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An Attempt Applied Chiefly to Solving Plane and Spherical Polygons 1797

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An Attempt Applied Chiefly to Solving Plane and Spherical Polygons 1797 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.
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