Understanding the Estimation of Circumference of the Earth by of Eratosthenes Based on the History of Science, for Earth Science Education
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Analytic Geometry
Guide Study Georgia End-Of-Course Tests Georgia ANALYTIC GEOMETRY TABLE OF CONTENTS INTRODUCTION ...........................................................................................................5 HOW TO USE THE STUDY GUIDE ................................................................................6 OVERVIEW OF THE EOCT .........................................................................................8 PREPARING FOR THE EOCT ......................................................................................9 Study Skills ........................................................................................................9 Time Management .....................................................................................10 Organization ...............................................................................................10 Active Participation ...................................................................................11 Test-Taking Strategies .....................................................................................11 Suggested Strategies to Prepare for the EOCT ..........................................12 Suggested Strategies the Day before the EOCT ........................................13 Suggested Strategies the Morning of the EOCT ........................................13 Top 10 Suggested Strategies during the EOCT .........................................14 TEST CONTENT ........................................................................................................15 -
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. -
Implementing Eratosthenes' Discovery in the Classroom: Educational
Implementing Eratosthenes’ Discovery in the Classroom: Educational Difficulties Needing Attention Nicolas Decamp, C. de Hosson To cite this version: Nicolas Decamp, C. de Hosson. Implementing Eratosthenes’ Discovery in the Classroom: Educational Difficulties Needing Attention. Science and Education, Springer Verlag, 2012, 21 (6), pp.911-920. 10.1007/s11191-010-9286-3. hal-01663445 HAL Id: hal-01663445 https://hal.archives-ouvertes.fr/hal-01663445 Submitted on 18 Dec 2017 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. Sci & Educ DOI 10.1007/s11191-010-9286-3 Implementing Eratosthenes’ Discovery in the Classroom: Educational Difficulties Needing Attention Nicolas De´camp • Ce´cile de Hosson Ó Springer Science+Business Media B.V. 2010 Abstract This paper presents a critical analysis of the accepted educational use of the method performed by Eratosthenes to measure the circumference of Earth which is often considered as a relevant means of dealing with issues related to the nature of science and its history. This method relies on a number of assumptions among which the parallelism of sun rays. The assumption of sun rays parallelism (if it is accurate) does not appear spontaneous for students who consider sun rays to be divergent. -
10-1 Study Guide and Intervention Circles and Circumference
NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 10-1 Study Guide and Intervention Circles and Circumference Segments in Circles A circle consists of all points in a plane that are a given distance, called the radius, from a given point called the center. A segment or line can intersect a circle in several ways. • A segment with endpoints that are at the center and on the circle is a radius. • A segment with endpoints on the circle is a chord. • A chord that passes through the circle’s center and made up of collinear radii is a diameter. chord: 퐴퐸̅̅̅̅, 퐵퐷̅̅̅̅ radius: 퐹퐵̅̅̅̅, 퐹퐶̅̅̅̅, 퐹퐷̅̅̅̅ For a circle that has radius r and diameter d, the following are true diameter: 퐵퐷̅̅̅̅ 푑 1 r = r = d d = 2r 2 2 Example a. Name the circle. The name of the circle is ⨀O. b. Name radii of the circle. 퐴푂̅̅̅̅, 퐵푂̅̅̅̅, 퐶푂̅̅̅̅, and 퐷푂̅̅̅̅ are radii. c. Name chords of the circle. 퐴퐵̅̅̅̅ and 퐶퐷̅̅̅̅ are chords. Exercises For Exercises 1-7, refer to 1. Name the circle. ⨀R 2. Name radii of the circle. 푹푨̅̅̅̅, 푹푩̅̅̅̅, 푹풀̅̅̅̅, and 푹푿̅̅̅̅ 3. Name chords of the circle. ̅푩풀̅̅̅, 푨푿̅̅̅̅, 푨푩̅̅̅̅, and 푿풀̅̅̅̅ 4. Name diameters of the circle. 푨푩̅̅̅̅, and 푿풀̅̅̅̅ 5. If AB = 18 millimeters, find AR. 9 mm 6. If RY = 10 inches, find AR and AB . AR = 10 in.; AB = 20 in. 7. Is 퐴퐵̅̅̅̅ ≅ 푋푌̅̅̅̅? Explain. Yes; all diameters of the same circle are congruent. Chapter 10 5 Glencoe Geometry NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 10-1 Study Guide and Intervention (continued) Circles and Circumference Circumference The circumference of a circle is the distance around the circle. -
Astronomy and Hipparchus
CHAPTER 9 Progress in the Sciences: Astronomy and Hipparchus Klaus Geus Introduction Geography in modern times is a term which covers several sub-disciplines like ecology, human geography, economic history, volcanology etc., which all con- cern themselves with “space” or “environment”. In ancient times, the definition of geography was much more limited. Geography aimed at the production of a map of the oikoumene, a geographer was basically a cartographer. The famous scientist Ptolemy defined geography in the first sentence of his Geographical handbook (Geog. 1.1.1) as “imitation through drafting of the entire known part of the earth, including the things which are, generally speaking, connected with it”. In contrast to chorography, geography uses purely “lines and label in order to show the positions of places and general configurations” (Geog. 1.1.5). Therefore, according to Ptolemy, a geographer needs a μέθοδος μαθεματική, abil- ity and competence in mathematical sciences, most prominently astronomy, in order to fulfil his task of drafting a map of the oikoumene. Given this close connection between geography and astronomy, it is not by default that nearly all ancient “geographers” (in the limited sense of the term) stood out also as astronomers and mathematicians: Among them Anaximander, Eudoxus, Eratosthenes, Hipparchus, Poseidonius and Ptolemy are the most illustrious. Apart from certain topics like latitudes, meridians, polar circles etc., ancient geography also took over from astronomy some methods like the determina- tion of the size of the earth or of celestial and terrestrial distances.1 The men- tioned geographers Anaximander, Eudoxus, Hipparchus, Poseidonius and Ptolemy even constructed instruments for measuring, observing and calculat- ing like the gnomon, sundials, skaphe, astrolabe or the meteoroscope.2 1 E.g., Ptolemy (Geog. -
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. -
Geometry Course Outline
GEOMETRY COURSE OUTLINE Content Area Formative Assessment # of Lessons Days G0 INTRO AND CONSTRUCTION 12 G-CO Congruence 12, 13 G1 BASIC DEFINITIONS AND RIGID MOTION Representing and 20 G-CO Congruence 1, 2, 3, 4, 5, 6, 7, 8 Combining Transformations Analyzing Congruency Proofs G2 GEOMETRIC RELATIONSHIPS AND PROPERTIES Evaluating Statements 15 G-CO Congruence 9, 10, 11 About Length and Area G-C Circles 3 Inscribing and Circumscribing Right Triangles G3 SIMILARITY Geometry Problems: 20 G-SRT Similarity, Right Triangles, and Trigonometry 1, 2, 3, Circles and Triangles 4, 5 Proofs of the Pythagorean Theorem M1 GEOMETRIC MODELING 1 Solving Geometry 7 G-MG Modeling with Geometry 1, 2, 3 Problems: Floodlights G4 COORDINATE GEOMETRY Finding Equations of 15 G-GPE Expressing Geometric Properties with Equations 4, 5, Parallel and 6, 7 Perpendicular Lines G5 CIRCLES AND CONICS Equations of Circles 1 15 G-C Circles 1, 2, 5 Equations of Circles 2 G-GPE Expressing Geometric Properties with Equations 1, 2 Sectors of Circles G6 GEOMETRIC MEASUREMENTS AND DIMENSIONS Evaluating Statements 15 G-GMD 1, 3, 4 About Enlargements (2D & 3D) 2D Representations of 3D Objects G7 TRIONOMETRIC RATIOS Calculating Volumes of 15 G-SRT Similarity, Right Triangles, and Trigonometry 6, 7, 8 Compound Objects M2 GEOMETRIC MODELING 2 Modeling: Rolling Cups 10 G-MG Modeling with Geometry 1, 2, 3 TOTAL: 144 HIGH SCHOOL OVERVIEW Algebra 1 Geometry Algebra 2 A0 Introduction G0 Introduction and A0 Introduction Construction A1 Modeling With Functions G1 Basic Definitions and Rigid -
Find the Radius Or Diameter of the Circle with the Given Dimensions. 2
8-1 Circumference Find the radius or diameter of the circle with the given dimensions. 2. d = 24 ft SOLUTION: The radius is 12 feet. ANSWER: 12 ft Find the circumference of the circle. Use 3.14 or for π. Round to the nearest tenth if necessary. 4. SOLUTION: The circumference of the circle is about 25.1 feet. ANSWER: 3.14 × 8 = 25.1 ft eSolutions Manual - Powered by Cognero Page 1 8-1 Circumference 6. SOLUTION: The circumference of the circle is about 22 miles. ANSWER: 8. The Belknap shield volcano is located in the Cascade Range in Oregon. The volcano is circular and has a diameter of 5 miles. What is the circumference of this volcano to the nearest tenth? SOLUTION: The circumference of the volcano is about 15.7 miles. ANSWER: 15.7 mi eSolutions Manual - Powered by Cognero Page 2 8-1 Circumference Copy and Solve Show your work on a separate piece of paper. Find the diameter given the circumference of the object. Use 3.14 for π. 10. a satellite dish with a circumference of 957.7 meters SOLUTION: The diameter of the satellite dish is 305 meters. ANSWER: 305 m 12. a nickel with a circumference of about 65.94 millimeters SOLUTION: The diameter of the nickel is 21 millimeters. ANSWER: 21 mm eSolutions Manual - Powered by Cognero Page 3 8-1 Circumference Find the distance around the figure. Use 3.14 for π. 14. SOLUTION: The distance around the figure is 5 + 5 or 10 feet plus of the circumference of a circle with a radius of 5 feet. -
3.1 the Robertson-Walker Metric
M. Pettini: Introduction to Cosmology | Lecture 3 RELATIVISTIC COSMOLOGY 3.1 The Robertson-Walker Metric The appearance of objects at cosmological distances is affected by the curvature of spacetime through which light travels on its way to Earth. The most complete description of the geometrical properties of the Universe is provided by Einstein's general theory of relativity. In GR, the fundamental quantity is the metric which describes the geometry of spacetime. Let's look at the definition of a metric: in 3-D space we measure the distance along a curved path P between two points using the differential distance formula, or metric: (d`)2 = (dx)2 + (dy)2 + (dz)2 (3.1) and integrating along the path P (a line integral) to calculate the total distance: Z 2 q Z 2 q ∆` = (d`)2 = (dx)2 + (dy)2 + (dz)2 (3.2) 1 1 Similarly, to measure the interval along a curved wordline, W, connecting two events in spacetime with no mass present, we use the metric for flat spacetime: (ds)2 = (cdt)2 − (dx)2 − (dy)2 − (dz)2 (3.3) Integrating ds gives the total interval along the worldline W: Z B q Z B q ∆s = (ds)2 = (cdt)2 − (dx)2 − (dy)2 − (dz)2 (3.4) A A By definition, the distance measured between two events, A and B, in a reference frame for which they occur simultaneously (tA = tB) is the proper distance: q ∆L = −(∆s)2 (3.5) Our search for a metric that describes the spacetime of a matter-filled universe, is made easier by the cosmological principle. -
Hyperbolic Geometry
Flavors of Geometry MSRI Publications Volume 31,1997 Hyperbolic Geometry JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON, AND WALTER R. PARRY Contents 1. Introduction 59 2. The Origins of Hyperbolic Geometry 60 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65 5. Generalizing to Higher Dimensions 67 6. Rudiments of Riemannian Geometry 68 7. Five Models of Hyperbolic Space 69 8. Stereographic Projection 72 9. Geodesics 77 10. Isometries and Distances in the Hyperboloid Model 80 11. The Space at Infinity 84 12. The Geometric Classification of Isometries 84 13. Curious Facts about Hyperbolic Space 86 14. The Sixth Model 95 15. Why Study Hyperbolic Geometry? 98 16. When Does a Manifold Have a Hyperbolic Structure? 103 17. How to Get Analytic Coordinates at Infinity? 106 References 108 Index 110 1. Introduction Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one of Euclid’s axioms. Einstein and Minkowski found in non-Euclidean geometry a This work was supported in part by The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc., by the Mathematical Sciences Research Institute, and by NSF research grants. 59 60 J. W. CANNON, W. J. FLOYD, R. KENYON, AND W. R. PARRY geometric basis for the understanding of physical time and space. In the early part of the twentieth century every serious student of mathematics and physics studied non-Euclidean geometry. -
1.1 Eratosthenes Measures the Earth (Copyright: Bryan Dorner All Rights Reserved)
1.1 Eratosthenes Measures the Earth (Copyright: Bryan Dorner all rights reserved) How big is the Earth? The mathematician Eratosthenes (276-195 BCE) lived in the city of Alexandria in northern Egypt near the place where the Nile river empties into the Mediterranean. Eratosthenes was chief librarian at the Alexandria museum and one of the foremost scholars of the day - second only to Archimedes (who many consider one of the two or three best mathematicians ever to have lived). The city of Alexandria had been founded about one hundred years earlier by Alexander the Great whose conquests stretched from Egypt, through Syria, Babylonia, and Persia to northern India and central Asia. As the Greeks were also called Hellenes, the resulting empire was known as the Hellenistic empire and the following period in which Greek culture was dominant is called the Hellenistic age. The Hellenistic age saw a considerable exchange of culture between the conquering Greeks and the civilizations of the lands they controlled. It is from about this period that the predominantly geometric mathematics of the Greeks shows a computational aspect borrowed from the Babylonians. Some of the best mathematics we have inherited comes from just such a blend of contributions from diverse cultures. Eratosthenes is known for his simple but accurate measurement of the size of the earth. The imprint of Babylon (modern Iraq) as well as Greece can be seen in his method. He did not divide the arc of a circle into 360 parts as the Babylonians did, but into 60 equal parts. Still, the use of 60 reveals the influence of the Babylonian number system - the sexagesimal system - which was based on the number 60 in the same way ours is based on the number 10. -
9 · the Growth of an Empirical Cartography in Hellenistic Greece
9 · The Growth of an Empirical Cartography in Hellenistic Greece PREPARED BY THE EDITORS FROM MATERIALS SUPPLIED BY GERMAINE AUJAe There is no complete break between the development of That such a change should occur is due both to po cartography in classical and in Hellenistic Greece. In litical and military factors and to cultural developments contrast to many periods in the ancient and medieval within Greek society as a whole. With respect to the world, we are able to reconstruct throughout the Greek latter, we can see how Greek cartography started to be period-and indeed into the Roman-a continuum in influenced by a new infrastructure for learning that had cartographic thought and practice. Certainly the a profound effect on the growth of formalized know achievements of the third century B.C. in Alexandria had ledge in general. Of particular importance for the history been prepared for and made possible by the scientific of the map was the growth of Alexandria as a major progress of the fourth century. Eudoxus, as we have seen, center of learning, far surpassing in this respect the had already formulated the geocentric hypothesis in Macedonian court at Pella. It was at Alexandria that mathematical models; and he had also translated his Euclid's famous school of geometry flourished in the concepts into celestial globes that may be regarded as reign of Ptolemy II Philadelphus (285-246 B.C.). And it anticipating the sphairopoiia. 1 By the beginning of the was at Alexandria that this Ptolemy, son of Ptolemy I Hellenistic period there had been developed not only the Soter, a companion of Alexander, had founded the li various celestial globes, but also systems of concentric brary, soon to become famous throughout the Mediter spheres, together with maps of the inhabited world that ranean world.