DOCSLIB.ORG

MST Topics in History of Mathematics Euclid’S Elements and the Works of Archimedes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
MST Topics in History of Mathematics Euclid’S Elements and the Works of Archimedes MST Topics in History of Mathematics Euclid’s Elements and the Works of Archimedes Paul Yiu Department of Mathematics Florida Atlantic University Summer 2014 June 25 Chapter 0 Greek geometry before Euclid Chronology -586 Babylonian captivity -585 Thales of Miletus; deductive geometry -580 Birth of Pythagoras -540 Pythagorean arithmetic and geometry -430 Elements of Hippocrates of Chios -427 Birth of Plato -420 Incommensurables -399 Death of Socrates -360 Eudoxus on proportion and exhaustion -347 Death of Plato -335 Eudemus: History of Geometry -332 Alexandria founded -323 Death of Alexander -322 Death of Aristotle -300 Euclid’s Elements -225 Apollonius: Conics -212 Death of Archimedes +75 Works of Heron of Alexandria +250 Diophantus: Arithmetica +320 Pappus: Mathematical Collections +485 Death of Proclus 2 Greek geometry before Euclid Thales of Miletus Theorem (Thales). The angle inscribed in semicircle is a right angle. 3 Other theorems attributed to Thales Eudemus’ History (ca -320): 1. A circle is bisected by a diameter. 2. The base angle of an isosceles triangle are equal. (Euclid I.5) 3. The pairs of vertical angles formed by two intersecting lines are equal. (Eu- clid I.15) 4. If two triangles are such that two angles and a side of one are equal respec- tively to two angles and a side of the other, then the triangles are congruent. (Euclid I.26) 4 Greek geometry before Euclid Pythaogas of Samos Proclus, on Euclid I: (p.298) Eudemus the Peripatetic ascribes to the Pythagoreans the discovery of this theorem, that any triangle has internal angles equal to right angles. He says they proved the theorem in question after this fashion. A D E B C Let ABC be a triangle, and through A let DE be drawn parallel to BC. Now since BC, DE are parallel, and the alternative angles are equal, the angle DAB is equal to the angle ABC, and EAC is equal to ACB. Let BAC be added to both. Then the angles DAB, BAC, CAE, that is, the angles DAB, BAE, that is, two right angles, are equal to the three angles of the triangle. Therefore the three angles of the triangle are equal to two right angles. Such is the proof of the Pythagoreans. 5 Iamblichus: On the Pythagorean Life (Thomas, 222–225): It is related of Hippasus that he was a Pythagoreans, and that, owing to his being the first to publish and describe the sphere from the twelve pentagons, he perished at sea for his impiety, but he received credit for the discovery, though really it all belonged to HIM (for in this way they refer to Pythagoras, and they do not call him by his name). 6 Greek geometry before Euclid Euclid, Elements X, Scholium i The Pythagoreans were the first to make inquiry into commensurability, having first discovered it as a result of their observation of numbers; for though the unit is a common measure of all numbers they could not find a common measure of all magnitudes. The reason is that all numbers, of whatsoever kind, however they be divided leave some least part which will not suffer further division; but all magnitudes are divisible ad infinitum and do not leave some part which, being the least possible, will not admit of further division, but that remainder can be divided ad infinitum so as to give an infinite number of parts, of which each can be divided ad infinitum; and in sum, magnitude partakes in division of the principle of the infinite, but in its entirety of the principle of the finite, while number in division partakes of the finite, but in its entirety of the infinite. ... There is a legend that the first of the Pythagoreans who made public the investigation of these matters perished in a shipwreck. 7 Democritus Archimedes, Method: ...incase of those theorems concerning the cone and pyramid of which Eudoxus first discovered the proof, the theorem that the cone is the third part of the cylinder, and the pyramind of the prism, having the same base and equal height, no small share of the credit should be given to Democritus, who was the first to make the assertion with regard to the said figure, though without proof. 8 Greek geometry before Euclid Duplication of the cube Theon of Smyrna (Thomas 257): In his work entitled Platonicus Eratosthenes says that, when the god announced to the Delians by oracle that to get rid of a plague they must construct an altar double of the existing one, their craftsmen fell into great perplexity in trying to find how a solid could be made double of another solid, and they went to ask Plato about it. He told them that the god had given this oracle, not because he wanted an altar of double the size, but he wished, in setting this task before them, to reproach the Greeks for their neglect of mathematics and their contempt for geometry. 9 Eutocius: Commentary on Archimedes’ Sphere and Cylinder, II (Thomas 257–259): It became a subject of inquiry among geometers in what manner one might double the given solid while it remained the same shape, and this problem was called the duplication of the cube; for given a cube, they sought to double it. When all were for a long time at a loss, Hippocrates of Chios first conceived that, if two mean proportionals could be found in continued proportion between two straight lines, of which the greater was double the lesser, the cube would be doubled, so that the puzzle was by him turned into no less a puzzle. After a time, it is related, certain Delians, when attempting to double a certain altar in accordance with an oracle, fell into the same quandary, and sent over to ask the geometers who were with Plato in the Academy to find what they sought. When these men applied themselves diligently and sought to find two mean proportionals between two given straight lines, Archytas of Taras is said to have found them by the half-cylinders, and Eudoxus by the so-called curved lines; but it turned out that all their solutions were theoretical, and they could not give a practical construction and it it to use, except to a certain small extent Menaechmus, and that with difficulty. An easy mechanical solution, was however, found by me, and by means of it I will find, not only two means to the given straight lines, but as many as may be enjoined. 10 Greek geometry before Euclid Nicomedes’ solution Given: Two lines AD and DC at right angles. To construct: Two continued mean proportionals between AD and CD. M A D F B E C K H T Z Construction: Complete the rectangle ADCB and take the midpoints E, F of BC and AB. Join DF to meet CB produced at H. Construct EZ perpendicular to BC so that CZ = AF . Join HZ and construct the parallel through C. On this parallel construct a point T so that ZT meet BC produced at K with TK = CZ. Then AD : AM = AM : CK = CK : CD, and AM, CK are two continued mean proportionals between AD and CD. 11 MAA Focus June/July 2014, 27. x and y are two continuous mean proportionals between 1 and 2: A x 1 1 Y 1 B 1 C y X 1 Z 12 Greek geometry before Euclid Cubic root of 2 by paper folding B. Casselman, If Euclid had been Japanese, Notices of AMS, 54 (2007) 626–628. A paper square ABCD is divided into three strips of equal area by the parallel lines PQand RS. The square is then folded so that C falls on AD and S falls on PQ (as C in the second√ diagram). AC = 3 2 Then CD . A B A B P Q P S C R S R D C D C 13 Trisection of an angle Archimedes [Book of Lemmas, Proposition 8] Given angle AOB with OA = OB (contained in a circle, center O), construct a line through A such that the intercept between the circle and the line BO has the same length as the radius of the circle. ∠AOC = 1 ∠AOB Then 3 . A A C O B 14 Greek geometry before Euclid Trisection of an angle To triangle an angle AOB, pass a line through O such that the intercept between the parallel and the perpendicular at A to the line OB is 2 · OA. Then this line is a trisector of angle AOB. A E M D O B 15 Angle trisection with the use of conics To trisect an angle AOB, construct the hyperbola with focus A, directrix OM, and eccentricity 2, to intersect the arc AB at C. Then ∠AOB =3∠AOC. C K B A M P O 16 Greek geometry before Euclid The quadratrix of Hippias A horizontal line HK (with initial position AB) falls vertically, and a radius OP (with initial position OA) rotates about O, both uniformly and arrive at OC at the same time. The locus of the intersection Q = HK ∩ OP is the quadratrix. A B P Q H K O C 17 Trisection of an angle by the quadratrix B P K Q A O To trisect angle AOB, let OB intersect the quadratrix at P . Trisect the segment OP at K. Construct the parallel through K to OA to intersect the quadratrix at Q. Then OQ is a trisector of angle AOB. 18 Greek geometry before Euclid Angle trisection by paper folding B. Casselman, If Euclid had been Japanese, Notices of AMS, 54 (2007) 626–628.
Load more

Recommended publications
  • A One-Page Primer the Centers and Circles of a Triangle
    1 The Triangle: A One-Page Primer (Date: March 15, 2004 ) An angle is formed by two intersecting lines. A right angle has 90◦. An angle that is less (greater) than a right angle is called acute (obtuse). Two angles whose sum is 90◦ (180◦) are called complementary (supplementary). Two intersecting perpendicular lines form right angles. The angle bisectors are given by the locus of points equidistant from its two lines. The internal and external angle bisectors are perpendicular to each other. The vertically opposite angles formed by the intersection of two lines are equal. In triangle ABC, the angles are denoted A, B, C; the sides are denoted a = BC, b = AC, c = AB; the semi- 1 ◦ perimeter (s) is 2 the perimeter (a + b + c). A + B + C = 180 . Similar figures are equiangular and their corresponding components are proportional. Two triangles are similar if they share two equal angles. Euclid's VI.19, 2 ∆ABC a the areas of similar triangles are proportional to the second power of their corresponding sides ( ∆A0B0C0 = a02 ). An acute triangle has three acute angles. An obtuse triangle has one obtuse angle. A scalene triangle has all its sides unequal. An isosceles triangle has two equal sides. I.5 (pons asinorum): The angles at the base of an isosceles triangle are equal. An equilateral triangle has all its sides congruent. A right triangle has one right angle (at C). The hypotenuse of a right triangle is the side opposite the 90◦ angle and the catheti are its other legs. In a opp adj opp adj hyp hyp right triangle, sin = , cos = , tan = , cot = , sec = , csc = .
    [Show full text]
  • Read Book Advanced Euclidean Geometry Ebook
    ADVANCED EUCLIDEAN GEOMETRY PDF, EPUB, EBOOK Roger A. Johnson | 336 pages | 30 Nov 2007 | Dover Publications Inc. | 9780486462370 | English | New York, United States Advanced Euclidean Geometry PDF Book As P approaches nearer to A , r passes through all values from one to zero; as P passes through A , and moves toward B, r becomes zero and then passes through all negative values, becoming —1 at the mid-point of AB. Uh-oh, it looks like your Internet Explorer is out of date. In Elements Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Pons asinorum Pythagorean theorem Thales's theorem Theorem of the gnomon. It might also be so named because of the geometrical figure's resemblance to a steep bridge that only a sure-footed donkey could cross. Calculus Real analysis Complex analysis Differential equations Functional analysis Harmonic analysis. This article needs attention from an expert in mathematics. Facebook Twitter. On any line there is one and only one point at infinity. This may be formulated and proved algebraically:. When we have occasion to deal with a geometric quantity that may be regarded as measurable in either of two directions, it is often convenient to regard measurements in one of these directions as positive, the other as negative. Logical questions thus become completely independent of empirical or psychological questions For example, proposition I. This volume serves as an extension of high school-level studies of geometry and algebra, and He was formerly professor of mathematics education and dean of the School of Education at The City College of the City University of New York, where he spent the previous 40 years.
    [Show full text]
  • 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.
    [Show full text]
  • Geometry by Its History
    Undergraduate Texts in Mathematics Geometry by Its History Bearbeitet von Alexander Ostermann, Gerhard Wanner 1. Auflage 2012. Buch. xii, 440 S. Hardcover ISBN 978 3 642 29162 3 Format (B x L): 15,5 x 23,5 cm Gewicht: 836 g Weitere Fachgebiete > Mathematik > Geometrie > Elementare Geometrie: Allgemeines Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. 2 The Elements of Euclid “At age eleven, I began Euclid, with my brother as my tutor. This was one of the greatest events of my life, as dazzling as first love. I had not imagined that there was anything as delicious in the world.” (B. Russell, quoted from K. Hoechsmann, Editorial, π in the Sky, Issue 9, Dec. 2005. A few paragraphs later K. H. added: An innocent look at a page of contemporary the- orems is no doubt less likely to evoke feelings of “first love”.) “At the age of 16, Abel’s genius suddenly became apparent. Mr. Holmbo¨e, then professor in his school, gave him private lessons. Having quickly absorbed the Elements, he went through the In- troductio and the Institutiones calculi differentialis and integralis of Euler. From here on, he progressed alone.” (Obituary for Abel by Crelle, J. Reine Angew. Math. 4 (1829) p. 402; transl. from the French) “The year 1868 must be characterised as [Sophus Lie’s] break- through year.
    [Show full text]
  • Euclid Transcript
    Euclid Transcript Date: Wednesday, 16 January 2002 - 12:00AM Location: Barnard's Inn Hall Euclid Professor Robin Wilson In this sequence of lectures I want to look at three great mathematicians that may or may not be familiar to you. We all know the story of Isaac Newton and the apple, but why was it so important? What was his major work (the Principia Mathematica) about, what difficulties did it raise, and how were they resolved? Was Newton really the first to discover the calculus, and what else did he do? That is next time a fortnight today and in the third lecture I will be talking about Leonhard Euler, the most prolific mathematician of all time. Well-known to mathematicians, yet largely unknown to anyone else even though he is the mathematical equivalent of the Enlightenment composer Haydn. Today, in the immortal words of Casablanca, here's looking at Euclid author of the Elements, the best-selling mathematics book of all time, having been used for more than 2000 years indeed, it is quite possibly the most printed book ever, apart than the Bible. Who was Euclid, and why did his writings have such influence? What does the Elements contain, and why did one feature of it cause so much difficulty over the years? Much has been written about the Elements. As the philosopher Immanuel Kant observed: There is no book in metaphysics such as we have in mathematics. If you want to know what mathematics is, just look at Euclid's Elements. The Victorian mathematician Augustus De Morgan, of whom I told you last October, said: The thirteen books of Euclid must have been a tremendous advance, probably even greater than that contained in the Principia of Newton.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • Lecture 18: Side-Angle-Side
    Lecture 18: Side-Angle-Side 18.1 Congruence Notation: Given 4ABC, we will write ∠A for ∠BAC, ∠B for ∠ABC, and ∠C for ∠BCA if the meaning is clear from the context. Definition In a protractor geometry, we write 4ABC ' 4DEF if AB ' DE, BC ' EF , CA ' F D, and ∠A ' ∠D, ∠B ' ∠E, ∠C ' ∠F. Definition In a protractor geometry, if 4ABC ' 4DEF , 4ACB ' 4DEF , 4BAC ' 4DEF , 4BCA ' 4DEF , 4CAB ' 4DEF , or 4CBA ' 4DEF , we say 4ABC and 4DEF are congruent Example In the Taxicab Plane, let A = (0, 0), B = (−1, 1), C = (1, 1), D = (5, 0), E = (5, 2), and F = (7, 0). Then AB = 2 = DE, AC = 2 = DF, m(∠A) = 90 = m(∠D), m(∠B) = 45 = m(∠E), and m(∠C) = 45 = m(∠F ), and so AB ' DE, AC ' DF , ∠A ' ∠D, ∠B ' ∠E, and ∠C ' ∠F . However, BC = 2 6= 4 = EF. Thus BC and EF are not congruent, and so 4ABC and 4DEF are not congruent. 18.2 Side-angle-side Definition A protractor geometry satisfies the Side-Angle-Side Axiom (SAS) if, given 4ABC and 4DEF , AB ' DE, ∠B ' ∠E, and BC ' EF imply 4ABC ' 4DEF . Definition We call a protractor geometry satisfying the Side-Angle-Side Axiom a neutral geometry, also called an absolute geometry. 18-1 Lecture 18: Side-Angle-Side 18-2 Example We will show that the Euclidean Plane is a neutral geometry. First recall the Law of Cosines: Given any triangle 4ABC in the Euclidean Plane, 2 2 2 AC = AB + BC − 2(AB)(BC) cos(mE(∠B)).
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]