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ISBN 978-90-393-4627-3 Between Theory and Observations Tobias Mayer’S Explorations of Lunar Motion, 1751–1755

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ISBN 978-90-393-4627-3 Between Theory and Observations Tobias Mayer’S Explorations of Lunar Motion, 1751–1755 ISBN 978-90-393-4627-3 Between Theory and Observations Tobias Mayer’s Explorations of Lunar Motion, 1751–1755 Tussen theorie en waarnemingen Tobias Mayers studies van de maanbeweging, 1751–1755 (met een samenvatting in het Nederlands) Proefschrift ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de rector magnificus, prof. dr. W.H. Gispen, ingevolge het besluit van het college voor promoties in het openbaar te verdedigen op dinsdag 25 september 2007 des middags om 12.45 uur door Steven Adriaan Wepster geboren op 18 maart 1964 te Rotterdam Promotores: Prof. dr. H.J.M. Bos Prof. dr. J.P. Hogendijk aan mijn vader die er zo naar uitkeek CONTENTS Samenvatting vi 1 Introduction 1 1.1 Subject . 1 1.2 Organization . 3 1.3 Conventions . 5 2 The quest for lunar theory 8 2.1 Scientific significance . 8 2.2 Application to navigation . 10 2.3 Astronomical prerequisites . 13 2.3.1 Celestial coordinates . 13 2.3.2 Unperturbed orbits . 17 2.3.3 The lunar orbit . 19 3 Biographical embedding 24 3.1 Youth and early career . 24 3.2 First lunar tables . 26 3.3 To Göttingen . 27 3.4 The first Göttingen tables . 28 3.5 A new course: multiple steps . 29 3.6 Contending for the Longitude Prize . 30 3.7 Further research . 33 3.8 The Board’s decision and beyond . 35 4 A manual to the tables 39 4.1 On tables . 39 4.2 Single-step and multistep procedures . 42 4.3 Calendar conventions . 43 4.4 An example computation . 44 4.4.1 To find the sun’s longitude . 44 4.4.2 To find the moon’s longitude . 45 4.5 A note on accuracy . 49 4.6 The wanderings of evection . 50 5 Theoria Lunae 52 5.1 Introduction . 52 5.2 Circumstances of composition . 54 ii Contents 5.3 Outline of Theoria Lunae ....................... 58 5.4 Extended survey of Theoria Lunae .................. 60 5.4.1 Notation . 60 5.4.2 Differential equations in spherical coordinates . 61 5.4.3 Two eliminations . 63 5.4.4 An inventive substitution . 65 5.4.5 Interpretation of p ...................... 67 5.4.6 Intermezzo: a link to Clairaut? . 69 5.4.7 Trigonometric series . 70 5.4.8 Working out the expansions . 72 5.4.9 Filling in the numbers . 76 5.4.10 Odds and ends . 79 5.5 The solution . 81 5.6 Concluding remarks . 81 6 The Horrocks Legacy 84 6.1 Introduction . 84 6.2 Newton on lunar motion, 1702 . 85 6.3 The equations of NTM ........................ 87 6.4 Horrocks’s variable orbit . 90 6.5 Old wine in new bottles . 92 6.6 Lemonnier’s version of NTM ..................... 96 6.7 ‘Mondtafeln (wahrscheinlich älterer Entwurf)’ ............ 99 6.7.1 Peering at the peers . 99 6.7.2 The first set of lunar tables . 102 6.7.3 ‘Entwurf neuer C Tafeln’................... 102 6.7.4 The second set of lunar tables . 103 6.8 Accuracy of theories compared . 105 6.9 The preface to the kil tables of 1753 . 107 6.10 Conclusion . 109 7 Multisteps in Theoria Lunae 112 7.1 Introduction . 112 7.2 Theoria Lunae’s breakpoint . 113 7.2.1 A change in procedure . 113 7.2.2 A change in notation . 116 7.2.3 A change in word meaning . 117 7.3 Failing assertions . 118 7.4 Substitution of arguments . 120 7.5 An approximation technique . 124 7.6 The complete transform . 126 7.7 Conclusions . 128 Contents iii 8 ‘Hausbackene Combinationen’ 132 8.1 Introduction . 132 8.2 A search in the literature . 133 8.2.1 Mayer about his own work . 133 8.2.2 Commentators of Mayer’s work . 135 8.3 A variety of manuscripts . 138 8.4 A pair of pliers . 141 8.4.1 Extraction of coefficients from position calculations . 142 8.4.2 Extraction of coefficients from tables . 145 8.4.3 Examples . 146 8.5 A role for each manuscript . 148 8.6 Spreadsheets . 151 8.6.1 Example: sheet 7 . 152 8.6.2 Data: origin and grouping . 155 8.6.3 Analysis . 159 8.7 Results . 161 8.7.1 Reflection . 163 9 Some aspects of model fitting 165 9.1 Introduction . 165 9.2 The quality of observations . 165 9.3 World temperature distribution . 168 9.4 Averaging and cancelling . 169 9.5 Libration: a case of model fitting . 172 9.5.1 Locating the rotational axis . 173 9.5.2 The ‘method’ of averages . 176 9.6 Euler’s lunar tables of 1746 . 180 10 Concluding observations 184 10.1 General conclusion . 184 10.1.1 Remarks on data analysis . 184 10.1.2 Remarks on the tables and their accuracy . 185 10.1.3 Remarks on the theory . 186 10.1.4 Remarks on the multistepped format . 187 10.2 Further research . 188 10.3 The development of Mayer’s lunar tables: a summary . 190 A Lunar equations: versions and aliases 193 B Spreadsheet contents 199 C Manuscript sources 211 Bibliography 214 Index 221 LIST OF FIGURES 2.1 Horizontal coordinates . 14 2.2 Ecliptic coordinates . 15 2.3 Parallax. 16 2.4 Elliptic orbit. 18 3.1 A portrait of Tobias Mayer . 34 4.1 Examples of tables . 41 5.1 Parameters in the lunar problem . 60 6.1 The variable lunar orbit . 90 6.2 Three states of the variable orbit . 91 6.3 Mayer’s treatment of the Horrocksian mechanism . 95 6.4 Comparison of coefficients . 100 7.1 Coefficients before and after the transform . 120 8.1 An example position calculation . 143 8.2 Schema of the process of improvement of coefficients . 149 8.3 Spreadsheet 7 . 156 8.4 Spreadsheet 2 . 158 9.1 Determination of the position of the moon’s polar axis . 174 9.2 The values of g − k plotted along the unit circle . 181 LIST OF DISPLAYS 4.1 The completed calculation of the solar longitude . 45 4.2 Calculation of mean arguments of the moon . 46 4.3 The minor equAtions in the computation of lunar longitude . 47 4.4 The completed calculation of lunar longitude . 48 4.5 EquAtion sequences in kil and rede ................. 50 5.1 Constants . 61 5.2 Variables . 62 5.3 The first twelve expressions for Kj, L j and λ j ............ 77 6.1 Lemonnier’s lunar tables . 97 6.2 Design of new lunar tables, alias geer ................ 104 6.3 Standard deviations of some table versions . 106 7.1 Theoria Lunae solutions, single- and multistep . 114 7.2 The multistep transform compared to Mayer’s result . 129 8.1 Mean arguments and initial erors of the data on sheet 7 . 153 8.2 Statistics of the spreadsheet mimic . 161 A.1 Versions of lunar equAtions found among Mayer’s papers . 196 A.2 Coefficients of the most prominent versions . 197 B.2 All amendments sorted by equAtion . 209 SAMENVATTING In het midden van de achttiende eeuw stond maantheorie—dat wil zeggen, de the- orie van de beweging van de maan aan de hemel—bijzonder in de belangstelling. Daar waren theoretische en praktische redenen voor. Vanuit theoretisch oogpunt was maantheorie belangrijk omdat ze.
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