41. Harmonic Series -- From MathWorld Divergence of the harmonic series was first demonstrated by nicole d Oresme (ca.13231382), but was mislaid for several centuries (Havil 2003, p. http://mathworld.wolfram.com/HarmonicSeries.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Calculus and Analysis Series General Series ... Unsolved Problems Harmonic Series The series is called the harmonic series. It can be shown to diverge using the integral test by comparison with the function . The divergence, however, is very slow. Divergence of the harmonic series was first demonstrated by Nicole d'Oresme (ca. 1323-1382), but was mislaid for several centuries (Havil 2003, p. 23; Derbyshire 2004, pp. 9-10). The result was proved again by Pietro Mengoli in 1647, by Johann Bernoulli in 1687, and by Jakob Bernoulli shortly thereafter (Derbyshire 2004, pp. 9-10). Progressions of the form are also sometimes called harmonic series (Beyer 1987). Oresme's proof groups the harmonic terms by taking 2, 4, 8, 16, ... terms (after the first two) and noting that each such block has a sum larger than 1/2, and since an infinite sum of 1/2's diverges, so does the harmonic series.

42. Search MathWorld The divergence, however, is very slow. Divergence of the harmonic series wasfirst demonstrated by nicole d Oresme (ca. 13231382), but was mislaid for sev http://mathworld.wolfram.com/search/index.cgi?num=10&q=gardner -Korteweg&start=6

43. életrajzok: O ORESME, nicole (Oresmicus) (1320?—1382. július 11.) francia matematikus.A középkori matematika utolsó nagy alakja. A franciaországi Caen mellett született http://members.iif.hu/visontay/ponticulus/eletrajzok/o.html

rovatok j¡t©k arch­vum jegyzetek mutat³k kitekintő v©lem©nyek inform¡ci³k ©letrajzok magyar¡zatok forr¡sok OMAR KHAJJM , al-Hajj¡m (1048. m¡jus 15.—1131. december 4.): k¶ltő, matematikus, csillag¡sz ©s filoz³fus. A perzs¡k nemzeti k¶ltője, neves matematikus. Tehets©g©re m¡r fiatal korban felfigyeltek ©s a szult¡n udvar¡ba h­vt¡k. Ezut¡n Iszfah¡nban ©lt a t¶r¶k szult¡n p¡rtfog¡sa alatt. 1079-ben megreform¡lta a perzsa napt¡rt. gy az csak 5000 ©venk©nt t©ved egy napot, m­g a mi napt¡runk 3330 ©venk©nt. A perzs¡k azonban k©sőbb ¡tt©rtek a mohamed¡n holdnapt¡rra. Algebr¡ja tºlhaladja AL-HVRIZMI‰T . Harmadfokº egyenleteket is old meg kºpszeletek seg­ts©g©vel. Foglalkozott az V. posztul¡tummal a k©sőbb SACCHERI ről elnevezett n©gysz¶g seg­ts©g©vel. Hozz¡j¡rult a t¶rtfogalom kialakul¡s¡hoz. ORESME, Nicole (Oresmicus) (1320?—1382. jºlius 11.): francia matematikus. A k¶z©pkori matematika utols³ nagy alakja. A franciaorsz¡gi Caen mellett sz¼letett. A p¡rizsi egyetemen tanult fizik¡t ©s matematik¡t, majd a teol³gi¡t v©gezte el. Ezut¡n teol³gi¡t tan­tott, majd kanonok, v©g¼l p¼sp¶k lett. Kor¡nak egyik legműveltebb ©s legsokoldalºbb elm©je volt. Jelentősek fizikai eredm©nyei. Az angol BRADWARDINE munk¡ss¡g¡t folytatva bevezette a t¶rtkitevős hatv¡nyokat.

44. Famous Left-Handers nicole d Oresme, mathematician Henry Ford, automobile manufacturer David Rockefeller,banker Dwight F. Davis, founder of the Davis Cup in tennis http://www.indiana.edu/~primate/left.html

in pig latin U.S. Presidents Politicos Miscellaneous ... submit a new name! This page is posted for entertainment purposes. Information pertaining to my research Hand Preference Questionnaire , my Curriculum vitae , or my Internet Underground article, " The World of Sinistral Subterfuge ", may be found at other sites. The top menu for these pages is: http://www.indiana.edu/~primate/index.html gauche! 13 August U.S. Presidents Politicos Senator Bill Bradley, Rhodes scholar, basketball star McGeorge Bundy, presidential advisor Benjamin Franklin, statesman/publisher/scientist Steve Forbes, businessman/publisher Ruth Bader Ginsberg, Supreme Court Justice Senator Daniel Inouye Anthony Kennedy, Supreme Court Justice Brigadier Gen. Lee Hsien Loon, Deputy Prime Minister of Singapore Robert S. McNamara, Secretary of Defense Col. Oliver North, White House aid H. Ross Perot, businessman William Perry, Secretary of Defense Nelson Rockefeller, Vice President Senator Hugh Scott Robert Wagner, New York mayor Henry Wallace, Vice President Joan of Arc (?), French heroine

45. B12 Occam's Razor For example, nicole d Oresme, a 14thcentury French physicist, and Galileo later,defended the Copernican heliocentric hypothesis (first impiously suggested http://geowords.com/histbooknetscape/b12.htm

Occam's razor < Occam is the Latinized spelling of Ockham; "principle of least amazement" in natural science > Scott: And so we thought we'd try it here for you. The feather happens to be, appropriately, a falcon feather for our Falcon. And I'll drop the two of them here and, hopefully, they'll hit the ground at the same time. (Pause) [Dave is holding the feather and hammer between the thumb and forefinger of his left and right hands, respectively, and has his elbows up and out the side. He releases the hammer and feather simultaneously and pulls his hands out of the way. The hammer and feather fall side by side and hit the ground at virtually the same time.] Scott: How about that! Mr. Galileo was correct! Allen: How about that! (Applause in Houston) [ transcript by Eric M. Jones, ©1996 omits the phrase in italics ] is to see if a component can be excised without altering conclusions that can be derived or its useful application. The component has no "substance" if it can be. Victor J. Ste n Reasonable Faith , 1994, is: "One ought not to try to prove the obvious via the less obvious." Aristotle, called by Dante "The master of those who knew," hopefully did not make that remark. What survives of his thoughts are some of his lecture notes and of his popular works we know some through the interpretive writings of Theophrastus his disciple and successor.Whatever of his makes contemporary sense may be our reading of it ignorant of his true thoughts. His poetic

46. Nicolas D'Oresme - Wikipédia Oresme fut un savant en avance sur son temps, Vois la bibliographie étendue de nicole Oresme www.nicole-oresme.com http://fr.wikipedia.org/wiki/Nicolas_d'Oresme

Wikim©dia a besoin de votre aide notre page de collecte de fonds pour plus de d©tails. Nicolas d'Oresme Un article de Wikip©dia, l'encyclop©die libre. N© en Allemagne (l'ancien nom de l'actuelle commune de Fleury-sur-Orne dans le Calvados Normandie France Nicolas d'Oresme fut un savant en avance sur son temps, dont les id©es annon§aient la p©riode de la Renaissance Toute sa vie se d©roula pendant la guerre de Cent Ans , la Normandie ©tant souvent occup©e par l' Angleterre . De naissance modeste, son d©sir d'apprendre le poussa   entrer dans les ordres. Remarqu© par l'entourage du roi de France , il commen§a ses ©tudes de th©ologie en . Il devint le pr©cepteur de Charles V qui lui offrit l'©vªch© de Lisieux en . C'est dans cette cit© qu'il d©c©da le 11 juillet Ses ©crits, partiellement en latin et fran§ais , couvrent deux types de sujets. Dans ses travaux tels que De l'origine, nature et mutation des monnaies, il travailla sur l'©conomie et dans ceux tels que De cœlo et mundo, Trait© du Ciel et du monde ), il se montre lui-mªme un pr©curseur de

47. Wiskundigen - Oresme nicole d Oresme (1323 11 juli 1382) was een Frans theoloog. nicole d Oresmewerd omstreeks 1325 geboren in Allemagne bij Caen. http://www.wiskundeweb.nl/Wiskundegeschiedenis/Wiskundigen/Oresme.html

Oresme Nicole d'Oresme Oresme was behalve Rooms Katholiek bisschop ook wetenschapper en econoom. Oresme is de uitvinder van de grafiek . Hij was de eerste die op het idee kwam om een variabele waarvan de waarde afhangt van bijvoorbeeld de tijd als een verticaal lijnstuk uit te zetten of de horizontale (tijd)-as. Hij begreep de gelijkwaardigheid van tabel en grafiek. Oresme overleed op 57-jarige leeftijd in Lisieux. Links naar anderstalige sites: Meer over Oresme en zijn tijd In 1377 werd Oresme gekozen tot bisschop van Lisieux, een ambt dat hij vervulde tot zijn dood in 1382. Oresme's belangrijkste werk Oresme's hield zich met allerlei takken van de wetenschap bezig. Hij publiceerde diverse geschriften. Op het gebied van de wiskunde was hij de grondlegger van de grafiek. Oresme bestudeerde de beweging van voorwerpen en bedacht een manier om die in een figuur weer te geven. Hij gaf de grootte van de snelheid aan met een lijnstuk: hoe groter de snelheid des te langer het lijnstuk. Die lijnstukken zette hij op een horizontale as die de tijd voorstelt. Als de snelheid gedurende een bepaalde periode hetzelfde blijft, zijn de lijnstukken allemaal even lang. Zo ontstaat een rechthoek waarvan de oppervlakte (zo begreep Oresme) de afgelegde weg voorstelt.

48. Cool Quiz! Trivia, Quizzes, Puzzles, Jokes, Useless Knowledge, FUN! Holy Roman emperor. nicole d Oresme Mathematician. H. Norman Schwartzkopf Military.Euell Gibbons Naturalist. Dr. Albert Schweitzer Physician / Missionary http://www.coolquiz.com/trivia/didyouknow/lefthand.asp

fwLoadMenus("#FFCCFF","#FEE4FA"); Search Cool Quiz! Trivia Quizzes Puzzles Humor ... Tell a Friend The Left Handed and Famous If the left side of your brain controls the right side of your body, and the right side of your brain controls the left side of your body, then left-handed people must be the only ones in their right minds. " - W.C. Fields In current society, "handedness" is simply defined as the hand you use to write with, but scientists measure handedness in many ways. Scientists will also use measures of grip strengh, hand preference in other tasks, and hand speed in manual tasks. The List: Art Male Female Historical ... U.S. Presidents Art Albrecht Dürer Bill Mauldin (Cartoonist) Cathy Guisewite (Cartoonist) Hans Holbein Leonardo da Vinci LeRoy Neiman M.C. Escher Matt Groening (Cartoonist) Michelangelo Milt Caniff (Cartoonist) Pablo Picasso Paul Klee Peter Paul Rubens Raphael Entertainment/Male Alan Funt Alan Thicke Allen Ludden Andrew McCarthy Anthony Perkins Bart Simpson Big Bird Brad Pitt Bronson Pinchot Bruce Boxleitner Bruce Willis Cary Grant Charlie Chaplin Christian Slater Chuck Conners Dan Aykroyd Danny Kaye David Duchovny David Letterman Dennis Quaid Dick Smothers Dick Van Dyke Don Adams Don Rickles Eddie Albert Frank Langella Fred Astaire Freddie Prinze Jr.

49. Canhotos Famosos nicole d Oresme, mathematician Pat Oliphand, political cartoonist Ronald Searle,cartoonist Pat Robertson, evangelist/politician Paul Prudhomme, chef http://members.tripod.com/~geniusbr/Avesso/famosos.html

setAdGroup('67.18.104.18'); var cm_role = "live" var cm_host = "tripod.lycos.com" var cm_taxid = "/memberembedded" Search: Lycos Tripod Dating Search Share This Page Report Abuse Edit your Site ... Brasileiros 1. Esportes 2. Escritores Internacionais 1. Políticos, Realezas e Líderes Presidentes Americanos Outros políticos e lideres 2. Artes e Entretenimento Autores Músicos Artistas Atores 3. Esporte 4. Diversos Brasileiros 1. Esportes Ayrton Senna (fórmula 1) Pelé - Edson Arantes do Nascimento (futebol) Romário (futebol) Roberto Carlos (futebol) 2. Escritores Machado de Assis Internacionais 1. Políticos, Realezas e Líderes Presidentes Americanos James A. Garfield (1831-1881) 20o Herbert Hoover (1874-1964) 31o Harry S. Truman (1884-1972) 33o Gerald Ford (1913- ) 38o Ronald Reagan (1911 - ) 40o George Bush (1924- ) 41o Bill Clinton (1946- ) 42o Outros políticos e líderes Alexander the Great Anthony Kennedy Ben Campbell Benjamin Franklin Benjamin Netanyahu Bill Bradley Bob Dole Charlemagne Colin Powell Daniel Inouye David Rockefeller Edward III Elizabeth II Fidel Castro Franz Josef Strauß George Bush George VI George II George IV Henry Ford Henry Wallace Henry Ford Jr.

50. Safety Razors And Shaving Collectibles nicole d Oresme, a 14thcentury French physicist, invoked the law of economy, asdid Galileo later, in defending the simplest hypothesis of the heavens. http://www.geocities.com/safetyrazors/occam.htm

The Philosophy of Razors, as exemplified by Ockham's razor also spelled OCCAM'S RAZOR, also called LAW OF ECONOMY, or LAW OF PARSIMONY, principle stated by William of Ockham (1285-1347/49), a scholastic, that non sunt multiplicanda entia praeter necessitatem ; i.e., entities are not to be multiplied beyond necessity. The principle was, in fact, invoked before Ockham by Durand de Saint-Pourçain, a French Dominican theologian and philosopher of dubious orthodoxy, who used it to explain that abstraction is the apprehension of some real entity, such as an Aristotelian cognitive species, an active intellect, or a disposition, all of which he spurned as unnecessary. Likewise, in science, Nicole d'Oresme, a 14th-century French physicist, invoked the law of economy, as did Galileo later, in defending the simplest hypothesis of the heavens. Other later scientists stated similar simplifying laws and principles. Ockham, however, mentioned the principle so frequently and employed it so sharply that it was called "Ockham's razor." He used it, for instance, to dispense with relations, which he held to be nothing distinct from their foundation in things; with efficient

51. ⊙ AntiQuark: Harmonic Series And Bricks This was proven by nicole d Oresme back in the Medieval times. (Believe it ornot, people were concerned about this stuff back then. http://www.antiquark.com/2005/04/harmonic-series-and-bricks.html

@import url("http://www.blogger.com/css/blog_controls.css"); @import url("http://www.blogger.com/dyn-css/authorization.css?blogID=3000598"); @import url(http://www.blogger.com/css/navbar/main.css); @import url(http://www.blogger.com/css/navbar/1.css); BlogThis! Truth, Beauty, Charm, Strange Harmonic Series and Bricks A summation of the form Is called the harmonic series. This series diverges. On other words, if you keep adding terms of the series, the sum will grow without a limit. This was proven by Nicole d'Oresme back in the Medieval times. (Believe it or not, people were concerned about this stuff back then.) He grouped the terms of the harmonic series like so: And noticed that every "chunk" was bigger than the chunks in the following series: And each of these chunks happen to add up to 1/2. Therefore, the harmonic series diverges, because it's like adding 1/2 (plus a bit) an infinite number of times. Now, this fact about the harmonic series produces a counterintuititive reality in the physical world. A mathematical stack of bricks.

52. Math Forum Electronic Newsletter Hipparchus Ptolemy s Geography Ancient Greek Geometry Apollonius and theConics Plane Loci - The Middle Ages nicole d Oresme and the Latitude of Forms http://mathforum.org/electronic.newsletter/mf.intnews8.37.html

Volume 8, Number 37 Back to Table of Contents http://mathforum.org/cgraph/history/ http://mathforum.org/cgraph/ http://isb.sccoe.org/mathmentor/TakeTheTest/ ... http://mathforum.org/dr.math/faq/faq.learn.multiply.html Math Forum Internet Mathematics Library: Multiplication http://mathforum.org/library/topics/multiplication/ Math Tools: Tool: Arithmetic Four T2T FAQ: Multiplication Help http://mathforum.org/t2t/faq/faq.multiplication.html Teacher Exchange: Practicing Arithmetic Skills http://mathforum.org/alejandre/frisbie/arithmetic.html http://mathforum.org/ Ask Dr. Math http://mathforum.org/dr.math/ Problems of the Week http://mathforum.org/pow/ Mathematics Library http://mathforum.org/library/ Math Tools http://mathforum.org/mathtools/ http://mathforum.org/t2t/ Discussion Groups http://mathforum.org/discussions/ Join the Math Forum http://mathforum.org/join.forum.html Send comments to comments@mathforum.org Donations http://www.drexel.edu/ia/mathforum/ Ask Dr. Math Books http://mathforum.org/pubs/dr.mathbooks.html

53. Math Forum Discussions nicole d Oresme and the Latitude of Forms Analytic Geometry Rene Descartes andLa Geometrie Pierre de Fermat - New Words, New Notation http://mathforum.org/kb/thread.jspa?threadID=486260&tstart=90

54. Famous Lefthanders List From Anything Left Handed d Oresme, nicole, Mathematician. Einstein, Albert, German, Professor of Physics,14Mar-1879. Pauling, Linus, Scientist. Pawlow, Iwan, Scientist http://www.anythingleft-handed.co.uk/fam_proff.html

Left handed products and left-handed information menus require JavaScript var DOCUMENTGROUP='Info'; //var DOCUMENTNAME=''; //var ACTION=''; Site Map View Basket Checkout US Visitors click here Any price Up to 3.00 Left Handed Shop Home Basket Summary View Basket Checkout Left Handers Club Free newsletters We value your privacy Famous left-handers - Professions If you are left-handed then you are in very good company. Throughout history left-handers have excelled as leaders, sportsmen, artists, musicians and in many other fields. This is our list of famous left-handers - with over 1,000 people we think it is the most comprehensive on the Web. The list is based on our research and reports from Club members, but we know we are not perfect. If you spot any errors in our list, can provide any further information or have any famous left-handers we should add to our list, please email us or use the form on our New Famous Lefties page. We will soon be adding web site links to most of our famous left-handers so that you can find out more about them. The list contains well over 1,000 left-handers

55. Eisenstein's Monster: Addition Symbol nicole d Oresme (13231382) may have used a figure which looks like a plussymbol ( ) as an abbreviation for the Latin et (meaning and ) in Algorismus http://babel.391.org/fz2020/eisensteinsmonster/addition.htm

frankenstein's monster eisenstein's monster eisenstein's montage Nicole d' Oresme (1323-1382) may have used a figure which looks like a plus symbol ( + ) as an abbreviation for the Latin et (meaning "and") in Algorismus proportionum , believed to have been written between 1356 and 1361. The symbol appears in a manuscript of this work believed to have been written in the fourteenth century, but perhaps by a copyist and not Oresme himself. Mercantile Arithmetic or Behende und hüpsche Rechenung auff allen Kauffmanschafft , by Johannes Widmann (born c. 1460), published in Leipzig in 1489. However, they referred not to addition or subtraction or to positive or negative numbers, but to surpluses and deficits in business problems (Cajori vol. 1, page 128). The plus and minus symbols were in use before they appeared in print. For example, they were painted on barrels to indicate whether or not the barrels were full.

56. New Page 1 tend to remember William of Ockham and nicole d Oresme from the period. In the fourteenth century nicole Oresme wrote a quite cogent argument that http://www.cerritos.edu/ladkins/a106/middle_ages/middle_ages.htm

The Medieval World (500 A.D. - 1500 A.D.) The span of time from the destruction of Rome by the Vandals in 475 A.D. to the Renaissance (the late fifteenth century) is usually referred to as the "medieval period" or the "middle ages". In the early part of this period (500 - 1000), learning and high culture virtually came to a halt in western Europe. However, in other parts of the world civilization was flourishing. Mayan civilization, for example, was reaching its pinnacle at this time. In the Eastern Mediterranean, the Byzantine Empire (Eastern Roman Empire) maintained an active intellectual life, and after 700 A.D. Islamic culture advanced to great heights. Western Europe Ruins of Coliseum in Rome San Gimignano, Italy Dante's Cosmos The ruins of the coliseum symbolize the end of the Roman Empire in the west and the end of classical culture as well. The walled Tuscan city of San Gimignano represents the feudal system which replaced it. When European learning began to revive (starting around 1000), scholarship was dominated by what is known as the "Scholastic" school of philosophy. One aspect of this philosophy was a fusion of Biblical scripture with Aristotle's beliefs concerning the natural world. Thus, there was some tendency to equate Aristotle's physics as authoritative (and as sacred) as the Bible. The greatest of the scholastic philosopher was

57. DOMAINES d Oresme et d Antoine de la Sale le Lexique des oeuvresde nicole Oresme, 530 pages; la publication sera assurée par les soins du http://www.univ-nancy2.fr/MOYENAGE/URFA/Domaines.html

URFA UMR 7002 MOYEN AGE Responsable : Charles Brucker Fax : (00 33) (0) 3 83 98 17 77 a) Prospective: concernant le livre V du Policratique b) Prospective: c) Prospective: d) Lexique de l'oeuvre d'Oresme et d'Antoine de la Sale Vient d'être achevée, dans la même perspective, par P. Demarolle le Lexique de "La Salade" d'Antoine de la Sale, 300 pages, dont la publication par Champion est prévue pour 2002 (CD-ROM et livre). Prospective: Prospective: - 2002-2002 : premier et second volume Prospective: Retour

58. Occam's Razor/Ockham's Razor Likewise, in science, nicole d Oresme, a 14thcentury French physicist, invokedthe law of economy, as did Galileo later, in defending the simplest http://occam.martinsewell.com/

Occam's Razor/Ockham's Razor Martin Sewell martin@martinsewell.com b fig . and in fig. context. Occam's ( also Ockham's) razor , the leading principle of the nominalism of William of Occam (see Occamism), that for purposes of explanation things not known to exist should not, unless it is absolutely necessary, be postulated as existing; usually called the Law of Parcimony.' Oxford English Dictionary Ockham's razor also spelled Occam 's razor, also called law of economy, or law of parsimony, Pluralitas non est ponenda sine necessitate; Encyclopædia Britannica "Coherent inference (as embodied by Bayesian probability) automatically embodies Occam's razor, quantitatively." MacKay (2003) p344 Wikipedia: Occam's Razor Papers DOMINGOS, Pedro, Occam's Two Razors: The Sharp and the Blunt "Occam's razor has been the subject of much controversy. This paper argues that this is partly because it had been interpreted in two quite different ways, the first of which (simplicity is a goal in itself) is essentially correct, while the second (simplicity leads to greater accuracy) is not." JEFFERYS, William H. and James O. BERGER

59. Prime Obsession It was produced in the late Middle Ages by a French scholar, nicole d Oresme (ca.13231382). D Oresme pointed out that 1/3+1/4 is greater than 1/2 ; so is http://www.math.wustl.edu/~freiwald/Math132/harmonic.htm

July 6, 2003 'Prime Obsession' By JOHN DERBYSHIRE ike many other performances, this one begins with a deck of cards. Take an ordinary deck of 52 cards, lying on a table, all four sides of the deck squared away. Now, with a finger slide the topmost card forward without moving any of the others. How far can you slide it before it tips and falls? Or, to put it another way, how far can you make it overhang the rest of the deck? The answer, of course, is half a card length, as you can see in Figure 1-1. If you push it so that more than half the card overhangs, it falls. The tipping point is at the center of gravity of the card, which is halfway along it. Now let's go a little further. With that top card pushed out half its length-that is, to maximum overhang-over the second one, push that second card with your finger. How much combined overhang can you get from these top two cards? The trick is to think of these top two cards as a single unit. Where is the center of gravity of this unit? Well, it's halfway along the unit, which is altogether one and a half cards long; so it's three-quarters of a card length from the leading edge of the top card (see Figure 1-2). The combined overhang is, therefore, three-quarters of a card length. Notice that the top card still overhangs the second one by half a card length. You moved the top two cards as a unit. If you now start pushing the third card to see how much you can increase the overhang, you find you can push it just one-sixth of a card length. Again, the trick is to see the top three cards as a single unit. The center of gravity is one-sixth of a card length back from the leading edge of the third card (see Figure 1-3).

60. O nicole d Oresme (ca. 1320 1382). Born in France. He introduced a mathematicalmethodology which eventually developed into analytical geometry. http://www.innvista.com/science/math/mathians/o.htm

O Home About Us Contact Us Home ... Mathematicians Search innvista.com Our Notice Board O Nicole d'Oresme (ca. 1320 - 1382) Born in France. He introduced a mathematical methodology which eventually developed into analytical geometry. He suggested the possibility of irrational proportion. William Oughtred Born in England. He introduced the symbol x for multiplication and the symbol of in proportion. He introduced the abbreviations for such tems as sine, cosine, tangent, cotangent, secant , and cosecant

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