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Around Caspar Wesse L and the Geometric Representation of Complex Numbers

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Around Caspar Wesse L and the Geometric Representation of Complex Numbers Around Caspar Wesse l and the Geometric Representation of Complex Numbers Proceedings of the Wessel Symposiu m at The Royal Danish Academy of Sciences and Letter s Copenhagen, August 1145 1998 Invited Papers Edited by Jesper Lützen Matematisk-fysiske Meddelelser 46:2 Det Kongelige Danske Videnskabernes Selska b The Royal Danish Academy of Sciences and Letters Commission Agent: C .A. Reitzels Forlag Copenhagen 2001 Abstract On March 10 1797 the Norwegian surveyor Caspar Wessel presented an essay On the Analytical Representation of Direction to The Royal Danish Academy of Sciences and Letters in which he described the geometric representa- tion of complex numbers that has since become standard . The paper was printed in the Academy's Journal two years later. In order to celebrate the 200th anniversary of this event the Academy arranged a Wessel Symposiu m on August 11-15 1998 . The contributions to the present volume are based on invited papers presented on that occasion . Their subjects range over a variety of historical themes related to Wessel and his family, to Wessel's wor k as a surveyor, to the geometric representation of complex numbers, and to the emergence of hyper-complex numbers . JESPER LÜTZEN Department of Mathematic s University of Copenhagen Universitetsparken 5 DK 2100 Copenhagen Ø lutzen©math .ku .dk © Det Kongelige Danske Videnskabernes Selskab 200 1 Printed in Denmark by Special-Trykkeriet Viborg a-s ISSN 0023-3323 . ISBN 87-7876-236-7 MflVI 46:2 1 Contents Preface 3 The Wessel family The naval hero Peter Tordenskiold & the Wessel famil y by Hans Christian Bjerg 5 Johan Herman Wesse l by Bjorn Linnestad 1 9 Technology Transfer to Denmark Thomas Bugge's journal of a voyage through Germany, Holland an d England, 1777 by Kurt Moller Pedersen 29 English instrument makers observed by predatory Dane s by Dan Ch. Christensen 47 Instrument maker on the run : A case of technology transfer by Olov Arvelin 65 Cartography and Astronomy in Denmar k Wessel as a cartographer by Leif Kahl Kristensen 8 1 Heinrich Christian Schumacher - mediator between Denmark an d Germany ; Centre of Scientific communication in astronom y by Jurgen Hamel 99 History of Complex Number s Viète 's generation of triangles by Otto B. Bekken 121 Argand and the early work on graphical representation : New source s and interpretations by Gert Schubring 125 Inexplicable? The status of complex numbers in Britain, 1750-185 0 by Adrian Rice 147 Bellavitis's equipollences calculus and his theory of complex number s by Paolo Freguglia 181 2 MfM 46 :2 History of Hypercomplex Number s Hypercomplex numbers in the work of Caspar Wessel and Herman n Günther Grassmann : are there any similarities ? by Karl-Heinz Schlote 205 Julius Petersen, Karl Weierstrass, Hermann Amandus Schwarz and Richard Dedekind on hypercomplex number s by Jesper Lützen 223 Priority claims and mathematical values : disputes over quaternions at the end of the nineteenth centur y by Tom Archibald 255 Imaginary Elements in Geometry On the role of imaginary elements in the 19th-century geometr y by David . E. Rowe 271 MfM 46:2 3 Preface This book arose as a result of a five day symposium held from the 11th to the 15th of August 1998 at the Royal Danish Academy of Sciences and Letter s in order to celebrate the bicentenary of the publication of Caspar Wessel' s essay Om Directionens analytiske Betegning, et Forsøg anvendt fornemmelig til plane og sph(Eriske Polygoners Opløsning . In this essay Wessel presented his famous geometric representation of the complex numbers and made a generalisation of this analytic formalism to three-dimensional space . Wessel presented his essay to the Royal Danish Academy on March 10th 1797 an d it was published two years later in the Academy's journal . At the symposium two key talks were given about Wessel and complex numbers : Bodil Branner and Nils Voje Johansen spoke about Caspar Wes- sel (1745-1818) . Surveyor and Mathematician, and Kirsti Andersen spoke about Wessel's Work on Complex Numbers and its Place in History . These talks were based on more extensive papers that have since been publishe d in volume 46:1 of these Matematisk-fysiske Meddelelser together with Flem- ming Damhus's new complete English translation of Wessel's essay1 . The present proceedings contain the majority of the remaining papers given at the Wessel Symposium . It can therefore be considered as a supplement to the above mentioned book, which in turn embodies the central themes around which these proceedings turn . The Wessel Symposium and this book deal with subjects from many differ- ent historical fields : literary history, military history, history of technology, history of astronomy and geodesy as well as history of mathematics . Such a variety of subjects may be appropriate in a publication by one of the few remaining academies of both sciences and letters . What binds the different papers together is their relation to the man Caspar Wessel, to his wor k as a surveyor, and to the subject of his mathematical essay : complex and hypercomplex numbers . The style of the papers varies as much as thei r content. A few papers are conversational in style whereas the majority ar e more scholarly. Kaspar Wessel: On the Analytical Representation of Direction . An attempt Applied Chiefly to Solving Plane and Spherical Polygons . 1797. Translated by Flemming Damhus. Introductory chapters by Bodil Branner, Nils Voje Johansen, and Kirsti Andersen . Edited by Bodil Branner and Jesper Liitzen. Matematisk-fysiske Meddelelser 46:1. Det Kongelige Danske Videnskabernes Selskab, Copenhagen 1999 . 4 MfM 46 : 2 I wish to express my gratitude to the following foundations that supporte d the Wessel Symposium financially : The Danish Research Councils, Knud Hojgaards Fond, Norske Selskab and last but not least The Carlsberg Foun- dation, which has also paid for the publication of these proceedings . I also wish to thank Pia Grüner and Flemming Lundgreen-Nielsen at the Roya l Academy for their help with the project and the authors who produced goo d scholarly works and conscientiously refereed each other's papers . Finally I am obliged to Jan Caesar at the Department of Mathematics, Copenhage n University, for having produced a nice and uniform LATEX layout of all th e contributions and to Knud Sorensen, Aarhus University, for his careful cor- rection of the language . Jesper Lützen MfM 46:2 5 The Naval Hero Peter Tordenskiold the Wessel Family Hans Christian Bjerg * 1 The Wessel Family The Wessel Family has contributed in several ways to the historical develop - ment of the Double-monarchy of Denmark-Norway which ended in the yea r 1814 when the two kingdoms separated . This volume is primarily dedicated to an example in the area of science, and it also contains a contributio n about Johann Herman Wessel, one of our famous poets . And in the Danish and Norwegian military history you will also find the name Wessel . One of our most famous naval heroes, Peter Tordenskiold, was christened Pete r Wessel and was a true member of the mentioned family . This paper will deal with this man and his merits in the beginning of the 18th Century . Everyone in Denmark and Norway knows the name Tordenskiold . Several anecdotes are told about him, but his fame is probably mainly due to th e fact that during this century his portrait has been reproduced on the mos t sold matchbox in Denmark . But first a little about the Wessel Family. In the middle of the 18th Century the family was of the opinion that they had immigrated from the Nether - lands, and indeed there is a Dutch locality called Wessel . But it is a fac t that in the 16th Century one finds the family name Wessel in a couple of th e *Chief Archivist, The Military Archives, The Danish National Archives, Rigsdagsgår- den, DK-1218 Copenhagen K, Denmark . 6 H. C. Bjerg MfM 46 :2 Hanseatic cities in the Northern part of Germany, where the trade agents had good connections with Norway, especially with Bergen, where Germa n merchants occupied a part of the city. It is also possible that the Wesse l family could have immigrated this way. We find for the first time a Wesse l from the Netherlands in Bergen in Norwegian files from 1593 . In any case, around 1620 a man with the name Jan Wessel lived in Bergen, and the grandfather of the naval hero Peter Tordenskiold named Henri k Jansson Wessel is traced in the archives from the middle of the 17th Centur y as a merchant and citizen of Bergen . Henrik Wessel's oldest son, Jan Wessel , was born in 1646 and went as a grown-up to Trondheim, a town in Norwa y north of Bergen . He married a very young girl, born 1656, Maren Scholler , who was out of one of the rich families in Trondheim. As a 16-year-old gir l she already gave birth to her first child . Within the next 26 year she becam e a mother of 18 children 12 sons and 6 daughters . An anecdote tells that at an advanged age Jan Wessel was on a trip sailing with a merchant shi p to Spain . The ship was boarded by French privateers . Investigating the passengers they were told that Jan Wessel was a father of 18 children . The privateers were so impressed with a man of such a "capacité" that they left the ship without doing any harm. Jan Wessel owned some buildings in Trondheim and some ships and wa s as a whole a respectable citizen . He became what we can call an alderman and a member of the Council of the City in 1693 .
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