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Squaring the Circle a Case Study in the History of Mathematics the Problem
Squaring the Circle A Case Study in the History of Mathematics The Problem Using only a compass and straightedge, construct for any given circle, a square with the same area as the circle. The general problem of constructing a square with the same area as a given figure is known as the Quadrature of that figure. So, we seek a quadrature of the circle. The Answer It has been known since 1822 that the quadrature of a circle with straightedge and compass is impossible. Notes: First of all we are not saying that a square of equal area does not exist. If the circle has area A, then a square with side √A clearly has the same area. Secondly, we are not saying that a quadrature of a circle is impossible, since it is possible, but not under the restriction of using only a straightedge and compass. Precursors It has been written, in many places, that the quadrature problem appears in one of the earliest extant mathematical sources, the Rhind Papyrus (~ 1650 B.C.). This is not really an accurate statement. If one means by the “quadrature of the circle” simply a quadrature by any means, then one is just asking for the determination of the area of a circle. This problem does appear in the Rhind Papyrus, but I consider it as just a precursor to the construction problem we are examining. The Rhind Papyrus The papyrus was found in Thebes (Luxor) in the ruins of a small building near the Ramesseum.1 It was purchased in 1858 in Egypt by the Scottish Egyptologist A. -
Squaring the Circle in Elliptic Geometry
Rose-Hulman Undergraduate Mathematics Journal Volume 18 Issue 2 Article 1 Squaring the Circle in Elliptic Geometry Noah Davis Aquinas College Kyle Jansens Aquinas College, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Davis, Noah and Jansens, Kyle (2017) "Squaring the Circle in Elliptic Geometry," Rose-Hulman Undergraduate Mathematics Journal: Vol. 18 : Iss. 2 , Article 1. Available at: https://scholar.rose-hulman.edu/rhumj/vol18/iss2/1 Rose- Hulman Undergraduate Mathematics Journal squaring the circle in elliptic geometry Noah Davis a Kyle Jansensb Volume 18, No. 2, Fall 2017 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 a [email protected] Aquinas College b scholar.rose-hulman.edu/rhumj Aquinas College Rose-Hulman Undergraduate Mathematics Journal Volume 18, No. 2, Fall 2017 squaring the circle in elliptic geometry Noah Davis Kyle Jansens Abstract. Constructing a regular quadrilateral (square) and circle of equal area was proved impossible in Euclidean geometry in 1882. Hyperbolic geometry, however, allows this construction. In this article, we complete the story, providing and proving a construction for squaring the circle in elliptic geometry. We also find the same additional requirements as the hyperbolic case: only certain angle sizes work for the squares and only certain radius sizes work for the circles; and the square and circle constructions do not rely on each other. Acknowledgements: We thank the Mohler-Thompson Program for supporting our work in summer 2014. Page 2 RHIT Undergrad. Math. J., Vol. 18, No. 2 1 Introduction In the Rose-Hulman Undergraduate Math Journal, 15 1 2014, Noah Davis demonstrated the construction of a hyperbolic circle and hyperbolic square in the Poincar´edisk [1]. -
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. -
Ruler and Compass Constructions and Abstract Algebra
Ruler and Compass Constructions and Abstract Algebra Introduction Around 300 BC, Euclid wrote a series of 13 books on geometry and number theory. These books are collectively called the Elements and are some of the most famous books ever written about any subject. In the Elements, Euclid described several “ruler and compass” constructions. By ruler, we mean a straightedge with no marks at all (so it does not look like the rulers with centimeters or inches that you get at the store). The ruler allows you to draw the (unique) line between two (distinct) given points. The compass allows you to draw a circle with a given point as its center and with radius equal to the distance between two given points. But there are three famous constructions that the Greeks could not perform using ruler and compass: • Doubling the cube: constructing a cube having twice the volume of a given cube. • Trisecting the angle: constructing an angle 1/3 the measure of a given angle. • Squaring the circle: constructing a square with area equal to that of a given circle. The Greeks were able to construct several regular polygons, but another famous problem was also beyond their reach: • Determine which regular polygons are constructible with ruler and compass. These famous problems were open (unsolved) for 2000 years! Thanks to the modern tools of abstract algebra, we now know the solutions: • It is impossible to double the cube, trisect the angle, or square the circle using only ruler (straightedge) and compass. • We also know precisely which regular polygons can be constructed and which ones cannot. -
Appendix I: on the Early History of Pi
Appendix I: On the Early History of Pi Our earliest source on 7r is a problem from the Rhind mathematical papyrus, one of the few major sources we have for the mathematics of ancient Egypt. Although the material on the papyrus in its present form comes from about 1550 B.C. during the Middle Kingdom, scholars believe that the mathematics of the document originated in the Old Kingdom, which would date it perhaps to 1900 B.C. The Egyptian source does not state an explicit value for 7r, but tells, instead, how to compute the area of a circle as the square of eight-ninths of the diameter. (One is reminded of the 'classical' problem of squaring the circle.) Quite different is the passage in I Kings 7, 23 (repeated in II Chronicles 4, 21), which clearly implies that one should multiply the diameter of a circle by 3 to find its circumference. It is worth noting, however, that the Egyptian and the Biblical texts talk about what are, a priori, different concepts: the diameter being regarded as known in both cases, the first text gives a factor used to produce the area of the circle and the other gives (by implication) a factor used to produce the circumference.1 Also from the early second millenium are Babylonian texts from Susa. In the mathematical cuneiform texts the standard factor for multiplying the diameter to produce the circumference is 3, but, according to some interpre tations, the Susa texts give a value of 31.2 Another source of ancient values of 7r is the class of Indian works known as the Sulba Sfitras, of which the four most important, and oldest are Baudhayana, .Apastamba, Katyayana and Manava (resp. -
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Arxiv:1908.01202V1 [Math.GM] 3 Aug 2019 113 of Π Correct to 6 Decimal Places
SQUARE THE CIRCLE IN ONE MINUTE HÙNG VIÊ. T CHU ABSTRACT. Since squaring the circle was proved to be impossible, mathematicians have picked up a new hobby to square the circle ’approximately’. The most famous work is due to Ramanujan who gave a construction correct to 8 decimal places. In his book Mathographics, Dixon gave constructions correct to 3 decimal places and stated that his method was quite complicated. In this paper, we improve Dixon’s method then give a new construction correct to 9 decimal places, a new record so far. 1. HISTORY Squaring the circle is a problem proposed by ancient geometers who asked whether it was possible to construct a square which has the same area with a given circle using only rulers and compasses. The method is called ruler-and-compass construction or classical construction. Nowadays, we know that this task is impossible, but the problem of squaring the circle had such a long history that many circle-squarers attempted and failed. Because it is possible to approximate the area of a circle with inscribing regular polygons to any degree of precision, and polygons can also be squared, it is reasonable to expect that a circle can be squared. While many Greek mathematicians and philoso- phers including Anaxagoras, Hippocrates of Chios, and Oenopides attempted a solution, mathematicians after the 16th century tried to prove its impossibility. However, it was not until the 19th century that considerable progress was made. In 1837, Pierre Wantzel [5] proved that lengths that can be constructed must be the zero of a polynomial. -
Delian Problem (Duplication/Doubling of the Cube)
HPS 322 Michael White History of Science Fall semester, 2013 The Delian Problem (Duplication/Doubling of the Cube) There are lots of (conflicting) stories about the so-called Delian Problem. But the gist of these stories are that the citizens of the Greek island of Delos were afflicted by the plague (alternatively, by internal political unrest) and asked the oracle of the god Apollo what they could do to alleviate the situation. The oracle said that they should construct an altar to Apollo in their temple that was twice the size of the altar, in the shape of a regular cube, that they already had. They tried doubling the side of the cube, but that didn’t work (the plague, or political conflict, continued unabated). So the Delians eventually decided (with the advice of Plato?) that they should construct a cubical altar with twice the volume of the old one. (Plutarch later [1st- 2nd centuries A.D.] maintains that Plato interpreted the oracle’s response in a metaphorical or ‘demythologized’ way–as a general injunction to the Delians to spend their time studying geometry rather than fighting with one another.) Interpreted literally, then the Delian Problem is the following: given a cube of side a, to find the linear distance x such that a cube of side x is twice the volume of a cube of side a. In algebraic notation (which the Greeks did not have): Given a, to find the length x such that x33 = 2a . According to ancient reports, the mathematician Hippocrates of Chios had, in the 5th century B.C., ‘reduced’ the Delian Problem of the Duplication of the Cube to the following problem: given two line segments of length a and b, to find the line segments of length x and y in continuing geometric proportion between a and b. -
Squaring the Circle: Marriage of Heaven and Earth
Rachel Fletcher Geometer’s Angle 113 Division St. Squaring the Circle: Great Barrington, MA 01230 USA Marriage of Heaven and Earth [email protected] Abstract. It is impossible to construct circles and squares of equal Keywords: Squaring the circle, areas or perimeters precisely, for circles are measured by the descriptive geometry, Leonardo incommensurable value pi (S) and squares by rational whole da Vinci, incommensurate numbers. But from early times, geometers have attempted to values reconcile these two orders of geometry. “Squaring the circle” can represent the union of opposing eternal and finite qualities, symbolizing the fusion of matter and spirit and the marriage of heaven and earth. In this column, we consider various methods for squaring the circle and related geometric constructions. I Introduction From the domed Pantheon of ancient Rome, if not before, architects have fashioned sacred dwellings after conceptions of the universe, utilizing circle and square geometries to depict spirit and matter united. Circular domes evoke the spherical cosmos and the descent of heavenly spirit to the material plane. Squares and cubes delineate the spatial directions of our physical world and portray the lifting up of material perfection to the divine. Constructing these basic figures is elementary. The circle results when a cord is made to revolve around a post. The right angle of a square appears in a 3:4:5 triangle, easily made from a string of twelve equally spaced knots.1 But "squaring the circle”—drawing circles and squares of equal areas or perimeters by means of a compass or rule—has eluded geometers from early times.2 The problem cannot be solved with absolute precision, for circles are measured by the incommensurable value pi (S= 3.1415927…), which cannot be accurately expressed in finite whole numbers by which we measure squares.3 At the symbolic level, however, the quest to obtain circles and squares of equal measure is equivalent to seeking the union of transcendent and finite qualities, or the marriage of heaven and earth. -
Squaring the Circle a Case Study in the History of Mathematics the Problem
Squaring the Circle A Case Study in the History of Mathematics The Problem Using only a compass and straightedge, construct for any given circle, a square with the same area as the circle. The general problem of constructing a square with the same area as a given figure is known as the Quadrature of that figure. So, we seek a quadrature of the circle. The Answer It has been known since 1822 that the quadrature of a circle with straightedge and compass is impossible. Notes: First of all we are not saying that a square of equal area does not exist. If the circle has area A, then a square with side √A clearly has the same area. Secondly, we are not saying that a quadrature of a circle is impossible, since it is possible, but not under the restriction of using only a straightedge and compass. These Notes In this set of notes I will leisurely trace the history of this problem. By “leisurely” I mean that we will take many detours and examine side issues that are of interest (to me at least), both mathematical and historical, as well as sociological and philosophical. Precursors It has been written, in many places, that the quadrature problem appears in one of the earliest extant mathematical sources, the Rhind Papyrus (~ 1650 B.C.). This is not really an accurate statement. If one means by the “quadrature of the circle” simply a quadrature by any means, then one is just asking for the determination of the area of a circle. This problem does appear in the Rhind Papyrus, but I consider it as just a precursor to the construction problem we are examining. -
Compass and Straightedge Applications of Field Theory
COMPASS AND STRAIGHTEDGE APPLICATIONS OF FIELD THEORY SPENCER CHAN Abstract. This paper explores several of the many possible applications of abstract algebra in analysing and solving problems in other branches of math- ematics, particularly applying field theory to solve classical problems in geom- etry. Seemingly simple problems, such as the impossibility of angle trisection or the impossibility of constructing certain regular polygons with a compass and straightedge, can be explained by results following from the notion of field extensions of the rational numbers. Contents 1. Introduction 1 2. Fields 2 3. Compass and Straightedge Constructions 4 4. Afterword 8 Acknowledgments 8 References 8 1. Introduction Trisecting the angle, doubling the cube, and squaring the circle are three famous geometric construction problems that originated from the mathematics of Greek antiquity. The problems were considered impossible to solve using only a compass and straightedge, and it was not until the 19th century that these problems were indeed proven impossible. Elementary geometric arguments alone could not explain the impossibility of these problems. The development of abstract algebra allowed for the rigorous proofs of these problems to be discovered. It is important to note that these geometric objects can be approximated through compass and straightedge constructions. Furthermore, they are completely constructible if one is allowed to use tools beyond those of the original problem. For example, using a marked ruler or even more creative processes like origami (i.e. paper folding) make solving these classical problems possible. For the purposes of this paper, we will only consider the problems as they were originally conceived in antiquity. -
The Beginnings of Greek Mathematics Greeks Were Not Confined to A
Chapter 3: The Beginnings of Greek Mathematics Greeks were not confined to a neighborhood or the Aegean and the Peloponnesis. They ranged far and wide including present day Italy and Sicily and many lived in North Africa including Egypt. This is of course before the expansion of the Roman Empire (Punic Wars, 3rd Century B.C. and the Second Punic War was 218-201 B.C.) The Greeks introduced abstraction and gave logical arguments. For example Greeks were interested in triangles and not simply triangular fields. Thales of Miletus (ca. 625-547) (Miletus is in Asia Minor). Thales is sometimes known as the father of geometry probably because he was the first to use the deductive method. He proved among other things (see page 83 of the text) that a triangle inscribed inside a semi-circle (with one side the diameter) must be a right triangle. He also observed that the sides similar triangles are proportional. We do not have any mathematical works directly attributable to Thales or Pythagoras. Euclid’s Elements (Euclid 323-285) and Nicomachus of Gerasa’s Introductio Arithmeticae written ca. 100 B.C. are two math texts which we depend on but neither gives credit to the originators and discoverers of the results. Introductio Arithmeticae unlike Elements was a popular textbook for teaching geometry. Thales is credited with using similar triangles to discover the height of a the Great Pyramid of Gizeh (2600 B.C.) He similarly estimated distance to a ship offshore from a tower above the shore. See page 84 of the text. Pythagoras of Samos (580-500 B.C.; or 569-475 B.C.) (Samos is an Island in the northern Aegean near Asia Minor) His father was given honorary citizenship for bringing corn (!) in a time of famine.