DOCSLIB.ORG

An Attempt Applied Chiefly to Solving Plane and Spherical Polygons 1797

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Gaspar Wessel On the Analytical Representation of Direction An Attempt Applied Chiefly to Solving Plane and Spherical Polygon s 1797 translated by Flemming Damhus Introductory chapters by Bodil Branner, Nils Voje Johansen and Kirsti Anderse n Edited by BODIL BRANNE R and JESPER LÜTZEN Matematisk-fysiske Meddelelser 46 : 1 Det Kongelige Danske Videnskabernes Selskab The Royal Danish Academy of Sciences and Letters Commission Agent: C.A. Reitzels Forlag Copenhagen 1999 Abstract This book contains the first complete English translation of Caspar Wessel's essay from 1797 »On the Analytical Representation of Direction« . The essay has become famous because it gives the first geometric interpretatio n of the complex numbers . As the complete translation shows, it is also remarkable for the analytical representatio n of directions (vectors) in space and the elegant analytic solution of plane and spherical polygons . The translation of Wesse l's essay is prefaced by two papers on Caspar Wessel the man and his work . They shed much new light on this first important Norwegian mathematician and set his work on 'directions<' in its proper historical context . Among other things it is shown that the idea of using complex numbers to represent directions in a plane occurred to Wes - sel as early as 1787 in connection with his work as a surveyor. Moreover the papers address the question as to why Wessel's essay has remained without influence . Bodil Branne r Jesper Lütze n Department of Mathematic s Department of Mathematic s Technical University of Denmark Copenhagen University Building 30 3 Universitetsparken 5 DK-2800 Kgs . Lyngby DK2100 Copenhagen ø E-mail : [email protected] E-mail : lutzen©math.ku.dk © Det Kongelige Danske Videnskabernes Selskab 1999 Printed in Denmark by Special-Trykkeriet Viborg a- s ISSN 0023-3323 ISBN 87-7876-158- 1 © Kort- & Matrikelstyrelsen MfM 46:1 3 Contents Preface 5 Caspar Wessel (1745-1818) . Surveyor and Mathematician by BodilBranner and Nils VojeJohansen 9 Introduction 9 1 Wessel's mathematical treatise presented to the Academy in 1797 10 2 The early years 1 2 2.1 Living in Norway 1 2 2 .2 At the university in Copenhagen 1 4 3 From assistant to cartographer and geographical surveyor 1 5 3.1 The beginning 15 3.2 Geographical surveying in Denmark 16 3.3 Wessel's geographical surveying in 1769 19 3.4 Promotion 22 4 Among Norwegians in Copenhagen 23 4.1 Brother and poet 23 4.2 The Norwegian Society 24 5 Towards trigonometrical surveyor 27 5.1 An unusual year -1778 27 5.2 Trigonometrical surveying 28 5.3 Wessel's first year as trigonometrical surveyor-1779 32 6 The changing eighties 40 6.1 The negotiator 40 6 .2 In Oldenburg 42 6 .3 The family- diminished 44 7 The mathematician 44 7.1 Wessel's trigonometrical calculations in 1787 45 7.2 Encouragement 49 7.3 The mathematical treatise 5 1 7.4 Promotion and retirement 52 8 The working pensioner 52 9 Acknowledgements 55 10 Index of names 56 11 References 57 11 .1 Abbreviations 57 11 .2 Caspar Wessel's work 57 Related to surveying 57 The mathematical treatise 58 Letters and other hand-written notes from Caspar Wessel 58 11.3 Caspar Wessel bibliography 59 Comtemporary references 59 Later references 59 11 .4 Literature related to surveying 61 4 MfM 46 :1 Wessel's Work on Complex Numbers and its Place in History by Kirsti Andersen 65 Introduction 65 1 Complex numbers before Wessel 65 1 .1 The theory of equations and complex numbers 65 1 .2 Extended use of complex numbers 68 2 Wessel's On the Analytical Representation ofDirection 71 2.1 Wessel's aim 71 2.2 Wessel's introduction of addition and multiplication 72 2 .3 Wessel's universal trigonometric formula 76 2 .4 Wessel's treatment of spherical triangles 77 3 Complex numbers from Wessel to the mid-nineteenth century 8 1 3.1 Geometric representations 81 3.2 Argand's ideas 84 3.3 Gauss and complex numbers 88 3.4 Cauchy and complex numbers 90 3.5 Hamilton and complex numbers 92 Conclusion 94 Acknowledgements 94 References 95 On the Analytical Representation of Direction. An Attempt Applied Chiefly to Solving Plane and Spherical Polygons 101 by Caspar Wessel. Translated from Danish by Flemming Damhus I. How to form new straight lines from given ones by the algebraic operations, and especially, to determine whic h directions and signs they should have §§ 1-16 104 II. Proof of Cote's theore m §§ 17-23 111 III. How to denote the direction of the radii in a sphere §§ 24-35 115 IV. On solving spherical polygon s §§ 36-37 119 V. The most important properties of the spherical triangle s §§ 38-71 122 Figures MfM 46:1 5 Preface The Norwegian surveyor Caspar Wessel is generally recognized as the first to have give n the now familiar geometric interpretation of the complex numbers and of their rules of composition. His paper in Danish on this subject was presented to the Royal Danish Academy of Sciences and Letters on March 1 0'h 1797 and published two years later in th e Academÿ s journal . It had no immediate effect and was only rediscovered a century later, when it was mentioned in 1894 by Sophus A. Christensen in his doctoral thesis on the de- velopment of mathematics in Denmark and Norway in the 18 "' century. Upon readin g this Christian Juel called attention to the importance of Wessel's achievements, afte r which Sophus Lie had Wessel's paper reprinted in the Archiv for Mathematik og Naturvidenskab of 1895. Two years later the Academy published a French translation of it . Finally, in 1929 David E . Smith included an English translation of the first 16 section s dealing with plane geometry and the complex numbers into his widely read Source Book i n Mathematics. On the occasion of the bicentenary of the publication of Wessel's paper the Academy is happy to present Flemming Damhus's translation into English of the whole paper, in- cluding the last less well known parts in which Wessel gave an analytic representation o f directions (vectors) in space, and applied his analytic machinery to solve plane an d spherical polygons . The translation of Wessel's paper is introduced by two papers on Wessel and the geo- metric representation of complex numbers . In the first one, Bodil Branner and Nils Voj e Johansen deal with the person Caspar Wessel and his work as a surveyor. Some papers, in particular by Viggo Brun, Asger Lomholt and Otto Harms, have previously dealt with Wessel, but he has remained a somewhat shadowy figure until Bodil Branner and Nil s Voje Johansen began their intensive search for Wesseliana . Their paper presents the re - sult of this research . It throws brighter light on Wessel as well as on his work and shows i n particular that the idea of using complex numbers to represent directions in a plane oc- curred to Wessel at least as early as 1787 in connection with his theoretical investigation s of geodesy. In the second paper Kirsti Andersen analyses Wessel's mathematical work and show s how it fits into a general history of complex numbers and their geometric representation . Again, many of the facts of this history have been recounted in various books and articles , but Kirsti Andersen's paper is the first to analyze the story in its full complexity . In particu- lar she addresses the question why Wessel's paper remained without any influence . Wessel's paper was the first important mathematical work published in the Collecte d Papers of the Royal Danish Academy of Sciences and Letters . With the present book the Academy wishes to make this contribution to the history of mathematics more accessible , to situate it into its historical and geodesic context, and to diffuse knowledge about its au- thor. We are indebted to professor Poul Lindegård Hjorth . In the early stages of preparing the book we discussed the project with him and we were fortunate to be able to draw o n his long experience with publications from the Academy. Unfortunately he passed away in May 1998 after a short period of illness. Bodil Branner andJesper Lützen MfM 46 :1 Front page of Det Kongelige Danske Videnskabernes Selskabs Skrifter, Nye Samling, V, 1799 in which Wessel's pa - per appeared on pages 469 -518. MfM 46 :1 9 Caspar Wessel (1745-1818) Surveyor and Mathematician by Bodil Branner and Nils Voje Johansen Introduction This biography is written in honour of Caspar Wessel on the occasion of the bicentenar y of the publication of his mathematical paper Om Directionens analytiskeBetegning, et Forsøg anvendt fornemmelig til plane ogsphæriske Polygoners Opløsning, published as a special print in 1798 and in the Collected Papers of The Royal Danish Academy of Sciences and Letter s in 1799. Our starting point for the investigation was Viggo Brun's chapter about Caspar Wessel in his book Regnekunsten i det gamle Norge, 1962, Asger Lomholt's fourth volume from 196 1 in the series Det Kongelige Danske Videnskabernes Selskab 1742-1942, Samlinger til Selskabets Historie, the volume dealing with surveying under the auspices of the Academy, and Ott o Harms' paper Die amtliche Topographie in Oldenburg und ihre kartographischen Ergebnisse, in Oldenburger Jahrbuch, 1961 . We have searched through a lot of different material, including letters, Caspar Wessel' s surveying diaries, trigonometrical reports and trigonometrical calculations, and other primary sources, and we have found many new pieces of information . Here we only wish to emphasise that our thorough study of Wessel's surveying reports unveiled that he ha d the idea of using the complex numbers to represent directions in a plane at least as early as 1787.
Recommended publications
  • G.C. Oeder's Conflict with Linnaeus and the Implementation of Taxonomic and Nomenclatural Ideas in the Monumental Flora Danica

    G.C. Oeder's Conflict with Linnaeus and the Implementation of Taxonomic and Nomenclatural Ideas in the Monumental Flora Danica

    Gardens’ Bulletin Singapore 71(Suppl. 2):53-85. 2019 53 doi: 10.26492/gbs71(suppl. 2).2019-07 G.C. Oeder’s conflict with Linnaeus and the implementation of taxonomic and nomenclatural ideas in the monumental Flora Danica project (1761–1883) I. Friis Natural History Museum of Denmark, University of Copenhagen, Universitetsparken 15, DK–2100 Copenhagen Ø, Denmark. [email protected] ABSTRACT. Hitherto unpublished parts of the history of the Icones … Florae Danicae (1761–1883), one of the largest illustrated botanical works published, are analysed; it covered the entire flora of the double monarchy of Denmark–Norway, Schleswig and Holstein and the North Atlantic dependencies. A study of the little noticed taxonomic and nomenclatural principles behind the Icones is presented. G.C. Oeder, founder of the project, approved the ideas of Buffon and Haller and rejected Linnaean binary nomenclature because of its lack of stability of genera. In the Icones …, Oeder cited all names used for each plant in chronological order, with the binary Linnaean name last, to which principle Linnaeus reacted. By the end of the 18th century, Linnaean nomenclature had become standard, apart from in Flora Danica and a very few other botanical works. Applying Linnaean nomenclature elsewhere, O.F. Müller, editor 1775–1782, and M. Vahl, editor 1787–1799, followed Oeder’s norm in the Icones. J.W. Hornemann, editor 1810–1840, followed Oeder in his first fascicles, but began experimenting with changes towards Linnaean nomenclature from 1810. After 1840, subsequent editors consistently applied Linnaean principles for accepted names and synonyms. Keywords. Accepted names, genera, natural classification, species, synonymy Introduction In his excellent monograph on how the Linnaean reforms gained general acceptance among botanists, Stafleu (1971: 260) specifically stated that he left out a discussion of C.G.
  • Michael Kühn Detlev Auvermann RARE BOOKS

    Michael Kühn Detlev Auvermann RARE BOOKS

    ANTIQUARIAT 55Michael Kühn Detlev Auvermann RARE BOOKS 1 Rolfinck’s copy ALESSANDRINI, Giulio. De medicina et medico dialogus, libris quinque distinctus. Zurich, Andreas Gessner, 1557. 4to, ff. [6], pp. AUTOLYKOS (AUTOLYCUS OF PYTANE). 356, ff. [8], with printer’s device on title and 7 woodcut initials; a few annotations in ink to the text; a very good copy in a strictly contemporary binding of blind-stamped pigskin, the upper cover stamped ‘1557’, red Autolyci De vario ortu et occasu astrorum inerrantium libri dvo nunc primum de graeca lingua in latinam edges, ties lacking; front-fly almost detached; contemporary ownership inscription of Werner Rolfinck on conuersi … de Vaticana Bibliotheca deprompti. Josepho Avria, neapolitano, interprete. Rome, Vincenzo title (see above), as well as a stamp and duplicate stamp of Breslau University library. Accolti, 1588. 4to, ff. [6], pp. 70, [2]; with large woodcut device on title, and several woodcut diagrams in the text; title a little browned, else a fine copy in 19th-century vellum-backed boards, new endpapers. EUR 3.800.- EUR 4.200.- First edition of Alessandrini’s medical dialogues, his most famous publication and a work of rare erudition. Very rare Latin edition, translated from a Greek manuscript at the Autolycus was a Greek mathematician and astronomer, who probably Giulio Alessandrini (or Julius Alexandrinus de Neustein) (1506–1590) was an Italian physician and author Vatican library, of Autolycus’ work on the rising and setting of the fixed flourished in the second half of the 4th century B.C., since he is said to of Trento who studied philosophy and medicine at the University of Padua, then mathematical science, stars.
  • Biologisk Mangfold I Spydeberg Kommune Handlingsplan 1995 - 2007

    Biologisk Mangfold I Spydeberg Kommune Handlingsplan 1995 - 2007

    O -----------~- -R-a p-po-n7_-_~~~* 1 Biologisk mangfold i Spydeberg kommune Handlingsplan 1995 - 2007 l l ; ..... , · Fylkesmannen i Østfold Milj,Øvemavdelliigen Fylkesmannen i Østfold Mllj~vemavdelingen POSTADRESSE: STATENS HUS, POSTBOKS 325, 1502 MOSS TLF: 69 24 71 00 Dato: 5. desember 1996 Rapport nr: 7, 1996 ISBN nr: 82 - 7395 - 120 - o Rtij2portens tittel Biologisk mangfold i Spydeberg kommune. Handlingsplan 1995 - 2007 Forfatter Ola M.Wergeland Krog Spydeberg kommune, i samarbeid med fylkesmannen i Østfold- miljøvernavdelingen Ekstrakt Rapporten gir oversikt over Spydeberg kommunes (143 kv.km) natur, basert på tilgjengelig litteratur og informasjon, supplert med tallrike befaringer. Rapporten stedfester viktig naturinformasjon og gir oversikt over naturområder utfra biologisk mangfold, der det i arealplanlegging og arealbruk må tas særlige hensyn. Sjeldne og sårbare naturtyper og arter med negativ utvikling er vektlagt. Planen består av en tekstdel, kart og en database, der bl.a. naturtyper og deres forekomster er registert. Utredningen gir forslag til konkrete tiltak for hvordan viktige forekomster bør forvaltes. Kommunens politiske behandling av planen er tatt inn foran i rapporten. Naturgrunnlag, påvirkninger, hovednaturtyper og ulike grupper planter og dyr samt litteratur er behandlet - og tiltakene er ansvars fordelt på ulike kommunale etater. Rapporten kan anvendes som mal for en kommunal handlingsplan for biologisk mangfold. Emneord: Natur, biologisk mangfold, handlingsplan, Spydeberg Forord Spydeberg var den første Østfold-kommune (egentlig prestegjeld), der det ble foretatt mer eller mindre systematiske registeringer av artsmangfold. Et pioner-arbeid ble utført av Jacob Nicolai Wilse (1735- 1801), som var sokneprest og bosatt på Spydeberg prestegård 1768 -85 QfWilse 1779 og 1796).- Vel200 år senere (1991) vedtok Spydeberg kommunestyre at kommunens visjon frem mot år 2016 skal være t<Miljøkommunen Spydeberg,.
  • Oslo Katedralskoles Historie 1153–1800 Skolen Og Tiden

    Oslo Katedralskoles Historie 1153–1800 Skolen Og Tiden

    Oslo katedralskoles historie 1153–1800 Einar Aas Oslo katedralskoles historie 1153–1800 Skolen og tiden Redaksjon: Anders Langangen, Vibeke Roggen, Hilde Sejersted, Tore Haakensen og Arild Eilif Aasbo Oslo 2016 (Skolens våpenskjold i mindre utførelse) ©Stiftelsen Oslo katedralskole, Oslo 2016 Redaksjon: Anders Langangen, Vibeke Roggen, Hilde Sejersted,Tore Haakensen og Arild Eilif Aasbo Boken er satt med Palatino Linotype 11 pkt/13 pkt Grafisk tilrettelegging: Bokproduksjon SA/Ove Olsen ISBN 978-82-992654-7-8 (e-bok) Redaksjonens forord På Oslo katedralskole lå det et 500 sider langt manuskript fra 1930-tallet. Det var et utkast til Oslo katedralskoles historie fra ca. 1150 til omkring 1800, skrevet av Einar Aas (1857–1941). Den som tok initiativet til å gjøre noe med dette var Anders Langangen, pensjonert lektor ved skolen, og gjennom mange år en drivkraft for beskjeftigelse med skolens historie. I 2012 fikk han med seg to tidligere kolleger, Hilde Sejersted og Tore Haakensen, samt Arild Eilif Aasbo, et barnebarn av Aas. Gjen- nom to år arbeidet denne gruppen med å omforme håndskrif- tet til en digital versjon, og, ettersom arbeidet skred frem: med å gjennomgå teksten. Det var på dette tidspunktet at Vibeke Roggen ble invitert med i redaksjonen; hun er førsteamanu- ensis i latin ved Universitetet i Oslo med forskningsarbeider relatert til Oslo katedralskole i eldre tid. Manuskriptet som redaksjonen har hatt som utgangspunkt for sitt arbeid, bærer preg av å være et utkast. Det er tydelig mindre gjennomarbeidet enn Aas’ publiserte skolehistorier: Stavanger katedralskoles historie 1243–1826 (1925) og Kristi- ansands katedralskoles historie 1642–1908 (1932), foruten Kristia- nia katedralskoles historie i det nittende århundre (1935).
  • Sjuttonhundratal Nordic Yearbook for Eighteenth-Century Studies Volume 14

    Sjuttonhundratal Nordic Yearbook for Eighteenth-Century Studies Volume 14

    Sjuttonhundratal Nordic Yearbook for Eighteenth-Century Studies Volume 14 2017 Sjuttonhundratal: Nordic Yearbook for Eighteenth-Century Studies An international multidisciplinary peer reviewed open access scholarly journal Published by Svenska sällskapet för 1700-talsstudier in cooperation with Suomen 1700-luvun tutkimuksen seura/Finska sällskapet för 1700-talsstudier Norsk selskap for 1700-tallsstudier Dansk selskab for 1700-talsstudier Félag um átjándu aldar fræði, Ísland Editor-in-chief: PhD My Hellsing, Uppsala University, Sweden Co-editors: Professor Johanna Ilmakunnas, University of Turku, Finland; PhD Per Pippin Aspaas, UiT The Arctic University of Norway, Tromsø, Norway; Associate Professor Søren Peter Hansen, Technical University of Denmark, Copenhagen, Denmark; Professor Lasse Horne Kjældgaard, Roskilde University, Denmark; PhD Hrefna Róbertsdóttir, University of Iceland, Reykjavík, Iceland Editors for this special issue: PhD Candidate Tim Berndtsson, Uppsala University, Sweden; Phd Annie Mattsson, Uppsala University, Sweden; Professor Otto Fischer, Uppsala University, Sweden; Professor Ann Öhrberg, Uppsala University, Sweden Review editors: PhD Alfred Sjödin, University of Copenhagen, Denmark; Associate Professor Henrika Tandefelt, University of Helsinki, Finland; PhD Thomas Ewen Daltveit Slettebø, UiT The Arctic University of Norway, Tromsø, Norway; PhD Jens Bjerring-Hansen, University of Copenhagen, Denmark; Research Professor Margrét Eggertsdóttir, The Árni Magnússon Institute for Icelandic studies, Reykjavík, Iceland Language
  • Gerald Murnane

    Gerald Murnane

    WORLD OF THE BOOK Books are mirrors of many worlds: worlds here and distant, past and present, real and imagined. Through text and image, they act as keepers of ideas, of knowledge and of stories. This exhibition showcases many of the rare, beautiful and historically significant books held by this Library on behalf of the Victorian community. It celebrates the unique place of books in our hearts and minds, taking you on a journey through the history of book production, design and illustration, from the medieval era to the present day. BOOKS AND IDEAS ‘[T]he book is an extension of the eye...’ Marshall McLuhan The history of ideas is mirrored in the history of the book. Books have altered the course of history itself, through the dissemination of ideas that have changed how we think about the world and ourselves. In many cultures across different eras, books have played a highly symbolic and iconic role. There was a time when it was thought that the world’s knowledge could be collected between the covers of a book. The information explosion of recent times now makes it impossible to contain the world’s knowledge within one library, let alone in one book, yet books continue to be a powerful means of informing and inspiring new generations. The Dome Galleries You are standing in what was once the book stacks for the whole library. This photograph shows the storage of ‘elephant folios’ (very large books) in this very spot. Librarians used the spiral metal stairs to retrieve the books requested by readers sitting at desks below in the Dome Reading Room.
  • Romantik 07 2018

    Romantik 07 2018

    van Gerven, Journal for the Study of Romanticisms (2018), Volume 07, DOI 10.14220/jsor.2018.7.1.17 Timvan Gerven (University of Amsterdam) WhoseTordenskjold?The Fluctuating Identities of an Eighteenth-Century Naval Hero in Nineteenth-Century Cultural Nationalisms Abstract The naval hero Peter Wessel Tordenskjold (1690–1720) wasone of the most celebrated historical figures in both nineteenth-century Norway and Denmark. This double national cultivation gavecause for an ongoing feud between Danish and Norwegian historians concerninghis true fatherland.Atthe same time, the uncertaintysurrounding his exact nationality offered awealth of material for narratives of Dano-Norwegian, and even pan- Scandinavian rapprochement. This article explores Tordenskjold’strack record as afigure of nationalcultivation by treating him as adynamic and transnationalmemory site (lieu de mémoire).Itwill be demonstrated that the contestation surrounding the ownership of his memory formed an importantmotivation for the rich artistic cultivation of this national hero, while the symbolic meaning attributedtohim wassubjected to the ideological needs of the individuals and groupsappropriatinghim. As such, Tordenskjold came to be alternately ingrained in Danish, Norwegian, Dano-Norwegian,and Scandinavian frameworks ac- cording to the relevant political and social circumstances. Keywords Peter Wessel Tordenskjold, Cultural memory, Cultivation of culture, National heroes, Lieu de mémoire Introduction Peter Wessel Tordenskjold (1690–1720) belongs to the most beloved and most cultivated national heroes in nineteenth-century Denmark and Norway. The list of songs, poems, stories, plays, novels, and biographies featuring the vice admiral in the leading role is aparticularly long one and includes works by such prominent authors as Adam Oehlenschläger, B. S. Ingemann, and Henrik Hertz.
  • Economy, Magic and the Politics of Religious Change in Pre-Modern

    Economy, Magic and the Politics of Religious Change in Pre-Modern

    1 Economy, magic and the politics of religious change in pre-modern Scandinavia Hugh Atkinson Department of Scandinavian Studies University College London Thesis submitted for the degree of PhD 28th May 2020 I, Hugh Atkinson, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that these sources have been acknowledged in the thesis. 2 Abstract This dissertation undertook to investigate the social and religious dynamic at play in processes of religious conversion within two cultures, the Sámi and the Scandinavian (Norse). More specifically, it examined some particular forces bearing upon this process, forces originating from within the cultures in question, working, it is argued, to dispute, disrupt and thereby counteract the pressures placed upon these indigenous communities by the missionary campaigns each was subjected to. The two spheres of dispute or ambivalence towards the abandonment of indigenous religion and the adoption of the religion of the colonial institution (the Church) which were examined were: economic activity perceived as unsustainable without the 'safety net' of having recourse to appeal to supernatural powers to intervene when the economic affairs of the community suffered crisis; and the inheritance of ancestral tradition. Within the indigenous religious tradition of the Sámi communities selected as comparanda for the purposes of the study, ancestral tradition was embodied, articulated and transmitted by particular supernatural entities, personal guardian spirits. Intervention in economic affairs fell within the remit of these spirits, along with others, which may be characterized as guardian spirits of localities, and guardian spirits of particular groups of game animals (such as wild reindeer, fish).
  • Professor I Hemmelig Tjeneste

    Professor I Hemmelig Tjeneste

    Professor i hemmelig tjeneste Matematikeren, datapioneren og kryptologen Ernst Sejersted Selmer Øystein Rygg Haanæs 1 Forord Denne populærvitenskapelige biografien er skrevet på oppdrag fra Universitetet i Bergen og Nasjonal sikkerhetsmyndighet for å markere at det er 100 år siden Ernst Sejersted Selmer ble født. La meg være helt ærlig. For ni måneder siden ante jeg ikke hvem Selmer var. Det gjorde nesten ingen av mine venner og bekjente heller. Jeg har faktisk mistanke om at svært få i Norge utenfor det matematiske miljøet kjenner til navnet. Det er en skam. Fra midten av forrige århundre ble høyere utdanning i Norge reformert og åpnet for massene. Universitetet i Bergen vokste ut av kortbuksene og begynte å begå virkelig seriøs vitenskap. De første digitale computerne ble bygd og tatt i bruk i forskning og forvaltning. Norge fikk en «EDB-politikk». Alle nordmenn fikk et permanent fødselsnummer. Forsvaret etablerte en kryptologitjeneste på høyt nivå, og norsk kryptoindustri ble kapabel til å levere utstyr til NATO-alliansen. Ernst Sejersted Selmer hadde minst én finger med i spillet i alle disse prosessene. Selmer hadde lange armer og stor rekkevidde. Han satte overveldende mange spor etter seg, og denne boken er et forsøk på å gå opp disse sporene og gi Selmer den oppmerksomheten han fortjener. Enda en innrømmelse når vi først er i gang. Jeg er ikke matematiker. Det har selvsagt sine sider når oppdraget er å skrive en biografi om nettopp en matematiker. Da jeg startet arbeidet, hadde jeg ikke det minste begrep om hva en diofantisk ligning eller et skiftregister var. Heldigvis har jeg fått uvurderlig hjelp av de tidligere Selmer-studentene Christoph Kirfel og Tor Helleseth.
  • Reposs #11: the Mathematics of Niels Henrik Abel: Continuation and New Approaches in Mathematics During the 1820S

    Reposs #11: the Mathematics of Niels Henrik Abel: Continuation and New Approaches in Mathematics During the 1820S

    RePoSS: Research Publications on Science Studies RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation and New Approaches in Mathematics During the 1820s Henrik Kragh Sørensen October 2010 Centre for Science Studies, University of Aarhus, Denmark Research group: History and philosophy of science Please cite this work as: Henrik Kragh Sørensen (Oct. 2010). The Mathematics of Niels Henrik Abel: Continuation and New Approaches in Mathemat- ics During the 1820s. RePoSS: Research Publications on Science Studies 11. Aarhus: Centre for Science Studies, University of Aarhus. url: http://www.css.au.dk/reposs. Copyright c Henrik Kragh Sørensen, 2010 The Mathematics of NIELS HENRIK ABEL Continuation and New Approaches in Mathematics During the 1820s HENRIK KRAGH SØRENSEN For Mom and Dad who were always there for me when I abandoned all good manners, good friends, and common sense to pursue my dreams. The Mathematics of NIELS HENRIK ABEL Continuation and New Approaches in Mathematics During the 1820s HENRIK KRAGH SØRENSEN PhD dissertation March 2002 Electronic edition, October 2010 History of Science Department The Faculty of Science University of Aarhus, Denmark This dissertation was submitted to the Faculty of Science, University of Aarhus in March 2002 for the purpose of ob- taining the scientific PhD degree. It was defended in a public PhD defense on May 3, 2002. A second, only slightly revised edition was printed October, 2004. The PhD program was supervised by associate professor KIRSTI ANDERSEN, History of Science Department, Univer- sity of Aarhus. Professors UMBERTO BOTTAZZINI (University of Palermo, Italy), JEREMY J. GRAY (Open University, UK), and OLE KNUDSEN (History of Science Department, Aarhus) served on the committee for the defense.
  • Reflections on the Pragmatics of the Illustrated Perspective Treatise Performative Failures and (Pre-) Romantic Innovations Eduardo Ralickas

    Reflections on the Pragmatics of the Illustrated Perspective Treatise Performative Failures and (Pre-) Romantic Innovations Eduardo Ralickas

    Document generated on 09/25/2021 8:06 a.m. Intermédialités Histoire et théorie des arts, des lettres et des techniques Intermediality History and Theory of the Arts, Literature and Technologies Reflections on the Pragmatics of the Illustrated Perspective Treatise Performative Failures and (Pre-) Romantic Innovations Eduardo Ralickas exposer Article abstract displaying This essay reconsiders visual demonstrations contained in a selection of Number 15, printemps 2010 illustrated perspective treatises. Based on a fundamental distinction made in the field of pragmatics, the author argues that the images designed to URI: https://id.erudit.org/iderudit/044680ar demonstrate/teach/instantiate the perspectival system are plagued by a DOI: https://doi.org/10.7202/044680ar contradiction between the conceptual “content” of perspective and the figurative means deployed to display such content. In all cases, this aporia, which defines the teaching of perspective by means of images, arises when the See table of contents figurative discourse of perspective attempts to integrate a representation of its user within the system itself. In closing, the author suggests that the perspective treatise’s “pragmatic unconscious” allows one to shed new light on Publisher(s) the pictorial innovations of German romanticism, particularly in the work of Caspar David Friedrich, which is tied to fundamental (and hitherto unforeseen) Revue Intermédialités (Presses de l’Université de Montréal) ways to the “failures” of the classical age. ISSN 1705-8546 (print) 1920-3136 (digital) Explore this journal Cite this article Ralickas, E. (2010). Reflections on the Pragmatics of the Illustrated Perspective Treatise: Performative Failures and (Pre-) Romantic Innovations. Intermédialités / Intermediality, (15), 163–185.
  • Medieval Monastery Gardens in Iceland and Norway

    Medieval Monastery Gardens in Iceland and Norway

    religions Article Medieval Monastery Gardens in Iceland and Norway Per Arvid Åsen Natural History Museum and Botanical Garden, University of Agder, 4604 Kristiansand, Norway; [email protected] Abstract: Gardening was an important part of the daily duties within several of the religious orders in Europe during the Middle Ages. The rule of Saint Benedict specified that the monastery should, if possible, contain a garden within itself, and before and above all things, special care should be taken of the sick, so that they may be served in very deed, as Christ himself. The cultivation of medicinal and utility plants was important to meet the material needs of the monastic institutions, but no physical garden has yet been found and excavated in either Scandinavia or Iceland. The Cistercians were particularly well known for being pioneer gardeners, but other orders like the Benedictines and Augustinians also practised gardening. The monasteries and nunneries operating in Iceland during medieval times are assumed to have belonged to either the Augustinian or the Benedictine orders. In Norway, some of the orders were the Dominicans, Fransiscans, Premonstratensians and Knights Hospitallers. Based on botanical investigations at all the Icelandic and Norwegian monastery sites, it is concluded that many of the plants found may have a medieval past as medicinal and utility plants and, with all the evidence combined, they were most probably cultivated in monastery gardens. Keywords: medieval gardening; horticulture; monastery garden; herb; relict plants; medicinal plants Citation: Åsen, Per Arvid. 2021. Medieval Monastery Gardens in 1. Introduction Iceland and Norway. Religions 12: Monasticism originated in Egypt’s desert, and the earliest monastic gardens were 317.