The History of Mathematics in the Nine- Teenth Century
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Canada Archives Canada Published Heritage Direction Du Branch Patrimoine De I'edition
Rhetoric more geometrico in Proclus' Elements of Theology and Boethius' De Hebdomadibus A Thesis submitted in Candidacy for the Degree of Master of Arts in Philosophy Institute for Christian Studies Toronto, Ontario By Carlos R. Bovell November 2007 Library and Bibliotheque et 1*1 Archives Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-43117-7 Our file Notre reference ISBN: 978-0-494-43117-7 NOTICE: AVIS: The author has granted a non L'auteur a accorde une licence non exclusive exclusive license allowing Library permettant a la Bibliotheque et Archives and Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par Plntemet, prefer, telecommunication or on the Internet, distribuer et vendre des theses partout dans loan, distribute and sell theses le monde, a des fins commerciales ou autres, worldwide, for commercial or non sur support microforme, papier, electronique commercial purposes, in microform, et/ou autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in et des droits moraux qui protege cette these. this thesis. Neither the thesis Ni la these ni des extraits substantiels de nor substantial extracts from it celle-ci ne doivent etre imprimes ou autrement may be printed or otherwise reproduits sans son autorisation. reproduced without the author's permission. -
Two Editions of Ibn Al-Haytham's Completion of the Conics
Historia Mathematica 29 (2002), 247–265 doi:10.1006/hmat.2002.2352 Two Editions of Ibn al-Haytham’s Completion of the Conics View metadata, citation and similar papers at core.ac.uk brought to you by CORE Jan P. Hogendijk provided by Elsevier - Publisher Connector Mathematics Department, University of Utrecht, P.O. Box 80.010, 3508 TA Utrecht, Netherlands E-mail: [email protected] The lost Book VIII of the Conics of Apollonius of Perga (ca. 200 B.C.) was reconstructed by the Islamic mathematician Ibn al-Haytham (ca. A.D. 965–1041) in his Completion of the Conics. The Arabic text of this reconstruction with English translation and commentary was published as J. P. Hogendijk, Ibn al-Haytham’s Completion of the Conics (New York: Springer-Verlag, 1985). In a new Arabic edition with French translation and commentary (R. Rashed, Les mathematiques´ infinitesimales´ du IXe au XIe siecle.´ Vol. 3., London: Al-Furqan Foundation, 2000), it was claimed that my edition is faulty. In this paper the similarities and differences between the two editions, translations, and commentaries are discussed, with due consideration for readers who do not know Arabic. The facts will speak for themselves. C 2002 Elsevier Science (USA) C 2002 Elsevier Science (USA) C 2002 Elsevier Science (USA) 247 0315-0860/02 $35.00 C 2002 Elsevier Science (USA) All rights reserved. 248 JAN P. HOGENDIJK HMAT 29 AMS subject classifications: 01A20, 01A30. Key Words: Ibn al-Haytham; conic sections; multiple editions; Rashed. 1. INTRODUCTION The Conics of Apollonius of Perga (ca. 200 B.C.) is one of the fundamental texts of ancient Greek geometry. -
History of Mathematics in Mathematics Education. Recent Developments Kathy Clark, Tinne Kjeldsen, Sebastian Schorcht, Constantinos Tzanakis, Xiaoqin Wang
History of mathematics in mathematics education. Recent developments Kathy Clark, Tinne Kjeldsen, Sebastian Schorcht, Constantinos Tzanakis, Xiaoqin Wang To cite this version: Kathy Clark, Tinne Kjeldsen, Sebastian Schorcht, Constantinos Tzanakis, Xiaoqin Wang. History of mathematics in mathematics education. Recent developments. History and Pedagogy of Mathematics, Jul 2016, Montpellier, France. hal-01349230 HAL Id: hal-01349230 https://hal.archives-ouvertes.fr/hal-01349230 Submitted on 27 Jul 2016 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. HISTORY OF MATHEMATICS IN MATHEMATICS EDUCATION Recent developments Kathleen CLARK, Tinne Hoff KJELDSEN, Sebastian SCHORCHT, Constantinos TZANAKIS, Xiaoqin WANG School of Teacher Education, Florida State University, Tallahassee, FL 32306-4459, USA [email protected] Department of Mathematical Sciences, University of Copenhagen, Denmark [email protected] Justus Liebig University Giessen, Germany [email protected] Department of Education, University of Crete, Rethymnon 74100, Greece [email protected] Department of Mathematics, East China Normal University, China [email protected] ABSTRACT This is a survey on the recent developments (since 2000) concerning research on the relations between History and Pedagogy of Mathematics (the HPM domain). Section 1 explains the rationale of the study and formulates the key issues. -
By the Persian Mathematician and Engineer Abubakr
1 2 The millennium old hydrogeology textbook “The Extraction of Hidden Waters” by the Persian 3 mathematician and engineer Abubakr Mohammad Karaji (c. 953 – c. 1029) 4 5 Behzad Ataie-Ashtiania,b, Craig T. Simmonsa 6 7 a National Centre for Groundwater Research and Training and College of Science & Engineering, 8 Flinders University, Adelaide, South Australia, Australia 9 b Department of Civil Engineering, Sharif University of Technology, Tehran, Iran, 10 [email protected] (B. Ataie-Ashtiani) 11 [email protected] (C. T. Simmons) 12 13 14 15 Hydrology and Earth System Sciences (HESS) 16 Special issue ‘History of Hydrology’ 17 18 Guest Editors: Okke Batelaan, Keith Beven, Chantal Gascuel-Odoux, Laurent Pfister, and Roberto Ranzi 19 20 1 21 22 Abstract 23 We revisit and shed light on the millennium old hydrogeology textbook “The Extraction of Hidden Waters” by the 24 Persian mathematician and engineer Karaji. Despite the nature of the understanding and conceptualization of the 25 world by the people of that time, ground-breaking ideas and descriptions of hydrological and hydrogeological 26 perceptions such as components of hydrological cycle, groundwater quality and even driving factors for 27 groundwater flow were presented in the book. Although some of these ideas may have been presented elsewhere, 28 to the best of our knowledge, this is the first time that a whole book was focused on different aspects of hydrology 29 and hydrogeology. More importantly, we are impressed that the book is composed in a way that covered all aspects 30 that are related to an engineering project including technical and construction issues, guidelines for maintenance, 31 and final delivery of the project when the development and construction was over. -
The What and Why of Whole Number Arithmetic: Foundational Ideas from History, Language and Societal Changes
Portland State University PDXScholar Mathematics and Statistics Faculty Fariborz Maseeh Department of Mathematics Publications and Presentations and Statistics 3-2018 The What and Why of Whole Number Arithmetic: Foundational Ideas from History, Language and Societal Changes Xu Hu Sun University of Macau Christine Chambris Université de Cergy-Pontoise Judy Sayers Stockholm University Man Keung Siu University of Hong Kong Jason Cooper Weizmann Institute of Science SeeFollow next this page and for additional additional works authors at: https:/ /pdxscholar.library.pdx.edu/mth_fac Part of the Science and Mathematics Education Commons Let us know how access to this document benefits ou.y Citation Details Sun X.H. et al. (2018) The What and Why of Whole Number Arithmetic: Foundational Ideas from History, Language and Societal Changes. In: Bartolini Bussi M., Sun X. (eds) Building the Foundation: Whole Numbers in the Primary Grades. New ICMI Study Series. Springer, Cham This Book Chapter is brought to you for free and open access. It has been accepted for inclusion in Mathematics and Statistics Faculty Publications and Presentations by an authorized administrator of PDXScholar. Please contact us if we can make this document more accessible: [email protected]. Authors Xu Hu Sun, Christine Chambris, Judy Sayers, Man Keung Siu, Jason Cooper, Jean-Luc Dorier, Sarah Inés González de Lora Sued, Eva Thanheiser, Nadia Azrou, Lynn McGarvey, Catherine Houdement, and Lisser Rye Ejersbo This book chapter is available at PDXScholar: https://pdxscholar.library.pdx.edu/mth_fac/253 Chapter 5 The What and Why of Whole Number Arithmetic: Foundational Ideas from History, Language and Societal Changes Xu Hua Sun , Christine Chambris Judy Sayers, Man Keung Siu, Jason Cooper , Jean-Luc Dorier , Sarah Inés González de Lora Sued , Eva Thanheiser , Nadia Azrou , Lynn McGarvey , Catherine Houdement , and Lisser Rye Ejersbo 5.1 Introduction Mathematics learning and teaching are deeply embedded in history, language and culture (e.g. -
History of Mathematics
Georgia Department of Education History of Mathematics K-12 Mathematics Introduction The Georgia Mathematics Curriculum focuses on actively engaging the students in the development of mathematical understanding by using manipulatives and a variety of representations, working independently and cooperatively to solve problems, estimating and computing efficiently, and conducting investigations and recording findings. There is a shift towards applying mathematical concepts and skills in the context of authentic problems and for the student to understand concepts rather than merely follow a sequence of procedures. In mathematics classrooms, students will learn to think critically in a mathematical way with an understanding that there are many different ways to a solution and sometimes more than one right answer in applied mathematics. Mathematics is the economy of information. The central idea of all mathematics is to discover how knowing some things well, via reasoning, permit students to know much else—without having to commit the information to memory as a separate fact. It is the reasoned, logical connections that make mathematics coherent. The implementation of the Georgia Standards of Excellence in Mathematics places a greater emphasis on sense making, problem solving, reasoning, representation, connections, and communication. History of Mathematics History of Mathematics is a one-semester elective course option for students who have completed AP Calculus or are taking AP Calculus concurrently. It traces the development of major branches of mathematics throughout history, specifically algebra, geometry, number theory, and methods of proofs, how that development was influenced by the needs of various cultures, and how the mathematics in turn influenced culture. The course extends the numbers and counting, algebra, geometry, and data analysis and probability strands from previous courses, and includes a new history strand. -
Can One Design a Geometry Engine? on the (Un) Decidability of Affine
Noname manuscript No. (will be inserted by the editor) Can one design a geometry engine? On the (un)decidability of certain affine Euclidean geometries Johann A. Makowsky Received: June 4, 2018/ Accepted: date Abstract We survey the status of decidabilty of the consequence relation in various ax- iomatizations of Euclidean geometry. We draw attention to a widely overlooked result by Martin Ziegler from 1980, which proves Tarski’s conjecture on the undecidability of finitely axiomatizable theories of fields. We elaborate on how to use Ziegler’s theorem to show that the consequence relations for the first order theory of the Hilbert plane and the Euclidean plane are undecidable. As new results we add: (A) The first order consequence relations for Wu’s orthogonal and metric geometries (Wen- Ts¨un Wu, 1984), and for the axiomatization of Origami geometry (J. Justin 1986, H. Huzita 1991) are undecidable. It was already known that the universal theory of Hilbert planes and Wu’s orthogonal geom- etry is decidable. We show here using elementary model theoretic tools that (B) the universal first order consequences of any geometric theory T of Pappian planes which is consistent with the analytic geometry of the reals is decidable. The techniques used were all known to experts in mathematical logic and geometry in the past but no detailed proofs are easily accessible for practitioners of symbolic computation or automated theorem proving. Keywords Euclidean Geometry · Automated Theorem Proving · Undecidability arXiv:1712.07474v3 [cs.SC] 1 Jun 2018 J.A. Makowsky Faculty of Computer Science, Technion–Israel Institute of Technology, Haifa, Israel E-mail: [email protected] 2 J.A. -
20. Geometry of the Circle (SC)
20. GEOMETRY OF THE CIRCLE PARTS OF THE CIRCLE Segments When we speak of a circle we may be referring to the plane figure itself or the boundary of the shape, called the circumference. In solving problems involving the circle, we must be familiar with several theorems. In order to understand these theorems, we review the names given to parts of a circle. Diameter and chord The region that is encompassed between an arc and a chord is called a segment. The region between the chord and the minor arc is called the minor segment. The region between the chord and the major arc is called the major segment. If the chord is a diameter, then both segments are equal and are called semi-circles. The straight line joining any two points on the circle is called a chord. Sectors A diameter is a chord that passes through the center of the circle. It is, therefore, the longest possible chord of a circle. In the diagram, O is the center of the circle, AB is a diameter and PQ is also a chord. Arcs The region that is enclosed by any two radii and an arc is called a sector. If the region is bounded by the two radii and a minor arc, then it is called the minor sector. www.faspassmaths.comIf the region is bounded by two radii and the major arc, it is called the major sector. An arc of a circle is the part of the circumference of the circle that is cut off by a chord. -
Finite Projective Geometries 243
FINITE PROJECTÎVEGEOMETRIES* BY OSWALD VEBLEN and W. H. BUSSEY By means of such a generalized conception of geometry as is inevitably suggested by the recent and wide-spread researches in the foundations of that science, there is given in § 1 a definition of a class of tactical configurations which includes many well known configurations as well as many new ones. In § 2 there is developed a method for the construction of these configurations which is proved to furnish all configurations that satisfy the definition. In §§ 4-8 the configurations are shown to have a geometrical theory identical in most of its general theorems with ordinary projective geometry and thus to afford a treatment of finite linear group theory analogous to the ordinary theory of collineations. In § 9 reference is made to other definitions of some of the configurations included in the class defined in § 1. § 1. Synthetic definition. By a finite projective geometry is meant a set of elements which, for sugges- tiveness, are called points, subject to the following five conditions : I. The set contains a finite number ( > 2 ) of points. It contains subsets called lines, each of which contains at least three points. II. If A and B are distinct points, there is one and only one line that contains A and B. HI. If A, B, C are non-collinear points and if a line I contains a point D of the line AB and a point E of the line BC, but does not contain A, B, or C, then the line I contains a point F of the line CA (Fig. -
Geometry: Euclid and Beyond, by Robin Hartshorne, Springer-Verlag, New York, 2000, Xi+526 Pp., $49.95, ISBN 0-387-98650-2
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 39, Number 4, Pages 563{571 S 0273-0979(02)00949-7 Article electronically published on July 9, 2002 Geometry: Euclid and beyond, by Robin Hartshorne, Springer-Verlag, New York, 2000, xi+526 pp., $49.95, ISBN 0-387-98650-2 1. Introduction The first geometers were men and women who reflected on their experiences while doing such activities as building small shelters and bridges, making pots, weaving cloth, building altars, designing decorations, or gazing into the heavens for portentous signs or navigational aides. Main aspects of geometry emerged from three strands of early human activity that seem to have occurred in most cultures: art/patterns, navigation/stargazing, and building structures. These strands developed more or less independently into varying studies and practices that eventually were woven into what we now call geometry. Art/Patterns: To produce decorations for their weaving, pottery, and other objects, early artists experimented with symmetries and repeating patterns. Later the study of symmetries of patterns led to tilings, group theory, crystallography, finite geometries, and in modern times to security codes and digital picture com- pactifications. Early artists also explored various methods of representing existing objects and living things. These explorations led to the study of perspective and then projective geometry and descriptive geometry, and (in the 20th century) to computer-aided graphics, the study of computer vision in robotics, and computer- generated movies (for example, Toy Story ). Navigation/Stargazing: For astrological, religious, agricultural, and other purposes, ancient humans attempted to understand the movement of heavenly bod- ies (stars, planets, Sun, and Moon) in the apparently hemispherical sky. -
SOME GEOMETRY in HIGH-DIMENSIONAL SPACES 11 Containing Cn(S) Tends to ∞ with N
SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES MATH 527A 1. Introduction Our geometric intuition is derived from three-dimensional space. Three coordinates suffice. Many objects of interest in analysis, however, require far more coordinates for a complete description. For example, a function f with domain [−1; 1] is defined by infinitely many \coordi- nates" f(t), one for each t 2 [−1; 1]. Or, we could consider f as being P1 n determined by its Taylor series n=0 ant (when such a representation exists). In that case, the numbers a0; a1; a2;::: could be thought of as coordinates. Perhaps the association of Fourier coefficients (there are countably many of them) to a periodic function is familiar; those are again coordinates of a sort. Strange Things can happen in infinite dimensions. One usually meets these, gradually (reluctantly?), in a course on Real Analysis or Func- tional Analysis. But infinite dimensional spaces need not always be completely mysterious; sometimes one lucks out and can watch a \coun- terintuitive" phenomenon developing in Rn for large n. This might be of use in one of several ways: perhaps the behavior for large but finite n is already useful, or one can deduce an interesting statement about limn!1 of something, or a peculiarity of infinite-dimensional spaces is illuminated. I will describe some curious features of cubes and balls in Rn, as n ! 1. These illustrate a phenomenon called concentration of measure. It will turn out that the important law of large numbers from probability theory is just one manifestation of high-dimensional geometry. -
Relativistic Spacetime Structure
Relativistic Spacetime Structure Samuel C. Fletcher∗ Department of Philosophy University of Minnesota, Twin Cities & Munich Center for Mathematical Philosophy Ludwig Maximilian University of Munich August 12, 2019 Abstract I survey from a modern perspective what spacetime structure there is according to the general theory of relativity, and what of it determines what else. I describe in some detail both the “standard” and various alternative answers to these questions. Besides bringing many underexplored topics to the attention of philosophers of physics and of science, metaphysicians of science, and foundationally minded physicists, I also aim to cast other, more familiar ones in a new light. 1 Introduction and Scope In the broadest sense, spacetime structure consists in the totality of relations between events and processes described in a spacetime theory, including distance, duration, motion, and (more gener- ally) change. A spacetime theory can attribute more or less such structure, and some parts of that structure may determine other parts. The nature of these structures and their relations of determi- nation bear on the interpretation of the theory—what the world would be like if the theory were true (North, 2009). For example, the structures of spacetime might be taken as its ontological or conceptual posits, and the determination relations might indicate which of these structures is more fundamental (North, 2018). Different perspectives on these questions might also reveal structural similarities with other spacetime theories, providing the resources to articulate how the picture of the world that that theory provides is different (if at all) from what came before, and might be different from what is yet to come.1 ∗Juliusz Doboszewski, Laurenz Hudetz, Eleanor Knox, J.