61. Number And Operation Session 6: Number Theory In this part, we ll continue a mathematics tradition begun by eratosthenes of cyrene(276194 BCE) the same person who s known for accurately estimating the http://www.learner.org/channel/courses/learningmath/number/session6/part_b/

Session 6, Part B: Looking for Prime Numbers (45 minutes) In This Part: Necessary and Sufficient Conditions Is This Number Prime? In this part, we'll continue a mathematics tradition begun by Eratosthenes of Cyrene (276-194 B.C.E.) the same person who's known for accurately estimating the diameter of the Earth based on the shadow cast by the Sun's light. Eratosthenes worked out a method, now called the "Sieve of Eratosthenes," to collect all the prime numbers and allow all composites (multiples of prime numbers) to "drain through." He used a grid that looked like what we now call the 100 board the first row is 1-10, the second row 11-20, etc. This grid does locate the prime numbers, but it does not help us understand where to look for them. If you try looking for prime numbers in this grid, you will discover that it's not so easy to locate them in a systematic way: In the following activity, we will use a different grid to locate the prime numbers. This grid has only six columns, starting with the numbers 2 through 7. As you will see, such positioning of numbers will make the patterns more noticeable and consequently will be more helpful in answering the question of where the prime numbers are located. Copy the grid above or print a PDF version Problem B1 Circle the 2, which is a prime number. Next, cross out all the multiples of 2, as they are not prime numbers.

62. Eratosthenes eratosthenes of cyrene 276 194 BC source. http://www.geomatics.ucalgary.ca/~sneeuw/hall_of_fame/seiten_e/eratosthenes.html

Eratosthenes of Cyrene 276 - 194 B.C. source

63. History Of Mathematics: Chronology Of Mathematicians Philo of Byzantium (fl. c. 250) *SB; Nicoteles of Cyrene (c. 250); Strato(c. 250); Persius (c. 250?); eratosthenes of cyrene (c. 276c. 195) *SB *MT; http://aleph0.clarku.edu/~djoyce/mathhist/chronology.html

Chronological List of Mathematicians Note: there are also a chronological lists of mathematical works and mathematics for China , and chronological lists of mathematicians for the Arabic sphere Europe Greece India , and Japan Table of Contents 1700 B.C.E. 100 B.C.E. 1 C.E. To return to this table of contents from below, just click on the years that appear in the headers. Footnotes (*MT, *MT, *RB, *W, *SB) are explained below List of Mathematicians 1700 B.C.E. Ahmes (c. 1650 B.C.E.) *MT 700 B.C.E. Baudhayana (c. 700) 600 B.C.E. Thales of Miletus (c. 630-c 550) *MT Apastamba (c. 600) Anaximander of Miletus (c. 610-c. 547) *SB Pythagoras of Samos (c. 570-c. 490) *SB *MT Anaximenes of Miletus (fl. 546) *SB Cleostratus of Tenedos (c. 520) 500 B.C.E. Katyayana (c. 500) Nabu-rimanni (c. 490) Kidinu (c. 480) Anaxagoras of Clazomenae (c. 500-c. 428) *SB *MT Zeno of Elea (c. 490-c. 430) *MT Antiphon of Rhamnos (the Sophist) (c. 480-411) *SB *MT Oenopides of Chios (c. 450?) *SB Leucippus (c. 450) *SB *MT Hippocrates of Chios (fl. c. 440) *SB Meton (c. 430) *SB

64. History Of Mathematics: Greece Philo of Byzantium (fl. c. 250); Nicoteles of Cyrene (c. 250); Strato(c. 250); Persius (c. 250?); eratosthenes of cyrene (c. 276c. 195); http://aleph0.clarku.edu/~djoyce/mathhist/greece.html

Greece Cities Abdera: Democritus Alexandria : Apollonius, Aristarchus, Diophantus, Eratosthenes, Euclid , Hypatia, Hypsicles, Heron, Menelaus, Pappus, Ptolemy, Theon Amisus: Dionysodorus Antinopolis: Serenus Apameia: Posidonius Athens: Aristotle, Plato, Ptolemy, Socrates, Theaetetus Byzantium (Constantinople): Philon, Proclus Chalcedon: Proclus, Xenocrates Chalcis: Iamblichus Chios: Hippocrates, Oenopides Clazomenae: Anaxagoras Cnidus: Eudoxus Croton: Philolaus, Pythagoras Cyrene: Eratosthenes, Nicoteles, Synesius, Theodorus Cyzicus: Callippus Elea: Parmenides, Zeno Elis: Hippias Gerasa: Nichmachus Larissa: Dominus Miletus: Anaximander, Anaximenes, Isidorus, Thales Nicaea: Hipparchus, Sporus, Theodosius Paros: Thymaridas Perga: Apollonius Pergamum: Apollonius Rhodes: Eudemus, Geminus, Posidonius Rome: Boethius Samos: Aristarchus, Conon, Pythagoras Smyrna: Theon Stagira: Aristotle Syene: Eratosthenes Syracuse: Archimedes Tarentum: Archytas, Pythagoras Thasos: Leodamas Tyre: Marinus, Porphyrius Mathematicians Thales of Miletus (c. 630-c 550)

65. Grécia Antiga Eratóstenes Geógrafo eratosthenes of cyrene (com referências);eratosthenes of cyrene; Malaspina Great Books Eratosthenes; World http://warj.med.br/cie/t-cie06d-2.asp

atlas abreviaturas contato GEOGRAFIA ... Ilhas do Egeu O ultramar Buscas Fonte grega New Athena Unicode e o sol atingia seu em Siena ( S A ), ambas no Egito, localizavam-se praticamente no mesmo meridiano. x A cartografia Para os gregos do século -III, o mundo habitado ("ecúmeno") era uma porção definida da superfície da terra, situada no que hoje chamamos de hemisfério norte. Ao norte, frio excessivo e, ao sul, calor intolerável tornavam essas regiões inacessíveis ao homem. A existência de terras e homens além dos limites conhecidos "era teoricamente possível, mas tratada como uma especulação inútil" (Siebold). fl calculou norte-sul leste-oeste , a palavra geografia (gr. ), que significa "descrição da terra". Mediu ainda a , o valor obtido foi 11/83 de 180°, i.e., 23° 51' 15", mas muitos eruditos modernos atribuem esse valor ao próprio Ptolomeu. Traçou a rota do rio Nilo até Khartoum e foi o primeiro autor antigo a supor que o rio tinha origem em um lago e o primeiro a explicar, quase corretamente, que as cheias periódicas do Nilo eram provocadas por pesadas chuvas que caíam próximas à sua nascente. Descreveu também a "Arábia Feliz" (o Iêmen atual) e suas raças. Homero , mais ou menos em anterior Data: URLs Eratosthenes Finds Diameter of Earth!

66. World InfoZone - Libya Facts century BC. eratosthenes of cyrene (276BC 197) was a famous Greekmathematician who worked on prime numbers and geometry. He made http://www.worldinfozone.com/facts.php?country=Libya

67. Title eratosthenes of cyrene Ca. 275 BCE to 200 BCE Originally from Cyrenein North Africa, Eratosthenes spent many years studying at http://www.math.uvic.ca/courses/math415/Math415Web/greece/gmen/erattext.html

ERATOSTHENES OF CYRENE Ca. 275 BCE to 200 BCE Originally from Cyrene in North Africa, Eratosthenes spent many years studying at Plato 's Academy in Athens before coming to Alexandria where he spent the rest of his life. Eratosthenes was more than just a skilled mathematician. He was gifted in all areas that Greeks held in high esteem, including astronomy, philosophy, poetry, geography, history, and sports. In Alexandria, Eratosthenes became the head of the Great Library, which at this point was growing very quickly and contained hundreds of thousands of documents. Eratosthenes' major contribution to mathematics was his work with prime numbers. He developed an algorithm for finding primes which would soon come to be known as the Sieve of Eratosthenes . Please click on the link to view an animated applet illustrating the Sieve of Eratosthenes. This algorithm starts with a list of positive numbers starting with the number two because the number one is defined as not being a prime number. Since two is prime, we circle it and eliminate all of its multiples from the list. We move to the next number on the list which is not crossed out, circle it, and cross out all remaining multiples of that number. We keep repeating this process until we are left with a list of circled numbers, all of which are prime.

68. Euclidswindow What, for example, could be more practical than measuring the size ofthe Earth? It was first done in 212 BC by eratosthenes of cyrene. http://scibooks.org/euclidswindow.html

Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace by Leonard Mlodinow, The Free Press, $26, 306 pages. The Chapel Hill News June 27, 2001 Einstein's debt to geometry By Phillip Manning; CHN Columnist Geometry is a high school subject that has few practical applications and is not important in modern science. At least that's what I thought before I read Leonard Mlodinow's captivating book "Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace" (The Free Press, $26). It turns out I was wrong on both counts. Geometry has important practical applications, and it played a starring role in one of the great scientific achievements of the 20th century. What, for example, could be more practical than measuring the size of the Earth? It was first done in 212 B.C. by Eratosthenes of Cyrene. He noticed that at noon of the summer solstice a stick stuck perfectly upright in the ground in his hometown cast no shadow. That meant that the sun's rays were parallel to the stick. If the earth is a sphere, Eratosthenes reasoned, then a stick in the ground a few miles away should cast a shadow. Eratosthenes then measured the angle of a shadow cast at noon in Alexandria. Employing perhaps the world's first graduate assistant (and treating him in a manner that all graduate assistants will understand), Eratosthenes had him walk between the two towns and measure the distance, a stroll that turned out to be 500 miles. Using the theorems of Euclid, the man who devised the geometry we study in high school, Eratosthenes calculated the circumference of the Earth. He came up with about 25,000 miles, very close to the correct value.

69. IMSA Astrophysics: Distance Ladder The actual size was calculated to good precision by eratosthenes of cyrene (284 192 BC), an ancient Greek and head of the famed library at Alexandria (Egypt http://staff.imsa.edu/science/astro/astrometry/ground1.html

Distance and Size of the Moon Distance to the Sun Distance Ladder The Ground Floor: Circumference of the Earth Introduction Contrary to popular legend about people during Christopher Columbus' time believing the world is flat, the circumference of the spherical world was well known for over 1500 years prior to that time. The actual size was calculated to good precision by Eratosthenes of Cyrene (284 - 192 BC), an ancient Greek and head of the famed library at Alexandria (Egypt). How did he do it? It all started when Eratosthenes read a story about a rather minor event that happened every year in Syene, a city located almost directly south of Alexandria, along the Nile River. Every year, on June 21st, the longest day of the year, at precisely noon, the sun's reflection could be seen in the water of the deepest wells. The author also noted that, at the same time, buildings and obelisks cast no shadows. Obviously the sun was directly overhead in Syene at that moment. Eratosthenes knew, also, that this phenomenon did not occur in Alexandria. This wouldn't make sense if the earth were flat. The ancient Greeks already knew that the sun was very far away, so far away that the rays of light striking the earth were essentially parallel. The sun could not be directly overhead in Syene and not be directly overhead in Alexandria if the world were flat. The only explanation for this difference was that the earth was round!

70. Earthradius Part 1 eratosthenes of cyrene. The dispute concerning the shape of the earth hasbeen settled by Eratosthenes (276 194 BC). He could demonstrate http://home.t-online.de/home/fam.zapp/earthradius/part1.htm

Eratosthenes of Cyrene The dispute concerning the shape of the earth has been settled by Eratosthenes (276 - 194 BC). He could demonstrate it is a sphere and additionally he determined the diameter of that sphere. This is natural science at it's best. Eratosthenes became aware of the difference in shadow between Syene (Aswan) and Alexandria on the day of summer solstice. At highnoon the sun is right overhead in Syene while at the same instant the obelisc in Alexandria casts a shadow. In addition he recognized, the zenith distance a of the sun in Alexandria at summer solstice is the difference in geographic lattitude of the two cities. He gave order to measure the distance b of the cities. The ratio of b to the whole circumference of the earth is the same as the ratio of a to the full angle of 360 o . So the knowledge of b and a allows the calculation of the circumference of the earth and division by p gives the diameter.

71. Bible History Links - Ancient Greece - 2. General Sites Texts http//eawc.evansville.edu/grpage.htm Epidemics and Military Battleshttp//www.ento.vt.edu/IHS/militaryEpidemics.html eratosthenes of cyrene (ca. http://www.bible-history.com/links.cfm?cat=3&sub=246

72. - Great Books - Brief Biography. eratosthenes was born in cyrene which is now in Libya in North Africa teachers included the scholar Lysanias of cyrene and the philosopher Ariston of Chios who had http://www.malaspina.com/site/person_515.asp

Eratosthenes (c. 276 BC-c. 196 BC) Brief Biography 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. Eratosthenes made a surprisingly accurate measurement of the circumference of the Earth. Details were given in his treatise On the measurement of the Earth which is now lost. However, some details of these calculations appear in works by other authors such as Cleomedes, Theon of Smyrna and Strabo . Eratosthenes compared the noon shadow at midsummer between Syene (now Aswan on the Nile in Egypt) and Alexandria. He assumed that the sun was so far away that its rays were essentially parallel, and then with a knowledge of the distance between Syene and Alexandria, he gave the length of the circumference of the Earth as 250,000 stadia. [ Adapted from MacTutor Detailed Biography Eratosthenes was the second librarian of the Great Library of Alexandria, so called to distinguish it from the smaller or "daughter" library in the Serapeum, was a foundation of the first Ptolemies for the purpose of aiding the maintenance of Greek civilization in the midst of the conservative Egyptians. If the removal of Demetrius Phalereus to Alexandria, in 296-295 B.C., was connected with the organization of the library, at least the plan for this institution must have been formed under Ptolemaios Soter (died c. 284 B.C.), but the completion of the work and its connection with the Museum was achievement of his successor, Ptolemaios Philadelphos. As

73. Eratosthenes Van Cyrene eratosthenes van cyrene. eratosthenes (276194 vC) van cyrene wasde derde bibliothecaris van Alexandriës bibliotheek. Hij volgde http://mediatheek.thinkquest.nl/~lla015/biografie/Eratosthenes.html

Eratosthenes van Cyrene Nadat hij blind werd, pleegde hij zelfmoord door zichzelf dood te hongeren.

74. Greek Scientists eratosthenes, One of the students of the great poet Callimachus was eratosthenesof cyrene (c.275192 BCE), who became librarian in the Museum, the scientific http://www.livius.org/gi-gr/greeks/scientists.html

home index ancient Greece portrait gallery Greek scientists Authors Historians Philosophers Politicians Scientists We know almost nothing about Thales of Miletus . Later generations told many anecdotes about this wise man, but it is difficult to verify the reliability of these stories. What seems certain is that he predicted the solar eclipse of 28 May 585, which was remembered because the Lydian king Alyattes and the Median king Cyaxares were fighting a battle on that day. Another reliable bit of information is that he did some geometrical research, which enabled him to measure the pyramids. However, his most important contribution to physics and philosophy is his attempt to give non-religious, rational explanations for physical phenomena. Behind the phenomena was not a catalogue of deities, but one single, original principle. Although his hypothesis that this principle was water is rather unfortunate, his idea to look for deeper causes was the true beginning of philosophy and science. One of the solutions to the problem postulated by Parmenides , was the hypothesis of Democritus of Abdera that matter was made up from atoms. There was no real evidence for this idea (which was not completely new), but is explained why change was possible. The atoms were always moving and clustering in various, temporary combinations. Therefore, things seemed to change, but 'not being' never changed into 'being'. (It was assumed that 'not being' was a vacuum, which means that it is in fact not a 'not being' because a vacuum exists in at least four dimensions.) The consequence of this idea is that we are allowed to use our senses, although Democritus warns us to be careful.

75. Eratosthenes Resources earth s circumference; Key West. picture of eratosthenes; eratosthenes sinventions and discoveries; eratosthenes cyrene; what high school http://www.4-menopause-symptoms-and-menopause-info.com/eratosthenes.html

eratosthenes CLICK HERE TO ENTER ERATOSTHENES RESOURCES Top 500 Related Search Terms - If you did not find the content you were looking for, scan through these commonly searched terms. Copy the term into the Google Search Box and see if this helps. eratosthenes Eratosthenes sieve of eratosthenes Sieve of Eratosthenes The Sieve of Eratosthenes Who was Eratosthenes ERATOSTHENES Eratosthenes biography Eratosthenes and the circumference of the earth Eratosthenes of Cyrene about Eratosthenes timeline of eratosthenes sieve of Eratosthenes the sieve of eratosthenes biography of Eratosthenes of Cyrene Information about Eratosthenes Eratosthenes picture ancient astronomers Eratosthenes eratosthenes' measurment of earths circumference program the sieve of eratosthenes with vb Eratosthenes and Science Eratosthenes experiment pictures of eratosthenes History information on eratosthenes map of Eratosthenes greek mathematician eratosthenes What does Eratosthenes have to deal with prime numbers pictures of Eratosthenes Accomplishments of Eratosthenes Eratosthenes influence today pictures of eratosthenes Eratosthenes and Marine Science sieb des eratosthenes eratosthenes theory eratosthenes pronouciation geometry eratosthenes seive of eratosthenes What is the size of the earth Eratosthenes method bio of eratosthenes eratosthenes elibrary sources eratosthenes picture Java Sieve of Eratosthenes how to program the sieve of eratosthenes with vb informtation on Eratosthenes eratosthenes prime number seive with javascript

76. Eratosthenes Eratosthenes Was Born In Cyrene... eratosthenes eratosthenes was born in cyrene which is now in Libyain North Africa. eratosthenes eratosthenes was born in cyrene http://www.instant-essays.com/mathematics/eratosthenes.shtml

Home Donate Search Links document.write("Contact"); Eratosthenes Eratosthenes was born in Cyrene... Eratosthenes 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. One of the important works of Eratosthenes was Platonicus which dealt with the mathematics which underlie Plato's philosophy. Theon of Smyrna tells us that Eratosthenes' work studied the basic definitions of geometry and arithmetic, as well as covering such topics as music. Eratosthenes also worked on prime numbers. He is remembered for his prime number sieve, the 'Sieve of Eratosthenes' which, in modified form, is still an important tool in number theory research.

77. Eratosthenes eratosthenes (276 BC 194 BC) was a Greek mathematician, geographer and astronomerwith (probably) Chaldean origins. He was born in cyrene (now Shahhat, Libya http://www.fact-index.com/e/er/eratosthenes.html

Main Page See live article Alphabetical index Eratosthenes The crater Eratosthenes on the Moon gives its name to the Lunar Eratosthenian period Eratosthenes 276 BC 194 BC ) was a Greek mathematician geographer and astronomer with (probably) Chaldean origins. He was born in Cyrene (now Shahhat, Libya ) and he died in Ptolemaic Alexandria . He is noted for devising a system of latitude and longitude and computing the size of Earth Eratosthenes studied at Alexandria and some years in Athens . In 236 BC he was appointed by Ptolemy III Euergetes I as a head and the third librarian of the Alexandrian library. He made several important contributions to mathematics and science . He was a good friend to Archimedes . Circa 255 BC he invented the armillary sphere , which was used till 17th century He believed the Earth was a sphere and circa 240 BC he calculated its circumference , using trigonometry and information on the altitude of the Sun at noon in Alexandria and Syene (now Aswan , Egypt). The calculation is based on the assumption that the Sun is so far away that its rays can be taken as parallel. Eratosthenes knew that on the summer solstice at local noon in Syene, the Sun would appear at the

78. PSIgate - Physical Sciences Information Gateway Search Results full record feedback - send details to friend. Page 1. eratosthenesof cyrene This site provides a biography of eratosthenes (276-194 BC). http://www.psigate.ac.uk/roads/cgi-bin/psisearch.pl?term1=Eratosthenes&subject=A

79. Encyclopedia4U - Eratosthenes - Encyclopedia Article eratosthenes (276 BC 194 BC) was a Greek mathematician, geographer and astronomerwith (probably) Chaldean origins. He was born in cyrene (now Shahhat, Libya http://www.encyclopedia4u.com/e/eratosthenes.html

ENCYCLOPEDIA U com Lists of articles by category ... Encyclopedia Home Page SEARCH : Eratosthenes Eratosthenes 276 BC 194 BC ) was a Greek mathematician geographer and astronomer with (probably) Chaldean origins. He was born in Cyrene (now Shahhat, Libya ) and he died in Ptolemaic Alexandria . He is noted for devising a system of latitude and longitude and computing the size of Earth Eratosthenes studied at Alexandria and some years in Athens . In 236 BC he was appointed by Ptolemy III Euergetes I as a head and the third librarian of the Alexandrian library. He made several important contributions to mathematics and science . He was a good friend to Archimedes . Circa 255 BC he invented the armillary sphere , which was used till 17th century He believed the Earth was a sphere and circa 240 BC he calculated its circumference , using trigonometry and information on the altitude of the Sun at noon in Alexandria and Syene (now Aswan , Egypt). The calculation is based on the assumption that the Sun is so far away that its rays can be taken as parallel. Eratosthenes knew that on the summer solstice at local noon in Syene, the Sun would appear at the

80. BBC - History - Eratosthenes (c.276 BC - C.194 BC) eratosthenes (c.276 BC c.194 BC). eratosthenes was born in cyrene on the northcoast of Africa but educated in Athens, and came to Egypt in his thirties. http://www.bbc.co.uk/history/historic_figures/eratosthenes.shtml

@import url('/includes/tbenh.css') ; Home TV Radio Talk ... A-Z Index MONDAY 31st May 2004 Text only BBC Homepage History Topics ... Help Like this page? Send it to a friend! Eratosthenes (c.276 BC - c.194 BC) Eratosthenes was born in Cyrene on the north coast of Africa but educated in Athens, and came to Egypt in his thirties. He was a contemporary of Archimedes, who would send him mathematical problems to solve, sending the solutions later in case of difficulties. Eratosthenes was a skilled geographer, astronomer, poet and literary critic. He drew a map of the world as it was then known and calculated the angle of tilt of the earth in space. For forty years he served as a librarian at the famous library of Alexandria, and as a renowned mathematician, he invented a way of finding prime numbers, known as the Sieve of Eratosthenes. Possibly his greatest feat, however, was to measure the size of the Earth. Therefore, if he could measure the distance between the two cities, he could estimate the size of the earth. Luckily he had the use of the Royal Pacers; soldiers trained to pace precisely the same distance in each stride. Counting their strides, he was able to measure the distance between the two cities, approximately 500 miles. As a result he calculated that the circumference of the Earth was 51 multiplied by 500, or 25 500 miles, accurate to within about one per cent of the true figure. Articles Echoes of Atlantis Critics and Critiques of Ancient Athenian Democracy Multimedia Zone A Brief History of Cosmology Animation Historic Figures

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