121. Lecture 11: Ptolemy ptolemy s geocentric model used epicycles to explain retrograde motion of theplanets. ptolemy s model was accepted, and continuously refined, http://www-astronomy.mps.ohio-state.edu/~ryden/ast161_3/notes11.html

Astronomy 161: Professor Barbara Ryden Monday, April 12 PTOLEMY ``It is always better to have no ideas than false ones; to believe nothing than to believe what is wrong.'' - Thomas Jefferson Key Concepts Ancient Greek models of the universe were based on erroneous assumptions. Ptolemy's geocentric model used epicycles to explain retrograde motion of the planets. Ptolemy's model was accepted, and continuously refined, from the 2nd century AD to the 16th century AD. (1) Ancient Greek models of the universe were based on erroneous assumptions. Many of the world's cultures have a tradition of accurate astronomical observations. The ancient Greeks, however, were the first culture to create scientific models of the universe, based on their knowledge of arithmetic and geometry. The Greek models were based on a number of assumptions, some of which were (alas) entirely erroneous. Bad assumptions of Greek astronomers: The Earth is stationary at the center of the universe. WRONG! The Earth is corrupt and subject to decay, while the heavens are perfect and incorruptible. WRONG!

122. Sine, Cosine, And Ptolemy's Theorem Proofs, the essence of Mathematics, ptolemy s Theorem, the Law of Sines. http://www.cut-the-knot.com/proofs/sine_cosine.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents Sine, Cosine, and Ptolemy's Theorem Ptolemy's theorem implies the theorem of Pythagoras . The latter serves as a foundation of Trigonometry, the branch of mathematics that deals with relationships between the sides and angles of a triangle. In the language of Trigonometry, Pythagorean Theorem reads sin (A) + cos (A) = 1, where A is one of the internal angles of a right triangle. If the hypotenuse of the triangle is of length 1, then sin(A) is the length of the side opposite to the angle A, cos(A) is the length of the adjacent side. Ptolemy's theorem also provides an elegant way to prove other trigonometric identities. In a little while, I'll prove the addition and subtraction formulas for sine: sin(A + B) = sin(A)cos(B) + cos(A)sin(B) sin(A - B) = sin(A)cos(B) - cos(A)sin(B) But first let's have a simple proof for the Law of Sines Proposition III.20 from Euclid's Elements says: In a circle the angle at the center is double of the angle at the circumference, when angles have the same circumference as base. The more common formulation asserts that an angle circumscribed in a circle is equal to half the central angle that subtends the same chord. (As a corollary, from here it follows that all circumscribed angles subtending the same arc are equal irrespective of their position on the circle. This is Proposition

123. Egypt, Ptolemy X - Ancient Greek Coinage - WildWinds.com Click here for the Egypt, ptolemy X page with thumbnail images. Diademed headof ptolemy X / PTOLEMAIOU BASILEWS, eagle standing on thunderbolt, http://www.wildwinds.com/coins/greece/egypt/ptolemy_X/i.html

Sponsored by Ancient Coinage of Egypt, Ptolemy X Click here for the Egypt, Ptolemy X page with thumbnail images. Ptolemy X Soter II AE30. 116-80 BC. Head right of Zeus Amon wairing tainia / ... PTOLEMAIOU , two eagles standing left on thunderbolt. Text Image SNGCop 359 PTOLEMAIOU BASILEWS , eagle standing on thunderbolt, Year IB before, mintmark PA behind. Text Obv Rev Svoronos 1671 PTOLEMAIOU BASILEWS , eagle standing on thunderbolt, Year IB before, mintmark PA behind. Text Image Svoronos 1672 PTOLEMAIOU BASILEWS PA behind. Text Obv Rev Svoronos 1674 HJB-Egypt; Ptolemy X, Tetradrachm Text Obv Rev Svoronos 1677 Ptolemy X AR Tetradrachm. Paphos, Year 19 = 95/94 BC. Diademed head of Ptolemy X right, as Zeus, aegis at shoulder / PTOLEMAIOU BASILEWS , eagle standing left, wings open, date L IØ before, mintmark PA behind. Text Obv Rev Svoronos 1680 Ptolemy X, Alexander I, Tetradrachm, Yr 20=95/94 BC, DAN SNG-369, SV-1680 Text Obv Rev Svoronos 1728 PA behind. Text Image Svoronos 1729 PA behind. Text Image Svoronos 1856 PTOLEMY X (or IX). 80-51 BC. Silver Tetradrachm. Head of Ptolemy I, right / Eagle standing, left. L Theta=Year 9. SNGCop 382. Text Obv Rev tetradrachm Malter- Tetradrachm of Ptolemy X VF Text Image Sear #7938 Ptolemy X, AR tetradrachm. Dated year A = 113/112 BC, Paphos Mint, Cyprus, Diademed head of Ptolemy I right, aegis at neck / RTOLEMAIOY BASILEWS Eagle standing left on thunderbolt, LA before, RA behind.

124. Egypt, Ptolemy VI - Ancient Greek Coinage - WildWinds.com Click here for the Egypt, ptolemy VI page with thumbnail images. Entry forEgypt, ptolemy VI on the Digital Historia Numorum Search for ptolemy VI in http://www.wildwinds.com/coins/greece/egypt/ptolemy_VI/i.html

Sponsored by Ancient Coinage of Egypt, Ptolemy VI Click here for the Egypt, Ptolemy VI page with thumbnail images. Svoronos- ARSINOHS FILADELFOU , double cornucopiae bound with fillet; LKG left, KI right. Text Image Svoronos 1205 Ptolemy VI Philometor AR Tetradrachm. Arados mint, dated CY 90 (170/169 BC). Diademed head of Ptolemy I right, in aegis / PTOLEMAIOU SWTHROS , eagle standing left on thunderbolt; retrograde P (date) to left. BMC 18, SNGCop 547. Text Image Svoronos 1208 Egypt, Ptolemy VI AR Tetradrachm. Arados mint, Year 92 (=168/7 BC). Diademed head of Ptolemy right / PTOLEMAIOU BASILEWS , eagle standing left on thunderbolt. Text Image Svoronos 1217cf Ptolemy VI AR Didrachm. Arados mint, Year 110 (150/149 BC). Diademed head of Ptolemy I right / PTOLEMAIOU BASILEWS , eagle standing left on thunderbolt, date P-I in fields. SNG Cop 557. Text Image Svoronos 1342 Ptolemy VI AR Tetradrachm. Paphos Mint, Cyprus, Year 19 = 162/161 BC, Diademed head of Ptolemy I right / PTOLEMAIOU BASILEWS , year LI Q to left, PA to right, eagle standing left on thunderbolt. SNGCop 576. Text Obv Rev Svoronos 1452cf Ptolemy VI AR Tetradrachm. Cyprus, Salamis Mint, Year 18 = 164/163 BC. Diademed head of Ptolemy I right, aegis at neck /

125. A Brief History Of Clocks From Thales To Ptolemy Although ptolemy was familiar with both the anaphoric clock and the astrolabe, Ecliptical coordinates were used by Hipparchos and ptolemy in their star http://www.perseus.tufts.edu/GreekScience/Students/Jesse/CLOCK1A.html

Please note: These papers were prepared for the Greek Science course taught at Tufts University by Prof. Gregory Crane in the spring of 1995. The Perseus Project does not and has not edited these student papers. We assume no responsibility over the content of these papers: we present them as is as a part of the course, not as documents in the Perseus Digital Library . We do not have contact information for the authors. Please keep that in mind while reading these papers. A Brief History of Clocks: From Thales to Ptolemy By: Jesse Weissman Look at the comments on this paper. The earliest example, in western culture, of a celestial sphere is attributed to the presocratic philosopher Thales. Unfortunately, little is known about Thales' sphere beyond Cicero's description in the De re publica For Gallus told us that the other kind of celestial globe, which was solid and contained no hollow space, was a very early invention, the first one of that kind having been constructed by Thales of Mileus, and later marked by Eudoxus with the constellations and stars which are fixed in the sky. (Price 56) This description is helpful for understanding the basic form of Thales' sphere, and for pinpointing its creation at a specific point in time. However, it is clearly a simplification of events that occurred several hundred years before Cicero's lifetime. Why would Thales' create a spherical representation of the heavens and neglect to indicate the stars? Of what use is a bowling ball for locating celestial bodies? Considering Eudoxus' preoccupation with systems of concentric spheres, a more logical explanation is that Thales marked his sphere with stars, and Eudoxus later traced the ecliptic and the paths of the planets on the exterior. The celestial sphere in question probably resembled this

126. Daily Kos :: Diaries CSISS Classics Claudius Ptolemaeus (ptolemy) Representation ptolemy s work in astronomy and geography have made him famous for the ages,despite the fact that many of his theories were in the following centuries http://www.dailykos.com/user/Ptolemy

Diary Comments Ratings Email: debowd01@popmail.med.nyu.edu Happily married to another liberal libertarian physician-scientist for 3 years. Ph.D. in Neuroscience and Physiology. Currently earning M.D. Howard Dean's disaster plan to help Katrina's victims by Ptolemy Tue Sep 6th, 2005 at 17:35:51 PDT The Democratic Party Dear Ptolemy, In the immediate aftermath of Hurricane Katrina, you mobilized to make sure that the Red Cross had the financial resources it needed to respond swiftly. The response was literally overwhelming so many donations poured in that their web site struggled to process them. Since then Americans have seen another kind of disaster unfold. The irresponsible lack of attention by our federal government has led directly to the devastation of communities and the loss of American lives. Ptolemy's diary Permalink There's more... (9 comments) Latest facts from BBC #5 - 2 pm EST 9/6 by Ptolemy Tue Sep 6th, 2005 at 11:24:01 PDT Let's get right to it. Iran is offering 20 million barrels of crude IF we lift our economic sanctions. That's kinda crass. The Superdome had knee-high piles of feces in the stairwells, because the toilets broke down. Urine was all over the floor.

127. Ptolemy S Commentary On The Gospel Of John Prologue In this commentary, ptolemy interpreted the prologue of John s gospel (Jn 1114) ptolemy falls in the second group. The full Gnostic Myth and its many http://www.pseudepigrapha.com/apocrypha_nt/ptl.htm

128. Bryn Mawr Electronic Resources Review - The House Of Ptolemy The House of ptolemy is a site with something for everyone. As noted on itsindex page, The House of ptolemy has been selected by the Discovery Channel http://ccat.sas.upenn.edu/bmcr/bmerr/1999/VenitHoPtoMay.html

Review of House of Ptolemy About this document House of Ptolemy A personal web site of annotated links ( http://www.houseofptolemy.org./ Author: created and maintained by Adam D. Philippidis. Audience: General audiences from middle school students onwards. Peer review, availability, permanence: This is a personal project that certainly deserves to be maintained into the distant future. That, in turn, means that some institutional connection would be desirable. Publication date: site initiated 1 April 1997. Reviewer: Marjorie Venit, Department of Art History and Archaeology, University of Maryland, http://www.inform.umd.edu/ARTH/arthfac/mvenit/ Review date: 2 May 1999. The House of Ptolemy is a site with something for everyone. It is a treasure trove of delights, far more extensive geographically and far less limited temporally than its title implies. The site takes its name from Edwyn R. Bevan's book on the Ptolemys, but although the electronic House of Ptolemy similarly concentrates on the Ptolemaic periodit ranges well beyond Bevan's temporal limit. True, more than 275 of its links are devoted to Ptolemaic Egypt, but an additional 200 or more address later Egypt, making The House of Ptolemy one-stop shopping for Egypt from ca. 330 BCE to the present. As noted on its index page, "

129. Ptolemy ptolemy and Regiomontanus shown on the frontispiece to Regiomontanus Epitome On the motions of the Sun, Moon, and planets, ptolemy again extended the http://abyss.uoregon.edu/~js/glossary/ptolemy.html

Ptolemy and Regiomontanus shown on the frontispiece to Regiomontanus' Epitome of the Almagest, 1496. The Epitome was one of the most important Renaissance sources on ancient astronomy. Ptolemy Ptolemy, Latin in full Claudius Ptolemaeus (fl. AD 127-145, Alexandria), ancient astronomer, geographer, and mathematician who considered the Earth the center of the universe (the "Ptolemaic system"). Virtually nothing is known about his life. Ptolemy's astronomical work was enshrined in his great book He mathematike syntaxis ("The Mathematical Collection"), which eventually became known as Ho megas astronomos ("The Great Astronomer"). During the 9th century, however, Arab astronomers used the Greek superlative Megiste to refer to the book. When the definite article al was prefixed to the term, its title then became known as the Almagest, the name still used today. The Christian Aristotelian cosmos, engraving from Peter Apian's Cosmographia, 1524 Ptolemy accepted the following order for celestial objects in the solar system: Earth (center), Moon, Mercury, Venus, Sun, Mars, Jupiter, and Saturn. He realized, as had Hipparchus, that the inequalities in the motions of these heavenly bodies necessitated either a system of deferents and epicycles or one of movable eccentrics (both systems devised by Apollonius of Perga, the Greek geometer of the 3rd century BC) in order to account for their movements in terms of uniform circular motion. In the Ptolemaic system, the plane of the ecliptic is that of the Sun's apparent annual path among the stars. The planes of the deferents of the planets were believed to be inclined at small angles to the plane of the ecliptic, while the planes of their epicycles were inclined by equal amounts to those of the deferents, so that the planes of the epicycles would always parallel that of the ecliptic. The planes of the deferents of Mercury and Venus were assumed to oscillate above and below the plane of the ecliptic, and likewise the planes of their epicycles were thought to oscillate with respect to the planes of the deferents.

130. The Galileo Project | Science | Ptolemaic System It is in the work of Claudius ptolemy, who lived in the second century CE, ptolemy s astronomy was taught as part of the undergraduate mathematical http://galileo.rice.edu/sci/theories/ptolemaic_system.html

The Galileo Project Science Ptolemaic System Ptolemy Ptolemaic System In his Dialogue Concerning the Two Chief World Systems, Ptolemaic and Copernican of 1632, Galileo attacked the world system based on the cosmology of Aristotle (384-322 BCE) and the technical astronomy of Ptolemy (ca. 150 CE). In his books On the Heavens, and Physics, Ptolemy The heavens, on the other hand, were made up of an entirely different substance, the aether or quintessence (fifth element), an immutable substance. Heavenly bodies were part of spherical shells of aether. These spherical shells fit tightly around each other, without any spaces between them, in the following order: Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn, fixed stars. Each spherical shell (hereafter, simply, sphere) had its particular rotation, that accounted for the motion of the heavenly body contained in it. Outside the sphere of the fixed stars, there was the prime mover (himself unmoved), who imparted motion from the outside inward. All motions in the cosmos came ultimately from this prime mover. The natural motions of heavenly bodies and their spheres was perfectly circular, that is, circular and neither speeding up nor slowing down. It is to be noted about this universe that everything had its natural place, a privileged location for bodies with a particular makeup, and that the laws of nature were not the same in the heavenly and the earthly regions. Further, there were no empty places or vacua anywhere. Finally, it was finite: beyond the sphere of the fixed stars and the prime mover, there was nothing, not even space. The cosmos encompassed all existence.

131. Klaudios Ptolemaios (Ptolemaeus) Geographie (Cosmographia) Home ptolemy. ptolemy and hisdescendants ruled Egypt for more than three hundred and fifty years. http://wwwuser.gwdg.de/~fhasele/ptolemaeus/

Geographie (Geographia; Cosmographia; Geography; Geographike hyphegesis) 1.Kegelentwurf Ptol. I.24 Marinos von Tyros Abb. von Teilbereichen der Oikumene Die sog. 'Donis-Projektion' ... Links Navigation mit Frames Martin Schaffner (1478-1549) : Ptolomeus. 'Tischplatte des Asymus Stedelin' (1533). VORWORT aus der Sicht des Geographen LINKS und die LITERATUR. Diese Seiten sind zur Zeit noch im Aufbau. Kritische Anmerkungen, Fragen und Anregungen sind stets willkommen.... (Frank Haselein 05.2005) FOREWORD Since there is many information about Ptolemy's Geographie available via the Internet in English, this page is currently in German language only. And it's under construction too. When I finished all the stuff in German, English translation will follow- sorry for that. You may have a look at the LINKS and LITERATURE pages to find other sources. Most of that You can read about Ptolemy's Geography focuses on critical text analysis in a philological-historical context. The aim of this site is to focus on some selected problems- predominately, but not exclusively concerning Ptolemy's Geography of Asia, which are not or different discussed elsewhere- from the Geographer's point of view. So, this pages are not an introduction in Ptolemy's "Geography", and they have to be studied in conjunction with other sources available. Critical comments, questions and suggestions are allways welcome..... (Frank Haselein 2005.05)

132. Ptolemy (Claudius Ptolemaeus) - Mathematics And The Liberal Arts Brendan, Brother T. How ptolemy Constructed Trigonometry Tables. Discusses howptolemy may have constructed his trigonometry tables, which in effect http://math.truman.edu/~thammond/history/Ptolemy.html

Ptolemy (Claudius Ptolemaeus) - Mathematics and the Liberal Arts To expand search, see Greece . Laterally related topics: Diophantus Aristotle Archimedes Euclid ... Philolaus , and Archytas The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews , published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet Brendan, Brother T. How Ptolemy Constructed Trigonometry Tables. Mathematics Teacher Discusses how Ptolemy may have constructed his trigonometry tables, which in effect give a table of sines for every quarter degree between

133. Comp.soft-sys.ptolemy Frequently Asked Questions 1.4 Demonstration Version (Ptiny ptolemy) 1.5 Current release 1.6 Who was ptolemy? 2.0 Getting ptolemy 2.1 ptolemy FTP site 2.2 ptolemy WWW site 2.3 Is http://www.faqs.org/faqs/soft-sys/ptolemy-faq/

Usenet FAQs Search Web FAQs Documents ... RFC Index comp.soft-sys.ptolemy Frequently Asked Questions comp.soft-sys.ptolemy ptolemy@eecs.berkeley.edu http://ptolemy.eecs.berkeley.edu/~cxh/nt-ptolemy.html http://ptolemy.eecs.berkeley.edu/tycho/Tycho.html ... http://ptolemy.eecs.berkeley.edu/ The Ptolemy WWW page contains information about Ptolemy, demonstrations of Ptolemy programs, and access to the Ptolemy FTP site, and a hypertext version of the Users's Manual and the Kernel Manual. 2.3 Is Ptolemy available on CDROM? Prime Time Freeware has in the past had Ptolemy available via CDROM. As of 9/95, the latest issue (Prime Time Freeware for UNIX, Issue 4-2) contains Ptolemy0.5.2 Prime Time Freeware can be reached at: Prime Time Freeware Tel: +1 408 433 9662 370 Altair Way, Suite 150 Fax: +1 408 433 0727 Sunnyvale, CA Email: info@ptf.com 94086 USA WWW: www.ptf.com 2.4 Documentation The Ptolemy Manual (The Almagest) is split into three volumes: User's Manual, Programmer's Manual, and Kernel Manual. The new user will only need the User's Manual to get started. All three volumes are available in PostScript, HTML and PDF formats from the Ptolemy FTP site at http://ptolemy.eecs.berkeley.edu/papers/almagest/index.html

134. Ptolemy V. The Columbia Encyclopedia, Sixth Edition. 2001-05 ptolemy V. The Columbia Encyclopedia, Sixth Edition. 200105. http://www.bartleby.com/65/pt/Ptlmy5.html

Select Search All Bartleby.com All Reference Columbia Encyclopedia World History Encyclopedia Cultural Literacy World Factbook Columbia Gazetteer American Heritage Coll. Dictionary Roget's Thesauri Roget's II: Thesaurus Roget's Int'l Thesaurus Quotations Bartlett's Quotations Columbia Quotations Simpson's Quotations Respectfully Quoted English Usage Modern Usage American English Fowler's King's English Strunk's Style Mencken's Language Cambridge History The King James Bible Oxford Shakespeare Gray's Anatomy Farmer's Cookbook Post's Etiquette Bulfinch's Mythology Frazer's Golden Bough All Verse Anthologies Dickinson, E. Eliot, T.S. Frost, R. Hopkins, G.M. Keats, J. Lawrence, D.H. Masters, E.L. Sandburg, C. Sassoon, S. Whitman, W. Wordsworth, W. Yeats, W.B. All Nonfiction Harvard Classics American Essays Einstein's Relativity Grant, U.S. Roosevelt, T. Wells's History Presidential Inaugurals All Fiction Shelf of Fiction Ghost Stories Short Stories Shaw, G.B. Stein, G. Stevenson, R.L. Wells, H.G. Reference Columbia Encyclopedia PREVIOUS NEXT ... BIBLIOGRAPHIC RECORD The Columbia Encyclopedia, Sixth Edition. Ptolemy V (Ptolemy Epiphanes) (t m p n z) ( KEY ) , d. 180

135. Encomium Of Ptolemy Philadelphus He masterfully demonstrates how in composing his Encomium of ptolemy Under ptolemy II Philadelphus, who ruled Egypt in the middle of the third century http://www.ucpress.edu/books/pages/9534.html

@import "style.css"; 238i pages, 5-1/2 x 8-1/4 inches Published November 2003 Available worldwide Entire Site Books Journals E-Editions The Press Theocritus Encomium of Ptolemy Philadelphus Hellenistic Culture and Society, XXXIX A Joan Palevsky Book in Classical Literature Text and Translation with Introduction and Commentary by Richard Hunter In stockships in 2-3 days Categories: Classics Poetry Classics Poetry ... MORE INFO AND CHOICES Email: Description Table of Contents About the Author Related Books "Current interest in interdisciplinary approaches to the Alexandrian culture is making it possible to understand Hellenistic texts in exciting new ways. Richard Hunter here brings to bear his superb skills as a commentator on one of the most neglected but culturally significant Hellenistic poems. He masterfully demonstrates how in composing his Encomium of Ptolemy Philadelphus Theocritus not only drew on the rich tradition of Greek encomiastic verse but also insinuated a double meaning for those knowledgeable in Egyptian culture. The result is a carefully nuanced study that will stand as a model for how to read texts from an intercultural perspective."Kathryn J. Gutzwiller, author of Poetic Garlands: Hellenistic Epigrams in Context "Richard Hunter demonstrates again his deftness as editor and commentator with this edition of Theocritus's Encomium of Ptolemy . He provides this text, usually overlooked in the Theocritean corpus, with thoughtful historicizing and sophisticated literary contextualizations. For scholars and students alike the style is accessible and informative, and as a result the poem has become much more valuable for understanding the poetic dynamics of the early Ptolemaic period."Susan Stephens, author of

136. Stephen Hawking's Universe: Universes Stephen Hawking s Universe The Web companion piece provides detailed explanationsof the most important ideas and developments in human understanding of http://www.pbs.org/hawking/universes/html/ptole.html

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