1. Theodorus theodorus of cyrene. theodorus of cyrene was a pupil of Protagoras and himself thetutor of Plato, teaching him mathematics, and also the tutor of Theaetetus. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Theodorus.html

Theodorus of Cyrene Born: 465 BC in Cyrene (now Shahhat, Libya) Died: 398 BC in Cyrene (now Shahhat, Libya) Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Theodorus of Cyrene was a pupil of Protagoras and himself the tutor of Plato , teaching him mathematics, and also the tutor of Theaetetus Plato travelled to and from Egypt and on such occasions he spent time with Theodorus in Cyrene. Theodorus, however, did not spend his whole life in Cyrene for he was certainly in Athens at a time when Socrates was alive. Theodorus, in addition to his work in mathematics, was [5]:- ... distinguished ... in astronomy, arithmetic, music and all educational subjects. A member of the society of Pythagoras , Theodorus was one of the main philosophers in the Cyrenaic school of moral philosophy. He believed that pleasures and pains are neither good nor bad. Cheerfulness and wisdom, he believed, were sufficient for happiness. Our knowledge of Theodorus comes through Plato who wrote about him in his work Theaetetus.

2. Theodorus Biography of Theodorus (465BC398BC) theodorus of cyrene. Born 465 BC in Cyrene (now Shahhat, Libya) http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Theodorus.html

Theodorus of Cyrene Born: 465 BC in Cyrene (now Shahhat, Libya) Died: 398 BC in Cyrene (now Shahhat, Libya) Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Theodorus of Cyrene was a pupil of Protagoras and himself the tutor of Plato , teaching him mathematics, and also the tutor of Theaetetus Plato travelled to and from Egypt and on such occasions he spent time with Theodorus in Cyrene. Theodorus, however, did not spend his whole life in Cyrene for he was certainly in Athens at a time when Socrates was alive. Theodorus, in addition to his work in mathematics, was [5]:- ... distinguished ... in astronomy, arithmetic, music and all educational subjects. A member of the society of Pythagoras , Theodorus was one of the main philosophers in the Cyrenaic school of moral philosophy. He believed that pleasures and pains are neither good nor bad. Cheerfulness and wisdom, he believed, were sufficient for happiness. Our knowledge of Theodorus comes through Plato who wrote about him in his work Theaetetus.

3. Chronology For 500BC To 1AD About 425BC theodorus of cyrene shows that certain square roots are irrational.This had been shown earlier but it is not known by whom. http://www-gap.dcs.st-and.ac.uk/~history/Chronology/500BC_1AD.html

Chronology for 500BC to 1AD Previous page Chronology index Full chronology Next page About 500BC The Babylonian sexagesimal number system is used to record and predict the positions of the Sun, Moon and planets. (See this History Topic About 500BC Panini 's work on Sanskrit grammar is the forerunner of the modern formal language theory. About 465BC Hippasus writes of a "sphere of 12 pentagons", which must refer to a dodecahedron About 450BC Greeks begin to use written numerals. (See this History Topic About 450BC Zeno of Elea presents his paradoxes. About 440BC Hippocrates of Chios writes the Elements which is the first compilation of the elements of geometry. About 430BC Hippias of Elis invents the quadratrix which may have been used by him for trisecting an angle and squaring the circle About 425BC Theodorus of Cyrene shows that certain square roots are irrational . This had been shown earlier but it is not known by whom. About 400BC Babylonians use a symbol to indicate an empty place in their numbers recorded in cuneiform writing. There is no indication that this was in any way thought of as a number. (See this History Topic Plato founds his Academy in Athens About 375BC Archytas of Tarentum develops mechanics. He studies the "classical problem" of

4. GO.HRW.COM Lesson 12.4 The Distance Formula theodorus of cyrene, theodorus of cyrene wasa mathematician, astronomer, musician, philosopher, and the tutor of Plato. http://go.hrw.com/ndNSAPI.nd/gohrw_rls1/pKeywordResults?MA1 Theodorus

5. References For Theodorus References for the biography of Theodorus L Giacardi, On theodorus of cyrene's problem, Arch. http://www-history.mcs.st-and.ac.uk/history/References/Theodorus.html

References for Theodorus Biography in Dictionary of Scientific Biography (New York 1970-1990). Articles: B Artmann, A proof for Theodorus' theorem by drawing diagrams, J. Geom. M S Brown, Theaetetus : Knowledge as Continued Learning, Journal of the History of Philosophy L Giacardi, On Theodorus of Cyrene's problem, Arch. Internat. Hist. Sci. T L Heath, A History of Greek Mathematics I (Oxford, 1921), 203-204, 209-212. R L McCabe, Theodorus' irrationality proofs, Math. Mag. A Wasserstein, Theaetetus and the History of the Theory of Numbers, Classical Quarterly Main index Birthplace Maps Biographies Index History Topics ... Anniversaries for the year JOC/EFR January 1999 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Theodorus.html

6. GO.HRW.COM The activity also provides links to interesting Internet sites related to physics.MA1 Theodorus Learn about theodorus of cyrene and the golden spiral. http://go.hrw.com/ndNSAPI.nd/gohrw_rls1/pKeywordResults?MA1 CH12

7. Spirals: From Theodorus To Chaos, By P. Davis around the study of a difference equation that Davis dubs theodorus of cyrene, the book takes us on an eclectic http://www.webbooks.net/books/_peters/Davis.html

Spirals: From Theodorus to Chaos Philip Davis In his introduction, the author writes, "To me, mathematics has always been more than its form, or its content, its logic, its strategies, or its applications. Mathematics is one of the greatest of human intellectual experiences, and as such merits and requires a rather liberal approach." He takes just such an approach in this book inspired by the Hedrick Lectures of the Seventy-Fifth Anniversary of the Mathematical Association of America. Although loosely organized around the study of a difference equation that Davis dubs Theodorus of Cyrene, the book takes us on an eclectic whirlwind tour of history, philosophy, anecdote and, of course, mathematics. Incorporating the old and the new, the proved and the conjectural, Davis examines Theodorus in light of mathematical concerns that have grown and changed over the past 2,500 years. A. K. Peters, Ltd., 1993 ISBN 1-56881-010-5 248 pages List price: $34.00 WebBooks Price: $30.50 Handling/shipping: Surface shipping included; express/overnight at cost. Jump to WebBooks Home Page

8. The Cosmologic Timeline c. 465 bce Hippasus writes of the sphere of 12 pentagons. 465 bce- theodorus of cyrene is born. 398 bce - theodorus of cyrene dies. http://www.panikon.com/cosmo/timeline.html

THE COSMOLOGIC TIMELINE This timeline is distilled down from a more general timeline originally created for Phurba. It is meant to give a better idea of the linear history of the topics discussed in Cosmologic. Some dates given are simply there to give a better grasp of the timeperiods, comparatively. This document will be expanded and fleshed out heavily in time. I have cut the timeline back to 1000 ce as the endpoint, as we will be concentrating on the timeperiod of the first few centuries of the common era for quite some time. Items related to current discussion are rendered in bold print. c. 1948 bce - Abram (later Abraham) was born. c. 1400 bce - Moses was born. c. 624 bce - Thales of Miletus is born. c. 610 bce - Anaximander or Miletus is born. 586 bce - The conquest of Judah by Babylon. c. 587-538 bce - The Babylonian Captivity. c. 569 bce - Pythagoras of Samos is born. c. 547 bce - Thales of Miletus dies. c. 546 bce - Anaximander or Miletus dies. c. 535 bce - Pythagoras goes to Egypt. c. 475 bce - Pythagoras of Samos dies.

9. History Of Mathematics: Greece c. 425); theodorus of cyrene (c. 425); Socrates (469399); Philolausof Croton (dc 390); Democritus of Abdera (c. 460-370); Hippasus 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)

10. A History Of Irrational Numbers Square roots. Pythagoras of Samos. Born about 569 BC in Samos, Ionia Someclaim he had proofed that the sqrt(2) is irrational. theodorus of cyrene. http://home.zonnet.nl/mathematics/Geschiedenis/Getallen/sub4.htm

A History of irrational numbers The history of irrational numbers is almost as complex as the history of rational numbers. I have diffided it in three sectens. 1. Square roots 2. Pi 3. e Square roots Pythagoras of Samos Born: about 569 BC in Samos, Ionia Some claim he had proofed that the sqrt(2) is irrational. Theodorus of Cyrene Born: 465 BC in Cyrene (now Shahhat, Libya) Theodorus of Cyrene was a pupil of Protagoras and himself the tutor of Plato, Our whole knowledge of Theodorus's mathematical achievements are given by this passage from Plato. Yet there are points of interest which immediately arise. The first point is that Plato does not credit Theodorus with a proof that the square root of two was irrational. This must be because 2 was proved irrational before Theodorus worked on the problem, as stated before, some claim this was proved by Pythagoras himself. Theaetetus of Athens Born: about 417 BC in Athens, Greece Euclid of Alexandria Born: about 325 BC Book ten deals with the theory of irrational numbers and is mainly the work of Theaetetus. Euclid changed the proofs of several theorems in this book so that they fitted the new definition of proportion given by Eudoxus. Pi e The number e was first studied by the Swiss mathematician Leonhard Euler in the 1720s, although its existence was more or less implied in the work of John Napier, the inventor of logarithms, in 1614. Euler was also the first to use the letter e for it in 1727 (the fact that it is the first letter of his surname is coincidental). As a result, sometimes e is called the Euler Number, the Eulerian Number, or Napier's Constant (but not Euler's Constant).

11. The Circle Is Not Simply Round Theatetus dialogue, Theatetus reports on an investigation of a still smaller intervalof action of this same curvature, by theodorus of cyrene, a Pythagorean http://www.geocities.com/antidummy/sub/circle.html

The Circle Is Not Simply Round by Bruce Director Double Your Mind - by Bruce Director The Means to Double Your Mind by Bruce Director

12. Spirals: From Theodorus To Chaos Although loosely organized around the study of a difference equation that Davisdubs theodorus of cyrene, the book takes us on an eclectic whirlwind tour of http://www.ibuki-trading-post.com/dir_akp/akp_spifro.html

Spirals: From Theodorus to Chaos Philip J. Davis ISBN 1-56881-010-5 AK Peters - 1993 Hardcover 248pp In his introduction to Spirals: From Theodorus to Chaos, Phil Davis writes, "To me, mathematics has always been more than its form, or its content, its logic, its strategies, or its applications. Mathematics is one of the greatest of human intellectual experiences, and as such merits and requires a rather liberal approach." He takes just such an approach in this book inspired by the Hedrick Lectures of the Seventy-Fifth Anniversary of the Mathematical Association of America. Although loosely organized around the study of a difference equation that Davis dubs Theodorus of Cyrene, the book takes us on an eclectic whirlwind tour of history, philosophy, anecdote and, of course, mathematics. Incorporating the old and the new, the proved and the conjectural, Davis examines Theodorus in light of mathematical concerns that have grown and changed over the past twenty-five hundred years. info@ibuki.com

13. Biography-center - Letter T Theodore II, www.knight.org/advent/cathen/14570b.htm; theodorus of cyrene,wwwhistory.mcs.st-and.ac.uk/~history/Mathematicians/Theodorus.html; http://www.biography-center.com/t.html

Visit a random biography ! Any language Arabic Bulgarian Catalan Chinese (Simplified) Chinese (Traditional) Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Norwegian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Turkish T 340 biographies Tabern, Donalee L. www.invent.org/hall_of_fame/142.html Tacca, Ferdinand www.getty.edu/art/collections/bio/a451-1.html Tacquet, Andrea www-history.mcs.st-and.ac.uk/~history/Mathematicians/Tacquet.html Taft, Helen Herron www.whitehouse.gov/WH/glimpse/firstladies/html/ht27.html Taft, William Howard odur.let.rug.nl/~usa/P/wt27/about/taftbio.htm Tagore, Rabindranath www.nobel.se/literature/laureates/1913/tagore-bio.html Tahir, ibn www-history.mcs.st-and.ac.uk/~history/Mathematicians/Al-Baghdadi.html Taiga, Marton www.kirjasto.sci.fi/taiga.htm Taillandier, Vincent www.getty.edu/art/collections/bio/a1075-1.html Tait, Peter Guthrie www-history.mcs.st-and.ac.uk/~history/Mathematicians/Tait.html Takacs, Karoly

14. Plato FAQ: The Allegory Of The Cave in a ditch while looking at the stars in the caricature of philosopher he putsin the mouth of Socrates for his friend theodorus of cyrene, the scientist, in http://plato-dialogues.org/faq/faq002.htm

Bernard SUZANNE Last updated May 1st, 1999 Plato and his dialogues : Home Biography - Works and links to them History of interpretation New hypotheses - Map of dialogues : table version or non tabular version . Tools : Index of persons and locations Detailed and synoptic chronologies - Maps of Ancient Greek World . Site information : About the author F requently A sked Q uestions about Plato The allegory of the cave "Could you tell me in what work of Plato I might find his "cave analogy". It is a story of men chained in a cave only able to see their own shadows and delude into thinking that this was all there was to reality." The story this letter refers to, usually called "the allegory of the cave", is found at the beginning of book VII of Plato's dialogue called The Republic . The Stephanus references (the universal way of quoting Plato, available in all editions of his works) for the section telling the allegory are Republic , VII, 514a-517a. It is followed by an interpretation of the allegory put by Plato in the mouth of Socrates, as is the allegory itself. The text of this section of the Republic is available in various English translations on the web, including :

15. Www.math.niu.edu/~rusin/known-math/98/sqrt_irrat 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

16. Who's Who In The Phaedrus Note This Theodorus is not theodorus of cyrene, the famous mathematician featuredin the dialogues Statesman and the Theaetetus . Thrasymachus of Chalcedon. http://www.anotherscene.com/phaedrus/whos.html

The Phaedrus Kit Earl Jackson, Jr. tomrip5@aol.com Who's Who in the Phaedrus Anacreon Flourished c. 650 BCE Anacreon important court poet. Serving both the court at Samos and the Athenian court under Hipparchus . For more background in Ancreon and other Greek Lyric Poets, see the Web site for Professor John Porter's course on The Greek Lyric Poets . And for examples of their work, see Professor Porter's selected translations If you need a change of pace from actual Platonic dialogues, you may want to read one of the more-than-likely-fake Platonic dialogues, such as the one to which Hipparchus leant his already dubious name. And see my Electronic Plato Map for the lowdown on the counterfeits and where to find them. Collect them and trade with your friends Cephalus Resident Alien (metus), father of the orator Lysias. The Republic one of Plato's longest and most celebrated dialogues, takes place in the home of Cephalus in the Piraeus. Socrates is in the vicinity on the day of a newly inaugurated festival is to take place. As he attempts to return to Athens, Lysias' s brother, Polemarchus and friends "kidnap" him and bring him to the home of Cephalus. Lysias is also in attendance there. Cephalus welcomes Socrates, and their conversation takes up a good part of the first book, and is one of the engaging in all the dialogues.

17. 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)

18. 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: 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. ``A scalpel, not a sledgehammer: Qualitative approach to numerical mathematics'' CWI Quarterly ZAMM ``Qualitative numerical analysis of ordinary differential equations'' , in The Mathematics of Numerical Analysis Lectures in Applied Maths , American Mathematical Society, Providence RI (1966), 421-442.

19. 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

20. The 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 : The 1989 K-State Undergraduate Lecture Series in Mathematics Printable Version The 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.

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