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1. Euclidean Geometry - Wikipedia, The Free Encyclopedia
euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid s text Elements is the earliest known
http://en.wikipedia.org/wiki/Euclidean_geometry
##### Euclidean geometry
Jump to: navigation search A representation of Euclid from The School of Athens by Raphael Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria . Euclid's text Elements is the earliest known systematic discussion of geometry . It has been one of the most influential books in history, as much for its method as for its mathematical content. The method consists of assuming a small set of intuitively appealing axioms , and then proving many other propositions theorems ) from those axioms. Although many of Euclid's results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could be fit together into a comprehensive deductive and logical system The Elements begin with plane geometry , still taught in secondary school as the first axiomatic system and the first examples of formal proof . The Elements goes on to the solid geometry of three dimensions , and Euclidean geometry was subsequently extended to any finite number of dimensions . Much of the Elements states results of what is now called number theory , proved using geometrical methods.

2. Euclidean Geometry -- From Wolfram MathWorld
A geometry in which Euclid s fifth postulate holds, sometimes also called parabolic geometry. Twodimensional euclidean geometry is called plane geometry,
http://mathworld.wolfram.com/EuclideanGeometry.html
 Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... General Geometry Euclidean Geometry A geometry in which Euclid's fifth postulate holds, sometimes also called parabolic geometry . Two-dimensional Euclidean geometry is called plane geometry , and three-dimensional Euclidean geometry is called solid geometry . Hilbert proved the consistency of Euclidean geometry. SEE ALSO: Elements Elliptic Geometry Geometric Construction Geometry ... Plane Geometry REFERENCES: Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, 1952. Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions with Numerous Examples, 2nd rev. enl. ed. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., 1967. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Dodge, C. W. Euclidean Geometry and Transformations. New York: Dover, 2004. Gallatly, W.

3. Introduction
The rest of euclidean geometry is based upon these basic building blocks. The fifth axiom, called the parallel postulate, has been very controversial.
http://math.youngzones.org/Non-Egeometry/index.html
 Euclidean Geometry Euclid (~ 325 to ~265 BC) is called the Father of Geometry, not because he invented it, but because his book Elements is the oldest geometry text we have in the western world today. Euclid believed each part of geometry should be based on previously proven parts. However, something has to be the starting point, and he proposed four undefined terms and five basic axioms. The rest of Euclidean geometry is based upon these basic building blocks. The fifth axiom, called the parallel postulate , has been very controversial. More complex than the other four, many mathematicians tried to prove it using the other axioms. Failure to do so developed into the existence of non-Euclidean geometries. (These pages best viewed in Internet Explorer)

4. NonEuclid: 1: Non-Euclidean Geometry
The geometry with which we are most familiar is called euclidean geometry. euclidean geometry was named after Euclid, a Greek mathematician who lived in 300
http://www.cs.unm.edu/~joel/NonEuclid/noneuclidean.html
##### 3: What is Non-Euclidean Geometry
1.1 Euclidean Geometry:
The geometry with which we are most familiar is called Euclidean geometry. Euclidean geometry was named after Euclid, a Greek mathematician who lived in 300 BC. His book, called "The Elements", is a collection of axioms, theorems and proofs about squares, circles acute angles, isosceles triangles, and other such things. Most of the theorems which are taught in high schools today can be found in Euclid's 2000 year old book. Euclidean geometry is of great practical value. It has been used by the ancient Greeks through modern society to design buildings, predict the location of moving objects and survey land.
1.2 Non-Euclidean Geometry: non-Euclidean geometry is any geometry that is different from Euclidean geometry. Each Non-Euclidean geometry is a consistent system of definitions, assumptions, and proofs that describe such objects as points, lines and planes. The two most common non-Euclidean geometries are spherical geometry and hyperbolic geometry. The essential difference between Euclidean geometry and these two non-Euclidean geometries is the nature of parallel lines: In Euclidean geometry, given a point and a line, there is exactly one line through the point that is in the same plane as the given line and never intersects it. In spherical geometry there are no such lines. In hyperbolic geometry there are at least two distinct lines that pass through the point and are parallel to (in the same plane as and do not intersect) the given line.

5. Non-Euclidean Geometry
Saccheri then studied the hypothesis of the acute angle and derived many theorems of noneuclidean geometry without realising what he was doing.
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Non-Euclidean_geometry.ht
##### Non-Euclidean geometry
Geometry and topology index History Topics Index
Version for printing
In about 300 BC Euclid wrote The Elements, a book which was to become one of the most famous books ever written. Euclid stated five postulates on which he based all his theorems:
• To draw a straight line from any point to any other.
• To produce a finite straight line continuously in a straight line.
• To describe a circle with any centre and distance.
• That all right angles are equal to each other.
• That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
It is clear that the fifth postulate is different from the other four. It did not satisfy Euclid and he tried to avoid its use as long as possible - in fact the first 28 propositions of The Elements are proved without using it. Another comment worth making at this point is that Euclid , and many that were to follow him, assumed that straight lines were infinite. Proclus (410-485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four, in particular he notes that
• 6. Euclidean Geometry
Smith, David A. euclidean geometry From Library of Math Online math organized by subject into topics. http//www.libraryofmath.com/euclideangeometry.
http://www.libraryofmath.com/euclidean-geometry.html
Online math organized by subject into topics. About Us Mission Statement Help
##### Cite this as
Smith, David A. "Euclidean Geometry" From Library of Math Online math organized by subject into topics.

7. Euclidean Geometry --  Britannica Online Encyclopedia
Britannica online encyclopedia article on euclidean geometry the study of plane and solid figures on the basis of axioms and theorems employed by the Greek
http://www.britannica.com/eb/article-9111070/Euclidean-geometry
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##### 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...

8. Question Corner -- Euclidean Geometry
euclidean geometry is just another name for the familiar geometry which is typically taught in grade school the theory of points, lines, angles,
http://www.math.toronto.edu/mathnet/questionCorner/euclidgeom.html
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##### Euclidean Geometry
Asked by a student at Lincolin High School on September 24, 1997 What is Euclidean Geometry? Can you also give me an example of it. Thank you very much. Euclidean geometry is just another name for the familiar geometry which is typically taught in grade school: the theory of points, lines, angles, etc. on a flat plane. It is given the name "Euclidean" because it was Euclid who first axiomatized it (rigorously described it). Another reason it is given the special name "Euclidean geometry" is to distinguish it from non-Euclidean geometries (described in the answer to another question The difference is that Euclidean geometry satisfies the Parallel Postulate (sometimes known as the Fifth Postulate). This postulate states that for every line l and every point p which does not lie on l , there is a unique line l ' which passes through p and does not intersect l (i.e., which is parallel to l Geometry on a curved surface, for example, may not satisfy this postulate, and hence is non-Euclidean geometry. Submit Your Own Question Create a Discussion Topic This part of the site maintained by (No Current Maintainers)
Last updated: April 19, 1999

9. Euclidean Geometry
This site provides as motivational introduction to geometry in a form (Euclidean) which is more accessible than nonEuclidean geometries.
http://www.geom.uiuc.edu/~crobles/hyperbolic/eucl/
Up: The Hyperbolic Geometry Exhibit
##### Euclidean Geometry
This site provides as motivational introduction to geometry in a form (Euclidean) which is more accessible than non-Euclidean geometries. Here we will establish definitions and concepts that we can apply, via analogy, to our discussion of hyperbolic geometry. This overview includes:
• A brief history of the parallel postulate . A familiarity with the parallel postulate is especially important as it is those geometries formed under the negation of Hilbert's parallel postulate that we define as hyperbolic geometries.
• Isometries of the plane.
• Reflection
• Translation
• Glide Reflection
• Rotation
• Isometries as products of reflections

10. The Ontology And Cosmology Of Non-Euclidean Geometry
That theory rests on the use of noneuclidean geometry. There are still many good questions to ask about non-euclidean geometry; but in treatment after
http://www.friesian.com/curved-1.htm
##### The Ontology and Cosmology of Non-Euclidean Geometry
Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence. David Hume, An Enquiry Concerning Human Understanding , Section IV, Part I, p. 20 [L.A. Shelby-Bigge, editor, Oxford University Press, 1902, 1972, p. 25] [ note
Until recently, Albert Einstein's complaints in his later years about the intelligibility of Quantum Mechanics often led philosophers and physicists to dismiss him as, essentially, an old fool in his dotage. Happily, this kind of thing is now coming to an end as philosophers and mathematicians of the caliber of Karl Popper and Roger Penrose conspicuously point out the continuing conceptual difficulties of quantum theory [cf. Penrose's searching discussion in The Emperor's New Mind reductio ad absurdum argument against A fine statement about all this can be found in Joseph Agassi's foreword to the recent Einstein Versus Bohr , by the dissident physicist Mendel Sachs (Open Court, 1991): It is amazing that such things need to be said, and it is particularly revealing that the responses Agassi got to his questions reminded him of the intolerance of religious dogmatism.

11. COMPUTING IN EUCLIDEAN GEOMETRY
This book is a collection of surveys and exploratory articles about recent developments in the field of computational euclidean geometry.
http://www.worldscibooks.com/compsci/2463.html
 Home Browse by Subject Bestsellers New Titles ... Browse all Subjects Search Bookshop New Titles Editor's Choice Bestsellers Book Series ... Lecture Notes Series on Computing - Vol. 4 COMPUTING IN EUCLIDEAN GEOMETRY edited by Ding-Zhu Du This book is a collection of surveys and exploratory articles about recent developments in the field of computational Euclidean geometry. Topics covered include the history of Euclidean geometry, Voronoi diagrams, randomized geometric algorithms, computational algebra, triangulations, machine proofs, topological designs, finite-element mesh, computer-aided geometric designs and Steiner trees. This second edition contains three new surveys covering geometric constraint solving, computational geometry and the exact computation paradigm. Contents: On the Development of Quantitative Geometry from Phythagoras to Grassmann (W-Y Hsiang) Computational Geometry: A Retrospective (B Chazelle) Randomized Geometric Algorithms (K L Clarkson) Voronoi Diagrams and Delaunay Triangulations (S Fortune) Geometric Constraint Solving in R and R Polar Forms and Triangular B-Spline Surfaces (H-P Seidel) Readership: Computer scientists and mathematicians.

12. Non-Euclidean (hyperbolic) Geometry Applet
This is analogous to ordinary sliding of objects in Euclidean space; however, in this noneuclidean geometry the Euclidean picture of it makes things
http://www.math.umn.edu/~garrett/a02/H2.html
##### Non-Euclidean Geometry
This applet allows click-and-drag drawing in the Poincare model of the (hyperbolic) non-Euclidean plane, and also motion . The circular arcs drawn by mouse drags are the geodesics (straight lines) in this model of geometry. In "move" mode, click-and-drag slides the whole picture in the direction of the mouse drag. This is analogous to ordinary "sliding" of objects in Euclidean space; however, in this non-Euclidean geometry the Euclidean picture of it makes things appear to become smaller as they move toward the edge. But, in fact, in terms of the non-Euclidean geometry, despite appearances, these motions preserve distances and angles. The preservation of angles should be detectable if one keeps in mind that the angles are angles between the arcs of circles at their point of intersection. Since the bounding circle is "infinitely far away", the motion of the picture does not exactly parallel the mouse drag motion, but instead moves about the same non-Euclidean distance as the Euclidean distance moved by the mouse. So the picture will appear to lag behind the mouse. The University of Minnesota explicitly requires that I state that "The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by the University of Minnesota."

13. Non-Euclidean Geometry -- From Eric Weisstein's Encyclopedia Of Scientific Books
Noneuclidean geometry, and The Theory of Parallels by Nikolas Lobachevski, with a Supplement Containing The Science of Absolute Space by John Bolyai.
http://www.ericweisstein.com/encyclopedias/books/Non-EuclideanGeometry.html
##### Non-Euclidean Geometry
see also Non-Euclidean Geometry Anderson, James W. Hyperbolic Geometry. New York: Springer-Verlag, 1999. 230 p. \$?. Bonola, Roberto. Non-Euclidean Geometry, and The Theory of Parallels by Nikolas Lobachevski, with a Supplement Containing The Science of Absolute Space by John Bolyai. New York: Dover, 1955. 268 p., 50 p., and 71 p. Borsuk, Karol. Foundations of Geometry: Euclidean and Bolyai-Lobachevskian Geometry. Projective Geometry. Amsterdam, Netherlands: North-Holland, 1960. 444 p. Carslaw, H.S. The Elements of Non-Euclidean Plane Geometry and Trigonometry. London: Longmans, 1916. Coxeter, Harold Scott Macdonald. Non-Euclidean Geometry, 6th ed. Washington, DC: Math. Assoc. Amer., 1988. 320 p. \$30.95. Greenberg, Marvin J. Euclidean and Non-Euclidean Geometries: Development and History, 3rd ed. San Francisco, CA: W.H. Freeman, 1994. \$?. Iversen, Birger. Hyperbolic Geometry. Cambridge, England: Cambridge University Press, 1992. 298 p. \$?. Manning, Henry Parker. Introductory Non-Euclidean Geometry.

14. 51M05: General Euclidean Geometry
An interesting problem in euclidean geometry show that a map which sends spheres to spheres must be an isometry. How many lines pass through four given
http://www.math.niu.edu/~rusin/known-math/index/51M05.html
Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields
POINTERS: Texts Software Web links Selected topics here
##### Introduction
We use this category to hold files concerning non-planar Euclidean geometry topics. The files on this page are more like samples of the techniques one may use for 3D problems (or n-dimensional: much of what is here is really independent of the number of dimensions.)
##### Applications and related fields
The actions of the point groups among the crystallographic groups are the basis for the construction of the Platonic solids and the regular divisions of the sphere in R^3. For more information, consult the polyhedra and spheres pages.
##### Subfields
Parent field: 51M - Real and Complex Geometry
##### Software and tables
For computational geometry see 68U05: Computer Graphics Pointer to Mesa , a 3-D graphics library (similar to OpenGL).
##### Selected topics at this site

noneuclidean geometry, branch of geometry in which the fifth postulate of euclidean geometry, which allows one and only one line parallel to a given line
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##### non-Euclidean geometry
non-Euclidean geometry, branch of geometry in which the fifth postulate of Euclidean geometry, which allows one and only one line parallel to a given line through a given external point, is replaced by one of two alternative postulates. Allowing two parallels through any external point, the first alternative to Euclid 's fifth postulate, leads to the hyperbolic geometry developed by the Russian N. I.

16. Forumgeom
Freeaccess electronic journal about elementary euclidean geometry.
http://forumgeom.fau.edu/
 Editorial Board About the Journal Instructions to Authors Submission of Papers Refereeing Statistics Links Forum Geometricorum indexed and reviewed by Mathematical Reviews Volume 1 (2001) Volume 2 (2002) Volume 3 (2003) ... Download and Viewing Instructions Subscription: If you want to receive email notifications of new publications, please send a blank email with subject line: Subscribe to FG. Editors' Corner Last modified by Paul Yiu, January 2, 2008.

17. Non-Euclidian Geometry
When the initial reading for Noneuclidean geometry has been completed. You should study the above video produced by the Open University .
http://www.scit.wlv.ac.uk/university/scit/modules/mm2217/neg.htm
##### Video "Non-Euclidean Geometry"
When the initial reading for Non-Euclidean geometry has been completed. You should study the above video produced by the "Open University". This video can be viewed in the University library when is open 9 am - 9 pm Monday to Thursday, 9 am - 5.15 pm Friday and 9 am - 12.30 pm Saturday. It is just necessary to ask for the video by name at the library enquiries.
##### Further Non-Euclidean Geometry
Before attempting the assessment, it may be necessary to read about Non-Euclidean Geometry in more detail. It is suggested that you look at the Non-Euclidean Geometry links and read the section on Non-Euclidean geometry from "The History of Mathematics - A Reader". (See reading list). D Thompson has a copy of this book which may be borrowed, if copies are unavailable in libraries.
Assessment
New Geometries, New Worlds
History of Mathematics Module
Links to other History of Mathematics sites ... Module Leader These pages are maintained by M.I.Woodcock.

18. Non-Euclidean Geometry By Henry Manning - Project Gutenberg
http://www.gutenberg.org/etext/13702
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##### Non-Euclidean Geometry by Henry Manning
Help Bibliographic Record Creator Manning, Henry Title Non-Euclidean Geometry Language English EText-No. Release Date Base Directory /files/13702/