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The Great Logarithmic and Trigonometric Tables of the French Cadastre: a Preliminary Investigation

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The Great Logarithmic and Trigonometric Tables of the French Cadastre: a Preliminary Investigation The great logarithmic and trigonometric tables of the French Cadastre: a preliminary investigation Denis Roegel 11 January 2011 This document is part of the LOCOMAT project: http://locomat.loria.fr 2 3 Important notice This report is supplemented by 47 volumes of tables, each as a PDF file. These tables replicate the structure of the Tables du cadastre and are provided to ease further research. They can be found on the LOCOMAT site, or at the location http://hal.inria.fr/inria-00543946/en. 4 Contents Introduction 9 1 The Tables du cadastre 11 1.1 The decimal metric system . 11 1.2 The need for more accuracy . 15 1.3 Prony and the cadastre . 17 1.4 History of the tables . 22 1.4.1 Work organization . 22 1.4.2 Computing (1793–1796) . 28 1.4.3 Printing . 33 1.4.4 Delays . 34 1.4.5 Revival under the Consulate . 38 1.4.6 Involvement of the British government (1819–1824) . 40 1.4.7 The waning of the project (1824–1833) . 45 1.4.8 Legalization of the decimal system . 46 1.4.9 The analysis of the tables (1858) . 47 1.4.10 Going beyond the Tables du cadastre .......... 51 1.5 Reduced tables . 52 1.6 The manuscripts . 55 1.7 Going further . 57 2 Computational methods and tables 59 2.1 Interpolation . 59 2.1.1 The method of differences . 59 2.1.2 Accuracy of the interpolation . 61 2.1.3 The influence on Babbage . 62 2.2 Lagrange’s formula for ∆nf(x) .................. 63 2.3 Logarithms of the numbers . 64 2.4 Sines . 71 2.5 Tangents . 75 2.5.1 Computation of tangents on 0q–0q:5000 ......... 75 5 6 CONTENTS 2.5.2 Computation of tangents on 0q:5000–0q:9400 ...... 76 2.5.3 Computation of tangents on 0q:9400–1q:0000 ...... 76 2.6 Logarithms of the sines . 78 2.7 Logarithms of the tangents . 82 2.8 Abridged tables . 87 2.9 Multiples of sines and cosines . 89 3 Practical interpolation and accuracy 91 3.1 The computers . 91 3.2 Forms for the interpolation . 93 3.2.1 Main forms . 93 3.2.2 Forms for the sines . 93 3.3 Interpolation methods . 102 3.3.1 Forward and retrograde interpolations . 102 3.3.2 Choosing a method of interpolation . 102 3.3.3 Interpolation types . 104 3.3.4 A note on rounding . 105 3.3.5 A classification of interpolation methods . 107 3.4 Structure of the differences . 108 3.4.1 Groups of numbers and dashed lines . 108 3.4.2 Vertical position of the constant ∆n ........... 109 3.5 Accuracy . 109 3.5.1 General considerations . 109 3.5.2 Log. 1–10000 . 111 3.5.3 Identity of the manuscripts and corrections . 111 3.5.4 Anomalies . 113 3.6 Strategies for retrograde interpolation . 113 3.7 Correction of errors . 114 4 Description of the manuscripts 115 4.1 Paper and binding . 115 4.1.1 Paper . 116 4.1.2 Binding . 116 4.1.3 Stamps . 117 4.2 Introductory volume . 118 4.3 Logarithms from 1 to 10000 . 120 4.4 Logarithms from 10000 to 200000 . 125 4.4.1 Truncation lines . 126 4.4.2 Comparison with Briggs’ tables . 126 4.4.3 Corrections by the Service géographique de l’armée .. 129 4.4.4 The pivots and their accuracy . 129 CONTENTS 7 4.4.5 Constant differences ∆i .................. 130 4.4.6 Accuracy of interpolated values . 132 4.4.7 Retrograde interpolations . 133 4.5 Sines . 137 4.5.1 Structure . 137 4.5.2 Retrograde (or backward) interpolation . 140 4.5.3 Truncation lines . 140 4.5.4 The last values of the table . 141 4.5.5 Accuracy . 142 4.5.6 Errors . 142 4.6 Logarithms of the arc to sine ratios . 144 4.6.1 Forms . 144 4.6.2 Truncation lines . 144 4.6.3 Positions of A and the ∆n ................ 144 4.6.4 Structure of the interpolation . 145 4.6.5 Pivots . 146 4.6.6 Errors . 146 4.7 Logarithms of sines from 0q:00000 to 0q:05000 ......... 149 4.8 Logarithms of sines after 0q:05000 ................ 151 4.8.1 Truncation lines . 151 4.8.2 Positions of ∆n ...................... 151 4.8.3 Constancy of ∆n ..................... 152 4.8.4 Pivots . 153 4.8.5 Discrepancy in 0q:51000 ................. 153 4.8.6 Retrograde interpolations . 153 4.8.7 Indication of degrees . 154 4.8.8 Interpolation corrections . 154 4.8.9 Errors . 154 4.8.10 Fragments . 155 4.9 Logarithms of the arc to tangent ratios . 156 4.9.1 Forms . 156 4.9.2 Truncation lines . 156 4.9.3 Positions of A0 and the ∆n ................ 156 4.9.4 Accuracy of A0 ...................... 157 4.9.5 Pivots . 157 4.9.6 Structure of the interpolation . 158 4.10 Logarithms of tangents from 0q:00000 to 0q:05000 ....... 159 4.11 Logarithms of tangents after 0q:05000 .............. 160 4.11.1 Truncation lines . 160 4.11.2 Pivots . 160 4.11.3 Position of the ∆n .................... 161 8 CONTENTS 4.11.4 Accuracy . 162 4.11.5 Interpolation adaptations . 162 4.11.6 Interpolated values . 162 4.11.7 Constancy of ∆n ..................... 163 4.11.8 Values of the logarithms . 163 4.11.9 Retrograde interpolations . 163 4.11.10 Unidentified interpolations . 164 4.11.11 Indication of degrees . 165 4.11.12 Interpolation corrections . 165 4.11.13 Errors . 165 4.12 Abridged tables of log. of sines/tangents . 167 4.12.1 Truncation . 167 4.12.2 Positions of the ∆n .................... 167 4.12.3 Structure . 168 4.12.4 Corrections . 170 4.12.5 Accuracy . 171 4.13 Multiples of sines and cosines . 172 5 Printing the tables 175 5.1 Planned structure . 175 5.1.1 Project 1 (1794) . 175 5.1.2 Project 2 (ca. 1794) . 176 5.1.3 Project 3 (1794–1795) . 177 5.1.4 Project 4 (1819) . 177 5.1.5 Project 5 (1825) . 178 5.2 Stereotyping . 178 5.3 Truncating the computations . 181 5.4 The 1891 excerpt . 181 5.5 Description of the 47 auxiliary volumes . 182 6 Conclusion and future research 185 7 Primary sources 189 Acknowledgements 193 References 195 Introduction Je ferai mes calculs comme on fait les épingles.1 Prony As part of the French reform in the units of weights and measures, an effort was undertaken at the beginning of the 1790s at the Bureau du cadastre to construct tables of logarithms which would not only be based on the more convenient decimal division of the angles, but also would become the most accurate such tables ever created. Gaspard de Prony had the task to implement this project, and he decided to split the computations among a number of computers. Use was made of only the simplest operations: additions and subtractions of differences. Begun in 1793, these tables were completed around mid-1796, but, al- though they were supposed to, they were never printed. Eventually, in the 1830s, the project was totally abandoned. This mythical endeavour of human computation nowadays lies forgotten in libraries in Paris and, apart from a 30-page description of the tables by Lefort in 1858,2 very little has been written on them. The work done cannot be merely described as interpolations using the method of differences. In fact, perhaps the main outcome of our investigation is that the picture is not as clear as the myth may have made it. It is actually much more complex. Additions and subtractions may seem simple operations, but so much appears to have been left unspecified. This has probably become clear to Prony and others, but only when it was too late. This document summarizes a preliminary investigation of these tables, but the task is much more daunting than it appears at first sight. This study does in fact barely scratch the surface. It tries to give some impetus, but much still lies ahead. 1[Edgeworth (1894)] 2[Lefort (1858b)] 9 10 Chapter 1 The Tables du cadastre 1.1 The decimal metric system The decimal logarithmic and trigonometric tables conceived by the French cadastre take their roots in the metric reform. The founding act was the law of 26 March 1791 which based the metric system on the measurement of the meridian.3 As pointed out by Gillispie, decimalization became incorporated, or even “smuggled,” into the metric system in corollary of that law, because the new unit was defined as the 10 millionth part of a quarter of a meridian and a sexagesimal division would have corrupted the unity of the system.4 In other words, the decimal metric system was made complete by substi- tuting a decimal or centesimal division to the old division of the quadrant.5 In turn, the decimal division of the quadrant made it necessary to compute new tables. Such arose the need for tables of logarithms of trigonometric functions using the decimal or centesimal division of the quadrant.6 Moreover, it was felt that the publication of new tables would help the propagation of the metric system.7 The first decimal8 tables made as a consequence of the new division of 3[Méchain and Delambre (1806–1810)] 4[Gillispie (2004), p. 244] 5[Carnot (1861), p. 552] 6It should be remarked that there had been at least one table with a partial decimal- ization, namely Briggs’ Trigonometria Britannica [Briggs and Gellibrand (1633)]. Briggs used the usual division of the quadrant in 90 degrees,.
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