History of Science and Technology Unit I
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Why Mathematical Proof?
Why Mathematical Proof? Dana S. Scott, FBA, FNAS University Professor Emeritus Carnegie Mellon University Visiting Scholar University of California, Berkeley NOTICE! The author has plagiarized text and graphics from innumerable publications and sites, and he has failed to record attributions! But, as this lecture is intended as an entertainment and is not intended for publication, he regards such copying, therefore, as “fair use”. Keep this quiet, and do please forgive him. A Timeline for Geometry Some Greek Geometers Thales of Miletus (ca. 624 – 548 BC). Pythagoras of Samos (ca. 580 – 500 BC). Plato (428 – 347 BC). Archytas (428 – 347 BC). Theaetetus (ca. 417 – 369 BC). Eudoxus of Cnidus (ca. 408 – 347 BC). Aristotle (384 – 322 BC). Euclid (ca. 325 – ca. 265 BC). Archimedes of Syracuse (ca. 287 – ca. 212 BC). Apollonius of Perga (ca. 262 – ca. 190 BC). Claudius Ptolemaeus (Ptolemy)(ca. 90 AD – ca. 168 AD). Diophantus of Alexandria (ca. 200 – 298 AD). Pappus of Alexandria (ca. 290 – ca. 350 AD). Proclus Lycaeus (412 – 485 AD). There is no Royal Road to Geometry Euclid of Alexandria ca. 325 — ca. 265 BC Euclid taught at Alexandria in the time of Ptolemy I Soter, who reigned over Egypt from 323 to 285 BC. He authored the most successful textbook ever produced — and put his sources into obscurity! Moreover, he made us struggle with proofs ever since. Why Has Euclidean Geometry Been So Successful? • Our naive feeling for space is Euclidean. • Its methods have been very useful. • Euclid also shows us a mysterious connection between (visual) intuition and proof. The Pythagorean Theorem Euclid's Elements: Proposition 47 of Book 1 The Pythagorean Theorem Generalized If it holds for one Three triple, Similar it holds Figures for all. -
Calendar of Roman Events
Introduction Steve Worboys and I began this calendar in 1980 or 1981 when we discovered that the exact dates of many events survive from Roman antiquity, the most famous being the ides of March murder of Caesar. Flipping through a few books on Roman history revealed a handful of dates, and we believed that to fill every day of the year would certainly be impossible. From 1981 until 1989 I kept the calendar, adding dates as I ran across them. In 1989 I typed the list into the computer and we began again to plunder books and journals for dates, this time recording sources. Since then I have worked and reworked the Calendar, revising old entries and adding many, many more. The Roman Calendar The calendar was reformed twice, once by Caesar in 46 BC and later by Augustus in 8 BC. Each of these reforms is described in A. K. Michels’ book The Calendar of the Roman Republic. In an ordinary pre-Julian year, the number of days in each month was as follows: 29 January 31 May 29 September 28 February 29 June 31 October 31 March 31 Quintilis (July) 29 November 29 April 29 Sextilis (August) 29 December. The Romans did not number the days of the months consecutively. They reckoned backwards from three fixed points: The kalends, the nones, and the ides. The kalends is the first day of the month. For months with 31 days the nones fall on the 7th and the ides the 15th. For other months the nones fall on the 5th and the ides on the 13th. -
Pappus of Alexandria: Book 4 of the Collection
Pappus of Alexandria: Book 4 of the Collection For other titles published in this series, go to http://www.springer.com/series/4142 Sources and Studies in the History of Mathematics and Physical Sciences Managing Editor J.Z. Buchwald Associate Editors J.L. Berggren and J. Lützen Advisory Board C. Fraser, T. Sauer, A. Shapiro Pappus of Alexandria: Book 4 of the Collection Edited With Translation and Commentary by Heike Sefrin-Weis Heike Sefrin-Weis Department of Philosophy University of South Carolina Columbia SC USA [email protected] Sources Managing Editor: Jed Z. Buchwald California Institute of Technology Division of the Humanities and Social Sciences MC 101–40 Pasadena, CA 91125 USA Associate Editors: J.L. Berggren Jesper Lützen Simon Fraser University University of Copenhagen Department of Mathematics Institute of Mathematics University Drive 8888 Universitetsparken 5 V5A 1S6 Burnaby, BC 2100 Koebenhaven Canada Denmark ISBN 978-1-84996-004-5 e-ISBN 978-1-84996-005-2 DOI 10.1007/978-1-84996-005-2 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2009942260 Mathematics Classification Number (2010) 00A05, 00A30, 03A05, 01A05, 01A20, 01A85, 03-03, 51-03 and 97-03 © Springer-Verlag London Limited 2010 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. -
The Worship of Augustus Caesar
J THE WORSHIP OF AUGUSTUS C^SAR DERIVED FROM A STUDY OF COINS, MONUMENTS, CALENDARS, ^RAS AND ASTRONOMICAL AND ASTROLOGICAL CYCLES, THE WHOLE ESTABLISHING A NEW CHRONOLOGY AND SURVEY OF HISTORY AND RELIGION BY ALEXANDER DEL MAR \ NEW YORK PUBLISHED BY THE CAMBRIDGE ENCYCLOPEDIA CO. 62 Reade Street 1900 (All rights reserrecf) \ \ \ COPYRIGHT BY ALEX. DEL MAR 1899. THE WORSHIP OF AUGUSTUS CAESAR. CHAPTERS. PAGE. Prologue, Preface, ........ Vll. Bibliography, ....... xi. I. —The Cycle of the Eclipses, I — II. The Ancient Year of Ten Months, . 6 III. —The Ludi S^eculares and Olympiads, 17 IV. —Astrology of the Divine Year, 39 V. —The Jovian Cycle and Worship, 43 VI. —Various Years of the Incarnation, 51 VII.—^RAS, 62 — VIII. Cycles, ...... 237 IX. —Chronological Problems and Solutions, 281 X. —Manetho's False Chronology, 287 — XI. Forgeries in Stone, .... 295 — XII. The Roman Messiah, .... 302 Index, ........ 335 Corrigenda, ....... 347 PROLOGUE. THE ABYSS OF MISERY AND DEPRAVITY FROM WHICH CHRISTIANITY REDEEMED THE ROMAN EMPIRE CAN NEVER BE FULLY UNDERSTOOD WITHOUT A KNOWLEDGE OF THE IMPIOUS WoA^P OF EM- PERORS TO WHICH EUROPE ONCE BOWED ITS CREDULOUS AND TERRIFIED HEAD. WHEN THIS OMITTED CHAPTER IS RESTORED TO THE HISTORY OF ROME, CHRISTIANITY WILL SPRING A LIFE FOR INTO NEW AND MORE VIGOROUS ; THEN ONLY WILL IT BE PERCEIVED HOW DEEP AND INERADICABLY ITS ROOTS ARE PLANTED, HOW LOFTY ARE ITS BRANCHES AND HOW DEATH- LESS ARE ITS AIMS. PREFACE. collection of data contained in this work was originally in- " THEtended as a guide to the author's studies of Monetary Sys- tems." It was therefore undertaken with the sole object of estab- lishing with precision the dates of ancient history. -
The Arabic Sources of Jordanus De Nemore
The Arabic Sources of Jordanus de Nemore IMPORTANT NOTICE: Author: Prof. Menso Folkerts and Prof. Richard Lorch All rights, including copyright, in the content of this document are owned or controlled for these purposes by FSTC Limited. In Chief Editor: Prof. Mohamed El-Gomati accessing these web pages, you agree that you may only download the content for your own personal non-commercial Deputy Editor: Prof. Mohammed Abattouy use. You are not permitted to copy, broadcast, download, store (in any medium), transmit, show or play in public, adapt or Associate Editor: Dr. Salim Ayduz change in any way the content of this document for any other purpose whatsoever without the prior written permission of FSTC Release Date: July, 2007 Limited. Publication ID: 710 Material may not be copied, reproduced, republished, downloaded, posted, broadcast or transmitted in any way except for your own personal non-commercial home use. Any other use Copyright: © FSTC Limited, 2007 requires the prior written permission of FSTC Limited. You agree not to adapt, alter or create a derivative work from any of the material contained in this document or use it for any other purpose other than for your personal non-commercial use. FSTC Limited has taken all reasonable care to ensure that pages published in this document and on the MuslimHeritage.com Web Site were accurate at the time of publication or last modification. Web sites are by nature experimental or constantly changing. Hence information published may be for test purposes only, may be out of date, or may be the personal opinion of the author. -
Astronomy Timeline
Astronomy Timeline Ancient Astronomy: 2800 B.C. to 1600 A.D. 2,800 B.C. – First phase of Stonehenge begins, it is used as a solar/lunar observatory 2,000 B.C. - Egypt and Mesopotamia build first solar/lunar calendars 280 B.C. - Greek astronomer, Aristarchus of Samos shows the Earth revolves around the Sun 240 B.C. - Greek mathematician Eratosthenes measures the circumference of the earth 130 B.C. - Greek astronomer Hipparchus develops the first accurate star map and star catalogue, and a reliable method to predict solar eclipses 46 B.C. - Julius Caesar, after consulting the astronomer Sosigenes of Alexandria, introduces the Julian Calendar, a regular year of 365 days divided into 12 months, and a leap day is added to February every four years 0 A.D. – Start of the Common Era 140 - Greek astronomer Ptolemy develops geocentric theory of the universe with Earth at the center 400 – 1200 – Pacific Islanders use constellations to travel across the Pacific Ocean 1050 – Mayan Pyramid of Kukulkan at Chichén Itzá is also used as calendar 1054 - Chinese astronomers observe supernova in Taurus 1120 - First astronomical observatory built in Cairo, Egypt 1259 - Persian astronomer Nasir al-Din al-Tusi builds Iran’s first astronomical observatory 1420 - Astronomer Ulugh Beg builds astronomical observatory built in Samarkand, Central Asia 1543 - Polish astronomer Nicolaus Copernicus publishes his heliocentric theory of the Universe 1572 - Danish astronomer Tycho Brahe discovers a supernova in constellation of Cassiopeia 1582 - Pope Gregory XIII introduces -
The Curious Case of the Milankovitch Calendar
Hist. Geo Space Sci., 10, 235–243, 2019 https://doi.org/10.5194/hgss-10-235-2019 © Author(s) 2019. This work is distributed under the Creative Commons Attribution 4.0 License. The curious case of the Milankovitch calendar Nenad Gajic Faculty of Technical Sciences, Trg Dositeja Obradovica´ 6, 21000 Novi Sad, Serbia Correspondence: Nenad Gajic ([email protected]) Received: 20 May 2019 – Revised: 11 August 2019 – Accepted: 23 August 2019 – Published: 26 September 2019 Abstract. The Gregorian calendar, despite being more precise than the Julian (which now lags 13 d behind Earth), will also lag a day behind nature in this millennium. In 1923, Milutin Milankovitch presented a calen- dar of outstanding scientific importance and unprecedented astronomical accuracy, which was accepted at the Ecumenical Congress of Eastern Orthodox churches. However, its adoption is still partial in churches and nonex- istent in civil states, despite nearly a century without a better proposition of calendar reform in terms of both precision and ease of transition, which are important for acceptance. This article reviews the development of calendars throughout history and presents the case of Milankovitch’s, explaining its aims and methodology and why it is sometimes mistakenly identified with the Gregorian because of their long consonance. Religious as- pects are briefly covered, explaining the potential of this calendar to unite secular and religious purposes through improving accuracy in both contexts. 1 Introduction global scientific project called “Climate: Long range Inves- tigation, Mapping, and Prediction” (CLIMAP, 1981), which aimed to reconstruct the worldwide climate history through Milutin Milankovic´ (1879–1958; see Fig. -
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. -
A Concise History of Mathematics the Beginnings 3
A CONCISE HISTORY OF A CONCISE HISTORY MATHEMATICS STRUIK OF MATHEMATICS DIRK J. STRUIK Professor Mathematics, BELL of Massachussetts Institute of Technology i Professor Struik has achieved the seemingly impossible task of compress- ing the history of mathematics into less than three hundred pages. By stressing the unfolding of a few main ideas and by minimizing references to other develop- ments, the author has been able to fol- low Egyptian, Babylonian, Chinese, Indian, Greek, Arabian, and Western mathematics from the earliest records to the beginning of the present century. He has based his account of nineteenth cen- tury advances on persons and schools rather than on subjects as the treatment by subjects has already been used in existing books. Important mathema- ticians whose work is analysed in detail are Euclid, Archimedes, Diophantos, Hammurabi, Bernoulli, Fermat, Euler, Newton, Leibniz, Laplace, Lagrange, Gauss, Jacobi, Riemann, Cremona, Betti, and others. Among the 47 illustra- tions arc portraits of many of these great figures. Each chapter is followed by a select bibliography. CHARLES WILSON A CONCISE HISTORY OF MATHEMATICS by DIRK. J. STRUIK Professor of Mathematics at the Massachusetts Institute of Technology LONDON G. BELL AND SONS LTD '954 Copyright by Dover Publications, Inc., U.S.A. FOR RUTH Printed in Great Britain by Butler & Tanner Ltd., Frame and London CONTENTS Introduction xi The Beginnings 1 The Ancient Orient 13 Greece 39 The Orient after the Decline of Greek Society 83 The Beginnings in Western Europe 98 The -
ARCHAEOLOGY the INLAND SITES Edited by Stefano Vassallo and Rosa Maria Cucco
TREASURE MAPS Twenty Itineraries Designed to Help You Explore the Cultural Heritage of Palermo and its Province Soprintendenza per i Beni culturali e ambientali di Palermo ARCHAEOLOGY THE INLAND SITES Edited by Stefano Vassallo and Rosa Maria Cucco REGIONE SICILIANA Assessorato dei Beni culturali e dell’Identità siciliana PO FESR Sicilia 2007-2013 Linea d’intervento 3.1.1.1. “Investiamo nel vostro futuro” Project TREASURE MAPS Twenty Itineraries Designed to Help You Explore the Cultural Heritage of Palermo and its Province project by: Ignazio Romeo R.U.P.: Claudia Oliva Soprintendente: Maria Elena Volpes Archaeology: The Inland Sites edited by: Stefano Vassallo and Rosa Maria Cucco texts by: Alba Maria Gabriella Calascibetta, Monica Chiovaro, Rosa Maria Cucco photographs: Soprintendenza per i Beni culturali e ambientali di Palermo editorial staff: Ignazio Romeo, Maria Concetta Picciurro, Riccardo Sapia photographic elaboration: Giancarlo Vinti graphics and printing: Ediguida Srl translations: Logoteum Language Services Treasure Maps: Twenty Itineraries Designed to Help You Explore the Cultural Heritage of Palermo and its Province. - Palermo: Regione siciliana, Assessorato dei beni culturali e dell’identità siciliana, Dipartimento dei beni culturali e dell’identità siciliana. – v. 709.45823 CDD-22 SBN Pal0274341 3. Archaeology: The Inland Sites / by Stefano Vassallo e Rosa Maria Cucco. - Palermo : Regione siciliana, Assessorato dei beni culturali e dell’identità siciliana, Dipartimento dei beni culturali e dell’identità siciliana, 2015. I. Vassallo, Stefano <1955>. II. Cucco, Rosa Maria <1966>. 937.845823 CDD-22 CIP - Biblioteca centrale della Regione siciliana “Alberto Bombace” © REGIONE SICILIANA Assessorato dei Beni culturali e dell’Identità siciliana Dipartimento dei Beni culturali e dell’Identità siciliana Soprintendenza per i Beni culturali e ambientali di Palermo Via Pasquale Calvi, 13 - 90139 Palermo Palazzo Ajutamicristo - Via Garibaldi, 41 - 90133 Palermo tel. -
The Empire Strikes: the Growth of Roman Infrastructural Minting Power, 60 B.C
The Empire Strikes: The Growth of Roman Infrastructural Minting Power, 60 B.C. – A.D. 68 A dissertation submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Classics of the College of Arts and Sciences by David Schwei M.A., University of Cincinnati, December 2012 B.A., Emory University, May 2009 Committee Chairs: Peter van Minnen, Ph.D Barbara Burrell, Ph.D. ABSTRACT Coins permeated the Roman Empire, and they offer a unique perspective into the ability of the Roman state to implement its decisions in Italy and the provinces. This dissertation examines how this ability changed and grew over time, between 60 B.C. and A.D. 68, as seen through coin production. Earlier scholars assumed that the mint at Rome always produced coinage for the entire empire, or they have focused on a sudden change under Augustus. Recent advances in catalogs, documentation of coin hoards, and metallurgical analyses allow a fuller picture to be painted. This dissertation integrates the previously overlooked coinages of Asia Minor, Syria, and Egypt with the denarius of the Latin West. In order to measure the development of the Roman state’s infrastructural power, this dissertation combines the anthropological ideal types of hegemonic and territorial empires with the numismatic method of detecting coordinated activity at multiple mints. The Roman state exercised its power over various regions to different extents, and it used its power differently over time. During the Republic, the Roman state had low infrastructural minting capacity. -
The Obelisk of Augustus – Part I Paolo Albèri-Auber (Trieste, Italy)
The Obelisk Of Augustus – Part I Paolo Albèri-Auber (Trieste, Italy) Preliminary - the Greek Science Scholars generally admit that Antique Greek Science grew out of Assyrian and Egyptian “science”. More correctly, it is generally known that the Assyrians and Egyptians developed some form of “science” but that their “science” was rather different from modern Science. Mainly, it dealt with observation, generally with a very simple mathematical speculation. Geometry, as we know it, was invented by Greek mathematicians: but not only ... Some geometrical-mathematical algorithms such as the “Analemma” and “Stereographical projection” were invented by Greek Scientists: using such mathematical instruments they investigated the deep rules of natural phaenomena (the motion of sun and stars). Aristarcus of Samos, Hipparcus, Ptolemy are only a few of these scientists. So Galileo Galilei many centuries after “...Philosophy (Science) has been written... in this great book of the Universe ...but you cannot read this book if you don’t first learn the language the book has been written with...It has been written in a mathematical language ...the letters are Triangles, Circles... if you don’t study this language it is impossible to understand it...”(Il Saggiatore, 1623) So Greek scientists radically investigated some concealed mathematical-geometrical rules of the “language” of Nature; in fact they did so for the first time in the history of mankind. We should then ask ourselves: did they therefore invent ...Theoretical Physics? It is important to accept these preliminary considerations if you want to comprehend why in the Mars Field in Rome a very important scientific instrument (a Meridian Line) was built in 9 BCE by Emperor Augustus with the seasonal descriptions (zodiacal signs and parapegmata1) written in greek letters2.