Sical Greek Geometry (The Other Two Problems Being the Suplication of the Cube and the Quadrature of the Circle)
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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. -
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. -
Patricia O'grady.Indd
HIPPIAS OF ELIS Hippias of Elis Patricia O’Grady Hippias of Elis cut an elegant fi gure as he strolled through the crowds at Olympia, dressed entirely in garments and accessories he had, him- self, made. But there is more to Hippias than the man Plato portrays as vain in Hippias Minor. Hippias was not only the exemplar of self-suffi ciency but ranks among the most talented and versatile of the sophists. He lectured on poetry, grammar, history, politics, and archaeology, he was a chronographer, and a prolifi c writer In this paper I will discuss Hippias’s work in geometry and the social contract he mentioned, and I will show that these two aspects of his work elevate him to another level. It will be seen that Hippias was more than an extremely rich and successful sophist, but that his work warrants his inclusion amongst the philosophers. Elegant and justifi ably proud, Hippias would have created a sensation as he strolled through the precincts of Olympia, dressed entirely in garments and accessories he had, himself, made. Th is is how Plato has Socrates speak of Hippias’s attire: You [Hippias] said that once, when you went to Olympia everything you had on your person was your own work; fi rst the ring—for you began with that—which you had was your own work, showing that you knew how to O'Grady, Patricia 2005. Hippias of Elis. In E. Close, M. Tsianikas and G. Frazis (eds.) "Greek Research in Australia: Proceedings of the Biennial International Conference of 17Greek Studies, Flinders University April 2003", Flinders University Department of Languages - Modern Greek: Adelaide, 17-38. -
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 . -
On the Quadratrix (Collection Book IV, Propositions 26-29)
HS 206r, Spring 2012, Connemara Doran Pappus (4th century AD) on the Quadratrix (Collection Book IV, Propositions 26-29) The quadratrix is a motion-generated curve used as a mathematical “tool” to extend what is constructible beyond what is possible with straight-edge and compass alone. As originally conceived (as early as 5th century BC), the quadratrix was the locus of points generated by the moving point of intersection of two line segments in synchronized motion – namely, a vertical side of a square pivoting about a lower vertex to trace a quadrant of a circle, and concurrently the top side of the square remaining horizontal while falling at a proportional rate, so that the two intersecting lines coincide in their arrival along the base of the square. If one accepts the legitimacy of the quadratrix as a constructive tool, it makes possible • trisecting the angle – constructing an angle one-third of a given angle (in fact, constructing any number of equal divisions of an angle) • doubling the cube – constructing the edge of a cube with volume double that of a given cube • squaring the circle – constructing the side of a square with area equal to that of a given circle The name quadratrix means “squarer” and was so named because its principle “symptom” is the generation – as the end product of the motions – of a unique point and hence “line segment” by means of which one can construct the square equal to the circle (of radius equal to the side of the original square). According to Heike Sefrin-Weis (250), Propositions 26-29 on the quadratrix -
ORIGAMI-CONSTRUCTIBLE NUMBERS by HWA
ORIGAMI-CONSTRUCTIBLE NUMBERS by HWA YOUNG LEE (Under the Direction of Daniel Krashen) Abstract In this thesis, I present an exposition of origami constructible objects and their associated algebraic fields. I first review the basic definitions and theorems of field theory that are relevant and discuss the more commonly known straightedge and compass constructions. Next, I introduce what origami is and discuss the basic single-fold operations of origami. Using the set of single-fold operations, I explain what it means for an object to be origami-constructible and show how to prove or disprove the constructibility of some origami objects. Finally, I present extensions of origami theory in literature and pose some additional questions for future studies. Index words: Field Theory, Compass and Straightedge Construction, Origami Construction, Constructible Numbers ORIGAMI-CONSTRUCTIBLE NUMBERS by HWA YOUNG LEE B.S., Ewha Womans University, 2004 M.Ed., Ewha Womans University, 2009 Ph.D., University of Georgia, 2017 A Thesis Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree MASTER OF ARTS Athens, Georgia 2017 c 2017 Hwa Young Lee All Rights Reserved ORIGAMI-CONSTRUCTIBLE NUMBERS by HWA YOUNG LEE Major Professor: Daniel Krashen Committee: Robert Rumely Pete Clark Electronic Version Approved: Suzanne Barbour Dean of the Graduate School The University of Georgia December 2017 Acknowledgments First, I would like to thank all the math teachers and students I had and have. Without these people, I would not be here. I would also like to thank my advisor and committee members who provided valuable feedback and support. -
ARCHIMEDES to PAPPUS Dramatis Personae Eudemus (Of Rhodes), fl
m3hw3.tex Week 3 25-30.1.18 THE GREEKS: ARCHIMEDES TO PAPPUS Dramatis Personae Eudemus (of Rhodes), fl. c. 320 BC: history of mathematics (now lost: W4) Archimedes (of Syracuse, c. 287-212 BC) Apollonius (of Perga, Asia Minor, c.262 { c.190 BC) Aristarchus (of Samos, fl. c. 280 BC) Eratosthenes (of Cyrene, 276-194 BC) Hipparchus (of Nicea, c. 180 { c. 125 BC) Menelaus (of Alexandria, c. 100 AD) Ptolemy (of Alexandria, fl. c. 127-150 AD) Heron (of Alexandria, 3rd C. AD) Diophantus (of Alexandria, fl. c. 250 AD) Pappus (of Alexandria, fl. c. 290 AD) Sources B Ch. 8-11, and other sources for Week 2; E. J. DIJKSTERHUIS, Archimedes, 2nd ed. (with an addendum by W. R. Knorr), PUP 1987 (1st ed. 1938-44, Dutch, Eng. tr. 1957) H. S. M. COXETER, Regular polytopes, 3rd ed., Dover, 1973 (1st ed. 1963). H. M. CUNDY & A. P. ROLLETT, Mathematical models, 2nd ed., OUP, 1961. Archimedes Archimedes lived in Syracuse, SE Sicily, then part of the Hellenistic world. He may have studied at Alexandria, and was in contact with mathematicians there. Syracuse sided with Carthage in the Punic Wars, and fell to Rome in 212 BC. Archimedes played a leading part in the defence of Syracuse during the long siege 214-212 BC, inventing engines of war: catapults to hurl stones, cranes of pull ships from the water and drop them, etc. (the siege is described in details by Plutarch, Livy and Polybius). The Roman general Marcellus admired Archimedes and ordered his life to be spared. -
Angle Trisection with Origami and Related Topics
ANGLE TRISECTION WITH ORIGAMI AND RELATED TOPICS CLEMENS FUCHS Abstract. It is well known that the trisection of an angle with compass and ruler is not possible in general. What is not so well known (even if it is folklore in the community of geometric constructions and mathemat- ical paper folding) is that angle trisection can be done with other tools, especially by an Origami construction. In this paper we briefly discuss this construction and give a geometric and an algebraic proof that the construction is correct. We also discuss which kind of problems Origami can solve and which other tools can be used to trisect an angle. 2000 Mathematics Subject Classification: 51M15, 11R04, 11R32. Keywords: Angle trisection, ruler and compass, mathematical paper folding, Origami geometry, Galois theory. 1. Introduction One of the famous old problems from antiquity is to trisect a given angle by using only a compass and a ruler. In other words: By using only a tool to draw a straight line segment through any two points and a tool to draw circles and arcs and duplicating lengths, one wants to construct in a finite number of steps two half-lines that comprise one third of an angle that itself is already given in terms of two half lines comprising it. The proof that this venture is indeed impossible in general is a prime example of how axiomatic mathematics works. We start with algebraization of this problem. Let a subset A2(R) of the affine plane and two different points 0, 1 be given. TheM⊆ fact that we have 0 and 1 is equivalent to the existence of∈M a coordinate system including the unity length by setting (0, 0) = 0, (1, 0) = 1.