Mysteries and Discoveries of Archaeoastronomy Mysteries and Discoveries of Archaeoastronomy
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The Loughcrew Hills and Passage Tomb Complex
Technological University Dublin ARROW@TU Dublin Books/Book Chapters Spatial Information Sciences Research Group 2011 The Loughcrew Hills and Passage Tomb Complex Frank Prendergast Technological University Dublin, [email protected] Follow this and additional works at: https://arrow.tudublin.ie/dsisbk Part of the Other Engineering Commons Recommended Citation Prendergast, F. (2011) ''The Loughcrew Hills and Passage Tomb Complex''. Stefanini, B. & Glynn, G.M. (Eds.) Field Guide No. 29 - North Meath. Irish Quaternary Association. pp 42 -54. This Book Chapter is brought to you for free and open access by the Spatial Information Sciences Research Group at ARROW@TU Dublin. It has been accepted for inclusion in Books/Book Chapters by an authorized administrator of ARROW@TU Dublin. For more information, please contact [email protected], [email protected]. This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License FIELD SITES The Loughcrew Hills and Passage Tomb Complex Frank Prendergast Introduction The Loughcrew Hills are a 5 km long by 1 km wide linear formation trending ENE-WSW and located 5 km southeast of the town of Oldcastle, County Meath. From the Ordnance Survey Ireland map of the area (Sheet 42 Discovery Series), six peaks can be discerned on its ridges. These range in altitude from 237 m AOD at the eastern end to ·the 254 m high summit of Carbane West at the other end of the range. Carbane East is the dominant peak with an altitude of 276 m AOD. Geologically, the main rock types of the formation are sandstone, greywacke and shale and are dated to the Silurian Period (440-410 million years ago). -
Ethnomathematics and Education in Africa
Copyright ©2014 by Paulus Gerdes www.lulu.com http://www.lulu.com/spotlight/pgerdes 2 Paulus Gerdes Second edition: ISTEG Belo Horizonte Boane Mozambique 2014 3 First Edition (January 1995): Institutionen för Internationell Pedagogik (Institute of International Education) Stockholms Universitet (University of Stockholm) Report 97 Second Edition (January 2014): Instituto Superior de Tecnologias e Gestão (ISTEG) (Higher Institute for Technology and Management) Av. de Namaacha 188, Belo Horizonte, Boane, Mozambique Distributed by: www.lulu.com http://www.lulu.com/spotlight/pgerdes Author: Paulus Gerdes African Academy of Sciences & ISTEG, Mozambique C.P. 915, Maputo, Mozambique ([email protected]) Photograph on the front cover: Detail of a Tonga basket acquired, in January 2014, by the author in Inhambane, Mozambique 4 CONTENTS page Preface (2014) 11 Chapter 1: Introduction 13 Chapter 2: Ethnomathematical research: preparing a 19 response to a major challenge to mathematics education in Africa Societal and educational background 19 A major challenge to mathematics education 21 Ethnomathematics Research Project in Mozambique 23 Chapter 3: On the concept of ethnomathematics 29 Ethnographers on ethnoscience 29 Genesis of the concept of ethnomathematics among 31 mathematicians and mathematics teachers Concept, accent or movement? 34 Bibliography 39 Chapter 4: How to recognize hidden geometrical thinking: 45 a contribution to the development of an anthropology of mathematics Confrontation 45 Introduction 46 First example 47 Second example -
Mathematics in African History and Cultures
Paulus Gerdes & Ahmed Djebbar MATHEMATICS IN AFRICAN HISTORY AND CULTURES: AN ANNOTATED BIBLIOGRAPHY African Mathematical Union Commission on the History of Mathematics in Africa (AMUCHMA) Mathematics in African History and Cultures Second edition, 2007 First edition: African Mathematical Union, Cape Town, South Africa, 2004 ISBN: 978-1-4303-1537-7 Published by Lulu. Copyright © 2007 by Paulus Gerdes & Ahmed Djebbar Authors Paulus Gerdes Research Centre for Mathematics, Culture and Education, C.P. 915, Maputo, Mozambique E-mail: [email protected] Ahmed Djebbar Département de mathématiques, Bt. M 2, Université de Lille 1, 59655 Villeneuve D’Asq Cedex, France E-mail: [email protected], [email protected] Cover design inspired by a pattern on a mat woven in the 19th century by a Yombe woman from the Lower Congo area (Cf. GER-04b, p. 96). 2 Table of contents page Preface by the President of the African 7 Mathematical Union (Prof. Jan Persens) Introduction 9 Introduction to the new edition 14 Bibliography A 15 B 43 C 65 D 77 E 105 F 115 G 121 H 162 I 173 J 179 K 182 L 194 M 207 N 223 O 228 P 234 R 241 S 252 T 274 U 281 V 283 3 Mathematics in African History and Cultures page W 290 Y 296 Z 298 Appendices 1 On mathematicians of African descent / 307 Diaspora 2 Publications by Africans on the History of 313 Mathematics outside Africa (including reviews of these publications) 3 On Time-reckoning and Astronomy in 317 African History and Cultures 4 String figures in Africa 338 5 Examples of other Mathematical Books and 343 -
Writing the History of Mathematics: Interpretations of the Mathematics of the Past and Its Relation to the Mathematics of Today
Writing the History of Mathematics: Interpretations of the Mathematics of the Past and Its Relation to the Mathematics of Today Johanna Pejlare and Kajsa Bråting Contents Introduction.................................................................. 2 Traces of Mathematics of the First Humans........................................ 3 History of Ancient Mathematics: The First Written Sources........................... 6 History of Mathematics or Heritage of Mathematics?................................. 9 Further Views of the Past and Its Relation to the Present.............................. 15 Can History Be Recapitulated or Does Culture Matter?............................... 19 Concluding Remarks........................................................... 24 Cross-References.............................................................. 24 References................................................................... 24 Abstract In the present chapter, interpretations of the mathematics of the past are problematized, based on examples such as archeological artifacts, as well as written sources from the ancient Egyptian, Babylonian, and Greek civilizations. The distinction between history and heritage is considered in relation to Euler’s function concept, Cauchy’s sum theorem, and the Unguru debate. Also, the distinction between the historical past and the practical past,aswellasthe distinction between the historical and the nonhistorical relations to the past, are made concrete based on Torricelli’s result on an infinitely long solid from -
Curriculum, Assessment, and Instruction Dr
May 2018 Curriculum, Assessment, and Instruction Dr. Fatima Morrell, Assistant Superintendent Our Vision: Through a commitment to equity and excellence, all students will receive a rigorous instructional program which prepares them to compete successfully, and contribute responsibly in a global society. Assistant Superintendent Highlights “Mathematics is Everywhere” Take a look around and math is everywhere. From finding the best deal while online shopping to calculating our students’ grades to cooking tonight’s dinner. When we think about math we tend to think about numbers and operations, fractions and decimals, standard algorithms and complicated formulas. But when was the last time you stopped to think about the stories behind the math? The who, the what, the where, the when? Stories connect us. They help us to create a deeper understanding and discover where we fit into the big picture. Have your students ever asked, or maybe you, yourself have wondered, “Why does this matter? When am I ever going to use this?” It’s these questions that show us how critical it is for students to understand both the history of math and the future of math. There are so many people that label the way students today are learning math as the “new way of doing math.” But how “new” is this math? The Ishango bone from Ancient Africa is one of the earliest artifacts of arithmetic, dating back 20,000 years ago. Many similarities can be found when comparing our base-10 number system to the base-20 number system Ancient Mayans used. Using shells to represent zero, dots to represent ones, and sticks to represent a groups of five, Mayans were able to calculate mathematical problems such as the length of the solar year. -
Introduction 7 the Ishango Bone 10 210 Holy Days 11 210 Frustrum 12
The Josephus Problem 69 239 The Miner 104 256 The Explorer’s Problem 70 239 The Blind Abbot 105 256 Monkey Nuts 71 240 The Captive Queen 106 257 The Book of Precious Things 72 240 A Spiral Walk 107 257 A Medieval Riddle 73 241 Eight Queens 108 258 Introduction 7 The Luo River Scroll 39 224 The Mariner 74 241 The Dinner Party 109 258 The Ishango Bone 10 210 Buridan’s Ass 40 224 The Memory Wheel 75 242 The Monkey and the Pulley 110 259 Holy Days 11 210 Hi Shi’s Third Paradox 41 225 Jia Xian’s Triangle 76 242 Kirkman’s Schoolgirls 111 259 Frustrum 12 211 The Zero Proof 42 225 The Old One 77 243 The Counterfeit Bill 112 260 Triangles of Babylon 13 211 Crocodile Tears 43 226 The Trouble With Rabbits 78 243 The Travelling Salesman 113 260 Ahmes' Loaves 14 212 The Ladder of Horus 44 226 The Ring Game 79 244 Ethiopian Mathematics 114 261 As I was going to Amenemhet III's 15 212 The Sieve of Eratosthenes 45 227 The Well 80 244 Cantor’s Infinities 115 261 A Question of Quantity 16 213 Archimedes’ Revenge 46 228 Tartaglia’s Wine 81 245 Nobody 116 262 A Fractional Issue 17 213 The Nine Chapters 47 229 Topsy-Turvy 82 245 Tesseract 117 262 Strong Grain 18 214 The Cistern Problem 48 229 The Wanderer 83 246 Bertrand’s Box 118 262 Progressive Loaves 19 214 Dog and Hare 49 230 The Hound 84 246 Nothing Lost 119 263 Dates 20 215 The Chickens 50 230 Regiomontanus’ Angle 85 246 Hilbert’s Hotel 120 263 The Rule of Three 21 215 Leg and Thigh 51 231 The Problem of Points 86 247 Wine/Water Problem 121 264 Progressive Shares 22 216 Men Buy a Horse 52 231 Modesty -
Discover Ireland's Rich Heritage!
Free Guide Discover Ireland’s rich heritage! FOR MORE INFORMATION GO TO WWW.DISCOVERIRELAND.IE/BOYNEVALLEY 1 To Belfast (120km from Drogheda) Discover Ireland’s Ardee rich heritage! N2 M1 Oldcastle 6 12 14 13 Slane 7 4 KELLS 8 M3 Brú na Bóinne 15 Newgrange NAVAN Athboy N2 9 11 10 TRIM M3 2 FOR MORE INFORMATION GO TO WWW.DISCOVERIRELAND.IE/BOYNEVALLEY KEY 01 Millmount Museum Royal Site 02 St Peter’s Church, Drogheda To Belfast (120km from Drogheda) Monastery 03 Beaulieu House Megalithic Tomb 04 Battle of the Boyne Church 05 Mellifont Abbey Battle Site 06 Monasterboice Castle Dunleer Slane Castle Tower 07 Period House 08 Brú na Bóinne (Newgrange) M1 09 Hill of Tara 10 Trim Castle 6 11 Trim Heritage Town 3 12 Kells Heritage Town Round Tower 5 2 DROGHEDA & High Crosses 16 13 Loughcrew Gardens 4 1 14 Loughcrew Cairns 15 Navan County Town Brú na Bóinne 16 Drogheda Walled Town Newgrange M1 Belfast N2 M1 The Boyne Area Dublin To Dublin (50km from Drogheda) 2 FOR MORE INFORMATION GO TO WWW.DISCOVERIRELAND.IE/BOYNEVALLEY Discover Ireland’s rich heritage! Map No. Page No. Introduction 04 Archaeological & Historical Timeline 06 01 Millmount Museum & Martello Tower 08 02 St. Peter’s Church (Shrine of St. Oliver Plunkett) 10 03 Beaulieu House 12 04 Battle of the Boyne Site 14 05 Old Mellifont Abbey 16 06 Monasterboice Round Tower & High Crosses 18 07 Slane Castle 22 08 Brú na Bóinne (Newgrange & Knowth) 24 09 Hill of Tara 26 10 Trim Castle 28 11 Trim (Heritage Town) 30 12 Kells Round Tower & High Crosses 32 13 14 Loughcrew Cairns & Garden 34 15 Navan (County Town) 36 16 Drogheda (Walled Town) 38 Myths & Legends 40 Suggested Itinerary 1,2 & 3 46 Your Road map 50 Every care has been taken to ensure accuracy in the completion of this brochure. -
Discover Ishango.Pdf
HAVE YOU HEARD OF ISHANGO? CONTENTS GEOGRAPHICAL CONTEXT 5 5 Virunga National Park 8 Lake Edward THE ISHANGO ARCHAEOLOGICAL SITE 9 10 Rock Layers (Stratigraphy) 12 Dating the Layers 12 Archaeological Remains 12 Human Remains 13 Who Lived in Ishango 25,000 Years Ago? 13 What was Life Like in Ishango? 14 Harpoons and Barbed Points 17 Quartz Tools THE ISHANGO CARVED BONE 18 18 What Is It? 19 What’s So Special about It? 19 Theories 20 Counting Sticks Jean de Heinzelin on the bank of Lake Edward in 1950 GEOGRAPHICAL CONTEXT 5 GEOGRAPHICAL CONTEXT THE VIRUNGA NATIONAL PARK Ishango, that’s a literary competition? The name of a small bone? An oratorio? A symbol of scientific research? It’s all of those things, but originally it was an African village! 3 2 1 4 Ishango is in the north-east of the Democratic Republic of Congo (formally known as Zaire), in the North-Kivu province, in the Virunga National Park Political map of Africa: (virunga.org). The park’s 790,000 hectares cover more than 13% of the province. The Virunga Park 1 Democratic Republic of Congo 2 Republic of Congo was created in 1925, making it the oldest protected 3 Uganda park in Africa. Formally known as the Albert Park, it 4 Rwanda was created to protect mountain gorillas who were threatened by poachers in the twenties. This agree- ment also ensured that the park was protected from deforestation. The park was classed as a UNESCO world heritage site in 1979, and was reclassified as ‘heritage in danger’ in 1994, because of the Rwandan genoci- de. -
Mathematics Through Millenia - V.L
MATHEMATICS: CONCEPTS, AND FOUNDATIONS – Vol. I - Mathematics Through Millenia - V.L. Hansen MATHEMATICS THROUGH MILLENIA V.L. Hansen Department of Mathematics, Technical University of Denmark, Denmark Keywords: History of mathematics, mathematics in various civilizations, mathematicians, applications of mathematics, abstraction in mathematics. Contents 1. Introduction 2. The dawn of mathematics 2.1. Egyptian Mathematics 2.2. Mesopotamian Mathematics 2.3. Mayan Mathematics 3. The Greek heritage in mathematics 3.1. Geometry 3.2. Number Theory 4. The golden period of the Hindus and the Arabs in mathematics 4.1. Hindu Mathematics 4.2. Islamic Mathematics 4.3. Mathematics in Europe in the Middle Ages 5. Mathematics in China 5.1. Ancient Chinese Mathematics 5.2. The “Nine Chapters on the Mathematical Art” 5.3. In the Shadows of the Great Masters 5.4. A Golden Century for Mathematics in China 6. European mathematics in the Renaissance 6.1. The Solution of Cubic Equations 6.2. Mathematics inspired by Applications 7. Mathematics and the scientific revolution 7.1. Analytic Geometry 7.2. Calculus gets off the Ground 7.3. Other Mathematical Discoveries from the Seventeenth Century 8. The tools of calculus are developed and consolidated 8.1. The UNESCOBirth of Mathematical Analysis – EOLSS 8.2. Further Remarks on Mathematics in the Eighteenth Century 9. Abstract mathematical structures emerge 9.1. New AlgebraicSAMPLE Structures CHAPTERS 9.2. Groundbreaking New Discoveries in Geometry 9.3. Rigor in Analysis 9.4. Further Developments in the Nineteenth Century 10. Mathematics in the twentieth century 10.1. Problems in the Foundations of Set Theory 10.2. Tendencies in Twentieth Century Mathematics 10.3. -
AMUCHMA LIVRE BIBLIO Fr Complet
Paulus GERDES & Ahmed DJEBBAR LES MATHEMATIQUES DANS L'HISTOIRE ET LES CULTURES AFRICAINES Une bibliographie annotée Union Mathématique Africaine 2004 Edition française : Unions Mathématique Africaine Commission Africaine d’Histoire des Mathématiques (AMUCHMA) U.F.R. de Mathématiques, Université des Sciences et des Technologies de Lille, 2007. Editions anglaises : Première édition : Africain Mathematical Union, Cape Town, Afrique du Sud, 2004. Seconde édition : Lulu Editeur, 2007. Les auteurs : Paulus GERDES Research Centre for Mathematics, Culture and Education, C.P. 915, Maputo, Mozambique. Tel. : +258 1 49 45 04 E-mail : [email protected] Ahmed DJEBBAR U.F.R. de Mathématiques, Bt. M2 Université des Sciences et des Technologies de Lille 59655 Villeneuve d’Asq Cedex, France Tel. : + 33 1 45 33 74 65 E-mail : [email protected] [email protected] 2 Table des matières PAGES Préface du Président de l'Union Mathématique Africaine (Pr. 5 Jan Persens) Présentation 6 Bibliographie A 9 B 30 C 49 D 58 E 77 F 87 G 92 H 123 I 131 J 136 K 139 L 149 M 160 N 172 O 175 P 180 R 186 S 196 T 214 U 221 V 223 W 228 Y 235 Z 237 Appendices 1 Sur des mathématiciens d'origine africaine/de la Diaspora 245 2 Publications d'auteurs africains sur l'histoire des mathématiques 250 en dehors de l'Afrique (incluant les comptes-rendus de ces publications) 3 PAGES 3 Sur le calcul du temps et l'astronomie dans l'histoire et les 253 cultures africaines 4 Les jeux de ficelle en Afrique 271 5 Exemples de livres et de livrets publiés par des mathématiciens -
Neolithic Passage Tomb Art Around the Irish Sea Iconography and Spatial Organisation
UNIVERSITÉ DE NANTES UFR HISTOIRE, HISTOIRE DE L'ART & ARCHÉOLOGIE Year 2008 Number assigned by the library esis to obtain the degree of DOCTEUR DE L'UNIVERSITÉ DE NANTES Discipline : Archaeology Presented and defended in public by Guillaume Robin on the 4th of November 2008 Title : Neolithic passage tomb art around the Irish Sea Iconography and spatial organisation Volume : text Supervisors: Mr Serge Cassen (Chargé de Recherche, CNRS) Mr Muiris O'Sullivan (Professor, University College Dublin) International PhD cotutelle convention JURY Mr Serge Cassen Chargé de Recherche, CNRS Supervisor Mr Muiris O'Sullivan Professor, University College Dublin Supervisor Mr André D'Anna Directeur de Recherche, CNRS Rapporteur Mr Julian omas Professor, University of Manchester Rapporteur Mrs Elizabeth Shee Twohig Senior Lecturer, University College Cork Examiner Mr Alasdair Whittle Professor, Cardiff University Examiner 2 Acknowledgements We address our most profound thanks to Mr Serge Cassen (CNRS) for having believed, four years ago, in our PhD project and for having given us daily an exceptional supervision. This work, the orientation of its questions of research and its results would have been quite different without the priceless scientiic inluence of the director of the Laboratoire de Recherches Archéologiques (LARA) in Nantes University. Our work owes much to Mr Muiris O'Sullivan (University College Dublin) who accepted to cosupervise it. We thank him for his scientiic investment, the personal documentation he put at our disposal and for allowing our work in the site of Knockroe. We thank the rapporteurs, Messrs André D’Anna (CNRS) and Julian Thomas (University of Manchester), and the examiners, Mrs Elizabeth Shee Twohig (University College Cork) and Mr Alasdair Whittle (Cardiff University), for accepting to evaluate this work. -
The History of Mathematics: Alternative Perspectives
Copyrighted Material Chapter One The History of Mathematics: Alternative Perspectives A Justification for This Book An interest in history marks us for life. How we see ourselves and others is shaped by the history we absorb, not only in the classroom but also from the Internet, films, newspapers, television programs, novels, and even strip cartoons. From the time we first become aware of the past, it can fire our imagination and excite our curiosity: we ask questions and then seek an- swers from history. As our knowledge develops, differences in historical perspectives emerge. And, to the extent that different views of the past affect our perception of ourselves and of the outside world, history becomes an important point of reference in understanding the clash of cultures and of ideas. Not surprisingly, rulers throughout history have recognized that to control the past is to master the present and thereby consolidate their power. During the last four hundred years, Europe and its cultural dependen- cies1 have played a dominant role in world affairs. This is all too often reflected in the character of some of the historical writing produced by Eu- ropeans in the past. Where other people appear, they do so in a transitory fashion whenever Europe has chanced in their direction. Thus the history of the Africans or the indigenous peoples of the Americas often appears to begin only after their encounter with Europe. An important aspect of this Eurocentric approach to history is the man- ner in which the history and potentialities of non-European societies were represented, particularly with respect to their creation and development of science and technology.