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The Discovery of the Series Formula for Π by Leibniz, Gregory and Nilakantha Author(S): Ranjan Roy Source: Mathematics Magazine, Vol
The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha Author(s): Ranjan Roy Source: Mathematics Magazine, Vol. 63, No. 5 (Dec., 1990), pp. 291-306 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2690896 Accessed: 27-02-2017 22:02 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to Mathematics Magazine This content downloaded from 195.251.161.31 on Mon, 27 Feb 2017 22:02:42 UTC All use subject to http://about.jstor.org/terms ARTICLES The Discovery of the Series Formula for 7r by Leibniz, Gregory and Nilakantha RANJAN ROY Beloit College Beloit, WI 53511 1. Introduction The formula for -r mentioned in the title of this article is 4 3 57 . (1) One simple and well-known moderm proof goes as follows: x I arctan x = | 1 +2 dt x3 +5 - +2n + 1 x t2n+2 + -w3 - +(-I)rl2n+1 +(-I)l?lf dt. The last integral tends to zero if Ix < 1, for 'o t+2dt < jt dt - iX2n+3 20 as n oo. -
Magnes: Der Magnetstein Und Der Magnetismus in Den Wissenschaften Der Frühen Neuzeit Mittellateinische Studien Und Texte
Magnes: Der Magnetstein und der Magnetismus in den Wissenschaften der Frühen Neuzeit Mittellateinische Studien und Texte Editor Thomas Haye (Zentrum für Mittelalter- und Frühneuzeitforschung, Universität Göttingen) Founding Editor Paul Gerhard Schmidt (†) (Albert-Ludwigs-Universität Freiburg) volume 53 The titles published in this series are listed at brill.com/mits Magnes Der Magnetstein und der Magnetismus in den Wissenschaften der Frühen Neuzeit von Christoph Sander LEIDEN | BOSTON Zugl.: Berlin, Technische Universität, Diss., 2019 Library of Congress Cataloging-in-Publication Data Names: Sander, Christoph, author. Title: Magnes : der Magnetstein und der Magnetismus in den Wissenschaften der Frühen Neuzeit / von Christoph Sander. Description: Leiden ; Boston : Brill, 2020. | Series: Mittellateinische studien und texte, 0076-9754 ; volume 53 | Includes bibliographical references and index. Identifiers: LCCN 2019053092 (print) | LCCN 2019053093 (ebook) | ISBN 9789004419261 (hardback) | ISBN 9789004419414 (ebook) Subjects: LCSH: Magnetism–History–16th century. | Magnetism–History–17th century. Classification: LCC QC751 .S26 2020 (print) | LCC QC751 (ebook) | DDC 538.409/031–dc23 LC record available at https://lccn.loc.gov/2019053092 LC ebook record available at https://lccn.loc.gov/2019053093 Typeface for the Latin, Greek, and Cyrillic scripts: “Brill”. See and download: brill.com/brill‑typeface. ISSN 0076-9754 ISBN 978-90-04-41926-1 (hardback) ISBN 978-90-04-41941-4 (e-book) Copyright 2020 by Christoph Sander. Published by Koninklijke -
Three Golf Real Estate Projects Nearing Joint Venture Phase
Vol. 22, No. 10 October 2014 In the News Three golf real estate projects A new tool in your box nearing joint venture phase The first economic index for Cuba is BY JOHANNES WERNER pany is listed on the Hong Kong Stock now available ......Page 3 Exchange. No details about the plans at tate company Grupo Empresarial Bellomonte have been released. Extrahotelero Palmares SA con- Original plans for the site by Coral Domingo on Cuba Scluded negotiations to establish a Capital Group Ltd. — a British company joint venture with a Chinese developer whose principals left the country after Church and opposition in Cuba are fac- for construction of a golf real estate proj- ing internal strife ..........Page 4 facing prison terms on allegations not re- ect in Bellomonte, just east of Havana, lated to Bellomonte — included a 1,100- and two more golf projects proposed unit golf community on a 628-acre site a Privates get a slice of the pie by Spanish companies are in “technical few miles from the beach, anchored by Foreign travel firms can now officially do preparation,” a tourism ministry official two 18-hole golf courses, plus a coun- business with private businesses...Page 6 said, according to news agency AIN. try club, spa, and 323,000 square feet of The Bellomonte SA joint venture with commercial space. On a nearby 20-acre Beijing Enterprise Group Real Estate Co. beach property, Coral was planning to Q&A: TV ads, hecho en Cuba is the second golf joint venture, following build a 160-room hotel and beach club. -
Bibliography of Map Projections
AVAILABILITY OF BOOKS AND MAPS OF THE U.S. GEOlOGICAL SURVEY Instructions on ordering publications of the U.S. Geological Survey, along with prices of the last offerings, are given in the cur rent-year issues of the monthly catalog "New Publications of the U.S. Geological Survey." Prices of available U.S. Geological Sur vey publications released prior to the current year are listed in the most recent annual "Price and Availability List" Publications that are listed in various U.S. Geological Survey catalogs (see back inside cover) but not listed in the most recent annual "Price and Availability List" are no longer available. Prices of reports released to the open files are given in the listing "U.S. Geological Survey Open-File Reports," updated month ly, which is for sale in microfiche from the U.S. Geological Survey, Books and Open-File Reports Section, Federal Center, Box 25425, Denver, CO 80225. Reports released through the NTIS may be obtained by writing to the National Technical Information Service, U.S. Department of Commerce, Springfield, VA 22161; please include NTIS report number with inquiry. Order U.S. Geological Survey publications by mail or over the counter from the offices given below. BY MAIL OVER THE COUNTER Books Books Professional Papers, Bulletins, Water-Supply Papers, Techniques of Water-Resources Investigations, Circulars, publications of general in Books of the U.S. Geological Survey are available over the terest (such as leaflets, pamphlets, booklets), single copies of Earthquakes counter at the following Geological Survey Public Inquiries Offices, all & Volcanoes, Preliminary Determination of Epicenters, and some mis of which are authorized agents of the Superintendent of Documents: cellaneous reports, including some of the foregoing series that have gone out of print at the Superintendent of Documents, are obtainable by mail from • WASHINGTON, D.C.--Main Interior Bldg., 2600 corridor, 18th and C Sts., NW. -
Gregory's Sixth Operation
Found Sci (2018) 23:133–144 https://doi.org/10.1007/s10699-016-9512-9 Gregory’s Sixth Operation 1 2 3,4 Tiziana Bascelli • Piotr Błaszczyk • Vladimir Kanovei • 5 5 6 Karin U. Katz • Mikhail G. Katz • Semen S. Kutateladze • 5 7 8 Tahl Nowik • David M. Schaps • David Sherry Published online: 20 December 2016 Ó Springer Science+Business Media Dordrecht 2016 Abstract In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathe- matics under scrutiny. Following Ian Hacking, we point out that in the history of mathe- matics the amount of contingency is larger than is usually thought. As a case study, we analyze the historians’ approach to interpreting James Gregory’s expression ultimate terms in his paper attempting to prove the irrationality of p. Here Gregory referred to the last or & Mikhail G. Katz [email protected] Tiziana Bascelli [email protected] Piotr Błaszczyk [email protected] Vladimir Kanovei [email protected] Karin U. Katz [email protected] Semen S. Kutateladze [email protected] Tahl Nowik [email protected] David M. Schaps [email protected] David Sherry [email protected] 1 Lyceum Gymnasium ‘‘F. Corradini’’, Thiene, Italy 2 Institute of Mathematics, Pedagogical University of Cracow, Cracow, Poland 3 IPPI, Moscow, Russia 4 MIIT, Moscow, Russia 123 134 T. Bascelli et al. ultimate terms of a series. More broadly, we analyze the following questions: which modern framework is more appropriate for interpreting the procedures at work in texts from the early history of infinitesimal analysis? As well as the related question: what is a logical theory that is close to something early modern mathematicians could have used when studying infinite series and quadrature problems? We argue that what has been routinely viewed from the viewpoint of classical analysis as an example of an ‘‘unrigor- ous’’ practice, in fact finds close procedural proxies in modern infinitesimal theories. -
Pretoria Centre Newsletter January 2005 Pages Final.Pub
Volume 1, Issue 1 Newsletter Date NEWSLETTER, JANUARY 2005 Date and time: Wednesday 26 January at 19h15 Chairperson: Johan Smit Beginner’s Corner: A Starry Sky Observing Course by Michael Poll What’s Up: by Wayne Mitchell ++++++++++ LEG BREAK - Library open +++++++++++++ Main Topic : “Bright stars - and some history” by Lorna Higgs The meeting will be followed by tea/coffee and biscuits as usual. The next social/practical evening will be held on Friday 21 January at the Pretoria Centre Observatory, which is also situated at CBC. Arrive anytime from 18h30 onwards. INSIDE THIS NEWSLETTER MEETING REPORT : NOVEMBER 24TH 2004..................... ………..……....2 OBSERVING EVENING REPORT: NOVEMBER 19TH 2004...………...…….2 LUNAR PARALLAX ..................................................................................... 3 CANIS MAJOR……………………………………………...…………………….4 ASTRONOMICAL WEBSITE ADDRESSES................................................. 6 BITS AND PIECES ....................................................................................... 7 THE THIRD PLANET FROM THE SUN ....................................................... 8 PRETORIA CENTRE COMMITTEE ............................................................. 8 PAGE 2 ASSA PRETORIA - OCTOBER 2004 NEWSLETTERPAGE 2 Meeting report : November 24th 2004 by A good turn out of more than 60 members in mid January the naked eye planets are all was present at a slightly different venue at visible in the morning sky, and from the CBC. Johann Swanepoel opened the horizon upwards along the ecliptic are in the proceedings with details about “Telescope same order as they are from the sun. Basics”. Johann described the configuration and light paths of refracting telescopes and The main topic was about the Cassini Huygens mission, given by Mike Haslam and various reflectors, and some properties of telescopes such as magnification and the Neville Young. They illustrated the trajectory of the craft from its launch, and showed how optical effects of the secondary mirror in a the Huygens probe was to be released for its reflector. -
A Companion to Astrology in the Renaissance Brill’S Companions to the Christian Tradition
A Companion to Astrology in the Renaissance Brill’s Companions to the Christian Tradition A series of handbooks and reference works on the intellectual and religious life of Europe, 500–1800 Editor-in-Chief Christopher M. Bellitto (Kean University) VOLUME 49 The titles published in this series are listed at brill.com/bcct A Companion to Astrology in the Renaissance Edited by Brendan Dooley LEIDEN • BOSTON 2014 Cover illustration: Luna and her children (detail). Tübingen, Univ.Bibl., Ms. M. d. 2, fol. 272r. Library of Congress Cataloging-in-Publication Data A companion to astrology in the Renaissance / edited by Brendan Dooley. pages cm. — (Brill’s companions to the Christian tradition, ISSN 1871-6377 ; VOLUME 49) Includes index. ISBN 978-90-04-18352-0 (hardback : alk. paper) — ISBN 978-90-04-26230-0 (e-book) 1. Astrology—History. 2. Renaissance. I. Dooley, Brendan Maurice, 1953- editor of compilation. BF1676.C66 2013 133.509—dc23 2013039553 This publication has been typeset in the multilingual “Brill” typeface. With over 5,100 characters covering Latin, IPA, Greek, and Cyrillic, this typeface is especially suitable for use in the humanities. For more information, please see www.brill.com/brill-typeface. ISSN 1871-6377 ISBN 978-90-04-18352-0 (hardback) ISBN 978-90-04-26230-0 (e-book) Copyright 2014 by Koninklijke Brill NV, Leiden, The Netherlands. Koninklijke Brill NV incorporates the imprints Brill, Brill Nijhoff, Global Oriental, Hotei Publishing and IDC Publishers. All rights reserved. No part of this publication may be reproduced, translated, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission from the publisher. -
11 · the Culmination of Greek Cartography in Ptolemy
11 · The Culmination of Greek Cartography in Ptolemy o. A. w. DILKE WITH ADDITIONAL MATERIAL SUPPLIED BY THE EDITORS By the time of Marinus of Tyre (fl. A.D. 100) and Clau about his work remain unanswered. Little is known dius Ptolemy (ca. A.D. 90-168), Greek and Roman in about Ptolemy the man, and neither his birthplace nor fluences in cartography had been fused to a considerable his dates have been positively established.4 Moreover, extent into one tradition. There is a case, accordingly, in relation to the cartographic component in his writings, for treating them as a history of one already unified we must remember that no manuscript earlier than the stream of thought and practice. Here, however, though twelfth century A.D. has come down to us, and there is we accept that such a unity exists, the discussion is fo no adequate modern translation and critical edition of cused on the cartographic contributions of Marinus and the Geography.5 Perhaps most serious of all for the stu Ptolemy, both writing in Greek within the institutions dent of mapping, however, is the whole debate about of Roman society. Both men owed much to Roman the true authorship and provenance of the general and sources of information and to the extension ofgeograph regional maps that accompany the several versions of ical knowledge under the growing empire: yet equally, the Byzantine manuscripts (pp. 268-74 below). AI- in the case of Ptolemy especially, they represent a cul mination as well as a final synthesis of the scientific tradition in Greek cartography that has been traced through a succession of writers in the previous three 1. -
Printing Natural Knowledge
Chapter 3: Printing Natural Knowledge “The purpose of writing may be to persuade the wise and intelligent; or else it may be to persuade everyone in town.” So wrote Giovanni Alfonso Borelli, pro- fessor at the university of Pisa and already the author of an important work on Euclid, to his ex-colleague Marcello Malpighi, just starting out on the faculty at the university of Bologna and about to publish his fundamental work revealing the microstructure of the lungs. Deciding what to publish, and for whom, was no easy matter, still some thirty years after the twin trials of Orazio Morandi and Galileo Galilei. “But if you choose to write to the common citizens of Bo- logna to persuade them about the calumnies made against truth,” Borelli con- tinued, “I conceive that in this case you may use every effort to deride any per- son who peddles a pile of idiocies just to offend virtuous and meritorious people.” 1 Of course, for Borelli, “virtuous and meritorious people” meant per- sons who followed empirical method for the study of nature, exemplified by Galileo; and “truth” meant the truth drawn from the use of that method. “Idio- cies” meant ideas conceived by those whom the Galileians conceived as dogma- tists and mystifiers. Still in the 1660s, whether Galileo’s disciples and their followers would triumph against the dogmatists and mystifiers was by no means clear. The story of their campaign to capture public attention, with all its fits and starts, is far less well known than the content of their ideas. It will occupy us here. -
Fermat's Dilemma: Why Did He Keep Mum on Infinitesimals? and The
FERMAT’S DILEMMA: WHY DID HE KEEP MUM ON INFINITESIMALS? AND THE EUROPEAN THEOLOGICAL CONTEXT JACQUES BAIR, MIKHAIL G. KATZ, AND DAVID SHERRY Abstract. The first half of the 17th century was a time of in- tellectual ferment when wars of natural philosophy were echoes of religious wars, as we illustrate by a case study of an apparently innocuous mathematical technique called adequality pioneered by the honorable judge Pierre de Fermat, its relation to indivisibles, as well as to other hocus-pocus. Andr´eWeil noted that simple appli- cations of adequality involving polynomials can be treated purely algebraically but more general problems like the cycloid curve can- not be so treated and involve additional tools–leading the mathe- matician Fermat potentially into troubled waters. Breger attacks Tannery for tampering with Fermat’s manuscript but it is Breger who tampers with Fermat’s procedure by moving all terms to the left-hand side so as to accord better with Breger’s own interpreta- tion emphasizing the double root idea. We provide modern proxies for Fermat’s procedures in terms of relations of infinite proximity as well as the standard part function. Keywords: adequality; atomism; cycloid; hylomorphism; indi- visibles; infinitesimal; jesuat; jesuit; Edict of Nantes; Council of Trent 13.2 Contents arXiv:1801.00427v1 [math.HO] 1 Jan 2018 1. Introduction 3 1.1. A re-evaluation 4 1.2. Procedures versus ontology 5 1.3. Adequality and the cycloid curve 5 1.4. Weil’s thesis 6 1.5. Reception by Huygens 8 1.6. Our thesis 9 1.7. Modern proxies 10 1.8. -
M3hw5.Tex Week 5. 8-13.2.18 from the ARABS to GALILEO Edward
m3hw5.tex Week 5. 8-13.2.18 FROM THE ARABS TO GALILEO Edward GIBBON, Decline and fall of the Roman Empire, 1776 Boyer Ch. 14; Dreyer Ch. X, Mediaeval cosmology Al-Kwarizmi's Algebra (concluded). Its strengths include logical exposition and proof (shared by the Greeks but not the Mesopotamian and Hindu in- fluences), and facility in passing between geometric (Greek) and numeric (Hindu) algebra. Its weaknesses include that only positive rots of equations were permitted, and that everything, even numerals, is written out in words. Astronomy The Caliph al-Mamun founded an observatory in Baghdad in 829, `where continuous observations were made and tables of planetary motions con- structed, while an important attempt was made to determine the size of the Earth' (Dreyer, 246). One of the leading astronomers here was al-Fargani (= Alfragamus), `whose Elements of Astronomy were translated into Latin in the 12th C., and contributed greatly to the revival of science in Europe.' Hindu astronomy was derived from the Greek through the conquest of Alexander the Great (B 12.12). Hindu astronomical texts, the Siddhanthas, reached Baghdad during the reign of Caliph al-Mansur `in 773) (Dreyer, 244), or `by 766' (B, 254). These were translated into Arabic as the Sindhind. Note. One of the main spurs behind Arab interest in astronomy derives from the Muslim religious year. Omar Khayyam (c. 1050-1123) (B 13.15). Kayyam wrote a text Algebra. Like al-Khwarizmi, he restricted himself to positive roots, but went beyond degree 2. He gave algebraic and geometric solutions for quadratics, and geometric solutions (involving intersecting con- ics) for cubics. -
De Sphaera of Johannes De Sacrobosco in the Early Modern
Matteo Valleriani Editor De sphaera of Johannes de Sacrobosco in the Early Modern Period The Authors of the Commentaries De sphaera of Johannes de Sacrobosco in the Early Modern Period Matteo Valleriani Editor De sphaera of Johannes de Sacrobosco in the Early Modern Period The Authors of the Commentaries Editor Matteo Valleriani Max Planck Institute for the History of Science Berlin, Germany Technische Universität Berlin Berlin, Germany University of Tel Aviv Tel Aviv, Israel ISBN 978-3-030-30832-2 ISBN 978-3-030-30833-9 (eBook) https://doi.org/10.1007/978-3-030-30833-9 © The Editor(s) (if applicable) and The Author(s) 2020. This book is an open access publication. Open Access This book is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The images or other third party material in this book are included in the book’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the book’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. The use of general descriptive names, registered names, trademarks, service marks, etc.