two right angles, then the lines meet on the side on which the angles are less than two right angles. A consistent logical system for which one of these postulates is modified in an essential way is non-Euclidean geometry. Although there are different types of Non-Euclidean geometry which do not use all of the postulates or make alterations of one or more of the postulates of Euclidean geometry, hyperbolic and elliptic are usually most closely associated with the term non-Euclidean geometry.

43. The Historical Importance Of Non-Euclidean Geometry The development of noneuclidean geometry caused a profound revolution, not just in mathematics, but in science and philosophy as well. http://www.dpmms.cam.ac.uk/~wtg10/historyetc.html

What is the historical importance of non-Euclidean geometry? I intend to write in more detail on this topic. For now, here is a brief summary. The development of non-Euclidean geometry caused a profound revolution, not just in mathematics, but in science and philosophy as well. The philosophical importance of non-Euclidean geometry was that it greatly clarified the relationship between mathematics, science and observation. Before hyperbolic geometry was discovered, it was thought to be completely obvious that Euclidean geometry correctly described physical space, and attempts were even made, by Kant and others, to show that this was necessarily true. Gauss was one of the first to understand that the truth or otherwise of Euclidean geometry was a matter to be determined by experiment, and he even went so far as to measure the angles of the triangle formed by three mountain peaks to see whether they added to 180. (Because of experimental error, the result was inconclusive.) Our present-day understanding of models of axioms, relative consistency and so on can all be traced back to this development, as can the separation of mathematics from science. The scientific importance is that it paved the way for Riemannian geometry, which in turn paved the way for Einstein's General Theory of Relativity. After Gauss, it was still reasonable to think that, although Euclidean geometry was not

44. Euclidean euclidean geometry. This page is created for my students to use in conjunction with Geometry for Enjoyment and Challenge , published by McDougal, http://www.mccallie.org/myates/Proofs/default.htm

Site Map EUCLIDEAN GEOMETRY This page is created for my students to use in conjunction with "Geometry for Enjoyment and Challenge", published by McDougal, Littell Inc., a division of Houghton Mifflin and is used with their permission. Many of the proofs are taken from this text and its support materials. Permission given by Ron Worman, Permissions Editor. This page will help you practice proofs, hopefully. Sometimes everything works beautifully. Sometimes everything is totally messed up. As with most computer glitches, I have no idea what is wrong, but I do know that if you exit and then reenter, it often works. This is the same technique as turning your computer off, then back on, and then finding out that your "problem" has been miraculously fixed. Good luck. You should try to complete the proof entirely on your own. If that is not possible, then you should click on one of the drop-down boxes for part of the answer. Then try to finish the rest of the proof on your own, and so on. Do not merely click on drop-down boxes and say to yourself "hmm that makes sense". Watching someone do something is an entirely different thing than doing it yourself. It is easy to fool yourself into thinking you know what you are doing. The best way to do math is to do it , the check yourself, then do another, and another, and another. It is ironic that students as athletes will practice a move thousands of times, but students as students will balk at only a few repetitions.

45. Key College Publishing: Posamentier/Advanced Euclidean Geometry Advanced euclidean geometry fills this void by providing a thorough review of the essentials of the high school geometry course and then expanding those http://www.keycollege.com/catalog/titles/advanced_euclidean_geom.html

Home Customer Service Ordering Information Contact Us ... Site Map Product Information Mathematics Products Statistics Products Mathematics Education Software Products ... Author Web Sites Resource Centers Data Matters DevMAP Fathom Dynamic Data The Geometer's Sketchpad ... Other Titles Resource Store for Future Teachers Supplemental Materials Core Resource and Secondary Texts Software Resources General Information Ordering Information Get Involved Professional Development Customer Service ... Press Room @ Keypress.com Other Key Sites Key Curriculum Press Keymath.com KCP Technologies Advanced Euclidean Geometry: Excursions for Students and Teachers With Illustrations in Alfred S. Posamentier, City College, The City University of New York Advanced Euclidean Geometry fills this void by providing a thorough review of the essentials of the high school geometry course and then expanding those concepts to advanced Euclidean geometry, to give teachers more confidence in guiding student explorations and questions. The text contains hundreds of illustrations created in Sketchpad This title is available through your college bookstore and is also available packaged with a specially priced Student Bundle Package of version 4. Contact your Key College Publishing sales representative.

46. Euclidean Geometry - Simple English Wikipedia, The Free Encyclopedia euclidean geometry is a system in mathematics. People think Euclid was the first person who described it. Therefore it bears his name. http://simple.wikipedia.org/wiki/Euclidean_geometry

Euclidean geometry From the Simple English Wikipedia, the free encyclopedia that anyone can change Jump to: navigation search Euclidean geometry is a system in mathematics . People think Euclid was the first person who described it. Therefore it bears his name. He first described it in his textbook Elements . The book was the first systematic discussion of geometryat it was known at the time. In the book, Euclid first assumes a few axioms . These form the base for later work. They are intuitively clear. Starting from those axioms, other theorems can be proven In the 19th century other forms of geometry were found. These are non-Euclidean. Carl Friedrich Gauss J¡nos Bolyai , and Nikolai Ivanovich Lobachevsky were some people that developed such geometries. change The axioms Euclid makes the following assumptions. These are axioms, and need not be proved. Any two points can be joined by a straight line Any straight line segment can be made longer (extended) to infinity, so it becomes a straight line. With a straight line segment it is possible to draw a circle, so that one endpoint of the segmenr is the center of the circle, and the other endpoint lies on the circle. The line segment becomes the radius of the circle. All right angles are congruent Parallel postulate . If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

47. Euclidean Geometry --  Britannica Online Encyclopedia euclidean geometry Study of points, lines, angles, surfaces, and solids based on Euclid s axioms. http://concise.britannica.com/ebc/article-9363978/Euclidean-geometry

var britAdCategory = "other"; Already a member? LOGIN Encyclopædia Britannica - the Online Encyclopedia Home Blog Advocacy Board ... Free Trial Britannica Online Content Related to this Topic This Article's Table of Contents Introduction Fundamentals Plane geometry Congruence of triangles Similarity of triangles Areas Pythagorean theorem ... Print this Table of Contents Linked Articles Carl Friedrich Gauss Nikolay Lobachevsky non-Euclidean, geometries Shopping New! Britannica Book of the Year The Ultimate Review of 2007. 2007 Britannica Encyclopedia Set (32-Volume Set) Revised, updated, and still unrivaled. New! Britannica 2008 Ultimate DVD/CD-ROM The world's premier software reference source. Euclidean geometry Page 1 of 12 the study of plane and solid figures on the basis of axioms and theorems employed by the Greek mathematician Euclid c. BC ). In its rough outline, Euclidean geometry is the plane and solid geometry commonly taught in secondary schools. Indeed, until the second half of the 19th century, when non-Euclidean geometries Euclidean geometry...

48. Non-Euclidean Geometry@Everything2.com It all starts with euclidean geometry. Euclid proposed various axioms or postulates in his Elements and starting from those deduced the familiar theorems of http://everything2.com/index.pl?node_id=1143021

49. [gr-qc/0407022] Spacetime And Euclidean Geometry Using only the principle of relativity and euclidean geometry we show in this pedagogical article that the square of proper time or length in a http://arxiv.org/abs/gr-qc/0407022

arXiv.org gr-qc Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text Help pages Full-text links: Download: PostScript PDF Other formats Citations SLAC-SPIRES HEP (refers to , cited by , arXiv reformatted) CiteBase gr-qc new recent General Relativity and Quantum Cosmology Title: Spacetime and Euclidean Geometry Authors: Dieter Brill Ted Jacobson (Submitted on 6 Jul 2004 ( ), last revised 4 Aug 2004 (this version, v2)) Abstract: Using only the principle of relativity and Euclidean geometry we show in this pedagogical article that the square of proper time or length in a two-dimensional spacetime diagram is proportional to the Euclidean area of the corresponding causal domain. We use this relation to derive the Minkowski line element by two geometric proofs of the "spacetime Pythagoras theorem". Comments: 11 pages, 9 figures; for a festschrift honoring Michael P. Ryan; v.2: References to related work added Subjects: General Relativity and Quantum Cosmology (gr-qc) ; High Energy Physics - Theory (hep-th); Physics Education (physics.ed-ph) Journal reference: Gen.Rel.Grav. 38 (2006) 643-651

50. Euclidean Geometry And Informal Geometry This outline details objectives for two courses euclidean geometry and Informal Geometry. Informal Geometry is a college preparatory course encompassing http://www.glc.k12.ga.us/pandp/math/infgeo.htm

Home QCC Standards and Resources Search GLC Teacher Resource Center ... Site Tour You are here: GLC Home Projects and Programs Mathematics Courses in Mathematics Introduction and Description for Courses in Mathematics Quality Core Curriculum 9-12 Euclidean Geometry and Informal Geometry Geometry provides students with a way to link their perceptions of the world with the mathematics that allow them to solve a variety of problems they will encounter not only in other disciplines but also in their lives. Geometry gives students a visual way to conceptualize or organize certain aspects of their environment, whereas algebra provides the tools for dealing with the quantitative aspects of their environment. Geometry should provide students with visual and concrete representations that help them gain insight into important areas of mathematics and their applications. The use of such tools as compass, straightedge, tracing and dot paper, mira, geoboard, calculator and computer is strongly recommended and encouraged for all geometry courses. High school geometry must extend beyond the traditional treatment of geometry as a deductive system and provide students with a broad view of geometry and its applications, including algebraic techniques associated with coordinates and transformations that reinforce important geometric concepts such as congruence, similarity, parallelism, symmetry, and perpendicularity. Thus, the integration of algebraic skills and concepts to solve geometric problems should be stressed throughout the course.

51. Computer-Generated Encyclopedia Of Euclidean Geometry Download - Computer-genera ComputerGenerated Encyclopedia of euclidean geometry download page. Computer-generated encyclopedia. http://www.topshareware.com/Computer-Generated-Encyclopedia-of-Euclidean-Geometr

contact us home software catalogs new downloads ... Computer-Generated Encyclopedia of Euclidean Geometry Computer-Generated Encyclopedia of Euclidean Geometry 1.01 COMPUTER-GENERATED ENCYCLOPEDIA OF EUCLIDEAN GEOMETRY SNAPSHOT Computer-generated encyclopedia. Windows 95/98/Me/NT/2000/XP Platform : USD $15 Price : 1.78 MB File Size : Screenshot Screenshot : Popularity : Date Added : Click Here for Support Technical Support : Write Your Review User Reviews : The first into the world computer-generated encyclopedia of mathematics and the first into the world computer-generated encyclopedia. The first part of the encyclopedia contains more than 3000 theorems in Euclidean Geometry, all of them produced by computer. The encyclopedia is the most complete source in Euclidean Geometry. Many of theorems are new, discovered by computer. DOWNLOADS FROM DEKOV SOFTWARE Geometric Constructions Mathematical program for middle and high students and teachers. Inverse Matrices Mathematical program for university students. Linear Systems Mathematical program for university students. Solving Triangles Mathematical program for high school students and teachers.

52. Janos Bolyai, Non-Euclidean Geometry, And The Nature Of Space An account of the major work of Janos Bolyai, a nineteenthcentury mathematician who set the stage for the field of non-euclidean geometry. http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=9626

53. Non-Euclidean Geometry The historical developments of noneuclidean geometry were attempts to deal with the fifth axiom. Mathematicians first tried to directly prove that the http://www.lycos.com/info/non-euclidean-geometry--axioms.html

var topic_urlstring = 'non-euclidean-geometry'; var topic = 'Non-Euclidean Geometry'; var subtopic_urlstring= ''; LYCOS RETRIEVER Retriever Home What is Lycos Retriever? Non-Euclidean Geometry built 1 day ago Retriever Science Math Geometry Sum of Angles of a Triangle in Non-Euclidean Geometry [08/01/2001] In a triangle on the surface of a sphere the sum of the angles is not 180 degrees. Is that possible? Why? Source: mathforum.org Coxeter's Non-Euclidean Geometry begins with a wonderful historical overview of the development of non-Euclidean geometry in the first chapter. Only a few proofs are given or sketched in this chapter. They flow with the prose and play an integral part in the understanding of the beginnings of hyperbolic, spherical, elliptic and differential geometry, among others. The mathematician, as well as the non-mathematician, is able to gain insights into these various types of geometry by the end of this chapter. Source: maa.org Non-Euclidean Geometry and discuss various models of the non-Euclidean space with an account of the more striking geometric theorems. This will lead to various topics in non-Euclidean analysis, some of relatively recent vintage.

54. Area Entrance-- Plane Geometry Before Coordinates This area on euclidean geometry on geometry before coordinates offers thoughtbased explanation of the following. Try to read them in sequence. http://whyslopes.com/Euclidean-Geometry-Introduction/

Appetizers and Lessons for Mathematics and Reason ( www.whyslopes.com Définition d'une variable Algèbre Arithmetique ... La raison basée sur les règles et modelés Online Volumes Elements of Reason. Pattern Based Reason Math Curriculum Notes Three Skills for Algebra (Optional Book Orders More Site Areas Help Your Child or Teen Learn Solving Linear Equations Euclidean Geometry Analytic Geometry/Functions ... More Calculus More Site Areas Complex Numbers Qc Maths Education Secondary IV(?) math s Real Analysis Electric Circuits Etc Français More Site Areas Math Education Essays 19. Quantitative Skills for home, shopping and work 20. Statistics Useful, or Not. Test the Twiddla Whiteboard Next Site Exit YOU are better than YOU think. Show yourself how: o o Read logic chapters 1 to 5 in online volume Three Skills for Algebra and study Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason , Bon Appetite.

55. JSTOR Non-Euclidean Geometry, Historical And Expository NONeuclidean geometry, HISTORICAL AND EXPOSITORY. By GEORGE BRUCE HALSTED, A.M., (Princenton); Ph.D., (Johns Hopkins); Member of the London Mathematioal http://links.jstor.org/sici?sici=0002-9890(189403)1:3<70:NGHAE>2.0.CO;2-0

56. Mark Solomonovich - Geometry euclidean geometry. Why Study euclidean geometry? Why is this textbook written the way it is? Table of Contents and Samples. Some interesting problems. http://www.solomonovich.com/geometry/

EUCLIDEAN GEOMETRY Why Study Euclidean Geometry? Why is this textbook written the way it is? Table of Contents and Samples. Some interesting problems.

57. Exploring Euclidean Geometry EXPLORING ADVANCED euclidean geometry with Geometer s Sketchpad by Gerard A. Venema may be downloaded as a sequence of pdf documents by clicking on the http://www.calvin.edu/~venema/eeg/eeg.html

EXPLORING ADVANCED EUCLIDEAN GEOMETRY with Geometer's Sketchpad by Gerard A. Venema EXPLORING ADVANCED EUCLIDEAN GEOMETRY with Geometer's Sketchpad by Gerard A. Venema may be downloaded as a sequence of pdf documents by clicking on the links below. The size of each file is 0.2MB or less. There are a total of 121 pages in all. Cover page (1 page, 84KB) Table of Contents (3 pages, 80KB) Preface (5 pages, 102KB) Chapter 0. A quick review of elementary Euclidean geometry (10 pages, 174KB) Chapter 1. The elements of GSP (8 pages, 146KB) Chapter 2. The classical triangle centers (6 pages, 147KB) Chapter 3. Advanced techniques in GSP (5 pages, 122KB) Chapter 4. Circumscribed circles, inscribed circles, and escribed circles (7 pages, 152KB) Chapter 5. The medial and orthic triangles (6 pages, 148KB) Chapter 6. Quadrilaterals (4 pages, 134KB) Chapter 7. The nine-point circle (4 pages, 140KB) Chapter 8. Ceva's Theorem (14 pages, 202KB) Chapter 9. The Theorem of Menelaus (4 pages, 132KB) Chapter 10. Circles and lines

58. Geometry - MSN Encarta While the postulates of euclidean geometry do seem plausible when applied to physical Twodimensional euclidean geometry is often called plane geometry; http://encarta.msn.com/encyclopedia_761569706_2/Geometry.html

var s_account="msnportalencarta"; MSN home Mail My MSN Sign in ... more Hotmail Messenger My MSN MSN Directory Air Tickets/Travel Autos City Guides Election 2008 ... More Additional Reference Materials Thesaurus Translations Multimedia Other Resources Education Resources Math Help Foreign Language Help Project Planner ... Help Editors' Picks Great books about your topic, Geometry , selected by Encarta editors Related Items Math Homework Help Mathematics more... Encarta Search Search Encarta about Geometry Also on Encarta 7 tips for funding an online degree How to succeed in the fashion industry without being a top designer Presidential Myths Quiz Advertisement Page 2 of 6 Geometry Encyclopedia Article Find Print E-mail Blog It Multimedia 10 items Article Outline Introduction Methodology Euclidean Geometry Analytic Geometry ... History of Geometry C Postulates Postulates, or axioms, are unproven but universally accepted assumptions, such as “there is one and only one line that passes through two distinct points.” A system consisting of a set of noncontradictory postulates concerning the undefined terms point, line, and plane, together with the theorems deduced from these postulates, is called a geometry. Different sets of postulates determine whole different systems of geometry. If the postulates selected are suggested by experience with physical space, then it is reasonable to expect that the conclusions will also correspond closely to experiences related to space. However, since any set of postulates must be selected on the basis of incomplete and approximate observation, they quite possibly apply only approximately to actual space. Thus, it is no surprise if any particular geometry should turn out to be inapplicable, or only approximately applicable, to problems in actual space.

59. KSEG KSEG is a Free (GPL) interactive geometry program for exploring euclidean geometry. It runs on Unixbased platforms (according to users, it also compiles http://www.mit.edu/~ibaran/kseg.html

KSEG Free Interactive Geometry Software Update: May 1, 2007 Mongolian Batnasan sent me a translation into Mongolian. Update: Februrary 3, 2006 KSEG 0.403 and Traditional Chinese I finally got around to a bit of maintenance on KSEG: it now uses qmake for building and should build on Qt 3.x without the compatibility headers. Yao Chang-Li sent me a Traditional Chinese translation and help file . It is included in the .403 file. Update: May 15, 2005 KSEG in Chinese Xu Xianghua kindly contributed Chinese translations of the KSEG UI and the help. Download the UI translation here and the help here Update: July 3, 2004 KSEG 0.401 for Windows! By popular demand, despite my dislike for microsoft, I've ported KSEG to windows using the old Qt noncommercial version 2.3 (I hope I'm not violating anything). The source is uglier and there are some bugs not in the Linux version. Download the whole thing here (it should run out of the box). Sample Output Images (not screenshotsthose are below) of a well-known theorem and a strange locus, both exported with KSEG: Description: KSEG is a Free (GPL) interactive geometry program for exploring Euclidean geometry. It runs on Unix-based platforms (according to users, it also compiles and runs on Mac OS X and should run on anything that Qt supports). You create a construction, such as a triangle with a circumcenter, and then, as you drag verteces of the triangle, you can see the circumcenter moving in real time. Of course, you can do a lot more than thatsee the feature list below.

60. Mathematics - Non-Euclidean Geometry Mathematics Non-euclidean geometry Product information, price comparisons and user reviews and ratings for Mathematics - Non-euclidean geometry. http://shopping.msn.com/results/mathematics-non-euclidean-geometry/bcatid2521/fo

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