The great logarithmic and trigonometric tables of the French Cadastre: a preliminary investigation Denis Roegel To cite this version: Denis Roegel. The great logarithmic and trigonometric tables of the French Cadastre: a preliminary investigation. [Research Report] 2010. inria-00543946 HAL Id: inria-00543946 https://hal.inria.fr/inria-00543946 Submitted on 6 Dec 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The great logarithmic and trigonometric tables of the French Cadastre: a preliminary investigation Denis Roegel 14 December 2010 This document is part of the LOCOMAT project: http://www.loria.fr/~roegel/locomat.html 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. Unfortunately, for reasons of space, the complete set is possibly not provided at the same location as this report. Consequently, we are currently (and for a limited time) making it available on CD by request, for a modest fee covering the cost of the CD, its burning, and its shipping. The CD contains all the tables from the LOCOMAT collection, that is, about 80 volumes. Contact the author for details. 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 and the ∆n . 144 4.6.4 Structure ofA 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 n 4.9.3 Positions of ′ and the ∆ . 156 A 4.9.4 Accuracy of ′ . 157 4.9.5 Pivots . .A . 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..