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43. Euclid's Algorithm euclid s Algorithm, Fundamental Theorem of Arithmetic. http://www.cut-the-knot.org/blue/Euclid.shtml

var MyPageLoc = document.location; var MyPageTitle = document.title; G o o g ... e Web CTK Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents Euclid's Algorithm Euclid's algorithm appears as the solution to the Proposition VII.2 in the Element's: Given two numbers not prime to one another, to find their greatest common measure. The algorithm is based on the following two observations: gcd (a, b) = b. This is indeed so because no number (b, in particular) may have a divisor greater than the number itself (I am talking here of non-negative integers.) If a = bt + r, for integers t and r, then gcd(a, b) = gcd(b, r). Indeed, every common divisor of a and b also divides r. Thus gcd(a, b) divides r. But, of course, . Therefore, gcd(a, b) is a common divisor of b and r and hence gcd(a, b) gcd(b, r). The reverse is also true because every divisor of b and r also divides a. Example Let a = 2322, b = 654. gcd(2322, 654) = gcd(654, 360) gcd(654, 360) = gcd(360, 294) gcd(360, 294) = gcd(294, 66) gcd(294, 66) = gcd(66, 30) gcd(66, 30) = gcd(30, 6)

44. Who Was Euclid? - History Fact Finder euclid (lived c. 300 B.C.) is remembered as one of the greatest mathematicians in the world. Historians believe that euclid probably studied at Athens under http://www.enotes.com/history-fact-finder/science-invention/who-was-euclid

utmSetVar('history_all'); Who Was Euclid? - History Fact Finder Welcome, guest! Login Join eNotes Entire Site Literature Science History Business Soc. Sciences Health Arts College Journals Literature Science History Business ... About Search History Fact Finder: Printable Version Download PDF Cite this Page Who was Euclid? Euclid (lived c. 300 B.C.) is remembered as one of the greatest mathematicians in the world. Historians believe that Euclid probably studied at Athens under students of Greek philosopher Plato (c. 428–348 or 347 B.C.) before he founded a school of mathematics in Alexandria, Egypt, during the reign of Ptolemy I (?–283 B.C.). Euclid is best known for his work titled Elements, a thirteen-volume textbook on the principles of mathematics. They include treatises on plane geometry (a branch of geometry dealing with plane figures), proportion (the relationship among parts), astronomy (the study of stars, planets, and heavenly bodies), and music. Although no one knows if all of the work in Elements was Euclid's or if he compiled the mathematical knowledge of his colleagues (such as Eudoxus), the work formed an important part of mathematics for 2,000 years. Even today, modified versions of the first few books of... [The entire page is 246 words long] Join eNotes The above is a free excerpt. Get total access to this content with the:

45. Welcome To Euclid SR Partners www.euclidsr.com/ 3k - Postulates from Wolfram MathWorldeuclid s fifth postulate cannot be proven as a theorem, although this was attempted by many people. euclid himself used only the first four postulates http://www.euclidsr.com/

46. Byrne's Edition Of Euclid An unusual and attractive edition of euclid was published in 1847 in England, edited by an otherwise unknown mathematician named Oliver Byrne. http://www.math.ubc.ca/~cass/Euclid/byrne.html

Main Euclid page Oliver Byrne's edition of Euclid 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.

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48. Euclid Managers euclid Managers underwrites, administers and markets a wide variety of insurance programs and products. We are privatelyheld, family-owned operation http://www.euclidmanagers.com/

Euclid Managers underwrites, administers and markets a wide variety of insurance programs and products. We are a privately-held, family-owned operation established over five decades ago. Our services are aligned into three unique areas: Employee Benefits Group Health, Dental, Life and Disability Programs Individual Professional Liability and Public Entity Internet, Tech, Media, Manufacturers, MPL ... and Professional Liability

49. Aleph Main Menu UCL LIBRARY SERVICES euclid ONLINE CATALOGUE. Signin Search Menu Results List Marked List Previous Searches Your Account ILL Store Request http://library.ucl.ac.uk/

50. Euclid And His Heritage: Conference At Oxford University A conference on the occasion of the publication, for the first time, of a complete digital edition of the oldest surviving manuscript of euclid s Elements. http://www.claymath.org/euclid/

Clay Mathematics Institute Dedicated to increasing and disseminating mathematical knowledge HOME ABOUT CMI PROGRAMS AWARDS ... PUBLICATIONS Euclid and His Heritage A Clay Mathematics Institute Conference at Oxford Bernard Sunley Lecture Theatre St. Catherine's College Oxford University October 7-8, 2005 A conference on Euclid and his Heritage is to be held on the occasion of the publication, for the first time, of a complete digital edition of the oldest surviving manuscript of Euclid's Elements. Written around 300 BC, this is the founding document of mathematics. The manuscript, dating from the year 888 AD, was copied by Stephen the Clerk in Constantinople, and since 1804 has been in the collection of the Bodleian Library. The digital edition was prepared by the Octavo corporation, which photographed the manuscript at the Bodleian in the fall of 2004. The conference will bring together classicists, historians, mathematicians and philosophers to provide a fresh look at Euclid's work, the transmission of Greek science from ancient to modern times, and the influence over twenty-three centuries of the Greek revolution in mathematics - the notion of proof, the systematic use of figures, and the organization of a complex and interconnected body of knowledge in the form of an axiomatic system. Abstracts of talks Registration information The conference is open to all, and there is no fee for attendees.. However, to facilitate planning, we ask you to fill out the online registration form below. Lunch will be served on October 7 and 8 in the elegant dining hall of St. Catherine's College, located next to the lecture hall where the conference is held. You are urged to make hotel reservations early in order receive a favorable rate. Two options for hotel accomodations are given below.

51. Damian Conway / Getopt-Euclid - Search.cpan.org s, 0.1.0......Getopteuclid-v0.0.9 26 Oct 2006, Getopt-euclid-v0.0.8 08 Oct 2006 Getopteuclid, Executable Uniform Command-Line Interface http://search.cpan.org/dist/Getopt-Euclid/

Home Authors Recent News ... Feedback in All Modules Distributions Authors permalink Damian Conway Getopt-Euclid This Release Getopt-Euclid-v0.1.0 Download Browse 02 Nov 2006 Other Releases Getopt-Euclid-v0.0.9 26 Oct 2006 Getopt-Euclid-v0.0.8 08 Oct 2006 Getopt-Euclid-v0.0.7 02 Oct 2006 Getopt-Euclid-v0.0.5 16 Feb 2006 Getopt-Euclid-0.0.4 04 Aug 2005 Getopt-Euclid-0.0.3 24 Jul 2005 Getopt-Euclid-0.0.2 22 Jul 2005 Links CPAN Testers Discussion Forum View/Report Bugs (11) Tools CPAN Testers PASS (105) FAIL (7) [ View Rating 2 Reviews Rate this distribution License Unknown Special Files Build.PL Changes MANIFEST META.yml ... README Modules Getopt::Euclid Executable Uniform Command-Line Interface Descriptions Getopt::Euclid::HierDemo hosted by perl.org , hardware provided by Shopping

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53. Epworth-Euclid UMC | Welcome! Epwortheuclid Church is located at University Circle in Cleveland, Ohio. http://www.epworth-euclid.org/

Providing a Beacon to God at University Circle Gathering to Worship This Sunday Music Ministry Our Church History Nurturing Our Faith Sunday School Study Groups - Lent 2008 Membership Development and Nurture Living Our Faith Church News - The Outlook Pastor's Message Neighborhood Lunch Mission Projects ... Realization of a Dream Fellowship Opportunities UM Youth Fellowship Explorers Contact Us staff@epworth-euclid.org Epworth-Euclid United Methodist Church 1919 E. 107 th Street Cleveland, Ohio 44106 Links: Check out North Coast District UMC activities at www.ncdistrictumc.org Information about East Ohio Conference of The United Methodist Church can be found at www.eocumc.com Reporting on national and world Church News is available at www.reporterinteractive.org , published by UMR Communications in Dallas. Prayer Room is open in Meditation Chapel for prayer and meditation any time the church is open. More Doors Are Opened With “Please” Than With Keys Success in God’s Kingdom Is Being Faithful Where You Are Inspirations on Our Lawn Join us in worship, Sundays 11 AM

54. Greek For Euclid: Contents Eu)klei/dou Stoixei~a, euclid s Elements, the classical textbook in geometry, is easy to read in the original ancient Greek, but its grammar and vocabulary http://www.du.edu/~etuttle/classics/nugreek/contents.htm

Reading Euclid This course combines Greek and Geometry to show how to read Euclid's Elements in the original language "I would make them all learn English; and then I would let the clever ones learn Latin as an honour, and Greek as a treat" Sir Winston Churchill Go immediately to Contents Introduction Eu)klei/dou Stoixei~a , Euclid's Elements, the classical textbook in geometry, is easy to read in the original ancient Greek, but its grammar and vocabulary are not those familiar from the usual course in elementary Greek, with peculiarities that make it difficult for the beginner. The text of the Elements that we have is written in the literary koinh/ typical of the 1st century AD. This course concentrates on exactly what is necessary to read Euclid, both in vocabulary and grammar. Its sole aim is to teach how to read this work, and similar texts in Greek mathematics, and not to compose Greek sentences, nor to read the Iliad or Plato. All necessary information is included in the course. A great amount of scholarship has been devoted to Euclid, mainly in Latin or German, and this course may expose some of it to a larger audience, to whom it has been largely inaccessible. For authoritative details, reference must be made to these sources, since the present one claims no expertise. There are many websites with information on Euclid and geometry. For example, look at the link to Euclid in the Seven Wonders website that is referenced in the Classics Index page, under the heading Pharos of Alexandria. As is typical of education on the Internet, many sites are poor, repetitive or childish, however.

55. Euclid's Proof Of The Infinitude Of Primes (c. 300 BC) euclid may have been the first to give a proof that there are infintely many primes. Even after 2000 years it stands as an excellent model of reasoning. http://primes.utm.edu/notes/proofs/infinite/euclids.html

Euclid's Proof of the Infinitude of Primes (c. 300 BC) (From the Prime Pages ' list of proofs Home Search Site Largest ... Submit primes Euclid may have been the first to give a proof that there are infinitely many primes. Even after 2000 years it stands as an excellent model of reasoning. Below we follow Ribenboim's statement of Euclid's proof [ , p. 3], see the page " There are Infinitely Many Primes " for several other proofs. Theorem. There are infinitely many primes. Proof. Suppose that p p p r are all of the primes. Let P p p p r +1 and let p be a prime dividing P ; then p can not be any of p p p r , otherwise p would divide the difference P p p p r =1, which is impossible. So this prime p is still another prime, and p p p r would not be all of the primes. It is a common mistake to think that this proof says the product p p p r +1 is prime. The proof actually only uses the fact that there is a prime dividing this product (see primorial primes The proof above is actually quite a bit different from what Euclid wrote. We now understand the integers as abstract objects, but the ancient Greeks understood them as lengths of line segments (multiples of some unit line segment). Where we talk of divisibility, Euclid wrote of "measuring," seeing length

56. * The Sisters Euclid * Kevin Breit * Folkalarm * www.kevinbreit.com/ 7k - Cached - Similar pages euclid - Definition from the Merriam-Webster Online DictionaryDefinition of euclid from the Merriam-Webster Online Dictionary with audio pronunciations, thesaurus, Word of the Day, and word games. http://www.kevinbreit.com/

57. Euclid, Elements (ed. Thomas L. Heath) As in the case of the other great mathematicians of Greece, so in euclid s case, we have only the most meagre particulars of the life and personality of the http://www.perseus.tufts.edu/cgi-bin/ptext?lookup=Euc. 1

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59. Euclid's Elements Of Geometry euclid s Elements is by far the most famous mathematical work of classical antiquity, and also has the distinction of being the world s oldest continuously http://farside.ph.utexas.edu/euclid.html

Euclid's Elements of Geometry Euclid's Elements is by far the most famous mathematical work of classical antiquity, and also has the distinction of being the world's oldest continuously used mathematical textbook. Little is known about the author, beyond the fact that he lived in Alexandria around 300 BCE. The main subjects of the work are geometry, proportion, and number theory. Most of the theorems appearing in the Elements were not discovered by Euclid himself, but were the work of earlier Greek mathematicians such as Pythagoras (and his school), Hippocrates of Chios, Theaetetus of Athens, and Eudoxus of Cnidos. However, Euclid is generally credited with arranging these theorems in a logical manner, so as to demonstrate (admittedly, not always with the rigour demanded by modern mathematics) that they necessarily follow from five simple axioms. Euclid is also credited with devising a number of particularly ingenious proofs of previously discovered theorems: e.g. , Theorem 48 in Book 1. The geometrical constructions employed in the Elements are restricted to those which can be achieved using a straight-rule and a compass. Furthermore, empirical proofs by means of measurement are strictly forbidden:

60. VMI Euclid New edition of the text of Heath, transformed into an extensively illustrated edition. http://www.visual-euclid.org/

Euclid Projects at the VMI Flash Animations of the Constructions of Book I (NEW September 18, 2003) VEE: The Visual Elements of Euclid (RESTORED September 18, 2003) VCE: The Visual Constructions of Euclid (RESTORED June 22, 2003) EVX: Euclid's Voyage into Chaos Byrne's Elements of Euclid The Chinese Proof of the Pythagorean Theorem Pythagorean Musical Arithmetic (new, December 2001) Note: Be sure and check out the online Elements of: D. E. Joyce at Clark University. Kepler Projects at the VMI Home page, visual-kepler.org (new, December 2001) Revised by Ralph Abraham, 18 September 2003

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