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Euclid Geometry:     more books (100)

The Figures Of Euclid: With The Enunciations, As Printed In Euclid's Elements Of Plane Geometry (1840) by Euclid, 2010-09-10 The Definitions, Postulates, Axioms, And Enunciations Of The Propositions Of The First Six, And The Eleventh And Twelfth Books Of Euclid's Elements Of Geometry (1848) by Euclid, 2010-05-23 The Commentary of al-Nayrizi on Books II-IV of Euclid's Elements of Geometry (Studies in Platonism, Neoplatonism, and the Platonic Tradition) by Anthony Lo Bello, 2009-04-30 Elements of Geometry: Containing the First Six Books of Euclid, with a Supplement On the Quadrature of the Circle, and the Geometry of Solids; to Which ... Elements of Plane and Spherical Geometry by Euclid, John Playfair, 2010-02-28 The Geometry of the Three First Books of Euclid, by Direct Proof from Definitions Alone, by H. Wedgwood by Euclid, Hensleigh Wedgwood, 2010-02-22 Euclid and His Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry (CSLI-Studies in the Theory and Applications of Diagrams) by Nathaniel Miller, 2008-07-08 The Father of Geometry: Euclid and His 3-D World (Great Minds of Ancient Science and Math) by Paul Hightower, 2010-09 The Elements of Geometry: In Which the Principal Propositions of Euclid, Archimedes, and Others Are Demonstrated After the Most Easy Manner. To Which is ... Arithmetical and Geometrical, Together Wit by William Emerson, 2009-04-27 The Elements Of Plane Geometry Part Two: Corresponding To Euclid, Books III-VI by Mathematical Association For The Improve, 2007-07-25 Euclid's Elements of Geometry, the First Six, the Eleventh and Twelfth Books; Translated into English from Dr. Gregory's Edition, with Notes and Additions for the Use of the British Youth by E. Stone. The Second Edition, with Corrections. by E[dmund], editor and translator. Stone, 1763 Euclid's Elements of Geometry: The School Edition by Robert Potts, 1868-01-01 Euclid's Elements of Geometry: The First Six Books, to Which Are Added Elements of Plain and Spherical Trigonometry, a System of Conick Sections, Elements ... According to the Newtonian System, and Elem by John Allen, 2010-02-26 Euclid's Elements of Geometry: The First Six Books, Chiefly from the Text of Dr. Simson ... by Anonymous, 2010-02-14 Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson...Together with a Selection of Geometrical Exercises from the Senate-House and College Examination Papers ... by Robert Potts, 2010-02-03

lists with details

41. Henri Poincare: Science And Hypothesis: Chapter 3: Non-Euclidean Geometries
    The geometry of Lobatschewsky. — If it were possible to deduce euclid s postulate from the several euclid s geometry would thus be a provisory geometry.  
    http://spartan.ac.brocku.ca/~lward/Poincare/Poincare_1905_04.html
    
    Extractions: Chapter 3: Non-Euclidean Geometries Citation: Henri Poincaré. "Non-Euclidean Geometries". Chapter 3 in Science and Hypothesis . London: Walter Scott Publishing (1905): 35-50. NON-EUCLIDEAN GEOMETRIES. EVERY conclusion presumes premisses. These premisses are either self-evident and need no demonstration, or can be established only if based on other propositions; and, as we cannot go back in this way to infinity, every deductive science, and geometry in particular, must rest upon a certain number of indemonstrable axioms. All treatises of geometry begin therefore with the enunciation of these axioms. But there is a distinction to be drawn between them. Some of these, for example, "Things which are equal to the same thing are equal to one another," are not propositions in geometry but propositions in analysis. I look upon them as analytical à priori intuitions, and they concern me no further. But I must insist on other axioms which are special to geometry. Of these most treatises explicitly enunciate three :—(1) Only one line can pass through two points; (2) a straight line is the (36) shortest distance between two points; (3) through one point only one parallel can be drawn to a given straight line. Although we generally dispense with proving the second of these axioms, it would be possible to deduce it from the other two, and from those much more numerous axioms which are implicitly admitted without enunciation, as I shall explain further on. For a long time a proof of the third axiom known as Euclid's postulate was sought in vain. It is impossible to imagine the efforts that have been spent in pursuit of this chimera. Finally, at the beginning of the nineteenth century, and almost simultaneously, two scientists, a Russian and a Bulgarian, Lobatschewsky and Bolyai, showed irrefutably that this proof is impossible. They have nearly rid us of inventors of geometries without a postulate, and ever since the

42. Read This: Euclid: The Creation Of Mathematics
    euclid s geometry, unlike ours, is not arithmetized . When we think of line segments, or of regions of the plane, or of space, we naturally think of their  
    http://www.maa.org/reviews/artmann.html
    
    Extractions: by Benno Artmann 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

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

44. GEOMETRY D FALL 2004
    Robin Hartshorne, geometry euclid and Beyond, Springer, Undergraduate Texts in Ch. 1, euclid s geometry. 1. A First Look at euclid s Elements  
    http://www.math.uu.se/~thomase/geometriD/geometriD2004.html
    
    Extractions: COURSE LITERATURE Robin Hartshorne, Geometry: Euclid and Beyond , Springer, Undergraduate Texts in Mathematics (2000) The book is available at Studentbokhandeln. LECTURES Mon Sept 13 Thomas Erlandsson, Introduction Wed Sept 15 Lars Lindberg, Axioms of Incidence and Fields (Hartshorne Sec 6 and 14) Mon Sept 20 Rodolfo Rios Z., Axioms of Betweenness and Ordered Fields (Hartshorne Sec 7 and 15) Thu Sept 23 Jonathan Mörndal, Congruence of Segments and Angles (Hartshorne Sec 8, 9 and 16) Mon Sept 27 Djalal Mirmohades, The Hilbert plane (Hartshorne Sec 10) Thu Sept 30 Djalal Mirmohades, Circles in the Hilbert plane (Hartshorne Sec 11) Thu Oct 14 Daniel Luna, Non-Archimedean Geometry Part I (Hartshorne Sec 18) Mon Oct 25 Daniel Luna, Non-Archimedean Geometry Part II (Hartshorne Sec 18) Thu Oct 28 Erik Andersson, The Area Problem (Hartshorne sec 22-24) Mon Nov 1 Jonathan Mörndal, Quadratura Circuli (Hartshorne sec 25) Wed Nov 3 Rodolfo Rios Z., Euclid's Theory of Volume and Hilbert's Third Problem (Hartshorne sec 26-27) Mon Nov 8 David Larsson, The Regular 17-Sided Polygon (Hartshorne sec 29)

45. Forward
    However, there are some problems with euclid s geometry. After he has a sufficient backlog of previously established theorems, the material rolls along  
    http://www.sonoma.edu/users/w/wilsonst/Papers/Geometry/Forward.html
    
    Extractions: Geometry is a very important course in the development of mathematics students. The material is ancient, having been first written down by Euclid c. 300 BC. Euclid is to be credited with establishing the method of mathematics used to the present day of organizing theories into sequences of theorems and proofs. A proof is a sequence of statements, each one requiring a reason, ordered so the the reason for each statement is a statement which has been previously established either in the theorem at hand or a previously proven theorem. However, there are some problems with Euclid's geometry. After he has a sufficient backlog of previously established theorems, the material rolls along brilliantly enough, but he has problems at the outset. For instance, the first theorem on the construction of an equilateral triangle does not follow from Euclid's axioms. While his axioms are all true in a two dimensional Cartesian space over the rational numbers, it can be shown that there is no equilateral triangle which has all rational coordinates. Unfortunately, in such a development, the beginning is the worst place to have logical difficulties. It should be clear today that Euclid's main problem was that he did not have algebra. The Arabs are credited with inventing algebra in Spain in the eighth century AD, but they did not use the symbology that we employ today. They spelled out the arithmetic to be done with unknowns longhand in prose. Algebra as we know it today was not developed until the end of the sixteenth century AD as a result of work with the radical formulas for the solution of cubic and quartic formulas which was developed by the renaissance Italian mathematicians.

46. Euclid - Wikipedia, The Free Encyclopedia
    In addition to providing some missing proofs, euclid s text also includes sections on Today, however, it is often referred to as euclidean geometry to  
    http://en.wikipedia.org/wiki/Euclid
    
    Extractions: Euclid of Alexandria Greek ) (ca. 325 BC 265 BC ) was a Greek mathematician who taught at Alexandria in Egypt almost certainly during the reign ( 323 BC 283 BC ) of Ptolemy I . Now known as "the father of geometry ," his most famous work is Elements , widely considered to be history's most successful textbook . Within it, the properties of geometrical objects and integers are deduced from a small set of axioms , thereby anticipating (and partly inspiring) the axiomatic method of modern mathematics Euclid also wrote works on perspective conic sections spherical geometry , and possibly quadric surfaces . Neither the year nor place of his birth have been established, nor the circumstances of his death. edit Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework. In addition to providing some missing proofs , Euclid's text also includes sections on number theory and three-dimensional geometry The geometrical system described in Elements was long known simply as "the" geometry. Today, however, it is often referred to as

47. Euclid's Elements - Wikipedia, The Free Encyclopedia
    euclid s books are in the fields of euclidean geometry, as well as the ancient Greek These basic principles reflect the constructive geometry euclid,  
    http://en.wikipedia.org/wiki/Euclid's_Elements
    
    Extractions: Euclid's Elements Greek Στοιχεία) is a mathematical and geometric treatise, consisting of 13 books, written by the Greek mathematician Euclid around 300 BC . It comprises a collection of definitions, postulates ( axioms ), propositions ( theorems ) and proofs thereof. Euclid's books are in the fields of Euclidean geometry , as well as the ancient Greek version of number theory . The Elements is one of the oldest extant axiomatic deductive treatments of geometry , and has proved instrumental in the development of logic and modern science It is considered the most successful textbook ever written: the Elements was one of the very first books to go to press, and is second only to the Bible in number of editions published (over 1000). For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students. Not until the 20th century did it cease to be considered something all educated people had read. It is still (though rarely) used as a basic introduction to geometry today. edit Euclid based his work in Book I on 23 definitions, such as

48. Untitled Document
    From euclid to Legendre, to name the most renowned of modern writers on A necessary sequel of this is that the propositions of geometry are not  
    http://wlym.com/antidummies/part34.html
    
    Extractions: Riemann for Anti-Dummies Part 34 POWER AND CURVATURE In his 1854 habilitation lecture, Bernhard Riemann spoke of the twofold task involved in lifting more than 2,000 years of darkness that had settled on science: Another principle is involved. As emphasized in last week's pedagogical discussion, the principle that generates the magnitude that doubles the cube, is expressed in a change of "curvature." As Riemann stated in his habilitation paper, the determination of extension is only the first step: "Now that the concept of an n-fold extended manifold has been constructed and its essential mark has been found to be this, that the determination of position therein can be referred to n determinations of magnitude, there follows as second of the problems proposed above, an investigation into the relations of measure that such a manifold is susceptible of, also, into the conditions which suffice for determining these metric relations." The curvature of these three surfaces can be measured by the sum of the angles of the triangles formed on each. On the spherical triangle, the sum of the angles is greater than 180 degrees. On the flat one, the sum of the angles is exactly 180 degrees. On the "catenary" triangle, the sum of the angles is less than 180 degrees. Now, think, as Gauss and Riemann did, of a manifold that encompasses all three curvatures. Begin first with a positively curved surface such as a sphere. Here the sum of the angles of a triangle is always greater than 180 degrees. The larger the triangle, the greater the sum, until a maximum is reached when the triangle covers the whole sphere. As these triangles become smaller, the sum of the angles approaches, but never reaches 180 degrees, for when the sum of the angles reaches 180 degrees, the surface becomes flat. On a negatively curved surface, just the opposite occurs. As the triangle becomes smaller, the sum of the angles of a triangle gets larger, approaching, but never reaching 180 degrees.

49. Proclus Diadochus Was A Neoplatonist And The Head Of Plato's Academy Who Wrote A
    Early geometry Section of Proclus Commentary on euclid s geometry. Proclus Biography of Proclus, with a look at his contributions to geometry, astronomy,  
    http://ancienthistory.about.com/od/proclusdiadochus/
    
    Extractions: zJs=10 zJs=11 zJs=12 zJs=13 zc(5,'jsc',zJs,9999999,'') About Homework Help Ancient / Classical History Homework Help ... Help zau(256,140,140,'el','http://z.about.com/0/ip/417/C.htm','');w(xb+xb+' ');zau(256,140,140,'von','http://z.about.com/0/ip/496/7.htm','');w(xb+xb); Sign Up Now for the Ancient / Classical History newsletter! See Online Courses Search Ancient / Classical History Proclus Diadochus was a neoplatonist and the head of Plato's Academy who wrote a commentary on Euclid's geometry. Alphabetical Recent Up a category Proclus Diadochus Glossary entry on Proclus explaining origin of his name and his philosophical career. Philosophers Timeline Chronological list of Greek and Roman philosophers and mathematicians with dates. Early Geometry Section of Proclus' Commentary on Euclid's Geometry. Proclus Biography of Proclus, with a look at his contributions to geometry, astronomy, physics and theology. Proclus Encyclopedia Britannica article on Proclus calls him the last major Greek philosopher. As a neoplatonist he taught that thoughts are reality, and concrete "things" are merely appearances, Topic Index Email to a Friend

50. Proclus Diadochus Was A Neoplatonist And The Head Of Plato's Academy Who Wrote A
    Section of Proclus Commentary on euclid s geometry. Philosophers Timeline Chronological list of Greek and Roman philosophers and mathematicians with dates.  
    http://ancienthistory.about.com/od/proclusdiadochus/index_a.htm
    
    Extractions: zJs=10 zJs=11 zJs=12 zJs=13 zc(5,'jsc',zJs,9999999,'') About Homework Help Ancient / Classical History Homework Help ... Help zau(256,140,140,'el','http://z.about.com/0/ip/417/C.htm','');w(xb+xb+' ');zau(256,140,140,'von','http://z.about.com/0/ip/496/7.htm','');w(xb+xb); Sign Up Now for the Ancient / Classical History newsletter! See Online Courses Search Ancient / Classical History Proclus Diadochus was a neoplatonist and the head of Plato's Academy who wrote a commentary on Euclid's geometry. Sort By: Guide Picks Recent Up a category Early Geometry Section of Proclus' Commentary on Euclid's Geometry. Philosophers Timeline Chronological list of Greek and Roman philosophers and mathematicians with dates. Proclus Biography of Proclus, with a look at his contributions to geometry, astronomy, physics and theology. Proclus Encyclopedia Britannica article on Proclus calls him the last major Greek philosopher. As a neoplatonist he taught that thoughts are reality, and concrete "things" are merely appearances, Proclus Diadochus Glossary entry on Proclus explaining origin of his name and his philosophical career.

51. A Formal Theory For Geometry
    euclid s geometry was still regarded as a model of logical rigor, a shining example of what Even euclid s geometry itself came under critical scrutiny.  
    http://www.math.psu.edu/simpson/papers/philmath/node15.html

52. The Geometry Of Euclid
    The geometry of euclid. The geometry of euclid. Above the gateway to Plato s academy appeared a famous inscription  
    http://www.math.psu.edu/simpson/papers/philmath/node13.html

53. Euclid's Geometry As An Attempt To Solve A Problem With Plato's Forms
    euclidean geometry provides the necessary connection. euclid discusses certain eidoi But, euclid explains this role of geometry in the universe. 21.  
    http://www.vivboard.net/doc/n0059.htm
    
    Extractions: Wed, 21 Sep 2005 09:46:41 GMT [The Topic] Metaphysical Review July 1996 - June 1997 Metaman [Date] [Copied From] [Orig. Author] Thomas O'Neill Post-Id Length 36930 bytes D.O.P Marking Post Reply [Follow-ups] HEADER Guest Name: (only for whom without registered name) sign in Main comment/Title Short note/Abstract: ATTACHMENT/MAIN Upload the text: allowed or Upload a file: (up to 100k bytes) or Refer to aother post: (to form a post that sharing an existing one's attachment/main content) SOURCE If it's a copy, please acknowledge the source by filling either or most of these. Copy from: URL Original author: Hosted by VivBoard Evaluation: Feel: Clear People's view:

54. Byrne's Edition Of Euclid
    It covers the first 6 books of euclid s Elements of geometry, which range through most of elementary plane geometry and the theory of proportions.  
    http://www.math.ubc.ca/people/faculty/cass/Euclid/byrne.html
    
    Extractions: Main Euclid page An unusual and attractive edition of Euclid was published in 1847 in England, edited by an otherwise unknown mathematician named Oliver Byrne. It covers the first 6 books of Euclid's Elements of Geometry , which range through most of elementary plane geometry and the theory of proportions. What distinguishes Byrne's edition is that he attempts to present Euclid's proofs in terms of pictures, using as little text - and in particular as few labels - as possible. What makes the book especially striking is his use of colour. Incidentally, at the time of its publication the first 6 books, which are the ones concerrned with plane geometry, made up the basic mathematics curriculum for many students. With the financial support of several undergraduate organizations at UBC - the Alma Mater Society of UBC, the Science Undergraduate Society at UBC, and the Undergraduate Mathematics Club - and the cooperation of the Special Collection Division of the UBC Library, we have had the entire edition photographed by Greg Morton at UBC Biomedical Communications We hope to mount eventually on this site digital images of all of the photographs. We imagine that it will serve as an interesting resource for geometry projects all over the world. We have mounted all of Byrne's book, but in the organization of the site is by no means final. We are still experimenting with the images to improve their quality, and sooner or later the structure found for Book VI, which is much better than the rest, will be transported to the other books. If you have any suggestions we'll be pleased to hear from you.

55. Non-Euclidean Geometries, Discovery
    community would not be able to accept a revolutionary denial of euclid s geometry. So Riemann modified euclid s Postulates 1, 2, and 5 to  
    http://www.cut-the-knot.org/triangle/pythpar/Drama.shtml
    
    Extractions: Drama of the Discovery Four names - C.F.Gauss N.Lobachevsky J.Bolyai (1802-1860), and B.Riemann (1826-1846) - are traditionally associated with the discovery of non-Euclidean geometries. In non-Euclidean geometries, the fifth postulate is replaced with one of its negations : through a point not on a line, either there is none (B) or more than 1 (C) line parallel to the given one. Carl Friedrich Gauss was apparently the first to arrive at the conclusion that no contradiction may be obtained this way. In a private letter of 1824 Gauss wrote: The assumption that (in a triangle) the sum of the three angles is less than 180 o leads to a curious geometry, quite different from ours, but thoroughly consistent, which I have developed to my entire satisfaction. From another letter of 1829, it appears that Gauss was hesitant to publish his research because he suspected the mediocre mathematical community would not be able to accept a revolutionary denial of Euclid's geometry. Gauss invented the term "Non-Euclidean Geometry" but never published anything on the subject. On the other hand, he introduced the idea of surface curvature on the basis of which Riemann later developed Differential Geometry that served as a foundation for Einstein's General Theory of Relativity.

56. Various Geometries
    Put another way, in euclid s geometry, some properties of figures (lengths, Absolute geometry is derived from the first four of euclid s postulates.  
    http://www.cut-the-knot.org/triangle/pythpar/Geometries.shtml
    
    Extractions: Various Geometries The term "Non-Euclidean Geometries" usually applies to the geometries of Riemann and Lobachevsky . However, once Euclid's postulates have been lowered from their lofty, 2300 years old pedestal, and brought into active mathematical investigation, many more geometries had evolved. Under close scrutiny, it became apparent that Euclid's Elements are not as solidly based on his postulates as one might have expected of the treatise known as the Mathematical Bible . Omissions were fundamental. For example, the first postulate A straight line may be drawn between any two points. would be true even if there were no points. For, as we know, falsity implies anything . We may stipulate that there are 2,3,4 point geometries. Note that line segments that appear on the diagrams are not elements of those geometries. They are there only to indicate the lines that pass through certain points. In the 2 point geometry, there exists a single line that contains exactly 2 points. (Without rigorous axiomatization, one may insist that, in addition, there are also two 1 point lines.) In the 4 point geometry, with additional stipulation that a line contains exactly two points we even have the Fifth postulate as annunciated by Euclid.

57. Euclid: Definition And Much More From Answers.com
    euclid s Elements is the basis for modern school textbooks in geometry. One of the basic statements, or postulates, of euclid s geometry is that if a line  
    http://www.answers.com/topic/euclid
    
    Extractions: Greek mathematician who applied the deductive principles of logic to geometry, thereby deriving statements from clearly defined axioms. var tcdacmd="cc=edu;dt"; Encyclopedia Euclid yÅ« klÄ­d ) , fl. 300 B.C. , Greek mathematician. Little is known of his life other than the fact that he taught at Alexandria, being associated with the school that grew up there in the late 4th cent. B.C. He is famous for his Elements, a presentation in thirteen books of the geometry and other mathematics known in his day. The first six books cover elementary plane geometry and have served since as the basis for most beginning courses on this subject. The other books of the Elements non-Euclidean geometry were deduced, one by Nikolai I. Lobachevsky (1826) and independently by J¡nos Bolyai (1832) and another by Bernhard Riemann (1854). A few modern historians have questioned Euclid's authorship of the Elements

58. Non-Euclidean: Definition And Much More From Answers.com
    Meaning 1 geometry based on axioms different from euclid s. Essay. Noneuclidean geometry. Even in Hellenistic times, euclid s Fifth Postulate was viewed  
    http://www.answers.com/topic/non-euclidean-geometry
    
    Extractions: showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Dictionary Encyclopedia WordNet Essay Wikipedia Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping non-Euclidean Dictionary non-Eu·clid·e·an nŏn yū-klĭd ē-ən Encyclopedia 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. Lobachevsky in 1826 and independently by the Hungarian Janos Bolyai in 1832. The second alternative, which allows no parallels through any external point, leads to the elliptic geometry developed by the German Bernhard Riemann in 1854. The results of these two types of non-Euclidean geometry are identical with those of Euclidean geometry in every respect except those propositions involving parallel lines, either explicitly or implicitly (as in the theorem for the sum of the angles of a triangle).

59. Euclid
    some easier way to learn geometry than by learning all the theorems. euclid replied, There is no royal road to geometry and sent the king to study.  
    http://www.crystalinks.com/euclid.html
    
    Extractions: Euclid of Alexandria is the most prominent mathematician of antiquity best known for his treatise on mathematics The Elements . The long lasting nature of The Elements must make Euclid the leading mathematics teacher of all time. For his work in the field, he is known as the father of geometry and is considered one of the great Greek mathematicians. Very little is known about the life of Euclid. Both the dates and places of his birth and death are unknown. It is believed that he was educated at Plato's academy in Athens and stayed there until he was invited by Ptolemy I to teach at his newly founded university in Alexandria. There, Euclid founded the school of mathematics and remained there for the rest of his life. As a teacher, he was probably one of the mentors to Archimedes Little is known of Euclid's life except that he taught at Alexandria in Egypt. According to Proclus (410-485 A.D.) in his Commentary on the First Book of Euclid's Elements , Euclid came after the first pupils of Plato and lived during the reign of Ptolemy I (306-283 B.C.). Pappus of Alexandria (fl. c. 320 A.D.) in his Collection states that Apollonius of Perga (262-190 B.C.) studied for a long while in that city under the pupils of Euclid. Thus it is generally accepted that Euclid flourished at Alexandria in around 300 B.C. and established a mathematical school there. Proclus also says that Euclid "belonged to the persuasion of Plato,'' but there exists some doubt as to whether Euclid could truly be called a Platonist. During the middle ages, Euclid was often identified as Euclid of Megara, due to a confusion with the Socratic philosopher of around 400 B.C.

60. Euclid (ca. 325-ca. 270 BC) -- From Eric Weisstein's World Of Scientific Biograp
    Allman, GJ Greek geometry from Thales to euclid. 1976. BulmerThomas, I. Selections Illustrating the History of Greek Mathematics, Vol.  
    http://scienceworld.wolfram.com/biography/Euclid.html
    
    Extractions: Greek geometer who wrote the Elements , the world's most definitive text on geometry. The book synthesized earlier knowledge about geometry, and was used for centuries in western Europe as a geometry textbook. The text began with definitions, postulates (" Euclid's postulates "), and common opinions, then proceeded to obtain results by rigorous geometric proof. Euclid also proved what is generally known as Euclid's second theorem the number of primes is infinite The beautiful proof Euclid gave of this theorem is still a gem and is generally acknowledged to be one of the "classic" proofs of all times in terms of its conciseness and clarity. In the Elements , Euclid used the method of exhaustion and reductio ad absurdum. He also discussed the so-called Euclidean algorithm for finding the greatest common divisor of two numbers, and is credited with the well-known proof of the Pythagorean theorem Neither the year nor place of his birth have been established, nor the circumstances of his death, although he is known to have lived and worked in Alexandria for much of his life. In addition, no bust which can be verified to be his likeness is known (Tietze 1965, p. 8). Elements

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