Introduction to the History of Mathematics Fall 2020 - R
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Differential Calculus and by Era Integral Calculus, Which Are Related by in Early Cultures in Classical Antiquity the Fundamental Theorem of Calculus
History of calculus - Wikipedia, the free encyclopedia 1/1/10 5:02 PM History of calculus From Wikipedia, the free encyclopedia History of science This is a sub-article to Calculus and History of mathematics. History of Calculus is part of the history of mathematics focused on limits, functions, derivatives, integrals, and infinite series. The subject, known Background historically as infinitesimal calculus, Theories/sociology constitutes a major part of modern Historiography mathematics education. It has two major Pseudoscience branches, differential calculus and By era integral calculus, which are related by In early cultures in Classical Antiquity the fundamental theorem of calculus. In the Middle Ages Calculus is the study of change, in the In the Renaissance same way that geometry is the study of Scientific Revolution shape and algebra is the study of By topic operations and their application to Natural sciences solving equations. A course in calculus Astronomy is a gateway to other, more advanced Biology courses in mathematics devoted to the Botany study of functions and limits, broadly Chemistry Ecology called mathematical analysis. Calculus Geography has widespread applications in science, Geology economics, and engineering and can Paleontology solve many problems for which algebra Physics alone is insufficient. Mathematics Algebra Calculus Combinatorics Contents Geometry Logic Statistics 1 Development of calculus Trigonometry 1.1 Integral calculus Social sciences 1.2 Differential calculus Anthropology 1.3 Mathematical analysis -
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Positional Notation Or Trigonometry [2, 13]
The Greatest Mathematical Discovery? David H. Bailey∗ Jonathan M. Borweiny April 24, 2011 1 Introduction Question: What mathematical discovery more than 1500 years ago: • Is one of the greatest, if not the greatest, single discovery in the field of mathematics? • Involved three subtle ideas that eluded the greatest minds of antiquity, even geniuses such as Archimedes? • Was fiercely resisted in Europe for hundreds of years after its discovery? • Even today, in historical treatments of mathematics, is often dismissed with scant mention, or else is ascribed to the wrong source? Answer: Our modern system of positional decimal notation with zero, to- gether with the basic arithmetic computational schemes, which were discov- ered in India prior to 500 CE. ∗Bailey: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. Email: [email protected]. This work was supported by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC02-05CH11231. yCentre for Computer Assisted Research Mathematics and its Applications (CARMA), University of Newcastle, Callaghan, NSW 2308, Australia. Email: [email protected]. 1 2 Why? As the 19th century mathematician Pierre-Simon Laplace explained: It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very sim- plicity and the great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions; and we shall appre- ciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity. -
Great Physicists
Great Physicists Great Physicists The Life and Times of Leading Physicists from Galileo to Hawking William H. Cropper 1 2001 1 Oxford New York Athens Auckland Bangkok Bogota´ Buenos Aires Cape Town Chennai Dar es Salaam Delhi Florence HongKong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Paris Sao Paulo Shanghai Singapore Taipei Tokyo Toronto Warsaw and associated companies in Berlin Ibadan Copyright ᭧ 2001 by Oxford University Press, Inc. Published by Oxford University Press, Inc. 198 Madison Avenue, New York, New York 10016 Oxford is a registered trademark of Oxford University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Cropper, William H. Great Physicists: the life and times of leadingphysicists from Galileo to Hawking/ William H. Cropper. p. cm Includes bibliographical references and index. ISBN 0–19–513748–5 1. Physicists—Biography. I. Title. QC15 .C76 2001 530'.092'2—dc21 [B] 2001021611 987654321 Printed in the United States of America on acid-free paper Contents Preface ix Acknowledgments xi I. Mechanics Historical Synopsis 3 1. How the Heavens Go 5 Galileo Galilei 2. A Man Obsessed 18 Isaac Newton II. Thermodynamics Historical Synopsis 41 3. A Tale of Two Revolutions 43 Sadi Carnot 4. On the Dark Side 51 Robert Mayer 5. A Holy Undertaking59 James Joule 6. Unities and a Unifier 71 Hermann Helmholtz 7. The Scientist as Virtuoso 78 William Thomson 8. -
The "Greatest European Mathematician of the Middle Ages"
Who was Fibonacci? The "greatest European mathematician of the middle ages", his full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa (Italy), the city with the famous Leaning Tower, about 1175 AD. Pisa was an important commercial town in its day and had links with many Mediterranean ports. Leonardo's father, Guglielmo Bonacci, was a kind of customs officer in the North African town of Bugia now called Bougie where wax candles were exported to France. They are still called "bougies" in French, but the town is a ruin today says D E Smith (see below). So Leonardo grew up with a North African education under the Moors and later travelled extensively around the Mediterranean coast. He would have met with many merchants and learned of their systems of doing arithmetic. He soon realised the many advantages of the "Hindu-Arabic" system over all the others. D E Smith points out that another famous Italian - St Francis of Assisi (a nearby Italian town) - was also alive at the same time as Fibonacci: St Francis was born about 1182 (after Fibonacci's around 1175) and died in 1226 (before Fibonacci's death commonly assumed to be around 1250). By the way, don't confuse Leonardo of Pisa with Leonardo da Vinci! Vinci was just a few miles from Pisa on the way to Florence, but Leonardo da Vinci was born in Vinci in 1452, about 200 years after the death of Leonardo of Pisa (Fibonacci). His names Fibonacci Leonardo of Pisa is now known as Fibonacci [pronounced fib-on-arch-ee] short for filius Bonacci. -
EVARISTE GALOIS the Long Road to Galois CHAPTER 1 Babylon
A radical life EVARISTE GALOIS The long road to Galois CHAPTER 1 Babylon How many miles to Babylon? Three score miles and ten. Can I get there by candle-light? Yes, and back again. If your heels are nimble and light, You may get there by candle-light.[ Babylon was the capital of Babylonia, an ancient kingdom occupying the area of modern Iraq. Babylonian algebra B.M. Tablet 13901-front (From: The Babylonian Quadratic Equation, by A.E. Berryman, Math. Gazette, 40 (1956), 185-192) B.M. Tablet 13901-back Time Passes . Centuries and then millennia pass. In those years empires rose and fell. The Greeks invented mathematics as we know it, and in Alexandria produced the first scientific revolution. The Roman empire forgot almost all that the Greeks had done in math and science. Germanic tribes put an end to the Roman empire, Arabic tribes invaded Europe, and the Ottoman empire began forming in the east. Time passes . Wars, and wars, and wars. Christians against Arabs. Christians against Turks. Christians against Christians. The Roman empire crumbled, the Holy Roman Germanic empire appeared. Nations as we know them today started to form. And we approach the year 1500, and the Renaissance; but I want to mention two events preceding it. Al-Khwarismi (~790-850) Abu Ja'far Muhammad ibn Musa Al-Khwarizmi was born during the reign of the most famous of all Caliphs of the Arabic empire with capital in Baghdad: Harun al Rashid; the one mentioned in the 1001 Nights. He wrote a book that was to become very influential Hisab al-jabr w'al-muqabala in which he studies quadratic (and linear) equations. -
Formulation of the Group Concept Financial Mathematics and Economics, MA3343 Groups Artur Zduniak, ID: 14102797
Formulation of the Group Concept Financial Mathematics and Economics, MA3343 Groups Artur Zduniak, ID: 14102797 Fathers of Group Theory Abstract This study consists on the group concepts and the four axioms that form the structure of the group. Provided below are exam- ples to show different mathematical opera- tions within the elements and how this form Otto Holder¨ Joseph Louis Lagrange Niels Henrik Abel Evariste´ Galois Felix Klein Camille Jordan the different characteristics of the group. Furthermore, a detailed study of the formu- lation of the group theory is performed where three major areas of modern group theory are discussed:The Number theory, Permutation and algebraic equations the- ory and geometry. Carl Friedrich Gauss Leonhard Euler Ernst Christian Schering Paolo Ruffini Augustin-Louis Cauchy Arthur Cayley What are Group Formulation of the group concept A group is an algebraic structure consist- The group theory is considered as an abstraction of ideas common to three major areas: ing of different elements which still holds Number theory, Algebraic equations theory and permutations. In 1761, a Swiss math- some common features. It is also known ematician and physicist, Leonhard Euler, looked at the remainders of power of a number as a set. All the elements of a group are modulo “n”. His work is considered as an example of the breakdown of an abelian group equipped with an algebraic operation such into co-sets of a subgroup. He also worked in the formulation of the divisor of the order of the as addition or multiplication. When two el- group. In 1801, Carl Friedrich Gauss, a German mathematician took Euler’s work further ements of a group are combined under an and contributed in the formulating the theory of abelian groups. -
Curren T Anthropology
Forthcoming Current Anthropology Wenner-Gren Symposium Curren Supplementary Issues (in order of appearance) t Humanness and Potentiality: Revisiting the Anthropological Object in the Anthropolog Current Context of New Medical Technologies. Klaus Hoeyer and Karen-Sue Taussig, eds. Alternative Pathways to Complexity: Evolutionary Trajectories in the Anthropology Middle Paleolithic and Middle Stone Age. Steven L. Kuhn and Erella Hovers, eds. y THE WENNER-GREN SYMPOSIUM SERIES Previously Published Supplementary Issues December 2012 HUMAN BIOLOGY AND THE ORIGINS OF HOMO Working Memory: Beyond Language and Symbolism. omas Wynn and Frederick L. Coolidge, eds. GUEST EDITORS: SUSAN ANTÓN AND LESLIE C. AIELLO Engaged Anthropology: Diversity and Dilemmas. Setha M. Low and Sally Early Homo: Who, When, and Where Engle Merry, eds. Environmental and Behavioral Evidence V Dental Evidence for the Reconstruction of Diet in African Early Homo olum Corporate Lives: New Perspectives on the Social Life of the Corporate Form. Body Size, Body Shape, and the Circumscription of the Genus Homo Damani Partridge, Marina Welker, and Rebecca Hardin, eds. Ecological Energetics in Early Homo e 5 Effects of Mortality, Subsistence, and Ecology on Human Adult Height 3 e Origins of Agriculture: New Data, New Ideas. T. Douglas Price and Plasticity in Human Life History Strategy Ofer Bar-Yosef, eds. Conditions for Evolution of Small Adult Body Size in Southern Africa Supplement Growth, Development, and Life History throughout the Evolution of Homo e Biological Anthropology of Living Human Populations: World Body Size, Size Variation, and Sexual Size Dimorphism in Early Homo Histories, National Styles, and International Networks. Susan Lindee and Ricardo Ventura Santos, eds. -
On Popularization of Scientific Education in Italy Between 12Th and 16Th Century
PROBLEMS OF EDUCATION IN THE 21st CENTURY Volume 57, 2013 90 ON POPULARIZATION OF SCIENTIFIC EDUCATION IN ITALY BETWEEN 12TH AND 16TH CENTURY Raffaele Pisano University of Lille1, France E–mail: [email protected] Paolo Bussotti University of West Bohemia, Czech Republic E–mail: [email protected] Abstract Mathematics education is also a social phenomenon because it is influenced both by the needs of the labour market and by the basic knowledge of mathematics necessary for every person to be able to face some operations indispensable in the social and economic daily life. Therefore the way in which mathe- matics education is framed changes according to modifications of the social environment and know–how. For example, until the end of the 20th century, in the Italian faculties of engineering the teaching of math- ematical analysis was profound: there were two complex examinations in which the theory was as impor- tant as the ability in solving exercises. Now the situation is different. In some universities there is only a proof of mathematical analysis; in others there are two proves, but they are sixth–month and not annual proves. The theoretical requirements have been drastically reduced and the exercises themselves are often far easier than those proposed in the recent past. With some modifications, the situation is similar for the teaching of other modern mathematical disciplines: many operations needing of calculations and math- ematical reasoning are developed by the computers or other intelligent machines and hence an engineer needs less theoretical mathematics than in the past. The problem has historical roots. In this research an analysis of the phenomenon of “scientific education” (teaching geometry, arithmetic, mathematics only) with respect the methods used from the late Middle Ages by “maestri d’abaco” to the Renaissance hu- manists, and with respect to mathematics education nowadays is discussed. -
The History of the Abel Prize and the Honorary Abel Prize the History of the Abel Prize
The History of the Abel Prize and the Honorary Abel Prize The History of the Abel Prize Arild Stubhaug On the bicentennial of Niels Henrik Abel’s birth in 2002, the Norwegian Govern- ment decided to establish a memorial fund of NOK 200 million. The chief purpose of the fund was to lay the financial groundwork for an annual international prize of NOK 6 million to one or more mathematicians for outstanding scientific work. The prize was awarded for the first time in 2003. That is the history in brief of the Abel Prize as we know it today. Behind this government decision to commemorate and honor the country’s great mathematician, however, lies a more than hundred year old wish and a short and intense period of activity. Volumes of Abel’s collected works were published in 1839 and 1881. The first was edited by Bernt Michael Holmboe (Abel’s teacher), the second by Sophus Lie and Ludvig Sylow. Both editions were paid for with public funds and published to honor the famous scientist. The first time that there was a discussion in a broader context about honoring Niels Henrik Abel’s memory, was at the meeting of Scan- dinavian natural scientists in Norway’s capital in 1886. These meetings of natural scientists, which were held alternately in each of the Scandinavian capitals (with the exception of the very first meeting in 1839, which took place in Gothenburg, Swe- den), were the most important fora for Scandinavian natural scientists. The meeting in 1886 in Oslo (called Christiania at the time) was the 13th in the series. -
All About Abel Dayanara Serges Niels Henrik Abel (August 5, 1802-April 6,1829) Was a Norwegian Mathematician Known for Numerous Contributions to Mathematics
All About Abel Dayanara Serges Niels Henrik Abel (August 5, 1802-April 6,1829) was a Norwegian mathematician known for numerous contributions to mathematics. Abel was born in Finnoy, Norway, as the second child of a dirt poor family of eight. Abels father was a poor Lutheran minister who moved his family to the parish of Gjerstad, near the town of Risr in southeast Norway, soon after Niels Henrik was born. In 1815 Niels entered the cathedral school in Oslo, where his mathematical talent was recognized in 1817 by his math teacher, Bernt Michael Holmboe, who introduced him to the classics in mathematical literature and proposed original problems for him to solve. Abel studied the mathematical works of the 17th-century Englishman Sir Isaac Newton, the 18th-century German Leonhard Euler, and his contemporaries the Frenchman Joseph-Louis Lagrange and the German Carl Friedrich Gauss in preparation for his own research. At the age of 16, Abel gave a proof of the Binomial Theorem valid for all numbers not only Rationals, extending Euler's result. Abels father died in 1820, leaving the family in straitened circumstances, but Holmboe and other professors contributed and raised funds that enabled Abel to enter the University of Christiania (Oslo) in 1821. Abels first papers, published in 1823, were on functional equations and integrals; he was the first person to formulate and solve an integral equation. He had also created a proof of the impossibility of solving algebraically the general equation of the fifth degree at the age of 19, which he hoped would bring him recognition. -
A Brief History of Microwave Engineering
A BRIEF HISTORY OF MICROWAVE ENGINEERING S.N. SINHA PROFESSOR DEPT. OF ELECTRONICS & COMPUTER ENGINEERING IIT ROORKEE Multiple Name Symbol Multiple Name Symbol 100 hertz Hz 101 decahertz daHz 10–1 decihertz dHz 102 hectohertz hHz 10–2 centihertz cHz 103 kilohertz kHz 10–3 millihertz mHz 106 megahertz MHz 10–6 microhertz µHz 109 gigahertz GHz 10–9 nanohertz nHz 1012 terahertz THz 10–12 picohertz pHz 1015 petahertz PHz 10–15 femtohertz fHz 1018 exahertz EHz 10–18 attohertz aHz 1021 zettahertz ZHz 10–21 zeptohertz zHz 1024 yottahertz YHz 10–24 yoctohertz yHz • John Napier, born in 1550 • Developed the theory of John Napier logarithms, in order to eliminate the frustration of hand calculations of division, multiplication, squares, etc. • We use logarithms every day in microwaves when we refer to the decibel • The Neper, a unitless quantity for dealing with ratios, is named after John Napier Laurent Cassegrain • Not much is known about Laurent Cassegrain, a Catholic Priest in Chartre, France, who in 1672 reportedly submitted a manuscript on a new type of reflecting telescope that bears his name. • The Cassegrain antenna is an an adaptation of the telescope • Hans Christian Oersted, one of the leading scientists of the Hans Christian Oersted nineteenth century, played a crucial role in understanding electromagnetism • He showed that electricity and magnetism were related phenomena, a finding that laid the foundation for the theory of electromagnetism and for the research that later created such technologies as radio, television and fiber optics • The unit of magnetic field strength was named the Oersted in his honor. -
John Napier's Life Napier's Logarithms KYLIE BRYANT & PAUL SCOTT
KYLIE BRYANT & PAUL SCOTT John Napier’s life Napier’s logarithms John Napier was born in 1550 in the Tower of Napier’s idea in inventing the logarithm was to Merchiston, near Edinburgh, Scotland. His save mathematicians from having to do large father, Archibald Napier, was knighted in 1565 calculations. These days, calculations and was appointed Master of the Mint in 1582. involving large numbers are done using calcu- His mother, Janet Bothwell, was the sister of lators or computers; however before these were the Bishop of Orkney. Napier’s was an impor- invented, large calculations meant that much tant family in Scotland, and had owned the time was taken and mistakes were made. Merchiston estate since 1430. Napier’s logarithm was not defined in terms Napier was educated at St Salvator’s of exponents as our logarithm is today. College, graduating in 1563. He was just 13 Instead Napier thought of his logarithm in years old when his mother died and he was terms of moving particles, distances and veloc- sent to St Andrew’s University where he ities. He considered two lines, AZ of fixed studied for two years. This is where he gained length and A'Z' of infinite length and points X his interest in theology. He did not graduate, and X' starting at A and A' moving to the right however, instead leaving to travel around with the same initial velocity. X' has constant Europe for the next five years. During this velocity and X has a velocity proportional to time he gained knowledge of mathematics and the distance from X to Z.