On Spherical Trigonometry in the Medieval Near East and in Europe
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1 Chapter 3 Plane and Spherical Trigonometry 3.1
1 CHAPTER 3 PLANE AND SPHERICAL TRIGONOMETRY 3.1 Introduction It is assumed in this chapter that readers are familiar with the usual elementary formulas encountered in introductory trigonometry. We start the chapter with a brief review of the solution of a plane triangle. While most of this will be familiar to readers, it is suggested that it be not skipped over entirely, because the examples in it contain some cautionary notes concerning hidden pitfalls. This is followed by a quick review of spherical coordinates and direction cosines in three- dimensional geometry. The formulas for the velocity and acceleration components in two- dimensional polar coordinates and three-dimensional spherical coordinates are developed in section 3.4. Section 3.5 deals with the trigonometric formulas for solving spherical triangles. This is a fairly long section, and it will be essential reading for those who are contemplating making a start on celestial mechanics. Sections 3.6 and 3.7 deal with the rotation of axes in two and three dimensions, including Eulerian angles and the rotation matrix of direction cosines. Finally, in section 3.8, a number of commonly encountered trigonometric formulas are gathered for reference. 3.2 Plane Triangles . This section is to serve as a brief reminder of how to solve a plane triangle. While there may be a temptation to pass rapidly over this section, it does contain a warning that will become even more pertinent in the section on spherical triangles. Conventionally, a plane triangle is described by its three angles A, B, C and three sides a, b, c, with a being opposite to A, b opposite to B, and c opposite to C. -
Spherical Trigonometry
Spherical trigonometry 1 The spherical Pythagorean theorem Proposition 1.1 On a sphere of radius R, any right triangle 4ABC with \C being the right angle satisfies cos(c=R) = cos(a=R) cos(b=R): (1) Proof: Let O be the center of the sphere, we may assume its coordinates are (0; 0; 0). We may rotate −! the sphere so that A has coordinates OA = (R; 0; 0) and C lies in the xy-plane, see Figure 1. Rotating z B y O(0; 0; 0) C A(R; 0; 0) x Figure 1: The Pythagorean theorem for a spherical right triangle around the z axis by β := AOC takes A into C. The edge OA moves in the xy-plane, by β, thus −−!\ the coordinates of C are OC = (R cos(β);R sin(β); 0). Since we have a right angle at C, the plane of 4OBC is perpendicular to the plane of 4OAC and it contains the z axis. An orthonormal basis −−! −! of the plane of 4OBC is given by 1=R · OC = (cos(β); sin(β); 0) and the vector OZ := (0; 0; 1). A rotation around O in this plane by α := \BOC takes C into B: −−! −−! −! OB = cos(α) · OC + sin(α) · R · OZ = (R cos(β) cos(α);R sin(β) cos(α);R sin(α)): Introducing γ := \AOB, we have −! −−! OA · OB R2 cos(α) cos(β) cos(γ) = = : R2 R2 The statement now follows from α = a=R, β = b=R and γ = c=R. ♦ To prove the rest of the formulas of spherical trigonometry, we need to show the following. -
Simon Stevin
II THE PRINCIPAL WORKS OF SIMON STEVIN E D IT E D BY ERNST CRONE, E. J. DIJKSTERHUIS, R. J. FORBES M. G. J. MINNAERT, A. PANNEKOEK A M ST E R D A M C. V. SW ETS & Z E IT L IN G E R J m THE PRINCIPAL WORKS OF SIMON STEVIN VOLUME II MATHEMATICS E D IT E D BY D. J. STRUIK PROFESSOR AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE (MASS.) A M S T E R D A M C. V. SW ETS & Z E IT L IN G E R 1958 The edition of this volume II of the principal works of SIMON STEVIN devoted to his mathematical publications, has been rendered possible through the financial aid of the Koninklijke. Nederlandse Akademie van Wetenschappen (Royal Netherlands Academy of Science) Printed by Jan de Lange, Deventer, Holland The following edition of the Principal Works of SIMON STEVIN has been brought about at the initiative of the Physics Section of the Koninklijke Nederlandse Akademie van Weten schappen (Royal Netherlands Academy of Sciences) by a committee consisting of the following members: ERNST CRONE, Chairman of the Netherlands Maritime Museum, Amsterdam E. J. DIJKSTERHUIS, Professor of the History of Science at the Universities of Leiden and Utrecht R. J. FORBES, Professor of the History of Science at the Municipal University of Amsterdam M. G. J. M INNAERT, Professor of Astronomy at the University of Utrecht A. PANNEKOEK, Former Professor of Astronomy at the Municipal University of Amsterdam The Dutch texts of STEVIN as well as the introductions and notes have been translated into English or revised by Miss C. -
Beginnings of Trigonometry
Beginnings of Trigonometry Trigonometry is an area of mathematics used for determining geometric quantities. Its name, first published in 1595 by B. Pitiscus, means "the study of trigons (triangles)" in Latin. Ancient Greek Mathematicians first used trigonometric functions with the chords of a circle. The first to publish these chords in 140 BC was Hipparchus, who is now called the founder of trigonometry. In AD 100, Menelaus, another Greek mathematician, published six lost books of tables of chords. Ptolemy, a Babylonian, also wrote a book of chords. Using chords, Ptolemy knew that sin(xy+= ) sin x cos y + cos x sin y and abc = = sinABC sin sin In his great astronomical handbook, The Almagest, Ptolemy provided a table of chords for steps of °, from 0° to 180°, that is accurate to 1/3600 of a unit. Ptolemy's degree of accuracy is due in part because the Hellenistic Greeks had adopted the Babylonian base- 60 (sexagesimal) numeration system. He also explained his method for constructing his table of chords, and in the course of the book he gave many examples of how to use the table to find unknown parts of triangles from known parts. Ptolemy provided what is now known as Menelaus's theorem for solving spherical triangles, as well, and for several centuries his trigonometry was the primary introduction to the subject for any astronomer. Sine first appeared in the work of Aryabhata, a Hindu. He used the word jya for sine. He also published the first sine tables. Brahmagupta, in 628, also published a table of sines for any angle. -
The Sphere of the Earth Description and Technical Manual
The Sphere of the Earth Description and Technical Manual December 20, 2012 Daniel Ramos ∗ MMACA (Museu de Matem`atiquesde Catalunya) 1 The exhibit The exhibit we are presenting explores the science of cartography and the geometry of the sphere. Althoug this subject has been widely treated on science exhibitions and fairs, our proposal aims to be a more comprehensive and complete treatment on the subject, as well as to be a fully open source, documented material that both museographers and public can experience, learn and modify. The key concept comes from the problem of representing the spherical surface of the Earth onto a flat map. Studying this problem is the subject of cartography, and has been an important mathematical problem in History (navigation, position, frontiers, land ownership...). An essential theorem in Geometry (Gauss' Egregium theorem) ensures that there is no perfect map, that is, there is no way of repre- senting the Earth keeping distances at scale. However, this is exactly what makes cartography a discipline: developing several different maps that try to solve well enough the problem of representing the Earth. A common activity is comparing an Earth globe with a (Mercator) map, observ- ing that distances are not preserved. Our module explores far beyond that. We present: • Several different maps, currently 6 map projections are used, each one featuring different special properties. Printed at nominal scale 1:1 of a model globe. • The script programs that generate these pictures. Generating a map takes less than 50 lines of code, anyone could generate his own. ∗E-mail: [email protected] 1 • A collection of tools and models: a flexible ruler to be used over the globe, plane and spherical protractors, models showing longitude and latitude.. -
Spherical Trigonometry Through the Ages
School of Mathematics and Statistics Newcastle University MMath Mathematics Report Spherical Trigonometry Through the Ages Author: Supervisor: Lauren Roberts Dr. Andrew Fletcher May 1, 2015 Contents 1 Introduction 1 2 Geometry on the Sphere Part I 4 2.1 Cross-sections and Measurements . .4 2.2 Triangles on the Sphere . .7 2.2.1 What is the Largest Possible Spherical Triangle? . .7 2.2.2 The Sum of Spherical Angles . .9 3 Menelaus of Alexandria 12 3.1 Menelaus' Theorem . 13 3.2 Astronomy and the Menelaus Configuration: The Celestial Sphere . 16 4 Medieval Islamic Mathematics 18 4.1 The Rule of Four Quantities . 18 4.2 Finding the Qibla . 20 4.3 The Spherical Law of Sines . 22 5 Right-Angled Triangles 25 5.1 Pythagoras' Theorem on the Sphere . 25 5.2 The Spherical Law of Cosines . 27 6 Geometry on the Sphere Part II 30 6.1 Areas of Spherical Polygons . 30 6.2 Euler's Polyhedral Formula . 33 7 Quaternions 35 7.1 Some Useful Properties . 35 7.1.1 The Quaternion Product . 36 7.1.2 The Complex Conjugate and Inverse Quaternions . 36 1 CONTENTS 2 7.2 Rotation Sequences . 37 7.2.1 3-Dimensional Representation . 39 7.3 Deriving the Spherical Law of Sines . 39 8 Conclusions 42 Bibliography 44 Abstract The art of applying trigonometry to triangles constructed on the surface of the sphere is the branch of mathematics known as spherical trigonometry. It allows new theorems relating the sides and angles of spherical triangles to be derived. The history and development of spherical trigonometry throughout the ages is discussed. -
Arxiv:1409.4736V1 [Math.HO] 16 Sep 2014
ON THE WORKS OF EULER AND HIS FOLLOWERS ON SPHERICAL GEOMETRY ATHANASE PAPADOPOULOS Abstract. We review and comment on some works of Euler and his followers on spherical geometry. We start by presenting some memoirs of Euler on spherical trigonometry. We comment on Euler's use of the methods of the calculus of variations in spherical trigonometry. We then survey a series of geometrical resuls, where the stress is on the analogy between the results in spherical geometry and the corresponding results in Euclidean geometry. We elaborate on two such results. The first one, known as Lexell's Theorem (Lexell was a student of Euler), concerns the locus of the vertices of a spherical triangle with a fixed area and a given base. This is the spherical counterpart of a result in Euclid's Elements, but it is much more difficult to prove than its Euclidean analogue. The second result, due to Euler, is the spherical analogue of a generalization of a theorem of Pappus (Proposition 117 of Book VII of the Collection) on the construction of a triangle inscribed in a circle whose sides are contained in three lines that pass through three given points. Both results have many ramifications, involving several mathematicians, and we mention some of these developments. We also comment on three papers of Euler on projections of the sphere on the Euclidean plane that are related with the art of drawing geographical maps. AMS classification: 01-99 ; 53-02 ; 53-03 ; 53A05 ; 53A35. Keywords: spherical trigonometry, spherical geometry, Euler, Lexell the- orem, cartography, calculus of variations. -
Preprint N°494
2019 Preprint N°494 The Reception of Cosmography in Vienna: Georg von Peuerbach, Johannes Regiomontanus, and Sebastian Binderlius Thomas Horst Published in the context of the project “The Sphere—Knowledge System Evolution and the Shared Scientific Identity of Europe” https://sphaera.mpiwg-berlin.mpg.de The Reception of Cosmography in Vienna: Georg von Peuerbach, Johannes Regiomontanus, and Sebastian Binderlius1 Abstract In this paper, the importance of the cosmographical activities of the Vienna astronomical “school” for the reception of the Tractatus de Sphaera is analyzed. First, the biographies of two main representatives of the Vienna mathematical/astronomical circle are presented: the Austrian astronomers, mathematicians, and instrument makers Georg von Peuerbach (1423– 1461) and his student Johannes Müller von Königsberg (Regiomontanus, 1436–1476). Their studies influenced the cosmographical teaching at the University of Vienna enormously for the next century and are relevant to understanding what followed; therefore, the prosopo- graphical introductions of these Vienna scholars have been included here, even if neither can be considered a real author of the Sphaera. Moreover, taking the examples of an impressive sixteenth-century miscellany (Austrian Na- tional Library, Cod. ser. nov. 4265, including the recently rediscovered cosmography by Sebastian Binderlius, compiled around 1518), the diversity of different cosmographical studies in the capital of the Habsburg Empire at the turning point between the Middle Ages and the early modern period is demonstrated. Handwritten comments in the Vienna edition of De sphaera (1518) also show how big the influence of Sacrobosco’s work remained as a didactical tool at the universities in the first decades of the sixteenth century—and how cosmographical knowledge was transformed and structured in early modern Europe by the editors and readers of the Sphaera. -
GEOMETRIC ANALYSIS of an OBSERVER on a SPHERICAL EARTH and an AIRCRAFT OR SATELLITE Michael Geyer
GEOMETRIC ANALYSIS OF AN OBSERVER ON A SPHERICAL EARTH AND AN AIRCRAFT OR SATELLITE Michael Geyer S π/2 - α - θ d h α N U Re Re θ O Federico Rostagno Michael Geyer Peter Mercator nosco.com Project Memorandum — September 2013 DOT-VNTSC-FAA-13-08 Prepared for: Federal Aviation Administration Wake Turbulence Research Program DOT/RITA Volpe Center TABLE OF CONTENTS 1. INTRODUCTION...................................................................................................... 1 1.1 Basic Problem and Solution Approach........................................................................................1 1.2 Vertical Plane Formulation ..........................................................................................................2 1.3 Spherical Surface Formulation ....................................................................................................3 1.4 Limitations and Applicability of Analysis...................................................................................4 1.5 Recommended Approach to Finding a Solution.........................................................................5 1.6 Outline of this Document ..............................................................................................................6 2. MATHEMATICS AND PHYSICS BASICS ............................................................... 8 2.1 Exact and Approximate Solutions to Common Equations ........................................................8 2.1.1 The Law of Sines for Plane Triangles.........................................................................................8 -
The Copernican System
The Copernican system Jan P. Hogendijk Dept of Mathematics, Utrecht University 8 june 2011 1 Prelude: the 15th century Copernicus: biography Copernicus’ new system: Main Points Copernicus’ new system: Dirty Details The reception of the new system 2 Renaissance Invention of Printing by Johannes Gutenberg (Mainz), ca. 1450: typesetting with movable type, printing press, paper. 3 Most important astronomer of the 15th century Johannes Muller¨ of K¨onigsberg (Bavaria), or Regiomontanus (1436-1476) 4 Biography of Regiomontanus (1436-1476) 1448: Entered University of Leipzig. 1460: Met cardinal Bessarion (from Trebizond), travelled with him to Italy. 1463: completed Epitome of the Almagest (of Ptolemy) [printed 1492]. ca. 1461: made observations of eclipses, etc. which diverged from the predictions by e.g. the Alfonsine tables. 5 Quotation from Regiomontanus (1464) “I cannot [but] wonder at the indolence of common astronomers of our age, who, just as credulous women, receive as something divine and immutable whatever they come upon in books, either of tables of their instructions, for they believe in writers and make no effort to find the truth.” Regiomontanus wanted to restore and “update” Ptolemaic astronomy in order to predict astronomical phenomena accurately. 1471: he founded printing press in Nurnberg,¨ producing mathematical and astronomical works with great accuracy. 6 Product of Regiomontanus’ Press (1472) Novae Theoricae Planetarum, Georg von Peuerbach (1423-1461) 7 Example of Regiomontanus’ own predictions solar and lunar eclipses, 1489, 1490 8 Nicolas Copernicus (1473-1543): biography 1473 Torun (Poland) - 1543 Frauenburg = Frombork (now in Poland, near coast and Russian border) Studied at the University of Krakow. Worked in service of the Catholic Church until his death. -
A Treatise on Spherical Trigonometry, and Its Application
A T REATISE SPHEEICAL T RIGONOMETKY. WORKSY B JOHN G ASEY, ESQ., LLD., F. R.8., FELLOWF O THE ROYAL UNIVERSITY OF IRELAND. Second E dition, Price 3s. A T REATISE ON ELEMENTARY TRIGONOMETRY, With n umerous Examples AND E&uesttfms f or Exammatttm. OKEY T THE EXERCISES IN THE TREATISE ON ELEMENTARY TRIGONOMETRY. Fifth E dition, Revised and Enlarged, Price 3s. 6d., Cloth. A S EQ,UEL TO THE FIRST SIX BOOKS OF THE ELEMENTS OF EUCLID, Containing- a n Easy Introduction to Modern Geometry . With a umerstts Examples. Seventh E dition, Price 4s. 6d. ; or in two parts, each 2s. 6d. THE E LEMENTS OE EUCLID, BOOKS I.-YL, AND PROPOSITIONS I.-XXI. OP BOOK XI. ; Together ' with an Appendix on the Cylinder, Sphere, Cone, &>c. @opdatt$ J tettotatiotts & aumeraus Exercises. Second E dition, Price 6s. AEY K TO THE EXERCISES IN THE FIRST SIX BOOKS OF CASEY'S "ELEMENTS OF EUCLID." Price 7 s. 6d. A T REATISE ON THE ANALYTICAL GEOMETRY OF THE POINT, LINE, CIRCLE, & CONIC SECTIONS, Containing a n Account of its most recent Extensions, Wxih t tttmeratts Examples. Price 7 s. 6d. A T REATISE ON PLANE TRIGONOMETRY, including THE THEORY OF HYPERBOLIC FUNCTIONS. LONDON : L ONGMANS? CO. DUBLIN: HODGES, FTGGIS & CO. A T REATISE ON- SPHERICAL T RIGONOMETRY, ANDTS I APPLICATION- TO GEODESYND A ASTRONOMY, ■WITH BY JOHN C ASEY, LL.D., F.E.S., Fellowf o the Royal University of Ireland; Memberf o the Council of the Royal Irish Academy ; Memberf o the Mathematical Societies of London and France ; Corresponding M ember of the Royal Society of Sciences of Liege; and Professorf o the Higher Mathematics and Mathematical Physics in t he Catholic University of Ireland. -
An Axiom of True Courses Calculation in Great Circle Navigation
Journal of Marine Science and Engineering Communication An Axiom of True Courses Calculation in Great Circle Navigation Mate Baric 1, David Brˇci´c 2,* , Mate Kosor 1 and Roko Jelic 1 1 Maritime Department, University of Zadar, 23000 Zadar, Croatia; [email protected] (M.B.); [email protected] (M.K.); [email protected] (R.J.) 2 Faculty of Maritime Studies, University of Rijeka, 51000 Rijeka, Croatia * Correspondence: [email protected] Abstract: Based on traditional expressions and spherical trigonometry, at present, great circle naviga- tion is undertaken using various navigational software packages. Recent research has mainly focused on vector algebra. These problems are calculated numerically and are thus suited to computer-aided great circle navigation. However, essential knowledge requires the navigator to be able to calculate navigation parameters without the use of aids. This requirement is met using spherical trigonometry functions and the Napier wheel. In addition, to facilitate calculation, certain axioms have been developed to determine a vessel’s true course. These axioms can lead to misleading results due to the limitations of the trigonometric functions, mathematical errors, and the type of great circle navigation. The aim of this paper is to determine a reliable trigonometric function for calculating a vessel’s course in regular and composite great circle navigation, which can be used with the proposed axioms. This was achieved using analysis of the trigonometric functions, and assessment of their impact on the vessel’s calculated course and established axioms. Citation: Baric, M.; Brˇci´c,D.; Kosor, Keywords: great circle; navigation; axiom M.; Jelic, R. An Axiom of True Courses Calculation in Great Circle Navigation.