61. Spring 2003 Sectional Meeting Of The Allegheny Mountain Section Of Abstract In the first century AD, the Introduction to Arithmetic, by Nicomachusof Gerasa and Mathematics Useful for Understanding Plato by theon of smyrna http://mallard.ds.psu.edu/~ecb5/MAA-Spring-03/tattersall.html

62. Child And Adult Portraiture, Martin Dace A reader from Greece thinks that theon of smyrna in ancient Greece could haveviewed the transit with a simple pinhole device, and another in the USA wants http://www.dace.co.uk/

@import url(z3ns.css); : www.dace.co.uk : child and adult portraits and strange but beautiful paintings what's new 4 August 2005: My little painting a small incident in fairyland is now finished. 20 July 2005: The picture on the right links to a sonnet for children, called The Broken-Winged Fairy . It is a little bit sad. 19 July 2005: A commissioned portrait. 7 July 2005: I believe in the religion Of love Whatever direction its caravans may take, For love is my religion and my faith. - Ibn 'Arabi (11th century CE) 6 July 2005: My essay from 30 October 2004 Towards a new art is now in printer-friendly format. The text is unchanged, but owing to the wonders of CSS, when you print the page it comes out without the menu or background, thus saving your ink cartridge. 5 July 2005: Girl grotesque. I thought this was finished, but someone's suggestion to add scales to the fishy tails caused me to develop this painting to a whole new level. As before, my more adventurous use of XHTML and CSS on this web page cannot be made to work fully in Internet Explorer 6 for Windows, although it displays correctly in all other browsers tested. The transparent overlay effect is lost because of Microsoft's incorrect implementation of CSS.

63. Prime Numbers theon of smyrna is also known for finding the approximate value of the radicalof a positive number a, but his finding was based on the use of a linear http://hypatia.math.uri.edu/~kulenm/diffeqaturi/m381f00fp/megan/meganmp.htm

Difference Equations and Recursive Relations Number theory index History Topics Index Major contributors to the theory of difference equations and recursive relations during this time period between and 400 were Heron Theon of Smyrna and Diophantus Heron is known for Heron's formula for finding the approximate value of the radical of a positive number a , given by: x n+1 = ( x n + a/x n This formula is a special case of Newton's formula: x n+1 = x n - f(x n )/f'(x n Theon of Smyrna is also known for finding the approximate value of the radical of a positive number a , but his finding was based on the use of a linear system and radical numbers. His greatest accomplishment was his procedure for obtaining closer and closer approximations of radical 2. Theon 's procedure working with radical 2 involved two sequences of numbers, the side numbers and the diagonal numbers We start with two numbers, one is the first side and is denoted by x and the other is the first diagonal and is indicated by y The second side (x ) and the second diagonal (y ) are formed from the first side and diagonal, then the third side (x

64. Pythagoras - Number Heath quotes theon of smyrna (1st century AD) as writing. of composite numbersthey call those which are contained by two numbers plane, http://www.mathgym.com.au/history/pythagoras/pythnum.htm

Return to MATHGYM Back P YTHAGORAS of S AMOS A Collection of Essays and Lessons for Junior and Senior High School Contents Early concepts of Number and Number Mysticism Figurate Numbers Number Patterns in Music Incommensurables ... Academic Introduction: The Pythagorean view of the universe rested squarely on the belief that Natural (counting) number was the key to the various qualities of mankind and matter. Since in their view everything was composed of number, the explanation for an objects existence could only be found in number. Elsewhere about this time, number existed for utilitarian purposes only, as a device for solving problems in calendar construction, building and commerce. It was the Pythagoreans who saw number as important in itself, the numbers themselves having "personality in a rustic landscape". The distinction was made between logistic (art of computation) and arithmetic (number theory). Kline [6 ] quotes the famous Pythagorean Philolaus (425 B.C.E.), as writing: "Were it not for number and its nature, nothing that exists would be clear to anybody either in itself or in its relation to other things...You can observe the power of number exercising itself ... in all acts and the thoughts of men, in all handicrafts and music." Pythagoras and the early Order initially treated number concretely, as patterns with pebbles, but over time the Pythagoreans developed and refined their concept of number into the same abstract entity which still exists today. Though it is difficult to separate fact from fancy in some of the surviving references to the Pythagoreans, it is generally conceded that they began number theory, and were responsible for the introduction and development of number mysticism in Western Society.

65. Ptolemy's Biography Another associate of his, a certain mathematician Theon, supplied him with to be the Platonic philosopher theon of smyrna (early second century AD, http://www.chass.utoronto.ca/~ajones/ptolgeog/biography.html

What we know of Ptolemy's biography The most definite information that we have about Ptolemy's life comes from his own writings. His astronomical treatise, the Almagest , contains references to several dated observations that he says that he made of the several heavenly bodies at Alexandria in Egypt from the mid 120s to the early 140s A.D. The full list is as follows: 125 April 5, lunar eclipse (Ptolemy does not say who made this observation) 127 March 26, opposition of Saturn 132 February 2, Mercury at greatest elongation 130 December 15, opposition of Mars 132 September 25, autumnal equinox 133 May 6, lunar eclipse 133 May 17, opposition of Jupiter 133 June 3, opposition of Saturn 134 February 18, Venus at greatest elongation 134 June 4, Mercury at greatest elongation 134 October 3, Mercury at greatest elongation 134 October 20, lunar eclipse 135 February 21, opposition of Mars 135 April 5, Mercury at greatest elongation 135 October 1, altitude of moon 136 March 6, lunar eclipse 136 July 8, opposition of Saturn 136 August 31, opposition of Jupiter

66. Square Root Algorithms -- From MathWorld is an approximation to sqrt(2) , then (a+2b)/(a+b) is a better one (the n2case) was known to theon of smyrna in the second century AD (Wells 1986, p. http://mathworld.wolfram.com/SquareRootAlgorithms.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 Special Functions Radicals Square Root Algorithms A sequence of approximations to can be derived by factoring (where -1 is possible only if -1 is a quadratic residue of ). Then and Therefore, and are given by the recurrence relations with . The error obtained using this method is The first few approximants to are therefore given by This algorithm is sometimes known as the Bhaskara-Brouckner algorithm, and the approximants are precisely those obtained by taking successive convergents to the continued fraction of . The fact that if is an approximation to , then is a better one (the case) was known to Theon of Smyrna in the second century AD (Wells 1986, p. 35). Another general technique for deriving this sequence, known as Newton's iteration , is obtained by letting . Then , so the sequence converges quadratically to the root. The first few approximants to

67. JMM HM DICIONÁRIO Translate this page Theon of Alexandria theon of smyrna Theaetetos Theodoros of Cyrene Theon ofSmyrna Theaetetus Theodorus of Cyrene Zeno of Elea, Thalès 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)

68. Nuptial Arithmetic the Florentine s commentary and explores its context, sources, and difficulties,especially its debts to Plato s Timaeus and to theon of smyrna. http://www.english.ucla.edu/RecentPubs/Nupt.html

Michael J. B. Allen's Nuptial Arithmetic: Marsilio Ficino's Commentary on the Fatal Number in Book VIII of Plato's Republic The latest of Michael Allen's distinguished studies of the leading Renaissance Neoplatonist, Marsilio Ficino (1433-1499), presents for the first time a difficult and fascinating text. Very late in his career Ficino wrote a commentary on the mathematical passage in Book VIII of Plato's Republic that concerns the mysterious geometric or "fatal" number. Since antiquity no one had interpreted this famous enigma; in doing so, Ficino addressed a variety of wide-ranging philosophical, psychological, numerological, astrological, and prophetic themes that are central to our understanding of his thought and of the mentalité of his age. In the first part of Nuptial Arithmetic, Allen introduces the Florentine's commentary and explores its context, sources, and difficulties, especially its debts to Plato's Timaeus and to Theon of Smyrna. He then analyzes Ficino's Pythagorean approach to figured numbers and their progressions and Ficino's determination of the fatal and the nuptial numbers. Allen next turns to Ficino's arresting speculations on eugenics, man's habitus

69. TIMELINE 2nd CENTURY Page Of ULTIMATE SCIENCE FICTION WEB GUIDE 117 theon of smyrna, so called to distinguish him from Theon of Alexandria 120129 The Expositio by theon of smyrna expands on Plato s theory of Music. http://www.magicdragon.com/UltimateSF/timeline2.html

TIMELINE 2nd CENTURY Return to Timeline Table of Contents Return to Ultimate SF Table of Contents TIMELINE 2nd CENTURY May be posted electronically provided that it is transmitted unaltered, in its entirety, and without charge. We examine both works of fiction and important contemporaneous works on non-fiction which set the context for early Science Fiction and Fantasy. There are hotlinks here to authors, magazines, films, or television items elsewhere in the Ultimate Science Fiction Web Guide or beyond. Most recently updated: 22 June 2003 [106 kilobytes (formerly 100)]. This web page draws heavily on FACTS as listed in " The Timetables of Science Facts were also checked against " The 1979 Hammond Almanac " [ed. Martin A. Bacheller et al., Maplewood, New Jersey, 1978], p.795. It also utilizes facts from Volume I of D.E. Smith's " History of Mathematics " [(c) 1921 by David Eugene Smith; (c) 1951 by May Luse Smith; New York: Dover, 1958]. Facts are also drawn from the 1911 Encyclopedia Brittanica , and the Wikipedia Executive Summary of the Second Century China in the Second Century Japan in the Second Century India in the Second Century ... Historians, and Where to Go for More Information

70. Greek Musical Writings - Cambridge University Press Minor authors quoted by Theon and Porphyry; Passages from theon of smyrna; Passagesfrom Porphyry; 10. Nicomachus Enchiridion; 11. http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521616972

71. Special Programs Authors Euclid, Plato, Aristotle, Nicomachus, theon of smyrna, Bacon, Descartes,Dedekind, and Newton. Priority given to students in the Experimental Study http://student.mit.edu/@1533813.14973/catalog/mSPa.html

Home Search Help Symbols Help ... My Selections Special Programs Women's Studies Program Project Interphase Program XL Experimental Study Group Concourse Program Terrascope Women's Studies Program Edgerton Center Freshman/Alumni Summer Internship Program Project Interphase: Pre-Freshman Summer Program SP.100 Project Interphase Prereq: Commitment to register as a freshman in the Fall Units arranged Enrollment by invitation only. A seven-and-a-half week academic and survival skills-honing program, Project Interphase is designed to enhance the preparedness of underrepresented minority students entering MIT in the fall semester. The program has a dual focus: academic readiness and general educational development. The program involves calculus, chemistry, physical education, physics, writing, and supporting academic activities, including small-group learning. The students can earn transcript credit for subjects taken in the program, sometimes resulting in advanced placement in corresponding subjects taken in the fall. Activities include day trips to area cultural, recreational, and business sites. Students participate in a range of personal and educational development seminars and activities designed to ensure their smooth transition to college life. The summer experience gives them increased opportunity to prepare for the rigors of an MIT full-subject load. A. N. Berker

72. Dictionary Of The History Of Ideas theon of smyrna (second century AD) claimed that Pythagoras had verified theseratios in all these circumstances. Boethius (fifth century http://etext.lib.virginia.edu/cgi-local/DHI/dhi.cgi?id=dv3-32

73. Chapter 8 Updates - Dissections: Plane & Fancy About 130 AD theon of smyrna gave the recursive formulas for ai and bi. They hadearlier been given in geometric form. (See page 341 of Dickson (1920). http://www.cs.purdue.edu/homes/gnf/book/Booknews/ch8.html

Updates to Chapter 8, "Step Right Up!", in , by Greg N. Frederickson Fibonacci to Diophantus Wrong figure referenced Penta-penta need not be integral ... Squares to one Fibonacci to Diophantus Katharina Huber pointed out that the roles of x and y are reversed when reducing Fibonacci's formula to Diophantus's formula on page 73. Wrong figure referenced Koji Miyazaki, Hirohisa Hioki, and Naoki Odaka pointed out a problem in the reference in the fourth and fifth lines from the bottom of page 75. It should state that Figure 8.4 closely resembles Figure 7.10, not Figure 7.5. Penta-penta need not be integral As suggested by Puzzle 8.3, the Penta class can include identities produced using nonintegral values for p and q. This observation also applies to the Penta-penta class. Typos in Method 5 There are two typos in the specification of Method 5. In the second line of step 2, the first expression should be y - q - 1 rather than z - q - 1 and the second expression should be y - x rather than z - x (These are corrected in the paperback edition.) An earlier dissection Using Method 2C, I gave 5-piece dissections for all squares in the PP-plus class. One solution in the PP-plus class is 9

74. New Page 0 theon of smyrna Thymaridas. go to index, go to index, go to index. Xenocrates Zeno of Elea Zeno of Sidon Zenodorus http://www.edfiles.com/top/GR6B/geometryP.htm

SCIENCE / GEOMETRY / MATH EDFILES SOCIAL STUDIES ANCIENT GREECE science math geometry index Ancient Greek mathematics greek contributions to science greek contributions to science ii ancient greek medicine ... euclids elements Ancient Greek mathematics Greek mathematics Anaxagoras Anthemius Antiphon ... Zenodorus greek contributions to science Ancient Greek Agriculture Botany Ancient Greek Astronomy Ancient Greek Earth Science Origins of Greek ScienCE ... go to index greek contributions to science ii (from the vatican) Vatican Exhibit Main Hall Greek Astronomy Greek Mathematics and Modern Heirs Mathematics Ancient Science Modern Fates ... go to index ancient greek medicine Ancient drugs BBC Medicine Asclepius (1200BC - 500AD) BBC Medicine - Greek Medicine BBC Medicine Hippocrates ... go to index on ancient medicine Part 1 Part 2 Part 3 Part 4 ... go to index on air waters and places Part 1 Part 2 Part 3 Part 4 ... go to index Articles about Greek mathematics Squaring the circle Doubling the cube Trisecting an angle Greek Astronomy ... Greek mathematics?

75. The Christ By John E. Remsberg (Chapter 2) Valerius Maximus. Arrian Petronius Dion Pruseus Paterculus Appian theon of smyrnaPhlegon Pompon Mela Quintius Curtius Lucian Pausanias Valerius Flaccus http://www.positiveatheism.org/hist/rmsbrg02.htm

The Christ John E. Remsberg [HTML and editing by Cliff Walker, 2000] Return to Historical Index Home to Positive Atheism Go to Table of Contents Chapter 2 Silence of Contemporary Writers Another proof that the Christ of Christianity is a fabulous and not a historical character is the silence of the writers who lived during and immediately following the time he is said to have existed. That a man named Jesus, an obscure religious teacher, the basis of this fabulous Christ, lived in Palestine about nineteen hundred years ago, may be true. But of this man we know nothing. His biography has not been written. A Renan and others have attempted to write it, but have failed have failed because no materials for such a work exist. Contemporary writers have left us not one word concerning him. For generations afterward, outside of a few theological epistles, we find no mention of him. The following is a list of writers who lived and wrote during the time, or within a century after the time, that Christ is said to have lived and performed his wonderful works: Josephus Philo-Judaeus Seneca Pliny the Elder Suetonius Juvenal Martial Persius Plutarch Justus of Tiberius Apollonius Pliny the Younger Tacitus Quintilian Lucanus Epictetus Silius Italicus Statius Ptolemy Hermogones Valerius Maximus Arrian Petronius Dion Pruseus Paterculus Appian Theon of Smyrna Phlegon Pompon Mela Quintius Curtius Lucian Pausanias Valerius Flaccus Florus Lucius Favorinus Phaedrus Damis Aulus Gellius

76. CHRONOLOGY OF RECREATIONAL MATHEMATICS By David Singmaster 130 theon of smyrna Biblion natural square often erroneously cited as magic.c150 Heron Peri Metron - Cistern Problem; Aristotle s Wheel Paradox. http://anduin.eldar.org/~problemi/singmast/recchron.html

Computing, Information Systems and Mathematics 87 Rodenhurst Road South Bank University London, SW4 8AF, England London, SE1 0AA, England Tel/fax: 0181-674 3676 Tel: 0171-815 7411 Fax: 0171-815 7499 E-mail: ZINGMAST@VAX.SBU.AC.UK CHRONOLOGY OF RECREATIONAL MATHEMATICS by David Singmaster last Web revision:December 22, 1998 Mario Velucchi's Web Index visitors since Dec. 22, 1998 Web page processed by Web Master - Mario Velucchi velucchi@bigfoot.com Mario Velucchi / Via Emilia, 106 / I-56121 Pisa - Italy Receive email when this page changes Click Here Powered by Netmind Resources provided by Brad Spencer

77. Aristarchus Of Samos 6, pp. 922F923A; cf. Heath, Aristarchus of Samos, p. 304. theon of smyrna (ed.Hiller) p. 200, 7-12. Cf. Heath, Aristarchus of Samos, p. 304. http://www.varchive.org/ce/orbit/arisam.htm

Aristarchus The first of the Greek philosphers and mathematicians to unravel the celestial plan and announce the discovery was Aristarchus of the isle of Samos. Others before him assumed that the Earth is a sphere and that it moves, but he was the first to formulate plainly the heliocentric theory, the scheme which has the Sun in the center. Aristarchus lived from about the year 310 before the present era to about 230, and among the geometers he succeeded Euclid and preceded Archimedes. In -288 or -287 he followed Theophrastus as the head of the Peripatetic School established by Aristotle. Whatever his scientific argument may have been, he accused Aristarchus of an act of impiety. Plutarch wrote in his book Of the Face in the Disc of the Moon (De facie in orbe lunae) We do not know whether there was any actual court action and verdict; however, we know that a verdict of judges, even if unanimous, could not make the Sun a satellite of the Earth. Not even a scientific tribunal can do this, not even if it is presided over by Archimedes and the most illustrious men of the generation sit as judges. Aristarchus had no followers in his generation, nor in the next generation. About a century after Aristarchus, Seleucus, a Chaldean of Seleucia on the Tigris, who lived and wrote about the year 150 before the present era, adopted the teaching of Aristarchus.

78. Esoteric Science theon of smyrna declares that this array of ten dots, the tetractys of Pythagoras,was a symbol of the greatest importance, to the discerning mind it http://www.prs.org/gallery-science.htm

Babbitt's Atom From Babbitt's Principles of Light and Color By far the most remarkable conception of the atom evolved during the last century is that produced by the genius of Dr. Edwin D. Babbitt. MPH The Problem of Diversity From Kircher's Ars Magna Sciendi In this diagram Kircher arranges 18 objects in two vertical columns and then determines the number of arrangements in which they can be combined. By the same method Kircher further estimates that fifty objects may be arranged in 1,273, 726,838,815,420,339, 851,343,083, 767,005,515,293, 749,454,795,473,408,000,000, 000,000 combinations. From this it will be evident that infinite diversity is possible, for the countless parts of the universe may be related to each other in an in-calculable number of ways. MPH Pythagoras, the First Philosopher

79. Harvard University Press/Translated By Ivor Thomas, Greek Mathematical Works, II (ii) theon of smyrna XXI. Trigonometry— 1. Hipparchus and Menelaus 2. Ptolemy— (a)General (b) Table of sines— (i) Introduction http://www.hup.harvard.edu/contents/L362_toc.html

FROM THE LOEB CLASSICAL LIBRARY TRANSLATED BY IVOR THOMAS Greek Mathematical Works, II, From Aristarchus to Pappus (a) General (b) Distances of the sun and moon (c) Continued fractions (?) (a) General (b) Surface and volume of the cylinder and sphere (c) Solution of a cubic equation (i) Preface (ii) Two lemmas (iii) Volume of a segment of a paraboloid of revolution (e) (i) Definitions (ii) Fundamental property (iii) A verging (iv) Property of the subtangent f ) Semi-regular solids (g) System of expressing large numbers (h) Indeterminate analysis : the Cattle Problem (i) Postulates (ii) Principle of the lever (iii) Centre of gravity of a parallelogram (j) Mechanical method in geometry (i) Postulates (ii) Surface of fluid at rest (iii) Solid immersed in a fluid (iv) Stability of a paraboloid of revolution (a) General (b) On means c ) The Platonicus (d) Measurement of the earth (a) (i) Relation to previous works (ii) Scope of the work (iii) Definitions (iv) Construction of the sections (v) Fundamental properties (vi) Transition to new diameter (i) General (ii) On the Cutting-off of a Ratio.

80. Theosophy Library Online - Great Teacher Series - ALBINUS Albinus was the contemporary of theon of smyrna, who wrote a mathematicalintroduction to Plato, a compilation of quotations on Platonic philosophy which http://theosophy.org/tlodocs/teachers/Albinus.htm

ALBINUS Plato's doctrine of Fate is something like this: all things, he says, are in Fate, but not everything is fated. For though Fate holds the rank of law, it does not decree that one person shall do this and another suffer that, for if it did, it would go on ad infinitum producing infinite things and infinite accidents associated with them. Moreover, there would be nothing in our power, and praise and blame as well as everything like that would vanish. Rather, Fate decrees that if a soul selects a life of this kind, and does certain acts, then certain things will follow upon the selection. The soul then is without a master, and it rests with itself to do or to refrain from an act: it is not forced to do this or that. Nonetheless, whatever follows from doing something is accomplished by Fate. For example, if a Paris carries off a Helen, he freely does so, but it follows that Greeks will war with Trojans for her sake. Didaskalikos XXVI ALBINUS The rise of Stoic philosophy, austere and admirable in its demanding way of life, but metaphysically limited from a Platonic perspective, eventually attracted the attention of the Academy. Whilst the Old Academy dealt with questions raised by Plato, the Middle Academy, marked by the election of Crates in 270 B.C., focussed on detailed responses to Stoic philosophy. This critical and even negative use of dialectic resulted in a steady drift towards scepticism, a reaction to all dogmatic schools which propounded specific doctrines. With the election of Carneades, who died in 128 B.C., the New Academy was rigidly sceptical; Carneades emphasized

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