Middlesex University Research Repository An open access repository of Middlesex University research http://eprints.mdx.ac.uk Delve, Janet (1999) The development of the mathematical department of the Educational Times from 1847 to 1862. PhD thesis, Middlesex University. Available from Middlesex University’s Research Repository at http://eprints.mdx.ac.uk/7993/ Copyright: Middlesex University Research Repository makes the University’s research available electronically. Copyright and moral rights to this thesis/research project are retained by the author and/or other copyright owners. The work is supplied on the understanding that any use for commercial gain is strictly forbidden. A copy may be downloaded for personal, non- commercial, research or study without prior permission and without charge. Any use of the thesis/research project for private study or research must be properly acknowledged with reference to the work’s full bibliographic details. This thesis/research project may not be reproduced in any format or medium, or extensive quotations taken from it, or its content changed in any way, without first obtaining permission in writing from the copyright holder(s). If you believe that any material held in the repository infringes copyright law, please contact the Repository Team at Middlesex University via the following email address: [email protected] The item will be removed from the repository while any claim is being investigated. MX 7226926 X Contents ABSTRACT Mathematics held an important place in the first twelve of years of the Educational Times (1847-1923), and in November 1848 a department of mathematical questions and solutions was launched. In 1864 this department was reprinted in a daughter journal: Mathematical Questions with Their solutions from The Educational Times (MQ). This thesis concentrates on the development of this department from its inception until 1862, when William John Clarke Miller became its editor; and is considered in terms of the editors, contributors and mathematics. To facilitate this research, a source-oriented database using KXEICO (kleio) software was constructed. It contains data taken from the questions and solutions and also miscellaneous items from the journal. Database analysis was used in conjunction with traditional, archival sources; for example, the respective, previously unknown correspondence of two of the main contributors, Thomas Turner Wilkinson and Miller. The development of the department fell into two main periods: the early 1850s when it was edited by Richard Wilson then James Wharton and had an educational bias; and the late 1850s when it was dominated by Miller and Stephen Watson who contributed moderately complex problems of a reasonably high standard on conic sections, probability and number theory. In 1850 Miller started contributing with a group of pupils and masters, including Robert Harley, from the Dissenters' College, Taunton. Another group of contributors which emerged was one of northern geometers, with whom Wilkinson was connected. He collaborated with Thomas Stephens Davies on geometry and this influenced his contributions to the department. Miller edited the department from 1862 to 1897 and MQ from 1863 to 1897 and made MO an international journal of renown for its original research. It contained contributions from some of the most eminent national and international mathematicians, including Cayley, Sylvester, Hirst and Clifford. The start of this new phase is briefly introduced and reviewed. 2 r Site MfDDLESEX UNIVERSITY LIBRARY Accer,sion ~) 1Z C=?°l 1 lo No. Class No. Special Collection Contents THE DEVELOPMENT OF THE MATHEMATICAL DEPARTMENT OF THE EDUCATIONAL TIMES FROM 1847 TO 1862 CONTENTS 0 INTRODUCTION Page 0.1 Scope of the thesis 13 0.2 Literature review 15 0.3 Outlineof the thesis 17 0.3.1 Methodology 17 0.3.2 Thesis contents 18 0.4 Acknowledgements 19 0.5 Abbreviations 20 1 BRITISH MATHEMATICS, 1840 TO 1865 1.0 Chapter overview 21 1.1 Prehistory of British and Continental mathematics 22 1.2 French mathematics after the Révolution of 1789 25 1.2.1 The Ecole Polytechnique 25 1.2.2 Major mathematicians connected with the Ecole Polytechnique 26 1.2.3 Journals connected with the Ecole Polytechnique or its staff 30 1.2.4 Other institutions 33 1.3 The Analytical Society and other reforms 34 1.4 British mathematics and higher éducation 37 1.4.1 Cambridge 37 1.4.2 The London collèges 43 1.4.3 The military establishments 46 Contents 1.5 British Journals and mathematical societies 49 1.5.1 Overview of the development of mathematical Journals 49 1.5.2 Archibald's work on British mathematical Journals 51 1.5.3 Wilkinson's work on mathematical periodicals 53 1.5.4 English and French scientific periodicals 55 1.5.5 Mathematical movements 59 1.6 A survey of British mathematics from around 1840 to the mid 1860s by subject area 60 1.6.1 Algebra 60 1.6.2 Symbolic logie 61 1.6.3 Quaternions 62 1.6.4 Déterminants and matrices 63 1.6.5 Invariant theory 64 1.6.6 Operator theory, differentiation and integration 65 1.6.7 Applied mathematics 65 1.6.8 Geometry 66 1.6.9 Probabilità 68 1.7 Summary 69 2 THE ORIGINS OF THE ET 2.0 Chapter overview 72 2.1 British elementary éducation 72 2.2 The inauguration of the College ofPreceptors 74 2.3 The launchofthe ET 78 2.4 The place of mathematics in the ET 86 2.4.1 Mathematics in the early years of the ET 86 2.4.2 Articles and books by Wharton 96 2.4.3 Articles concerning De Morgan 101 2.4.4 Articles concerning Whewell 103 2.5 Later educational reforms 104 2.6 Summary 106 4 Contents 3 THE LAUNCHING OF THE MATHEMATICAL DEPARTMENT AND ITS EARLY EDITORS 3.0 Chapter overview 108 3.1 The inception of the mathematical department 108 3.1.1 A survey of the first questions posed 108 3.1.2 The initial structure of the mathematical department 113 3.2 Survey of éditorial policy 114 3.2.1 Space considérations 115 3.2.2 Junior policy 116 3.2.3 General rationale 118 3.2.4 Original problems 119 3.2.5 Prize questions 120 3.2.6 Letters 121 3.2.7 Other items 123 3.3 The succession of editors 126 3.3.1 Identifying the editors 126 3.3.2 Summary of the editors 130 3.4 Thomas Turner Wilkinson 132 3.4.1 Biography 132 3.4.2 Mathematical periodicals 135 3.4.3 Wilkinson and the ET 136 3.4.4 The Lancashire Geometers 138 3.5 William John Clarke Miller 139 3.5.1 Biography 139 3.5.2 The Miller corresponde ne e 140 3.6 Summary 141 4 THE ET DATABASE 4.0 Chapter overview 144 4.1 Databases related to the history of mathematics 144 5 Conients 4.2 The selectíon of a datábase management System (DBMS) 147 4.2.1 The relational model 148 4.2.1.1 The ET data 149 4.2.1.2 Functional dependencies and normalization 151 4.2.2 Source-oriented data modelling and the KA-EICO DBMS 160 4.3 The KA,SICO £7* datábase 163 4.3.1 The ETstructure diagram 167 4.3.2 The £7"KXEICO structure file 169 4.4 Data entry in the ET datábase 175 4.4.1 The codification of the questions/solutions into mathematical categories 176 4.4.2 A detailed analysis of data entry into KXEICO of one question 183 4.4.3 Selected data entry examples 186 4.5 Error checking and data integrity in the £7"database 188 .4.5.1 Revision of the datábase structure 189 4.5.2 Revising the mathematical categories 191 4.5.3 Data entry methods 192 4.5.4 Spotchecks 192 4.5.5 Compilation errors 193 4.5.6. 'Data position errors' 193 4.5.7 'Data logic errors' 194 4.5.8 Miscellaneous errors 194 4.6 Analysing the £7" datábase 194 4.6.1. KA.EUO querying 194 4.6.2. Excel™ spreadsheets and Charts 195 4.7 Summary 195 5 OVERALL QUANTITATIVE ASSESSMENT OF THE DEPARTMENT 5.0 Chapter overview 197 5.1 Survey of the headings and sub-headings of the mathematical department 197 6 Contents 5.2 Summary of the questions 199 5.3 Summary of the solutions 200 5.4 Survey of solved questions 203 5.4.1 Length of time to solve questions 203 5.4.2 Total number of questions and the number of solutions per question 205 5.5 Unsolved questions and their proposers 206 5.5.1 Unsolved questions 206 5.5.2 Main contributors of unsolved questions 207 5.5.3 Matthew Collins 209 5.5.4 Mortimer Collins 210 5.5.5 Unpublished solutions 211 5.6 Contributors who answered their own questions 211 5.6.1 Northern geometers 212 5.6.2 Thomas Gaskin 213 5.6.3 Summary of questions solved by proposer 217 5.7 Contributors associated with trends in the department 217 5.8 Junior mathematics 219 5.8.1 Overall mathematical catégories for junior questions and solutions 219 5.8.2 Junior mathematical catégories by mathematical branch 220 5.8.3 Major junior contributors 221 5.8.4 Richard Wilson 222 5.8.5 Nemo's junior contribution catégories 225 5.8.6 Junior questions, an example period, October 1851 with solutions, December 1851 226 5.8.7 Summary of junior section 228 5.9 Summary 229 6 CONTRIBUTORS TO THE DEPARTMENT 6.0 Chapter overview 231 6.1 Summary tables of contributor information 231 7 Contents 6.2 Wilkinson 237 6.2.1 Overview of Wilkinson's contributions 237 6.2.2 Early questions and solutions 238 6.2.3 Pure geometry 242 6.2.4 Prize questions 243 6.2.5 Thomas Stephens Davies 246 6.2.6 Historical and contemporary book and journal références 254 6.2.7 Modem geometry 256 6.2.8 Summary of Wilkinson's mathematics 257 6.2.9 Rutherford, Fenwick and other contributors 257 6.3 The Dissenters' Collège, Taunton 260 6.3.1 Overview of contributions from the Dissenters' Collège 260 6.3.2 Miller's early contributions 262 6.3.3 Harley's contributions 265 6.3.4 The Dissenters in society 268 6.4 Miller and Watson 269 6.4.1 Overview of Miller's and Watson's contributions 269 6.4.2 Miller's renewed