Contexts and Constructions of Ottoman Science with Special Reference to Astronomy
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A Genetic Context for Understanding the Trigonometric Functions Danny Otero Xavier University, [email protected]
Ursinus College Digital Commons @ Ursinus College Transforming Instruction in Undergraduate Pre-calculus and Trigonometry Mathematics via Primary Historical Sources (TRIUMPHS) Spring 3-2017 A Genetic Context for Understanding the Trigonometric Functions Danny Otero Xavier University, [email protected] Follow this and additional works at: https://digitalcommons.ursinus.edu/triumphs_precalc Part of the Curriculum and Instruction Commons, Educational Methods Commons, Higher Education Commons, and the Science and Mathematics Education Commons Click here to let us know how access to this document benefits oy u. Recommended Citation Otero, Danny, "A Genetic Context for Understanding the Trigonometric Functions" (2017). Pre-calculus and Trigonometry. 1. https://digitalcommons.ursinus.edu/triumphs_precalc/1 This Course Materials is brought to you for free and open access by the Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) at Digital Commons @ Ursinus College. It has been accepted for inclusion in Pre-calculus and Trigonometry by an authorized administrator of Digital Commons @ Ursinus College. For more information, please contact [email protected]. A Genetic Context for Understanding the Trigonometric Functions Daniel E. Otero∗ July 22, 2019 Trigonometry is concerned with the measurements of angles about a central point (or of arcs of circles centered at that point) and quantities, geometrical and otherwise, that depend on the sizes of such angles (or the lengths of the corresponding arcs). It is one of those subjects that has become a standard part of the toolbox of every scientist and applied mathematician. It is the goal of this project to impart to students some of the story of where and how its central ideas first emerged, in an attempt to provide context for a modern study of this mathematical theory. -
International Journal for Scientific Research & Development| Vol. 5, Issue 12, 2018 | ISSN (Online): 2321-0613
IJSRD - International Journal for Scientific Research & Development| Vol. 5, Issue 12, 2018 | ISSN (online): 2321-0613 A Case Study on Applications of Trigonometry in Oceanography Suvitha M.1 Kalaivani M.2 1,2Student 1,2Department of Mathematics 1,2Sri Krishna Arts and Science College, Coimbatore, Tamil Nadu, India Abstract— This article introduces a trigonometric field that bisected to create two right-angle triangles, most problems extends in the field of real numbers adding two elements: sin can be reduced to calculations on right-angle triangles. Thus and cos satisfying an axiom sin²+cos²=1.It is shown that the majority of applications relate to right-angle triangles. assigning meaningful names to particular elements to the One exception to this is spherical trigonometry, the study of field, All know trigonometric identities may be introduced triangles on spheres, surfaces of constant positive curvature, and proved. The main objective of this study is about in elliptic geometry (a fundamental part of astronomy and oceanography how the oceanographers guide everyone with navigation). Trigonometry on surfaces of negative curvature the awareness of the Tsunami waves and protect the marine is part of hyperbolic geometry. animals hitting the ships and cliffs. Key words: Trigonometric Identities, Trigonometric Ratios, II. HISTORY OF TRIGONOMETRY Trigonometric Functions I. INTRODUCTION A. Trigonometry Fig. 2: Hipparchus, credited with compiling the first trigonometric, is known as "the father of trigonometry". A thick ring-like shell object found at the Indus Fig. 1: Valley Civilization site of Lothal, with four slits each in two All of the trigonometric functions of an angle θ can be margins served as a compass to measure angles on plane constructed geometrically in terms of a unit circle centered at surfaces or in the horizon in multiples of 40degrees,upto 360 O. -
Asian Contributions to Mathematics
PORTLAND PUBLIC SCHOOLS GEOCULTURAL BASELINE ESSAY SERIES Asian Contributions to Mathematics By Ramesh Gangolli Ramesh Gangolli Dr. Ramesh Gangolli was born in India, and educated at the University of Bombay, Cambridge University and MIT. He held the Cambridge Society of India Scholarship while at Cambridge University, and was a College Scholar at Peterhouse. During 1966-68 he was a Sloan Foundation Fellow. In 1967, he was awarded the Paul Levy Prize by a Committee of the French Academy of Sciences. At present he is a Professor of Mathematics at the University of Washington, Seattle. He has been active in mathematical research as well as in mathematics education, and has held visiting positions at many universities in the U.S. and abroad. He has served professionally in several capacities: as Chair of the mathematics of the Committee on Education of the American Mathematical Society, and as a member of several committees of the U.S. National Academy of Sciences and the National Research Council. He is also a musician specializing in the Classical Music of India, and is an Adjunct Professor of Music in the School of Music at the University of Washington. Version: 1999-12-22 PPS Geocultural Baseline Essay Series AUTHOR: Gangolli SUBJECT: Math Contents 1. INTRODUCTION. 1 1.1 Purpose of this essay. 1 1.2 Scope. 2 1.3 Acknowledgments. 3 2. SOME VIEWS OF THE DEVELOPMENT OF MATHEMATICS. 5 3. THE PLACE OF MATHEMATICS IN ANCIENT CULTURES. 10 3.1 Agriculture, commerce, etc. 11 3.2 Ritual, worship, etc. 12 3.3 Intellectual curiosity and amusement. -
Stjepan/Ahmed- Paša Hercegović (1456.?-1517.) U Svjetlu Dubrovačkih, Talijanskih I Osmanskih Izvora
STJEPAN/AHMED- PAšA HERCEGOVić (1456.?-1517.) U SVJETLU DUBROVAčKIH, TALIJANSKIH I OSMANSKIH IZVORA Kontroverzne teme iz života Stjepana/Ahmed-paše Hercegovića Petar VRANKIć UDK: 929.7 Stjepan/Ahmed-paša Kath.-Theologische Fakultät der Hercegović Universität Augsburg Izvorni znanstveni rad Privatadresse: Primljeno: 14. ožujka 2017. Kardinal-Brandmüller-Platz 1 Prihvaćeno: 5. travnja 2017. D - 82269 Geltendorf E-pošta: [email protected] Sažetak Knez Stjepan Hercegović Kosača, prvotno najmlađi i si- gurno najomiljeniji sin hercega Stjepana Vukčića Kosa- če, postavši kasnije Ahmed-paša Hercegović, vrlo ugled- ni i uspješni osmanski zapovjednik, upravitelj, ministar, veliki vezir, diplomat, državnik i pjesnik, ostaje i dalje djelomično kontroverzna ličnost u južnoslavenskim, ta- lijanskim i turskim povijesnim prikazima. Nije ni danas lako točno prikazati njegov životni put. Povjesnici se ne slažu u podrijetlu i imenu njegove majke, godini rođenja, slanju ili odlasku na dvor sultana Mehmeda Osvajača, sukobu s polubratom, hercegom Vlatkom, uspjelom ili neuspjelom preuzimanju oporukom utvrđenoga nasli- 9 PETAR VRANKIĆ — STJEPAN/AHMED-PAŠA HERCEGOVIĆ... jeđa njegova oca Stjepana i majke Barbare, založenog u Dubrovniku, godini ženidbe, broju žena i broju vlastite djece, te o broju i mjestu najvažnijih državnih službi iz njegova četrdesetogodišnjeg osmanskog životnog puta. U ovom prilogu autor pokušava unijeti više svjetla u gotovo sva sporna pitanja služeći se prvenstveno, danas dostu- pnim, objavljenim i neobjavljenim vrelima te dostupnom bibliografijom u arhivima i knjižnicama Dubrovnika, Venecije, Milana, Firence, Riminija, Sarajeva i Istanbula. Ključne riječi: Stjepan Hercegović/Ahmed-paša Herce- gović; herceg Stjepan; hercežica Jelena; hercežica Bar- bara; hercežica Cecilija; Dubrovnik; Carigrad; talaštvo; konverzija na islam; herceg Vlatko; sultan Mehmed II.; sultan Bajazid II.; borba za obiteljsko naslijeđe; djeca Ahmed-paše. -
Ii Inci Bayezid'in O~Ullarindan Sultan Korkut
II INCI BAYEZID'IN O~ULLARINDAN SULTAN KORKUT Ord. Prof. ISMAIL HAKKI UZUNÇAR~ILI Sultan Bayezid'in o~ullar~ : Sultan Ikinci Bayezid'in Abdullah, ~ehin~ ah, Alem~ ah, Ahmed, Korkut, Selim, Mehmed, Mahmud adlar~nda sekiz O~lu vard~. Bunlardan en büyük o~lu Abdullah 5 ~evval 888 ve 6 Kas~m 1483'de Konya valisi bulunurken ve ~ehin~ ah 5 Rebiul- âhir 917 ve 2 Temmuz ~ 5~~ 'de yine Konya valisi iken ve ~ehzade Alem~ ah gog h 1503 Manisa'da (Çelebi Sultan) yani vali bulun- du~u s~rada ve ~ehzade Mehmed g ~ o h 1504 m'de Kefe Sancak beyi iken ve ~ehzade M ahmud ise 913 h 1507 m'de Manisa'da San- cak beyi bulundu~u esnada vefat ettiklerinden Sultan Bayezid'in son zamanlar~nda büyü~ü Ahmed, ortancas~~ Korkut ve en küçükleri Selim olmak üzere hayatta üç o~lu kalm~~ t~ '. Sultan Korkut bir kayda göre 872 h 1467 m ve di~er kayda göre de 874 h 1469'da Amasya'da babas~n~ n orada vali bulundu~u zaman do~mu~tur. Korku t'un validesi, Nigâr Hatun olup Bayezid'in k~z~~ Fatma Sultan'la ana, baba bir karde~dirler 2. ~ehzadelerin Sancak Beylikleri : Eyalet ve sancaklara tayin edilen Osmanl~~ ~ehzadelerinin Devlet divan~ndaki memurlar ve hizmetler gibi kendilerinin maiyyetlerinde 1 Ldtift tezkiresinde (matbu nüsha) s. 65, 66 Ahmed'in Korkut'tan büyük oldu~u zikrediliyor; Artus Tomas'da Ahmed'in büyük oldu~unu, K~nal~zâde Korkut'un Ahmed'ten büyük oldu~unu Tdc-iit-Tevaril~~ He~t Bihi~eden naklen (c. -
Pomponazzi on Identity and Individuation Han Thomas Adriaenssen Abstract. Aristotle Defines Growing As a Process in Which An
Final version forthcoming in Journal of the History of Philosophy Pomponazzi on Identity and Individuation Han Thomas Adriaenssen Abstract. Aristotle defines growing as a process in which an individual living being persists as it accumulates new matter. This definition raises the question of what enables an individual to persist as its material composition continuously changes over time. This paper provides a systematic account of Pietro Pomponazzi’s answer to this question. In his De nutritione et augmentatione, Pomponazzi argues that individuals persist in virtue of their forms. Forms are individuated in part by their material, causal, and temporal origins, which commits Pomponazzi to the view that individuals necessarily have the material, causal, and temporal origins they do. I provide an account of why Pomponazzi was willing to this view. While his opponents remain unnamed, I argue that his arguments for this view are best read as addressing, among others, Paul of Venice and Gregory of Rimini. In On Generation and Corruption, Aristotle describes growing as a process in which “any and every part of the growing magnitude is made bigger . by the accession of something, and thirdly in such a way that the growing thing is preserved and persists” (321a18–22).1 For instance, when an animal grows as the result of the intake of food, all of its limbs grow larger as the result of the intake of new matter, in such a way that the same individual animal persists throughout the process. 1 All Aristotle translations are taken from the Jonathan Barnes edition of the Complete Works. 1 Final version forthcoming in Journal of the History of Philosophy Simple though it may perhaps appear, this definition confronted Aristotle’s later followers and commentators with a problem. -
The Aeolipile As Experimental Model in Early Modern Natural Philosophy
7KH$HROLSLOHDV([SHULPHQWDO0RGHOLQ(DUO\0RGHUQ1DWXUDO3KLORVRSK\ &UDLJ0DUWLQ 3HUVSHFWLYHVRQ6FLHQFH9ROXPH1XPEHU0D\-XQHSS $UWLFOH 3XEOLVKHGE\7KH0,73UHVV )RUDGGLWLRQDOLQIRUPDWLRQDERXWWKLVDUWLFOH KWWSVPXVHMKXHGXDUWLFOH Access provided by Oakland University (22 Aug 2016 16:29 GMT) The Aeolipile as Experimental Model in Early Modern Natural Philosophy Craig Martin Oakland University What causes winds was regarded as one of the most difficult questions of early modern natural philosophy. Vitruvius, the ancient Roman architectural au- thor, put forth an alternative to Aristotle’s theory by likening the generation of wind to the actions of the aeolipile, which he believed made artificial winds. As Vitruvius’s work proliferated during the sixteenth and seventeenth centu- ries, numerous natural philosophers, including Descartes, used the aeolipile as a model for nature. Yet, interpretations of Vitruvius’s text and of the relation of the aeolipile to natural winds varied according to definitions and concep- tions of air, wind, rarefaction, condensation, and vapor. 1. Introduction Numerous early modern natural philosophers, Aristotelians as well as their detractors, invoked the workings of the aeolipile to explain the origin and matter of winds. An aeolipile, in its simplest form, is a hollow metal ball or vessel that has one or two small openings. An external fire, or other source of heat, warms water placed within the ball until jets of steam and air shoot forth from the aperture or apertures. From antiquity to the seventeenth century, natural philosophers referred to this artificial device as evidence for specific theories of the wind. These writings typically cited Vitruvius, the ancient Roman author of De architectura, who gave the earliest Prelimary versions of the paper were presented at seminars sponsored by the Department of History and Science and Technology at Johns Hopkins University and by the Graduate School Field Committee in Medieval and Early Modern Studies at the University of Maryland, College Park. -
18 Medidas.Indd
Número 17 - 18. Nueva época 1.er y 2.º semestre de 2018 AWRAQRevista de análisis y pensamiento sobre el mundo árabe e islámico contemporáneo AWRAQRevista de análisis y pensamiento sobre el mundo árabe e islámico contemporáneo DIRECCIÓN Pedro Martínez-Avial, director general de Casa Árabe CONSEJO DE REDACCIÓN Karim Hauser Elena González Nuria Medina Olivia Orozco Javier Rosón SECRETARÍA DE AWRAQ [email protected] WEB Y SUSCRIPCIÓN www.awraq.es EDITORES Casa Árabe. c/ Alcalá, 62. 28009 Madrid (España) www.casaarabe.es Nota: Los artículos de la parte central de este número de Awraq son resultado del encuentro multidisciplinar que tuvo lugar en la sede de Casa Árabe en Córdoba del 20 al 22 de septiembre de 2017, en colaboración con la Fundación Ramón Areces y bajo la dirección académica de Mònica Rius Piniés (Universidad de Barcelona) y Cristina de la Puente (CSIC), bajo el título «Ciencia en al- Ándalus». El presente volumen cuenta con la coordinación académica de la profesora de la sección de Estudios Árabes del Departamento de Filología Clásica, Románica y Semítica de la Universidad de Barcelona Mònica Rius-Piniés. Copyright © Casa Árabe © de los textos: sus autores. © de los anuncios: los anunciantes. Todos los derechos reservados. Gráfica: Hurra! Estudio ISSN: 0214-834X Depósito legal: M-40073-1978 Imprenta: Imprenta Tecé Número 17-18. Nueva época 1.er y 2.º semestre de 2018 CARTA DEL DIRECTOR 3 EL TEMA: CIENCIA EN AL-ÁNDALUS Introducción. Mònica Rius 5 La ciencia en al-Ándalus y su papel como puente entre la ciencia árabe y la europea. Julio Samsó 9 Los sabios de origen andalusí y su aportación a la ciencia otomana. -
Christian Allies of the Ottoman Empire by Emrah Safa Gürkan
Christian Allies of the Ottoman Empire by Emrah Safa Gürkan The relationship between the Ottomans and the Christians did not evolve around continuous hostility and conflict, as is generally assumed. The Ottomans employed Christians extensively, used Western know-how and technology, and en- couraged European merchants to trade in the Levant. On the state level, too, what dictated international diplomacy was not the religious factors, but rather rational strategies that were the results of carefully calculated priorities, for in- stance, several alliances between the Ottomans and the Christian states. All this cooperation blurred the cultural bound- aries and facilitated the flow of people, ideas, technologies and goods from one civilization to another. TABLE OF CONTENTS 1. Introduction 2. Christians in the Service of the Ottomans 3. Ottoman Alliances with the Christian States 4. Conclusion 5. Appendix 1. Sources 2. Bibliography 3. Notes Citation Introduction Cooperation between the Ottomans and various Christian groups and individuals started as early as the beginning of the 14th century, when the Ottoman state itself emerged. The Ottomans, although a Muslim polity, did not hesitate to cooperate with Christians for practical reasons. Nevertheless, the misreading of the Ghaza (Holy War) literature1 and the consequent romanticization of the Ottomans' struggle in carrying the banner of Islam conceal the true nature of rela- tions between Muslims and Christians. Rather than an inevitable conflict, what prevailed was cooperation in which cul- tural, ethnic, and religious boundaries seemed to disappear. Ÿ1 The Ottomans came into contact and allied themselves with Christians on two levels. Firstly, Christian allies of the Ot- tomans were individuals; the Ottomans employed a number of Christians in their service, mostly, but not always, after they had converted. -
Investigating Gifted Middle School Students' Images About
Science Education International Vol. 27, Issue 1, 2016, 136-150 Investigating Gifted Middle School Students’ Images about Scientists: A Cultural Similarity Perspective N. BAYRI*, M. S. KOKSAL†, P. ERTEKIN‡ ABSTRACT: The purpose of this study is to investigate gifted middle school students’ images about scientists in terms of cultural similarity. Sample of the study is 64 gifted middle school students taking courses from a formal school for gifted students. The data were collected by using Draw-a-scientist (DAST) instrument and was analysed by two researchers using Draw-a-scientist C form. The data involved two different drawings for the scientist from similar culture and the popular scientists respectively, and explanations of the drawings. When looked at the findings from cultural similarity perspective, it was seen that the findings about the drawings changed for the scientists in terms of cultural similarity with the students. They wrote mostly Einstein and Edison for the popular scientists while they gave names of Cahit Arf, Avicenna and Ali Qushji for scientists from similar culture. Also they did draw lab coat and eyeglasses more frequently for the popular scientists than those for scientists from similar culture. As another finding, they drew more number of tools for research of common scientists than those for scientists from similar culture. Finally, they drew messy, noisy, non-sterile places for the scientists from similar culture while they imagined the scientists from similar culture as lazy, ignorant inattentive and clumsy. These findings show importance of cultural similarity perception of gifted students on the scientists when they think about the scientists, this situation asks new questions about culture-dependent scientist images of gifted students to gifted science education researchers using DAST. -
The History of Arabic Sciences: a Selected Bibliography
THE HISTORY OF ARABIC SCIENCES: A SELECTED BIBLIOGRAPHY Mohamed ABATTOUY Fez University Max Planck Institut für Wissenschaftsgeschichte, Berlin A first version of this bibliography was presented to the Group Frühe Neuzeit (Max Planck Institute for History of Science, Berlin) in April 1996. I revised and expanded it during a stay of research in MPIWG during the summer 1996 and in Fez (november 1996). During the Workshop Experience and Knowledge Structures in Arabic and Latin Sciences, held in the Max Planck Institute for the History of Science in Berlin on December 16-17, 1996, a limited number of copies of the present Bibliography was already distributed. Finally, I express my gratitude to Paul Weinig (Berlin) for valuable advice and for proofreading. PREFACE The principal sources for the history of Arabic and Islamic sciences are of course original works written mainly in Arabic between the VIIIth and the XVIth centuries, for the most part. A great part of this scientific material is still in original manuscripts, but many texts had been edited since the XIXth century, and in many cases translated to European languages. In the case of sciences as astronomy and mechanics, instruments and mechanical devices still extant and preserved in museums throughout the world bring important informations. A total of several thousands of mathematical, astronomical, physical, alchemical, biologico-medical manuscripts survived. They are written mainly in Arabic, but some are in Persian and Turkish. The main libraries in which they are preserved are those in the Arabic World: Cairo, Damascus, Tunis, Algiers, Rabat ... as well as in private collections. Beside this material in the Arabic countries, the Deutsche Staatsbibliothek in Berlin, the Biblioteca del Escorial near Madrid, the British Museum and the Bodleian Library in England, the Bibliothèque Nationale in Paris, the Süleymaniye and Topkapi Libraries in Istanbul, the National Libraries in Iran, India, Pakistan.. -
The Persian-Toledan Astronomical Connection and the European Renaissance
Academia Europaea 19th Annual Conference in cooperation with: Sociedad Estatal de Conmemoraciones Culturales, Ministerio de Cultura (Spain) “The Dialogue of Three Cultures and our European Heritage” (Toledo Crucible of the Culture and the Dawn of the Renaissance) 2 - 5 September 2007, Toledo, Spain Chair, Organizing Committee: Prof. Manuel G. Velarde The Persian-Toledan Astronomical Connection and the European Renaissance M. Heydari-Malayeri Paris Observatory Summary This paper aims at presenting a brief overview of astronomical exchanges between the Eastern and Western parts of the Islamic world from the 8th to 14th century. These cultural interactions were in fact vaster involving Persian, Indian, Greek, and Chinese traditions. I will particularly focus on some interesting relations between the Persian astronomical heritage and the Andalusian (Spanish) achievements in that period. After a brief introduction dealing mainly with a couple of terminological remarks, I will present a glimpse of the historical context in which Muslim science developed. In Section 3, the origins of Muslim astronomy will be briefly examined. Section 4 will be concerned with Khwârizmi, the Persian astronomer/mathematician who wrote the first major astronomical work in the Muslim world. His influence on later Andalusian astronomy will be looked into in Section 5. Andalusian astronomy flourished in the 11th century, as will be studied in Section 6. Among its major achievements were the Toledan Tables and the Alfonsine Tables, which will be presented in Section 7. The Tables had a major position in European astronomy until the advent of Copernicus in the 16th century. Since Ptolemy’s models were not satisfactory, Muslim astronomers tried to improve them, as we will see in Section 8.