Patricia O'grady.Indd
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Some Curves and the Lengths of Their Arcs Amelia Carolina Sparavigna
Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna To cite this version: Amelia Carolina Sparavigna. Some Curves and the Lengths of their Arcs. 2021. hal-03236909 HAL Id: hal-03236909 https://hal.archives-ouvertes.fr/hal-03236909 Preprint submitted on 26 May 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna Department of Applied Science and Technology Politecnico di Torino Here we consider some problems from the Finkel's solution book, concerning the length of curves. The curves are Cissoid of Diocles, Conchoid of Nicomedes, Lemniscate of Bernoulli, Versiera of Agnesi, Limaçon, Quadratrix, Spiral of Archimedes, Reciprocal or Hyperbolic spiral, the Lituus, Logarithmic spiral, Curve of Pursuit, a curve on the cone and the Loxodrome. The Versiera will be discussed in detail and the link of its name to the Versine function. Torino, 2 May 2021, DOI: 10.5281/zenodo.4732881 Here we consider some of the problems propose in the Finkel's solution book, having the full title: A mathematical solution book containing systematic solutions of many of the most difficult problems, Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical Journals of the United States, and Many Original Problems and Solutions. -
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. -
Greek Mathematics Recovered in Books 6 and 7 of Clavius’ Geometria Practica
Introduction – Clavius and Geometria Practica Book 6 and Greek approaches to duplication of the cube Book 7 and squaring the circle via the quadratrix Conclusions Greek Mathematics Recovered in Books 6 and 7 of Clavius’ Geometria Practica John B. Little Department of Mathematics and CS College of the Holy Cross June 29, 2018 Greek Mathematics in Clavius Introduction – Clavius and Geometria Practica Book 6 and Greek approaches to duplication of the cube Book 7 and squaring the circle via the quadratrix Conclusions I’ve always been interested in the history of mathematics (in addition to my nominal specialty in algebraic geometry/computational methods/coding theory, etc.) Want to be able to engage with original texts on their own terms – you might recall the talks on Apollonius’s Conics I gave at the last Clavius Group meeting at Holy Cross (two years ago) So, I’ve been taking Greek and Latin language courses in HC’s Classics department The subject for today relates to a Latin-to-English translation project I have recently begun – working with the Geometria Practica of Christopher Clavius, S.J. (1538 - 1612, CE) Greek Mathematics in Clavius Introduction – Clavius and Geometria Practica Book 6 and Greek approaches to duplication of the cube Book 7 and squaring the circle via the quadratrix Conclusions Overview 1 Introduction – Clavius and Geometria Practica 2 Book 6 and Greek approaches to duplication of the cube 3 Book 7 and squaring the circle via the quadratrix 4 Conclusions Greek Mathematics in Clavius Introduction – Clavius and Geometria Practica Book 6 and Greek approaches to duplication of the cube Book 7 and squaring the circle via the quadratrix Conclusions Clavius’ Principal Mathematical Textbooks Euclidis Elementorum, Libri XV (first ed. -
A Short History of Greek Mathematics
Cambridge Library Co ll e C t i o n Books of enduring scholarly value Classics From the Renaissance to the nineteenth century, Latin and Greek were compulsory subjects in almost all European universities, and most early modern scholars published their research and conducted international correspondence in Latin. Latin had continued in use in Western Europe long after the fall of the Roman empire as the lingua franca of the educated classes and of law, diplomacy, religion and university teaching. The flight of Greek scholars to the West after the fall of Constantinople in 1453 gave impetus to the study of ancient Greek literature and the Greek New Testament. Eventually, just as nineteenth-century reforms of university curricula were beginning to erode this ascendancy, developments in textual criticism and linguistic analysis, and new ways of studying ancient societies, especially archaeology, led to renewed enthusiasm for the Classics. This collection offers works of criticism, interpretation and synthesis by the outstanding scholars of the nineteenth century. A Short History of Greek Mathematics James Gow’s Short History of Greek Mathematics (1884) provided the first full account of the subject available in English, and it today remains a clear and thorough guide to early arithmetic and geometry. Beginning with the origins of the numerical system and proceeding through the theorems of Pythagoras, Euclid, Archimedes and many others, the Short History offers in-depth analysis and useful translations of individual texts as well as a broad historical overview of the development of mathematics. Parts I and II concern Greek arithmetic, including the origin of alphabetic numerals and the nomenclature for operations; Part III constitutes a complete history of Greek geometry, from its earliest precursors in Egypt and Babylon through to the innovations of the Ionic, Sophistic, and Academic schools and their followers. -
Geometry in the Age of Enlightenment
Geometry in the Age of Enlightenment Raymond O. Wells, Jr. ∗ July 2, 2015 Contents 1 Introduction 1 2 Algebraic Geometry 3 2.1 Algebraic Curves of Degree Two: Descartes and Fermat . 5 2.2 Algebraic Curves of Degree Three: Newton and Euler . 11 3 Differential Geometry 13 3.1 Curvature of curves in the plane . 17 3.2 Curvature of curves in space . 26 3.3 Curvature of a surface in space: Euler in 1767 . 28 4 Conclusion 30 1 Introduction The Age of Enlightenment is a term that refers to a time of dramatic changes in western society in the arts, in science, in political thinking, and, in particular, in philosophical discourse. It is generally recognized as being the period from the mid 17th century to the latter part of the 18th century. It was a successor to the renaissance and reformation periods and was followed by what is termed the romanticism of the 19th century. In his book A History of Western Philosophy [25] Bertrand Russell (1872{1970) gives a very lucid description of this time arXiv:1507.00060v1 [math.HO] 30 Jun 2015 period in intellectual history, especially in Book III, Chapter VI{Chapter XVII. He singles out Ren´eDescartes as being the founder of the era of new philosophy in 1637 and continues to describe other philosophers who also made significant contributions to mathematics as well, such as Newton and Leibniz. This time of intellectual fervor included literature (e.g. Voltaire), music and the world of visual arts as well. One of the most significant developments was perhaps in the political world: here the absolutism of the church and of the monarchies ∗Jacobs University Bremen; University of Colorado at Boulder; [email protected] 1 were questioned by the political philosophers of this era, ushering in the Glo- rious Revolution in England (1689), the American Revolution (1776), and the bloody French Revolution (1789). -
Quadratrix of Hippias -- from Wolfram Mathworld
12/3/13 Quadratrix of Hippias -- from Wolfram MathWorld Search MathWorld Algebra Applied Mathematics Geometry > Curves > Plane Curves > Polar Curves > Geometry > Geometric Construction > Calculus and Analysis Interactive Entries > Interactive Demonstrations > Discrete Mathematics THINGS TO TRY: Quadratrix of Hippias quadratrix of hippias Foundations of Mathematics 12-w heel graph Geometry d^4/dt^4(Ai(t)) History and Terminology Number Theory Probability and Statistics Recreational Mathematics Hippias Quadratrix Bruno Autin Topology Alphabetical Index Interactive Entries Random Entry New in MathWorld MathWorld Classroom About MathWorld The quadratrix was discovered by Hippias of Elias in 430 BC, and later studied by Dinostratus in 350 BC (MacTutor Contribute to MathWorld Archive). It can be used for angle trisection or, more generally, division of an angle into any integral number of equal Send a Message to the Team parts, and circle squaring. It has polar equation MathWorld Book (1) Wolfram Web Resources » 13,191 entries with corresponding parametric equation Last updated: Wed Nov 6 2013 (2) Created, developed, and nurtured by Eric Weisstein at Wolfram Research (3) and Cartesian equation (4) Using the parametric representation, the curvature and tangential angle are given by (5) (6) for . SEE ALSO: Angle trisection, Cochleoid REFERENCES: Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 223, 1987. Law rence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 195 and 198, 1972. Loomis, E. S. "The Quadratrix." §2.1 in The Pythagorean Proposition: Its Demonstrations Analyzed and Classified and Bibliography of Sources for Data of the Four Kinds of "Proofs," 2nd ed. -
3.2 Archimedes' Quadrature of the Parabola
3.2 Archimedes’ Quadrature of the Parabola 111 mere points for other functions to be defined on, a metalevel analysis with applications in quantum physics. At the close of the twentieth century, one of the hottest new fields in analysis is “wavelet theory,” emerging from such applications as edge de- tection or texture analysis in computer vision, data compression in signal analysis or image processing, turbulence, layering of underground earth sed- iments, and computer-aided design. Wavelets are an extension of Fourier’s idea of representing functions by superimposing waves given by sines or cosines. Since many oscillatory phenomena evolve in an unpredictable way over short intervals of time or space, the phenomenon is often better repre- sented by superimposing waves of only short duration, christened wavelets. This tight interplay between current applications and a new field of math- ematics is evolving so quickly that it is hard to see where it will lead even in the very near future [92]. We will conclude this chapter with an extraordinary modern twist to our long story. Recall that the infinitesimals of Leibniz, which had never been properly defined and were denigrated as fictional, had finally been banished from analysis by the successors of Cauchy in the nineteenth century, using a rigorous foundation for the real numbers. How surprising, then, that in 1960 deep methods of modern mathematical logic revived infinitesimals and gave them a new stature and role. In our final section we will read a few passages from the book Non-Standard Analysis [140] by Abraham Robinson (1918– 1974), who discovered how to place infinitesimals on a firm foundation, and we will consider the possible consequences of his discovery for the future as well as for our evaluation of the past. -
The Project Gutenberg Ebook #31061: a History of Mathematics
The Project Gutenberg EBook of A History of Mathematics, by Florian Cajori This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: A History of Mathematics Author: Florian Cajori Release Date: January 24, 2010 [EBook #31061] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK A HISTORY OF MATHEMATICS *** Produced by Andrew D. Hwang, Peter Vachuska, Carl Hudkins and the Online Distributed Proofreading Team at http://www.pgdp.net transcriber's note Figures may have been moved with respect to the surrounding text. Minor typographical corrections and presentational changes have been made without comment. This PDF file is formatted for screen viewing, but may be easily formatted for printing. Please consult the preamble of the LATEX source file for instructions. A HISTORY OF MATHEMATICS A HISTORY OF MATHEMATICS BY FLORIAN CAJORI, Ph.D. Formerly Professor of Applied Mathematics in the Tulane University of Louisiana; now Professor of Physics in Colorado College \I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history."|J. W. L. Glaisher New York THE MACMILLAN COMPANY LONDON: MACMILLAN & CO., Ltd. 1909 All rights reserved Copyright, 1893, By MACMILLAN AND CO. Set up and electrotyped January, 1894. Reprinted March, 1895; October, 1897; November, 1901; January, 1906; July, 1909. Norwood Pre&: J. S. Cushing & Co.|Berwick & Smith. -
On the Value Equivalent to π in Ancient Mathematical Texts. A
On Value Equivalent to in Ancient Mathematical Texts. A New Interpretation A. J. E. M. SMEUR Communicated by J. E. HOFMANN 1. Studying the ancient history of mathematics, one sometimes comes across calculations of the circumference or the area of a circle or the area or the volume of a sphere or of some part of them. To us such calculations are approximations only. In works on history of mathematics those calculations can be found "trans- lated" into modern wording, which means that the now well known formulas 2~ R, ~ R 2, 4z R 2 and ~-0zR 3 are employed. As a result of such translation one meets with an assertion running something like this: "the calculation is equiv- alent to the assumption that z has the value ...". But when stated in this absolute way, such an assertion can give rise to misunderstanding, for the con- stant a, occurs in different formulas. Thus if we learn, for instance, that the circumference of a circle is found by taking three times the diameter, we might conclude that ~ has the value 3. However, we must understand that such an assertion refers to calculation of the circumference only, with not even a hint that the area of a circle should be calculated by taking three times the square on the radius. In what follows we first discuss an example of such a false interpretation, together with its consequences, from MORITZ CANTOI~'SVorlesungen ~ber Geschichte der Mathematik. We next propose a modified notation for some formulas in which the constant 7g is met with. -
Squaring the Circle
Squaring the Circle A Case Study in the History of Mathematics Part II π It is lost in the mists of pre-history who first realized that the ratio of the circumference of a circle to its diameter is a constant. All the ancient civilizations knew this fact. Today we call this ratio π and express this relationship by saying that for any circle, the circumference C and the diameter d satisfy: C = πd. The use of the symbol “π” for this ratio is of relatively recent origin; the Greeks did not use the symbol. “π was first used by the English mathematicians Oughtred (1647), Isaac Barrow (1664) and David Gregory (1697) to represent the circumference of a circle. The first use of “π” to represent the ratio of circumference to diameter was the English writer William Jones (1706). However, it did not come into common use until Euler adopted the symbol in 1737.1 Archimedes Euclid XII.2 says that the ratio of the area of any circle to the square of its diameter is also a constant, but does not determine the value of this constant. It was Archimedes (287 – 212 B.C.) who determined the constant in his remarkable treatise Measurement of a Circle. There are only three propositions in this short work (or at least, that is all of that work that has come down to us) and the second proposition is out of place – indicating that what we have is probably not the original version.2 We shall look at the first and third proposition. -
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 History of PI
Eastern Illinois University The Keep Plan B Papers Student Theses & Publications 1-1-1966 The History of PI John Allan Sanders Follow this and additional works at: https://thekeep.eiu.edu/plan_b Recommended Citation Sanders, John Allan, "The History of PI" (1966). Plan B Papers. 502. https://thekeep.eiu.edu/plan_b/502 This Dissertation/Thesis is brought to you for free and open access by the Student Theses & Publications at The Keep. It has been accepted for inclusion in Plan B Papers by an authorized administrator of The Keep. For more information, please contact [email protected]. THE HISTORY OF PI (TITLE) BY JOHN ALLAN SANDERS PLAN B PAPER SUBMITIED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE MASTER OF SCIENCE IN EDUCATION AND PREPARED IN COURSE MATH 570 IN THE GRADUATE SCHOOL, EASTERN ILLINOIS UNIVERSITY, CHARLESTON, ILLINOIS 1966 YEAR I HEREBY RECOMMEND THIS PLAN B PAPER BE ACCEPTED AS TABLE OF CONTENTS Page INTRODUCTION 1 Chapter I. THE EARLIEST KNOWN VALUES OF PI UP TO THE TIME 3 OF THE GREEKS II. CONTRIBUTION OF THE GREEKS 6 III. THE AGE OF CLASSICAL METHOD 12 IV. THE EVALUATION OF PI BY THE SERIES 15 V. EVALUATION OF PI BY PROBABILITY 22 VI. INTERESTING ANECDOTE 33 CONCLUSIONS 34 BIBLIOGRAPHY 35 INTRODUCTION One of the best known mathematical constants is rr . This constant is helpful in expressing many well- known mathematical relationships , and men throughout history have tried to find the correct value of TI . The correct value of rr is 3.1415926535897932384626433832795028841971693993751058 ···. This paper is about the history of TI and how man, through increased knowledge , has devised new and more intelligent ways to evaluate the true value of TI .