DOCSLIB.ORG

Brittiny Tran MATH 4388- Dr. Ji Essay 1: the Evolution of Euclidean Geometry Geometry Is Ancient and Universal. the History of G

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Brittiny Tran MATH 4388- Dr. Ji Essay 1: the Evolution of Euclidean Geometry Geometry Is Ancient and Universal. the History of G Brittiny Tran MATH 4388- Dr. Ji Essay 1: The Evolution of Euclidean Geometry Geometry is ancient and universal. The history of geometry goes back to about nearly 4,000 years ago in Mesopotamia, Egypt, India, and China. The geometry that is well known today, started back in 600 BC, which was when Thales of Miletus brought logic and logical arguments into geometry. This later became known as Euclidean Geometry. Since Euclid was the only geometer to explain Greek geometry in a way that we can understand, he is often referred to as the “Father of Geometry1.” Euclidean Geometry is the geometry that ‘makes sense’ and the one that everyone is more familiar with. In fact, most of the geometry textbooks used in high schools were most likely based off his book, Elements. It is common knowledge that to construct any geometric shape, lines are necessary. In Euclidean geometry, lines are made up of actual Euclidean points whereas in other geometries they are made of vague ‘line stuff’. Euclidean Geometry is one of three geometries that are known as “The Big Three2.” The other two geometries are Spherical and Hyperbolic Geometries. These three have many similarities and differences. In fact, all three have a distance function and vertical angles are congruent. “The Big Three” are axiomatic structures. Therefore, they have all undefined terms, axioms, theorems and definitions. The Angle-Side Angle Theorem applies, and Side- Angle-Side is an axiom for the space of the geometries. ______________________________ 1. Martyn Shuttleworth, Explorable.com 2. Gerard A. Venema, Foundations of Geometry Second Edition However, there are numerous properties and characteristics that sets each of these geometries apart from each other. For example, the measure of an exterior angle is the sum of the measures of the remote interior angles is only true in Euclidean! The statement, “the Pythagorean Theorem is really a theorem about side lengths3” only applies to Euclidean geometry. The only geometry that is unbounded is Euclidean. Euclid proposed that “things equal to the same thing are equal4”. Euclid’s Elements, may be “the most influential non-religious book of all time.5” Shuttleworth states that Elements is organized by subjects that pertain to “every area of mathematics developed by the Greeks.” These ranged from basic geometry to whole numbers. Euclidean Geometry is “the most typical expression of general mathematic thinking.6” Euclid had begun with explaining some undefined terms such as a point and line. He defined a point as “that which has no part.4” He considered a line as “a length without breadth.4” From there, he started to define what angles and various shapes were such as circles and triangles. Euclid explained angles as the incline of two straight lines; circles were “plane figure(s) consisting of all points that have a fixed distance (radius) from a given centre.4” For more aspects that required more logical thinking, Euclid provided five principles that would be the basis otherwise known as axioms. The first axiom or postulate states that a line joins two points together. The second says that a straight line can be growing infinitely. The third axiom defines that a circle is made when two things are given: a center point and distance for the radius. The ______________________________ 3. Gerard A. Venema, Foundations of Geometry Second Edition 4. Benno Artmann, Encyclopaedia Britannica, inc., https://www.britannica.com/science/Euclidean-geometry 5. Martyn Shuttleworth, Explorable.com next declares that “all right angles are equal.6” Lastly, “a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, will meet on that side on which the angles are less than the two right angles.6” In other words, if two lines don’t intersect, then they are parallel lines. This fifth axiom or postulate later became known as the parallel postulate. Now, even though, Euclid is considered the “Father of Geometry,7” there were other geometers that opposed some of his teachings. The ancient geometers had a deeper, “more advanced understanding of geometry.8” They believed that geometry was “tied to the physical world.8” On the other hand, the ‘more modern’ geometers are complacent and more receptive to the newer methods/teachings. They typically think that Euclid ‘is right.’ Nowadays, geometry and mathematics is more abstract due to the axiomatic systems. These systems help us to comprehend the basis of each geometry such as “The Big Three8”. Throughout history, it has been discovered that both geometry and mathematics utilize logic and can be applicable to real-world situations. Out of “The Big Three8,” Euclidean geometry is hands down the easiest to grasp. In the class textbook, it is stated that “Euclidean geometry has been regarded as the unique way to observe our universe.9” However, since then, the “Geometry Explosion8” occurred in the 19th Century and “Non-Euclidean Geometry” was introduced. ______________________________ 6. Benno Artmann, Encyclopaedia Britannica, inc., https://www.britannica.com/science/Euclidean-geometry 7. Martyn Shuttleworth, Explorable.com 8. Gerard A. Venema, Foundations of Geometry Second Edition 9. Dr. Shanyi Ji, History of Mathematics, p. 68 Works Cited Artmann, Benno. “Euclidean geometry.” Encyclopaedia Britannica Online. Encyclopaedia Britannica inc., 2016. Web. 15 Sept. 2018. Shuttleworth, Martyn. (May 12, 2010). Euclid, the Father of Geometry. Retrieved Sep 15, 2018 from Explorable.com: https://explorable.com/euclid Venema, Gerard A. Foundations of Geometry Second Edition. 2012 .
Load more

Recommended publications
  • 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.
    [Show full text]
  • A CONCISE MINI HISTORY of GEOMETRY 1. Origin And
    Kragujevac Journal of Mathematics Volume 38(1) (2014), Pages 5{21. A CONCISE MINI HISTORY OF GEOMETRY LEOPOLD VERSTRAELEN 1. Origin and development in Old Greece Mathematics was the crowning and lasting achievement of the ancient Greek cul- ture. To more or less extent, arithmetical and geometrical problems had been ex- plored already before, in several previous civilisations at various parts of the world, within a kind of practical mathematical scientific context. The knowledge which in particular as such first had been acquired in Mesopotamia and later on in Egypt, and the philosophical reflections on its meaning and its nature by \the Old Greeks", resulted in the sublime creation of mathematics as a characteristically abstract and deductive science. The name for this science, \mathematics", stems from the Greek language, and basically means \knowledge and understanding", and became of use in most other languages as well; realising however that, as a matter of fact, it is really an art to reach new knowledge and better understanding, the Dutch term for mathematics, \wiskunde", in translation: \the art to achieve wisdom", might be even more appropriate. For specimens of the human kind, \nature" essentially stands for their organised thoughts about sensations and perceptions of \their worlds outside and inside" and \doing mathematics" basically stands for their thoughtful living in \the universe" of their idealisations and abstractions of these sensations and perceptions. Or, as Stewart stated in the revised book \What is Mathematics?" of Courant and Robbins: \Mathematics links the abstract world of mental concepts to the real world of physical things without being located completely in either".
    [Show full text]
  • A Survey of the Development of Geometry up to 1870
    A Survey of the Development of Geometry up to 1870∗ Eldar Straume Department of mathematical sciences Norwegian University of Science and Technology (NTNU) N-9471 Trondheim, Norway September 4, 2014 Abstract This is an expository treatise on the development of the classical ge- ometries, starting from the origins of Euclidean geometry a few centuries BC up to around 1870. At this time classical differential geometry came to an end, and the Riemannian geometric approach started to be developed. Moreover, the discovery of non-Euclidean geometry, about 40 years earlier, had just been demonstrated to be a ”true” geometry on the same footing as Euclidean geometry. These were radically new ideas, but henceforth the importance of the topic became gradually realized. As a consequence, the conventional attitude to the basic geometric questions, including the possible geometric structure of the physical space, was challenged, and foundational problems became an important issue during the following decades. Such a basic understanding of the status of geometry around 1870 enables one to study the geometric works of Sophus Lie and Felix Klein at the beginning of their career in the appropriate historical perspective. arXiv:1409.1140v1 [math.HO] 3 Sep 2014 Contents 1 Euclideangeometry,thesourceofallgeometries 3 1.1 Earlygeometryandtheroleoftherealnumbers . 4 1.1.1 Geometric algebra, constructivism, and the real numbers 7 1.1.2 Thedownfalloftheancientgeometry . 8 ∗This monograph was written up in 2008-2009, as a preparation to the further study of the early geometrical works of Sophus Lie and Felix Klein at the beginning of their career around 1870. The author apologizes for possible historiographic shortcomings, errors, and perhaps lack of updated information on certain topics from the history of mathematics.
    [Show full text]
  • The Project Gutenberg Ebook #31061: a History of Mathematics
    The Project Gutenberg EBook of A History of Mathematics, by Florian Cajori This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: A History of Mathematics Author: Florian Cajori Release Date: January 24, 2010 [EBook #31061] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK A HISTORY OF MATHEMATICS *** Produced by Andrew D. Hwang, Peter Vachuska, Carl Hudkins and the Online Distributed Proofreading Team at http://www.pgdp.net transcriber's note Figures may have been moved with respect to the surrounding text. Minor typographical corrections and presentational changes have been made without comment. This PDF file is formatted for screen viewing, but may be easily formatted for printing. Please consult the preamble of the LATEX source file for instructions. A HISTORY OF MATHEMATICS A HISTORY OF MATHEMATICS BY FLORIAN CAJORI, Ph.D. Formerly Professor of Applied Mathematics in the Tulane University of Louisiana; now Professor of Physics in Colorado College \I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history."|J. W. L. Glaisher New York THE MACMILLAN COMPANY LONDON: MACMILLAN & CO., Ltd. 1909 All rights reserved Copyright, 1893, By MACMILLAN AND CO. Set up and electrotyped January, 1894. Reprinted March, 1895; October, 1897; November, 1901; January, 1906; July, 1909. Norwood Pre&: J. S. Cushing & Co.|Berwick & Smith.
    [Show full text]
  • Timeline for Euclidean Geometry and the “Fifth Postulate”
    Timeline For Euclidean Geometry and the “Fifth Postulate” There is a lot of history of geometry before Euclid can onto the scene. The Babylonians, Egyptians, Greeks, and Pythagoreans did a great deal to advance the subject of geometry. The area of geometry that is taught most widely in high school mathematics classes is Euclidean Geometry. The focus for my project will be Euclidean Geometry. ~2000-500 BC- The Babylonians and Egyptians created many rules that became the basis for Geometry. They created rules, from experimentation, that were used only by the engineers within the civilizations. They had little previous knowledge on the subject of Geometry and both civilizations spent much time assuming rules and then confirming or refuting what they had declared to be true. ~500 BC- Pythagoras of Samos began to use logical thinking to make geometrical conclusions. He also used basic geometrical knowledge to make conclusions regarding advanced geometrical properties. ~300 BC (Before Euclid and The Elements)- Plato developed the precise idea of what a proof should entail. He also pushed for all conclusions, such as definitions and hypothesis, to be clear and able to be understood by most educated individuals. ~300 BC- Euclid wrote what is know as The Elements. This work is a collection of thirteen different books that are collection of geometric knowledge that was decided even before Euclid’s existence. This was knowledge expressed by Pythagoras, Plato, Hippocrates, and other mathematicians that were the pioneers in expressing geometrical conclusions. The first of these books was composed of 23 definitions, 5 postulates, 5 common notions, and 48 propositions.
    [Show full text]
  • Mathematics History in the Geometry Classroom an Honor Thesis
    - Mathematics History in the Geometry Classroom An Honor Thesis (HONRS 499) by Lana D. smith Thesis Advisor Dr. Hubert Ludwig Ball State University Muncie, Indiana April, 1993 Graduation: May 8, 1993 ,- s~rl 1he.slS t...D ~HS9 ."Zlf I '1ct 3 - Abstract ,. SC:>51f This thesis is a collection of history lessons to be used in a mathematics classroom, specifically the geometry class, but many of the topics and persons discussed will lend themselves to discussion in other mathematics class. There are fourteen lessons to correlate with chapters of most geometry texts. The introduction contains several useful suggestions for integrating history in the classroom and use of the material. The goal of these lessons is to add interest in enthusiasm in an area that generally receives negative reactions from many students--math. - "In most sciences one generation tears down what another has - built, and what one has established another undoes. In mathematics alone each generation builds a new story to the old structure." -Hermann Hankel (Boyer and Merzbach, p. 619). Why use history? As Posamentier and Stepelman put it, in reference to the mathematics classroom fl ••• to breathe life into what otherwise might be a golem." using the lives, loves, successes, and failures of the people that created mathematics gives interest to a sometimes boring topic. History can be integrated into the classroom in a light and lively way (Posamentier and Stepelman, p. 156). The use of history can attract student interest and enthusiasm as it did with me. The following are ways to use history in the classroom: -mention past mathematicians anecdotally -mention historical introductions to concepts that are new to students -encourage pupils to understand the historical problems which the concepts they are learning are answers -give "history of mathematics" lessons -devise classroom/homework exercises using mathematical texts from the past -direct dramatic activity which reflects mathematical interaction -encourage creation of poster displays or other projects with a historical theme.
    [Show full text]
  • A Brief Historical Antecedents to the Evolution of Geometry Education
    A Brief Historical Antecedents to the Evolution of Geometry Education Mohammad A. Yazdani, Ph.D. † Abstract The contributions of geometry to the development and advancement of the other branches of mathematics, physics, astronomy, chemistry, biology, engineering, architecture, and arts are recognized by historians, mathematicians, scientists, artists, and educators. Perhaps, the need for learning geometric concepts and skills and their applications in the 21st century is more crucial than any other time in the history of mankind. We observe the implications of geometric skills and concepts in our every day lives, whether we are in need of a simple estimation such as the number of shingles to replace the roof in our house or a complicated subject such as defining the curvature of space and the path of moving bodies to learn the laws of universe. Utilizing geometric concepts and skills in learning and problem solving in other scientific disciplines require an efficient level of proficiency in a large number of geometric concepts or skills. Considering the importance of geometry in school curriculum, an effective method of teaching and learning geometry is desirable for mathematics educators. A brief review of geometry textbooks published in the United States between 1854 and 2001 helps us to explore and examine the instructional design and methods utilized for teaching and learning of this discipline. Background From 8000 B.C. (the beginning of the development of agriculture) when Sumerians used simple plain clay tokens in the shapes of disks, spheres, cylinders, and cones to keep track of their different kinds of goods (before the invention of writing) to the 20th century when Einstein's new geometry of space and time attempted to explain the natural motions of the objects in the universe, geometry has helped mankind to organize, classify, describe, represent, explain, interpret, define, and express in a logical way the world surrounding us.
    [Show full text]
  • Geometry and Empirical Science Author(S): C
    Geometry and Empirical Science Author(s): C. G. Hempel Source: The American Mathematical Monthly, Vol. 52, No. 1 (Jan., 1945), pp. 7-17 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2304830 . Accessed: 03/02/2015 13:33 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org This content downloaded from 140.180.251.176 on Tue, 3 Feb 2015 13:33:42 PM All use subject to JSTOR Terms and Conditions GEOMETRY AND EMPIRICAL SCIENCE C. G. HEMPEL, QueensCollege 1. Introduction. The most distinctive characteristic which differentiates mathematics from the various branches of empirical science, and which ac- counts forits fame as the queen of the sciences,is no doubt the peculiar certainty and necessityof its results. No propositionin even the most advanced parts of empiricalscience can ever attain this status; a hypothesisconcerning "matters of empirical fact" can at best acquire what is loosely called a high probability or a high degree of confirmationon the basis of the relevant evidence available; but however well it may have been confirmedby careful tests, the possibility can never be precluded that it will have to be discarded later in the light of new and disconfirmingevidence.
    [Show full text]
  • From Geometry to Topology - J.J
    HISTORY OF MATHEMATICS - From Geometry To Topology - J.J. Gray and V.L. Hansen FROM GEOMETRY TO TOPOLOGY J.J. Gray Department of Mathematics, Open University, UK V.L. Hansen Department of Mathematics, Technical University of Denmark, Denmark Keywords: Geometry, topology, surface, Riemann surface, manifold, global differential geometry, curvature, dense subset, nowhere dense subset, Cantor set, dimension, homotopy, homology, metric space, compact, connected, topological space. Contents 1. Introduction 2. Three Visions of Geometry in the late Nineteenth Century 2.1. Projective Geometry as a Fundamental Geometry 2.2. Non-Euclidean Geometry and Physical Space 2.3. The Kleinian View of Geometry 2.4. Geometry in Spaces of any Number of Dimensions 2.5. The Search for an Axiomatic Foundation of Geometry 3. Main Roots to Topology 3.1. The Classification of Surfaces 3.2. Complex Function Theory and the Birth of Manifolds 3.3. Fourier Series and Topology of Point Sets 3.4. The Cantor Set 3.5. Construction of the Real Number System 3.6. The Dirichlet Problem and Topology of Planar Domains 4. Creating a Theory of Geometric Topology 4.1. The Alexander Horned Sphere 4.2. The Brouwer Fixed Point Theorem 4.3. Shaping the Notion of Dimension 4.4. Topology of Manifolds 5. Differential Geometry and Topology 5.1. Vector Fields on Manifolds 5.2. CurvatureUNESCO-EOLSS of Curves and Surfaces 5.3. Theorema Egregium 5.4. Surfaces of Constant Curvature and the Uniformization Theorem 5.5. The Gauss-SAMPLEBonnet Theorem and Generalizations CHAPTERS 6. Creating a Theory of Point-set Topology 6.1. The Beginnings to Point-set Topology 6.2.
    [Show full text]
  • II.2 Geometry
    i II.2. Geometry 83 Menninger, K. 1992. Number Words and Number Symbols: of geometry from most other kinds of science is its A Cultural History of Numbers. New York: Dover. (Trans- highly deductive nature. It really seems that by tak- lated by P. Broneer from the revised German edition of ing the simplest of concepts and thinking hard about 1957/58: Zahlwort und Ziffer. Eine Kulturgeschichte der them one can build up an impressive, deductive body Zahl. Göttingen: Vandenhoeck und Ruprecht.) of knowledge about space without having to gather Reid, C. 2006. From Zero to Infinity: What Makes Numbers Interesting. Natick, MA: A. K. Peters. experimental evidence. But can we? Is it really as simple as that? Can we have genuine knowledge of space without ever leaving our II.2 Geometry armchairs? It turns out that we cannot: there are other Jeremy Gray geometries, also based on the concepts of length and angle, that have every claim to be useful, but that dis- 1 Introduction agree with Euclidean geometry. This is an astonishing discovery of the early nineteenth century, but, before it The modern view of geometry was inspired by the novel could be made, a naive understanding of fundamental geometrical theories of hilbert [VI.63] and Einstein in concepts, such as straightness, length, and angle, had the early years of the twentieth century, which built in to be replaced by more precise definitions—a process their turn on other radical reformulations of geometry that took many hundreds of years. Once this had been in the nineteenth century. For thousands of years, the done, first one and then infinitely many new geometries geometrical knowledge of the Greeks, as set out most were discovered.
    [Show full text]
  • A History of Greek Mathematics
    CORNELL UNIVERSITY LBRAaY Cornell University Library QA 22.H43 V.1 A history of Greek mathematics, 3 1924 008 704 219 A HISTORY OF GREEK MATHEMATICS VOLUME I A HISTORY OF GKEEK MATHEMATICS BY SIR THOMAS HEATH K.C.B., K.C.V.O.. F.R.S. Se.D. CAMI). ; HON. D.SC. OXFORD HONORARV FEt.r.OW (FORMFRLV FELLOw) OF TRI>fITY COLI.FHF, CAAIBRIDGE ' . An independent world, Created out of pnre intelligence.' Wordsworth. VOLUME I FROM THALES 'JO EUCIJD OXFORD AT THE CLARENDON PRESS 1921 OXFORD UNIVERSITY PRESS London Edinburgh Glasgow Copenhagen New York Toronto Melbourne Cape Town Bombay Calcutta Madras Shanghai HUMPHREY MILFORD Publisher to the University PREFACE The idea may seem quixotic, but it is nevertheless the author's confident hope that this book will give a fresh interest to the story of Greek mathematics in the eyes both of mathematicians and of classical scholars. For the mathematician the important consideration is that the foundations of mathematics and a great portion of its content are Greek. The Greeks laid down the first principles, invented the methods ah initio, and fixed the terminology. Mathematics in short is a Greek science, whatever new developments modern analysis has brought or may bring. The interest of the subject for the classical scholar is no doubt of a different kind. Greek mathematics reveals an important aspect of the Greek genius of which the student of Greek culture is apt to lose sight. Most people, when they think of the Greek genius, naturally call to mind its master- pieces in literature and art with their notes of beauty, truth, freedom and humanism.
    [Show full text]
  • A Quarta Dimens˜Ao: Da Sua Natureza Espacial Na Geometria Euclidiana
    Revista Brasileira de Ensino de F´ısica, vol. 43, e20210034 (2021) Artigos Gerais www.scielo.br/rbef c b DOI: https://doi.org/10.1590/1806-9126-RBEF-2021-0034 Licenc¸aCreative Commons A quarta dimens˜ao:da sua natureza espacial na geometria euclidiana `acomponente tipo tempo de variedades n˜aoeuclidianas The fourth dimension: from its spatial nature in Euclidean geometry to a time-like component of non-Euclidean manifolds Jos´eMaria Filardo Bassalo1,2, Francisco Caruso*3,4 , Vitor Oguri4 1Academia Paraense de Ciˆencias, Bel´em,PA, Brasil. 2Funda¸c˜aoMinerva, Bel´em,PA, Brasil. 3Centro Brasileiro de Pesquisas F´ısicas,Coordena¸c˜aode F´ısicade Altas Energias, 22290-180, Rio de Janeiro, RJ, Brasil. 4Universidade do Estado do Rio de Janeiro, Instituto de F´ısica Armando Dias Tavares, 20550-900, Rio de Janeiro, RJ, Brasil. Recebido em 27 de janeiro de 2021. Revisado em 12 de fevereiro de 2021. Aceito em 13 de fevereiro de 2021. Neste artigo, apresenta-se a evolu¸c˜aodas ideias sobre a quarta dimens˜aoespacial, partindo daquelas que surgem da geometria euclidiana cl´assicae abordando, em seguida, as que resultam do ˆambito das geometrias n˜ao euclidianas, como as de Riemann e Minkowski. Particular aten¸c˜ao´edada ao momento no qual o tempo real passa efetivamente a ser considerado como uma quarta dimens˜ao,conforme introduzido por Einstein. Palavras-chave: Quarta Dimens˜ao,Espa¸co,Geometrias n˜aoEuclidianas, Hist´oriada Geometria, Hist´oriada F´ısica. In this article, the evolution of the ideas about the fourth spatial dimension is presented, starting from those which come out within classical Euclidean geometry and going through those arose in the framework of non- Euclidean geometries, like those of Riemann and Minkowski.
    [Show full text]