21. À§´ëÇѼöÇÐÀÚ ¸ñ·Ï Born about 415 BC in Athens, Greece Died about 369 BC in Athens, Greece Theodorus,theodorus of cyrene Born 465 BC in Cyrene (now Shahhat, Libya) Died 398 http://www.mathnet.or.kr/API/?MIval=people_seek_great&init=T

22. 1989 K-State Undergraduate Lecture Series In Mathematics Davis, a renowned author and mathematician from Brown University, visited KStateand gave a lecture entitled From the spiral of theodorus of cyrene to chaos. http://www.math.ksu.edu/main/lectures/freshmanseminar/frnews89.htm

KSU Math Home Lectures Freshman Seminars : 1989 K-State Undergraduate Lecture Series in Mathematics Printable Version 1989 K-State Undergraduate Lecture Series in Mathematics Lectures by a distinguished visiting mathematician and an alumnus were the highlights of our Undergraduate Lecture Series at Kansas State University in the fall of 1989. Philip Davis, a renowned author and mathematician from Brown University, visited K-State and gave a lecture entitled From the spiral of Theodorus of Cyrene to chaos. Michael Anderson, a 1969 K-State mathematics alumnus originally from Hays, gave a lecture entitled Combined hierarchial prioritization and mathematical programming. He discussed the allocation of new resources, using some matrix algebra and operations research techniques. Dr. Anderson currently is working for the TRADOC Analysis Command at Fort Leavenworth, Kansas. Three of our own faculty gave lectures in the Undergraduate Lecture Series. Andrew Bennett spoke on Algebraic Geometry and the historical development of calculus. Thomas Muenzenberger spoke on Career opportunities in mathematics.

23. ThinkQuest : Library : A Taste Of Mathematic c. 450); Hippocrates of Chios (c. 450); Meton (c. 430) *SB; Hippiasof Elis (c. 425); theodorus of cyrene (c. 425); Socrates (469399 http://library.thinkquest.org/C006364/ENGLISH/history/historygreece.htm

Index Math A Taste of Mathematic Welcome to A Taste of Mathematics.You will find the taste of mathematics here.The history of Mathematics,famous mathematicians,cxciting knowledge,the world difficult problems and also mathematics in our life... Browsing,thinking,enjoying,and have a good time here! Visit Site 2000 ThinkQuest Internet Challenge Languages English Chinese Students fangfei Beijing No.4 High School, Beijing, China ziyan Beijing No.4 High School, Beijing, China Coaches Tife Zesps3 Szks3 Ogslnokszta3c9cych Numer 1, Beijing, China xueshun Beijing No.4 High School, Beijing, China Want to build a ThinkQuest site? The ThinkQuest site above is one of thousands of educational web sites built by students from around the world. Click here to learn how you can build a ThinkQuest site. Privacy Policy

24. Chapter 16: Archimedes were Archytas of Tarentum, Plato s geometry teacher, Hippocrates of Chios, whotried to fit together all the rules, and theodorus of cyrene, who discovered http://www.anselm.edu/homepage/dbanach/arch.htm

Selections from Julia E. Diggins, String, Straightedge, and Shadow Viking Press, New York , 1965. (Illustrations by Corydon Bell) 16. A ROYAL ROAD, AFTER ALL During the 4th century B.C., Greek geometry burst its bonds and went on to the tremendous discoveries of the "age of giants." And Greek culture, too, burst from the mainland of Hellas and spread to most of the eastern Mediterranean. Both developments were connected with the romantic figure of Alexander the Great. After Plato's time, teachers and alumni from the Academy had gone on to found schools of their own. In particular, Plato's most famous associate, the great philosopher Aristotle, had set up the Lyceum in Athens, and started the systematic classification of human knowledge. And Aristotle's most renowned pupil was the warrior king Alexander of Macedon, who tried to conquer the world. In thirteen years, Alexander extended his rule over Greece proper, and Ionia, Phoenicia, Egypt, and the vast Persian domains as far as India. Then he died, and his empire broke up. But throughout those far-flung lands, he had founded Greek cities and planted the seeds of Greek civilization-the Greek language, Greek art, and, of course, Greek mathematics. Mathematicians traveled with his armies. And there is even a

25. JMM HM DICIONÁRIO Translate this page of Alexandria Theon of Smyrna Theaetetos Theodoros of Cyrene Zeno of Elea, ThalesTheon of Alexandria Theon of Smyrna Theaetetus theodorus of cyrene Zeno of http://phoenix.sce.fct.unl.pt/jmmatos/HISTMAT/HMHTM/HMDIC.HTM

Bibliografia Recursos na rede bem vindos em latim Anaximandro (-611-545) Antifonte Aristarco de Samos (-310-230?) Aristeo (c. -330) Arquimedes de Siracusa (-287?-212) Arquitas de Tarento (c. -375) Apollonius Archimedes Boetius Apollonios of Perga Aristarchos Aristaeus Aristotle Archimedes of Syracuse Archytas Apollonius of Perga Aristarchus Aristaeus Aristotle Archimedes of Syracuse Archytas Boethius Apollonios Diofanto de Alexandria (c. 250) Diophantus Democritos Dinostratos Diophantos Diocles Democritos Dinostratus Diophantus Diocles Diogenes Laertius Euclides de Alexandria (c. -300) Filolaos Endemus Eudoxus Philolaus Eratosthenes Euclid of Alexandria Endemos Eudoxos of Cnidos Eratosthenes Euclid of Alexandria Endemus Eudoxus of Cnidos Philolaus Euclide Hiparco de Alexandria (-190-120) Hipasos Hipsicles Herodotus Hipparchus Hero Herodotos Hypatia Hipparchos Hippocrates of Chios hekat Heron Herodotus Hypatia Hipparchus Hippocrates of Chios Iamblichus Iamblichos Iamblichus Menecmo (c. -350)

26. CHRONOLOGY Dialogue by that name), teaches about incommensurability . Also taughtby theodorus of cyrene. Circa 365 BC, Eudoxus (c. 408355 http://members.fortunecity.com/jonhays/chronology.htm

web hosting domain names email addresses CHRONOLOGY OF THE SQUAREGATE (MATHEMATICS BEHIND "CANDY MISER'S PUZZLE") Mathematicians and students today extensively apply the right triangle in geometry and trigonometry using the "Pythagorean Formula" to relate lengths of the triangle's SIDES ( a, b ) and DIAGONAL ( c ): a + b = c . However, Pythagoras did not discover this formula, but apparently learned it in his travels through Egypt and Mesopotamia. Nevertheless, Knowledge of Egyptian mathematics derives from two artifacts, The Rhynd Papyrus (named for its donor to The British Museum) and The Moscow Papyrus (now in Museum of Fine Arts, Moscow, Russia). Both apparently derive from a period between 2000 B.C. and 1580 B.C. They describe ability to solve linear (first degree) equations in one unknown , but no evidence of solving second degree equations or formulas such as "The Pythagorean". The Egyptians deal with fractions by adding unit fractions , such as 1/2, 1/3, 1/4, 1/5, etc. (You use "Egyptian fractions" nearly every day in "making change in commercial transactions". Thus, a penny = $1/100; a nickel = $1/20; a dime = $1/10; a quarter = $1/4; a fifty-cent piece = $1/2.) Later we'll see how the "Egyptian fraction" format extends to the best test of irrationality of numberness The "Pythagorean" formula was discovered circa 2000 B.C. in Babylonia or Egypt or both. We know about its Babylonian origin from approximately half of a million clay Babylonian tablets engraved in their written

27. The History Of Freethought And Atheism Finally we can say with confidence that theodorus of cyrene was an atheistfrom the contents of his work On the Gods. Roman Skeptics. http://www.positiveatheism.org/india/s1990c25.htm

Atheist Centre 50+ Golden Jubilee (1940-1990) International Conference on "Future of Atheism Humanism" Vijayawada, December 29-31, 1990 [OCR, HTML, editing, Cliff Walker] The History of Freethought and Atheism Dr. Gordon Stein Ph.D. from "An Anthology of Atheism and Rationalism" Index: International Conference Souvenir Index: Atheism in India Home to Positive Atheism The idea that atheism has a long and colorful history may strike some people as unlikely. It is true, however, and those few people who have made an effort to study that history have found enough there to fill several thick volumes (and most of them barely scratched the surface). There have undoubtedly always been unbelievers in the current religious belief of an area or a people. Before the invention of writing, however, we know of no specific unbelievers who left any record of their unbelief. In ancient India, there was a group of freethinkers known as the Lokayatas (before 600 B.C., although remnants of this group are found in India as late as the 14th century.) In China, both Confucius and Lao Tsze can be viewed as freethinkers in relation to the religion(s) prevalent in China at the time. Early Greek Freethinkers The first real freethinkers or atheists who seem to have produced work specifically dealing with religion in a negative way were found in ancient Greece. Although most of these works have not survived to the present day (and charges of atheism were often politically motivated and really not based upon fact), we know that Anaxagoras was accused of impiety and forced to leave Greece. He supposedly held that the sun was a red-hot body and that the moon was a physical object which was larger than Greece. He did not, however, apparently make an attack on the popular religious beliefs.

28. Ce-Cl: Positive Atheism's Big List Of Quotations Protagoras doubted whether there were any. Diagoras the Melian andtheodorus of cyrene entirely believed there were no such beings. http://www.positiveatheism.org/hist/quotes/quote-c1.htm

Positive Atheism's Big List of Quotations Ce-Cl No-Frames Quotes Index Load This File With Frames Index Home to Positive Atheism French author With two thousand years of Christianity behind him a man can't see a regiment of soldiers march past without going off the deep end. It starts off far too many ideas in his head. : Ferdinand Bardamu, in Journey to the End of the Night (1932; 1934; 1966, p. 123) Aulus Cornelius Celsus (C.E. First Century) Roman writer who wrote an encyclopedia on the subjects of medicine, rhetoric, history, philosophy, warfare, and agriculture Before accepting any belief one ought to follow reason as a guide, for credulity without enquiry is a sure way to deceive oneself. Celsus (ca. C.E. 170), quoted from Antony Flew, Atheistic Humanism, p. 17 This is one of their [the Christians'] rules. Let no man that is learned, wise, or prudent come among us: but if they be unlearned, or a child, or an idiot, let him freely come. So they openly declare that none but the ignorant, and those devoid of understanding, slaves, women, and children, are fit disciples for the God they worship. Celsus , quoted from John E. Remsberg

29. Body Ethics 5.15 Hippocrates, On Regimen 5.16 Ps.Aristotle, Problems 5.17 Diogenes theCynic 5.18 Bion of Borysthenes 5.19 theodorus of cyrene 5.20-5.25 Zeno of http://www.utexas.edu/courses/cc348hubbard/toc.html

Table of Contents Chapter 1. Archaic Greek Lyric: Introductory Matter. Chapter 1. Archaic Greek Lyric: Bibliographic Note. 1.1-1.3 Archilochus 1.4 Alcman, First Maidens' Song ... 2.28 gymnastic law from Beroea Chapter 3. Greek Comedy 3.1-3.10 Aristophanes, Knights 3.11 Aristophanes, Clouds 3.12 Aristophanes, Wasps 3.13 Aristophanes, Birds 3.14 Aristophanes, Thesmophoria Women 3.15-3.16 Aristophanes, Frogs 3.17 Aristophanes, Wealth 3.18 Aristophanes, fr. 556 3.19-3.20 Cratinus 3.21 Eupolis 3.22 Strattis 3.23 Pherecrates 3.24 Theopompus 3.25 Ephippus 3.26-3.28 Alexis 3.29 Amphis 3.30 Anaxandrides 3.31 Antiphanes 3.32 Archedicus 3.33 Damoxenus 3.34 Diphilus Chapter 4. Greek Oratory 4.1 Andocides, On the Mysteries 4.2 Lysias, Against Teisias 4.3 Lysias, Against Alcibiades 4.4 Lysias, Against Simon 4.5 Isaeus, On the Estate of Aristarchus 4.6 Demosthenes, Against Androtion

30. PHILTAR - Compendium Of Philosophers/T life and work. theodorus of cyrene (465398 BC) A brief introductionto his life and work. Theon of Smyrna (c70-c135) An introduction http://philtar.ucsm.ac.uk/compendium_of_philosophers/t/

Compendium of Philosophers T Links to materials by and/or about over a thousand philosophers from thousands of years from all over the world from A to Z This compendium contains entries large and small, single or multiple, on hundreds of philosophers. Links vary in size from a few lines of biography to the whole of the Summa Theologica. Sometimes you are directed to a site which has further links. In that case there is no guarantee that all the further links will work, but enough work to make a visit worthwhile. This compendium does not provide links to philosophers’ own home pages. A list of them can be found here A B C ... Z Tagore, Rabindranath (1861-1941) A small Tagore website with a few links Paramahansa Yogananda’s memories Tagore An account of his life Taparelli, Aloysius (1793-1862) A brief introduction to his work Tarski, Alfred (1902-1983) A very brief introduction to his life and work The Semantic Conception of Truth and the Foundations of Semantics Tatarkiewicz, Wladislaw (1886-1980) A brief introduction to his work Tauler, John (c1300-1361) An introduction to his life and work Taylor, Charles (b 1931)

31. Theaetetus --  Encyclopædia Britannica Theaetetus was a disciple of Socrates and studied with theodorus of cyrene. Hetaught at some time in Heraclea (located in presentday southern Italy). http://www.britannica.com/eb/article?eu=73825

32. Guthrie : Life Of Plato And Philosophical Influences That Plato knew and respected the mathematician theodorus of cyrene appears fromthe role which he assigned to him in the Theaetetus; and Cyrene was also the http://www.ellopos.net/elpenor/greek-texts/ancient-greece/guthrie-plato.asp

page url : http://www.ellopos.net/elpenor/greek-texts/ancient-greece/guthrie-plato.asp HOME LANGUAGE LIBRARIES HELP ... CONTACT W.K.C. Guthrie Life of Plato and philosophical influences From, A History of Greek Philosophy, vol. IV, Plato: the man and his dialogues, earlier period , Cambridge University Press, 1989 , pp. 8-38. (Ι) LIFE (a) Sources (b) Birth and family connexions (c) Early years ... Greek Fonts (Ι) LIFE (a) Sources In his own writings Plato keeps himself firmly out of sight, and they reveal little or nothing about his life. He never writes in his own person, We also hear of lives by pupils of Aristotle, Clearchus (an ‘encomium’), Dicaearchus and Aristoxenus. Plato was also a favourite butt of the poets of the Middle Comedy, from whom we have a number of satirical quotations. All these early writings are lost, and the earliest extant life is by Apuleius in the second century A.D., who followed the earlier encomiasts in making his subject a typical hero-figure. Not much later is the book devoted to Plato in the

33. Lunes.html By some at the time, it was considered a foundational crisis when theodorus of cyreneand his pupil Taetetus discovered that most of the square roots from the http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Williford/Lunes/lunes.html

Lunes, Lunes and More Lunes by Dixie Williford Journey Through Genius, pg. 12). In Hippocrates' day (460-380 B.C.), the quadrature of irregular polygons had been accomplished. Though this was respected, it was overshadowed by the fact that these were rectilinear objects. The more challenging issue was whether or not curvilinear figures were able to be squared ( Journey Through Genius, pg. 17). Through Hippocrates did not square the circle, he did successfully square the lunule, from which he falsely believed the quadrature of the circle could be derived. Hippocrates methods for squaring the lunulea plane figure bounded by two circular arcs (a crescent)is on what I will focus. His argument relied on three preliminary results: (1) The Pythagorean theorem (2) an angle inscribed in a semicircle is right (3) the areas of two circles or semicircles are to each other as the squares of their diameters. Though Pythagorus was able to state his famous theorem relating the hypotenuse and legs of right triangles, he was probably not able to prove it. Euclid, however, records the theorem's proof in Book VII of his Elements.

34. Table Of Contents For Gazalé, M.: Number: From Ahmes To Cantor. 162 Pythagoras s Theorem 162 Pythagorean Triples 164 The Plimpton 322 Tablet 167The Ladder of theodorus of cyrene and Diophantine Equations 169 A Variation on http://pup.princeton.edu/TOCs/c6794.html

PRINCETON University Press SEARCH: Keywords Author Title More Options Power Search Search Hints E-MAIL NOTICES NEW IN PRINT E-BOOKS ... HOME PAGE Number: From Ahmes to Cantor Book Description TABLE OF CONTENTS: Preface xi INTRODUCTION 3 CHAPTER 1 The Genesis of Number Systems 9 Foundations 9 Matching 10 Naming 11 Counting 12 Grouping 15 Archaic Number Systems 16 The Egyptians 17 The Mesopotamians 28 The Greeks 34 The Mayas 38 Two Current Number Systems 41 The Hindus 42 The Arabs 43 The Decimal Number System 45 Fractional Numbers 50 Uttering versus writing 52 Units 54 The Binary Number System 55 CHAPTER 2 Positional Number Systems 59 The Division Algorithm 59 Codes 61 Mixed-Base Positional Systems 64 Finding the Digits of an Integer 69 Addition 72 Uniform-Base Multiplication 75 Mixed-Base Multiplication 77 Construction 1: A Parallel Adder 78 Construction 2: A Digital-to-Analog Converter 80 Construction 3: A Reversible Binary-to-Analog Converter 80 Positional Representation of Fractional Numbers 82 Going to Infinity 89 How Precise Is a Mantissa? 93

35. Gazalé, M.: Number: From Ahmes To Cantor. What was the Ladder of theodorus of cyrene and how did the ancient Greekscalculate square roots with such extraordinary proficiency? http://pup.princeton.edu/titles/6794.html

PRINCETON University Press SEARCH: Keywords Author Title More Options Power Search Search Hints E-MAIL NOTICES NEW IN PRINT E-BOOKS ... HOME PAGE Winner of the 2000 Association of American Publishers Award for Best Professional/Scholarly Book in Mathematics and Statistics One of Choice 's Outstanding Academic Books of 2000 Number: From Ahmes to Cantor Shopping Cart Table of Contents We might take numbers and counting for granted, but we shouldn't. Our number literacy rests upon centuries of human effort, punctuated here and there by strokes of genius. In his successor and companion volume to Gnomon: From Pharaohs to Fractals This abundantly illustrated book, remarkable for its coherency and simplicity, will fascinate all those who have an interest in the world of numbers. Number will be indispensable for all those who enjoy mathematical recreations and puzzles, and for those who delight in numeracy. Table of Contents Gnomon: From Pharaohs to Fractals. Subject Areas: Mathematics History of Science and Medicine, Philosophy of Science VISIT OUR MATH WEBSITE Main Selection of the Library of Science Book Club Shopping Cart: For customers in the U.S., Canada, Latin America, Asia, and Australia

36. In Proceedings And Similar Publications A. Iserles, ``The dynamics of the Theodorus spiral , Supplement B to Spirals Fromtheodorus of cyrene to MetaChaos (PJ Davis), Hedrick Lectures 1990, Math. http://www.damtp.cam.ac.uk/user/na/people/Arieh/Publications/Proceedings.html

In proceedings and similar publications: ``Applications of radial basis functions: Sobolev-orthogonal functions, radial basis functions and spectral methods'' , in Algorithms for Approximation IV A. Iserles, ``Numerical analysis in Lie groups'' , in Foundations of Computational Mathematics, Oxford 1999 (R. DeVore, A. Iserles and E. Suli, eds), Cambridge University Press, Cambridge (2001), 105-123. ``Geometric integration: Numerical solution of differential equations on manifolds'' Phil. Trans Royal Soc. A A. Iserles, ``Lie groups and the computation of invariants'' Self-Similar Systems ``Linear ODEs in Lie groups'' Proceedings of the 15th IMACS World Congress A. Iserles, ``Insight, not just numbers'' Proceedings of the 15th IMACS World Congress A. Iserles, ``Beyond the classical theory of computational ordinary differential equations'' , in State of the Art in Numerical Analysis A. Iserles, ``Numerical methods on (and off) manifolds'' , in Foundations of Computational Mathematics (F. Cucker and M. Shub, eds), Springer-Verlag, New York (1997), 180-189.

37. MathFiction Thomas Gray Philosopher Cat (Philip J. Davis) of mathematics with his research. The central mathematical puzzle is what a medievalmanuscript says about the problem of theodorus of cyrene and the square http://math.cofc.edu/faculty/kasman/MATHFICT/mfview.php?callnumber=mf274

38. From Ken.Pledger@vuw.ac.nz (Ken Pledger) Newsgroups Sci.math Shortly after then, theodorus of cyrene generalized the result by proving theirrationality of the square roots of 3, 5, , 15, but got stuck at sqrt(17 http://www.math.niu.edu/~rusin/known-math/98/sqrt_irrat

From: Ken.Pledger@vuw.ac.nz (Ken Pledger) Newsgroups: sci.math Subject: Irrationals (was Re: please don't laugh...) Date: Wed, 09 Dec 1998 15:05:55 +1200 In article Newsgroups: sci.math Subject: Re: how to prove: if x is not a perfect square, sqr (x) is irrational, x positive integer Date: Sat, 12 Dec 1998 23:12:34 -0500 On Sat, 12 Dec 1998, TS wrote: :Date: Sat, 12 Dec 1998 20:40:37 GMT :From: TS

39. Index Of Ancient Greek Mathematicians And Astronomers theodorus of cyrene (4th century BC). . Pythagorean. Plato s teacherin mathematics. Shows that the square roots of 3, 5, 6, 7, 8 http://www.ics.forth.gr/~vsiris/ancient_greeks/classical_period.html

Classical period (5th - 4th century B.C.) Within this period Athens flourishes under Pericles, the Parthenon is built on the Acropolis, the tragedies of Sophocles, Aeschylus and Euripides are created, the phisolophical schools of Socrates and Plato (known as Academy) are established, and the Lyceum of Athens is founded by Aristotle. In science, the importance of the experimental method is accepted. Philosophers-Scientists Socrates (Athens, 470-399 B.C.). Died from poison after the state found him guilty for corrupting the youth. Theodorus of Cyrene (4th century B.C.). . Pythagorean. Plato's teacher in mathematics. Shows that the square roots of 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, and 17 are irrational. Archytas of Tarentum (420-350 B.C.). Greek mathematician, astronomer and engineer. Last of the Pythagorians. Plato and Eudoxus was his pupils. Built a series of toys, among them a mechanical pigeon propelled by a steam jet. Developed the theory for the pulley. Plato (Athens, 430-350 B.C.) . Greek philosopher. He was the founder of the Academy (named from the hero Academos owner of the grove where the Academy was built). Believed that mathematics played an important role in education. Disregarded practicality, a belief he passed to his students such as Eucledes. He started a three part trilogy :

40. Index Of Ancient Greek Scientists theodorus of cyrene. . Pythagorean. Plato s teacher in mathematics. Shows that thesquare roots of 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, and 17 are irrational. http://www.ics.forth.gr/~vsiris/ancient_greeks/whole_list.html

not complete Agatharchos. Greek mathematician. Discovered the laws of perspectives. Anaxagoras of Clazomenae (480-430 B.C.). Greek philosopher. Believed that a large number of seeds make up the properties of materials, that heavenly bodies are made up of the same materials as Earth and that the sun is a large, hot, glowing rock. Discovered that the moon reflected light and formulated the correct theory for the eclipses. Erroneously believed that the Earth was flat. Links: Anaxagoras of Clazomenae, MIT Anaximander (610-545 B.C.). Greek astronomer and philosopher, pupil of Thales. Introduced the apeiron (infinity). Formulated a theory of origin and evolution of life, according to which life originated in the sea from the moist element which evaporated from the sun ( On Nature ). Was the first to model the Earth according to scientific principles. According to him, the Earth was a cylinder with a north-south curvature, suspended freely in space, and the stars where attached to a sphere that rotated around Earth. Links: Anaximander, Internet Encyclopedia of Philosophy

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