61. Geometry CHANGING SYSTEMS OF geometry FROM euclid TO KLEIN S PROGRAMME. Introduction geometry Origins of geometry; The Golden Age of Greek http://www.scit.wlv.ac.uk/university/scit/modules/mm2217/geometry.htm

CHANGING SYSTEMS OF GEOMETRY FROM EUCLID TO KLEIN'S PROGRAMME Introduction Geometry: Origins of Geometry The Golden Age of Greek Mathematics Projective Geometry New Geometry, New Worlds ... Module Leader These pages are maintained by M.I.Woodcock.

62. New Geometries alternatives produced contradiction. Main point there are geometries different from euclid s. How can new geometry be correct? What about http://www.scit.wlv.ac.uk/university/scit/modules/mm2217/ng.htm

NEW GEOMETRIES, NEW WORLDS Euclid's geometrical system was used and developed century after century and no one doubted its truth. However, a few thinkers including Euclid himself were disturbed by 2 of his axioms. a) A line segment can be extended as far as one pleases in either direction. b) Parallel axiom - says that through a point P not on a line L there passes one and only one line M (in the plane of P and L ) and does not meet L no matter how far M and L are extended. These axioms are open to doubt because we cannot say we know what happens in physical space - it may be true around us but what about 10 miles up in space? Even limited space in which we move they might not be true - projective geometry. Euclid doesn't use the parallel axiom unless he has to. Also he uses line segments not lines. Many mathematicians tried to work with the parallel axiom - they sought to either deduce it from other axioms or find a more acceptable substitute. 1800 - parallel axiom labelled the scandal of Geometry. Saccheri had a brand new idea - he argued given a line and a point P then either: a) There is exactly one parallel to L through P , or b) there are no parallels to L through P , or c) there are at least 2 parallels to L through P Alternative (a) was Euclid's parallel axiom. Suppose it was replaced by (b) and the latter together with the other 9 axioms of Euclid were shown to lead to contradictory theorems. Then (b) couldn't be correct, similarly with (c) - then it would follow that Euclid's axiom is the only one, - so Saccheri said.

63. Reference Pieces On Space One interesting question about the assumptions for euclid s system of geometry is the difference between the axioms and the postulates. Axiom is from http://www.friesian.com/space.htm

Euclid's Axioms and Postulates One interesting question about the assumptions for Euclid's system of geometry is the difference between the "axioms" and the "postulates." "Axiom" is from Greek , "worthy." An axiom is in some sense thought to be strongly self-evident. A "postulate," on the other hand, is simply postulated , e.g. "let" this be true. There need not even be a claim to truth, just the notion that we are going to do it this way and see what happens. Euclid's postulates, indeed, could be thought of as those assumptions that were necessary and sufficient to derive truths of geometry, of some of which we might otherwise already be intuitively persuaded. As first principles of geometry, however, both axioms and postulates, on Aristotle's understanding, would have to be self-evident. This never seemed entirely quite right, at least for the Fifth Postulate hence many centuries of trying to derive it as a Theorem. In the modern practice, as in Hilbert's geometry, the first principles of any formal deductive system are "axioms," regardless of what we think about their truth which in many cases has been a purely conventionalistic attitude. Given Kant's view of geometry, however, the Euclidean distinction could be restored: "axioms" would be

64. Euclid of Sir Thomas Heath s translation of The Elements, I have graphically glossed Books I IV to produce a reader friendly version of euclid s Plane geometry. http://www.furman.edu/~jpoole/mth15hp/mathematicians/euclid.htm

Euclid The Elements , Books I - IV Book I Definitions Postulates and Common Notions Propositions Book II Definitions Propositions Book III Definitions Propositions Book IV Definitions Propositions Using the text of Sir Thomas Heath's translation of The Elements , I have graphically glossed Books I - IV to produce a reader friendly version of Euclid's Plane Geometry. The four books contain 115 propositions which are logically developed from five postulates and five common notions . In the first proposition, Proposition 1, Book I, Euclid shows that, using only the postulates and common notions, it is possible to construct an equilateral triangle on a given straight line. In the hundred fifteenth proposition, Proposition 16, Book IV, he shows that it is possible to inscribe a regular 15-gon in a circle. And along the way he develops many beautiful, interesting, captivating, and pleasing results. You are invited to read this part of one of the world's great books. This presentation grew out of material developed for a mathematics course, Ideas in Mathematics , offered for liberal arts students at Furman. Every interested person, ninth grade student to ninety year old retiree, should be able to read most, if not all, of the material; that is the intended audience.

65. HighBeam Research: ELibrary Search: Results geometry became more general, and euclid s geometry took on the appearance difference Porisms. euclid s geometry formed the basis http://www.highbeam.com/library/search.asp?FN=AO&refid=ency_refd&search_dictiona

66. HighBeam Research: ELibrary Search: Results Fields of study included mathematics ( euclid s geometry, c 300 bc); astronomy (heliocentric theory of Aristarchus, 310230 BC; Julian calendar, 45 BC http://www.highbeam.com/library/search.asp?FN=AO&refid=ency_refd&search_almanacs

67. A Formal Theory For Geometry euclid s geometry was still regarded as a model of logical rigor, a shining example of what a wellorganized scientific discipline ideally ought to look like. http://www.math.psu.edu/simpson/papers/philmath/node15.html

68. The Geometry Of Euclid The geometry of euclid. The logic of Aristotle and the geometry of euclid are universally recognized as towering scientific achievements of ancient Greece. http://www.math.psu.edu/simpson/papers/philmath/node13.html

69. High School Euclid Paper can be proven. 7 Lobachevsky s geometry grew out of his unsuccessful attempts to prove euclid s parallel postulate. 8 Zeno of http://www.obkb.com/dcljr/euclidhs.html

High school Euclid paper jump to... text of paper Endnotes Bibliography bottom of page Euclid and his Elements One of the most influential mathematicians of ancient Greece, Euclid flourished around 300 B.C. Not much is known about the life of Euclid. One story which reveals something about Euclid's character concerns a pupil who had just completed his first lesson in geometry. The pupil asked what he would get from learning geometry. So Euclid told his slave to give the pupil a coin so he would be gaining something from his studies. Included in the many works of Euclid is Data , concerning the solution of problems through geometric analysis, On Divisions (of Figures) , the Optics , the Phenomena , a treatise on spherical geometry for astronomers, several lost works on higher geometry, and the Elements , a thirteen volume textbook on geometry. The Elements , which surely became a classic soon after its publication, eventually became the most influential textbook in the history of civilization. In fact, it has been said that apart from the Bible , the Elements is the most widely read and studied book in the world.

70. Discussion The type of noneuclidean geometry where axiom 5 is the opposite of euclid s fifth axiom (there is more than one line passing through a given point, which is http://cvu.strath.ac.uk/courseware/msc/jgraves/

71. The Origins Of Geometry euclid based his geometry on five fundamental assumptions Postulate I For every point P and for every point Q not equal to P there exists a unique line that http://www.math.uncc.edu/~droyster/math3181/notes/hyprgeom/node4.html

!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 3.0//EN"> Next: Spherical Geometry Up: Neutral and Non-Euclidean Geometries Previous: List of Tables The Origins of Geometry In the beginning geometry was a collection of rules for computing lengths, areas, and volumes. Many were crude approximations derived by trial and error. This body of knowledge, developed and used in construction, navigation, and surveying by the Babylonians and Egyptians, was passed to the Greeks. The Greek historian Herodotus (5th century B C .) credits the Egyptians with having originated the subject, but there is much evidence that the Babylonians, the Hindu civilization, and the Chinese knew much of what was passed along to the Egyptians. The Babylonians of 2,000 to 1,600 B C knew much about navigation and astronomy, which required a knowledge of geometry . They also considered the circumference of the circle to be three times the diameter. Of course, this would make a small problem. This value for carried along to later times. The Roman architect Vitruvius took . Prior to this it seems that the Chinese mathematicians had taken the same value for . This value for was sanctified by the ancient Jewish civilization and sanctioned in the scriptures. In

72. Who Was Euclid ? Einstein s theory of General Relativity like euclid s geometry was found to have limits in it applicability. Today we use Pythagoras s http://www.surferz.net/~marina/euclid.html

Who Was Euclid ? During Alexandria's heyday Eretosthenese had calculated the diameter of the earth to within 1% by measuring the difference in the angle of the noonday sun in distant cities. It would take centuries and the persecution of Galileo before west would again understand that the earth was spherical. All in all Alexandria was a shining light of learning for almost 700 years. It was this city that Euclid called "home". Euclid's lived from 325 BCE to around 265 BCE. His contemporaries included Eretosthenese, Eudemus of Rhodes, Autolycus of Pitane. He was too young to have studied with Plato, but many of Plato's students lived at the same time as Euclid. For the most part Euclid's though is Platonic. For a Platonist the reality we see around us is merely a shadow of the real truth which lies in the realm of pure thought. The Mac history archives at the School of Mathematics and statistics says this about Euclid: "In his aim he was a Platonist, being in sympathy with this philosophy, whence he made the end of the whole "Elements" the construction of the so-called Platonic figures. " (1) http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Euclid.html To the same ends Euclid and Plato were very close to being Pythagoreans. Pythagoras is known for his teaching the equation a^2+b^2=c^2 a and b being the legs of a right angle triangle c being the hypotenuse. This equation has applications in all sorts of physical phenomenon. Einstein's famous E=MC^2 equation can be derived from a^2+b^2=c^2.(8) Pythagoras is less well known for his belief that mathematics is the basis of reality.

73. Edward Tufte, William Penn Hotel, Pittsburgh PA, 4/14/2000 euclid s geometry—Edward Tufte owns Ben Johnson s (signed) copy of euclid s geometry. (This was just one of the highlights of his presentation. http://www.ideawatch.org/tufte414.htm

back to guru watch Idea Watch Home Edward Tufte, William Penn Hotel, Pittsburgh PA, April 14, 2000 Notes compiled by Theresa Marchwinski I first saw and heard Edward Tufte at the 1997 STC Annual Conference where he spoke as that year's STC Honorary Fellow. When I discovered he was speaking in Pittsburgh, I gladly made the five-hour trip from Cincinnati to see him. He is one of the most polished speakers you will ever see. His presentation was so practiced and his content so clear and relevant that everything he said fit into mental contexts and pictures stronger and more lasting than the ubiquitous bulleted-list style of presentations that most presenters rely on. (He used no overheads or presentation software.) All attendees were given a print of the Charles Joseph Minard's classic "Napoleon's March to Moscow." (I framed the print and hung it in my office where it provides great inspiration to me as one of the most elegant works of information design.) The following three books by Edward Tufte were also included as part of the registration fee. The Visual Display of Quantitative Information

74. GEOMETRY D FALL 2004 CONTENTS. Ch. 1, euclid s geometry. 1. A First Look at euclid s Elements. Ch. 5, Area. 22. Area in euclid s geometry. 23. Measure of Area Functions. 24. Dissection. 25. http://www.math.uu.se/~thomase/geometriD/geometriD2004.html

GEOMETRY D FALL 2004 COURSE LITERATURE Robin Hartshorne, Geometry: Euclid and Beyond , Springer, Undergraduate Texts in Mathematics (2000) From the cover: "The text is intended for junior- to senior-level mathematics majors." Explanation: The text is intended for third and fourth year students specializing in mathematics. My review of the text: I think this text deals with difficult topics in a most sophisticated way, in other words, it is truly D-level. I placed an order for 15 copies at Studentbokhandeln and they should be available by the end of April. CONTENTS Ch. 1 Euclid's Geometry A First Look at Euclid's Elements Ruler and Compass Constructions Euclid's Axiomatic Method Construction of the Regular Pentagon Some Newer results Ch. 2 Hilbert's Axioms Axioms of Incidence Axioms of Betweenness Axioms of Congruence for Line Segments Axioms of Congruence for Angles Hilbert planes Intersection of Lines and Circles Euclidean planes Ch. 3 Geometry over Fields The Real Cartesian Plane Abstract Fields and Incidence Ordered Fields and Betweenness Congruence of Segments and Angles Rigid Motions and SAS Non-Archimedean Geometry Ch. 4

75. Various Geometries Put another way, in euclid s geometry, some properties of figures (lengths, angles, areas) remain invariant under the group of rigid motions. http://www.cut-the-knot.org/triangle/pythpar/Geometries.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site 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.

76. Non-Euclidean Geometries, Discovery to publish his research because he suspected the mediocre mathematical community would not be able to accept a revolutionary denial of euclid s geometry. http://www.cut-the-knot.org/triangle/pythpar/Drama.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Non-Euclidean Geometries 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.

77. Euclid Of Alexandria player shape ideas? From their heads, which they learned along time ago, thanks to euclid, who invented geometry. His ideas change http://asijonline.net/math/euclid/euclid.htm

Welcome to Euclid's Home Page! Euclid was a very famous mathematician. He was called the father of geometry. The best thing he was known for was his 13 volume The Elements Almost nothing is known about Euclid's life, but they do know what he was like as a historical figure, and here it is. Without him the world might not be as far in the world of mathematics or developed. As a mathematician he is ranked probably one of the highest, for writing The Elements , the base of geometry. He was also a teacher and when he founded a school in Alexandria, Egypt he taught there. He was a nice person, but he was often sarcastic. He most likely came from a rich family because when he was little he went to Plato's school and in ancient Greece only the rich had education. After he finished school he most likely went to Alexandria where he discovered geometry and then wrote The Elements which he based on what he discovered. Euclid of Alexandria was confused with Euclid of Megara for a while, who was also a mathematician. Euclid was a popular name back then, so he was confused with a lot of Euclid's of that time. But what set him apart from all the others was his amazing inventions. The great contribution to the world my mathematician made was basically geometry. In his book

78. 4. Voting And Elections: Enter Kenneth Arrow the other axioms of euclid except that one uses the negation of the euclidean 5th postulate as an axiom, which is no better or worse than euclid s geometry. http://www.ams.org/new-in-math/cover/voting-arrow.html

Voting and Elections: Enter Kenneth Arrow Feature Column Archive 4. Enter Kenneth Arrow The fact that different seemingly appealing methods can elect different winners suggests a change in perspective, from that of an election system delivering the will of the people to one in which the results are as consistent, fair, or equitable over a range of possible election patterns that the voters might provide to the system. What rules should an election system obey so that we will think it is a good system? What rules should an election system obey so that we will think it is better than some other system? Insight using this approach was provided by Kenneth Arrow, who developed a collection of fairness, consistency or reasonability conditions (axioms) that any fair election method was to obey. What is an example of such a fairness axiom? Suppose that one has an election decision procedure based on preference ballots. Suppose a particular election where candidate A is ranked at the top of 9 schedules, as shown below: Suppose the decision method assigns A as victor in the election. Now imagine an election in which all the ballots are the same except that instead of having 9 votes for the schedule shown above, one has 10 votes for this schedule. Would it make any sense that the decision procedure applied to these ballots now elect someone other than A? If it seems unreasonable, we might state that we require any fair election method to obey this rule. Arrow developed a variety of fairness conditions that he thought any reasonable election method should obey. He then proceeded to show that for elections where there were more than 2 candidates no election decision method obeyed all of the rules! Since Arrow's original work many investigators have developed a wide array of desirable fairness rules and showed results similar to Arrow's Theorem.

79. Writing Physics In one paragraph Einstein reminds us of one of the great temples in intellectual history — euclid s geometry — but also raises some doubts about how solid http://www.cas.muohio.edu/~devriepl/phy286/writing/WritingPhysics.htm

Writing Physics Albert Einstein gazes out at me from the cover of a small paperback book called Relativity: The Special and the General Theory , looking not at all scary-looking, in fact, like a benign uncle who is about to tell me a story. The blurb on the cover says: A CLEAR EXPLANATION THAT ANYONE CAN UNDERSTAND. Albert Einstein is going to explain the theory of relativity to me. To me? That's what it says. I've wanted to get the book ever since two science professors at Gustavus Adolphus College mentioned that it was a model of clear linear writing. At first that surprised me; I hadn't expected Einstein's theory to be reducible to plain English. But on second thought it made sense. If clear writing is clear thinking, a mind clear enough to think of the theory of relativity would be likely to express itself simply and well. The burden was therefore on me. Could a lifelong science boob follow Einstein's train of thought? The only way to find out was to find out. I opened the book, which Einstein wrote, incidentally, in 1916, and plunged in: In paragraph two he tells us more about Euclid's "noble building": Geometry sets out from certain conceptions such as "plane," "point" and "straight line," with which we are able to associate more or less definite ideas, and from certain simple propositions (axioms) which, by virtue of these ideas, we are inclined to accept as "true." Then, on the basis of a logical process, the justification of which we feel ourselves compelled to admit, all remaining propositions are shown to follow from those axioms

80. WannaLearn: Geometry: Euclid And Beyond geometry euclid and Beyond. A WannaLearn Recommended Instructional Book Pick. Hardcover geometry euclid and Beyond by Robin Hartshorne. List http://www.wannalearn.com/Academic_Subjects/Mathematics/Geometry/0387986502.shtm

Geometry: Euclid and Beyond A WannaLearn Recommended Instructional Book Pick Hardcover 08 June, 2000 Springer Verlag ISBN: 0387986502 Geometry: Euclid and Beyond by Robin Hartshorne List Price: Amazon.com's Price: You Save: Usually ships within 24 hours Find Out More About This Book Buy from Amazon.com Buy from Amazon.ca Buy from Amazon.co.uk Comments from people who have read this book Rating: 4/5 - This book and course is not for the faint of heart! This is without exception the hardest math course I have ever taken. Your understanding of the concepts is pertinent. I had to read the 1st chapter over five times just to understand projective geometry. Hartshorne tries to simplify the material but only so much can be done. It is just a hard course, period. The book does contain many example and logical proofs but be ready to burn the midnight candle on this one. Rating: 5/5 - A stunning book Hartshorne is a leading mathematician known for work in rather abstract geometry (see his book ALGEBRAIC GEOMETRY). He takes Euclid's ELEMENTS as great mathematics, no mere genial precursor, and collates it with Hilbert's FOUNDATIONS OF GEOMETRY. Of course Harshorne proves that Euclid needed the parallel postulate, by exhibiting a non-Euclidean geometry. He gives a very pretty compass and straight-edge Euclidean theory of circles, which then turns into the Poincare plane model for hyperbolic geometry. He also proves that Euclid needed the method of exhaustion for volumes of solids: he gives the agreeably simple Dehn invariant proof that even a cube and a tetrahedron of equal volumes are not decomposable into congruent parts. It is a famous proof, rarely seen, and a beautiful use of the modern algebraic viewpoint in classical geometry. I had always supposed it must be hard but it is not.

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