41. Sieve Of Eratosthenes Repeatedly take the next unmarked integer as the next prime and mark every multipleof the prime. Note Invented by eratosthenes of cyrene (276 BC 194 BC). http://www.nist.gov/dads/HTML/sieve.html

sieve of Eratosthenes (algorithm) Definition: An algorithm to find all prime numbers up to a certain N. Begin with an (unmarked) array of integers from 2 to N. The first unmarked integer, 2, is the first prime. Mark every multiple of this prime. Repeatedly take the next unmarked integer as the next prime and mark every multiple of the prime. Note: Invented by Eratosthenes of Cyrene (276 BC - 194 BC). For example, here's a beginning array. Since 2 is unmarked, it is our first prime. We mark every second integer, that is, 4, 6, 8, 10, 12, etc. The next unmarked integer is 3, so it is prime, and we mark every third integer, i.e., 6, 9, 12, etc. Note that we mark 6, 12, 18, etc. again. Now 5 is the next prime, and we mark every fifth integer. The only new integer marked in range is 25. From here we find the primes 7, 11, 13, 17, etc. To optimize, when we find the prime n, we can begin marking at n , since any composite less than that is a multiple of a lesser prime, and so will have been marked earlier. As a corollary, we can stop marking when n is greater than our range. That is, any unmarked numbers greater than the square root of the range are primes.

42. Zeal.com - United States - New - Library - Sciences - Mathematics - Mathematicia 4. eratosthenes of cyrene http//wwwgroups.dcs.st-and.ac.uk/~history/Mathematicians/E Biography of Eratosthenes, who very accurately measured the http://www.zeal.com/category/preview.jhtml?cid=572631

43. Number Patterns, Curves & Topology itself. eratosthenes of cyrene (276194 BC) conceived a method of identifyingprime numbers by sieving them from the natural numbers. http://www.camosun.bc.ca/~jbritton/jbfunpatt.htm

Investigating Patterns Number Patterns Fun with Curves TOPIC LINKS TOPIC 1 (Prime Numbers / Magic Squares) Title: Sieve of Eratosthenes Comment: A natural number is prime if it has exactly two positive divisors, 1 and itself. Eratosthenes of Cyrene (276-194 BC) conceived a method of identifying prime numbers by sieving them from the natural numbers. Web page uses the sieve to find all primes less than 50. Includes a link to a Sieve of Eratosthenes Applet which also begins with a size or upper boundary of 50. Eratosthenes' Sieve contains a similar applet preset to find all primes less than 200. Both applets require a JAVA-capable browser. Title: Prime Number List Comment: Once you have entered the lower bound and upper bound, this javascript applet will display all prime numbers within the selected range. Title: Prime Factorization Machine Comment: A positive integer (natural number) is either prime or a product of primes. This applet decomposes any positive integer less than 1,000,000 into its prime factors. The bigger the number, the longer it will take. Requires a JAVA-capable browser. Title: Comment: Includes a link to Mini-Lessons demonstrating how to find the Common Divisor Factor (GCF) or Greatest Common Divisor (GCD) and the Least Common Multiple (LCM) of two or more natural numbers using prime factorization. Features an interactive applet with detailed explanations and solutions.

44. Sieve Of Eratosthenes eratosthenes of cyrene (276194 BC) was the first to estimate accuratelythe diameter of the earth. For several decades, he served http://www.camosun.bc.ca/~jbritton/jberatosthenes.htm

Sieve of Eratosthenes A natural number is prime if it has exactly two positive divisors, 1 and itself. Otherwise it is composite . The number 1 is usually regarded as being neither prime nor composite. Eratosthenes of Cyrene (276-194 BC) was the first to estimate accurately the diameter of the earth. For several decades, he served as the director of the famous library in Alexandria. He was highly regarded in the ancient world, but unfortunately only fragments of his writing have survived. Eratosthenes devised a 'sieve' to identify prime numbers. A sieve is like a strainer that you drain spaghetti through when it is done cooking. The water drains out, leaving your spaghetti behind. The Sieve of Eratosthenes drains out composite numbers and leaves prime numbers behind. Make a list of all the integers less than or equal to n (and greater than one). Strike out the multiples of all primes less than or equal tothe square root of n Then the numbers that are left are the primes. Note that if one divisor or factor of a number (other than a perfect square) is greater than its square root, then the other factor will be less than its square root. Hence all multiples of primes greater than the square root of n need not be considered.

45. Eratosthenes . eratosthenes of cyrene Born 276 BC in Cyrene, North Africa, now Liberia. Hewas a highly regarded mathematician who worked in the Museum in Alexandria. http://alexandrias.tripod.com/eratosthenes.htm

Don't forget to visit our discussion forum!!! Eratosthenes How Eratosthenes measured the diameter of Earth. (Taken from Cambridge Latin Course pg. 157) " At noon, when Eratosthenes had calculated that the sun was directly overhead in Syene, he measured the length of the shadow of an object in Alexandria. From this he could calculate the angle A between the sun's rays and the object. Since the sun's rays are parallel, by simple geometry angle B is the same size as angle A. Knowing angle B and the distance between Syene and Alexandria (which was north of Syene), he was able to calculate the circumference of the earth. " Eratosthenes of Cyrene Born: 276 BC in Cyrene, North Africa, now Liberia. He was a highly regarded mathematician who worked in the Museum in Alexandria. He was the director of the library of Alexandria for couple of decades. He was the first person to accurately calculate the diameter of Earth using only simple geometry. He also created "Sieve of Eratosthenes" a method of identifying prime number. Eratosthenes also measured the tilt of the Earths' axis by 23.5, which gave us the seasons. He was bind at an old age and died of voluntary starvation in 195 B.C. to scientists_of_alexandria@altavista.com

46. Bibliography Alfeld, Peter. eratosthenes of cyrene. 02Jun-1998. http//www html. eratosthenes of cyrene. Discovering World History. 2000. http http://alexandrias.tripod.com/bibliography.htm

Don't forget to visit our discussion forum!!! Alfeld, Peter. Eratosthenes of Cyrene . 02-Jun-1998. http://www.math.utah.edu/~alfeld/Eratosthenes.html " Hero of Alexandria ." Funk and Wagnalls Online. http://www.funkandwagnalls.com/encyclopedia/low/articles/h/h011000819f.html "Ctesibius." Encarta Online Deluxe. 1999. "Eratosthenes." Encarta Online Deluxe. 1999 . " Ctesibius of Alexandria." Britannica Online. http://www.britannica.com/bcom/eb/article/0/0,5716,28547+1+28098,00.html "Hero of Alexandria." Britannica Online. http://www.britannica.com/bcom/eb/article/0/0,5716,41048+1+40189,00.html "Eratosthenes of Cyrene." Discovering World History. "Ctesibius of Alexandria" Discovering World History. "Hero of Alexandria" Discovering World History. Gillispie, Charles Coulston. Dictionary of scientific biography . New York: Scribner, 1980. Cambridge Latin Course Unit 2 *. University of Cambridge. *most pictures taken from Cambridge Latin Course Revised: May 07, 2000

47. Florentina Georgeta Stancu-Soare (1971) eratosthenes of cyrene. London Oxford University Press. O’Connor, JJ Robertson E, F. (1999) eratosthenes of cyrene . Off the World Wide Web. http://www.astro.utoronto.ca/~seaquist/sci199y/biblio/stancu1.html

Florentina Georgeta Stancu-Soare E. R. Seaquist SCI 199Y1 – L0111 October 28, 2002 Eratosthenes – A triumph of Mathematics Works Used Asimov, Isaac. (1994) Asimov’s Chronology of Science and Discovery . New York: Harper Collins Publishers. Coulston, Charles. (1980) Dictionary of Scientific Biography . (Vol.3) New York: Charles Scribner’s Sons. Donovan, Dennis P. (1996) Eratosthenes Finds Diameter of Earth! .Off the World Wide Web http://math.rice.edu/~ddonovan/Lessons/eratos.html Daintith, John and Sarah Mitchell. (1994) Biographical Encyclopedia of Scientists nd edition, Vol.1) London: Market House Book Ltd. Institute of Physics Publishing Bristol and Philadelphia. Fraser, Peter Marshall. (1971) Eratosthenes of Cyrene . London: Oxford University Press. Lasky, Kathryn. (1994). The Librarian who measured the earth st edition. Boston: Joy Street Books. Yong, Robyn and Zoran Minderovic. (1998) Notable Mathematicians From Ancient Times to the Present . Detroit: Gale Research. Eratosthenes of Cyrene . Off the World Wide Web www-gap.dcs.st-and.ac.uk/~history/ Mathematicians/Eratosthenes.html

48. Online Book rotate around a stationary Earth. eratosthenes of cyrene Eratosthenesof Cyrene (ca. 284ca. 192 BC) Among Eratosthenes' accomplishments http://www.physics.sfasu.edu/astro/astronomylinks/all1.html

Ancient Astronomy A Brief Introduction to Archaeoastronomy The study of the astronomical practices, celestial lore, mythologies, religions and world-views of all ancient cultures we call archaeoastronomy. America's Stonehenge Along with it's astronomical alignments, America's Stonehenge continues to be one of the most fascinating archaeological discoveries of the century. Ancient Astronomy Links Some selected links to pages with information on Ancient Astronomy Calendars and Ancient Astronomy in Egypt, India, Maya, and Mesopotamia. Ancient Astronomy Timeline A timeline of ancient astronomy from 4000 B.C. to around 140 A.D. Ancient Mesopotamia: Astronomy and Calendry Part of a history of astronomy course with links to photographs and explanations. Babylonian Planetary Theory and the Heliocentric Concept Gathered ideas regarding the view of the cosmos by ancient Babylonians as well as new interpretations are on display here. With data on periods and planetary motions. Chinese Astronomy Astronomy truly is an ancient science in China. In fact, mankind's first record of an eclipse of the Sun was made in China in 2136 BC. Egyptian Estimates of the Size and Shape of the Earth How did ancient Egyptians measure the size of the Earth?

49. ERATOSTHENES ERATOSTHENES c.274 c.194 BC Greek Scholar eratosthenes of cyrene studied inAlexandria and Athens. In Alexandria he was director of the great library. http://www.hyperhistory.com/online_n2/people_n2/persons2_n2/eratosthenes.html

ERATOSTHENES c.274 - c.194 BC Greek Scholar Eratosthenes of Cyrene studied in Alexandria and Athens. In Alexandria he was director of the great library. This scholar of natural history did his most outstanding work in mathematics and geography. Eratosthenes calculated the circumference of the earth correctly. This he did by observing the different angle's that the sun's rays fall in two cities 500 miles apart. He correctly assumed the Sun's distance to be so great that the rays are practically parallel when they reach the earth. www link : From the University of St. Andrews, Scotland School of Mathematics Biography

50. Mappa.Mundi Magazine - Locus - Triangulation earth. » A bio of eratosthenes of cyrene from the School of Mathematicaland Computational Sciences, University of St Andrews. http://mappa.mundi.net/locus/locus_008/

Stephanie Faul is director of public relations for the AAA Foundation for Traffic Safety. A long-time writer and editor, she has been a frequent contributor to on-line and old-line media. Locus is a series about words about places and how language, culture and society have created and used these words down through the ages. Locus Archives Links Relating to this edition Triangulation - Geog 121: Partitioning the Land from the Geographic Information Science department at Penn State University. Geodesy : geodetic surveyors and the experiences of those who surveyed America. A publication of the NOAA Central Library and the Office of High Performance Computing and Communications. How Eratosthenes measured the earth A bio of Eratosthenes of Cyrene from the School of Mathematical and Computational Sciences, University of St Andrews. The History of Surveying at the Purdue University Department of Agricultural and Biological Engineering. A bio of Jean Picard from the MacTutor History of Mathematics. A bio of Giovanni Domenico Cassini from the MacTutor History of Mathematics.

51. Enciclopédica Geodésica | Erathostenes Thus may it be, and let any one who sees this offering say Thisis the gift of eratosthenes of cyrene . eratosthenes of cyrene. http://geodesia.ufsc.br/Lexicon/e/Erathostenes.htm

ERATOSTHENES Nacido em Cyrene 275 BC Estudos em Alexandria e Atenas Diretor da biblioteca de Alexandria em 236 AC desde 195 AC sego † 194 BC Erathostenes Erathostenes de Cyrene Literatura Eratosthenes was born in Cyrene which is now in Libya in North Africa. His teachers included the scholar Lysanias of Cyrene and the philosopher Ariston of Chios who had studied under Zeno, the founder of the Stoic school of philosophy. Eratosthenes also studied under the poet and scholar Callimachus who had also been born in Cyrene. Eratosthenes then spent some years studying in Athens. The library at Alexandria was planned by Ptolemy I Soter and the project came to fruition under his son Ptolemy II Philadelphus. The library was based on copies of the works in the library of Aristotle . Ptolemy II Philadelphus appointed one of Eratosthenes' teachers Callimachus as the second librarian. When Ptolemy III Euergetes succeeded his father in 245 BC and he persuaded Eratosthenes to go to Alexandria as the tutor of his son Philopator. On the death of Callimachus in about 240 BC, Eratosthenes became the third librarian at Alexandria, in the library in a temple of the Muses called the Mouseion. The library is said to have contained hundreds of thousands of papyrus and vellum scrolls. Despite being a leading all-round scholar, Eratosthenes was considered to fall short of the highest rank.

52. Math World: Math Central Gr. 6, Ch. 3 eratosthenes of cyrene Find out about the Greek astronomer, mathematician, and librarianEratosthenes at this site, created by Peter Alfeld, a University of http://www.eduplace.com/math/mathcentral/grade6/603m.html

Division Around the World Internet Links Following are links to Internet resources that can be used to support instruction of the Math World feature for this chapter. Eratosthenes of Cyrene Find out about the Greek astronomer, mathematician, and librarian Eratosthenes at this site, created by Peter Alfeld, a University of Utah mathematics professor. The site also offers links for more information about Eratosthenes and a version of the Sieve of Eratosthenes, which can be used on Java-enabled browsers. http://www.math.utah.edu/~alfeld/Eratosthenes.html The Prime Pages What a prime location! This Web page provides information about all things prime, including research, records, and resources. http://www.utm.edu/research/primes/ Mathematics Center Math Central Grade 6 Education Place ... Terms and Conditions of Use

53. Web Articles: 3. People eratosthenes of cyrene (hyperlinked biography, from the MacTutor Historyof Mathematics Archive, University of St Andrews) {March, 2002}; http://www.ihrinfo.ac.uk/maps/textpeople.html

WWW-Virtual Library: History Map History / History of Cartography: THE Gateway to the Subject HOME (main menu) WEB ARTICLES menu INDEX complete SITEMAP What the site is ABOUT WHAT'S NEW Web articles and commentaries on specific topics in the History of Cartography People (The only online bibliography for the history of cartography. The monthly additions are indicated thus, e.g. will find entries added at any time in that year) Make sure to consult its before using any of the texts below Back to the Web Articles Main Menu online exhibitions works with a theoretical (often literary) dimension Battista AGNESE AL-BIRUNI. Abu Arrayhan Muhammad ibn Ahmad al-Biruni (hyperlinked biography, from the MacTutor History of Mathematics Archive, University of St Andrews) ALBUQUERQUE. Luís de Albuquerque (1917-1992) [biographical notes and a bibliography of his writings by Francisco Contente Domingues, 1999] ' ALEPH': Geographical Fun Atlas 'László ALMASY: The real Hungarian desert explorer' (a short illustrated article, also in German - Zsolt Török; and the same author's 'Desert Love' - Mercator's World 2:6, 1997)

54. Eratosthenes From FOLDOC Recommended Reading PM Fraser, eratosthenes of cyrene (Oxford, 1972).A Dictionary of Philosophical Terms and Names. 200110-29 . http://www.swif.uniba.it/lei/foldop/foldoc.cgi?Eratosthenes

55. HistoryCenter.net eratosthenes of cyrene studied in Alexandria and Athens. In Alexandriahe was director of the great library. This scholar of natural http://www.historycenter.net/science-detail1.asp?ID=8&TimeZone=5

56. COMP203 (2002) Lab Project 4: Caches eratosthenes of cyrene. eratosthenes of cyrene was born around 276BC. He made a number of contributions towards science, including http://www.mcs.vuw.ac.nz/courses/COMP203/2002/Labs/lab4/

Up to COMP203 (2002) Home Page COMP203 (2002) Lab Project 4: Caches Contents What this lab project is about Eratosthenes of Cyrene The Sieve Algorithm How to prepare for the lab ... Marking guide What this lab project is about In this lab you will modify an algorithm to get the best performance from the cache. The algorithm in question is the Sieve of Eratosthenes, which is a method for finding prime numbers without doing any division operations . The algorithm works by making repeated passes through a large array of numbers, 'sieving' out non-primes on each pass. It is this array that is brought into the cache piece by piece. What you have to do is to find ways of modifying the basic algorithm given to you so that it produces fewer caches misses and consequently has better performance. Eratosthenes of Cyrene Eratosthenes of Cyrene was born around 276 BC. He made a number of contributions towards science, including the first accurate method for measuring the diameter of the earth (his calculations were out by a mere 50 miles). He was also director of the library in Alexandria, he produced the map of the world shown below (I can see Cyrene on the map, but where is New Zealand?!), and he came up with the algorithm for finding prime numbers that we use in this lab. Eratosthenes died around 196 BC of voluntary starvation, induced by despair at his blindness.

57. XGC - Links Problems. URLs related to Eratosthenes and his Sieve Eratosthenes'Biography eratosthenes of cyrene The Sieve of Eratosthenes. URLs http://members.tripod.com/~aercolino/goldbach/xgc_links.html

Short Descriptions of the Goldbach Conjecture Mathematical mysteries: the Goldbach conjecture The Prime Glossary: Goldbach's conjecture Prime Conjectures and Open Questions Partially Annotated Prime References URLs related to Goldbach and his Conjecture Christian Goldbach's Biography Verifying Goldbach's Conjecture up to Unsolved Mathematics Problems Homepage for full Goldbach's Sequence Hunt ... Unsolvable and Unsolved Problems URLs related to Eratosthenes and his Sieve Eratosthenes' Biography Eratosthenes of Cyrene The Sieve of Eratosthenes URLs related to Prime Numbers Prime numbers Number Theory 11 - Number theory The Prime Machine ... Search Amazon books for Subject=Numbers,Prime URLs related to Maths Mathematics on the Web CAHRC's Homepage For Applied Sequence Algebra And Symbolic Computations Permutations and Combinations Mathematics web sites around the world ... Mathematical Constans and Interesting Numbers Research Papers New experimental results concerning the Goldbach conjecture by Jean-Marc Deshouillers, Yannick Saouter, Herman te Riele in 3rd Annual Number Theory Symposium (ANTS III), Portland, juin 1998.

58. Prime Or Not on. eratosthenes of cyrene. On which continent is Cyrene located?_.What great discovery did Eratosthenes make in Egypt?. http://www.indyschools.com/PITCrew/LessonPlans/6-12lp/math/primeornot.htm

Prime or not ? By Alice Douglas Grade Level: 7 th Objectives: TSW know the definition of Prime and Composite TSW know the list of Primes less than 100 TSW know the divisibility test rules for 1-10 TSW use technology to learn these objectives Materials: A list of the divisibility test and worksheets to practice using them. Worksheet: Rectangles and Factors and a bucket of square tiles. Number Munchers: Primes (a computer game) Power Point to make the Sieve of Eratosthenes Vocabulary: Prime, composite, divisible, Eratosthenes Sieve, factor, Procedures: 1. Teach a lesson on the Divisibility Rules for the numbers 1-10. If some of the rules are not included in your text, assign them as a computer research activity. 2. Complete the rectangles and factors activity with the square tiles. 3. In the Computer Lab, have the students follow the directions to complete the Power Point Activity. 4. Test the students over the Divisibility rules and the list of prime numbers. 5. Assign a journal activity: Explain the relationship between the number of rectangles in the activity and whether the number is prime or composite.

59. LookSmart - Eratosthenes eratosthenes of cyrene Biography of Eratosthenes, who very accuratelymeasured the circumference of the Earth more than 2000 years ago. http://canada.looksmart.com/eus1/eus302562/eus317836/eus317914/eus328800/eus5187

60. The Transit Of Venus And The Quest For The Solar Parallax The Greek astronomer, eratosthenes of cyrene (c276 195 BC), was able to deducethe Earth's radius by means of a strikingly simple set of observations. http://www.dsellers.demon.co.uk/venus/ven_ch1.htm

T HE T RANSIT OF V ENUS The Quest for the Solar Parallax by David Sellers (Leeds, England) Appears Winter 2001 as a full-length book: Order Now from Amazon! (in association with Amazon.co.uk) (Published by MagaVelda Press Almost every High School child knows that the Sun is 93 million miles (or 150 million Kilometres) away from the Earth. Despite the incredible immensity of this figure in comparison with everyday scales - or perhaps even because it is so hard to grasp - astronomical data of this kind is accepted on trust by most educated people. Very few pause to consider how it could be possible to measure such a distance - the 'Astronomical Unit' - and few are aware of the heroic efforts which attended early attempts at measuring it. Unfortunately, even most popular astronomy text books give insufficient information to allow one to see precisely how the task was accomplished. Sizing the Yardstick: the Diameter of the Earth In the history of astronomy, the quest for the Astronomical Unit (AU) has normally been regarded as the quest for a more fundamental quantity: namely, the 'Solar Parallax'. The solar parallax is not a distance at all. It is an angle: the angle subtended at the centre of the Sun by the Earth's radius (see Figure 1 ). If this angle is known and the radius of the Earth can be measured, then the distance to the Sun can be deduced by simple calculation. Clearly, the first piece of information which was needed in order to size the AU was the radius of the Earth.

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