81. Read This: Geometry: Euclid And Beyond Read This! The MAA Online book review column review of geometry euclid and Beyond, by Robin Hartshorne. geometry euclid and Beyond by Robin Hartshorne. http://www.maa.org/reviews/euclidbeyond.html

Search MAA Online MAA Home Read This! The MAA Online book review column Geometry: Euclid and Beyond by Robin Hartshorne Reviewed by Morteza Seddighin This is a well written book on geometry and its history. The author starts with Euclid's postulates and gradually presents geometry from a modern standpoint and links the subject to advanced theories such as Field Theory, Galois Theory and Inversion Theory. The book is divided into eight chapters.  In chapter 1 the author discusses Euclid's system of axioms. This chapter also contains some newer results, such as the Euler line and the nine-point circle. In chapter 2 the author discusses Hilbert's axioms and how they complete Euclid's axioms, and defines Hilbert's plane as an abstract set of objects (points) together with an abstract set of subsets (lines) which satisfy the axioms. In chapter 3 the author introduces geometry over fields and proves several theorems of Euclidean geometry using coordinate techniques. In chapter 4 the reader is introduced to segment arithmetic, in which one can define addition and multiplication of line segments in a Hilbert plane that satisfies the parallel axiom. Using the equivalence classes of line segments the author presents rigorous proofs for some theorems regarding similarity of triangles. In chapter 5 the author very clearly explains the difference between the notion of Area as it was conceived by Euclid and the modern conception of Area as defined by a "measure of area" function on the Hilbert plane. To understand chapter 5 of this book one has to be familiar with some topics from modern abstract algebra such as ordered abelian groups and tensors. Here one finds an elegant modern solution to Hilbert's Third Problem (first solved by Max Dehn in 1900). The problem is to find two solid figures of equal volume that are not equivalent by dissection even after possibly adding on other figures that are equivalent by dissection.

82. Read This: Euclid: The Creation Of Mathematics recently been emphasized by David Fowler. euclid s geometry, unlike ours, is not arithmetized . When we think of line segments, or http://www.maa.org/reviews/artmann.html

Read This! The MAA Online book review column Euclid: The Creation of Mathematics by Benno Artmann Reviewed by Stacy G. Langton Forty years ago, Jean Dieudonné declared "Euclid must go!" Our school system, remarkably, seems to have taken up this slogan, though it is hard to imagine that Dieudonné would have been pleased by what the schools have put in Euclid's place. But of course, Euclid has not gone. There seems to be an endless fascination with Euclid's Elements , the greatest textbook ever written. Since Dieudonné's call for his ouster, a great deal of work has been done which deepens our understanding of Euclid -notably, in recent years, the work of the late Wilbur Knorr and of David Fowler. "Euclid", in Benno Artmann's new book Euclid: The Creation of Mathematics , refers exclusively to the Euclid of the Elements ; none of Euclid's other works is discussed. There is nothing wrong with this, of course, though it might be a little misleading in suggesting to the reader that the Elements is the only work of Euclid which we possess. There is a tendency among modern authors to take the style of the

83. Euclid euclid The Father of geometry. euclid of Alexandria. euclid (ca. 325ca. 270 BC). euclid. Introduction to the Works of euclid. euclid biography. euclid and Proofs. http://www.khsd.k12.ca.us/district/instruct/technology/workshops/euclid/

Euclid: The Father of Geometry Euclid of Alexandria Euclid (ca. 325-ca. 270 BC) Euclid Introduction to the Works of Euclid ... The Elements Comments? Feedback?

84. Math Lair - Euclid There is no royal road to geometry euclid. euclid started the book by stating his five (actually there were ten, but only five were geometric) postulates. http://www.stormloader.com/ajy/euclid.html

Euclid "There is no royal road to geometry" - Euclid Euclid (330 - 275 B.C.) was the first major scholar at the Library of Alexandria . His most important work was his Elements , a collection of the most important results of the previous three centuries of Greek mathematics. This book was divided into thirteen volumes, the first six of which dealt with plane geometry ("Euclidean geometry"), and the last seven dealing with solid (three-dimensional) geometry, number theory (for example, perfect numbers ), and other things. More copies have been made of the Elements than of any other work, with the exception of the Bible. It had been the primary geometry textbook for two millennia. Clearly, it was a very important work. The work of many Greek mathematicians that otherwise would have been lost has been preserved in the Elements Euclid started the book by stating his five (actually there were ten, but only five were geometric) postulates. These form the basis of all his theorems. These well-known postulates are: A straight line can be drawn between any two points.

85. Saccheri S Solution To Euclid S BLEMISH Beltrami, Bolyai, Poincaré. Here we will take a look at these two new geometries which challenge our unquestioning reliance on euclid s geometry. http://www.faculty.fairfield.edu/jmac/sj/sacflaw/sacflaw.htm

Girolamo Saccheri, S.J. and his solution to Euclid's blemish Saccheri's Flaw while eliminating Euclid's "Flaw" The Evolution of Non-Euclidean Geometry Summary Non-Euclidean geometry is one of the marvels of mathematics and even more marvelous is how it gradually evolved through a process of eliminating flaws in logical reasoning. It is misleading to think that non-Euclidean geometry was similar to suddenly finding a precious jewel. Rather, it has may likened to the discovery of America by bold adventurers who would not be silenced by the complacent savants of the "known world". Complacent philosophies attempted to stifle mathematicians harboring suspicions concerning Euclid's Postulate #V. Nevertheless, America was discovered and so was non-Euclidean geometry. The story began with the Jesuit Girolamo (Jerome) Saccheri who undertook the fearsome task of removing Euclid's "flaw" concerning postulate #V. By introducing ingenious methods and rigorous logic (except for his own "flaw") he opened the way for geometers succeeding him to discover incredible geometries which provide a better model for our universe than Euclid ever dreamt of. Proofs of some of Saccheri's theorems can be viewed by moving to the page Hyperbolic Geometry Theorems of Girolamo Saccheri, S.J.

86. Euclid's Fifth Postulate(Intuiitve Geometry?) euclid s Fifth Postulate(Intuiitve geometry?). http://superstringtheory.com/forum/dualboard/messages12/629.html

String Theory Discussion Forum String Theory Home Forum Index Euclid's Fifth Postulate(Intuiitve geometry?) Follow Ups Post Followup String and M Theory Duality XII FAQ Posted by sol on July 19, 2003 at 13:06:12: In Reply to: Is Mathematics Invented or Discovered? posted by sol on July 19, 2003 at 12:12:14: Intuitive geometry (Report this post to the moderator) Follow Ups: (Reload page to see most recent) Post a Followup Follow Ups Post Followup String and M Theory Duality XII FAQ

87. Ayn Rand & Objectivism - Euclid euclid s geometry came to Europe via Spain and the Arabic language. The English scholastic philosopher Adelard of Bath traveled http://www.dailyobjectivist.com/Heroes/Euclid.asp

Areas of TDO Support TDO Amazon Laissez Faire General Info Search Net General News Sports Weather ... Health Popular with Objectivists Objectivist Center CATO Reason.org Free-Market.net ... Chris Sciabarra TDO Info Contact TDO TDO Policies TDO Staff More Links Connection Extrospection Spirituality Reciprocal Links Euclid You've heard of Euclidean geometry. Euclid (c. 330-c. 275 BC) is the guy who invented it; or, perhaps, recorded its invention in his book Elements . His architectonic remained unchallenged for almost 2000 years until, in the 19th century, a non-Euclidean geometry was devised. Among Euclid's other works are "Data," "On Divisions," "Optics," and "The Elements of Music." Euclid's geometry came to Europe via Spain and the Arabic language. The English scholastic philosopher Adelard of Bath traveled to Spain disguised as a Muslim student and returned with a copy of The Elements . Adelard's Latin translation was published in 1120. A sample from The Elements Definition 1. A point is that which has no part. Definition 2. A line is breadthless length.

88. History Of Geometry He later did the most influential work in geometry since euclid, publishing Grundlagen der Geometrie (1899) which put geometry in a formal axiomatic setting http://geometryalgorithms.com/history.htm

History Home Overview [History] Algorithms Books Gifts Web Sites A Short History of Geometry Ancient This is a short outline of geometry's history, exemplified by major geometers responsible for it's evolution. Click on a person's picture or name for an expanded biography at the excellent: History of Mathematics Archive (Univ of St Andrews, Scotland) Also, Click these links for recommended: Greek Medieval Modern History Books ... History Web Sites Ancient Geometry (2000 BC - 500 BC) Babylon Egypt The geometry of Babylon (in Mesopotamia) and Egypt was mostly experimentally derived rules used by the engineers of those civilizations. They knew how to compute areas, and even knew the "Pythagorian Theorem" 1000 years before the Greeks (see: Pythagoras's theorem in Babylonian mathematics ). But there is no evidence that they logically deduced geometric facts from basic principles. Nevertheless, they established the framework that inspired Greek geometry. A detailed analysis of Egyptian mathematics is given in the book: Mathematics in the Time of the Pharaohs India (1500 BC - 200 BC) The Sulbasutras Baudhayana (800-740 BC) Apastamba (600-540 BC) Greek Geometry (600 BC - 400 AD) Time Line of Greek Mathematicians Major Greek Geometers (listed cronologically) [click on a name or picture for an expanded biography].

89. 20th WCP: Two Traditions Of Western And Chinese Cultures ABSTRACT In European atomic theory, euclid s geometry and Aristotle s logic complement each other and are generally acknowledged sources of Western science. http://www.bu.edu/wcp/Papers/Cult/CultLi.htm

Philosophy of Culture Two traditions of Western and Chinese Cultures Li Ya-ning Sichuan University ABSTRACT: In European atomic theory, Euclid's geometry and Aristotle's logic complement each other and are generally acknowledged sources of Western science. In China, the book Zhou Yi Philosophy atom theory of ancient Greece, Euclid's geometry and Aristotle's logic complement each other and are generally acknowledged the source of Western science. But in China it was obviously unilateral that air - monism theory was used to explain the source of Chinese science in a long time and the careful inquiry from the viewpoints of logic and mathematics were ignored. In my opinion the book Zhou Yi is the starting point in the inquiry because its system contains ideologies of philosophy, logic and mathematics and a unity of them. Notes (1) Sarton's essay Scientific History and New Humanism in Science and Philosophy, 4th issue of 1984,edited by the Magazine Office of Natural Dialectics of China Science Academy. (2) Collected Works of jose li, edited by Pan li-xin, page 54,1986,Shenyang.

90. Lobachevskian Geometry Home Page Click here for a brief history of how euclid s geometry eventually gave rise to noneuclidean geometry, including the geometry of Lobachevsky. http://www.southernct.edu/~grant/nicolai/

Lobachevskian Geometry with Interactive Drawing-Pad The main feature of this web-site is an interactive "drawing-pad" for constructing geometric figures that faithfully correspond to the axioms of Lobachevskian Geometry. This is done via the unit-disk model of Henri Poincaré. To put this in some context, the site also contains a brief history of the development of Lobachevskian Geometry and of the use and function of mathematical models like Poincaré's. Click here for a brief history of how Euclid's geometry eventually gave rise to non-Euclidean geometry, including the geometry of Lobachevsky. Click here for a brief description of mathematical models in general, and the Poincaré model in particular. Click here for a brief description of the java applet accessible from this page: "The Lobachevskian Plane." Click here to go to "The Lobachevskian Plane," the interactive drawing-pad for Lobachevskian figures. (Maximize the window after opening.) (Please wait: it may take several minutes to download this sizeable applet and its link.)

91. Untitled And so persuasive and logically compelling was euclid s geometry that it took over two thousand years before mathematicians began to suspect that there might http://www.southernct.edu/~grant/nicolai/history.html

A Brief History of Geometry Leading to non-Euclidean Geometry Much of the following is excerpted, with permission, from A Mathematics Sampler: Topics for the Liberal Arts, Fourth Edition About 300 BC, Euclid organized all of the geometry and much of the arithmetic and number theory that was known (some 300 years of work by earlier Greek mathematicians) into a single, cohesive work, The Elements . Euclidean geometry is the geometry we learn informally as children and somewhat more formally in high school geometry; for most, it is just "geometry." Euclid's goal was to systematize the various relationships that had been observed among spatial figures, which he, like Plato and Aristotle, regarded as ideal representations of physical entities. Euclid's organizational scheme, the axiomatic method, was so ingenious that it has remained the paradigm for all of mathematics and much of science. And so persuasive and logically compelling was Euclid's geometry that it took over two thousand years before mathematicians began to suspect that there might be other ways of looking at geometry, that perhaps Euclid's geometry was not the geometry, but

92. Comparison Of Axiomatic Systems For Geometry: Euclid, Hilbert, And Birkoff December 9. Comparison of Axiomatic Systems for geometry euclid, Hilbert, and Birkoff. http://www.beva.org/math323/asgn7/dec12.htm

December 9 Comparison of Axiomatic Systems for Geometry: Euclid, Hilbert, and Birkoff Problems Euclid's 28 Axioms Hilbert's Axioms Birkoff's Axioms Problems During the 20th century, mathematicians, who worked in the field called the foundation of mathematics, pointed out flaws again and again in Euclid's assumptions. Euclid assumed and used assumptions that had not been proven or stated. Here is a list of some of the logical problems: 1) the need for a list of undefined terms - do points and lines exist? 2) the need for a definite statement about the continuity of lines and circles - does every line have at least two points on it? 3) the need for statements about the order of points on a line - if there was no order of points on a line, then would not the object be a circle? 4) the need for statement about the concept of betweenness - just because a diagram shows a bisector of an angle intersecting the opposite side in a triangle "between" two endpoints, is it true that there is a point between those two endpoints?

93. Euclid euclid s most famous work is his treatise on geometry The Elements . euclid s decision to make this an axiom led to euclidean geometry. http://intranet.woodvillehs.sa.edu.au/pages/resources/maths/History/cld.htm

Euclid of Alexandria Born: about 365 BC in Alexandria, Egypt Died: about 300 BC Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Welcome page Euclid is the most prominent mathematician of antiquity best known for his treatise on geometry The Elements . The long lasting nature ofThe Elements must make Euclid the leading mathematics teacher of all time. Little is known of Euclid's life except that he taught at Alexandria in Egypt. Euclid's most famous work is his treatise on geometry The Elements . The book was a compilation of geometrical knowledge that became the centre of mathematical teaching for 2000 years. Probably no results in The Elements were first proved by Euclid but the organisation of the material and its exposition are certainly due to him. The Elements begins with definitions and axioms, including the famous fifth, or parallel, postulate that one and only one line can be drawn through a point parallel to a given line. Euclid's decision to make this an axiom led to Euclidean geometry. It was not until the 19th century that this axiom was dropped and non-euclidean geometries were studied. Zeno of Sidon , about 250 years after Euclid wrote the Elements , seems to have been the first to show that Euclid's propositions were not deduced from the axioms alone, and Euclid does make other subtle assumptions.

94. Project Euclid Journals Journals in euclid. sort by title. Bulletin of Symbolic Logic. Canadian Applied Mathematics Quarterly. Communications in Analysis and geometry – Forthcoming. http://projecteuclid.org/Dienst/UI/1.0/TitleShort

Journals in Euclid sort by title sort by title - details sort by publisher sort by discipline Abstract and Applied Analysis ... Advances in Applied Probability Advances in Theoretical and Mathematical Physics Forthcoming The Annals of Applied Probability Annals of Mathematics The Annals of Probability The Annals of Statistics Asian Journal of Mathematics Forthcoming Bernoulli Bulletin of the Belgian Mathematical Society-Simon Stevin Bulletin of Symbolic Logic Canadian Applied Mathematics Quarterly Communications in Analysis and Geometry Forthcoming Forthcoming Communications in Mathematical Physics (1965-1996) Forthcoming Communications in Mathematical Sciences Forthcoming Current Developments in Mathematics Forthcoming Duke Mathematical Journal Experimental Mathematics Homology, Homotopy and Applications Forthcoming International Statistical Review Internet Mathematics Journal of Applied Mathematics Journal of Applied Probability Journal of Differential Geometry Forthcoming Journal of Symbolic Logic Journal of Symplectic Geometry Forthcoming Kodai Mathematical Journal Methods and Applications of Analysis Forthcoming The Michigan Mathematical Journal Notre Dame Journal of Formal Logic Review of Modern Logic Revista Matemática Iberoamericana ... home

95. Mathematics Archives - Topics In Mathematics - Geometry The geometry Applet ADD. KEYWORDS euclid s Elements, history, interactive pages; geometry Bibliography, articles from Mathematics Teacher; http://archives.math.utk.edu/topics/geometry.html

Topics in Mathematics Geometry Alice Meets the 4th Dimension ADD. KEYWORDS: Four Dimensional Cube AMS's Materials Organized by Mathematical Subject Classification ADD. KEYWORDS: Electronic Journals, Preprints, Web Sites, Databases AskERIC Lesson Plans - Geometry Border Pattern Gallery ADD. KEYWORDS: Symmetry Cabri Java Project ADD. KEYWORDS: Examples, Animations, 3D geometry, Loci, Conics, Hyperbolic Geometry SOURCE: TECHNOLOGY: Java applets ADD. KEYWORDS: Cabri Geometry resources Cercle d'Euler ADD. KEYWORDS: Euler Circles, Malfatti Circles, Poncelet Triangles, Morley Triangles, Podaires Triangles, Fermat Points, Feuerbach Points, Nagel Points, Gergonne Points Combinatorial Tiling Theory: Theorems - Algorithms - Visualization ADD. KEYWORDS: Two Dimensional Tilings, Three Dimensional Tilings, Face-Transitive Tilings, Periodic Delone Tilings, Orbifold Graphs, Classification of 3-Dimensional Tilings, Tiling Space by Platonic Solids, Enumeration of Polyhedral Crystal Structures The Computation of Certain Numbers Using a Ruler and Compass ADD. KEYWORDS:

96. A Science As A Geometry Lets briefly review the evolution of this geometrical conception of science. Before Descartes, algebra and euclid s geometry were separate topics. http://www.ensc.sfu.ca/people/grad/brassard/personal/THESIS/node20.html

Next: Klein`s Ordering of the Up: STRUCTURALISM Previous: Hierarchy is History in A Science as a Geometry A science is a simple description of the regularities in a phenomenal domain. It needs a framework of expression for the forms (or structures) of this phenomenal domain. Such a framework of expression constitutes a geometry. A science has to be based on a geometry. The simplest regularities of a science constitute its geometry. All the other regularities are expressed in terms of these simplest ones. Lets briefly review the evolution of this geometrical conception of science. After completing a detailed geodetic survey of the kingdom of Hanover, Carl Friedrich Gauss (1777-1855) wrote in 1827 a paper called General Investigations of Curved Surfaces . In this paper, he used calculus to show that some of the basic notions of surface theory, including the notion of curvature, were intrinsic to the surface and did not depend on how the surface was situated in three-dimensional space. Differential geometry follows the basic intuitions of Gauss and Riemann (1826-1866) of building up a complete geometrical system on the basis of concepts and axioms that only affect the immediate neighborhood of each point. From this concept, the notion of field arises. The points of interest in this field are not necessarily the origin but the critical points (or singularities). In 1830, Evarist Galois conceived the notion of the permutation group and discovered the relation between the theory of algebraic equations and the theory of group (see appendices

97. Geometry Tutorial the most distinguished, who doubt whether the whole universe, or to speak more widely, the whole of being, was only created in euclid’s geometry; they even http://www.gbt.org/geo.html

Euclidean Geometry Tutorial This tutorial covers the Geometry in Euclid's Elements . Euclid's famous text was "the" book for the study of Geometry until the 19th century. It has been studied by a host of intellectual greats. His systematic approach to Geometry is not a only a tremendous study in how to think and reason, but it became the paradigm that later philosophers would attempt to follow in setting up their own systems of thought. There is really no other mathematical text that rivals its impact on intellectual history. This tutorial is highly recommended not only for its tremendous historical value, but also as a fine addition to the Geometry-starved Saxon program. The only fault I see with the Saxon programs is its meager treatment of Geometric proofs. Along with Saxon, most modern math texts are downplaying Geometric proofs because they are teaching to the SAT and it does not require proofs. But after going through the magnificent proofs of Euclid, you will see why his work is truly a mathematical classic. In order to enroll in the third year of the Great Books Tutorial, all students must take my Euclidean Geometry course. The course requires 5-10 hours a week of preparation and should be seen as a college level mathematics course.

98. [FOM] On Euclidean And Hilbertian Geometry And Predicativeness euclid s geometry is clearly predicative This could be paraphrased by saying that euclid s geometry is constructivistic (in the present sense of the term). http://www.cs.nyu.edu/pipermail/fom/2003-May/006667.html

[FOM] On Euclidean and Hilbertian geometry and predicativeness Aatu Koskensilta aatu.koskensilta at xortec.fi Wed May 21 16:52:01 EDT 2003 Previous message: [FOM] Semantical realism without ontological realism in mathematics Next message: [FOM] FOM Potential Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] As I told before, comparison is not easy, and I do not agree completely with Aatu's opinions. For example: In Euclid's Element's the axioms and postulates are stated in terms of constructions; This is not true for axioms, which concern with equality, and also the V postulate concerns with parallels and does not seem a construction. There has been a interesting discussion about how far the construction is actual or simply reveals implicit properties of the diagram (Mugler, Cambiano). Maybe in the IV century B.C. there was a keen discussion between a more theoretical (Plato-Speusippos) and a more problematic (Eudoxus-Menaechmus) approach. Perhaps. I faintly recall reading a book on the development of the deductive method in greek mathematics which argued that in fact the diagrams are part of the semiformal system (other part contain rigid syntactic proof-elements expressed by repeated and recursive use of a very limited set of expressions), and that some of the propositions in Euclid which have been taken to contain tacit premises don't in fact cotnain anything hidden provided we assume that the diagram is also to be considered part of the set up for the theorem and not as a mere "visualisation aid". I'm not sure but perhaps the book was Reviel Netz's The shaping of deduction in Greek mathematics : a study in cognitive history. >>

99. WORKVIEW 3D for visualization. WORKVIEW3D can save euclid geometry in one of the supported formats (VRML2, DXF, STL). The application for http://gil.home.cern.ch/gil/WORKVIEW3D_EN.htmL

WORKVIEW 3D DOCUMENTATION INTRODUCTION WORKVIEW3D is a visualization tool for various CAD exchange formats. The "open" design of WORKVIEW3D allows the direct connexion to user-specific server applications for a particular CAD/CAM system. WORKVIEW3D can be used to display models, to create assemblies from parts and to produce drawings with multiple views for annotations. WORKVIEW3D is accessible from the WWW Browser, EDMS or from a DB application to whitch on have sent the filename of the geometrie as parameter. WORKVIEW3D may be used to exchange 3D geometrical data for models with a polygonal boundary representation or triangular meshes representation. WORKVIEW3D has two modes : 3D model views and 2D layout views. WORKVIEW3D is a multi-plateform application. It may be used under Windows95, NT, Unix, Motif, and MacOS. By default it uses OpenGL but it can be used without. In WORKVIEW3D, the removal of hidden lines is realized by means of an accelerating algorithm developed by the society Delta Concept GENERALITIES Input formats Polygonal boundary representation : VRML-2 OBJ STL Direct Connexion to a Server Exact boundary representation : IGES SAT Output formats WORKVIEW3D can write a polygonal boundary representation to : VRML-2 DXF STL Direct Connexion to a Server Assembly management WORKVIEW3D uses its own WVL file in ASCII format to store the position, the name and the references of input files. The WVL file may be created manually or by a program. WVL files may contain references to other WVL files.

100. NonEuclid Interactive Constructions In Hyperbolic Geometry NonEuclid Interactive Constructions in Hyperbolic geometry Joel Castellanos, a graduate student at the University of New Mexico, in cooperation with a professor and another graduate student, is http://rdre1.inktomi.com/click?u=http://cs.unm.edu/~joel/NonEuclid/&y=029119

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