While In College, Why Did Euler Work Through Advanced Math Books On His Own? His Father Required It. The Quality Of Education Offered Was Not Satisfactory. His Teacher Advised It Because He Did Not Have Time To Tutor Euler Privately. He Wanted To Come Up (2023)

1. While in college, why did Euler work through advanced math - Kunduz

  • 11 Jul 2022 · O The quality of education offered was not satisfactory. O His teacher advised it because he did not have time to tutor Euler privately.

  • While in college, why did Euler work through advanced math books on his own? O His father required it. O The q > Receive answers to your questions

While in college, why did Euler work through advanced math - Kunduz

2. Solved: While in college, why did Euler work through advance[algebra]

Solved: While in college, why did Euler work through advance[algebra]

3. Leonhard Euler: The Catalyst behind the Revolution of Mathematics

  • Missing: advanced books quality satisfactory. advised privately.

  • Leonhard Euler Is one of the most influential mathematicians of all time.

Leonhard Euler: The Catalyst behind the Revolution of Mathematics

4. [PDF] Reconceptualizing Mathematics Education - CORE

  • During the same period of time, teachers who were not qualified to teach mathematics taught the lower level mathematics classes. This problem is perpetuated.

5. [PDF] All-attainment Secondary Mathematics Teaching in England

  • The research on which this thesis is based is focussed on the grouping of students in secondary school mathematics by a conception of their “ability”. Focussing ...


  • The findings in mathematics education showed that the teacher had a significant role on students' achievements. Moreover, the way they conceived mathematics ...

7. [PDF] John Wallis's 1685 Treatise of algebra - Open Research Online

  • modesty, for although he was reluctant to claim the book as his own, he had no ... work, which he himself admitted did not really belong in his book, immediately.

8. ElizabethLewisPhDThesis.pdf.txt - St Andrews Research Repository

  • PETER GUTHRIE TAIT NEW INSIGHTS INTO ASPECTS OF HIS LIFE AND WORK; AND ... books for students of mathematics. His books proved remarkably popular: in effect ...

  • PETER GUTHRIE TAIT NEW INSIGHTS INTO ASPECTS OF HIS LIFE AND WORK; AND ASSOCIATED TOPICS IN THE HISTORY OF MATHEMATICS Elizabeth Faith Lewis A Thesis Submitted for the Degree of PhD at the University of St Andrews 2015 Full metadata for this item is available in St Andrews Research Repository at: http://research-repository.st-andrews.ac.uk/ Please use this identifier to cite or link to this item: http://hdl.handle.net/10023/6330 This item is protected by original copyright PETER GUTHRIE TAIT NEW INSIGHTS INTO ASPECTS OF HIS LIFE AND WORK; AND ASSOCIATED TOPICS IN THE HISTORY OF MATHEMATICS ELIZABETH FAITH LEWIS This thesis is submitted in partial fulfilment for the degree of Ph.D. at the University of St Andrews. 2014 1. Candidate’s declarations: I, Elizabeth Faith Lewis, hereby certify that this thesis, which is approximately 59,000 words in length, has been written by me, and that it is the record of work carried out by me, or principally by myself in collaboration with others as acknowledged, and that it has not been submitted in any previous application for a higher degree. I was admitted as a research student in September 2010 and as a candidate for the degree of Ph.D. in September 2010; the higher study for which this is a record was carried out in the University of St Andrews between 2010 and 2014. Signature of candidate ...................................... Date .................... 2. Supervisor’s declaration: I hereby certify that the candidate has fulfilled the conditions of the Resolution and Regulations appropriate for the degree of Ph.D. in the University of St Andrews and that the candidate is qualified to submit this thesis in application for that degree. Signature of supervisor ...................................... Date .................... 3. Permission for publication: In submitting this thesis to the University of St Andrews I understand that I am giving permission for it to be made available for use in accordance with the regulations of the University Library for the time being in force, subject to any copyright vested in the work not being affected thereby. I also understand that the title and the abstract will be published, and that a copy of the work may be made and supplied to any bona fide library or research worker, that my thesis will be electronically accessible for personal or research use unless exempt by award of an embargo as requested below, and that the library has the right to migrate my thesis into new electronic forms as required to ensure continued access to the thesis. I have obtained any third-party copyright permissions that may be required in order to allow such access and migration, or have requested the appropriate embargo below. The following is an agreed request by candidate and supervisor regarding the publication of this thesis: PRINTED COPY Embargo on all or part of print copy for a period of two years on the following ground(s): publi- cation would preclude future publication. Supporting statement for printed embargo request: We submit a request for an embargo on both all of printed copy and electronic copy of this thesis for the same fixed period of two years as publication would preclude future publication. Chapter 2 of this thesis has already been accepted for publication by O.U.P., as a chapter in Mathematicians and their gods, eds. Mark McCartney and Snezana Lawrence. An embargo is also necessary in order to realize plans to publish other material from the thesis after submission. ELECTRONIC COPY Embargo on all or part of electronic copy for a period of two years on the following ground(s): publication would preclude future publication. Supporting statement for electronic embargo request: See statement for printed copy above. Signature of candidate ...................................... Date .................... Signature of supervisor ...................................... Date .................... Embargo granted: on both the printed and electronic copies of this thesis for the requested period of two years from the date the thesis is logged in the university library. Contents Abstract viii Acknowledgements ix Foreword xi 1 An introduction to P. G. Tait 2 1.1 Concise biography 2 1.2 Selected biographical highlights 9 1.2.1 Reputation as a lecturer . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 The Tait Memorial Movement . . . . . . . . . . . . . . . . . . . . . . 13 1.2.3 Contribution to the Royal Society of Edinburgh . . . . . . . . . . . . 16 1.2.4 Involvement in controversy . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2.5 Family life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.6 Spiritual life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2 The Unseen Universe (1875) 24 2.1 Introduction 24 2.1.1 The anonymity game . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1.2 Balfour Stewart (1828–1887): a biographical sketch . . . . . . . . . . 29 2.2 The Christian man of science 32 2.2.1 The science versus religion debate: Tait and Stewart’s contribution . 34 2.2.2 Tyndall’s Belfast address . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3 The Principle of Continuity 39 2.4 The authors’ hypotheses 40 2.4.1 Great First Cause, beginning of the universe and origin of life . . . . 40 i 2.4.2 The end of the visible universe . . . . . . . . . . . . . . . . . . . . . . 43 2.4.3 The existence of an unseen universe . . . . . . . . . . . . . . . . . . . 46 2.4.4 Superior and angelic intelligences . . . . . . . . . . . . . . . . . . . . 49 2.4.5 Immortality and the spiritual body . . . . . . . . . . . . . . . . . . . 50 2.4.6 Divine action: miracles, the incarnation and the resurrection . . . . . 52 2.4.7 The authors’ practical conclusion . . . . . . . . . . . . . . . . . . . . 52 2.5 Reception of The Unseen Universe 52 2.6 String theory and M-theory anticipated 56 2.7 Closing remarks 56 3 Tait’s statistical models 58 3.1 Introduction 58 3.1.1 Tait’s pocket notebook . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.1.2 Tait’s contribution to statistics and probability . . . . . . . . . . . . . 60 3.1.3 James Matthews Duncan (1826–1890): a biographical sketch . . . . . 62 3.2 Fecundity, Fertility and Sterility (1866) 65 3.2.1 Fertility and fecundity of the mass of wives . . . . . . . . . . . . . . . 66 3.2.2 Fertility and fecundity of the average individual . . . . . . . . . . . . 71 3.2.3 Relative fertility and fecundity of different races . . . . . . . . . . . . 73 3.2.4 Tait’s appreciation of good data . . . . . . . . . . . . . . . . . . . . . 75 3.2.5 The second edition (1871) . . . . . . . . . . . . . . . . . . . . . . . . 75 3.2.6 Why Tait was writing in French . . . . . . . . . . . . . . . . . . . . . 79 3.3 Concluding remarks 79 3.3.1 Influence of Tait’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . 80 4 Tait’s schoolboy introduction to complex numbers 82 ii 4.1 Introduction 82 4.1.1 Schoolboy association with Maxwell . . . . . . . . . . . . . . . . . . . 83 4.1.2 The Tait–Maxwell school-book . . . . . . . . . . . . . . . . . . . . . . 85 4.1.3 Gloag’s influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.1.4 Significance of a particular school-book entry . . . . . . . . . . . . . . 87 4.2 Bishop Terrot 88 4.2.1 Charles Hughes Terrot (1790–1872): a biographical sketch . . . . . . 88 4.2.2 Devotion to mathematics . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.2.3 Reputation in Edinburgh, contribution to the R.S.E. . . . . . . . . . 92 4.2.4 Associations with the Edinburgh Academy . . . . . . . . . . . . . . . 93 4.3 Bishop Terrot’s 1847 paper 95 4.3.1 Summary of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.3.2 Tait’s notes on Bishop Terrot’s paper . . . . . . . . . . . . . . . . . . 98 4.3.3 Tait’s interest in the paper . . . . . . . . . . . . . . . . . . . . . . . . 99 4.3.4 On priority: John Warren’s influence . . . . . . . . . . . . . . . . . . 101 4.4 Associated historical insights 105 4.4.1 The discovery of quaternions . . . . . . . . . . . . . . . . . . . . . . . 105 4.4.2 Warren’s reference to Bue´e and Mourey . . . . . . . . . . . . . . . . . 109 4.4.3 Tait’s account of the developments . . . . . . . . . . . . . . . . . . . 110 4.5 Coming full circle 110 5 C.-V. Mourey’s single science of algebra & geometry 111 5.1 Introduction 111 5.2 C.-V. Mourey: a biographical enigma 112 5.2.1 Self-funded publication . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2.2 Mourey’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 iii 5.3 Mourey’s mathematics 127 5.3.1 Motivated by algebra, seeking algebraic reform . . . . . . . . . . . . . 127 5.3.2 Definitions and fundamental principles . . . . . . . . . . . . . . . . . 130 5.3.3 Applications, including a proof of the F.T.A. . . . . . . . . . . . . . . 144 5.3.4 Characteristic features of Mourey’s mathematics . . . . . . . . . . . . 165 5.4 Notable references to Mourey 166 5.4.1 Lefe´bure de Fourcy (1787–1869): a biographical sketch . . . . . . . . 166 5.4.2 Joseph Liouville (1809–1882): a biographical sketch . . . . . . . . . . 167 5.4.3 Dissemination of knowledge of Mourey’s contribution . . . . . . . . . 168 5.4.4 Hamilton and De Morgan’s correspondence on Mourey, 1852–1862 . . 180 5.5 Benefits of the algebraic perspective 194 Overall final remarks 196 Bibliographical essay 198 Appendices 201 A Family tree report for the Tait family 201 B Tait’s poem on the Franco–Prussian War (1870) 225 C Schooldays at the Edinburgh Academy 230 D Transcription of Tait’s notes from Terrot’s paper 238 E Bue´e’s 1806 paper and Gergonne’s 2D table 255 F Hamilton’s unpublished paper on the F.T.A. 264 G Mourey’s terminology and notation: a look-up table 270 iv List of Figures 0.1 Peter Guthrie Tait: an etching by William Brassey Hole, 1884 . . . . 1 1.1 Lieutenant Freddie Tait: a photograph by Marshall Wane, 1896 . . . 22 2.1 Tait’s annotation of a review published in The Glasgow Herald . . . . 27 2.2 Tait’s annotation of a review published in The Nation . . . . . . . . . 28 2.3 Balfour Stewart: a sketched portrait . . . . . . . . . . . . . . . . . . 29 2.4 Tait and Stewart’s concentric model of the Great Whole . . . . . . . 48 3.1 Tait’s pocket notebook . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2 An entry in French in Tait’s pocket notebook . . . . . . . . . . . . . 59 3.3 Dr. Matthews Duncan: a sketched portrait . . . . . . . . . . . . . . . 63 3.4 Tait’s graph of fecundity at different ages . . . . . . . . . . . . . . . . 68 4.1 Charles Hughes Terrot: a portrait by Mason & Co. . . . . . . . . . . 90 4.2 Tait’s drawing of the figure to accompany Cotes’ theorem . . . . . . . 97 5.1 Title page of the first edition of Mourey . . . . . . . . . . . . . . . . 114 5.2 Mourey’s signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.3 Technical drawings for Mourey’s timber-profiling machine . . . . . . . 120 5.4 Technical drawings for Mourey’s saw to cut standing trees . . . . . . 121 5.5 Technical drawings for Mourey’s saw to cut felled trees . . . . . . . . 122 5.6 Acte de de´ce`s for Claude Mourey . . . . . . . . . . . . . . . . . . . . 126 5.7 Journeys along the real line . . . . . . . . . . . . . . . . . . . . . . . 131 5.8 Addition of directed lines . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.9 The directed angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.10 Mourey’s proof of the distributive laws using similar triangles . . . . 139 5.11 Relating trigonometric lines and paths . . . . . . . . . . . . . . . . . 145 5.12 Mourey’s geometrical interpretation of the F.T.A. . . . . . . . . . . . 153 5.13 AP taken along AN . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.14 The curve δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.15 Discontinuity at m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 v 5.16 Application to plane curves, polar co-ordinates . . . . . . . . . . . . . 163 5.17 Application to plane curves, rectilinear co-ordinates . . . . . . . . . . 164 5.18 Dissemination of knowledge of Mourey’s contribution . . . . . . . . . 179 A.1 Ancestors of P. G. Tait . . . . . . . . . . . . . . . . . . . . . . . . . . 218 A.2 Descendants of P. G. Tait . . . . . . . . . . . . . . . . . . . . . . . . 219 A.3 1851 Census: Tait resident at Peterhouse, Cambridge . . . . . . . . . 220 A.4 1851 Census: others also resident at Peterhouse, Cambridge . . . . . 221 A.5 Tripos examination results, Cambridge, 1852 . . . . . . . . . . . . . . 222 A.6 Tait and Steele, graduates at Cambridge, 1852 . . . . . . . . . . . . . 223 A.7 Notification of Tait’s honorary degree from Edinburgh . . . . . . . . . 224 C.1 Academical Club Prize results, 1846 . . . . . . . . . . . . . . . . . . . 233 D.1 Tait’s notes from Terrot’s paper . . . . . . . . . . . . . . . . . . . . . 245 E.1 Bue´e’s application of Pythagoras’ theorem . . . . . . . . . . . . . . . 259 E.2 Gergonne’s 2D table of real and imaginary magnitudes . . . . . . . . 262 vi List of Tables 3.1 Age at marriage and the advent of sterility . . . . . . . . . . . . . . . 70 3.2 Influence of the advent of sterility on whole fertility of marriage . . . 70 3.3 Comparative fertility of wives in England and Scotland, taking age at marriage into account . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.4 Average fertility of married women of 15–19 for four nations . . . . . 78 4.1 Terrot’s use of material from Warren’s 1828 treatise . . . . . . . . . . 104 5.1 Positions of P on δ for A inside δ . . . . . . . . . . . . . . . . . . . . 157 5.2 Positions of P on δ for A outside δ . . . . . . . . . . . . . . . . . . . 158 5.3 Positions of P on δ for A on δ . . . . . . . . . . . . . . . . . . . . . . 159 5.4 Contributions of AP to the sum of the prime-directeurs . . . . . . . . 160 A.1 Family group report for the Tait family . . . . . . . . . . . . . . . . . 201 C.1 Syllabus, 6th and 7th classes, the Edinburgh Academy . . . . . . . . 231 C.2 Prizes won by Tait at the Edinburgh Academy . . . . . . . . . . . . . 232 D.1 Editorial corrections made to Tait’s notes from Terrot’s paper . . . . 254 G.1 Mourey’s terminology and notations . . . . . . . . . . . . . . . . . . . 270 vii Abstract In this thesis I present new insights into aspects of Peter Guthrie Tait’s life and work, derived principally from largely-unexplored primary source material: Tait’s scrapbook, the Tait–Maxwell school-book and Tait’s pocket notebook. By way of associated historical insights, I also come to discuss the innovative and far-reaching mathematics of the elusive Frenchman, C.-V. Mourey. P. G. Tait (1831–1901) F.R.S.E., Professor of Mathematics at the Queen’s Col- lege, Belfast (1854–1860) and of Natural Philosophy at the University of Edinburgh (1860–1901), was one of the leading physicists and mathematicians in Europe in the nineteenth century. His expertise encompassed the breadth of physical science and mathematics. However, since the nineteenth century he has been unfortunately overlooked—overshadowed, perhaps, by the brilliance of his personal friends, James Clerk Maxwell (1831–1879), Sir William Rowan Hamilton (1805–1865) and William Thomson (1824–1907), later Lord Kelvin. Here I present the results of extensive research into the Tait family history. I explore the spiritual aspect of Tait’s life in connection with The Unseen Universe (1875) which Tait co-authored with Balfour Stewart (1828–1887). I also reveal Tait’s surprising involvement in statistics and give an account of his introduction to complex numbers, as a schoolboy at the Edinburgh Academy. A highlight of the thesis is a re-evaluation of C.-V. Mourey’s 1828 work, La Vraie The´orie des quan- tite´s ne´gatives et des quantite´s pre´tendues imaginaires, which I consider from the perspective of algebraic reform. The thesis also contains: (i) a transcription of an unpublished paper by Hamilton on the fundamental theorem of algebra which was inspired by Mourey and (ii) new biographical information on Mourey. viii Acknowledgements I am pleased to record my special thanks to those who have helped make my post- graduate studies both pleasant and productive. At the University of St Andrews, my supervisors, Professor Edmund Robertson and Dr. John O’Connor : for generously sharing their expertise; for always know- ing the path ahead and the best way to proceed; for their patience as I developed and their encouragement along the way. Professor Peter Jupp: for his valuable comments on Tait’s contribution to statistics, and my interpretation of it, and his continued interest in my research. Dr. Colin Campbell : for his company on our conference trip to Hungary, his tales of the Edinburgh Academy and his support throughout. Dr. Isobel Falconer : for her enthusiasm about Tait as a research topic. And Stefanie Eminger, my friend and fellow researcher: for sharing the experience with me. At the University of Dundee, Ross MacDonald : for explaining aspects of Scots succession law in relation to Tait’s assets. In Edinburgh, the trustees of the James Clerk Maxwell Foundation: for arranging access to the archives preserved in India Street; for kindly giving their permission to publish material from Tait’s scrapbook and the Tait–Maxwell school-book; and for providing me with contact details for living members of the Tait family, including Susan Rutherford (Tait), the paternal great-granddaughter of P.G.T., who gener- ously shared information on her family history. Also Andrew McMillan, Honorary Archivist at the Edinburgh Academy: for facilitating a very pleasant visit to the Academy archives in November 2013; and for giving his kind permission to publish some material from the archive, especially the mathematics paper Tait and Maxwell attempted in 1847 as part of the Academical Club Prize competition. All those associated with the British Society for the History of Mathematics. Dr. Mark McCartney of the University of Ulster: for the opportunity to contribute to Mathematicians and their gods (accepted for publication by O.U.P.). In Oxford, Dr. Jackie Stedall of the Queen’s College: for the opportunity to publish my first journal article. And Dr. Peter Neumann O.B.E., also of the Queen’s College, Ox- ix ford: for his warm hospitality in Oxford at the RiP conference in March 2013 and his invaluable suggestions relating to Mourey. In Cambridge, Nathalie and John Hamilton: for kindly checking my translations of Mourey’s French into English. Further afield, in Dublin, the staff at the Manuscripts Library of Trinity Col- lege, Dublin: for facilitating a productive research trip in February 2014; and for their kind permission to publish some of the correspondence between Hamilton, De Morgan and Salmon, and an unpublished paper by Hamilton on the fundamental theorem of algebra. In Paris, the staff at the Archives de Paris : for their expert assistance in looking for biographical information on Mourey during my visit to the archives, also in February 2014. And in Hungary, Professor Pe´ter Ko¨tesi from the University of Miskolc: for organising memorable conferences in Sa`rospatak, Hungary in May 2012 and in Cluj-Napoca, Romania in May 2014. Finally at home, my family: for their love, support and prayers. x Foreword Peter Guthrie Tait (1831–1901) was one of the most eminent physicists and math- ematicians of the nineteenth century. Few could claim a greater reputation than Professor Tait: for his original researches, in physical science (experimental and theoretical) and in mathematics; for his contribution to the Royal Society of Edin- burgh and for his powers of exposition in the lecture theatre. Beyond the nineteenth century, however, Tait has been unfortunately overlooked. Consequently, the life and work of P. G. Tait has vast scope as a research topic. I first encountered Tait during my undergraduate days at St Andrews. My senior honours dissertation was on knot theory and Tait featured, as a key figure, in the account that I gave of its historical developments. The benefits of choosing Tait as a Ph.D. topic were numerous. First, it seemed that there was something to be learned about every aspect of his life, personal and professional—a multitude of avenues to original research. He was a fascinat- ing character—known to be blinkered, stubborn, fiercely patriot and prejudiced but at the same time, a loyal friend and family man. He was also a first-class teacher and a solid contributor to the corpus of scientific knowledge. His expertise and interests were truly diverse, covering the full breadth of physical science and mathematics. He was also well connected, being intimately associated with James Clerk Maxwell (1831–1879), Sir William Rowan Hamilton (1805–1865) and William Thomson (1824–1907), later Lord Kelvin. Second, there was the practical aspect: Tait was local and largely-unexplored primary source material was available nearby. However rich Tait as a research topic proved to be, it was always my intention to explore any avenue of research which promised to lead to something of historical interest or significance. Tait was to be used as a springboard to any topic in the history of mathematics worthy of further research. The first chapter in this thesis provides the reader with a concise introduction to P.G.T.—a biographical sketch covering various aspects of his personal and profes- sional life. It is supplemented by a family tree report for the Tait family in Appendix A. The second chapter discusses a remarkable book, co-authored by Tait and the xi Scottish physicist and meteorologist, Balfour Stewart (1828–1887). Therein, Tait and Stewart proposed hypotheses in an attempt to unite the latest scientific theo- ries with the established doctrines of Christianity. Reviews of The Unseen Universe, which was originally penned anonymously, have been sourced from Tait’s scrapbook. The third chapter reveals Tait’s surprising involvement in statistics. Knowledge of Tait’s contribution to statistics has come from an entry in his pocket notebook. An unpublished poem by Tait on the Franco–Prussian War, which was also dis- covered in the notebook, is transcribed in Appendix B. The fourth chapter in this thesis is an account of Tait’s first introduction to complex numbers, based on an entry from the Tait–Maxwell school-book. This particular school-book entry is tran- scribed in Appendix D. Supplementary material associated with Tait’s schooldays at the Academy appears in Appendix C, including a transcription of a mathemat- ics paper attempted by Tait and Maxwell as part of the Academical Club Prize competition in 1847. In Appendix E, I provide additional insights into historical developments regarding the geometrical representation of complex numbers, focus- ing on the contributions of Adrien-Quentin Bue´e (1748–1826) and Joseph Gergonne (1771–1859). By associated historical insights we are lead on to the final chapter, on C.-V. Mourey’s 1828 work, La Vraie The´orie des quantite´s ne´gatives et des quan- tite´s pre´tendues imaginaires. After exhaustive research, I am able to provide new biographical information on Mourey who has remained an unknown to historians of mathematics for the past 186 years. In Appendix F, I provide a transcription of an unpublished paper by Hamilton on the fundamental theorem of algebra which was inspired by Mourey. I discovered the paper in one of Hamilton’s notebooks in the Manuscripts Library of Trinity College, Dublin. A guide to Mourey’s notation and terminology is given in Appendix G. xii Principal unpublished sources This thesis is based primarily on three unpublished sources: Tait’s scrapbook, the Tait–Maxwell school-book and Tait’s pocket notebook. Other principal sources are discussed in a bibliographical essay at the end of the thesis (page 198). The scrap- book and the school-book are preserved by the James Clerk Maxwell (J.C.M.) Foun- dation at Maxwell’s birthplace, 14 India Street, Edinburgh.1 Tait’s pocket notebook has been in my possession since January 2011. Prior to that it was kept by the Ed- inburgh Mathematical Society. After submission I intend to hand it over to the J.C.M. Foundation. Tait’s scrapbook is a collection of miscellaneous documents once belonging to Tait which includes: newspaper cuttings of reviews, letters to editors, political car- toons, articles written by Tait on golf science, obituary tributes, etc.; a limited amount of correspondence (some from Maxwell, Thomson and Forbes); original poems, some by Tait, others by Maxwell; copies of examinations set by Tait and syllabuses he taught; records of Tait’s school and university examination results; copies of addresses given by Tait, Thomas Andrews etc.; edited drafts for submis- sion for publication; and so on. Some material has been inserted by Tait and some must have been put in after his death—obituaries, for instance—presumably by the family.2 The scrapbook is signed on the inside cover: ‘Edith Tait, 1878’. Edith was Tait’s eldest child. She would have been eighteen in 1878. The scrapbook was presented to the Foundation by Murray Tait (P.G.T.’s paternal great-grandson) in 2003. The Tait–Maxwell school-book dates from 1846–1847. It belonged to Tait and was originally intended as a fair-copy book: some entries are written carefully in ink, and are signed and dated; however, there is also an abundance of rough pencil 1For more information on Maxwell’s birthplace see [1]. For a biographical sketch of P.G.T., sup- plemented with extracts from the scrapbook, see [2]. 2In some places Tait’s annotations appear on the actual pages of the scrapbook, rather than on the cuttings pasted in, which is evidence that Tait started the scrapbook. See Figure 2.1 (page 27) and Figure 2.2 (page 28) for instance. xiii work, with workings-out and schoolboy sketches subsequently fitted into available space. Entries in the school-book include: (factually dubious) notes on the history of enumeration; a table recording the positions of the satellites of Jupiter, as ob- served by Tait at the age of thirteen; a number of problems and solutions on the mensuration of heights and distances, which I have traced to a contemporary text- book [3]; a summary of a paper published in the Transactions of the Royal Society of Edinburgh on the geometrical representation of complex numbers (see Chapter 4); and copies of the MSS. which Tait and Maxwell exchanged during their final year at the Edinburgh Academy; but the bulk of the school-book is taken up with notes which Tait abstracted from the Encyclopaedia Britannica (7th edition, 1842) on ‘Algebra’ and ‘Fluxions’. Dr John W. Arthur, a trustee of the James Clerk Maxwell Foundation, drew my attention to the possibility that Maxwell’s entries in the school-book might be written in Tait’s hand: I had originally thought that Tait and Maxwell passed the school-book between them and that they had each written in their own entries; but careful analysis of the handwriting suggests that in reality it was Tait who had produced a fair copy of Maxwell’s manuscripts in the school-book, imitating Maxwell’s handwriting. Tait’s pocket notebook dates from 1870. All the entries are written in pencil. Very few pages have been utilized. One of the pages is signed: ‘P. G. Tait, College, Edinburgh’. The notebook contains: Tait’s quaternion version of Green’s theorem; an unpublished poem by Tait on the Franco–Prussian War; an entry written in French and a reminder to make amendments to an address which Tait was to give to Section A of the British Association for the Advancement of Science (B.A.A.S.) in 1871. xiv Figure 0.1: Peter Guthrie Tait, an etching by William Brassey Hole, 1884. From the “Quasi Cursores”, the gallery of portraits of the Principal and Professors of Edinburgh at the time of the Tercentenary in 1884. Reproduced with the kind permission of the National Portrait Gallery, London. 1 CHAPTER 1 AN INTRODUCTION TO P. G. TAIT This chapter provides a biographical sketch of P. G. Tait, covering various aspects of his personal and professional life. It is supplemented by a family tree report for the Tait family in Appendix A. 1.1 Concise biography Peter Guthrie Tait (Figure 0.1, page 1) was, in his time, ‘one of the most renowned scientists and mathematicians in Europe’.3 He was born in Dalkeith, Midlothian on 28 April 1831 and educated at the Edinburgh Academy—where he established a life- long friendship with fellow student, James Clerk Maxwell (1831–1879)—and at the Universities of Edinburgh and Cambridge.4,5 At Edinburgh he studied mathemat- 3 [4, p120] 4For a book on the history of the Edinburgh Academy see [5]. The Academy’s Register [4] is another useful resource, providing concise biographies of former pupils: Tait appears on page 120. 5James Clerk Maxwell (1831–1879): Professor of Natural Philosophy at Marischal College, Ab- erdeen (1856–1860) and King’s College, London (1860–1865); then Cavendish Professor of Physics at the University of Cambridge (1871–1879); author of the celebrated A Treatise on Electricity and Magnetism (1873). From the Edinburgh Academy, Tait and Maxwell both went on to the Uni- versity of Edinburgh: Tait remained at Edinburgh for only one session, while Maxwell remained for three sessions before following Tait to Cambridge. Maxwell originally entered Peterhouse but transferred to Trinity, advised by Forbes that it would be easier to obtain a fellowship at Trin- ity. [6] After Cambridge, the pair maintained their friendship principally through correspondence, rather than face-to-face meetings. Through this correspondence they were of influence to one another, in the sharing of expertise and in the sounding of ideas. Forfar and Pritchard write: ‘each had the highest regard for the abilities of the other which were freely put at the other’s disposal’. [7] For an obituary tribute to Maxwell written by Tait see [8]. For a contemporary biography of Maxwell see [9]. 2 ics under Philip Kelland (1808–1879) and natural philosophy under James David Forbes (1809–1868).6,7,8 He received his B.A. from Cambridge in 1852. He came out as Senior Wrangler (the second Scot on record) and First Smith’s Prizeman.9 A fellowship at Peterhouse kept him in Cambridge until 1854. From Cambridge Tait went to Belfast as Professor of Mathematics at the Queen’s College, where he established a number of happy and profitable associations.10 Through Thomas Andrews (1813–1885), Vice-President and Professor of Chemistry, he was introduced to experimental work and to Sir William Rowan Hamilton (1805– 6Philip Kelland (1808–1879): Professor of Mathematics at the University of Edinburgh (1838– 1879), the first Scot to hold the position. See [10] for an obituary tribute to Kelland, written by Tait and George Chrystal, Professor of Mathematics at the University of Edinburgh (1879–1911) and previously Regius Professor of Mathematics at St Andrews (1877–1879). 7James David Forbes (1809–1868): Professor of Natural Philosophy at the University of Edinburgh (1833–1860); Principal of the United College, St Andrews (1860–1868); invented the seismometer in 1842; remembered for his work on glaciers. For a biography of Forbes in which Tait deals with his scientific work see [11]. 8For a book on the history of the University of Edinburgh see [12]: pages 247–250 are on Tait. 9 [13] Senior Wrangler : the student who placed first amongst graduates taking first-class degrees in mathematics in a given year at Cambridge. According to Craik, the first Scot to be Senior Wrangler was Alexander Ellice in 1833. [14, p45f(no.70)] Smith’s Prizeman: recipient of the Smith’s Prize, an annual £25 prize awarded to two students who had excelled in examinations in mathematics and natural philosophy at Cambridge. The influence of the Cambridge mathematical Tripos on the contribution to physics made by students and professors at Cambridge during the nineteenth century is discussed in [15]: note, there is more emphasis on Thomson and Maxwell than Tait. 10In 1908 the Queen’s College in Belfast—along with the Queen’s Colleges in Cork and Galway and the Royal University—became the Queen’s University of Belfast and the National University of Ireland. For a book on the history of the Queen’s College, Belfast see [16]: Tait is mentioned briefly on pages 164–165 of volume I. 3 1865), the discoverer of quaternions.11,12 After Hamilton, Tait is considered to have been the leading expounder of the quaternionic theory and the foremost advocate for its use in physics.13 In 1860 Tait returned to Edinburgh to take up the Chair 11Tait and Andrews worked together investigating the density of ozone and the action of electric discharge on oxygen and other gases. In fact, Tait’s first published papers were written in conjunction with Andrews and came out of their work on the density of ozone. These papers were presented to the Royal Society of London and printed in Andrews’ Scientific Papers [17], where there is a biographical memoir on Andrews written by Tait and Alexander Crum Brown (1838–1922), who was Tait’s brother-in-law and Professor of Chemistry at the University of Edinburgh (1869–1908). 12William Rowan Hamilton (1805–1865): Andrews Professor of Astronomy at Trinity College, Dublin, Astronomer Royal of Ireland and Director of Dunsink Observatory; remembered as the discoverer of quaternions and for his work on optics and dynamics; knighted in 1835. Tait’s correspondence with Hamilton began in August 1858; an introduction having been secured by Thomas Andrews. Tait had bought a copy of Hamilton’s Lectures on Quaternions in 1853, while at Cambridge. [18, p13] Tait was keen to understand the ‘physical applications of the method’, which was his primary interest in the quaternionic theory. [18, p119] The correspondence also proved beneficial to Hamilton: encouraged by Tait’s enthusiasm, Hamilton resumed his interest in quaternions and began work on his Elements of Quaternions (published posthumously in 1866). Hamilton acknowledged his debt to Tait in a letter to Tait dated 21 January 1859. See [18, p131]. Unfortunately, a misunderstanding over Tait’s plans to publish his own treatise on quaternions threatened the cordial relationship. Hamilton had misunderstood the planned scope of Tait’s Elementary Treatise on Quaternions: he had given Tait permission to publish a set of examples on the application of the quaternionic method but he believed Tait was to publish, instead, a full exposition of the theory which was to appear before Hamilton’s Elements. Hamilton became suspicious of Tait’s motives and in a letter to Augustus De Morgan (dated 14 November 1860), Hamilton expressed his fear of having been deceived by Tait. See [19, pp361–362]. To allay Hamilton’s fears, Tait agreed to delay the publication of his book until Hamilton’s Elements had appeared. Tait’s Elementary Treatise on Quaternions was finally published in 1867. Tait and Hamilton met for the first time in 1859, at the meeting of the B.A.A.S. in Aberdeen. For an obituary tribute to Hamilton written by Tait see [20]. For a contemporary biography of Hamilton see [19]. For selected correspondence between Hamilton and Tait, in relation to the application of quaternions, see [21]. 13In his biography of Hamilton, Hankins describes Tait as ‘Hamilton’s chief disciple’. [19, p316] Tait’s biographer, Knott writes: ‘Tait was one of the very few who really appreciated the immense 4 of Natural Philosophy at the university there. He was elected a Fellow of the Royal Society of Edinburgh (R.S.E.) in 1861. Amongst Tait’s chief contributions to mathematics was his work on quaternions and knot enumeration.14 In his experimental researches, he investigated thermal and electric conductivity and thermo-electricity, devising the first thermoelectric diagram in 1873.15 His experimental work in connection with the Challenger Expe- dition in the late 1870s led him to his invention of a pressure-measuring instrument called the Tait Gauge and to further important scientific work on: the compress- ibility of water, glass, mercury, etc.; the physical properties of fresh water and value of Hamilton’s work.’ [18, p14] 14It was Thomson’s theory of vortex atoms which brought about Tait’s mathematical work on knots: persuaded by Thomson, Tait chased a full classification of the forms of knotted vortex rings, under the assumption that a unique form of vortex ring existed for each of the elements. Working with the two-dimensional projection of the knots, Tait took an intuitive approach to investigating the number of different knots which exist with the same number of crossings and how these knots are represented by the scheme of the knot, i.e. a sequence of letters recording the order in which the labelled crossings are encountered when traversing the projection. Tait’s work was taken further by the Revd. Thomas P. Kirkman (1806–1895) as far as eleven crossings. Tait, Thomson and Maxwell all worked on the topology of knots and many new concepts were developed in the correspondence exchanged between them. [22] Some of Tait’s papers on knots were reprinted in volume I of his Scientific Papers (Cambridge : at the University Press, 1898). For Chris Pritchard’s paper on Tait’s knot theory see [23]: in addition to explaining the roles of Thomson and Kirkman and the previous work done by Listing, Pritchard also describes the knot-specific terminology invented by Tait and highlights the modern-day applications of knot theory. The ‘Third Man’ of the title is a reference to Tait: Pritchard has Tait as the ‘third man of natural philosophy’ (after Maxwell and Thomson). 15During the 1870s Tait undertook experiments in his Edinburgh laboratory to determine the effect of temperature on the thermo-electric properties of metals. His thermo-electric researches had been inspired by the earlier work of J. D. Forbes and William Thomson. Tait’s thermo-electric diagram was a graphical representation of the relationship between thermo-electric-motive force and temperature: Tait found that for most metals electro-motive force followed a parabolic law, while thermo-electric power followed a linear law. For Tait’s papers on his thermo-electric diagram see [24] first, then [25]. 5 sea water; the effects of pressure on the maximum density point of water; and so on.16 In 1875, in collaboration with the physicist and chemist, Sir James Dewar (1842–1923), he conducted experiments on Crookes’ radiometer which led him to the true dynamical explanation of the phenomena which Thomson described as ‘one of the most interesting and suggestive of all the scientific wonders of the nineteenth century’.17,18 Between 1885 and 1892 Tait published a series of five papers on the kinetic theory of gases, giving in the fourth, according to Thomson, the first proof of the Waterston–Maxwell equipartition theorem of the average equal partition of energy in a mixture of two different gases.19 Tait, who was especially fond of golf, combined mathematics and experimental work when he undertook research into golf 16The Challenger Expedition was a four-year voyage of scientific discovery of the oceans, conducted between 1872 and 1876. Tait was asked by the scientific leader of the expedition, Sir Charles Wyville Thomson (1830–1882) to determine the corrections that would need to be applied to the temperature readings taken by the self-recording deep-sea thermometers used during the expedition, in order to compensate for the high-pressure conditions. Tait had known Wyville Thomson at Queen’s, Belfast: Thomson was Professor of Mineralogy and Geology at Queen’s when Tait was Professor of Mathematics; and in 1870 Thomson came to the University of Edin- burgh as Professor of Natural History. [26] For Tait’s published papers relating to the Challenger Expedition consult his Scientific Papers, especially [27]. 17 [28, p367] Thomson’s explanation of the Crookes’ radiometer phenomena is this: ‘The phenom- ena to be explained is that in highly rarefied air a disc of pith or cork or other substance of small thermal conductivity, blackened on one side, and illuminated by light on all sides, even the cool light of a wholly clouded sky, experiences a steady measurable pressure on the blackened side.’ [Ibid.] Tait and Dewar’s results were communicated to the R.S.E. on 5 July 1875; however, the only published record is an article, ‘Charcoal Vacua’ published in Nature on 15 July 1875. 18Sir James Dewar (1842–1923): Lecturer in Chemistry (1869) and later Professor of Chemistry (1875) at the Royal (Dick) Veterinary College; Jacksonian Professor of Natural Experimental Philosophy at Cambridge (1875–1923) and Fullerian Professor of Chemistry at the Royal Insti- tution, London (1877–1923). [29] 19 [28, p366] For Tait’s papers on the kinetic theory of gases see volume II of his Scientific Papers (Cambridge : at the University Press, 1900). 6 ball aerodynamics.20,21 During his career he published some 365 papers and twenty- two books.22 He remains best remembered, however, as the co-author with William Thomson (1824–1907) of the epoch-making Treatise on Natural Philosophy—a con- 20Tait published on golf science in Nature, Golf and the Proc. and Trans. Roy. Soc. Edinburgh. Some of his papers were republished in volume II of his Scientific Papers. Denley and Pritchard have published a paper [23] on Tait’s golf science in which they explain aspects of Tait’s research on the subject and go through some of his early papers. They also recount a charming anecdote in which Tait’s son, Freddie, who was a championship golfer and used to take part in Tait’s experiments, proved Tait’s results wrong by exceeding the maximum distance a golf ball could be made to travel by an experienced golfer as stated by Tait. 21Chris Pritchard discusses various aspects of Tait’s scientific work in [30]: the topology of knots, golf ball aerodynamics and quaternions. 22For Tait’s published papers, consult the Proc. and Trans. R.S.E. and his Scientific Papers, published in two volumes by Cambridge University Press (1898 and 1900). For a detailed bib- liography see [31]. Tait’s principal publications include: Dynamics of a Particle (1856) with Steele, Sketch of Elementary Dynamics (1863) with Thomson, Treatise on Natural Philosophy (1867) with Thomson, Elementary Dynamics (1867) with Thomson, Elementary Treatise on Quaternions (1867), Sketch of Thermodynamics (1868), Elements of Natural Philosophy (1873) with Thomson, Introduction to Quaternions (1873) with Kelland, Recent Advances in Physi- cal Science (1876), Heat (1884), Light (1884), Properties of Matter (1885), Dynamics (1895) and Newton’s Law of Motion (1899). For the Encyclopaedia Britannica, he contributed the arti- cles: ‘Light’, ‘Mechanics’, ‘Quaternions’, ‘Radiation and Convection’ and ‘Thermodynamics’. He wrote biographical essays on: Hamilton, Maxwell, Balfour Stewart, Kelland, Forbes, Rankine, Kirchhoff, Stokes and Listing. He also published papers on atmospheric, meteorological and astronomical phenomena; graph theory and recreational mathematics; and education, including the Cambridge mathematical Tripos. 7 temporary treatment of the subject in terms of the new physics of energy.23,24,25 In recognition of his achievements, Tait received the Keith Prize and the Gun- ning Victoria Jubilee Prize of the R.S.E., and the Royal Medal of the Royal Society 23William Thomson (1824–1907), later Lord Kelvin: Professor of Natural Philosophy at the Univer- sity of Glasgow (1846–1899); remembered for his work in the areas of thermodynamics, electricity and submarine telegraphy and for his success in patenting a number of inventions (a mirror gal- vanometer and a mariner’s compass, to name but two); he was knighted in 1866 in recognition of the part he had played in the installation of transatlantic telegraph cables; he took the title Baron Kelvin of Largs in 1892. Tait’s association with Thomson began in 1860, when Tait took up the Chair of Natural Philosophy at Edinburgh. Tait was already acquainted with William’s brother, the engineer, James Thomson (1822–1892) who he had known at Queen’s, Belfast. Although William Thomson was based in Glasgow, he visited Edinburgh regularly for the meetings of the R.S.E. It was soon after meeting that Tait and Thomson decided that they ought to produce a book together on natural philosophy. [28, p364] Treatise on Natural Philosophy, also known as “Thomson and Tait” or T and T ′, was pronounced “‘one of the greatest books which have appeared since the Principia—a book not only profound, but full of original methods of treat- ment”’. [32, pp203–204] Knott said of the work: ‘The publication of Thomson and Tait’s Natural Philosophy was an event of the first importance in the history of physical science. No more mo- mentous work had been given to the world since the days of the brilliant French mathematicians, Laplace, Lagrange, and Fourier.’ [18, p176] The ambitious project was a huge undertaking which demanded eighteen years of collaboration and in that time only the first of a series of planned volumes ever saw publication. In fact, it is said that the work would never have been published had it not been for Tait’s ‘dogged persistence’: Thomson disliked book writing and over the years he became much wearied by the project and his enthusiasm waned. [33, p35] For a time Tait used T and T ′ as a teaching guide at Edinburgh. [18, p21] It was referred to by his students as the “‘Student’s First Glimpse of Hades”’. [34, p45] For a contemporary biography of Thomson see [35]. 24See [36] for Crosbie Smith’s contextual history of the science of energy in the nineteenth century. Special emphasis is put on the network of the “‘North British” physicists and engineers’: William Thomson; Macquorn Rankine (1820–1872), Professor of Civil Engineering and Mechanics at Glasgow; James Clerk Maxwell; Tait and Fleeming Jenkin (1833–1885), Professor of Engineering at the University of Edinburgh. [36, p1] Chapter 10 in this book is on Tait and Thomson’s Treatise on Natural Philosophy. 25T and T ′ is one of a select number of books to feature in Grattan-Guinness’ Landmark Writings in Western Mathematics 1640–1940 : see [37]. 8 of London; in addition to a number of honorary degrees and fellowships.26 He died on 4 July 1901, at the age of seventy, shortly after his retirement from the Edin- burgh chair. As a lasting memorial to Tait, a second chair in the natural philosophy department was instituted in the University of Edinburgh in 1922.27 1.2 Selected biographical highlights In this section I cover, in brief, a variety of aspects of Tait’s life and character. 1.2.1 Reputation as a lecturer In 1860, aged twenty-seven, Tait returned to Edinburgh from Belfast—with a wife, Margaret and a daughter, Edith—to take up the Chair of Natural Philosophy in the University of Edinburgh, succeeding Forbes who had gone to St Andrews as Principal of the United College. Tait had successfully outrivaled a number of strong candidates for the post, including: James Clerk Maxwell (Marischal College, Ab- erdeen), E. J. Routh (Peterhouse, Cambridge) and Frederick Fuller (King’s College, Aberdeen).28 Tait stood out from the group for combining abilities in original sci- entific investigation with remarkable powers of exposition in the lecture theatre. Tait’s biographer, C. G. Knott (1856–1922) described Tait as being ‘unsurpassed’ as a lecturer.29 Professor Robert Flint (1838–1910), Professor of Moral Philosophy at St Andrews (1864–1876) and Professor of Divinity at Edinburgh (1876–1903), described Tait as ‘one universally recognised to have had not only a genius of the 26Tait received honorary degrees from the Universities of Ireland (Sc.D., 1875), Glasgow (LL.D., 1901) and Edinburgh (LL.D., 1901), and honorary fellowships of the Edinburgh Mathematical Society (1883) and Peterhouse College, Cambridge (1885). He was an honorary member of the academies of Denmark, Holland, Sweden and Ireland. 27 [12, p252] 28 [18, p16] 29 [18, p19] 9 first order for research, but rare gifts as a teacher’.30 Several of Tait’s former students at Edinburgh have produced written accounts bearing witness to his gifts as an expositor and to his presence in the lecture theatre. J. M. Barrie (1860–1937) recalled:31 Never, I think, can there have been a more superb demonstrator. I have his burly figure before me. The small twinkling eyes had a fascinating gleam in them; he could concentrate them until they held the object looked at; when they flashed round the room he seemed to have drawn a rapier. I have seen a man fall back in alarm under Tait’s eyes, though there were a dozen benches between them. These eyes could be merry as a boy’s, though, as when he turned a tube of water on students who would insist on crowding too near an experiment, for Tait’s was the humour of high spirits. I could conceive him at marbles still, and feeling annoyed at defeat. He could not fancy anything much funnier than a man missing his chair.32 Preserved in Tait’s scrapbook is an article published in the University of Edin- burgh’s newspaper, The Student, in which two former students of Tait’s at Edin- burgh recorded their experience when they returned to his natural philosophy class in 1888. The following is an extract from their account: We were not long seated before the door leading to the Physical Laboratory opened, and the tall figure of the Professor slipped round it. The noises, more pronounced than formerly we thought, soon ceased under the steady gaze which followed the peculiar and well-remembered bow. This seems to be made when the motion towards the audience is combined with that in a direction parallel to it; but the full resolution into its components still remains for some bright young student. The old gown was yet extant, and the dark jacket, fastened by a single button at the neck, still seemed as far as ever from the toga virilis.33 30 [38, p60] 31Sir James Matthew Barrie (1860–1937): Scottish playwright and novelist; creator of Peter Pan; Rector of the University of St Andrews (1919–1922) and Chancellor of the University of Edin- burgh (1930–1937). [39] 32 [34, pp46–47] 33Toga virilis: ‘the toga of manhood, assumed by boys at puberty; hence in figurative context’; white in colour. [40] 10 The lecturer has lost nothing of his ancient power of graphic illustration and charming style. The conclusions are obtained from various trains of reasoning, and apt reference to everyday commonplaces clinches the argument, while the subtle humour of the man is every now and then revealed, and serves to keep his audience in good fettle. Details are skilfully subordinated, and the principles stand out in bold relief. The brilliant experiments are all necessary, and none are shown for effect. Performed at the right moment, they never fail, and make all seem clear as at noon.34 One feels that the students are not the only ones who thoroughly enjoy the Natural Philosophy lectures, and indeed it is said that sometimes when dealing with the more complex parts of Dynamics the lecturer alone seems perfectly contented. Listening to Professor Tait, one feels that it is quite impossible ever to forget the facts talked about and their relations and consequences. But sad experience, gained in the examination hall, has made it quite plain that it is the lucidity of the lecturer who makes everything transparent to us for the time, not we who can see through Carnot’s Reversible Cycle and Simple Harmonic Motion on first hearing about them.35 Tait’s presence in the lecture theatre was no doubt partly on account of his height and his uncommonly strong physique. The physicist and mathematician, Alexander Macfarlane (1851–1913), who was a former student of Tait’s at Edinburgh and who became a disciple of the quaternionic system, described Tait’s physical appearance in the following terms: Tait was not only an intellectual, but likewise a physical, giant. I am nearly six feet high, but standing beside Tait, I used to feel diminutive. He was well-built, and muscular. He wore a long beard, the hair on the top of his head had disappeared at an early date, and left exposed a massive forehead. To protect his head while lecturing it was his custom to wear a skull cap. On the street he wore a sack-coat and a soft felt hat, and with cane in hand, was always walking rapidly.36 34According to Barrie, Tait’s experiments were not always successful. On hand, should the equip- ment fail, was Tait’s mechanical assistant, Thomas Lindsay, son of James Lindsay who was mechanical assistant to both Forbes and Tait. [34, pp50–51] 35 [41] 36 [42, p42] 11 Professor Tait is captured in lecturing mode in Figure 0.1 (page 1). It appears, from the apparatus set up on the bench and the formulae written on the chalk-board, that he is explaining the fundamental principles of electrostatics. From Knott we learn that the apparatus Tait is operating is the Holtz machine.37 In 1868 Tait further contributed to the natural philosophy department, when he undertook the development of laboratory facilities and secured a grant for this purpose with the support of the Principal of the university, Sir David Brewster (1781–1868). With the benefit of the new laboratory, Tait could offer students the opportunity to undertake their own original investigations, after some initial training in research methods provided by Tait and his assistants. Many of these students were actually put to work helping Tait in his experimental researches. Tait’s laboratory at Edinburgh was modelled on William Thomson’s Glasgow laboratories.38 Explaining 37 [18, pp49–50] The Holtz machine converts mechanical energy into static electricity by a process of induction. The static electricity is then stored in the Leyden jar capacitors, also shown in the sketch (Figure 0.1, page 1). The formula Q = CV gives the voltage-charge relationship for the capacitor: Q is the charge on the capacitor, C is a constant called the capacitance and V is the voltage across the capacitor. The formula W = Q 2 2C gives the energy stored in the capacitor W , in terms of charge and capacitance. 38See Graeme Gooday’s Ph.D. thesis [43]: chapters 2,3,7 are especially relevant to this thesis. Goo- day argues that Thomson’s Glasgow laboratory (his chapter 2) became the authoritative example in laboratory teaching and precision measurement practices for subsequent laboratories, includ- ing Tait’s in Edinburgh (his chapter 3) and Balfour Stewart’s at Owens College in Manchester (his chapter 7). Gooday details the efforts of Thomson, Tait and Stewart to secure—in spite of problems with funding and accommodation—an experimental environment in which students could have: a practical illustration of the theoretical laws introduced in lectures, instruction in research methods and the opportunity to engage in their own research, working on customized experiments (which were often at the centre of their professor’s own current researches) with the chance to make a real contribution to science. Gooday describes the approach of Thomson, Tait and Stewart in their respective laboratories as non-hierarchical; and thus they operated in a manner true to the ‘Scottish “democratic” context’. [43, p2:19] In these laboratories there was little distinction between teaching and research, which Gooday characterizes as a ‘uniquely Scot- tish form of natural philosophy pedagogy’. [43, p2:25] Thomson’s was a largely organic approach, while Tait and Stewart followed a more organized plan of study in their laboratories. 12 the importance of the physical laboratories at Edinburgh and Glasgow, and the significance of Tait and Thomson’s influence, Macfarlane writes: Prior to the founding of the Cavendish Laboratory at Cambridge, the facilities at Edinburgh and Glasgow for gaining an experimental knowledge of physics were the best in Great Britain and this was due to the circumstance that in these twin cities of the North, the chairs of physics were occupied by twin giants in physical science.39 1.2.2 The Tait Memorial Movement Following Tait’s death, his friends and former students expressed a desire to con- tribute something to a memorial to him. Unfortunately, the raising of funds had to be postponed, as the University of Edinburgh was already running an Exten- sion Fund and they were not prepared to run the Tait Memorial Fund alongside. Consequently, fund-raising for Tait began in 1911, when the university’s Extension Scheme was completed. Fortunately, this coincided with the publication of Knott’s biography of Tait, which helped renew enthusiasm for the cause. A Tait Memorial Committee was established in November 1911, at a meeting held in the physical laboratory at the university, with the Principal, Sir William Turner (1832–1916) presiding. Serving as Honorary Secretary to the committee was Prof. J. G. MacGregor (1852–1913), Tait’s successor.40 When MacGregor died sud- denly in May 1913, Knott took on the role of Honorary Secretary. At a subsequent meeting of the committee, it was decided that a fitting memo- rial to Tait would be the institution of a second chair in the natural philosophy department: ‘The proposed chair would be connected with the department of the Professor’s work in which he achieved especially conspicuous success, namely, the 39 [42, p41] 40Prof. James Gordon MacGregor (1852–1913): born in Nova Scotia, Canada; Professor of Natural Philosophy at the University of Edinburgh (1901–1913); a student at the university during the 1870s; he undertook research in Tait’s laboratory on the electric conductivity of saline solutions, c.1872. [12, p250] 13 application of mathematics to the solution of physical problems, including those which bear upon engineering and other departments of applied science’.41 MacGre- gor had recalled that Tait himself had ‘strongly urged the founding of a second chair in his department’.42 For this purpose, the committee recommended raising a sum of £20–25,000. Appeals for funds were made to former students of Tait’s (some ten thousand in number), men of science, businessmen and men of industry.43 Amongst the documents relating to the Tait Memorial Movement, preserved in the National Library of Scotland, there is an official letter from Sir James Alfred Ewing (1855–1935), Turner’s successor as Principal, dated 9 November 1918.44 He was writing to formally lend his support to the work of the committee. The following is an extract from his letter: The University of Edinburgh ought certainly to have a memorial of Tait. He was for many years a great figure in it as a teacher, a thinker, and founder of a school of research. During his Professorship here he exerted an influence that has profoundly affected the development of Physical Science. No memorial to him could be more fitting than the endowment of a professorship in the branch of science with which his name has and will always have a world-wide association, namely, the Mathematical 41 [44] 42[Ibid.] Tait was writing in Macmillan’s Magazine in 1872. 43Those who contributed to the Tait Memorial Fund, according to [45], include: Prof. P. R. Scott Lang (1850–1926), who was assistant to Tait in the natural philosophy department at Edinburgh in the early 1870s, later Regius Professor of Mathematics at St Andrews (1879–1921) (£25); the physicist and mathematician, Alexander Macfarlane (1851–1913), who was a former student of Tait’s at Edinburgh and a disciple of the quaternionic system (£10); the oceanographer, Sir John Murray (1841–1914), who is remembered for his work in connection with the Challenger Expedition, having first worked on constructing the apparatus for use in the expedition in Tait’s laboratory in the 1870s, his home in Wardie, Leith was where Tait died in July 1901 (£100); J. S. Porter, Tait’s brother-in-law (£100), the Revd. H. S. Reid and Mrs. Reid, Tait’s son-in-law and daughter (£25); and Mrs. Tait, P.G.T.’s wife (£100). 44Sir James Alfred Ewing (1855–1935): Principal of the University of Edinburgh (1916–1929); pre- viously a student of engineering at the university; he worked with MacGregor in Tait’s laboratory during the 1870s. [12, pp250,273] 14 side of Physics. [. . . ] As myself an old pupil of Tait, grateful of his inspiration and cherishing his mem- ory, I earnestly hope that the efforts of your Committee will secure for him this most appropriate memorial.45 The title of the proposed second chair was discussed at a meeting of the executive committee on 28 May 1920.46 While the decision would ultimately be made by the University Court, the committee recommended: ‘the Tait Chair of Natural Philosophy on the mathematical and theoretical side, which might be shortly termed Mathematical Physics’.47 Raising the requisite sum proved difficult. By May 1920 the total subscriptions amounted to just under £3000 and the committee realized that there was no prospect of them raising all of the money themselves—they would need to join forces with 45Ewing to Convener of the Tait Memorial Committee (9 Nov. 1918) in [46]. 46The Tait Memorial Committee: Members of the executive sub-committee, according to [44]. (1) W. B. Blaikie (1847–1928): a civil engineer, astronomer and printer; a former pupil at the Edinburgh Academy (1858–1864). (2) B. Hall Blyth (1849–1917): a civil engineer and a student at the University of Edinburgh in the 1860s. (3) George A. Gibson (1858–1930): extra-academical lecturer in medicine at the University of Edinburgh; Prof. of Mathematics at the University of Glasgow (1909–1927) and President of the Edinburgh Mathematical Society (1888–1889). (4) Alexander Crum Brown (1838–1922): Prof. of Chemistry at the University of Edinburgh (1869– 1908) and Tait’s brother-in-law. (5) C. G. Knott (1856–1922): Tait’s biographer; a former student of Tait’s at Edinburgh; assistant to Tait in the natural philosophy department (1876–1883); appointed lecturer in mathematics (1891) and reader in applied mathematics at the University of Edinburgh. (6) J. G. MacGregor (1852–1913): Honorary Secretary of the Tait Memorial Committee; see footnote no.40. (7) Sir David Paulin (1847–1930): an actuary who founded the Scottish Life Assurance Co. (8) William Peddie (1861–1946): in charge of the natural philosophy laboratories at Edinburgh; assistant (1883) and lecturer (1892–1907) in the natural philosophy department; Professor of Physics at University College, Dundee (1907–1942) which was then a constituent college of St Andrews, and President of the Edinburgh Mathematical Society (1896–1897). [All eight were Fellows of the R.S.E.] And (9) Sir George M. Paul (1839– 1926): Honorary Treasurer for the Tait Memorial Committee; an Edinburgh lawyer who studied law at the University of Edinburgh during the 1860s. 47 [47] 15 other persons or bodies with the same aim. Funds were distributed to the R.S.E. (£1550) and the University of Edinburgh (£640). An amount was invested in war stock, with dividends presumably being dispersed periodically in support of the second chair. The Tait Chair of Natural Philosophy was eventually founded in 1922 and in 1923 the physicist, Charles Galton Darwin (1887–1962), grandson of Charles Darwin, the naturalist, became its first incumbent.48 1.2.3 Contribution to the Royal Society of Edinburgh Tait was proposed for fellowship of the R.S.E. by Philip Kelland in December 1860 and was elected on 7 January 1861. He served the Society as Councillor (1861–1864), Secretary to the Ordinary Meetings (1864–1879) and General Secre- tary (1879–1901).49 Campbell and Smellie, in their history of the R.S.E., describe Tait as a ‘devoted servant of the Society’.50 They also credit Tait, together with Thomson, as having ‘played a great role in maintaining the prestige of the Society’.51 Knott, who was himself a Fellow of the Society, described Tait’s role in the R.S.E. thus: ‘To many of the frequenters of the meetings in the seventies and eighties, Tait was in fact the Royal Society [of Edinburgh]; and there is no doubt that he guided its affairs with consummate skill.’52 Following Tait’s death, the Council of the Society formally expressed their feel- ings of loss in the minutes of the meeting of the Council held on 19 July 1901. The 48 [12, p252] The title of the chair changed in 1966 to the Tait Chair of Mathematical Physics. According to [48], the current Tait Professor of Mathematical Physics is Professor R. D. Kenway, O.B.E., F.R.S.E. 49 [29] 50 [33, p45] 51 [33, p46] 52 [18, p30] Knott served the R.S.E. as Councillor (1894–1897, 1898–1901, 1902–1905), Secretary to the Ordinary Meetings (1905–1912) and General Secretary (1912–1922). He was proposed for fellowship by Tait, and others, in 1880. He received the Society’s Keith Prize in 1893. [29] 16 original handwritten document, of which the following is an extract, is preserved in Tait’s scrapbook. It is signed by Kelvin (William Thomson), who was serving as President of the Society at the time. During his forty years of Fellowship he [Tait] contributed a very large number of Papers, all of them original and interesting, some of them of the very highest impor- tance, with a place assured for ever in the history of the development of science. His loss will be felt in the Society not only as a contributor, but perhaps even more as a wise counsellor and guide. The Council always felt that in his hands the affairs of the Society were safe—nothing would be forgotten, everything that ought to be done would be brought before them at the right time and in the right way. [. . .] [. . .] What the Council now feel is that a great man has been removed, a man great in intellect and in the power of using it, in clearness of vision and purity of purpose, and therefore great in his influence, always for good, on his fellow men; they feel that they and many in the Society and beyond it have lost a strong and true friend.53 The Society awarded Tait its Keith Prize on two occasions (1867–1869, 1871– 1873) and once, their Gunning Victoria Jubilee Prize (1887–1890). The Keith Prize was awarded “‘for the most important discoveries in Science made in any part of the world, but communicated by their author to the Royal Society of Edinburgh and published for the first time in the Transactions”’.54 The Gunning Victoria Jubilee Prize was awarded to those with Scottish connections, ‘in recognition of original work in Physics, Chemistry, or Pure or Applied Mathematics’.55 Tait won the Keith Prize in 1867 for his paper, ‘On the Rotation of a Rigid Body About a Fixed Point’ and in 1871 for a paper entitled, ‘First Approximation to a Thermoelectric Diagram’. The Gunning Victoria Jubilee Prize was awarded to Tait for his researches connected 53 [49] Published in Proc. Roy. Soc. Edinburgh, XXIV (1901–1902), pp. 2–4. 54 [33, p153] The Keith Prize was awarded every two years (biological and physical sciences alter- nately); winners received a solid gold medal, which had been gifted to the Society by its first Treasurer, Alexander Keith of Dunottar. [Ibid.] 55 [33, p152] The Gunning Victoria Jubilee Prize was a monetary award, presented every four years; it was founded in 1887 by His Excellency, Dr. R. H. Gunning (1818–1900). [Ibid.] 17 with the Challenger Expedition and in the wider context of his contribution to physical science. 1.2.4 Involvement in controversy Aspects of Tait’s character often led him into bitter disputes with other leading scientists. Knott believed that it was Tait’s straight-forward approach and his loyalty to his friends and colleagues that often led to his involvement in controversy—‘always on behalf of others’.56 ‘Tait was one who reacted sharply to anything he considered unfair or unjust either to himself or to his colleagues’, writes Campbell and Smellie.57 Following Tait’s death in 1901, his personal friend, Professor Flint acknowledged: I am quite aware that great as he [Tait] was, he had his own limitations, and some- times looked at things and persons from one-sided and exaggerated points of view, but the consequent aberrations of judgement were of a kind which did no one much harm and only made himself the more interesting. His strong likes and dislikes, although generally in essentials just, were apt to be too strong.58 Tait entered public disputes with the Irish natural philosopher, John Tyndall (1820–1893) over priority in energy physics and Forbes’ glacier work.59 Over Thom- son’s thermodynamic discoveries, he came into conflict with the Prussian, Rudolf Clausius (1822–1888). And his intense dislike of the vector calculus which threat- ened Hamilton’s quaternions brought him into conflict with the Englishman, Oliver Heaviside (1850–1925) and the American, J. Willard Gibbs (1839–1903).60 56 [18, p37] 57 [33, p69] 58 [38, pp60–61] 59John Tyndall (1820–1893): Professor of Natural Philosophy at the Royal Institution; remembered for his work on glacial movement, light, sound, magnetism and radiant heat in the context of gases and vapours. [50] 60See Chris Pritchard’s two papers, [51] and [52]. The first covers: Tait’s initial interest in quater- nions and his correspondence with Hamilton; the challenges faced by Tait during his campaign for 18 Tait’s involvement in the controversy which arose in the 1860s over energy physics showed him in an especially bad light. Fiercely patriotic, Tait took to public forums to assert British priority in the development of the science of energy; in the discovery of the equivalence of work and heat, in particular. In defence of the Englishman, James Prescott Joule’s (1818–1889) priority in the discovery, Tait criticized the contribution of the German, Julius von Mayer (1814–1878) and diminished his pri- ority.61 Understandably, accusations of chauvinism were levelled at Tait following his unreasonable conduct in the affair. His response was this: “I cannot pretend to absolute accuracy, but I have taken every means of ensuring it, to the best of my ability, though it is possible that circumstances may have led me to regard the question from a somewhat too British point of view. But, even supposing this to be the case, it appears to me that unless contemporary history be written with some little partiality, it will be impossible for the future historian to compile from the works of the present day a complete and unbiased statement. Are not both judge and jury greatly assisted to a correct verdict by the avowedly partial statements of rival pleaders? If not, where is the use of counsel?”62 1.2.5 Family life Early home life Peter Guthrie Tait was born in Dalkeith, Midlothian on 28 April 1831. His parents, John Tait (Private Secretary to the fifth Duke of Buccleuch, Walter Francis Scott) and Mary Ronaldson (daughter of John Ronaldson, a tenant farmer) were married on 27 June 1829 in Dalkeith. P.G.T. was the eldest of their three children: he had the acceptance of quaternions and Maxwell’s position on quaternions. The second covers Tait’s disputes with Arthur Cayley (over Cartesian versus quaternion methods) and with Heaviside and Gibbs. 61See [36] for an excellent account of the disputes which arose over energy physics during the nineteenth century. 62 [42, pp43–44] 19 younger sisters, Anne Margaret and Mary. P.G.T. lost both of his parents in his childhood: at the age of six he lost his father and at the age of fifteen he lost his mother. When his father died, the family moved to Warriston Crescent, Edinburgh. When his mother died, the children went to live in Somerset Cottage, Raeburn Place (also in Edinburgh); with Mary’s bach- elor brother, John Ronaldson who was a clerk in the National Bank of Scotland and Mary’s maiden sister, Margaret Ronaldson. Presumably, John and Margaret as- sumed parental responsibility for the children. According to Knott, although John Ronaldson was a banker by profession, he had a keen interest in scientific inves- tigation which influenced P.G.T.’s enthusiasm for science: Ronaldson spent much quality time with P.G.T.—on geological rambles, making astronomical observations and getting to grips with photography which had recently been invented.63 Wife and children Tait married Margaret Archer Porter on 13 October 1857 in Shankill, Antrim. Mar- garet was sister to the Porter brothers, William Archer and James, who Tait had known at Cambridge.64 Together, Tait and his wife Margaret had six children: Edith, John Guthrie, Mary Guthrie, William Archer, Frederick Guthrie and Alexan- der Guthrie. With the exception of Edith, who was born in Belfast, all the children were born in Edinburgh. Edith married Harry Seymour Reid, later Bishop of Edinburgh.65 Mary mar- ried Charles Walker Cathcart, an Edinburgh surgeon. Of all Tait’s children, his eldest son, John Guthrie seems to have followed most closely in his father’s foot- steps. He was Cambridge educated (a student of Peterhouse) and became a Fellow of the R.S.E. Unlike Tait, however, after Cambridge he trained as a barrister and 63 [18, p3] 64James Porter (1827–1900) went on to become Master of Peterhouse, Cambridge (1876–1900) and Vice-Chancellor (1881–1884). [13] Tait grew close to the Porter family during his time in Belfast. 65H. S. Reid officiated, with the Revd. Canon Cowley Brown, at Tait’s funeral which took place at St John’s Episcopal Church, Edinburgh. [18, p40] 20 was called to the bar in 1888. Then in 1890 he left for India to take up a posi- tion in the government education department; and in 1908 he became Principal of the Central College, Bangalore, where previously he was Professor of Languages and Vice-Principal. In 1904 he married the daughter of his predecessor as Princi- pal, John Cook who was originally a mathematics teacher of Arbroath, Scotland. William was a civil engineer who received his training at the University of Edin- burgh and under Sir J. Wolfe Barry and H. M. Brunel, son of Isambard Kingdom Brunel (1806–1859). He was involved in building the Talla Reservoir in the Scottish Borders. He was also a Fellow of the R.S.E. and served as Vice-President between 1921 and 1924. Freddie (Figure 1.1, page 22) was a Lieutenant in the Black Watch. He was killed in action in the South African War, in the battle of Koodoosberg Drift: he was shot in the heart, leading his men in a reconnaissance mission under Gen- eral Macdonald.66 He is remembered today, especially in St Andrews, as Scotland’s amateur golf champion.67 More information on his military career and his golfing successes can be found in John Low’s F. G. Tait: A Record [53], which is a most precious resource for those desiring an insight into Tait family life: excerpts from Freddie’s letters home portray beautifully P.G.T. in his beloved role as head of the Tait family. Alexander, the youngest of Tait’s children, was a glass merchant and spent a period of time in the port city of Liverpool. All Tait’s boys were educated at the Edinburgh Academy and were noted for their sporting prowess: in golf, shooting and rugby. For further details about the lives of Tait’s children see Appendix A. A family tragedy The end of Tait’s long and remarkable career followed shortly after Freddie’s death in February 1900. Tait was profoundly affected by the tragic event and from the 66Following Freddie’s death, a memorial fund was started in his name to put towards a new wing at the Cottage Hospital in St Andrews. [53, p226f] The new wing was opened by his mother Margaret, according to an exhibit in the St Andrews Preservation Trust Museum on North Street in St Andrews. 67Winner (1896, 1898); runner-up (1899). 21 time of his bereavement his own health began to fail. The change in him was noted by his life-long friend and collaborator, William Thomson in his obituary tribute to Tait: The cheerful brightness which I found on our first acquaintance forty-one years ago remained fresh during all these years, till first clouded when news came of the death in battle of his son Freddie in South Africa, on the day of his return to duty after recovery from wounds received at Magersfontein. The cheerful brightness never quite returned.68 Tait stopped teaching at the university in December 1900 and on 30 March 1901 he formally retired from the natural philosophy chair. He died less than four months later on 4 July 1901. Figure 1.1: Lieutenant Freddie Tait, a photograph by Marshall Wane, 1896.69 68 [28, p369] 69 [53, pfacing 78] 22 1.2.6 Spiritual life Tait was known to have been a member of the Scottish Episcopal Church.70 Other than this, little information is available on the spiritual aspect of his life. Knott, in his biography of Tait, offers the following insights. He writes: Tait was indeed a close student of the sacred records. The Revised Version of the New Testament always lay conveniently to hand on his study table; and frequently alongside of it lay the Rev. Edward White’s book on Conditional Immortality. I am not aware that he distinctly avowed himself a believer in this doctrine, as Stokes did, but he often expressed the high opinion he held of Edward White and his writings. His reverence for the undoubted essentials of the Christian Faith was deep and unmovable; and nothing pained him so much as a flippant use of a quotation from the Gospel writings.71 Robert Flint, in his obituary tribute to Tait, described Tait as someone who had managed to reconcile, in his own life, the scientific and religious aspects: Our departed friend had no sympathy with theological dogmatism, and little with anti-religious scepticism, and consequently held in contempt discussions on the so- called incompatibility of religion and science. At the same time, he had a steady yet thoughtful faith in God, and in that universe which no mere eye of sense, aided by any material instrument, can see. This faith must have made his life richer, stronger, and happier than it would otherwise have been.72 Tait challenged ‘the so-called incompatibility of religion and science’ in The Unseen Universe, which is the subject of chapter 2. 70Tait is known to have attended St John’s Episcopal Church in Edinburgh, where his body is interred in the family grave. The Scottish Episcopal Church is a Scottish Christian Church in communion with, but historically distinct from, the Church of England; it is a member of the Anglican Communion; comprising seven diocese, each with a bishop. 71 [18, pp36–37] 72 [38, p62] Reproduced in [18, pp44–46]. 23 CHAPTER 2 THE UNSEEN UNIVERSE (1875) This chapter is about a remarkable book which Tait co-authored with the Scottish physicist and meteorologist, Balfour Stewart. In The Unseen Uni- verse—which was originally penned anonymously—Tait and Stewart proposed hypotheses which they hoped would serve to unite science and Christian doc- trine. 2.1 Introduction In April 1875 an anonymous publication appeared, in which the possibility of im- mortality and the existence of an unseen universe were argued for on a scientific basis. The authors—revealed to be two leading figures in nineteenth century physi- cal science—believed that they had found unity in the latest scientific theories and the established doctrines of Christianity. The authors approached their task humbly, as ‘reverent students of the Scrip- tures’, hoping to communicate that science, properly understood, does not stand in opposition to religion.73 Plainly, in their own words: ‘Our object, in this present work, is to endeavour to show that the presumed incompatibility of Science and Re- ligion does not exist’.74 Guided by their fundamental Principle of Continuity, they believed that they could show science to be, in fact, the ‘most efficient supporter’ of Christian doctrines.75 Their chief concern was to remove the scientific objection to immortality. 73 [54] 74 [55, pxi] 75 [55, p209] 24 They aimed their argument to those for whom scientific objections to man’s immortality and the existence of an invisible world—raised by some in the scien- tific community—were proving to be insurmountable hurdles to a belief in these doctrines. 2.1.1 The anonymity game The Unseen Universe; Or, Physical Speculations on a Future State ran through three anonymous editions before authorship was revealed officially in April 1876. With each edition came the critics’ reviews and conjectures on the mysterious authorship issue. Publishing anonymously using the third person plural prompted many to as- sume a joint or collaborative effort. Rumours of two distinguished physicists were supported by evidence from the book of the authors’ ‘thorough knowledge of the highest and most recent developments of natural philosophy’.76 To some, however, the book’s title and the nature of some of the authors’ hypotheses were suggestive of spiritualist authors. In May 1875 The Spiritualist Newspaper was able to quash rumours of spiritualist authors, by reporting the Athenaeum’s revelation of the authors’ identities: ‘Dr. Balfour Stewart, of Manchester, and Mr. P. G. Tait, Professor of Natural Philosophy at the University of Edinburgh.’77 In the preface to their fifth edition, Tait and Stewart admonished the ‘London “Weekly,”’ who stated their names as facts, and did so without authorization, barely days after publication of the first edition.78 Other critics played fairer and refused to publish names, in respect of the authors’ desire to remain anonymous. Despite such an early outing, which no doubt came as a blow to Tait and Stew- art, there were pockets of individuals who remained unaware of the authors’ true 76 [56] 77 [57] 78 [58, pxxiii] 25 identities and the curious among them continued to search the text for clues left unwittingly by the authors. Naturally, those familiar with Tait and Stewart’s sci- entific interests and their connections within the scientific community would have easily deduced authorship. There are references to William Rowan Hamilton and William Thomson, later Lord Kelvin, who were both associated with Tait. There are detailed references to sunspots and J. D. Forbes’ work on glaciers, which both relate to Stewart. There are also many explicit references to Tait and Stewart’s ear- lier publications, for instance: Tait’s Thermodynamics ; Tait and Thomson’s Natural Philosophy ; Tait and Steele’s Dynamics of a Particle and Stewart’s Conservation of Energy. References to the experiments carried out jointly by Tait and Stewart, on a disk rotating in vacuo, are also very suggestive. As late on as December 1875, William Thomson’s name remained associated with the book: as one of three authors (Tait, Thomson and Stewart) or as half of the already known collaborative team, Tait and Thomson. Tait responded to this succession of candidates by annotating a review published in The Glasgow Herald, with a reference in Greek to Book VI of Homer’s Iliad : ‘φυλλων γνη’.79 See Figure 2.1 (page 27). I quote from an 1866 translation by the English mathematician and astronomer, Sir John F. W. Herschel (1792–1871): Man’s generations flourish and fall, like the leaves of the forest. Leaves on the earth by winds are strown, yet others succeed them, Ever renewed with returning spring. So fares it with mortals: One generation decays and its place is filled by another.80 The consensus view as reported by The Glasgow Herald was that Thomson and 79A quotation taken from line 146. I am grateful to Daniel Mintz (Ph.D. 2011, St Andrews, History of Mathematics) for recognizing the quotation’s original source. Elsewhere in Tait’s scrapbook, lines 146–149 are written out in full in the original Greek. Unfortunately, the entry is undated, however, it is possible that the entry dates from Tait’s time at the Edinburgh Academy: the handwriting appears naive and, according to the Directors’ Report for 1847 [59, p5], Tait would have studied Book VI of Homer’s Iliad in his final year at the Edinburgh Academy. 80 [60, p121] Lines 146–149 from Book VI of Homer’s Iliad. 26 ‘another Professor of repute’ had co-authored The Unseen Universe, which was con- tradicted by the newspaper, who claimed to have it on ‘the best authority’ that Thomson was not one of the authors.81 Figure 2.1: Tait’s annotation of a review published in The Glasgow Herald, 3 December 1875. Sourced from Tait’s scrapbook. Reproduced with the kind permission of the J.C.M. Foundation. ‘φυλλων γνη’ is a quotation from Book VI of Homer’s Iliad, line 146. It is interesting to speculate on Tait and Stewart’s reasoning behind choosing anonymity. By withholding authorship, their readers would approach the work without preconceived ideas of what they might expect from a well known, named author: the book would be judged solely on the quality of the authors’ speculations. Anonymity would also serve as a tactical measure, to promote discussion which would draw in a readership. Since Tait and Stewart invited criticism of their hy- potheses and since they dealt with the criticism they received robustly and publicly, it is inconceivable that they opted for anonymity in order to shield themselves from 81 [61] 27 harsh criticism and to protect their reputations. Tait, certainly, was not averse to the cut and thrust of public debate. Evidence exists which suggests that Tait, in particular, was prepared to resist the draw of recognition so that he might enjoy participating in the intellectual spec- ulations surrounding anonymity. In May 1875 a critic from The Nation newspaper brought to light a communication which had been published in Nature magazine on 15 October 1874. The communication, signed ‘West’, gave a proposition in the form of an anagram: A8 C3 D E12 F 4 G H6 I6 L3 M3 N5 O6 P R4 S5 T 14 U6 V 2 W X Y 2 With its letters correctly arranged it reads: “‘Thought conceived to affect the matter of another universe simultaneously with this may explain a future state.”’ The critic recognized The Unseen Universe as ‘the full elucidation and expansion’ of this proposition and went on to reason that West was in fact ‘one of the two reputed authors of “The Unseen Universe,” and presumably the senior partner’.82 Figure 2.2: Tait’s annotation of a review published in The Nation, 27 May 1875. Sourced from Tait’s scrapbook. Reproduced with the kind permission of the J.C.M. Foundation. 82 [62] 28 Again a cutting of the review is preserved in Tait’s scrapbook. Figure 2.2 (page 28) is a photograph of the original. The name West has been circled and a hand- written annotation, signed by Tait, has been inserted: ‘Note. This was my joke:– We (viz) S & T !’ So the name West stood for both authors, “Stewart & Tait”, who had written the communication in Nature so as to cryptically reveal authorship well in advance of publication and thereby establish priority when the first edition appeared.83 2.1.2 Balfour Stewart (1828–1887): a biographical sketch Balfour Stewart (Figure 2.3), a distinguished Scottish physicist and meteorologist, is remembered for his dedication to observation and experimental research, and for his ability to make from these, incisive deductions in the areas of radiant heat and solar phenomena in particular. Figure 2.3: Balfour Stewart, a sketched portrait.84Original source unknown. 83To establish priority Tait and Stewart referred to the Nature article in the book. See [55, p159]. 84 [63, pfacing 257] The same source [63, pp359–361] gives a biographical sketch of Stewart. 29 Stewart and Tait became acquainted in 1861, when Stewart was appointed an Additional Examiner in Mathematics for the University of Edinburgh. They collab- orated from 1866 onwards; initially, working together on experiments investigating the heating of a disk rotating in vacuo.85 Stewart was born in Edinburgh on 1 November 1828, to parents William Stew- art, a tea merchant of Leith, and Jane Clouston, daughter of William Clouston (a minister of Stromness, Orkney). Stewart’s university education began, at the age of thirteen, at St Andrews. From St Andrews he went on to Edinburgh (1845–1846) where he studied natural philosophy under Forbes. Encouraged by his parents, he embarked on a career as a merchant; but after a brief period spent in Leith and Australia, he returned to Edinburgh to resume his interest in physical science. In February 1856 he was appointed as assistant observer to John Welsh (1824– 1859) at Kew Observatory and in October of that year, assistant to Forbes at Ed- inburgh.86 In 1859 he succeeded Welsh as Director of Kew Observatory. In 1862 he was elected a Fellow of the R.S.E.; honorary fellowship followed in 1878.87 Stewart remained at Kew until 1870, when he took up a position as Professor of Physics at Owens College, Manchester.88 He died in 1887, while on holiday in Drogheda, Ireland. He had married in 1863 and had three children. Stewart might have had a distinguished career as a mathematician—he had shown promise as a mathematician under the influence of Kelland at Edinburgh, 85For the results of Tait and Stewart’s experiments see [64] and [65]. Follow-up papers on the subject were published in 1867, 1873 and 1878. 86 [66] Stewart’s biographer, Schuster has 1853 for the date Stewart became Forbes’ assistant. See [67, p253]. 87Stewart was proposed for fellowship in 1862 by Forbes. [29] Stewart also served as: Secretary to the Government Meteorological Committee (1867); President of the Society for Psychical Research (1885); President of the Manchester Literary and Philosophical Society (1887) and President of the Physical Society (1887). From the University of Edinburgh Stewart received the honorary degree of LL.D. 88Owens College was founded in 1851. In 1880 it became the Victoria University of Manchester and in 2004 it amalgamated with UMIST and became the University of Manchester. 30 and had published a paper on the theory of numbers—but he chose to devote him- self to experimental science.89 His interests—particularly in heat, meteorology and terrestrial magnetism—were influenced by his association with Forbes.90 Stewart’s most important contribution was his work on radiant heat: in Forbes’ laboratory in Edinburgh he undertook researches which led him to an extension of Pre´vost’s Law of Exchanges. For these researches he was awarded the Rumford Medal by the Royal Society in 1868. In his obituary tribute to Stewart, Tait explains the significance of Stewart’s contribution: His paper (which was published in the Transactions of the Royal Society of Edin- burgh) contained the greatest step which had been taken in the subject since the early days of Melloni and Forbes. The fact that radiation is not a mere surface phenomenon, but takes place like absorption throughout the interior of bodies, was seen to be an immediate consequence of the new mode in which Stewart viewed the subject.91 Stewart was the author of a number of other noteworthy papers, including those in which he presented the results of his experiments on: radiant light emitted through glass, tourmaline and uniaxial crystals; and, with Tait, the heating of a disk rotating in vacuo.92 At Kew Stewart was actively engaged in research into meteorology and terrestrial magnetism. Under his guidance Kew became a centre for the standardization and 89Stewart published one paper on mathematics: ‘On a Proposition in the Theory of Numbers’. See [68]. 90 [69, pix] 91[Ibid.] 92Amongst Stewart’s principal publications were the textbooks: Treatise on Heat (1866); Lessons in Elementary Practical Physics (1870) with W. W. Haldane Gee; Conservation of Energy (1872) and Primer in Physics (1872). For the ninth edition of the Encyclopaedia Britannica he con- tributed an article, ‘Terrestrial Magnetism’. He also presented papers on a variety of subjects to the Royal Societies of London and Edinburgh. An extensive summary of his work is given by his former student and successor at Owens College, Arthur Schuster (1851–1934) in [67]. 31 testing of the instruments used in experiments in these fields.93 In the area of solar physics he did much to establish the connection between sun spots, planetary configurations and terrestrial meteorology. At Owens College Stewart had as his students, the Nobel Prize winner and dis- coverer of the electron, J. J. Thomson (1856–1940) and the distinguished physicist, John H. Poynting (1852–1914). Little information is available on the religious aspect of Stewart’s life. Bap- tismal records reveal that he was baptised into the Scottish Presbyterian Church.94 Otherwise, the best there is in this regard is the following description of him: A devoted and fervent Churchman, who in later years was a member of a committee appointed by Lambeth Conference to promote interchange of views between scientific men of orthodox opinions in religious matters, he [Stewart] maintained throughout his career a deep interest in the more mysterious problems of existence, and became one of the founders of the Society for Psychical Research, over which he presided from the year 1885 until his death.95 2.2 The Christian man of science As Christian men, the authors of The Unseen Universe believed that they encoun- tered God both in the study of nature and in the Scriptures. For them there was no conflict between these two sources: they believed them to be of the same divine origin and to constitute the Two Books of Revelation. Referring to the Book of Nature metaphor, Tait and Stewart write: ‘In fine, the physical properties of matter form the alphabet which is put into our hands by God, the study of which will, if 93 [66] 94Stewart was baptised at the Tron Church in Edinburgh in 1828. The Scottish Presbyterian Church is not a member of the Anglican Communion and is different from the Scottish Episcopal Church in its administration in that it does not have bishops. 95 [70, p163] 32 properly conducted, enable us more perfectly to read that Great Book which we call the Universe.’96 Drawing on these two sources, Tait and Stewart would inevitably face criticism. Their appeal to Scriptures to unite physical theories and the spiritual was regarded by many as unscientific; they were criticized on points of religious licence and accused of venturing beyond their own areas of expertise. One critic applied the warning of the German botanist, Schleiden (1804–1881) who was one of the first to accept Darwin: The first rule which the exact investigator of nature should observe is, that he should not allow himself to pronounce an opinion, either in affirmation or negation, on subjects which do not fall and cannot fall within the sphere of his observation and experience. [. . .] If the natural philosopher comes, not in his special capacity, but in that of man merely, to speak of these matters (as every man has a right to do), then he must have before his eyes the second rule, which is, that he must not pass opinion, form his judgement, nor utter it, upon matters of any science to the present level of which he has not brought himself.97 To those that accused Tait and Stewart of ‘invading the province of religion’ they responded: we do not write for those who are so assured of the truth of their religion that they are unable to entertain the smallest objection to it. We write for honest inquirers—for honest doubters, it may be, who desire to know what science, when allowed perfect liberty of thought and loyally followed, has to say upon those points which so much concern us all.98 Consistent with Christian teachings, our authors had faith in a Creator who cre- ated us in His own image so that we might be capable of knowing Him. This doctrine encourages an honest inquiry into those fundamental questions which demand both 96 [55, 185] 97 [71] Originally Ueber den Materialismus (Leipzig : W.S.W., 1863). 98 [55, p162] 33 a scientific and theological approach. With an intimate knowledge of Christian doctrine, acquired through personal faith, and with a profound understanding of contemporary science, few would have been better placed than Tait and Stewart to ponder these questions and to identify the ‘connecting link between Revelation and Science’.99 Their scientific investigations, functioning as an inlet to Revelation, might even be considered to constitute religious service. Tait and Stewart expected opposition to the theories which they were putting forward, yet they invited reactions from within both the scientific and religious camps: ‘Entertaining these views we shall welcome with sincere pleasure any remarks or criticism on these speculations of ours, whether by the leaders of scientific thought or by those of religious inquiry.’100 A bold invitation indeed. 2.2.1 The science versus religion debate: Tait and Stewart’s contribution The Unseen Universe was regarded, by some at the time, as a model of how to engage in the science versus religion debate, which was at its height in that era— ‘according to the properly exacting conditions of Science’.101 In their own minds, the authors’ particular contribution to the debate was to identify, once and for all, the true antagonists. Tait and Stewart’s hypotheses were intended to demonstrate that there is nothing inherent in science which cannot accommodate Christian doctrines. Accomplishing this, materialism might be exposed as the true foe of religion. Tait, in particular, expressed a fervent dislike of materialists, referring in his address to Section A of the B.A.A.S. in 1871, to the ‘ignorance’ which shows itself in the ‘pernicious nonsense of the Materialist’.102 Clearly, he thought materialism abhorrent and a dangerous 99 [72, p419] 100 [55, p210] 101 [73] 102 [74, p7] 34 deviation from real truth. On this basis, it was inevitable that readers of The Unseen Universe would make the connection with John Tyndall’s Belfast address, which Tyndall had delivered in August 1874, as President of the B.A.A.S. Although Tait and Stewart made no reference to the address and no explicit references to Tyndall, it was determined that The Unseen Universe had been written as a refutation of Tyndall’s address. 2.2.2 Tyndall’s Belfast address John Tyndall was a self-professed materialist, who argued, with characteristic di- rectness, for the freedom of scientific inquiry from religious authority and an end to religious intrusion into the domain of science; maintaining the ‘superior authority of science over religious or non-rationalist explanations’.103 His philosophy was shared by fellow members of the X-Club; a group of friends and eminent scientists, who formed themselves into a London society. One of its members, the mathematician, Thomas A. Hirst (1830–1892) explained: ‘the bond that united us was devotion to science, pure and free, untrammelled by religious dogmas’.104 The theme of Tyndall’s address was the historical development of man’s intel- lect, with an emphasis on how well he realizes the natural impulse, inherent in men, to consider the ‘sources of natural phenomena’.105 Primeval man, rightly and nat- urally, drew on his experiences but erring in his focus—looking to man and not to nature—his theories took on an ‘anthropomorphic form’ so that ‘supersensual beings [. . .] were handed over the rule and governance of natural phenomena’.106 From the relationship between these capricious gods and mankind developed a sub-theme in 103 [75] 104 [76, p311] Members of the X-Club, in addition to Tyndall and Hirst: Joseph Hooker (1817– 1911), Thomas Huxley (1825–1895), Herbert Spencer (1820–1903), Edward Frankland (1825– 1899), George Busk (1807–1886), John Lubbock (1834–1913) and William Spottiswoode (1825– 1883). The X-Club met for the first time in November 1864. [76, p307] 105 [77, p1] 106[Ibid.] 35 Tyndall’s address—the association of religion with fear, superstition and restricted freedom. Whenever in the course of the history given by Tyndall, scientific develop- ment is found to be slow, halting or practically non-existent, the Christian influence is blamed. Some modern commentators have preferred to see Tyndall as a misunderstood pantheist rather than a materialist.107 They have associated with him a love of nature and a belief in a Power which is ‘immanent in or identical with the universe’, according to the definition of pantheism.108 Whatever the label, it is clear from his address that Tyndall’s views would have challenged the traditional doctrines of the Christian faith; for instance, his insistence on a materialistic explanation for the universe and the origin of life. Giving an account of Darwin’s theory of evolution, Tyndall considered the impli- cations of Darwin’s primordial germ—the common origin of all life. His conclusion: if we are to abandon the idea of creative acts and say instead that the primordial germ developed from matter, then a new and very different understanding of mat- ter is required, since the traditional conception cannot admit of life coming out of matter. It was a real frustration for Tyndall that science was unable to prove ex- perimentally that life can develop from anything other than life. Keen to apportion blame for science’s inability to understand the relationship between matter and life, he found fault with those who were the first to define matter—the mathematicians. While Tyndall insisted on the necessity of materialism, he accepted its insuffi- ciency: Understanding alone cannot satisfy man and for this reason, ‘physical science cannot cover all the demands of his nature’.109 Therefore, Understanding must be supplemented, with: passion, ‘Awe, Reverence, Wonder’; ‘love of the beautiful, phys- ical, and moral, in Nature, Poetry, and Art’ and ‘religious sentiment’.110 Tyndall’s 107Barton, for instance, in [78]. 108 [79] 109 [77, pp6–7] 110 [77, p60] 36 use of the term ‘religious sentiment’, which is loaded with offence, no doubt served to distinguish between faith, and philosophical or scientific reasoning. Although Tyndall was prepared to tolerate religious sentiment, he mistrusted its development into doctrine and religious practice; but they too might even be permitted, so long as they adapted to fall in line with all other evolving forms of knowledge: ‘The facts of religious feeling are as certain to me as the facts of physics. But the world, I hold, will have to distinguish between the feeling and its forms, and to vary the latter in accordance with the intellectual condition of the age.’111 Undoubtedly, Tyndall’s address had the potential to offend and antagonize Chris- tian scientists who maintained traditional beliefs. The biographers of William Robertson Smith (1846–1894), who Tait and Stewart consulted in the course of putting the book together, testify to this. Smith was a Scottish theologian and Semitic scholar and, between 1868 and 1870, assistant to Tait at Edinburgh.112 His involvement in The Unseen Universe is evidenced in a series of letters he received from Tait.113 His role was that of consultant: he was asked to give his opinion on proofs and to suggest improvements. His biographers recalled: Tyndall’s address created a sensation both in the theological and in the scientific world which was quite out of proportion to its importance as a serious attack on the orthodox position. It gave special offence to a distinguished group of scientific men who, like Lord Kelvin and Clerk Maxwell and their great predecessor, Faraday, were staunch upholders of the truths of revealed religion. This feeling of irritation was probably the immediate occasion of The Unseen Universe, a work of some celebrity in its day, which may be regarded as an elaborate counterblast to Dr. Tyndall’s provocative manifesto.114 A further connection between Tyndall’s address and The Unseen Universe is the 111 [77, pvi] 112For a biographical note on Smith written by Tait see [18, pp291–292]; originally Nature, 12 April 1894. 113See [70, pp162–166]. 114 [70, pp162–163] 37 correspondence in themes. Both touch upon: the continuity of nature; the practice of reaching beyond the bounds of the senses with intellect; a history of atomic theory; the mind-body problem; molecular processes and consciousness; and the mysterious link between matter and life, and energy and life. Tait’s biographer, Knott established the chronological link between the two: In the winter of 1874, a few months after the delivery by Tyndall of his famous presidential address before the B.A.A.S. at Belfast, it began to be whispered among the students of Edinburgh University that Tait was engaged on a book which was to overthrow materialism by a purely scientific argument. When, in the succeeding spring, The Unseen Universe appeared it was at once accepted as the fulfilment of this rumour.115 All this suggests that The Unseen Universe was written as a refutation of Tyn- dall’s address. But while we might admit that the book was hastily compiled in order to produce a timely response to Tyndall, we should realize that much of the substance of The Unseen Universe is present in the authors’ earlier work.116 Tait and Stewart themselves anticipated accusations alleging a hasty, ill-thought-out con- tribution. In the preface to their first edition: ‘We may state that the ideas here developed—very imperfectly, of course, as must always be the case in matters of the kind—are not the result of hasty guessing, but have been pressed on us by the reflections and discussions of several years.’117 115 [18, p236] 116Stewart’s earlier writings in the periodicals are discussed by Gooday in [80]; together with Stewart’s further work in the periodicals post publication of The Unseen Universe, which he produced largely in response to criticism of the book. I am grateful to Isobel Falconer for this reference. 117 [55, pxii] 38 2.3 The Principle of Continuity The book’s most difficult concept to understand is the Principle of Continuity—the thread by which all the arguments hang.118 The authors promise to define it but never do so explicitly—an illustration in terms of astronomy has to suffice—and each time the Principle is invoked there is some re-modelling of it. So we appeal to the critics for clarification, whom, in their frustration, went to great efforts to formulate a precise definition. The interpretation which is closest to what the authors appear to have had in mind offers a dual definition. In the spirit of science, the Principle constitutes a belief in the uniformity of the laws of nature: ‘The government of the universe has proceeded on a certain plan, ruled by certain fixed laws, we may therefore infer that it will continue to be so’.119 In the spirit of religion, the Principle is an expression of trust: ‘God has endowed us with certain capacities which enable us to dwell safely in the world and serve Him according to His laws. He will not distress or alarm His children by capriciously suspending or setting aside the laws which guide His universe.’120 Another critic, in effect, unites the two definitions by describing the uniformity of natural law as ‘the steady expression of the unchanging Will of the Creator’.121 In the preface to the fourth edition, the authors (who felt some clarification was warranted) asserted that the Principle ‘has solely reference to the intellectual faculties’.122 So perhaps the best interpretation of the Principle is as some sort of intellectual process—a process which makes sense of a ‘continuous chain of cause 118According to [81], the Principle of Continuity was ‘first enunciated by Sir William Grove in his inaugural address to the B.A.A.S. at Nottingham’. 119 [72, p422] Heimann discusses various philosophies of the uniformity of nature held in mid- Victorian Britain in [82]. 120 [72, p422] 121 [81] 122 [83, pvi] 39 and effect, of antecedent and consequent’.123 Tait and Stewart were more explicit about what constitutes a breach of con- tinuity: ‘Continuity, in fine, does not preclude the occurrence of strange, abrupt, unforeseen events in the history of the universe, but only of such events as must finally and for ever put to confusion the intelligent beings who regard them.’124 We will encounter the authors’ application of the Principle as we follow their arguments. Tait and Stewart proposed the following working hypotheses on the basis of the latest scientific investigations and discoveries. These hypotheses attempt to explain the beginning and end of the universe, and the production of life, and to realize the possibility of immortality. 2.4 The authors’ hypotheses 2.4.1 The Great First Cause, the beginning of the universe and the origin of life From the outset, Tait and Stewart identify themselves as believers in a Creator God: ‘Let us begin by stating at once that we assume, as absolutely self-evident, the existence of a Deity who is the Creator of all things.’125 Still, they are men of science and as such they believe that they are required to adopt a particular philosophy when speculating on the origins of natural phenomena: We think it is not so much the right or privilege as the bounden duty of the man of sci- ence to put back the direct interference of the Great First Cause—the unconditioned— as far as he possibly can in time. This is the intellectual or rather theoretical work which he is called upon to do—the post that has been assigned to him in the economy of the universe. If, then, two possible theories of the production of any phenomenon are presented 123 [81] 124 [55, p60] 125 [55, p47] 40 to the man of science, one of these implying the immediate operation of the uncondi- tioned, and the other the operation of some cause existing in the universe, we conceive that he is called upon by the most profound obligations of his nature to choose the second in preference to the first.126 Maintaining this philosophy throughout, Tait and Stewart consider first, the beginning of the universe, and the origins and evolution of life. Following the development hypothesis of Kant and Laplace they explain how the universe was formed: initially there was a ‘diffused or chaotic’ state; then, when the gravitating matter condensed and coalesced, potential energy was converted into heat and visible motion; and at the centre of what would become our solar system, a swirling mass threw out satellites and ‘planetary attendants’ as it cooled.127 Thus, the process of evolution had two principal elements—the integration of matter and the dissipation of energy.128 When the earth had matured sufficiently to produce favourable conditions, the first forms of life appeared and from these, complex biological life-forms developed. In formulating their hypotheses on the origins of life, Tait and Stewart do not offer any scientific objections to Darwin’s primordial germ: they cannot, of course, deny the staggering explanatory power of Darwin’s hypotheses. Still they have to account for the germ’s existence. They assume that this first form of life requires a living antecedent, for life cannot develop from anything other than life and the act of its creation would constitute a breach of continuity. And this antecedent must be conditioned, for the Principle of Continuity requires ‘an endless development of the conditioned’.129 To be ‘conditioned’ is to be subject to the laws of the universe—‘laws according to which the beings of the universe are conditioned by the Governor thereof, as regards 126 [55, pp131–132] 127 [55, p125] 128[Ibid.] 129 [55, p169] 41 time, place and sensation’.130 Rejecting the hypotheses of abiogenesis and spontaneous generation, Tait and Stewart turn to Scripture for the conditioned, living antecedent of Darwin’s primor- dial germ:131 If we now turn once more to the Christian system, we shall find that it recognises such an antecedent as an agent in the universe. He is styled the Lord and Giver of Life. The third Person of the Trinity is regarded in this system as working in the universe, and therefore in some sense as conditioned, and as distributing and developing this principle of life, which we are forced to regard as one of the things of the universe, in the same manner as the second Person of the Trinity is regarded as developing that other phenomenon, the energy of the universe.132 In their eagerness to provide cohesive and comprehensive hypotheses, Tait and Stewart are somewhat presumptuous in their interpretations of the Christian God and the specifics of Trinitarian doctrine. Having said this, the routes to this hy- pothesis are well sign-posted: the Holy Ghost is the Spirit residing in the souls of the faithful, who works in preparation for everlasting life; the Son is the developer of the Will of the Father which is expressed in the laws of the universe in which energy plays a fundamental role. In accounting for the variety of species, Tait and Stewart rule out separate acts of creation—maintaining the scientific man’s philosophy. Informed by Darwin, Huxley and Wallace, they cite natural variation, and natural and artificial selection as the probable causes. Still, the initial creative force remains the same: ‘We have driven the creative operation of the Great First Cause into the durational depths of the 130 [55, p47] 131Abiogenesis: the hypothesis that living matter can be produced from non-living matter; a term used first by T. H. Huxley in 1870. [84] Spontaneous generation: ‘the development of living organisms without the agency of pre-existing living matter, usually considered as resulting from changes taking place in some inorganic substance’. [85] 132 [55, p179] 42 universe,—into the eternity of the past,—but for all that we have not got rid of God.’133 2.4.2 The end of the visible universe Scripture and Science both point to a coming catastrophe, the one in language a child can understand, the other in the wordless eloquence of Nature’s changeless laws.134 Central to the science behind Tait and Stewart’s grand scheme is the objective existence of matter and energy. The objective existence of matter is a conviction based on the conservation of matter; a law which enshrines the ‘experimental truth’ that matter is not susceptible to changes in quantity.135 Tait and Stewart reason that if we admit the objective existence of matter, then we must afford an objective existence to anything which is conserved ‘in the same sense’ as matter.136 Examining a number of possibilities from abstract dynamics, they find that energy is alone conserved in this very particular sense: while energy may undergo transformation into a variety of forms, the sum of all the various energies in a closed system remains constant, according to the conservation of energy. The characteristic natures of matter and energy are described in a novel fashion in the following quotation from the book: matter is always the same, though it may be masked in various combinations, energy is constantly changing the form in which it presents itself. The one is like the eternal, unchangeable Fate or Necessitas of the ancients; the other is Proteus himself in the variety and rapidity of its transformations.137 133 [55, pp185–186] 134 [72, p428] 135 [55, p72] 136 [55, p73] 137 [55, p81] 43 Now, if in the universe there exists only matter and energy, and if matter is merely passive, we must conclude, Tait and Stewart argue, that all physical changes, including the thoughts and actions of living things, are transformations of energy. Accordingly, the following question is ‘of the very utmost importance’: ‘Are all forms of energy equally susceptible of transformation?’138 For if there exists some grade of energy which is less capable of transformation, after successive transformations, while the quantity of energy in the system remains the same, the majority will have degraded into the form which is least susceptible of transformation and will be unusable. Heat is the least transformable form of energy and unless a temperature gradient exists no work can be obtained from it. So while energy may be present in a system in the form of heat, none may be available for transformation. The process of transforming heat into work takes place in the thermodynamic operations of Carnot’s perfect heat engine.139 Tait and Stewart explain that the whole purpose of such an engine—which operates on the reversible Carnot cycle—is to transform heat into work and that its operations, being reversible, enable the greatest amount of work to be obtained from a given amount of heat. The process involves taking the heat which is not converted into work to the condenser and then reinvesting this heat, together with the heat-equivalent of the work done, back into the boiler. But even a perfect heat engine cannot transform all of the heat which passes through the system into useful work, for the condition which would enable this efficiency—that the temperature of the condenser is absolute zero—can never be achieved. So at each conversion attempt, only a portion of heat is available for transformation into work, the remainder is degraded. And with each successive 138 [55, p82] 139The perfect heat engine was a concept developed by the French engineer, Sadi Nicolas Le´onard Carnot (1796–1832) in Re´flexions sur la puissance motrice du feu et sur les machines propres a` de´velopper cette puissance (1824), a book on the thermodynamic workings of the steam engine. S. N. L. Carnot was the eldest son of Lazare Nicolas Margue´rite Carnot (1753–1823), author of Ge´ome´trie de position (1803). 44 transformation the heat becomes more degraded, or more dissipated, and so less and less is available for transformation into work. The analogy of the heat engine helps Tait and Stewart to explain why the present visible universe will one day come to an end. The sun functions as the furnace in the vast heat engine which we call the present visible universe. It radiates energy: a portion reaches earth, supplying life-giving energy; a larger portion still is released into the universe in the form of heat. The loss of the sun’s heat causes it to cool. From the analogy of the heat engine we know that the availability of energy in the universe will continue to diminish. Other events are also taking place. Tait and Stewart propose that one day the planets will be drawn into the sun by something like ‘ethereal friction’.140 They will lose their orbital energy, spin into the sun and merge into its mass. Upon impact, the power of the cooling sun will temporarily be restored as visible energy is converted into heat. The sun will then resume cooling, until the restoration of the next collision, and so on . . . Still, within this process is the possibility of the formation of new solar systems, formed from the nebulous dust surrounding some of the new coalesced masses. But the process will not continue indefinitely: in general, the total number of masses is still decreasing. A time will come when the matter of the universe is but a solitary cooling mass. It will exhibit no visible motion and will amount to nothing more than a useless store of energy—heat at a uniform temperature. Thus, according to Tait and Stewart, a combination of processes will affect the end of the present visible universe—the dissipation and degradation of energy, and the aggregation of masses.141 140 [55, p91] 141In [36, pp253–255] Crosbie Smith puts the The Unseen Universe in the context of developments in the science of energy during the nineteenth century. See also [86, pp63–66]. 45 2.4.3 The existence of an unseen universe Guided by the Principle of Continuity, Tait and Stewart arrive at the existence of something other than the visible—‘an invisible order of things’.142 They cannot suppose that only the visible universe exists, for both its beginning and end would constitute breaks of continuity. Instead, they must conclude that ‘the visible system is not the whole universe, but only, it may be, a very small part of it’.143 Tait and Stewart propose that the visible, both in matter and energy, evolved out of an unseen universe and that into this unseen universe it will ultimately re- treat. Though the two universes are currently independent, they remain ‘intimately connected’, with exchanges of energy taking place between them, facilitated by a less-than-perfectly-transparent ether.144 The energy that is stored in the unseen is held for the purposes of new creation. This conception of the unseen provides a novel way of coming to terms with the wasteful loss of the sun’s energy through dissipation: the law of conservation of energy applies to the whole system. The scientific theory which is put forward to explain how the visible evolved out of the invisible is that of William Thomson’s vortex atom. The theory, which was communicated to the R.S.E. in February 1867, was the latest speculation on the theory of matter. It regarded the universe’s primordial atoms as vortices developed from a pre-existing perfect fluid filling all space. The idea that motion could be a basis for matter was a well-established one, with the first real contribution being the theory of vortex motion suggested by the German physicist, Hermann von Helmholtz (1821–1894).145 142 [55, p157] 143[Ibid.] 144 [55, p158] 145Hermann von Helmholtz (1821–1894): initially, he was a doctor in the Prussian army; then, successively, Professor of Physiology at Ko¨nisberg, Bonn and Heidelberg; in the late 1860s he made the transition from physiology to physics and in 1871 he was appointed to the Chair of Physics at Berlin. For Tait, the appeal of Hamilton’s quaternionic theory was the promise of its application in the theory of vortex motion which Tait had encountered in Helmholtz’s 1858 46 Tait and Stewart’s only real objection to Thomson’s vortex atom theory was that Thomson had assumed a perfect fluid. In a perfect fluid there is no viscosity and in the absence of viscosity there can be no rotation without outside influence. Therefore, a perfect fluid necessitates a creative act in time which constitutes a break in continuity. Hence, in order to have both unbroken continuity and a vortex atom theory, the invisible universe cannot be a perfect fluid. In a perfect fluid, as Helmholtz proved, the rotating portions of fluid are arranged in knotted filaments which once set in motion forever ‘maintain their identity ’.146 In contrast, in a fluid which is not perfect, the permanence of the structures is no longer guaranteed. Our authors saw in this, evidence of the non-permanence of the visible order of things. Thomson’s theory impressed and inspired Tait and Stewart. They claimed that Thomson had ‘gone further than any one else’ in accounting for the origin of life.147 Indeed, during the 1870s Tait began to teach Thomson’s vortex atom theory in his Edinburgh lectures. In his address to the B.A.A.S. in 1871, Tait shared his hopes for the promising theory: ‘Our President’s [Thomson’s] splendid suggestion of Vortex-atoms, if it be correct, will enable us thoroughly to understand matter, and mathematically to investigate all its properties.’148 Tait was to invest much of himself in these mathematical investigations during the latter years of his career. He was chasing a full classification of the forms of knotted vortex rings, believing in the existence of a unique form of vortex ring for each of the elements. Eventually, Tait admitted, in Properties of Matter (1885), that ‘the discovery of the ultimate nature of matter is probably beyond the range of human intelligence’.149 paper [87]. Captivated by quaternions, Tait maintained a life-long fascination with them. For Tait’s translation of Helmholt’z paper into English see [88]. 146 [89, p20] 147 [55, p186] 148 [74, p6] 149 [89, p15] 47 Pursuing the chain of continuity in a backwards direction, our authors are led to the possibility of an endless number of invisible universes. In form, their model (Figure 2.4) resembles the Ptolemaic model. Each universe is represented by one of the concentric circles in the diagram. Figure 2.4: Tait and Stewart’s concentric model of the Great Whole.150 At the centre of the model is an evanescent smoke-ring; produced here on earth, by a smoke box for instance, as in Tait’s experiments in 1867. Tait’s experiments with smoke rings were conducted for the benefit of William Thomson in order to verify Helmholtz’s claims regarding the interaction of vortex atoms. The permanence of form of the smoke rings suggested the vortex atom theory to Thomson. The molecules of this smoke ring—being part of the visible universe—are vortex atoms developed from the unseen. And the entities which constitute the invisible universe are themselves vortex atoms developed from another invisible universe and so on. Pursuing the chain of continuity infinitely far back, Tait and Stewart reach a universe of infinite energy, which has an intelligent developing agency of infinite energy. Together these universes form ‘the Great Whole’—a self-contained system, which is ‘infinite in energy, and will last from eternity to eternity’.151 150 [55, p171] 151 [55, p172] 48 2.4.4 Superior and angelic intelligences Our authors do not rely on their multiverse theory, as is often done today with the modern equivalent, as a means of explaining how special our universe is.152 For them our universe has not been endowed with such fruitfulness on account of the laws of probability but on account of the generosity of the Creator. Tait and Stewart appreciate the ‘delicacy of construction’.153 In the development of complexity from that which is simple: be that the development of compounds from rudimentary elements, or human life from Darwin’s primordial germ. In the delicate balance of unstable forces: be that the regularity of the planets, or the abrupt meteorological changes of the sun and earth. And in the uniformity of the construction of atoms: evidence, for Herschel and Clerk Maxwell, that atoms are ‘manufactured articles’.154 The possibility that other intelligences besides humankind may exist in the uni- verse is not ruled out, which is surprising. Tait and Stewart are open, for example, to the possibility of life on Mars—in agreement with astronomers and physicists of the time, according to Tait and Stewart—though they understand that relevant knowledge will not be forthcoming in their own lifetime. A discussion on the reality of angelic intelligences is not entered into. Tait and Stewart say only that such beings would not belong to the present visible universe; for we cannot perceive them, nor do we imagine that their fate depends on that of the visible universe. 152In the multiverse theory, there exists a multitude of unobservable universes but, owing to the fine-tuning of natural constants, only one with suitable conditions for our survival. The historical development of the theory is given in [90]. I am grateful to Mark McCartney for this reference. 153A phrase used throughout The Unseen Universe. 154 [55, p167] 49 2.4.5 Immortality and the spiritual body Tait and Stewart do not speculate on the likelihood of personal immortality, having nothing similar to the Principle of Continuity to apply to it, but they do cite two sources in favour of immortality: statements about Christ and man’s ‘intense longing for immortality’.155 Whether the latter, being need-driven, constitutes real evidence is questioned by one critic, who regards a materialist accepting this as evidence as ‘a creature of the imagination’.156 Speculations on immortality are put into three categories of doctrine: (i) the ethereal state, (ii) the bodily state and (iii) the inconceivability and/or impossi- bility of a future state. The authors’ view is grounded in Scriptural revelation concerning Christ’s death and resurrection and the specific nature of His physical transformation. Thus, the nature of our future physical state is bodily but spiritual, or angelic, rather than natural. Tait and Stewart suppose that immortality is a transference. They propose the following three suppositions: It may be regarded as a transference from one grade of being to another in the present visible universe; or secondly, as a transference from the visible universe to some other order of things intimately connected with it; or lastly, we may conceive it to represent a transference from the present visible universe to an order of things entirely unconnected with it.157 To these suppositions they apply the Principle of Continuity. The first suppo- sition is discarded on the grounds that the visible universe had its beginning in time and will eventually come to an end. The third supposition is discounted for the following reason. Our authors state two conditions of continued intelligent exis- tence. An individual must: (1) possess an organ of memory, in order to maintain a 155 [55, p166] 156 [91] 157 [55, pp66–67] 50 connection with the past, and (2) be capable of action, or varied movement, in the present. If such an individual, with a connection to the past in one order of things, was to enter an entirely unconnected order of things, they would suffer permanent intellectual confusion which would constitute a breach of continuity. Therefore, only the second supposition remains; hence, immortality must represent a ‘transference from the visible universe to some other order of things intimately connected with it’. Regarding the nature of the transference, the authors’ hypotheses follow from their discussion on the latest theories—those of Huxley, especially—on the connec- tions between mind and matter, in terms of brain traces and the physical foundations of memory. Tait and Stewart construct a ‘frame’ for each individual so that they might be connected with the unseen—a ‘spiritual body’ which receives into it the molecular displacements of the brain.158 In this conception, the meaning of the anagram in Nature, noted earlier, becomes clear. In the conception of the spiritual body both conditions of continued intelligent existence are fulfilled. The second condition, the capability of action or movement, follows from the author’s premise that the unseen is to be full of energy when the visible universe comes to an end. In satisfying both conditions in the conception of the spiritual body, Tait and Stewart feel that they have demonstrated the possibility of the continuance of life beyond the death of the perishable material body. Tait and Stewart admit that their conception of the spiritual body, as an instru- ment for personal immortality, may be ‘detached from all conceptions regarding the Divine essence’.159 They maintain, however, that we are logically bound to accept some kind of spiritual body if we accept both the doctrine of immortality and the Principle of Continuity. 158 [55, p159] 159 [55, p198] 51 2.4.6 Divine action: miracles, the incarnation and the resurrection Tait and Stewart believe that in the light of their work there need no longer be a scientific objection to miracles. They are to be regarded as ‘transmutations of energy from the one universe into the other’; ‘the result of a peculiar action of the invisible upon the visible universe’.160 The incarnation of Christ presents no problem either: there is no breach of the Principle of Continuity because, in traditional Christian doctrine, Christ submitted Himself to the laws of the universe which are an expression of the Will of the Father. Tait and Stewart suppose that the resurrection of Christ could also have been accomplished without a break in continuity, by an infinite intelligent agency who is capable of developing the visible universe from the unseen. 2.4.7 The authors’ practical conclusion The analogy of Carnot’s perfect heat engine inspires the practical conclusion of Tait and Stewart’s hypotheses: And just as reversibility is the stamp of perfection in the inanimate engine, so a similar reversibility may be the stamp of perfection in the living man. He ought to live for the unseen—to carry into it something which may not be wholly unacceptable. But, in order to enable him to do this, the unseen must also work upon him, and its influences must pervade his spiritual nature. Thus a life for the unseen through the unseen is to be regarded as the only perfect life.161 2.5 Reception of The Unseen Universe Early editions of The Unseen Universe were widely known and subject to unusu- ally close scrutiny: the authors were successful in securing a readership that would 160 [55, p189] 161 [55, p192] 52 normally have turned away from an inquiry into unseen worlds.162 Despite being of reduced significance in the scientific world, having incorporated religious doctrine, The Unseen Universe made a notable impact on the ‘enlightened portion of the pub- lic’.163 Its appeal: the authors’ ‘real and intimate’ knowledge of the latest scientific theories.164 The dialogue which continued through subsequent responsive editions ensured the longevity of its appeal. Demand led to a sequel, Paradoxical Philosophy (1878), in which the hypotheses of The Unseen Universe were reworked into the form of a dialogue conversation between a German mathematician and a select group of the religious and social orthodoxy. It was dedicated to the members of the Paradoxical Society. Reactions to The Unseen Universe were truly diverse. One described the work as: ‘212 pages of the most hardened and impenitent nonsense that ever called itself “original speculation”’.165 Another declared: ‘nothing as original, and, so far as we judge, satisfactory, whether as regards respect for science, or as giving a logical basis for the belief in man’s immortality and the divine rule, has been written this century’.166 Criticism targeted many areas. (i) The authors’ ability to reason philosophically: the book’s philosophical sections were weaker than its scientific sections, making for a ‘curious compound of severe science and third-rate poetry’.167 (ii) The legitimacy of their scientific inferences: critics recognized illegitimate and unscientific speculations and cautioned against reliance on analogy.168 (iii) The effective communication of their hypotheses: the unscientific reader would understand little of the involved 162 [73] 163 [92] 164 [81] 165 [93] Quoted by Tait and Stewart in the preface to the second edition, [55, pvi]. 166 [94] 167 [95, p3] 168 [96, p1426] 53 science. (iv) Their authority to speak on matters of faith. (v) Issues of religious licence: in order to tie up a philosophical and theological Trinity, Tait and Stewart had produced a Trinity unconnected with the orthodox and ‘without warrant from Scripture’.169 And (vi) their application of science: inappropriately, for the purposes of authenticating the Bible and validating the teachings of Scripture. Issues with the authors’ conception of a spiritual body were numerous and com- plex, and thoroughly understandable. On a practical level: How is it that spiritual bodies do not interact with one another? Can consciousness exist in two places at once? Does the physical body retire entirely into the spiritual body upon death? . . . The English mathematician and philosopher, Professor William K. Clifford (1845– 1879), writing in The Fortnightly Review, seemed set to demolish the integrity of The Unseen Universe, having particular issue with the concept of a spiritual body.170 Tait and Stewart dealt robustly with the criticism they received. Later editions were revised and enlarged to make use of criticism and reply to objections; however, they refused to recall any statements. In the preface to the second edition they listed the charges brought against them: Some call us infidels, while others represent us as very much too orthodoxly credulous; some call us pantheists, some materialists, others spiritualists. As we cannot belong at once to all these varied categories, the presumption is that we belong to none of them. This, by the way, is our own opinion.171 Braced for an attack from religious leaders, Tait and Stewart were ‘delightfully perplexed’, and encouraged, by their response: the two parties agreed on many points and where there were differences of opinion, they were pointed out ‘with the utmost courtesy’ so as to safeguard the Church’s independence and show due regard 169 [97, p416] 170A much-edited version of Clifford’s review appears in [98]. For Tait and Stewart’s robust re- sponse to the review see the preface to their second edition, [55, ppviii–x]. 171 [55, pv] 54 for the authors.172 Inevitably, Tait and Stewart were accused of treating others badly in the course of their inquiry, particularly theologians, materialists and spiritualists. It is of little doubt that the work ‘wields a very heavy blow at materialism’, in the words of one commentator, and that the views expressed about contemporary spiritualists are extremely provocative; however, the claim has no real foundation as regards theologians.173 Another challenge faced by the authors was the misrepresentation of their work. Commentators evidently felt qualified, and at liberty, to report on the book’s sci- entific content. They lifted extracts from the book with no regard for context, often inserting terms of their own invention. This careless malpractice infuriated the authors, who addressed the issue publicly in new editions.174 Commentators who reacted positively to the book praised its boldness and orig- inality, and applauded the authors’ novel ideas which were laid out, without preju- dice, in an honest search for the truth and for a noble purpose. Critics recognized an ‘earnest religious spirit’ which ran through the work.175 The authors were com- mended for their applications of ‘sound scientific reasoning’.176 And for their ‘many clear expositions of scientific truth’.177 A number of commentators hoped and be- lieved that there may be some truth in their major propositions. 172 [83, pv] 173 [92] 174For instance, in [83, pviii] Tait and Stewart object to the term ‘luminiferous force’ which was misquoted by The Christian Treasury in [97, p414]. 175 [56] 176 [99] 177 [100] 55 2.6 String theory and M-theory anticipated In The Unseen Universe, Tait and Stewart went far beyond attempting to reconcile science and religion on a few points of conflict: they had proposed a theory of everything; and they were, without question, overambitious in attempting to work out every detail of the unity they saw. While we cannot remark on the verity of their hypotheses regarding immortality, we might note that some of the scientific hypotheses that they advanced have since reappeared, in modern physics’ approaches to a single scientific theory of everything: the concept of an infinite number of universes features in the modern multiverse theory; and Thomson’s vortex atom, which was finally abandoned following the acceptance of the theory of relativity around 1910, appears recast in modern string theory.178 Having said this, expect to find in the book evidence of imperfections in the con- temporary knowledge of science, recognized today as false theories and antiquated notions. The long-forgotten ether is probably the best example but even this has value again, reinterpreted in quantum theory as the energy of the vacuum. Address- ing the B.A.A.S. in 1871, Tait spoke of the value of “incorrect” scientific hypotheses: ‘in science nothing of value can ever be lost; it is certain to become a stepping-stone on the way to further truth’.179 2.7 Closing remarks Tait and Stewart—men of both scientific perception and religious instinct—maintained that science and religion are complementary aspects of the same truth. They were 178Knotted strings are at the heart of both Thomson’s vortex-atom theory and modern string theory. In modern string theory, vibrating strings are considered to be the fundamental build- ing blocks of matter; the resonant frequencies of the strings determining the type of particle produced. Both theories are discussed in their historical contexts in [90]. 179 [74, p5] 56 described by one critic as modern day examples of Newton and Faraday: ‘princes in science, and yet humble, believing Christian men’.180 Still, they recognized as two distinct groups, those who study the How of the universe and those who study the Why: A division as old as Aristotle separates speculators into two great classes,—those who study the How of the Universe, and those who study the Why. All men of science are embraced in the former of these, all men of religion in the latter. The former regard the Universe as a huge machine, and their object is to study the laws which regulate its working; the latter again speculate about the object of the machine, and what sort of work it is intended to produce.181 In our final quotation from the book, Tait and Stewart express their understand- ing of their roles as men of science, accepting humbly the limits of their own human intellect: the position of the scientific man is to clear a space before him from which all mystery shall be driven away, and in which there shall be nothing but matter and certain definite laws which he can comprehend. There are however three great mysteries (a trinity of mysteries) which elude, and will for ever elude his grasp [. . .]—they are the mystery of matter and energy; the mystery of life; and the mystery of God,—and these three are one.182 180 [101] 181 [55, p2] 182 [55, p183] 57 CHAPTER 3 TAIT’S STATISTICAL MODELS This chapter reveals Tait’s surprising involvement in statistics. 3.1 Introduction 3.1.1 Tait’s pocket notebook In January 2011 a pocket notebook once belonging to Tait (Figure 3.1) came into my possession. It had been retained prior by the Edinburgh Mathematical Society. I believe few are aware of the existence of the notebook and perhaps none have made a careful study of its contents. Although Tait utilized only a very small number of pages in the notebook, its contents reveal much about the scope of this remarkable polymath’s expertise and interests. Figure 3.1: Tait’s pocket notebook. 58 Occupying the first three pages of the notebook is Tait’s quaternion version of Green’s theorem.183 There is also a reminder to add in some references—presumably to a draft of Tait’s address to Section A of the B.A.A.S., 1871—and a draft of an unpublished poem by Tait (transcribed in Appendix B). But the most significant find in the notebook (Figure 3.2) comprises just six lines: Figure 3.2: An entry in French in Tait’s pocket notebook. Mariages par aˆges 15–25, 25–35, &c., . . . Les accouche´es par aˆges 15–20, 20–25, . . . Ne´s vivants le´gitimes De´ce`s par aˆges Femmes marie´es pour un ne´ vivant le´gitime Rapports des naissances dans quelques autres pays And so it appears that Tait, at some stage, was concerned with the collection and/or 183This entry may relate to Tait’s paper, ‘On Green’s and Other Allied Theorems’, Trans. Roy. Soc. Edinburgh XXVI (1870). 59 analysis of data; data on the number of children born to women of various ages, tak- ing into account the age at which they were married. Immediately, the following questions spring upon us: What was the nature and circumstance of Tait’s in- volvement in statistics? Did he contribute anything significant, including published papers? Who was (and is) aware of his involvement? And why was he writing in French? I aim to address each of these questions during the course of this chapter. 3.1.2 Tait’s contribution to statistics and probability As a preliminary step in trying to understand the significance of this notebook en- try, I compiled a list from Tait’s published work of contributions which might come under the umbrella of statistics and looked for a correspondence with the notebook entry. One paper listed in Knott’s bibliography of Tait’s works in [18] looked partic- ularly promising: ‘Note on the Formulae Representing the Fecundity and Fertility of Women’ [102], published in the R.S.E.’s Transactions in 1867. On this paper, Knott made the following intriguing remark: ‘Fecundity is found to depend linearly and fertility parabolically on age [. . .] The formulae are known to Statisticians as Tait’s Laws.’184 From this I inferred that in his paper on fecundity and fertility, Tait had modelled the regularity he had observed in some data and that his results were well known. An extended version of this paper appeared as part VI of James Matthews Dun- can’s book, Fecundity, Fertility, Sterility and Allied Topics (1866) [103]. Comparing the notebook entry with Tait’s contribution to Duncan’s book, it was immediately obvious that the two corresponded; in relation to data categories and the grouping of ages. And there was a further correspondence; between dates recorded in the notebook (August, October and November 1870) and the time when Tait would be preparing for the second edition of the book, published in 1871. Further investigation revealed the full extent of Tait’s contribution to probability and statistics: 184 [18, p353] 60 1865 ‘Probability’ [104]: an article published in the first edition of Chambers’ En- cyclopaedia.185 And ‘On the Law of Frequency of Error’ [105]: a paper in which Tait devised an intuitive and probabilistic approach to deriving the error law.186 1866 Part VI of James Matthews Duncan’s Fecundity, Fertility, Sterility and Al- lied Topics [103]: Part VI is an extended version of Tait’s paper, ‘Note on the Formulae Representing the Fecundity and Fertility of Women’ [102]. In 1871 a second edition of Duncan’s book [106] was published, with additions, amendments and a review of the first edition. 1873 ‘On a Question of Arrangement and Probabilities’ [107]: a paper on a combi- natorial problem related to the game of golf.187 185In this article, Tait provides an introduction to the elementary definitions and principles behind the mathematical theory, using familiar examples from games of chance: coins, dice, balls drawn from a bag, horse racing odds, etc. An indication of the applications of the theory is also given: to life assurance; in the application of the method of least squares; and in ascertaining the value of evidence given in a trial and the probability of correctness of a jury’s verdict. 186It occurred to Tait, while working on his probability article, that the best approach to the concept of error is to take, as the logical foundation for your investigations, something simple and intuitive; thereby, avoiding the ‘unnecessarily elaborate analysis’ of Laplace and Poisson. [105, p139] Adopting this philosophy, Tait bases his investigations in his paper on error on the following problem: ‘To find the relative probabilities of different combinations of mutually exclusive simple events in the course of a large number of trials.’ [Ibid.] He looks at the possible combinations of black and white balls drawn from a bag. The error associated with a particular combination is its deviation from the most probable combination. The law of error y = Ae−M 2x2 is the ratio of the probability of that particular combination to the probability of the most probable combination. 187The problem Tait investigates in this paper is suggested by the game of golf: ‘When a player is x holes “up,” and y “to play,” in how many ways may he win?’ [107, p37] (a reincarnation of the Problem of Points). Tait considers both a geometrical and analytical approach to the problem. His fundamental equation is the recurrence relation Px+1,y+1 = Px+2,y + Px+1,y + Px,y, where Px,y is the number of ways a player may win when x up, with y to play. At any given hole he may win, draw (halve) or lose the hole, so that the number of holes he is ahead may increase by 1, stay the same or decrease by 1. Chris Pritchard has published on this paper: see [108]. He 61 1885 Five papers on the kinetic theory of gases in which Tait applied probability to –1892 the behaviour of particles.188 It was a delight to realize, after some research, that Tait’s contributions to prob- ability and statistics do in fact serve to sign-post developments in these areas up until around 1900. It was in reading of the progress of probability and statistics that I came to understand how natural Tait’s involvement was and the significance of his contributions in their proper context.189 In this chapter, the focus is Tait’s contribution to Duncan’s book, Fecundity, Fertility, Sterility and Allied Topics. 3.1.3 James Matthews Duncan (1826–1890): a biographical sketch James Matthews Duncan M.A. M.D. (Aberdeen) (Figure 3.3, page 63) was a leading physician, obstetrician and surgeon. He was born in Aberdeen, in April 1826. He was educated at the grammar school in Aberdeen and at Marischal College, and studied medicine at Edinburgh and Paris. He was regarded as the most promising student of Sir James Simpson (1811–1870), the internationally celebrated physician and obstetrician, with whom he had worked on the introduction of chloroform as an anaesthetic. Duncan established himself in Edinburgh during the first half of his career: he was instrumental in the founding of Edinburgh’s Sick Children’s Hospital (1860); he served on the ward for diseases of women at the Edinburgh Royal Infirmary; and he lectured on midwifery and was an examiner on the subject in various universi- gives some historical background to the Problem of Points, explains how Tait solves the problem by analysing the coefficients of a trinomial expansion (a + 1 + a−1)n and devises an extension of the problem, assigning unequal probabilities to the three possible results at each hole. 188Tait’s papers on the kinetic theory of gases were reprinted in volume II of his Scientific Papers. 189See Porter’s The Rise of Statistical Thinking 1820–1900 [109] for example. 62 ties and colleges, including St Andrews.190 From Edinburgh he went to London, as Physician Accoucheur and lecturer on midwifery at St Bartholomew’s Hospital (1877). In 1883 he delivered the Goulstonian lectures, ‘On Sterility in Women’ to the Royal College of Physicians of London. These lectures were based on Fecundity, Fertility and Sterility.191 Duncan’s approach—which was founded on extensive sta- tistical work—was ‘widely regarded as a breakthrough in the study of fecundity and sterility’.192 Figure 3.3: Dr. Matthews Duncan, a newspaper clipping of a sketched portrait from The Modern Athenian, 13 October 1877. Sourced from Tait’s scrapbook. Duncan was a Fellow of the Royal College of Physicians of Edinburgh (1851), the Royal Societies of Edinburgh (1863) and London (1883), and the Royal College 190University College, Dundee was founded in 1881. In 1897 it became a constituent college of the University of St Andrews. In 1954 University College was renamed Queen’s College; becoming the University of Dundee in 1967, when it gained independence from St Andrews. Pre-1967, midwifery was taught on the Dundee site. 191The lectures were reproduced in the British Medical Journal in March 1883. 192 [110] 63 of Physicians of London (1883). He served as President of the Obstetrical Societies of Edinburgh (1873–1875) and London (1883). He received honorary degrees from the Universities of Edinburgh and Cambridge (LL.D.), and the University of Dublin (M.D.). Duncan married in 1860 and the couple had at least ten children. He died on 1 September 1890 at Baden-Baden, aged sixty-four.193 Association with Tait Duncan and Tait were associated through the R.S.E. and through an Edinburgh society called the Evening Club. It is possible that they also knew one another through Duncan’s Edinburgh practice. —Through the Royal Society of Edinburgh. Around the time of collaboration, both Tait and Duncan were prominent members of the R.S.E.: Tait had been elected a Fellow in 1861 and as one of the Secretaries to the Ordinary Meetings in 1863; Duncan had been elected a Fellow in 1863 and during the periods 1866–1868 and 1875–1877 he served the Society as Councillor.194 —Through membership of the Evening Club. Both men were among the founding members of an Edinburgh society called the Evening Club. The society was mod- elled on London’s Century and Cosmopolitan Clubs, and was a thriving society for some twenty-five years. According to Knott, it could boast of such members as: ‘prominent Edinburgh lawyers, artists, physicians, clergymen, teachers both in college and school, bankers, commercial men, publishers, engineers etc.’.195 From Knott we also learn that Tait personally introduced some seventy guests to the Club between 1870 and 1884, and that members of the society would ‘recall Tait as one of the great personalities, taking his full share in the talk, and enjoying the relaxation 193Uncited sources of biographical information on Duncan: [29] and [111]. 194 [29] 195 [18, p348] According to the same source [18, pp347–348] Mr. A. Findlater, editor of Chambers’ Encyclopaedia, also enjoyed membership at the Evening Club. 64 from the hard thinking in which he usually passed his evenings’.196 The group met, generally twice weekly, ‘for purely social intercourse, cards and serious subjects of debate being taboo’.197 Unfortunately, Duncan and Tait’s collaboration pre-dates their association at the Evening Club which was founded in 1869. —Through Duncan’s private practice in Edinburgh. It is possible that Tait, aware of Duncan’s reputation as a leading obstetrician, had him to tend to his wife Mar- garet during her pregnancies: Duncan had a private practice in Edinburgh between 1851 and 1877, and Tait’s children were born in Edinburgh between 1861 and 1873, excluding his eldest child, Edith who was born in Belfast in 1860. 3.2 Fecundity, Fertility and Sterility (1866) Tait opens his chapter in Fecundity, Fertility and Sterility with the following state- ment, in which he explains his objective and qualifies the extent of his involvement: Dr. Matthews Duncan having requested me to point out to him some simple method of comparing the fertility of different races, I endeavoured, as a preliminary step, to represent by formulae some of the chief results which he has obtained in his very lucid and elaborate papers recently read to this Society [R.S.E.] and printed in their Transactions for 1863–4 and for the present session. Some of the formulae which I have obtained are so simple, and accord so well with the tables, that I have thought them worth bringing before the Society. Of course it must be understood that I advocate no theory, and pretend to no physiological knowledge of the question. I merely try to present, in a simple analytical form, the contents of some of Mr. Duncan’s tables.198 Tait proposes to work towards his objective by considering: (i) the fertility and fecundity of the mass of wives; (ii) the fertility and fecundity of the average indi- vidual; and (iii) the relative fertility and fecundity of different races. He defines 196 [18, p349] 197 [18, p348] 198 [103, p207] 65 fecundity and fertility as follows:199 By fecundity at a given age we mean the probability that during the lapse of one year of married life, at that age, pregnancy, producing a living child, will ensue. By fertility, at any age, we mean the number of children which a married woman of that age is likely to have during the rest of her life, or some numerical multiple of it. Note, Tait’s definition of fecundity is easy to misinterpret: the event that Tait is concerned with is pregnancy, rather than the birth itself. 3.2.1 Fertility and fecundity of the mass of wives Tait’s data sets I: • Data set A—Fecundity values for the wives of Edinburgh and Glasgow, as a whole— [103, p19] • Data set B—Fertility values for the mass of wives in one district in Edinburgh, St George’s-in-the-East— [103, p139] Linear model of fecundity Working with his definitions, Tait formulates an expression for fertility at age t. It is given as the sum of the fecundities from age t until the onset of sterility, which Tait assumes takes place when a woman reaches 50. Thus Ft = ft + ft+1 + · · ·+ f49 = Σ 49 i=tfi (3.1) To verify the formula, Tait takes the fecundity values in data set A and, using (3.1), calculates fertility values for the Edinburgh and Glasgow wives. He then compares these values with the observed fertility values from data set B for married women in one district in Edinburgh. Mindful of problems with the data—including the non-comparability of the two data sets in terms of age groupings— Tait is pleased 199 [103, pp208, 208] 66 with the comparison, writing: ‘These numbers agree as well as could possibly be expected.’200 To derive his linear model of fecundity (3.2), Tait fits a straight line to the continuous curve (Figure 3.4, page 68) comprising the fecundity values from data set A. The formula for the straight line is easily deduced once the value of the x-intercept has been estimated. This is Tait’s linear model of fecundity: he has fecundity at age t ft = k(50− t) (3.2) where k is a constant to be found. Tait describes his linear model as ‘a simple formula very closely representing the tabulated results’.201 He remarks on the closeness of the fit between ages 17 and 40 (Figure 3.4) and suggests, in places where the curve clearly departs from the line, that inaccuracies and omissions in the data are to blame. Analysis of the same data (Table LXXVII [103, p209]) using the statistical package R suggests that Tait provided very precise estimates of the y-intercept and slope of the fitted line.202 200 [103, p211] 201 [103, p213] 202The coefficient of determination (the R2 statistic) is 0.9944, meaning that 99% of the total variation in fecundity is explained by age. R produces the following linear regression model: E(fecundity) = 78.2−1.63(age) (3 s.f.). Tait has ft = 32 (50−t). Tait’s values for the y-intercept and slope fall within the 95% confidence interval R generates for the parameters, i.e. a range of values for β1 and β2 such that 95% of the data is explained by the model E(Y ) = β1 + β2x. 67 F ig ur e 3. 4: T ai t’ s gr ap h of ‘f ec un di ty at di ff er en t ag es ’. [1 03 , p2 12 ] N ot e th e ab se nc e of a ve rt ic al ax is . 68 Quadratic model of fertility Tait derives a quadratic model of fertility (3.3) by substituting his linear model (3.2) into (3.1) for each fecundity. He has fertility at age t Ft = 1 2 k(50− t)(51− t) ≈ 1 2 k(50− t)2 (3.3) which is a reasonable approximation to make. Once again, a tabular comparison follows; this time between observed and model- generated fertility values. Putting the quadratic formula to use, Tait estimates the total number of children a woman can expect to have (i.e. her completed family size) based on the number of children she has had to date.203 Taking age at marriage into account Tait finds evidence within Duncan’s tables to suggest that there is a relationship between the age at which a woman is no longer able to have children and her age at marriage: ‘the age of sterility is uniformly later as the age at marriage is greater’.204 See Table 3.1 (page 70) for the data. Tait adjusts his linear and quadratic models accordingly. His linear model (3.2) becomes ft = k(C − t) where C is the age at which sterility arrives. However, since the value of the param- eter C is dependent on the age at marriage, Tait has, in effect, fta = k(Ca − t) where fta is the fecundity at age t for women married at age a and Ca is the corre- sponding age of sterility for women married at age a. 203See [103, p214]. 204 [103, p216] 69 Age at Marriage Age of Sterility 15–19 43 20–24 46 25–29 47.5 30–34 48.5 Table 3.1: ‘showing the age at marriage, and, of the advent of sterility’. Repro- duced from Table LXXXIV [103, p216]. Age at Marriage Whole Fertility Ft 15–19 10 10 20–24 7.7 7.4 25–29 5.5 5.0 30–34 3.4 3.1 Table 3.2: ‘showing the influence of the advent of sterility upon the whole fertility of marriage’. Reproduced from Table LXXXV [103, p217]. The values in the third column are those produced by the original quadratic model (3.3) in which age at marriage is not taken into account. Adjusting his quadratic model of fertility (3.3), Tait derives an expression for what he terms “whole fertility”. He has whole fertility = Fa = 1 2 k(Ca − a) 2 where a is the age at marriage and Ca is the corresponding age of sterility.205 We 205Tait uses t for both the current age of a woman and her age at marriage. To avoid confusion, I 70 can interpret whole fertility as the total number of children any woman marrying at age a can expect to have before the onset of sterility which is determined by her age at marriage; that is, her completed family size as determined by her age at marriage. The figures are given in Table 3.2 (page 70). Geometric distribution? Of particular interest is a remark Tait makes in connection with his interpretation of the expression f17 = 1013 . He writes: ‘It may be well to notice that the interpretation of the expression f17 = 1013 is, that a wife of 15–19 will, on the average, become pregnant at 1.3 years after marriage—that is, she will have a child within about two years of marriage.’206 It appears that Tait had an intuitive understanding of the geometric distribution. He has given, in effect, the expected value 1p of a geometrically distributed random variable, p being the probability of success in each trial. In this case, the random variable is the time interval between marriage and the first success, i.e. pregnancy resulting in a living child. 3.2.2 Fertility and fecundity of the average individual Tait’s distinction between “for the mass of women” and “for the average individual” is unclear. I can only think that for the average individual he is considering age-at- marriage subgroups. Tait’s data sets II: • Data set C—Tables of sterility, giving the percentages of sterile and not-sterile wives from Edinburgh and Glasgow, taken as a whole— [103, p193] have chosen to use a to denote age at marriage. 206 [103, p217] Tait is working with midpoint values because the data for each separate age is unavailable. 71 Testing the linear and quadratic models against new data This new data set C provides Tait with a further opportunity to test his models. He supposes that for each five-year age group, the figures for the percentage of not-sterile wives will be proportional to the average fecundities and so will form a straight line when plotted.207 Instead, however, they produce the dotted curve in Figure 3.4 (page 68). Tait suspects problems with the data. For the depressions in the curve between the ages 25–30 and 40–45, Tait offers a remarkable explanation, citing ‘the loose way in which women from 30 to 40 call themselves 30, and those from 40 to 50 call themselves 40’.208 To substantiate his bold claim, Tait quotes some figures from the 1851 Census Report. He calculates that 140,000 women aged less than ten years in the period between the censuses of 1841 and 1851. This is evidence to Tait of ‘how strong is the desire [amongst women] to be considered as remaining under the magic limit of thirty years of age’.209 Of course, possible alternative explanations might include: (i) the questionable reliability of the early censuses and (ii) immigration, remembering, especially, the displacement of people by the potato famine in Ireland during the period 1845–1852. A tabular comparison follows, comparing (i) the observed values of percentages of not-sterile women with the model-generated fecundity values produced by the linear model (3.2) for a suitable k (Table LXXXVII [103, p220]) and (ii) fertility values, derived from equation (3.1) using observed percentages of not-sterile women, and model-generated fertility values obtained from the quadratic model (3.3) (Table LXXXVIII [103, p221]). Foremost in Tait’s mind is the importance of deriving a simple model. He is willing to sacrifice a little in terms of the amount of data explained by the model for the sake of its simplicity: ‘It is easy, of course, to construct a formula to represent 207In this context, sterility means an inability thus far to produce a living child. 208 [103, p219] 209[Ibid.] 72 any series of numbers, but unless it be simple it is of little use’.210 Tait is pleased with the results of the tabular comparisons, bearing in mind the following problems with the sterility data. First, there are insufficient numbers of women in the age groups 40–44, 45–49. Second, there is an issue with the 15–19 year old group: women in this group may have been married for such a short time that pregnancy and children will fall within the next age group; and this effect is amplified within the first age group as there is no group below (of younger women) feeding into it, i.e. the 15–19 group will not include mothers married at 14. Third, plural births are not eliminated. Here is a simple example to illustrate the problem with plural births. Consider three women A, B, C. Woman A gives birth to two children, woman B gives birth to one child and woman C is childless. Without due care, it may appear that three women have each given birth to one child, which will result in a 0% sterility figure. Tait notes the implausibility of a 0% sterility figure, an outlier, for the 20–24 age group (Table LXXXVII [103, p220]).211 3.2.3 Relative fertility and fecundity of different races Applying the above results, Tait is able to compare the fertility of different races. Tait’s data sets III: • Data setD—The Registrar General’s Reports for England and Scotland (1866), extracted into Tables LXXXIX and XC— [103, p224] Tait’s ratio Working towards a means of comparing the fertility of two given races, Tait has to assume a number of ‘postulates’ (or assumptions). First, that for a period of ten or fifteen years, the number of marriages at any age remains the same—a stationary variable in modern terminology—so that we can represent the number of births in a 210 [103, p220] 211See [Ibid.]. 73 single year by the total fertility of those married in that year. Second, that the two countries obey the same law of fertility; or, the ratio of the fertility of the second race to that of the first at every age t, t+ 1, . . . is a constant e. Tait has e = β′ β ΣµtFt Σµ′tFt for t from 15 to 49, where µt is the number of marriages of women at t years of age in any one year, β is the number of legitimate births in a year and Ft is the fertility at age t. Dashes distinguish between the nations.212 Comparing the fertilities of England and Scotland, Tait finds e = 0.812 or that the fertility in England is 810 of that in Scotland. On this result, he makes the following remark: It is to be observed that if the insinuations we sometimes hear about Scottish mar- riages have any foundation in fact, their consideration would tend to make the dif- ference in fertility between the two countries even greater than that just given; for legitimation per subsequens matrimonium does not put a child’s name on the Regis- trar’s books.213 By per subsequens matrimonium, Tait means the legitimization upon marriage of children born before their parents married: it had been a law in Scotland for cen- turies before it was introduced in England in the Legitimacy Act of 1926.214 Tait’s fertility values for England and Scotland, taking age at marriage into ac- count, are those shown in Table 3.3 (page 75). 212For details of how Tait forms his ratio see [103, pp222–224]. 213 [103, p225] 214 [112, pp5–6] 74 15–19 20–24 25–29 30–34 35–39 40–44 45–49 Scotland 7.44 5.54 3.77 2.30 1.24 0.37 0.06 England 6.04 4.49 3.02 1.87 1.01 0.30 0.05 Table 3.3: ‘showing comparative fertility of a mass of wives in England and Scotland, taking account of the age at marriage’. Reproduced from Table XCI [103, p226]. 3.2.4 Tait’s appreciation of good data Throughout, Tait has been plagued by issues associated with poor quality data. In his concluding remarks to Part VI, he alludes to the ineffectiveness of contempo- rary information gathering which, of course, has a bearing on the validity of the conclusions drawn from analysis of the data. He writes: As in all questions of average, the value of our deductions in this matter is mainly dependent on the extent and accuracy of our data, and it is sad to think that the enormous blue-books which load our shelves contain so much painfully-elaborated information which is of no use, and so little of those precious statistics which would at once be easy of acquirement and invaluable to physiologists.215 3.2.5 The second edition (1871) The first edition of Fecundity, Fertility and Sterility was so well received that a second edition [106] was published in 1871. Included in this revised and enlarged second edition is an anonymous review of the first edition. Regarding the critic’s identity, we know only that he is not known to Duncan, for the critic remarks on Duncan’s kindness in sending his tables to him, who is ‘a complete stranger’.216 I discovered the identity of the critic in an unusual context—in 215 [103, p227] 216 [106, p256] 75 Memoir of Fleeming Jenkin (1912) by Robert Louis Stevenson (1850–1894). Before becoming a celebrated author, Stevenson had studied engineering at the University of Edinburgh under Professor Fleeming Jenkin (1833–1885).217 Stevenson reveals in the Memoir that it was Jenkin who had reviewed Duncan and Tait’s first edition and that Duncan had included the review in the second edition.218 Fleeming Jenkin’s review Jenkin’s position is that he believes that factual information on fertility and fecun- dity issues ought to be available to ordinary people, so that they might evaluate the risks associated with pregnancy and childbirth, and make informed decisions about the size of their families. Here is a brief summary of some of the key points he makes in the review, in relation to Tait’s contribution especially. — Duncan and Tait’s approach is straightforward. They have presented their information honestly and intelligently, without looking to manipulate their findings or draw inferences. — Insufficient data was available to Duncan and Tait. — Duncan has been unable to produce information on the risks associated with bearing children in rapid succession. An optimum interval between births needs to be ascertained. — Duncan and Tait’s definitions of fecundity and fertility are inconsistent. The difference in their interpretation of fecundity is responsible for the difference 217 [5, p186] For a period Stevenson had worked on scientific investigations in Tait’s laboratory. [33, p47] See [113] for Stevenson’s recollections. Jenkin and Stevenson were both former pupils at the Edinburgh Academy: Henry Charles Fleeming Jenkin (1833–1885) was a classmate of Tait’s during the 1840s; Robert Louis Stevenson (1850–1894) attended the Academy between 1861 and 1863. [4, pp117, 263–264] 218See [114, p91]. 76 in the conclusions they draw from the data. When Jenkin redefines the terms for himself he follows Tait more closely than Duncan.219 — Tait’s linear model (3.2) is a ‘general law of great importance’.220 It produces fecundity values which are close to the observed figures. Its simplicity is im- pressive: ‘It is very rare to find a very simple result derived from complex elements’.221 From Tait’s linear model we understand that women are pos- sessed of different degrees of fecundity (high, low and intermediate), the levels exhibiting themselves in the number and spacing of children produced. An awareness of the degrees of fecundity may lead to an explanation of what has been observed without the need to consider, for instance, sterility as depend- ing on age at marriage. However, Tait’s linear model has been ‘proved for a mass of women only’ and so the law must be re-interpreted as follows: ‘The average number of children per annum born to a mass of women of any age is proportional to the difference between that age and 50.’222 It is likely the law will hold for an individual but to prove so would mean verifying that the law holds for large subgroups of women, all married at the same age. — Tait’s application of his formulae to the comparison of nations is worthwhile. Similar analyses should be carried out on the ‘inferior races’.223 Comparing the fertility of England, Scotland, Ireland and Sweden In the second edition—with access to new data from Sweden and from the Registrar- General’s Reports for England, Scotland and Ireland—Tait compares the fertility of 219See [106, p245]. 220 [106, p246] 221 [106, p247] 222 [106, p246] 223 [106, p261] Surely a reference to the developing nations. 77 these four nations. The results of his comparison are presented in Table XCV [106, p239] and in Table 3.4 where the average fertility of married women of 15–19 in each of the four countries is given. Nation Average fertility of married women of 15–19 Fertility relative to Scotland England 6.21 86% Scotland 7.23 100% Ireland 7.14 98% Sweden 8.48 117% Table 3.4: Average fertility of married women of 15–19 for four nations. Repro- duced from Table XCVI [106, p240] with an additional third column showing how Tait interprets each nation’s figures as a percentage of Scotland’s figure. It is likely that Tait chose to include Sweden in his cross-national comparison because of the wealth of high-quality demographic data available in that country. Indeed, professional demographers have since made frequent use of Sweden’s demo- graphic records, either to demonstrate new methodologies or to test new models.224 Sweden boasted an unusually complete set of parish registers, recording vital statis- tics (births, marriages and deaths) from the sixteenth century onwards. Yet fur- ther improvements in the quality of Swedish demographic data took place because of a reorganization of data-collection and data-recording procedures, following the establishment of a central statistical commission and office in 1858. The central statistical office had exclusive charge of the statistics of population. Its statistical library contained some 4000 volumes c.1860. In 1860 Dr. Berg—Vice-President of the commission and head of the statistical office who worked on examining and tab- ulating extracts of Sweden’s parish registers—attended the International Statistical 224 [115, pp8–9] 78 Congress in London as the official delegate for Sweden.225 He presented a detailed report on the present state of statistical inquiry in Sweden.226 3.2.6 Why Tait was writing in French The further work done by Tait in the second edition provides a possible explanation for why he was writing in French in his pocket notebook. The data that he had ac- cess to from Sweden was made available to him by Dr. Berg, Chief of the Statistical Bureau in Stockholm.227 In correspondence with Dr. Berg, I believe that Tait would have used French as a common language.228 Therefore, I suggest that Tait’s French entry in his pocket notebook was perhaps a draft list of the data he wished to obtain from Dr. Berg. I have already remarked (page 60) that the dates recorded in the notebook correspond with the time when he would have been preparing for the sec- ond edition of Duncan’s book. Of course, it is possible that Tait was corresponding with someone else, attempting to get hold of some data from France perhaps. What is less likely is that Tait was writing in French purely as an intellectual exercise, just for fun or to keep the contents of his notebook private. 3.3 Concluding remarks Ultimately, Tait’s “digression” into probability and statistics is not surprising if we bear in mind the following: — Tait’s credentials. (i) Being a “combinatorial man” Tait would have had no 225 [116, p384] 226 [117, pp45–49] The preceding information on Sweden has come from this source. 227 [106, p237] 228French was the language of the International Statistical Congress, which was held in London in 1860: German was permitted at the Congress but English was not. [118, p377] It is very unlikely that Tait attended the International Statistical Congress in 1860. He certainly does not appear on the list of delegates. 79 difficulty in overcoming the logic of the subject.229 (ii) He had experience in handling data and would treat Duncan’s data as he would any data coming from experiment or observation. — Tait could be counted amongst those driving forward statistical developments in the nineteenth century—not mathematicians, especially, at this stage—but social scientists, biologists, physicists and generalists.230 3.3.1 Influence of Tait’s Laws Earlier in this chapter (page 60) I included a quote from Knott in which he referred to “Tait’s Laws”. Evidence suggests that the first of Tait’s Laws (his linear model) was of influence well into the twentieth century. In 1906 the statistician, G. Udny Yule (1871–1951) referred to Tait’s first law in a paper [119] published in the Journal of the Royal Statistical Society entitled, ‘On the Changes in the Marriage- and Birth- Rates in England and Wales During the Past Half Century’.231 Yule used Tait’s linear model of fecundity on age to calculate expected birth-rates and his coefficient k as a measure of fertility. It seems that Yule’s 1906 paper led to a number of subsequent authors citing Tait’s Laws.232 229For evidence of Tait’s remarkable intuition when tackling combinatorial problems, consider his work on knot enumeration: without the benefit of rigorous methods, Tait was able to determine whether or not two knot diagrams were equivalent; and in this way, he was able to produce the very first knot tables. 230 [109, pxi] 231G. Udny Yule (1871–1951) was a Fellow of the Royal Statistical Society of London, the Royal Society and St John’s College, Cambridge. He came to statistics, through the influence of Karl Pearson (1857–1936), from a background of engineering and experimental physics. His area of expertise was correlation and regression. [120] 232Including the American biophysicist, Alfred J. Lotka (1880–1949) who is remembered, in par- ticular, for having developed—in 1925, independently of Vito Volterra—the Lotka–Volterra predator-prey model. He refers to “‘Tait’s Law” of linear fertility decrease with age’ in [121]: see page 160f for the reference to Tait. 80 References to Tait’s Law(s) continued until the 1970s.233 Yule referred to Tait’s Laws again in 1920.234 233From the 1970s onwards, an ‘enriched environment for research on populations’ developed; due to ‘technological advances in the publication of census information, as well as vital statistics’ and the ‘growth in the range and frequency of population surveys’. [122, p27] This in turn led to a period of statistical innovation, which included the development of new methodologies in data processing and analysis, model building and parameter estimation. [115, p1] In particular, the methods of exploratory data analysis (E.D.A.), developed by Tukey and Mosteller, enabled fertility models to admit the influence of: natural fertility; patterns of marriage, divorce and widowhood; the pace/timing of childbearing and efforts to limit fertility. For more information see [115]. 234Yule refers to Tait’s Laws again in [123]. 81 CHAPTER 4 TAIT’S SCHOOLBOY INTRODUCTION TO COMPLEX NUMBERS This chapter describes Tait’s first encounter with complex numbers and their geometrical representation. 4.1 Introduction From age ten to sixteen, Tait was educated at the celebrated Edinburgh Academy. He was a pupil in Dr. Cumming’s class during his first four years; thereafter, the class came under the care of the Rector, Archdeacon John Williams (1792–1858).235 Dr. James Cumming (1800–1875) instructed the boys in classics; the three R’s was taught by Robert Hamilton (“Hammy”); mathematics by James Gloag and French by Franc¸ois Sene´bier (“Snibby”). During his time at the Edinburgh Academy, Tait won a number of prizes and medals, including Dux annually throughout his school career.236 For the full list of his schoolboy achievements see Appendix C (pages 232–233). Lieutenant-Colonel 235In 1850 the surviving members of Dr. Cumming’s class formed themselves into the Cumming Club, with the intention of maintaining or re-establishing those early friendships and sharing in their mutual affection for Dr. Cumming. Dr. James Cumming had a reputation as a first-class educator and as a master who was popular with the boys: he was known to be fair, good- humoured and kind. [5, p146] For a book on the Cumming Club, written by a classmate of Tait’s, see [32]. 236Tait’s school medals are on display at 14 India Street, Edinburgh. A gold medal was awarded to the Dux in the Rector’s class and a silver medal to the Dux in all other classes; winners of lower prizes received maps and books. [5, p95] The gold Dux medal was engraved with the Louvre bust of Virgil; the silver medal featured the Townley bust of Homer from the British Museum. [5, p106] 82 Alexander Fergusson (1831–1892)—a classmate of Tait’s and author of Chronicles of the Cumming Club—recalled that such sustained success won for Tait a reputation amongst his schoolfellows as ‘Our permanent Dux’.237 He writes: ‘Through all the classes, from the First to the Sixth, when he left the Academy, Tait was easily our leader.’238 4.1.1 Schoolboy association with Maxwell When Tait was in the fourth class at the Academy, he became friends with fellow student, James Clerk Maxwell. This association is well known.239 Both boys were the same age but Maxwell was placed in the class above Tait (Mr. Carmichael’s class) because Dr. Cumming’s class was full when Maxwell came to enrol. Tait remembered his early acquaintance with Maxwell in a short biographical note he wrote for the R.S.E. upon Maxwell’s death: When I first made Clerk-Maxwell’s acquaintance about thirty-five years ago, at the Edinburgh Academy, he was a year before me, being in the fifth class while I was in the fourth. At school he was at first regarded as shy and rather dull; he made no friendships, and he spent his occasional holidays in reading old ballads, drawing curious diagrams, and making rude mechanical models. His absorption in such pursuits, totally unintel- ligible to his schoolfellows (who were then quite innocent of mathematics), of course procured him a not very complimentary nickname [“Dafty”240], which I know is still remembered by many Fellows of this Society. About the middle of his school career, however, he surprised his companions by suddenly becoming one of the most bril- liant among them, gaining high, and sometimes the highest, prizes for Scholarship, 237 [32, p22] 238 [32, p202] 239Indeed, for some, Tait’s association with Maxwell is all they know of Tait. Amongst those who know of Tait through Maxwell, it seems that the majority prefer Maxwell: they think of him, in later years, as a more remarkable talent, as being more modest and as having a less aggressive personality than Tait. 240See [5, p149] and [32, pp25–26]. 83 Mathematics, and English verse composition. From this time forward I became very intimate with him, and we discussed together, with school-boy enthusiasm, numerous curious problems, among which I remember particularly the various plane sections of a ring or tore, and the form of a cylindrical mirror which should show one his own image unperverted. I still posses some of the MSS. which we exchanged in 1846 and early in 1847. Those by Maxwell are on “The Conical Pendulum,” “Descartes’ Ovals,” “Meloid and Apioid,” and “Trifocal Curves.” All are drawn up in strict ge- ometrical form and divided into consecutive propositions.241 The three latter are connected with the first published paper, communicated by Forbes to this Society and printed in our “Proceedings,” vol. ii, under the title “On the Description of Oval Curves, and those having a plurality of foci” (1846). At the time when these papers were written he had received no instruction in Mathematics beyond a few books of Euclid, and the merest elements of Algebra.242 241For Maxwell’s manuscripts see The Scientific Letters and Papers of James Clerk Maxwell edited by P. M. Harman: “The Conical Pendulum” (text 6 [124, pp64–67]); “Descartes’ Ovals”, and “Meloid and Apioid” (texts 3(1) [124, pp47–54] and 3(2) [124, pp55–61]); and “Trifocal Curves” (text 2 [124, pp43–46]). Text 2 has been transcribed from the original in the University Library, Cambridge. Texts 3 and 6 have been reproduced from Campbell and Garnett’s Life of Maxwell (1882). Harman explains [124, p43f(1)]: The manuscript ‘On Trifocal Curves’, [Number 2] which is annotated in pencil by Tait, and dated ‘March 1847’, is clearly the paper referred to by Tait. The manuscript on the conical pendulum dated 25 May 1847 (Number 6), and the propo- sitions on ‘Oval’ and ‘Meloid and Apioid’ (Number 3) which are reproduced from the Life of Maxwell, are possibly the other papers referred to, or are drafts of these papers. The papers ‘On Trifocal curves’, ‘Oval’ and ‘Meloid and Apioid’ are closely related in content; the order in which they are reproduced here [in Harman (1990)] may not be the chronological order of their composition. 242 [8, p332] 84 4.1.2 The Tait–Maxwell school-book Tait copied the ‘MSS. which [he and Maxwell] exchanged in 1846 and early in 1847’— referred to in the above extract—into his school-book. It seems that the school-book was originally intended as a fair-copy book: some entries are written carefully in ink, and are signed and dated; however, there is also an abundance of rough pencil work, with workings-out and schoolboy sketches subsequently fitted into available space. Entries in the school-book include: (factually dubious) notes on the history of enumeration; a table recording the positions of the satellites of Jupiter, as observed by Tait at the age of thirteen; a number of problems and solutions on the mensura- tion of heights and distances, which I have traced to a contemporary textbook [3]; and copies of the MSS. which Tait and Maxwell exchanged; but the bulk of the school-book is taken up with Tait’s notes which he abstracted from the Encyclopae- dia Britannica (7th edition, 1842).243 Tait made copious notes on a range of math- ematical topics covered in the articles, ‘Algebra’ and ‘Fluxions’. From the article on algebra Tait made notes on the arithmetic of sines, produc- ing a comprehensive list of trigonometric formulae. The article on fluxions is in two parts: Part I details the direct method of fluxions; Part II explains the inverse method of fluxions, otherwise known as the integral calculus. Tait’s notes on Part I cover: successive differentiation, Taylor’s theorem, Maclaurin’s theorem, differenti- ation of equations of two variables, vanishing fractions, the greatest and least values of a function, determination of tangents to curves, generation of curves by evolution and contact of curves. Part II covers: integration of rational functions involving one variable, integrals of irrational fractions, integration of binomial differentials and integration of angular or circular functions. Tait’s approach was to copy down key results/formulae and some illustrative examples. Perhaps this wider reading was in preparation for university or perhaps Tait did not find the mathematics syl- 243I imagine that Tait would have had a set of encyclopaedias at home. 85 labus at school challenging enough.244 Certainly, the material extracted from the encyclopaedia ventures beyond the Academy’s syllabus.245 4.1.3 Gloag’s influence For an indicator of the nature of the mathematical instruction given at the Academy, there is none better than a profile of Tait’s mathematics master, James Gloag.246 Knott writes: ‘Gloag was a teacher of strenuous character and quaint originality [. . .] With him mathematics was a mental and moral discipline’.247 By all accounts, Gloag was strict but fair, and the boys, once over their initial terror, seemed to progress well under his tutelage: The fact that he [Gloag] was much the strictest disciplinarian in the school, and deadly with a tawse withal, was offset by his indisputable fairness and impartiality, a gruff kindliness under the irascibility, and the growing evidence that the boys learned well under him—once they had got over their initial terror of him.248 On Gloag’s technique with the tawse, Tait is reported to have said: “To use a well- known cricketing phrase, Gloag could get “more work” on the tawse than any of the other masters. His secret was in great part a dynamical one.”249 It was in Gloag’s lessons that young Maxwell’s abilities in mathematics became apparent: growing in confidence, as a result of Gloag’s influence, Maxwell’s perfor- mance improved, not only in mathematics but across the board.250 Indeed, at the 244It is known that Tait entered the higher division of Prof. Forbes’ natural philosophy class at Edinburgh. According to [124, p4] the higher division course required knowledge of the calculus. 245See Table C.1 (page 231) for the syllabus for classes 6–7 in 1846–1847. 246James Gloag: Master of the Arithmetical and Geometrical School at the Edinburgh Academy (1824–1864). [4, pxlix] 247 [18, pp4–5] 248 [5, p99] 249 [5, p102] 250 [5, p150] 86 age of fourteen, Maxwell achieved the rare distinction of having an academic paper read before the R.S.E. The paper on ovals—which is referred to in Tait’s biographi- cal note (page 84)—was communicated to the Society by Professor J. D. Forbes and published in the Proceedings.251 Gloag revelled in his pupils’ success and in 1852 when news of Tait’s achievements at Cambridge reached the Academy, he took great pride in the part he had played early on.252 4.1.4 Significance of a particular school-book entry On 18 January 1847 Charles Hughes Terrot, Bishop of the Scottish Episcopal Diocese of Edinburgh, read a paper [126] before the R.S.E. on the geometrical representation of complex numbers.253 On 29 January Terrot’s paper was handed over to Professor Kelland who was to produce a report on it and on 12 February, based on Kelland’s report, it was ordered that the paper be printed in the Society’s Transactions.254 The paper was published in the Transactions with the title, ‘An Attempt to Elu- cidate and Apply the Principles of Goniometry, as published by Mr Warren, in his Treatise on the Square Roots of Negative Quantities’.255 On 27 May Tait entered into his school-book an abstract of Terrot’s published paper under the heading, ‘On the imaginary roots of negative quantities. By the Right Reverend Bishop Terrot. 1847’. In May 1847 Tait was sixteen, in his final year at the Academy, and headed for the University of Edinburgh in the autumn. 251For more on Maxwell’s paper on ovals see: [125, pp74–79] and [124, p2–3]. 252 [32, pp85–86] To celebrate Tait’s success at Cambridge, members of the Cumming Club arranged a banquet to be held in Tait’s honour. The event took place at Archers’ Hall in Edinburgh on 22 March 1852, with Dr. Cumming, Dr. Gloag, Mr. Hamilton and M. Sene´bier in attendance. See [32, pp84–88] for Fergusson’s account of the high-spirited event. 253See [127] for the record of the communication in the Proceedings. 254According to the R.S.E.’s Council minute books: National Library of Scotland, Acc.10000/19, Nov. 1846 – Feb. 1859. 255goniometry : the measurement of angles; derived from the word goniometer, which is ‘an instru- ment used for measuring angles’. [128] 87 Terrot’s paper was surely Tait’s first introduction to complex numbers and their geometrical representation. Quadratic equations were on the syllabus at the Academy for classes 6–7; however, it is extremely unlikely that the schoolboys, during the normal course of their lessons, would have been exposed to anything other than real roots.256 In Davidson’s System of Practical Mathematics [3]—a con- temporary textbook from which Tait’s worked examples in the school-book on the mensuration of heights and distances are taken—only real roots are covered, ques- tions having been carefully selected to avoid complex roots. The fact that the Bishop’s paper was Tait’s first introduction to complex numbers and their geometrical representation makes the find in the school-book significant but further research has added weight to this significance: it will be through this particular entry in the school-book that we will come to learn something further about the discovery of quaternions and extend the scope of the history of the Ar- gand diagram, going beyond Wessel, Argand and Gauss. First, an introduction to the Right Revd. Bishop Terrot. 4.2 Bishop Terrot 4.2.1 Charles Hughes Terrot (1790–1872): a biographical sketch Charles Hughes Terrot (1790–1872) (Figure 4.1, page 90) was born at Cuddalore, India on 19 September 1790. He was the son of Elias Terrot, a Captain of the 52nd Regiment in the Indian Army. His great grandfather, Monsieur de Terotte—a protestant (Huguenot) exile from France—had fled from La Rochelle and sought refuge in England on the revocation of the Edict of Nantes in 1685.257 Upon Elias’ death at the siege of Bangalore in 1790, Terrot’s mother, Mary Fonteneau left India with her infant son and settled in Berwick. Young Terrot was 256Again, see Table C.1 (page 231) for the syllabus for classes 6–7 in 1846–1847. 257A visual indication of Terrot’s ancestry is given in [129, p227]. 88 placed under the care of the Revd. John Fawcett of Carlisle and was educated at Carlisle Grammar School. It was at Cambridge that Terrot earned a reputation for scholarship, particularly in mathematics. He entered Trinity College in 1808, gaining his B.A. in 1812, gradu- ating with mathematical honours, despite disappointing Tripos examination results. The problem had not been a lack of intellect, rather an unwillingness to apply it in tedious activity. From the R.S.E.’s Proceedings, with information provided by Prof. Kelland: The fact is that Terrot’s mind revolted at the drudgery of acquiring branches of the science [mathematics] towards which he felt no inclination. It was characteristic of him to tread a small circle, but to tread it well; and he was constitutionally unfitted for stowing away in his memory, for temporary purposes, facts and figures in which he took no interest.258 Fortunately for Terrot, he had had ample opportunity outwith the Tripos exam- inations to prove his abilities and on this basis he was elected a Fellow of Trinity College in 1813. The same year he was ordained a deacon, with ordination to the priesthood following in 1814. In 1815 he settled in Haddington, taking up a position held previously by his uncle, the Revd. William Terrot, as Minister of the Episcopal congregation. Terrot’s move to Edinburgh took place in 1817: he was to assist the Revd. James Walker at St Peter’s in Roxburgh Place. In 1833 Terrot joined two other clergy at St Paul’s in York Place. During the next twenty years, his appointments grew in prestige: he was appointed Dean of Edinburgh and Fife in 1837; Rector of St Paul’s in 1839; Pantonian Professor at the theological college, and Bishop of Edinburgh, in 1841 and Primus of the Scottish Episcopal Church in 1857. He remained as Primus until 1862, when a paralytic stroke forced his resignation. He married twice: his first wife, with whom he had fourteen children, left him a widower.259 He died in 258 [130, p9] 259According to [131] Terrot’s eldest daughter accompanied Florence Nightingale to the Crimea and for her service there she was awarded a Royal Red Cross. 89 Stockbridge on 2 April 1872, aged eighty-two. Figure 4.1: Charles Hughes Terrot by Mason & Co (Robert Hindry Mason), albumen carte-de-visite, 1860s. Reproduced with the kind permission of the National Portrait Gallery, London. The handwritten text reads ‘Bp [Bishop] of Edinburgh’. The size of this reproduction is representative of the typical size of a carte-de-visite, 54mm × 89mm mounted on a card 64mm × 100mm. 4.2.2 Devotion to mathematics Understandably, Terrot sought relief from the heavy burden of his responsibilities. As Kelland recalled, Terrot turned to mathematics for refuge: ‘To mathematics, when harassed by the cares and vexations incident to his position, he had recourse as a retreat from irritating thoughts. His passion for the science was strong enough to take possession of his mind, and soothing enough to settle it down to repose.’260 260 [130, p11] 90 The extent of Terrot’s reliance on mathematics and its implications, both benefi- cial and otherwise, were discussed by the Revd. Walker—whom Terrot had assisted at St Peter’s—in his biographical work, Three Churchmen (1893). Walker writes: Absorbed in the depths of original research, the bishop found that which can, it is said, be always found in the depths of the ocean, viz., calm, in the midst of storm.261 Readers will probably take very different views of Bishop Terrot’s occasionally ardent devotion to mathematical study. Some will think that whenever he had any spare time for investigation and research it ought to have been devoted exclusively to professional subjects, such as the theological and biblical problems of the day. Others will hold that the one study was a help rather than a hindrance to the other; the occasional subjection of the mind to the vigorous mathematical discipline being the best corrective of loose thinking and illogical reasoning. Those who take this view will believe that the bishop’s addiction to mathematical research was an advantage to the Church as well as to himself; not only assuring him an occasional refuge from worry, but also maintaining in him that judicial frame of mind which never deserted him in the hottest controversies, and which extorted the admiration of his opponents.262 The only evil effect of the bishop’s mathematics was probably a little intolerance of the loose talk and inconsequential reasoning which often prevail in general society.263 Apart from mathematics, Terrot spent time writing poetry and much of his leisure time while at Haddington was devoted to it. His poem ‘Hezekiah and Sen- nacherib, Or the Destruction of Sennacherib’s Host’—about the destruction of Sen- nacherib’s army before Jerusalem—won him the Seatonian Prize in 1816.264 He also had a love of architecture and enjoyed membership of the Architectural Society of Scotland. 261 [132, p156] 262 [132, p161] 263 [132, p163] 264The Seatonian Prize has been awarded by the University of Cambridge annually since 1750, for a poem written in English on a sacred subject. 91 4.2.3 Reputation in Edinburgh, contribution to the Royal Society of Edinburgh Terrot was elected a Fellow of the R.S.E. in 1840, proposed for fellowship by J. D. Forbes. Terrot’s role in the Society was described by Kelland thus: For many years of his life he was one of the regular attendants at our meetings; and when not actively engaged in the work going on, he was an active listener, and, when occasion called for it, and unsparing critic. He had a real love for the Society. As he left the building for the last time, he expressed himself to the effect, that henceforth his heart would be with us, but that the work of his hands was done. The only part of the proceedings which he did not relish was the tea-drinking after the meeting.265 Terrot served as a Councillor for the Society (1841–1844) and as their Vice-President between 1844 and 1860. Looking through the Proceedings, there are numerous in- stances of the ‘Right Rev. Bishop Terrot, Vice-President, in the Chair’. Terrot delighted in good conversation and, according to the Revd. Walker, in Edinburgh he had developed a ‘very high reputation as a talker of the Johnsonian type’.266 By this, Walker meant: precision of thought and language, ready wit, repartee and love of argument—all set off to advantage by a distinct voice and deliberate utterance. He [Terrot] was also almost as impatient as Johnson himself was of twaddle and of pretence—“humdrum and humbug”—and thus to weak reasoners and pretentious talkers he appeared to be, and doubtless sometimes was, severe and sarcastic. But to men of like mind with himself—deep and just thinkers and earnest talkers—his conversation was very highly prized, and his society much courted.267 Terrot’s contributions to the R.S.E. comprise the following papers (ordered by date of communication to the Society): 265 [130, pp12–13] 266 [132, p167] Dr. Samuel Johnson (1709–1784): author and lexicographer; remembered as the compiler of Johnson’s Dictionary, published in two volumes in 1755. 267 [132, p182] 92 1845 ‘On the Sums of the Digits of Numbers’ 1847 ‘An Attempt to Elucidate and Apply Mr Warren’s Doctrine Respecting the Square Root of Negative Quantities’ 1848 ‘On Algebraical Symbolism’ 1849 ‘An Attempt to Compare the Exact and Popular Estimates of Probability’ 1850 ‘On Probable Inference’ 1853 ‘On the Summation of a Compound Series, and its Application to a Problem in Probabilities’ 1856 ‘On the Possibility of Combining Two or More Independent Probabilities of the Same Event, so as to Form One Definite Probability’ 1858 ‘On the Average Value of Human Testimony’ According to Kelland in [130], Terrot’s 1856 paper was his ‘best contribution to mathematical science’: it had inspired Boole’s paper, ‘On the Application of the Theory of Probabilities to the Question of the Combination of Testimonies or Judgements’, for which Boole was awarded the R.S.E.’s Keith Prize in 1858.268 4.2.4 Associations with the Edinburgh Academy An obituary in the British Medical Journal suggests that Terrot, at some stage, taught at the Academy: ‘Spencer Thomson, MD., Torquay [. . .] Educated at the Edinburgh Academy, under the late Rev. Dr. Terrot, afterwards Bishop of Edin- burgh’.269 Yet there is no evidence in the Academy’s Register [4] to support such a theory: the three references to ‘Terrot’ in the Register reveal only that Terrot had educated his children at the Academy between 1825 and 1842.270 It seems unlikely, 268 [130, p12] 269 [133, p442] 270Terrot’s children educated at the Edinburgh Academy: Charles Samuel John, Elias Charles and William H. [4, pp35,63,84] 93 therefore, that Terrot was ever a member of staff at the Academy. Perhaps the obituarist had misinterpreted Terrot’s influence on young Thomson. I suggest that Terrot may have acted as a mentor or tutor to the boy since Thomson was in the same class at the Academy as one of Terrot’s sons, Elias Charles, and he resided in the same street as the Terrot family, Northumberland Street.271 In Cassell’s Old and New Edinburgh, a description is given of Northumberland Street and Bishop Terrot, as one of its residents: In the narrow and somewhat sombre thoroughfare named Northumberland Street have dwelt some people who were of note in their time. [. . .] No. 19 in the same street was for some years the residence of the Right Rev. Charles Hughes Terrot, D.D., elected in 1857 Primus of the Scottish Episcopal Church, and whose quaint little figure, with shovel-hat and knee-breeches, was long familiar in the streets of Edinburgh. [. . .] few men were more esteemed and respected by others than Dr. Terrot of the Episcopal Church.272 It is possible that Terrot provided Thomson with private tuition in mathematics. From the time of his fellowship at Trinity, Terrot had supplemented his income by taking on pupils for private tuition.273 Indeed, a substantial increase in salary had to be arranged so that Terrot no longer needed to tutor and could devote himself to his ecclesiastical duties: ‘The minute of the vestry states that “so large an advance has been at once made to Mr. Terrot’s salary, making it higher than any clergyman of the chapel ever received before with the view of securing his undivided attention to his duties of minister of the chapel.”’274 Terrot was known to have had a close association with Archdeacon William’s suc- cessor as Rector, Dr. John Hannah (1818–1888) who was a former Fellow of Lincoln 271 [4, p35] 272 [134, p198] 273 [132, pp104–105] 274 [132, pp110–111] 94 College, Oxford.275 The Revd. Walker describes Dr. Hannah as one of the Bishop’s ‘most intimate friends’.276 The two men lived near to each other and Dr. Hannah took great pleasure in familiarizing Terrot with ‘Oxford forms of thought’ at a time when the philosopher, Sir William Hamilton (1805–1865) was promoting commu- nication between Edinburgh and Oxford.277 The two were also connected through the R.S.E.: Hannah was elected a Fellow in March 1848, having been proposed by Terrot in the January.278,279 4.3 Bishop Terrot’s 1847 paper A summary of the Bishop’s paper now follows. For a transcription of Tait’s notes on the paper see Appendix D. 4.3.1 Summary of the paper Terrot begins his paper by explaining that while √ −1 is called “impossible” or “imaginary”, since any number squared must be positive, with a geometrical inter- pretation √ −1 is no more impossible than +1 or −1, for each is capable of being represented by a directed line in the plane. A line of length a, fixed at one end to the origin and inclined at an angle ϑ, he symbolizes by a ϑ 2rpi . He then asks us to consider lines of equal length, or radii of a circle expressed as R × 1 ϑ 2rpi , where R is the length of the radius and 1 ϑ 2rpi is the “coefficient of di- rection”. He explains that the radii of a circle, with equal angles between them, actually represent the nth roots of unity and he sets out a method which enables us 275Archdeacon John Williams was Rector at the Academy during the periods 1824–1828 and 1829– 1847, and Dr. John Hannah was Rector at the Academy between 1847 and 1854. [4, pxliii] 276 [132, p178] 277 [132, p168] 278According to Hannah’s entry in [29]. 279Uncited sources of biographical information on Terrot: [13] and [29]. 95 to properly order the roots: the nth roots of unity appear in pairs, a + √ −b with a− √ −b, either side of the original radius, with equal angles between them and the original radius. Note, Terrot does not use the term complex conjugates. Keen to show how to multiply lines together, Terrot then introduces an alterna- tive expression for 1 ϑ 2rpi ; written in polar form as cos (ϑ) + √ −1 sin (ϑ). Similarly, a line inclined at an angle pϑ he expresses in the form cos (pϑ) + √ −1 sin (pϑ). By thinking of a line inclined at an angle pϑ as the result of p rotations through an angle ϑ, he establishes De Moivre’s theorem. As a corollary, he gives a means of obtaining the quadratic factors of the polynomial xp−1 = 0, by multiplying together conjugate pairs of the pth roots. Bishop Terrot also explains in his paper that from the algebraic expression of a line a ϑ 2rpi we know the line’s length and its direction; and, therefore, we might think of a ϑ 2rpi as representing the transference of a point in space, moving from A to C say. Within the triangle ABC there are two routes from A to C: the direct route; or, via B, journeying a distance of |AB| in the direction of AB and then a distance of |BC| in the direction BC. Vectors in all but notation. Terrot establishes one side of the triangle as a line in an original position and expresses all other sides as rotations of this original. Note, Terrot uses round brackets to indicate the length of a line. Having applied his symbolism to various elementary propositions in plane trigonom- etry, Terrot then uses it in a proof of Cotes’ theorem. In the R.S.E.’s Proceedings Terrot’s proof is described as ‘a new demonstration of Cotes’ properties of the cir- cle’.280 Terrot’s proof of Cotes’ theorem is the summit of his paper. Theorem: Cotes’ Properties of the Circle.281 Let the circumference of the circle be divided into n equal parts; and to the extremities of these let lines be drawn from the centre [Figure 4.2, page 97], as OP1, OP2, &c., and from any other point C in the diameter. Then 280 [127, p111] 281Wording abstracted from Terrot’s paper [126, p353]. 96 |CP1| × |CP2| × |CP3| · · · |CPn| = |OA| n − |OC|n [and variants of this, depending on the position of C.] Proof. For Terrot’s proof of the theorem see [126, pp353–354] and page 242 of this thesis. Figure 4.2: Tait’s drawing of the figure to accompany Cotes’ theorem. Sourced from the Tait–Maxwell school-book. Reproduced with the kind permission of the J.C.M. Foundation. A remark on Wessel’s proof of Cotes’ theorem The above theorem was formulated by Roger Cotes (1682–1716) in 1716 and was published posthumously in Harmonia Mensurarum in 1722; however, Cotes left no demonstration of the theorem. In 1797 the Norwegian surveyor, Caspar Wessel (1745–1818) gave a proof of the theorem in his report to the Royal Academy of Denmark. The report was published in the Academy’s memoirs in 1799. In his proof of Cotes’ theorem [135, p111] Wessel recognizes the vertices of the regular n-gon as the solutions of the cyclotomic equation zn−rn = 0. He writes each of the CPi (Terrot’s notation, not Wessel’s) in the form (z−a root), so that their product is given by zn− rn. He calls on a previous result to show it is legitimate to 97 take the modulus of both sides. In Terrot’s proof, the product of the CPi is written as a sum where each term is a product: multiplied together are the product of the coefficients of direction for the OPi (since CPi = OPi − OC) and the product of the lengths. In the first term there are n OPi taken together; in the second, n − 1 OPi are taken together, etc. Recognizing the coefficients of direction as roots of xn − 1 = 0 Terrot reasons on the values of the various products of these coefficients: all but the first and the last terms in the sum disappear. He has then only to divide through by the product of the coefficients of direction so as to consider length alone. 4.3.2 Tait’s notes on Bishop Terrot’s paper In summarizing the Bishop’s paper, Tait copies down only what constitutes essential information: he sifts out and records the key mathematical concepts and results; no- tably, he chooses to omit the Bishop’s references to the Revd. John Warren’s earlier work on the subject, to Peacock’s Algebra and to a paper on symbolical geometry by Sir William Rowan Hamilton published in the Cambridge and Dublin Mathematical Journal.282 It is interesting to note that Tait’s preference is to use mathematical notation as a shorthand where possible, employing the symbols ∵ for ‘because’, ∴ for ‘therefore’ and ∠ for ‘angle’. He translates Terrot’s words into the equivalent mathematical expressions; for instance, from Terrot’s ‘Let BAC represent a right-angled triangle . . .’, Tait writes ‘BCA = 90◦’.283 Another point of departure from Terrot’s paper is 282References to Warren: Terrot explains how he will take Warren’s work further and he gives some of Warren’s key definitions and propositions. See [126, pp346,348,351]. References to Peacock’s Treatise on Algebra: Terrot writes that Peacock dealt with coefficients of direction in his treatise; Terrot takes his unordered list of roots of x6 = 1 as one of his examples from the same source; and he remarks that he has derived the trig. formulae for sin(A+B) and cos(A+B) in a similar fashion to Peacock. See [126, pp346,350]. Reference to Hamilton: Terrot writes that Hamilton had written something similar to him with respect to the symbolical sum of lines. See [126, p348f]. 283 [126, p349] 98 that Tait uses Arabic numerals 1, 2, 3, ... to denote his sections, where the Bishop opts for Roman numerals I, II, III, ... As one would expect, Tait writes his summary in language which is more famil- iar to him—which he is more comfortable with. Thus ‘procure’ becomes ‘obtain’, ‘extremity’ becomes ‘end’ and ‘algebraic’ becomes ‘algebraical’. Now and then he shows real flair, when, for instance, he adds in phrases such as ‘described on the ra- dius AD’. As he gets further into the Bishop’s paper, Tait begins to follow Terrot’s wording more closely. This is, of course, the natural tendency when summarizing an extended piece of writing but it is true that the mathematics is more involved later in the paper. Tait is methodical. He makes neat copies of Terrot’s figures and places them near to where they are referenced in the text, unlike Terrot who—presumably because of type-setting constraints—sometimes has them positioned in less sensible places. Tait clearly understands Terrot’s paper. Indeed, there are numerous instances when Tait corrects typographical errors which appear in Terrot’s printed mathematics, for instance, Terrot has: (i) sin (A+B) = A × cosB + cosA × sinB and (ii) (CD21) which should be (CD1)2.284 4.3.3 Tait’s interest in the paper So why did Tait take a special interest in this particular paper? He copied no other papers into the school-book. In actual fact, it was Cotes’ theorem which drew Tait to Bishop Terrot’s paper. The evidence for this exists elsewhere in the school-book: within Tait’s notes on the section, ‘Arithmetic of sines’ from the article on algebra in the Encyclopaedia Britannica, Tait has added in a cross-reference to Terrot’s paper. He writes: ‘These are the analytical expressions of the Theorem of Cotes. (See proceeding paper.)’ and the paper follows after Tait has finished recording his notes on that particular section. Presumably, Tait’s cross-reference was added in at 284 [126, pp352,353] 99 a later date.285 So Tait, wanting to learn more about Cotes’ theorem, had consulted Bishop Terrot’s paper. But how did Tait know that Terrot had published a paper in which he had given a proof of the theorem? There are a number of possible scenarios, one of which is this: that someone, aware of Tait’s interest in Cotes’ theorem and with knowledge of the Bishop’s paper, suggested to Tait that he read the Bishop’s paper; perhaps they also furnished him with a copy of the Transactions. Tait’s uncle, John Ronaldson—though not a Fellow of the R.S.E.—took an active interest in science. Archdeacon John Williams, Rector of the Edinburgh Academy, was a Fellow of the R.S.E.286 There was also Maxwell’s father, John Clerk Maxwell (c.1790–1856).287 He was a Fellow of the R.S.E. and on a number of occasions he had taken young James along to the R.S.E. meetings.288 Indeed, it was John who had written out a fair copy of James’ paper on ovals so that it could be presented to the Society.289 Perhaps Tait had mentioned his interest in Cotes’ theorem to Maxwell and through Maxwell he learned of the Bishop’s paper and got a copy of the Transactions. However events unfolded, this episode is of interest because it is evidence of Tait’s early engagement with R.S.E. publications and the Encyclopaedia Britannica. Tait, later in life, would become a prolific contributor to the R.S.E.’s Proceedings and Transactions and the author of a number of articles for the Encyclopaedia Britannica on a variety of scientific subjects and on the life and work of Sir William Rowan Hamilton in a biographical piece. 285Visual indications suggest that this is the case: the text appears bolder and some of the lettering goes through a horizontal line which Tait had drawn in to mark the end of the section. 286Archdeacon John Williams was elected a Fellow of the R.S.E. on 7 February 1825. He served the Society as Councillor between 1835 and 1838. [29] 287John Clerk Maxwell was elected a Fellow of the R.S.E. on 5 February 1821. [29] 288According to Campbell and Garnett in [125, p73]. 289According to Harman in [124, p35f(2)]. 100 4.3.4 On priority: John Warren’s influence Unfortunately, Bishop Terrot has no claim to priority as the first discoverer of the ge- ometrical representation of complex numbers. This honour falls to Caspar Wessel, whose 1797 report, referred to above (pages 97–98), serves to establish his prior- ity. Therein, Wessel has the geometrical representation of complex numbers framed within a wider system of vector analysis. The regrettable fact that Wessel’s con- tribution went unnoticed by the mathematical community for close to a century enabled subsequent workers to rediscover his results and claim for themselves some share in the priority of the discovery.290 I am thinking of Jean Robert Argand (1768–1822), a Swiss book-keeper resident in Paris, who presented his results in a privately-published pamphlet in 1806; the Revd. John Warren (1796–1852), a Fellow of Jesus College, Cambridge, who published his treatise [136] in 1828; and the German, Johann Carl Friedrich Gauss (1777–1855), who had the geometrical representation of complex numbers explicit for the first time in his 1848 proof of the fundamental theorem of algebra (F.T.A.).291 In 1843, of course, there came the pivotal discovery of Hamilton’s quaternions.292,293 290Wessel’s work was rediscovered ninety-eight years later and republished in French in 1897, on its one-hundredth anniversary, at the request of the Danish Academy. 291Gauss invented the term “complex numbers”. He was also the first to use the letter i to represent √ −1; and the first to represent a complex number in the form a + ib, in Theoria residuorum biquadraticorum (1831). 292In fact, even as early as 1833 Hamilton had, in effect, removed the conceptual difficulties sur- rounding imaginaries by expressing complex numbers a+ib as algebraic couples or ordered pairs of real numbers (a, b). 293For more information on the historical developments which preceded the discovery of quater- nions, and on the adoption of the modern system of vector analysis, see [137]: a timeline of the developments is given on pages 256–259; the contributions of Wessel, Bue´e, Argand, Mourey, Warren and Gauss are discussed on pages 5–11. 101 John Warren’s influence on Terrot For his Treatise on the Geometrical Representation of the Square Roots of Negative Quantities (1828), the Revd. John Warren is regarded as the English source of the Argand diagram. Warren was, like Terrot, Cambridge educated; admitted to Jesus College in 1814 and graduating (B.A.) as fifth Wrangler in 1818 (M.A. in 1821). He remained, until 1829, a Fellow and tutor of the College and during the period 1825– 1826 he functioned as moderator and examiner. As the son of the Dean of Bangor, perhaps it was inevitable that Warren would be ordained, firstly as a deacon in 1819 and as a priest the following year. He served the communities of Caldecott, Huntingdonshire between 1822 and 1852, and Graveley, Cambridgeshire between 1828 and 1852. Warren married in 1835 but had no children. He was elected a Fellow of the Royal Society of London in 1830.294 Comparing Terrot’s paper with Warren’s 1828 treatise, it is clear that Terrot’s work was heavily influenced by Warren: chapter 1 of Warren’s treatise provided Terrot with the fundamentals of his theory; and in chapter 2, Terrot was given all the tools he needed for his work on nth roots.295 For specific examples of Terrot’s use of material from Warren’s treatise see Table 4.1 (page 104). Tait made no mention of Warren in his notes but Terrot did readily acknowledge Warren’s contribution. In his paper Terrot stated that his role was to explain Warren’s results in greater detail and explore new applications of Warren’s theory. He writes: On some points, however, Mr Warren has been too sparing of his words, and has thus apparently used the common symbols of algebra in a sense very different from their ordinary acceptation. In the following paper I have endeavoured to supply this deficiency of explanation; and then to apply the system of symbols so established to some important problems of goniometry to which, as far as I know, it has not yet been applied.296 294Sources of biographical information on Warren: [138] and [13]. 295Chapters 3 and 4 of Warren’s 1828 treatise bear little resemblance to the material covered in Terrot’s paper. 296 [126, p346] 102 So while Warren should be credited with having established the fundamentals, the Bishop surely deserves some recognition: he seems to have had a sound grasp of the mathematics involved, which he was able to apply in valid and novel ways, and it is through his paper that Tait had his first introduction to a relatively new math- ematical discovery of huge import. Certainly, the R.S.E. held Terrot’s mathematical researches in this area in the highest regard: The subject [. . .] had been floating somewhat dimly before the eyes of mathematicians for half a century, and was just then beginning to assume a living form in the mind, and a living exponent, though a somewhat obscure one, in the writings of Sir W. R. Hamilton. It was not until six year later that the doctrine of Quaternions of the great master, as developed in his “Lectures,” swallowed up in its vast amplitude all that had preceded it. Terrot accordingly must be considered as one of the pioneers of the science.297 The Revd. Walker believed that Terrot might have had a remarkable career in mathematics had he chosen to commit himself to it exclusively: had he [Terrot] cared to devote himself to “research,” living chiefly on his fellowship, he might have made important discoveries in some branches of the higher mathemat- ics, probably anticipating Sir William Rowan Hamilton in his discovery of Quater- nions. But he had other views; and doubtless he took the wiser course.298 297 [130, pp11–12] 298 [132, p102] 103 Chapter 1 of Warren’s 1828 treatise Arts. 1, 2 A line has both direction and length. It is drawn in the plane, anchored to the origin, inclined at an angle. Art. 3 The sum of two lines is the diagonal of their parallelogram. Art. 4 The subtraction of lines is the reverse process of addition. Art. 8 Positive and negative lines are opposite in direction. Arts. 51, 52 Two propositions relating to the addition of angles when multiply- ing lines together. Chapter 2 of Warren’s 1828 treatise Art. 54 A definition of the nth root of a quantity. Art. 58 corr. The nth roots of a quantity are co-equal in length. Art. 61 There are n nth roots of a quantity. Arts. 105–107, 110 A method of calculating, and representing geometrically, the 4th roots of unity. Arts. 109, 113 A proof that ‘Any quantity may be expressed in the form±a±b √ −1 where a and b are positive quantities’; with an example, expressing the 6th roots of unity in this form. Art. 119 The length of a complex number given by a = ±b ± c √ −1 is√ b2 + c2. [Used in Terrot’s proof of Pythagoras’ theorem.] Table 4.1: Examples of Terrot’s use of material from Warren’s 1828 treatise [136]. 104 4.4 Associated historical insights 4.4.1 The discovery of quaternions As well as being a source of inspiration for the Bishop’s mathematical researches, John Warren’s treatise of 1828 was of influence to Sir William Rowan Hamilton in his discovery of quaternions. Hamilton wrote to the editors of the Philosophical Magazine on 20 November 1844. He enclosed a copy of a letter he had sent to the mathematician and jurist, John T. Graves (1806–1870) in October 1843.299 In this letter to Graves, Hamilton had reported a break-through in the theory of quaternions and given an account of the thought processes which had led him to the discovery. Some of the letter had been published in the July and October editions of the magazine; but Hamilton hoped publication in full would provide an opportunity to publicly acknowledge John Warren’s contribution and might encourage Graves—with whom Hamilton had been in fruitful correspondence for many years—to make public how he had been able to extend Hamilton’s work and bring to light his own results. Hamilton and Graves had maintained a close friendship from the time they began their studies together at Trinity College, Dublin in 1823: For many years Graves had been Hamilton’s sympathetic friend and mathematical confidant, and the two men maintained an active correspondence, in which they com- peted with each other in their attempts to produce a full and coherent interpretation of imaginaries. Graves worked at perfecting algebraic language; Hamilton had the higher object of arriving at the meaning of the science and its operations.300 It was Graves’ work on imaginary logarithms that eventually led Hamilton to his discovery of quaternions: from conjugate functions to the theory of triplets; onto the sets of moments, steps and numbers; all building to the discovery of quaternions. This source of inspiration was acknowledged by Hamilton but Graves ‘modestly 299John T. Graves was the brother of R. P. Graves, Hamilton’s chief biographer. 300 [139] 105 disclaimed the credit of suggestion’.301 Hamilton’s letter to the editors ran as follows:302 To the Editors of the Philosophical Magazine and Journal Gentlemen, I have been induced to think that the account contained in the following letter [H to Graves (17 Oct. 1843)], of the considerations which led me to conceive that theory of quaternions, a part of which you have done me the honour to publish in two recent Numbers (for July and October) of your Magazine, might not be without interest to some of your readers. Should you think proper to insert it, a public acknowledgement (very pleasing to my own feelings) will have been rendered, on the one hand to the Rev. Mr. Warren, whose work on the Geometrical Representation of the Square Roots of Negative Quantities (printed at Cambridge in 1828) long since attracted my attention and influenced my thoughts; and on the other hand to the gentleman (John T. Graves, Esq.) to whom the letter was addressed, and with whom I had been engaged, at intervals, for many years in a correspondence, very instructive and suggestive to me, on subjects connected therewith. Nor am I without hope that Mr. Graves may thus be led to communicate through you to mathematicians some of the extensions which he has made of results of mine, with some of those other specula- tions which are still more fully his own. On some future occasion I may perhaps be allowed to mention any other quarters from which I may be conscious of having de- rived more recent assistance, in my investigations on the same mathematical subject, many of which are hitherto unpublished. I have the honour to be, Gentlemen, Your obedient Servant, William Rowan Hamilton Observatory of Trinity College, Dublin, November 20, 1844 301[Ibid.] 302H to Phil. Mag. (20 Nov. 1844) in [140, pp489–490]. 106 Clearly, Hamilton owed a debt to Warren. In fact, Hamilton had not known about the geometrical representation of complex numbers until 1829 when Graves had encouraged him to read Warren’s treatise.303 Hamilton generously acknowl- edged his debt to Warren in the preface to his Lectures on Quaternions, writing: ‘To suggestions from that Treatise [(Warren 1828)] I gladly acknowledge myself to have been indebted, although the interpretation of the symbol √ −1, employed in it, is entirely distinct from that which I have since come to adopt in the geometrical applications of the quaternions.’304 Hamilton’s letter to Graves is of great value as it paints a very honest picture of the road to mathematical discovery.305 Given our present focus, it is highly signifi- cant that therein Hamilton explained his investigations had sprung from geometrical considerations. The letter began: Observatory, October 17, 1843 My Dear Graves,—A very curious train of mathematical speculation occurred to me yesterday, which I cannot but hope will prove of interest to you. You know that I have long wished, and I believe that you have felt the same desire, to possess a Theory of Triplets, analogous to my published Theory of Couplets, and also to Mr. Warren’s geometrical representation of imaginary quantities. Now I think that I discovered yesterday a theory of quaternions which includes such a theory of triplets. My train of thought was of this kind. Since √ −1 is in a certain well-known sense, a line perpendicular to the line 1, it seemed natural that there should be some other imaginary to express a line perpendicular to the former; and because the rotation from 1 to this also being doubled conducts to −1, it also ought to be a square root of negative unity, though not to be confounded with the former. Calling the old root, as the Germans often do, i, and the new one j, I inquired what laws ought to be assumed for multiplying together a+ ib+ jc and x+ iy + jz.306 303 [19, p262] Also page 191 of this thesis. 304 [141, p(31)f] 305For more on Hamilton’s discovery of quaternions see the largely historical preface to [141]. 306H to Graves (17 Oct. 1843) in [140, p490]. 107 The break-throughs which led Hamilton to the now familiar algebraic relations i2 = j2 = k2 = ijk = −1 are worth noting. They were to come out of his reasoning on the ij that appeared in the product (a+ ib+ jc) · (x+ iy+ jz). Hamilton asked: ‘but what are we to do with ij?’307 He began by considering the simplest case of a product, a square, and noted that the law of multiplication of moduli holds if we ignore the term in ij altogether.308 Thus if (a+ ib+ jc) · (a+ ib+ jc) = [a2 − b2 − c2] + [2ab]i+ [2ac]j +XXXX[2bc]ij then (a2 + b2 + c2)2 = [a2 − b2 − c2]2 + [2ab]2 + [2ac]2 But to Hamilton it seemed ‘odd and uncomfortable’ to set ij = 0.309 Instead, he set ij = −ji = k, with the value of k (zero or otherwise) still to be determined. This assumption achieved the same desired effect—the suppression of the ij term—but seemed ‘less harsh’ to Hamilton than requiring ij = 0.310 In order to find k, he considered the law of multiplication of moduli again, this time for the triplets (a + ib + jc) and (x + iy + jz); imagining that the term in k might, once again, require suppression for the law to hold. Thus if (a+ ib+ jc) · (x+ iy + jz) = [ax− by − cz] + [ay + bx]i+ [az + cx]j + XXXXX[bz − cy]k then . . . but this time (a2 + b2 + c2)(x2 + y2 + z2) 6= [ax− by − cz]2 + [ay + bx]2 + [az + cx]2 as the right hand side is too small by [bz− cy]2, which is the square of the coefficient of k. Seeing this, Hamilton realized that k cannot be written as a linear combination 307[Ibid.] 308Law of multiplication of moduli : if (a1, a2, a3) · (b1, b2, b3) = (c1, c2, c3) then (a12 + a22 + a32) · (b1 2 + b2 2 + b3 2) = (c12 + c22 + c32). 309H to Graves (17 Oct. 1843) in [140, p491]. 310[Ibid.] 108 of i and j, and that in order to work successfully with triplets he would need to work in four dimensions: And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triplets; or transferring the paradox to algebra, must admit a third distinct imaginary symbol k, not to be confounded with either i or j, but equal to the product of the first as multiplier, and the second as multiplicand; and therefore was led to introduce quaternions, such as a+ ib+ jc+ kd, or (a, b, c, d).311 The fact that Hamilton was open to the possibility of non-commutativity is remarkable. In not conforming absolutely to all the laws of algebra formerly thought inviolable, Hamilton had given himself room to manoeuvre and consequently made his discovery. 4.4.2 Warren’s reference to Bue´e and Mourey Warren followed his 1828 treatise with two related papers: in the first [142] he countered objections raised against the geometrical representation of imaginaries and in the second [143] he extended the treatise. It is in the first of these follow-up papers that Warren brought to light two contributions to the geometrical representation of complex numbers: Adrien-Quentin Bue´e’s 1806 paper [144] and C.-V. Mourey’s 1828 work [145].312 Warren stated—for the sake of priority—that he became aware of these contributions only after his own treatise was written: Bue´e’s in November 1827 and Mourey’s in December 1828. Biographical information on l’abbe´ Bue´e and further information on his contribution is given in Appendix E, where Gergonne’s 1811 conception of a two-dimensional table of real and imaginary magnitudes is also discussed. Mourey and his mathematics are discussed at length in Chapter 5. 311H to Graves (17 Oct. 1843) in [140, pp491–492]. 312See [142, pp251–254]. 109 4.4.3 Tait’s account of the developments Some forty years after making his notes on the Bishop’s paper, Tait—in an article entitled, ‘Quaternions’ in the ninth edition of the Encyclopaedia Britannica—gave a typically thorough, yet concise, history of developments in the geometrical represen- tation of √ −1 in the lead-up to the discovery of quaternions. His account is confined to the contributions of those whose interpretations had ‘geometrical applications in view’, as Hamilton’s had.313 Thus, his chronicle excludes anything that might be termed “the calculus of complex numbers”. According to Tait, Hamilton ‘was led to his great invention by keeping geometrical applications constantly before him while he endeavoured to give a real significance to √ −1 ’.314 Tait recognizes the contributions of Wallis, Bue´e, Argand, Franc¸ais, Warren, Mourey and, of course, Hamilton, who achieved that which the others could not—the ability to work in three dimensions. Note, Wessel’s 1799 contribution is missing from Tait’s account. 4.5 Coming full circle With Tait’s chronicle our record of these associations is complete. We began with Bishop Terrot’s paper introducing Tait to the geometrical representation of complex numbers, whilst Tait was still a schoolboy at the Edinburgh Academy; the Bishop had been inspired by Warren, as had Hamilton, and it would be Hamilton’s quater- nions which would captivate Tait and inspire a lifetime of mathematical research. 313 [146, p445] 314[Ibid.] 110 CHAPTER 5 C.-V. MOUREY’S SINGLE SCIENCE OF ALGEBRA AND GEOMETRY This chapter is on C.-V. Mourey’s 1828 work, La Vraie The´orie des quantite´s ne´gatives et des quantite´s pre´tendues imaginaires. 5.1 Introduction In 1828 C.-V. Mourey shared his results relating to the difficulties presented by the theory of algebra, in a book published in Paris under the title, La Vraie The´orie des quantite´s ne´gatives et des quantite´s pre´tendues imaginaires. Seeking algebraic re- form, Mourey had set out to discover a brand new set of definitions and fundamental principles as a basis for algebra. To this end, he developed a theory of directed lines, which constituted a single science of algebra and geometry; and, as an application of the theory, he gave a proof of the fundamental theorem of algebra (F.T.A.). In this chapter I reconsider Mourey’s motivations, re-evaluate his mathematics and present new biographical information on Mourey who has remained an unknown to historians of mathematics for the past 186 years. There will also be a survey of the reception of Mourey’s work, a highlight of which is the ten-year correspondence on Mourey between Hamilton and De Morgan. Language is an obvious barrier to understanding so, where appropriate, I have provided bi-lingual quotations from Mourey’s text. The original French appears in the footnotes. Currently, there is no English translation of the work. A second and significant barrier to understanding is the over-abundance of Mourey’s original terms and notations. Their sheer number confuses and distracts the reader and obscures the development of Mourey’s approach. So, as a further aid to the reader, I have produced a look-up table in Appendix G which provides a sum- mary of Mourey’s original terms and notations. Its entries are given in the order in 111 which they appear in Mourey (1861) so that it might be used as a companion-guide to the text. In itself the table provides a good summary of Mourey’s approach. In the commentary occasional reference is made to terms in the table. These references are given within square brackets in the footnotes, e.g. [See angle directif.]. 5.2 C.-V. Mourey: a biographical enigma On the title page of Mourey’s 1828 first edition (Figure 5.1, page 114) the author’s address is given as ‘Paris [. . .] rue des Quatre-Vents, no. 8’. This address consti- tutes the only biographical information on Mourey currently available.315 In 1861 a reprint of the book was produced by the same publishers, Bachelier.316 In this later edition no address is given for the author, which suggests that by 1861 Mourey was deceased.317 I assume that Bachelier published the reprint at their own expense in response to demand for the first edition. As early as 1846, an appeal for biographical information on Mourey was pub- lished in Nouvelles annales de mathe´matiques [147]. The editors write that copies of Mourey (1828) have become extremely rare and that to their knowledge, amongst mathematicians in Paris, only Lefe´bure de Fourcy (1787–1869) has a copy.318 They ask that those with biographical information on Mourey make contact. Since there was no published response to the appeal—notably, Mourey himself failed to make contact with the editors—it seems likely that Mourey had left Paris, or died, a short time after the publication of his book. It seems that Mourey was truly an unknown in Paris’ academic circles. Cer- 315I have found no other author who has made reference to this address—perhaps owing to the rarity of the first edition. 316By then Mallet–Bachelier. 317It appears that no other additions or amendments were made to this second edition; notably, typographical errors, which presumably were in the first edition, remain uncorrected. 318At that time Lefe´bure was Examiner for Admissions at the E´cole Polytechnique and Chair of Differential and Integral Calculus at the Faculte´ de Sciences. 112 tainly, there is no indication, in either edition of his work, of his affiliation with any academic or scientific institution. He is not remembered as a student at any of the well known educational establishments in the city and has never been referred to in connection with the great mathematicians who were known to have been in Paris during that period; thinking of the young talents of Abel and Galois, but also of Cauchy, Poisson, Legendre, Hachette, Dirichlet, Fourier and Lacroix. While the work is scholarly, it was not written for the purposes of instruction, which supports the theory that Mourey held no fixed teaching position in mathematics. The text itself offers no information on its author; notably, it is without reference to Mourey’s mathematical influences and in this respect it is similar to the work of the great Greek authors. The most probable reason for the absence of references is that Mourey simply had no instinct to cite other authors. Consequently, we have no information on which mathematics books Mourey had read. Likewise, we have no information on the nature of Mourey’s mathematical training. Presumably, Mourey did receive some training in mathematics: the strong focus on trigonometry and mechanics in the book suggests that he did receive some training in the practical application of mathematics.319 319See page 138 of this thesis for Mourey’s references to trigonometry and mechanics. 113 Figure 5.1: Title page of the first edition of Mourey. Reproduced with the kind permission of Collections E´cole Polytechnique, Palaiseau (France). The author’s address is given as ‘Paris [. . .] rue des Quatre-Vents, no. 8’. This first edition copy is now in very poor condition. 114 5.2.1 Self-funded publication From the preface of his book, we learn that Mourey’s 1828 publication was in fact an abridgement of a larger manuscript which Mourey had not been able to publish in full because of certain undisclosed circumstances. Mourey writes: Until now I have only dealt, as you understand, with the fundamental principles, and still I have written quite a considerable manuscript. However, as circumstances do not allow me now to have such a voluminous work printed, I have decided to publish this booklet first, which is but a small abstract.320 From this I infer that Mourey had published at his own expense and that he had restricted funds. The theory of a self-funded publication is also supported by the lack of evidence to suggest that Mourey’s work was subject to peer review prior to publication. But if finances were an issue for Mourey, why did he not submit his work to be published in a journal, such as: Gergonne’s Annales, Memoire de l’Institut de France or Ferrusac’s Bulletin? There are a number of possible reasons. From indications given by Mourey in his preface, it appears that the work pub- lished in 1828—his only publication—was the culmination of many years of private study: his mathematical researches had remained a private project until such a time when he considered his ideas to be sufficiently developed to share with others or un- til he had acquired the means to publish. Consequently, the impact of his work would have been a key consideration for Mourey. Had he published his results in a journal, he would have had to divide his work into a number of small contributions, on account of the amount of material, which might have had a negative bearing on impact. Mourey may have also faced difficulties in getting his work published, as an un- known in academic circles with no-one to recommend him for publication. Mourey 320‘Jusqu’ici je n’ai pu m’occuper, comme on le pense bien, que des principes fondamentaux, et cependant j’ai compose´ un manuscrit assez conside´rable. Mais, les circonstances ne me permettant pas de faire imprimer actuellement un ouvrage aussi volumineux, j’ai pris le parti de publier d’abord cet opuscule, qui n’en est qu’un faible abre´ge´.’ [145, pix] 115 probably considered self-funded publication to be the quickest and easiest route to getting his work noticed by the mathematical community. Mourey dedicated his work to the ‘Amis de l’Evidence’ (or ‘the friends of evidence’), which was probably not an actual group but a motto of Mourey’s: he dedicated his work to all those who, like him, search for truth. See Figure 5.1 (page 114) for the dedication. 5.2.2 Mourey’s identity Taking everything into account, it seems reasonable to state the following null hy- pothesis regarding Mourey’s identity: Mourey was not a professional academic but a first-class amateur mathematician whose trade involved the practical application of mathematics. And I have found a candidate who fits this profile perfectly: Claude-Victor Mourey, me´canicien a` Paris Claude-Victor Mourey, me´canicien a` Paris, held five-year patents for two inventions: a timber-profiling machine and a tree saw. The patents were granted on 12 July 1822 and 3 August 1822 respectively.321 Two files containing documents relating to Mourey’s patents are extant in the archives of the Institut National de la Proprie´te´ Industrielle.322 Both files contain: a detailed specification of the invention and tech- nical drawings (Figures 5.3–5.5, pages 120–122) which were submitted by Mourey; and the correspondence between Mourey and the administration, the Comite´ con- sultatif (advisory committee) des Arts et Manufactures. These documents are all handwritten. Mourey signs and dates the documents and gives his address. He signs himself ‘Victor Mourey. Me´canicien’ (Figure 5.2, page 119). The address he gives on 6 May 1822 is rue Saint-Honore´, no. 130, Seine, Paris. On 21 May, a 321Mourey’s patent for his tree saw lapsed in November 1824, possibly because the maintenance payments had not been kept up. 322I.N.P.I. file nos. 1BA1681 and 1BA1691. Available to view online: see [148] and [149]. 116 different address: rue Saint-Maur, no. 84, Faubourg du Temple, Seine, Paris.323 In some published lists of granted patents, Mourey’s full name is used: ‘Mourey (Claude-Victor), me´canicien a` Paris’.324 MM. Hacks and Co. The reports of the advisory committee, for both machines, indicate that before Mourey had submitted his patent applications, another me´canicien resident in Paris, M. Jean-Pierre Hacks, had designed and built the same or similar machines and had shown them to several members of the Socie´te´ d’Encouragement pour l’Industrie Nationale. Hacks’ own timber-profiling machine was given a favourable mention in the Society’s general meeting on 3 October 1821. His tree saw was presented to the Society on 28 March 1822; was given a favourable report in a meeting on 15 May and featured in the June issue of the Society’s Bulletin.325 Hacks submitted a patent application for his timber-profiling machine on 17 April 1823 and a five-year patent was granted on 22 May.326 From indications given in a Paris directory of the period, it appears that Jean- Pierre Hacks was the proprietor of MM. Hacks et compagnie—manufacturers of machines to saw timber, trees etc., who operated out of a workshop on the grande rue du Faubourg Saint-Antoine, no. 47.327 In July 1823 MM. Hacks et compagnie 323See [150] for an 1830 map of Paris by Girard. 324For instance: [151] and [152, p209]. 325For the feature on Hacks’ saw in the Society’s Bulletin, see [153] and [154]: the report by M. Molard (a member of the advisory committee) was approved in the meeting of the Society on 15 May 1822. 326I.N.P.I. file no. 1BA1799. Available to view online: see [155]. The documents relating to Hacks’ patent application contain no reference to Mourey, however, the committee report is missing from the file. 327The directory featured top artists and manufacturers in Paris. See [156, pp259–260] for Hacks’ entry in the directory, which includes a description of a number of Hacks’ machines, including his saw for standing trees. Note, Mourey is not listed in the directory. 117 won a silver medal for mechanically-produced wooden mouldings at an exhibition of French products of industry that was held at the Louvre Palace.328 This series of coincidences strongly suggests an association between Mourey and Hacks. I suggest the possibility that Mourey was employed by Hacks and Co. as a draughtsman and that the designs for both machines were originally his. I have found no evidence to suggest that Mourey had his own workshop. 328See [157, p419] for the jury’s report on Hacks’ entry at the exhibition. 118 Figure 5.2: Mourey’s signature: ‘Victor Mourey. Me´canicien. Rue St. Maur no. 84 f.b. [Faubourg] du Temple, a` Paris, le 21 Mai, 1822.’ Reproduced from the original patent application for his tree saw. Source: Archives I.N.P.I. 119 F ig ur e 5. 3: M ou re y’ s te ch ni ca ld ra w in gs fo r hi s ti m b er -p ro fil in g m ac hi ne . So ur ce : A rc hi ve s I. N .P .I .I t op er at es by fe ed in g ti m b er th ro ug h th e de vi ce , w hi le th e pr ofi le is cu t in to th e w oo d by th re e he ig ht -a dj us ta bl e ir on cu tt er s. 120 Figure 5.4: Technical drawings for Mourey’s machine to saw standing trees. Source: Archives I.N.P.I. 121 Figure 5.5: Technical drawings for Mourey’s machine to saw felled trees, adapted from his designs for his machine to saw standing trees. Source: Archives I.N.P.I. 122 Mourey’s Paris addresses Looking to prove that Mourey the me´canicien was the author of the mathemat- ics book in 1828, the only way forward was to follow up on the three addresses: from the title page of the first edition and the two addresses given in the patent applications. Fortunately, ledger volumes are extant in the Archives de Paris that contain information on property owners in Paris (their name, occupation etc.), to- gether with property valuations and details of when properties changed hands and for what reason (e.g. through sale or succession).329 The search for Mourey, through his Paris addresses, was made more difficult by the fact that the house number for each of Mourey’s addresses had changed twice: each property has an old, current and new number. Unfortunately, following up on the address drew a blank: I found no reference at all to Mourey; however, the search did rule out the possibility that the address given in the first edition was a holding address for the publisher. It is hard to believe that there is no information on record about Mourey. How- ever, if Mourey was not born in Paris, for instance, then there would be no record of his birth in the Paris archives; and if he did not have the means to afford to purchase his own property but instead rented rooms as a private tenant—as is suggested by the fact that he moved to a number of different addresses—he would not appear in the records of property owners in Paris. In any case, there would be little chance of finding any official record of Mourey because almost all the civil records for Paris prior to 1860 were destroyed by a fire at the Hoˆtel de Ville (where the paper records were stored) in May 1871 during the Paris Commune uprising: between five and eight million records were destroyed. Claude Mourey (1791–1830) There are, however, two records extant in the Paris archives for a Claude Mourey (note the absence of a middle name): an acte de de´ce`s (death certificate) and an 329One or two streets would normally be in each volume and each building would usually comprise around ten apartments. 123 acte de mariage (marriage certificate).330 According to his death certificate (Figure 5.6, page 126) Claude Mourey was born in the Valay department in the Haute-Saoˆne region in the east of France but was resident in Paris, at the time of his death, at rue du Faubourg, Saint-Antoine, no. 255; he was married to Marie Claire Klein and he died in Paris on 10 July 1830, aged thirty-nine. The term ‘employe´e’, which appears on his death certificate, indicates that Claude Mourey was not a gentleman; that is, he did not have the means to support himself without engaging in some sort of trade. The marriage certificates for Claude and his wife Marie Claire provide further information: the couple were married in Paris, in the parish of St Marguerite on 21 October 1829; Claude was the son of Claude Joseph Mourey and Anne Franc¸oise Fontaine; and Marie Claire was the daughter of Henry Klein and Marie Franc¸oise Gre´gorie. It is quite possible that our three Moureys are in fact one and the same individual. Certainly, I have found no evidence which rules out the possibility: the chronology makes sense and the various indications given of Mourey’s occupation, style of living etc., are all consistent. If indeed they are the same person, then assimilating all the biographical information available gives the overview of his life on page 125; but we move on now to consider Mourey’s mathematics. 330Found amongst the microfilmed copies of the reconstituted civil records for the period between the C16th and 1859. 124 Claude-Victor Mourey (1791–1830), me´canicien a` Paris, was born in the Valay department in the Haute- Saoˆne region in the east of France; to parents Claude Joseph Mourey and Anne Franc¸oise Fontaine. In 1822, having moved to Paris, he took out five-year patents for two ma- chines he had invented: a timber-profiling machine and a tree saw. He may have been employed as a draughtsman by MM. Hacks et compagnie, whose workshops were on the grand rue du Faubourg, Saint-Antoine, no. 47. In 1828, aged thirty-seven, Mourey had made sufficient money from his inventions to enable him to publish a mathematics book entitled, La Vraie The´orie des quantite´s ne´gatives et des quantite´s pre´tendues imaginaires: he was a first-class ama- teur mathematician. In Paris, on 21 October the following year, he married Marie Claire Klein, daughter of Henry Klein and Marie Franc¸oise Gre´gorie. He died in Paris on 30 July 1830, aged thirty-nine; just two years after his book was published and after only nine months of marriage. By 1846 there were very few copies of his work in Paris and his identity, as its author, had fallen into obscurity. 125 Figure 5.6: Acte de de´ce`s for Claude Mourey. Reproduced from microfilm. Source: Archives de Paris. 126 5.3 Mourey’s mathematics 5.3.1 Motivated by algebra, seeking algebraic reform Mourey explains in the opening to his preface that his motivation was algebra. He writes: Trained, almost from childhood, in analytical reasoning, I can resist no longer the desire to submit to the judgement of scholars some of the results that I have reached. These are the difficulties presented by the theory of Algebra, which have, for many years, been the principal subject of my reflections. Eager for clarity, my mind could not be satisfied with this science as it has been presented to this day.331 Dissatisfied with algebra—with the theories of negatives and imaginaries in par- ticular—Mourey sought improvements. He was concerned over two problems in particular: (1) how the operation of subtraction would translate from arithmetical algebra into a more symbolical form of algebra and (2) the practice of treating imag- inaries in the same way as reals without proof that it is legitimate to do so. Problem 1 Mourey understood that persisting with the definition of subtraction in arithmetic was causing problems for algebraists who were dealing with unknown or arbitrary quantities represented by algebraic symbols: if both the magnitude and the sign of the quantities involved in the subtraction a − b are unknown, it cannot be determined whether the operation is “possible” (for b < a) or “impossible” (for b > a). Mourey pronounced: it is impossible to express the difference [between two quantities] by means of the sign −, when one of the terms is unknown or arbitrary. [. . .] 331‘Entraˆıne´, presque de`s l’enfance, dans les me´ditations analytiques, je ne puis re´sister plus longtemps au de´sir de soumettre au jugement des eˆtres pensants quelques-uns des re´sultats aux-quels je suis parvenu. Ce sont les difficulte´s que pre´sente la the´orie de l’Alge`bre, qui ont fait, pendant de longues anne´es, le sujet principal de mes re´flexions. Avide de clarte´, mon esprit ne pouvait eˆtre satisfait de cette science, telle qu’elle a e´te´ pre´sente´e jusqu’a` ce jour.’ [145, pv] 127 It follows that the sign −, considered to express subtraction, cannot be admitted in Algebra. Because Algebra is supposed to deal only with unknown or arbitrary quantities, it cannot admit of subtraction.332 And so Mourey resolved ‘to find the means of expressing the difference between two quantities, without having recourse to subtraction’.333 Problem 2 Mourey acknowledged that it was encouraging that the practice of manipulating imaginaries as if they were reals had not been shown to lead to false results; yet, he demanded more of algebra. He wanted algebra to have a logical struc- ture and a basis similar to that of geometry: ‘It must be agreed that the science would be far more satisfactory, if we could base all its parts on rigorous reasoning, on first-rate evidence, on simple and palpable ideas, like those of the elements of Geometry.’334 Mourey had realized that the issues relating to negative and imaginary quanti- ties arose from deficiencies in algebra—deficiencies in its definitions and fundamental principles. It was the persistent appearance of negatives and imaginaries that had revealed to Mourey that the operations of algebra were capable of being more com- prehensive. John Warren had observed the same. He wrote of imaginaries: One thing was evident respecting them; that they were quantities capable of under- going algebraic operations analogous to the operations performed on what are called 332‘Il suit de la` qu’il est impossible d’exprimer la diffe´rence par le moyen du signe −, lorsque l’un des termes est inconnu ou arbitraire. [. . .] Il suit de la` que le signe −, conside´re´ comme exprimant la soustraction, ne peut pas eˆtre admis en Alge`bre. L’Alge`bre, e´tant cense´e ne s’occuper que de quantite´s inconnues ou arbitraires, ne peut point admettre de soustraction.’ [145, pp1–2] 333‘Il faut donc trouver le moyen d’exprimer la diffe´rence de deux quantite´s, sans recourir a` la soustraction; autrement l’Alge`bre resterait imparfaite.’ [145, p2] 334‘On doit convenir que la science serait beaucoup plus satisfaisante, si l’on pouvait en baser toutes les parties sur des raisonnements rigoureux, sur une e´vidence du premier ordre, sur des ide´es simples, palpables, comme celles des e´le´ments de Ge´ome´trie.’ [145, pvii] 128 possible quantities, and of producing correct results: thus it was manifest, that the operations of algebra were more comprehensive than the definitions and fundamental principles; that is, that they extended to a class of quantities, viz. those commonly called impossible roots, to which the definitions and fundamental principles were inap- plicable. It seemed probable, therefore, that there was a deficiency in the definitions and fundamental principles of algebra; and that other definitions and fundamental principles might be discovered of a more comprehensive nature, which would extend to every class of quantities to which the operations of algebra were applicable; that is, both to possible and impossible quantities, as they are called.335 There is a striking similarity between Warren and Mourey’s approaches to their researches: both intended to establish a new set of definitions and fundamental principles in algebra that were applicable to all classes of quantities; both focused on algebraic operations and both used geometry as a way in. Mourey claimed not only to have succeeded in establishing a basis for algebra that was similar to geometry but also to have discovered—unexpectedly, at the same time—a new system of geometry: Not only have I reached this goal [to establish a basis for algebra], but I have en- countered at the same time another result, which is perhaps no less valuable; with a new system of Algebra, which I was looking for, I found a new system of Geometry, which I was not expecting. They are not, however, two sciences; it is one and the same science, one and the same theory, which has two sides, one algebraic and the other geometric. It is an Algebra which emanates from Geometry; it is a Geometry which is generalized and made algebraic.336 Mourey developed this dual system of algebra and geometry under a unifying concept of directed lines. 335 [142, p241] 336‘Non-seulement j’ai atteint ce but, mais j’ai rencontre´, en meˆme temps, un autre re´sultat qui n’est peut-eˆtre pas moins pre´cieux; avec un nouveau syste`me d’Alge`bre, que je cherchais, j’ai trouve´ un nouveau syste`me de Ge´ome´trie, auquel je ne m’attendais pas. Ce ne sont cependant pas deux sciences; ce n’est qu’une seule science, une seule the´orie, laquelle a deux faces, l’une alge´brique, et l’autre ge´ome´trique. C’est une Alge`bre e´mane´e de la Ge´ome´trie; c’est une Ge´ome´trie ge´ne´ralise´e et rendue alge´brique.’ [145, ppvii–viii] 129 Mourey anticipated fierce opposition to his new radical system; and yet, he was confident that one day it would come to be recognized as the one true theory. He writes: The system which I am putting forward is altogether new; it will meet with many powerful enemies: ideas which have prevailed for centuries, inveterate habits, species of vested interests . . . Without any doubt it will eventually prevail, as everything that is good and true must do; but will its author enjoy this triumph?337 In the commentary which follows, I will highlight evidence of the far-sightedness of Mourey’s work, in view of the way mathematics has since developed. 5.3.2 Definitions and fundamental principles Recall that Mourey hoped to discover some means of expressing the difference be- tween two quantities without resorting to subtraction. For real quantities there was, of course, already a standard approach: real quantities could be represented as points on the real line (Figure 5.7, page 131) with addition interpreted as a journey in a positive direction and subtraction as a journey in the opposite direction. But for non-real quantities, a more general theory of directed lines was required. Mourey promised a theory in which all algebraic quantities whatever could be represented geometrically by directed lines in the plane; lines which would be determined by the two parameters of length and direction. Mourey defines a directed line or path as a line which leads in a particular direc- tion. It is, in modern terminology, a vector.338 To denote the path which leads from an origin A to a terminus B he writes AB and, similarly, BA for the path which leads from B to A.339 He notes, in the first instance, that AB and BA represent the 337‘Le syste`me que je propose est tout neuf; il rencontrera des ennemis nombreux et puissants: des ide´es qui re`gnent depuis des sie`cles, des habitudes inve´te´re´es, des espe`ces de droits acquis. . . . Sans doute il finira toˆt ou tard par triompher, comme doit le faire tout ce qui est bon et vrai; mais est-il re´serve´ a` son auteur de jouir de ce triomphe?’ [145, px] 338[See ligne directive.] 339[See origine, terme.] 130 same non-directed line but different paths: to be equal as non-directed lines, two lines need only have the same length; to be equal as paths, however, ‘it is necessary and sufficient that they be concurrent and have the same length’.340 With direction taken into account, AB and BA are termed inverses and BA = −AB.341 Figure 5.7: Journeys along the real line.342 Having defined his set of directed lines, Mourey looks to establish the rules for their combination. He defines addition first, according to the following ‘fundamental principle’343 AB +BC = AC That is: ‘The sum of two consecutive paths is the single path which leads from the origin of the first to the terminus of the second’.344 See 4ABC, for example, in Figure 5.8 (page 132). The principle extends, of course, to sums of more than two 340‘Donc, pour que deux chemins soient e´gaux, il faut et il suffit qu’ils soient concurrents et qu’ils aient meˆme longueur.’ [145, p8] 341[See (chemins) inverses.] 342 [145, p2] 343‘L’e´quation AB +BC = AC sera notre principe fondamental.’ [145, p5] 344‘La somme de deux chemins de suite est le chemin simple qui conduit de l’origine de premier au terme du second’. [145, pp5–6] 131 directed lines, which Mourey demonstrates by successive addition of paths.345 Figure 5.8: Addition of directed lines.346 Next, Mourey considers addition for paths that do not follow on from one another—that are not de suite (consecutive)—such as the paths AB and DE in Figure 5.8.347 His approach is to trace from the point B, a path BC that is parallel to, and of the same length as, DE (translation of the vector DE in other words). Thus AB +DE = AC. Mourey describes addition as a ‘fundamental principle’ because he believed that it provided the only means by which we might substitute for subtraction in algebra. Consider the addition of two inverse paths AB and −AB: if the sign − expressing subtraction is interpreted as “+ inverse”, then subtraction can be performed by changing the sign of the quantity to be subtracted and proceeding as in addition. Mourey writes: we have proved that the sign−, considered to express subtraction, cannot be admitted in Algebra; it is therefore appropriate to give it some other interpretation, in order 345See [145, pp6–7] for details. 346 [145, p2] 347[See (chemins) de suite.] 132 to use it. The above equation [AB + inverse AB = AB − AB] provides us with the means; we have only to admit that this sign − will signify + inverse. Or else, we can admit that − will simply mean inverse, and we will write AB + −AB = 0. We will use either one or the other of these two interpretations; but, in either case, the term will be the same, plus inverse.348 I would suggest that Mourey has re-defined subtraction by introducing the additive inverse into his system. He also has − − AB = −BA = AB, or −(−x) = e.x in modern notation, which is a property deducible from group axioms. Continuing with the concepts of direction and opposition, Mourey explains how they relate to abstract numbers. A directed number is an abstract number, consid- ered as a quantity measured relative to an abstract unit; and for this abstract unit, Mourey takes a path of arbitrary length and direction. Thus, the positive integers 1, 2, 3, . . . are defined as numbers with the same direction as the unit 1; the negative integers −1,−2,−3, . . . as numbers with a direction opposite to 1. This relationship between the integers and the unit suggests to Mourey new names for positives and negatives: comme´triques for positives and antime´triques for negatives.349 Mourey recognizes an advantage of defining the integers in this way—that these definitions serve to dispel some of the myths surrounding negative quantities. He writes: Several consequences follow from these definitions which are important to emphasize in teaching, in order to keep the pupils from the false ideas naturally implied by the words positive and negative, as well as the sign − when referred to by the word 348‘Mais nous avons de´montre´ que le signe −, conside´re´ comme exprimant la soustraction, ne peut pas eˆtre admis en Alge`bre; il est donc convenable de lui donner quelque autre interpre´tation, afin de l’utiliser. L’e´quation ci-dessus [AB + inverse AB = AB − AB] nous en fournit le moyen; nous n’avons qu’a` admettre que ce signe − signifiera + inverse. Ou bien encore, nous pouvons admettre que − signifiera simplement inverse, et nous e´crirons AB +−AB = 0. Nous emploierons tantoˆt l’une et tantoˆt l’autre de ces deux interpre´tations; mais, dans l’un et l’autre cas, l’e´nonce´ sera le meˆme, plus inverse.’ [145, p10] 349[See unite´ relative, nombre relatif, concret, abstrait, nombre directif, (nombre) positif, (nombre) ne´gatif.] 133 minus. [. . .] The same directed quantity will be either positive or negative as we please, depending on whether we would like to take the unit in one direction or the other. [. . .] Every negative quantity is as large as its positive inverse. [. . .] The negative quantities are just as real and palpable as the positive ones. [. . .] The negative quantities are not at all the result of impossible subtractions.350 Staying with the integers, Mourey considers how they are related to one another by the signs < and >. While he recognizes that the same relation exists between, for instance, the numbers 1 and 0, and 0 and −1 +1 = 0 + a positive 0 = −1 + a positive he yet concludes—at odds with some contemporary algebraists—that −1 > 0,−2 > −1,−3 > −2, etc., He is, therefore, ultimately unsuccessful in introducing an order relation onto the integers. But perhaps Mourey’s intention was not to introduce an order relation onto the integers: perhaps he was looking to introduce an order relation onto the complex numbers; remember, Mourey sees the integers as a special case of the com- plex numbers. An ordering of the complex numbers is impossible of course. The very best we can do in this regard is to order them by their magnitudes. Doing this, we do have, in agreement with Mourey, −1 > 0,−2 > −1,−3 > −2, etc. In order to depart from the real line, Mourey introduces the notion of the di- rected angle. Suppose that the directed line AB (Figure 5.9, page 135) is made to rotate about its origin A through an angle r to become the directed line AC. This geometrical operation is expressed algebraically by ABr = AC 350‘Des ces de´finitions re´sultent plusieurs conse´quences qu’il est important de signaler dans l’enseignement, pour pre´server les e´le`ves des fausses ide´es que pre´sentent naturellement les mots positif et ne´gatif, ainsi que le signe −, lorsqu’on l’e´nonce par le mot moins. [. . .] La meˆme quantite´ directive sera positive ou ne´gative a` notre gre´, selon qu’il nous plaira de prendre l’unite´ dans un sens ou dans l’autre. [. . .] Toute quantite´ ne´gative est aussi grande que son inverse positive. [. . .] Les quantite´s ne´gatives sont tout aussi re´elles et aussi palpables que les positives. [. . .] Les quantite´s ne´gatives ne sont point les re´sultats de soustractions impossibles.’ [145, p13] 134 where the angle r is termed the verseur of AB. Figure 5.9: The directed angle.351 Now, the angle leading from AB to AC is necessarily a directed angle, as it must lead in a particular direction: either clockwise (measured by the major arc BHFDC) or anti-clockwise (measured by the minor arc BxC). Mourey adopts the convention that angles described by an anti-clockwise rotation are positive. Mourey indicates angles in a peculiar fashion. He denotes the included angle between AB and AC by ABC: the first letter symbolizes the vertex, the second identifies the leading side (the side from which the angle originates which Mourey terms the coˆte´ dirigeant) and the last identifies the facing side (the side towards which the angle approaches which Mourey terms the coˆte´ dirige´).352 If no convention is established as to the sense in which the angle turns, then additional letters are placed between the letters representing the leading and facing side. For example, the directed angle leading from AB to AC via x (Figure 5.9) is expressed by ABxC. For the unit of the directed angle, Mourey chooses an anti-clockwise rotation of 90◦ which he denotes by a subscript 1. Thus AB90◦ = AB1 = AD 351 [145, p19] 352[See coˆte´ dirigeant, coˆte´ dirige´.] 135 and similarly AB180◦ = AB2 = AF, AB−90◦ = AB−1 = AH, AB60◦ = AB 2 3 = AC, etc. Frequently, the path AB is given unit length and is taken along the positive real axis so that AB = 1, AD = 11, AF = 12 = −1, AH = 13 = −11, AC = 1 2 3 , etc. This is the realization of a statement made by Mourey in his preface: in his system, ‘the algebraic (or even arithmetic) expression of a path will determine its direction, in relation to another path taken as a term of comparison, or, if we like, as unity’.353,354 Now, even if we were to make AB perform any number of full revolutions in addition to r, it would invariably arrive at AC. Mourey’s understanding of the phenomenon is this: Thus, the directed angle is likely to grow to infinity; but any directed angle which is not smaller than 4 right angles, may be replaced by the excess over the largest multiple of 4 right angles which it contains. We can therefore say that two angles are equal as directed angles, if their difference is exactly 4q or a multiple of 4q.355 Note, the notation 4q is a little misleading: it denotes four right angles rather than powers of 4. Mourey is not explicit about this and does not define q. Mourey terms this species of equality, which he represents by the sign =´, as the 353‘L’e´xpression alge´brique (ou meˆme arithme´tique) d’un chemin en de´terminera la direction, rela- tivement a` un autre chemin pris pour terme de comparaison, ou, si l’on veut, pour unite´.’ [145, pviii] 354[See verser, verseur, version, rapport directeur, angle directif.] 355‘Ainsi, l’angle directif est susceptible de croˆıtre a` l’infini; mais tout angle directeur qui n’est pas plus petit que 4 angles droits, peut eˆtre remplace´ par son exce´dant sur le plus grand multiple de 4 angles droits qu’il contienne. On peut donc dire que deux angles sont e´gaux en tant qu’angles directeurs, si leur diffe´rence est exactement 4q ou un multiple de 4q.’ [145, p24] 136 special equality of directed angles. It is expressed in his notation, in terms of units of right angles, as r =´ r + 4 =´ r + 8 =´ r + 12 =´ · · · =´ r + 4n (5.1) where n is an integer.356 Mourey also has, for angle measured contrary to convention, −r =´ 4− r Those with a background in modern mathematics might notice two things: (i) Mourey is identifying as equal, angles that are congruent modulo 2pi and is, therefore, describing an equivalence relation and (ii) his equation (5.1) appears remarkably similar in form to the following equation [r] = [r + 4] = [r + 8] = [r + 12] = . . . = [r + 4n] which expresses that all numbers congruent modulo 2pi to r belong to the equiva- lence class [r], the representatives of which may be indifferently 0, 4, 8, 12, . . . , 4n. The concept of the congruence of integers with respect to a modulus, and the as- sociated notation, was introduced by Johann Carl Friedrich Gauss (1777–1855) in Disquisitiones Arithmeticae in 1801. At the same time as rotating a directed line ar through a given angle s, we might wish to stretch its length by a given multiple b. To this end, we multiply the directed line ar by bs according to Mourey’s definition of multiplication ar × bs = (ar × b)s = [(ab)r]s = abr+s Mourey defines the directed number bs as an operator, which multiplies the length of ar by b and rotates it through an angle s. The concept of a geometrical operator was known to Hamilton in the form of a quaternion from 1844.357 In several places Mourey explains the significance of multiplication. (i) ‘Every 356[See e´galite´ spe´ciale des angles directifs.] 357For verification of this, see the correspondence between Hamilton and De Morgan dated 14 July 1854 (page 183 of this thesis). 137 directed number is derived from the directed unit by multiplication, division and version’.358 Version is the operation of rotation implicit in multiplication. (ii) ‘[The definition of multiplication] is very useful, because it follows from it [. . .] that all Algebraic formulae express real quantities, and that they apply very favourably to Geometry (and therefore to Mechanics)’.359 This is not Mourey’s only reference to mechanics. In the preface he writes: ‘In its application to Mechanics, the advantages are not least; it even seems that the system in question is made specially for it.’360 For the operations of addition and multiplication, Mourey is aware that certain properties hold. For addition: commutativity and associativity. For multiplication: commutativity, associativity and distributivity over addition. Mourey proves the commutative property for addition and multiplication, and the distributive laws; but he omits the details of a proof of associativity for both operations.361 Mourey’s proof of commutativity for multiplication is entirely algebraic in nature. He has ar × bs = bs × ar Since, the first = abr+s and the second = bas+r: now, ab = ba, and r + s = s+ r; therefore, etc. His proof of the distributive laws is based on vector methods. 358‘Tout nombre directif se forme de l’unite´ directive, par multiplication, division et version’. [145, p30] 359‘On doit admettre cette convention si elle est utile; or, elle est tre`s-utile, car il en re´sulte, comme on le verra, que toutes les formules de l’Alge`bre expriment des quantite´s re´elles, et qu’elles s’appliquent tre`s-avantageusement a` la Ge´ome´trie (et par conse´quent a` la Me´canique).’ [145, p31] 360‘Dans l’application a` la Me´canique, on ne rencontrera pas de moindres avantages; il semble meˆme que ce soit spe´cialement pour cette partie que soit fait le syste`me dont il s’agit.’ [145, pix] 361Mourey’s proof of commutativity for addition relies heavily on the equality of triangles: see [145, pp8–9]. 138 Mourey’s proof of the distributive laws To prove the distributive laws, Mourey first constructs the similar triangles ABC and ADE (Figure 5.10). He has AD = AB × p , AE = AC × p and DE = BC × p, where p is positive. Figure 5.10: Mourey’s proof of the distributive laws using similar triangles.362 But as vector sums, AC = AB +BC AE = AD +DE therefore (AB +BC)× p = AB × p+BC × p Next, Mourey rotates the triangle ADE through an angle r, about the vertex A, so that it comes to rest at APQ. By construction Mourey has ADP = r and AP = ADr. But AE has also moved through an angle r to become AQ, therefore, AEQ = r and AQ = AEr. Mourey wants to prove that PQ = DEr. Since PQ and DE are clearly the same length, all Mourey needs to prove is that PQ ∵ DE = r = ADP , where PQ ∵ DE denotes the angle which takes DE to PQ. This is new notation introduced by Mourey to denote the angle between two directed lines that do not share the same origin.363 362 [145, p32] 363[See dige`ne.] See also [145, pp26–27]. 139 Now, ∠PQA = DEA, therefore, PA ∵ PQ = DA ∵ DE ∠PQA being the ratio of PA to PQ and likewise ∠DEA being the ratio of DA to DE.364 Switching the position of the means, Mourey has PA ∵ DA = PQ ∵ DE but PA ∵ DA = AP ∵ AD = ADP = r therefore PQ ∵ DE = r therefore PQ = DEr as required. Now AQ = AP + PQ = ADr +DEr but AQ = AEr = (AD +DE)r therefore (AD +DE)r = ADr +DEr From commutativity and associativity of multiplication, it follows that (AB +BC)× pr = AB × pr +BC × pr To simplify, Mourey lets AB = c, BC = d and pr = q, so that (c+ d)× q = c× q + d× q which is the right distributive law. From this, using the commutative property of multiplication, Mourey deduces the left distributive law q × (c+ d) = q × c+ q × d 364[See e´qui-dige`ne.] See also [145, pp29–30]. 140 Then, letting q = a + b, Mourey derives an equation which shows how to mul- tiply out brackets and which, he remarks, can be extended to the multiplication of polynomials (a+ b)× (c+ d) = a× c+ b× c+ a× d+ b× d Note, that what Mourey says in relation to the notation ∵ suggests that he is close to the notion of the scalar product u ·v = uv cos θ. From an algebraic equation involving ∵ he deduces the relative position of two paths: The equation a ∵ b = 0 shows us that a and b are concurrent. The equation a ∵ b = 2 shows us that a and b are opposite. The equation a ∵ b = 0 or 2 shows us a and b are parallel. The equation a ∵ b = 1 or 3 shows us a and b are perpendicular. The difference is that Mourey is not taking magnitudes into account. From the rule for multiplication Mourey deduces the rule for division bs = (d a ) z−r and the rule for taking powers (amr ) n = am×nr×n Note, amr signifies (a m)r and not (ar)m.365 And from this principle Mourey deduces the nth roots of unity 1, 1 4 n , 1 8 n , 1 12 n , . . . 1 4(n−1) n or more generally 1σ n where σ represents the set of multiples of 4 right angles {0, 4, 8, 12, . . .}. The set notation is mine. Mourey notes that this is a periodic sequence of n values and 365[See amr .] 141 reasons, on the basis of this observation, that there are exactly n distinct nth roots of unity. He accounts for the plurality of the roots as follows. Setting x = 1z the equation xn = 1 becomes (1z) n = 1z×n = 10 hence z × n = 0 or more generally z × n =´ 0 This last equation has n roots z = σ n where σ again represents the set of multiples of 4 right angles {0, 4, 8, 12, . . .}. There- fore, the roots of xn = 1 are the set of solutions x = { 1σ n } Similarly, he gives the nth roots of xn = 1r as x = n √ 1r+σ = { 1r+σ } Again the set notation is mine. Note, Mourey has n √ 1r+σ rather than n √ 1r as the algebraic formula which produces the nth roots of 1r. This is because he determines that n √ 1r expresses uniquely the root 1 rn , which in turn has a single value; and therefore, in effect, n different algebraic formulae are required in order to produce the n nth roots of 1r. Mourey also establishes a special convention for determining the values of the formula n √ −1. Outside of the radical sign in his system −1 = 12 = 16 = 110 = · · · . If this were allowed to be the case under the radical sign, then the formula n √ −1 would represent multiple values 1 2 n , 1 6 n , 1 10 n , etc.; in general 1 2+σ n . Believing this to be not very useful, Mourey restricts the formula n √ −1 to having a single value 1 2 n . Thus, under the radical sign −1 denotes 12 and not 12±σ, −a denotes a2 and not a2±σ, 142 and −ar denotes a2+r and not a2+r±σ. A consequence of admitting this convention is that in Mourey’s system certain equalities from ordinary algebra fail to hold. For instance, in ordinary algebra 3 √ −1 = −1, 5 √ −1 = −1, 7 √ −1 = −1, etc.; however, in Mourey’s system 3 √ −1 = 1 2 3 , 5 √ −1 = 1 2 5 , 7 √ −1 = 1 2 7 , etc., and 1 2 3 6= 1 2 5 6= 1 2 7 6= −1. In order to accommodate his peculiar conventions for taking nth roots, Mourey introduces new notation to express the equality of two quantities under the radical sign. This species of equality, expressed by the sign := , is termed super-equality. 366 For two directed lines to be equal under the radical sign, they must have the same length and direction, and make exactly the same angle with the positive real axis (it is not good enough that their angles are congruent modulo 2pi).367 Mourey’s example is 14 and 10. Although these two directed lines have the same length and direction, and their angles are congruent modulo 2pi, they are not equal under the radical sign since √ 14 = 1 4 2 = 12 = −1 6= √ 10 = 10. Continuing with the difficulties associated with the calculus of radicals, Mourey considers what happens when polynomials, rather than monomials, appear under the radical sign. Mourey runs into difficulty when it is discovered that equality is not preserved. His example is (a+ √ a2 − c2)× (a− √ a2 − c2) := a 2 + a √ a2 − c2 − a √ a2 − c2 − (a2 − c2) which he reduces to : = a 2 − (a2 − c2) := a 2 − a2 −−c2 := c 2 4 Now, taking the square root of both sides gives √ a+ √ a2 − c2 × √ a− √ a2 − c2 = √ c24 = c2 = −c If we let a = c we have √ c+ √ c2 − c2 × √ c− √ c2 − c2 = √ c× √ c = +c 366[See (signe) super-e´gal, (chemins) super-e´gaux.] 367[See prime-directeur.] 143 however, if we let a = 0 we have √√ −c2× √ − √ −c2 = √ c1× √ −c1 = √ c1× √ c3 = ( √ c) 1 2 × ( √ c) 3 2 = c 4 2 = c2 = −c Mourey concludes that there is no general method for multiplying polynomials under a radical sign. Let A and B be any two polynomials and let A×B = P . We should not conclude, he says, that n √ A × n √ B = n √ P but only that n √ A × n √ B = n √ P σ n ; that is, that the product is any one of the roots of the equation xn = P . These few examples evidence some very serious consequences of Mourey’s pecu- liar conventions. Although Mourey was aware of the problems in his system, and knew that they arose from the conventions he had admitted for taking nth roots, it seems that he was willing to accept that they might surface occasionally. Mourey concludes his introductory section by showing how logarithms can make the process of multiplying directed lines easier. His example is x = 7 1 2 ×−15×10×1 2 3 . He calculates the logarithm of the length of each path, sums the result and then finds the antilogarithm using tables. This gives him the length of x. The direction of x, or its prime-directeur, is found by adding the prime-directeurs of all the factors of x. 5.3.3 Applications, including a proof of the fundamental theorem of algebra Having established the preliminaries, Mourey attempts to show how his theory of directed lines in the plane relates to trigonometry. With reference to Figure 5.11 (page 145), he establishes an association between trigonometric lines and paths. His approach is to express trigonometric lines in terms of their position with respect to the real axis. First, he establishes AB (Figure 5.11) as a line drawn in a positive direction AB = posit. = 1 Next, he identifies the angles ∠ABC = C ∠ABC ′ = C ′ ∠ABCC ′C ′′ = C ′′ ∠ABCC ′C ′′C ′′′ = C ′′′ 144 Figure 5.11: Relating trigonometric lines and paths.368 Thus he has for sine, for example, sinC = sinC ′ = pC = p′C ′ = pc1 = (pos.)1 = pos.× √ −1 sinC ′′ = sinC ′′′ = p′C ′′ = p′C ′′′ = pc3 = (pos.)3 = pos.×− √ −1 or = (pc′)1 = (neg.)1 = neg.× √ −1 and, therefore, in general sinx = (posit.)1 or (neg.)1 = (paral.)1 that is, the sine of an angle corresponds with a path that is perpendicular to the real axis. 368 [145, p58] 145 Similarly cosx = posit. or negat. = parallel tanx = (posit.)1 or (neg.)1 = (paral.)1 secx = (posit.)x or (neg.)x = (paral.)x sin vers x = posit. Note, sin vers x is the versed sine of x, that is, the sine of x rotated through an angle of 90◦. It is related to sinx by the formula vers x = 1−cosx, which Mourey deduces from Figure 5.11.369 The versed sine featured historically in the mathematics of engineering and surveying. Mourey also deduces from Figure 5.11, the formula for sec x in terms of tanx (secx)=x = 1 + (tanx)=1 and Euler’s identity 1x = (cosx)= + (sinx)=1 (5.2) In the new notation a= denotes a path of length a which is parallel to unity and a=1 denotes a path of length a which is perpendicular to unity.370 Mourey also establishes—from Figure 5.11, by means of similar triangles—that cosx : 1 :: sinx : tan x :: 1x : secx, from which the trigonometric formulae for tanx and sec x follow. Equating the first two ratios, we have tanx = sinx cosx Equating the first and third, we have secx = 1x cosx 369See [145, p60] for details. 370[See a=, a=1, mi-de´verseur.] 146 He then goes on to show how we can derive the compound angle formulae for cos (x+ z) and sin (x+ z) using the familiar method of equating two expressions for 1x+z.371 Application of trigonometry to the calculus of equations Next, Mourey outlines a method whereby we can use Euler’s identity to write the nth roots of the equation xn = 1 in the form a+ b √ −1. Recall, the roots of xn = 1 are { 1σ n } . Substituting σn for x in (5.2) gives 1σ n = cos σ n + (sin σ n )=1 = cos σ n + (sin σ n )= √ −1 where σ = 0, 4, 8, 12, . . . From Euler’s identity, Mourey establishes the polar form of a directed line ar = a× 1r = a cos r + a sin r Note, he frequently drops the subscript notation. Next, Mourey considers three problems: in the first two he explains how to approach the geometrical operation of addition algebraically; in the third he works through a method for solving the cubic. In order to add two paths pr and qs we need to solve the following equation for the unknowns x and u pr + qs = xu which becomes, when pr and qs are in polar form, xu = (p cos r + q cos s) + (p sin r + q sin s) Again, Mourey drops the subscript notation. Identifying the real and imaginary parts, so that if xu = ±a± b1, we have ±a = p cos r + q cos s ±b = p sin r + q sin s 371See [145, pp62–64] for details. 147 From here we can find tanu using the formula Mourey deduces from Figure 5.11 ±b1 ±a = sinu cosu = tanu and from there, find u using tables. We can then find x using the following formula, which Mourey deduces from Figure 5.11 by considering similar triangles, x = ±a cosu Alternatively, Mourey suggests we might use the following formulae x = √ a2 + b2 (sinu)= = ±b x Mourey’s method for solving the cubic is as follows. He begins with the reduced general cubic equation x3 + px+ q = 0 (5.3) and proceeds initially according to the method of Tartaglia given in Girolamo Car- dano’s (1501–1576) Ars Magna (1545).372 He substitutes y + z for x into (5.3) and lets y × z = −p 3 (5.4) which gives y3 = − q 2 + √ q2 4 + p3 27 = A (5.5) z3 = − q 2 − √ q2 4 + p3 27 = B (5.6) from which it follows that y = ( 3 √ A)σ′ 3 z = ( 3 √ B)σ′′ 3 372The issue of priority in the solution of the cubic is discussed in [158]. 148 since any root of xn = a is x = 1σ n × n √ a = ( n √ a)σ n . Next, he substitutes for y and z into (5.4), which gives ( 3 √ A 3 √ B)σ′+σ′′ 3 = 1 3 p2 (5.7) Now, the signs of A,B, p are unknown. If A,B, p are positive then the values of n √ A, n √ B, n √ p will be the positive numbers for which A,B, p are the nth powers. If A,B, p are negative, Mourey says make A = A+α, B = B+β and p = p+pi. The notation A+ signifies a directed line of the same length as A but always taken in a positive direction. For example, −2 = 2+2.373 Proceeding under the assumption that A,B, p might be negative, Mourey has n √ A = ( n √ A+)αn n √ B = ( n √ B+) β n n √ p = ( n √ p+)pin so that y = ( 3 √ A+)α+σ′ 3 z = ( 3 √ B+)β+σ′′ 3 Equation (5.7) then becomes ( 3 √ A+ 3 √ B+)α+β+σ′+σ′′ 3 = 1 3 p+pi+2 from which it follows that two conditions must hold: 1. Respecting lengths: 3 √ A+ 3 √ B+ = 13p+ 2. Respecting angles: α+σ ′ 3 + β+σ′′ 3 =´ pi + 2 To verify that condition (1) is satisfied, Mourey suggests we multiply together the expressions for A and B given in equations (5.5) and (5.6). Condition (2): Next, Mourey deduces the values of α, β, pi for which the second condition is satisfied. He takes σ′ as arbitrary and writes σ′′ in terms of σ′, so that we have x = y + z = ( 3 √ A+)α+σ′ 3 + ( 3 √ B+) 3pi−α+6−σ′ 3 373[See AB+, de´verseur.] 149 Substituting in σ′ = 0, 4, 8, 12, . . . gives x = ( 3 √ A+)α3 + ( 3 √ B+) 3pi−α+6 3 x = ( 3 √ A+)α+4 3 + ( 3 √ B+) 3pi−α+2 3 x = ( 3 √ A+)α+8 3 + ( 3 √ B+) 3pi−α−2 3 Mourey explains that there are only three distinct x values because σ ′ 3 has only three distinct values. Mourey considers a number of particular cases, in which various combinations of the signs of p and q appear. In each, he deduces the values of α, β and pi. The most interesting case is the irreducible case, where p is negative, q is positive or negative, and ∣ ∣ ∣ p3 27 ∣ ∣ ∣ > q 2 4 . 374 The irreducible case Recall y3 = − q 2 + √ q2 4 + p3 27 = A z3 = − q 2 − √ q2 4 + p3 27 = B To simplify, Mourey writes − q 2 = ±m q2 4 + p3 27 = −n = n2 where m and n are positive paths, so that we have A+α = ±m+ ( √ n)1 B+β = ±m− ( √ n)1 Next, he identifies ±m and ( √ n)1 in Figure 5.11 (page 145) ±m = AG(or AG′) ( √ n)1 = GL(or G ′L′) where A+α = AL(or AL ′) and ∠α = ABC(or ABCC ′) 374Typographical error: Mourey has p 3 27 < q2 4 which is incorrect. See [145, p72]. 150 from which it follows that −( √ n)1 = GL ′′′(or G′L′′) B+β = AL ′′′(AL′′) where β = ABCC ′C ′′C ′′′(ABCC ′C ′′) therefore B+ = A+ β = 4− α and since p is negative, pi = 2 and p = p+2 therefore x = ( 3 √ A+)α3 + ( 3 √ A+)−α3 x = ( 3 √ A+)α+4 3 + ( 3 √ A+)−α+43 x = ( 3 √ A+)α+8 3 + ( 3 √ A+)−α+83 Since these expressions are all of the form ar + a−r and since ar + a−r = 2a cos r, Mourey deduces x = 2 3 √ A+ cos α 3 x = 2 3 √ A+ cos α + 4 3 x = 2 3 √ A+ cos α + 8 3 These x values are all clearly real, which Mourey duly notes. Now, the values of A+ and α can be deduced from A+α = ±m+ ( √ n)1 using the formulae tanα = √ n ±m = √ −q2 4 − p3 27 − q2 151 A+ = ±m cosα = − q2 cosα or, alternatively, using375 A2+ = m 2 + n = + q2 4 + ( − q2 4 − p3 27 ) = −p3 27 from which it follows that A+ = √ −p3 27 = √ p3+ 27 Finally, to find α substitute for A+ in the following equation cosα = − q 2 A+ Mourey remarks that this method for solving the cubic can be applied to equa- tions with imaginary or complex coefficients and that it might be extended to the solution of quartic equations. Mourey’s proof of the fundamental theorem of algebra A modern statement of the theorem is, as we find in [159, p52]: Fundamental theorem of algebra If p(z) is a polynomial in z with coefficients in C then there is a number c ∈ C with p(c) = 0. Mourey’s statement of the theorem is: Every equation has at least one root.376 On his proof, Mourey writes in the preface: ‘The proof, which is elementary and which seems to me very rigorous, is so general that it even encompasses the case 375Typographical error: Mourey has A2+ = m 2 + n2 which is incorrect. The rest of the equation, however, is correct. See [145, p74]. 376‘toute e´quation a au moins une racine’. [145, pviii] 152 where the coefficients are what we call imaginary.’377 His proof is as follows.378 It begins with a re-statement of the problem in terms of geometry. The problem stated geometrically Given n fixed points situated in the plane A,B,C,D,E, . . . (Figure 5.12) find a point P also in the plane, such that the prod- uct of the paths AP,BP,CP,DP,EP, . . . is equal to a given path g. Figure 5.12: Mourey’s geometrical interpretation of the F.T.A.379 Mourey writes AP = x+u BP = x ′ +u′ CP = x ′′ +u′′ etc., and g = g+r where x+ denotes a positive path of the same length as x. With notation in place, he 377‘La de´monstration, qui est e´le´mentaire, et qui me paraˆıt tre`s-rigoureuse, est tellement ge´ne´rale, qu’elle embrasse meˆme le cas ou` les coefficients sont ce qu’on appelle imaginaries.’ [145, ppviii-ix] 378A detailed summary of Mourey’s proof of the F.T.A. also appears in [160, pp58–62]. 379 [145, p76] 153 is able to reduce the geometrical problem to the following system of two equations x+ × x ′ + × x ′′ + × · · · = g+ (i) u+ u′ + u′′ + · · · =´ r (ii) The first equation says simply that the product of the lengths of AP , BP , CP , etc., must equal the length of g. The second requires that the sum of the prime- directeurs of AP , BP , CP , etc., must equal (modulo 2pi) the prime-directeur of g. The prime-directeur of a directed line is the angle the line makes with the positive real axis. Writing each of the paths BP , CP , etc., as a vector sum involving AP = x, Mourey has BP = BA+ AP = b+ x CP = CA+ AP = c+ x DP = DA+ AP = d+ x ... etc. which enables him to combine equations (i) and (ii) to form an algebraic equation of degree n in one unknown, x x× (b+ x)× (c+ x)× (d+ x)× · · · = g or, multiplied out, xn + (b+ c+ d+ · · · )xn−1 + (bc+ cd+ bd+ · · · )xn−2 + · · ·+ (bcd · · · )x− g = 0 (A) The solution of the geometrical problem is equivalent to the solution of this algebraic equation of degree n. To prove the fundamental theorem Mourey has to prove: First — ‘every problem of this kind can be solved at least in one way ’380 380‘Tout proble`me de cette nature peut eˆtre re´solu, au moins d’une manie`re’. [145, p78] 154 Second — ‘every equation with only one unknown quantity is the translation of a problem of this kind ’381 It is a proof in two parts. Following Mourey, we will consider his proof of each statement in turn. Every problem of this kind can be solved at least in one way In order to prove this first part of the proof, Mourey needs to show that it is always possible to satisfy the conditions expressed in equations (i) and (ii). He addresses condition (i) first, which concerns the product of the lengths. Condition (i): x+ × x′+ × x ′′ + × · · · = g+ This condition is satisfied easily by fixing u (the direction of AP ) in the direction of an arbitrary path AN . In modern vector notation −−→ AN = k −→ AP . See Figure 5.13. Figure 5.13: AP taken along AN .382 As P may be situated anywhere on AN , it is afforded continuous movement along the line. When P is at A, the product of the lengths is zero. When P is infinitely far from A, the product of the lengths is infinitely large. In between, 381‘Toute e´quation a` une seule inconnue est la traduction d’un proble`me de cette nature’. [145, p84] 382Figures 5.13–5.15 do not appear in Mourey. 155 the value of the product increases continuously: as the product—being a product of continuous functions of P—is itself a continuous function of P . Therefore, (by implicit use of the intermediate value theorem) there is some intermediate stage at which the product of the lengths x+ × x′+ × x ′′ + × · · · is equal to g+. Note, Mourey proves neither the continuity of sums and products of continuous functions, nor that the paths AP , BP , etc., are continuous functions of P . Now, since u is arbitrary, AN may be taken as any one of the infinitely many radii emerging from A (Figure 5.14). Along each radius there is a position for P such that condition (i) is satisfied. Joining all such points P on the radii, Mourey forms a curve δ (represented by nszmn in Figures 5.12 and 5.14) which surrounds A. Thus, all points on δ satisfy condition (i). Figure 5.14: The curve δ. Condition (ii): u+ u′ + u′′ + · · · =´ r Mourey wants to show that there is at least one point on the curve δ which satisfies condition (ii) which relates to the sum of the prime-directeurs. Again, his argument involves the non-algebraic concept of continuity. He intends to move P in a continuous manner, in a positive sense, around the curve δ and show that during its progress, the sum of the prime-directeurs u+ u′ + u′′ + · · · increases continuously from a value less than r to a value greater than r. From this, he will conclude—again by implicit use of the intermediate value theorem—that at some intermediate stage the sum u+ u′ + u′′ + · · · equals r. Mourey considers the following three cases, which relate to the position of A (the 156 origin of the path AP ) in relation to the curve δ. In each case, Mourey is looking for the change in the value of the prime-directeur of AP after a full revolution of P around δ. This is the contribution of the path AP to the sum of the prime-directeurs. Case 1: A lies inside δ Suppose that the position of A is fixed inside δ. The positions of P as it moves around δ are recorded in Table 5.1. When P returns to m (posn. 5), having trav- elled via n, s and z, the prime-directeur of AP will have described 4 positive right angles. Position no. Position of P 1 m 2 n 3 s 4 z 5 m Table 5.1: Positions of P on δ for A inside δ.383 Case 2: A lies outside δ Suppose that the position of A is fixed outside δ. (If referring to Figure 5.12 on page 153, take E as the point A.) The positions of P on δ are recorded in Table 5.2 (page 158) along with some indication of the value of the prime-directeur of AP (or EP ) at that position. It is evident that after one full revolution, the prime-directeur has gained a balance of positive and negative contributions, which amounts to zero 383Tables 5.1–5.4 do not appear in Mourey. 157 overall. Position no. Position of P Prime-directeur 1 m negative 2 n zero 3 s positive 4 z zero 5 m negative Table 5.2: Positions of P on δ for A outside δ. Case 3: A is on δ Suppose that the position of A is fixed at m on δ and the distance between l and i is infinitely small. The positions of P as it moves around δ are recorded in Table 5.3 (page 159). As P moves from i to l (posn. 2 → 6), AP describes 2 positive right angles, so that the prime-directeur of AP increases from v say, to v + 2. But as P returns to i via m (posn. 6 → 7), a sudden jump occurs in the value of the prime-directeur, from v + 2 to v (or v + 4, as P has made a full revolution around δ). See Figure 5.15 (page 159). A gradual transition cannot be made from l to i without causing a jump in the value of the prime-directeur of AP , which in turn would affect a jump in the sum of the prime-directeurs. Fortunately, this situation cannot arise, as Mourey explains. Suppose that one of the given points, C for instance, lies on δ. When P , in the course of its journey around δ, sits at C, the length of CP will be zero. Thus, condition (i) is satisfied for the trivial case, g = 0. Now, since CP has no length its direction u′′ is unde- termined, therefore, the sum of the prime-directeurs is also undetermined. Hence, case 3 can be discarded. 158 Position no. Position of P 1 m 2 i ... ... 6 l 7 i Table 5.3: Positions of P on δ for A on δ. Figure 5.15: Discontinuity at m. It still remains to prove that the sum of the prime-directeurs passes from a neg- ative value to an infinitely large positive value. Mourey considers the case where A is alone inside δ. Table 5.4 (page 160) summarizes the contributions of AP to the sum of the prime-directeurs for each of the three cases discussed above, for A alone in δ. 159 Case Contribution to u+ u′ + u′′ + · · · After 1 revolution of δ After n revolutions of δ 1 4 right angles 4n right angles 2 0 0 3 ——–discarded——— Table 5.4: Contributions of AP to the sum of the prime-directeurs, for A alone in δ. Mourey’s approach to ensuring that the sum of prime-directeurs u+u′+u′′+ · · · is negative, and therefore less than r, is to measure the prime-directeurs from left to right (contrary to convention) so that each prime-directeur is negative and u+ u′ + u′′ + · · · = −s After n revolutions u+ u′ + u′′ + · · · = −s+ 4n→∞ (n→∞) (My notation.) Thus, the sum of the prime-directeurs will pass from −s < r to infinity with n and, therefore, will be equal to r at some intermediate stage (again by implicit use of the intermediate value theorem). It follows that there is at least one point Q on δ such that if P is at Q condition (ii) is satisfied; and since Q lies on δ condition (i) is also satisfied. Hence, the geometrical problem has at least one solution and the proposition is proved. Moreover, Mourey argues that since the procedure of constructing a curve δ can be repeated around each of the n fixed points A, B, C, etc., there are n solutions in total and equation (A) has n roots. 160 every equation with only one unknown quantity is the translation of a problem of this kind To prove this second statement, Mourey sets out to prove the equivalent statement: any equation of degree n in 1 unknown, x xn + pxn−1 + qxn−2 + · · ·+ sx+ t = 0 (B) can be transformed into equation (A). Recall equation (A) xn + (b+ c+ d+ · · · )xn−1 + (bc+ cd+ bd+ · · · )xn−2 + · · ·+ (bcd · · · )x− g = 0 (A) Now, to transform (B) into (A) we need to find b, c, d, . . . etc., such that they satisfy the following system of n− 1 equations b+ c+ d+ · · · = p bc+ cd+ bd+ · · · = q ... ... bcd · · · = s and set t = −g. Mourey forms the following equation of degree n− 1 in one unknown, z (z − b)× (z − c)× (z − d)× · · · = zn−1 − pzn−2 + qzn−3 − · · · ± s = 0 (C) He reasons that if this equation has n − 1 roots these roots will be b, c, d, . . . etc., the n− 1 solutions of the system of n− 1 equations; and having solved the system of equations, equation (B) can be transformed into equation (A) which completes the proof. Mourey believes that he has proved something further: that any equation of degree n has n roots. He has shown that if an equation (C) of degree n − 1 has 161 n − 1 roots, then an equation (A) of degree n has n roots; and by recalling that any equation of degree 1 has 1 root, he proves—by implicit use of mathematical induction—that any equation of degree n has n roots for all values of n. Of course, if he proves this, then he also proves the lesser statement that any equation of de- gree n has at least one root. He explains that he did not set out to prove that any equation has as many roots as dimensions but that it came out naturally in his proof of the existence of at least one root. Remark about Mourey’s proof: lack of rigour in analysis It is disappointing, perhaps, that Mourey is not as rigorous in analysis as he is in algebra; that his proof on the existence of roots relies on insufficiently precise notions of continuity. Perhaps our disappointment is borne out of our own unrealistic expectations. The problem common to early proofs of the fundamental theorem is explained by Stillwell in his Elements of Algebra: ‘Early attempts to prove the theorem were incomplete, mainly because they failed to reckon with the existential part of the proof—usually an application of the extreme value theorem or the intermediate value theorem.’384 He cites among such early proofs, those by Laplace (1795) and Gauss (1816), which required the intermediate value theorem, and those of d’Alembert (1746) and Argand (1806), which required the extreme value theorem. It was close to half a century after Mourey (in 1874) when Karl Weierstrass (1815–1897) was able to provide rigorous proofs of the intermediate and extreme value theorems—once a definition of real numbers was secured—thus enabling complete and rigorous proofs of the fundamental theorem.385 The definition of real numbers came by virtue of Richard Dedekind (1831–1916), in terms of Dedekind cuts, and Georg Cantor (1845– 1918), in terms of Cauchy sequences. 384 [159, p55] 385[Ibid.] 162 Application of directed algebra to plane curves In this section Mourey takes a kinematic approach to constructing curves. Imagine a curve drawn in the plane. From an arbitrary fixed point A in the plane (Figure 5.16) draw the path AQ, such that the point Q sits somewhere on the curve. Q is allowed to move but it must remain on the curve. As Q moves, the distance between A and Q will vary according to the parameter r which is the prime directeur of AQ. The curve (or, equivalently, the locus of Q) can be expressed in polar co-ordinates or the rectilinear equivalent. The simplest curve, the circle, is expressed in polar co-ordinates as AQ = ar. Now, AQ may be split up into a succession of paths (Figure 5.16). As the sum of two paths AQ = AP + PQ or q+r = x+s + y+t To determine the nature of the given curve, Mourey says that we must establish the relationship between x and s, t and s, and y and t. Figure 5.16: Application to plane curves, polar co-ordinates.386 386 [145, p86] 163 In terms of rectilinear co-ordinates (Figure 5.17) Mourey has AQ = AP + PQ or q = x+ y The procedure here is to fix the line AP = x and find an expression for y in terms of x. Mourey considers a linear relationship between x and y and one expressed by the complete quadratic equation y2 + Ay +Bxy + Cx2 +Dx+ E = 0 which he simplifies, with a change of co-ordinates, to give y2 = a2x2 + b2x+ c2 He considers three cases: for a = 0, b = 0, c = 0. Now, when the y’s are mutually parallel the equation of second degree expresses a conic section; so for each of the three cases Mourey determines the conditions for parallel y’s and identifies the conic section described by the locus of Q. Figure 5.17: Application to plane curves, rectilinear co-ordinates.387 387 [145, p89] 164 Of course, there is nothing new in a kinematic approach to generating curves. We need only think of the geometrical work of Descartes (1596–1650), l’Hoˆpital (1661–1704) and de Witt (1625–1672) two centuries prior. Supplementary section In his final section Mourey works through some additional examples to show how his theory might be applied to a variety of problems involving quantities of a variety of species. In the main body of the work he has already shown: how to track the gains and losses of a gentleman’s fortune, by means of a path with a moving terminal point, and how to calculate durations of time, by representing indivisible instants of time as points on the real line. The problems covered in this supplementary section are of a similar nature. 5.3.4 Characteristic features of Mourey’s mathematics If called to identify two characteristic features of Mourey’s mathematics, I might advance: (i) the interaction of algebra and geometry, and (ii) the modernity of his approach and his mathematical concepts. Evidence of the interaction between algebra and geometry is everywhere in Mourey’s work. We recall: he devises an algebraic expression for the geometri- cal operation of rotation, and algebraic definitions of addition and multiplication; he proves the distributive laws using vector methods; he translates the geometrical problem of addition into the solution of an algebraic equation, which is solved by trigonometric formulae deduced from a geometric figure; from algebraic equations he deduces the relative positions of two paths, etc. We encounter further examples still, in the applications of his theory. Mourey’s approach is a modern-algebra approach: he defines his set of directed lines and then his operations; he identifies the properties of these operations and then proves that they hold. Some of the concepts introduced or developed by him are sophisticated, innovative and far-reaching: the additive inverse, the equivalence relation, the geometric operator, etc. Sometimes he comes across as pedantic, often his mathematics is too involved 165 and not everything he does is rigorous enough by modern standards. However, Mourey believed earnestly that he was working things out properly for the first time. If we are mindful of this, then his shortcomings are easier to understand. 5.4 Notable references to Mourey This section surveys the most notable references to Mourey. From the nineteenth century, I have collated all the references to Mourey that I have been able to find. From the twentieth and twenty-first centuries, I have only included those references which venture beyond a brief reference to Mourey in terms of the Argand diagram. Of particular significance are the references made by Warren in England, Lefe´bure de Fourcy and Liouville in Paris, and Hamilton and De Morgan in Ireland and England. The information is also presented graphically in a timeline in Figure 5.18 (page 179). The timeline runs up until the end of the nineteenth century. It gives some indication of geography and of the noticeable gaps between references so that we might better understand the dissemination of knowledge of Mourey’s contribution. The Revd. John Warren, a Fellow of Jesus College, Cambridge, appears to have been the first to notice Mourey’s contribution. The first individuals in Paris to refer to Mourey were associated with the E´cole Polytechnique. They were: Louis-Etienne Lefe´bure de Fourcy (1787–1869), Examiner for Admissions, and Joseph Liouville (1809–1882), Professor of Analysis and Mechanics. 5.4.1 Louis-Etienne Lefe´bure de Fourcy (1787–1869): a biographical sketch Lefe´bure studied at the E´cole Polytechnique between 1803 and 1805. In 1807 he took up a position at the E´cole as assistant re´pe´titeur in analysis, progressing to re´pe´titeur in analysis in 1813. Later, he taught applied analysis and descriptive geometry there. Between 1826 and 1861 he functioned as Examiner for Admissions. In December 1828 he was one of three examiners who refused Evariste Galois (1811– 1832) entry into the E´cole. Lefe´bure was also employed as a mathematics teacher 166 at Louis-le-Grand, where he taught Victor Hugo, and as a professor of mathematics at the Colle`ge Royal de Saint-Louis, where he taught Joseph Liouville. In 1825 he became assistant to Lacroix at the Faculte´ des Sciences, succeeding him in the Chair of Differential and Integral Calculus in 1843.388 5.4.2 Joseph Liouville (1809–1882): a biographical sketch Joseph Liouville is remembered at the founder and editor of the Journal de mathe´- matiques pures et applique´es. He established the journal in 1836 and remained as editor until 1874. The journal—which now serves as an important record of math- ematical activity during the period—became known as the Journal de Liouville. Liouville is also remembered for his connection with Evariste Galois. Liouville encountered Galois’ mathematical writings in 1843. Galois had died eleven years prior, at the age of twenty-one; in a dual, at the hands of an ‘unknown adversary’.389 Liouville published some of Galois’ work in the Journal de mathe´matiques in 1846. This brought the significance of Galois’ work on the solvability of algebraic equa- tions to the public’s attention. Liouville thereby ‘participated, indirectly, in the elaboration of modern algebra and of group theory’.390 Several parallels can be drawn between Mourey and Galois, with respect to: (i) chronology, (ii) geographical proximity and (iii) Liouville’s publicity of their little- known work. In addition, it is not unreasonable to consider both as innovators in mathematics but to a much lesser extent in Mourey’s case, of course. Liouville was educated at the Colle`ge Royal de Saint-Louis, where he was taught by Lefe´bure; the E´cole Polytechnique (1825–1827) and the E´cole des Ponts et Chausse´es (1827–1830). In 1831 he took up a position as re´pe´titeur in analysis and mechanics at the E´cole Polytechnique. In 1838 he succeeded Mathieu as Pro- fessor in Analysis and Mechanics. He remained in this post until 1851. During this 388Sources of biographical information on Lefe´bure de Fourcy: [161] and [162]. 389 [163] 390 [164] 167 period he also taught at a number of other institutions in the city: at the E´cole Centrale des Arts et Manufactures, where he taught mechanics and mathematics (1833–1838), and at the Colle`ge de France, where he substituted for Biot (between 1837 and 1843). In 1851 he took up Libri’s chair at the Colle`ge de France and in 1857 he succeeded Sturm as Professor of Mechanics at the Faculte´ des Sciences. He was elected to the Astronomy Section of the Acade´mie des Sciences (1839) and to the Bureau des Longitudes (1840). His mathematical interests were truly diverse. Encompassing both pure and applied mathematics, they included: mathematical physics, mechanics, mathematical analysis, algebra, number theory and geometry.391 5.4.3 Dissemination of knowledge of Mourey’s contribution This is a survey of the reception of Mourey’s work, which includes details of the author, publication and reference to Mourey. 1828 C.-V. Mourey [145] The first edition of Mourey’s work is published in Paris by Bachelier. On the title page Mourey’s address is given as ‘Paris [. . .] rue des Quatre-Vents, no. 8’. 1829 John Warren [166] [142, pp251–254] [143, pp339–340] In December 1828 Warren communicates Mourey’s proof of the F.T.A., with a cor- rection, to the Cambridge Philosophical Society. His correction takes into account the possibility that two or more points might be enclosed by a single curve δ and, instead of proving the existence of n roots by geometry, he prefers to use the fact that the existence of n roots follows by polynomial induction once the existence of one root is proved.392 In 1829 Warren publishes two follow-up papers to his 1828 treatise. In the first 391Uncited sources of biographical information on Liouville: [165]. 392If a polynomial p(z) of degree n has a root c, then we can factorize p(z) into (z − c)q(z), where q(z) is a polynomial of degree n− 1, and then repeat the process with q(z), and so on. 168 he brings to light the contributions of Bue´e and Mourey: he remarks on the simi- larity between his and Mourey’s approaches to the subject and describes Mourey’s proof of the F.T.A. as ‘remarkably clear and satisfactory, and an example of the advantages which mathematicians may derive from a knowledge of the true theory of the quantities improperly called impossible or imaginary’.393 Warren’s second paper is inspired by Mourey. In his 1828 work [145, pp94–95] Mourey had alluded to the fact that—in a larger unpublished manuscript—he had represented certain forms geometrically as directed lines in the plane, namely a √ −1, a√−1, sin √ −1, etc. Warren realized that if this was the case, then Mourey had discovered a geometrical representation of all algebraic quantities. As circumstances had prevented Mourey from publishing his manuscript in full, Warren was provided with an opportunity to further develop his own researches. 1834 Moritz Wilhelm Drobisch [167, pxvi] Drobisch (1802–1896)—Professor of Mathematics at the University of Leipzig— publishes a textbook on the analytic and geometric properties of higher numerical equations. In the preface he cites Mourey (he writes ‘Mouray’) within a list of math- ematicians who gave meaning to the expression a+ b √ −1.394 1835 Louis-Etienne Lefe´bure de Fourcy [169, pii] Lefe´bure recalls Mourey’s work in the preface to the 2nd edition of Lec¸ons d’alge`bre, in the context of the difficulties associated with the calculus of imaginaries. He reports Mourey’s aim as being to free analysis from imaginary quantities entirely and remarks on Mourey’s success in introducing into the calculus the two species of quantities referred to in analytical geometry as “polar co-ordinates”, as a means of representing imaginaries. 393 [142, p254] 394Knowledge of Drobisch was acquired through [168]. I am grateful to Stefanie Eminger for translating Drobisch’s preface into English. 169 1839 Joseph Liouville [170] Liouville publishes a paper on Mourey in his Journal de mathe´matiques. He writes that mathematicians will have been reminded of Mourey’s work by Lefe´bure’s in- clusion of a proof (by Liouville and Sturm) of Cauchy’s theorem in the new edition of Lec¸ons de ge´ome´trie analytique (1840). He explains that another proof of the theorem by Sturm [171] is based on the same lemma used by Mourey in his proof of the F.T.A. The lemma relates to the gains and losses, after a complete revolution, of the prime-directeurs of radii drawn from any point within, without or on, a closed circuit drawn in the plane. He goes on to provide a brief exposition of Mourey’s proof of the F.T.A. which, for the benefit of the reader, he writes in language more familiar to algebraists: he believes that Mourey’s original terms and notation are unworthy of replacing the established terminology and notation. His only other criticism of Mourey is in regard to the incompleteness of his proof: Mourey ought to have proved that the points P form a true curve, which would require him to prove that the radius vector AP varies continuously with its prime-directeur (i.e. the angle between the radius vector and the positive real axis). In addition, Liouville provides an alternative proof of the F.T.A. It is, however, nothing other than Gauss’ third proof which was published in 1816.395 Liouville does make reference to Gauss but by no means does he make priority clear.396 1840 Joseph Liouville [173] Liouville publishes an addition to his 1839 paper, in which he attempts to deal with the technical issues which relate to the curve δ in Mourey’s proof.397 1840 Louis-Etienne Lefe´bure de Fourcy [174, ppv–vi] Lefe´bure refers to Mourey in the preface to the 4th edition of Lec¸ons de ge´ome´trie analytique. He is writing to inform the reader about an addition made to the sec- 395For verification see [172, p99–102]. 396A summary of Liouville’s 1839 paper is given in [160, pp63–65]. 397A summary of Liouville’s 1840 paper is given in [160, pp65–66]. 170 tion on analytical geometry in two dimensions: it is to include, for the first time, a proof of Cauchy’s theorem on the number of complex roots of an algebraic equation situated inside a given contour. The proof is by Sturm and Liouville and it was published in the Journal de mathe´matiques in August 1836. On Mourey, Lefe´bure writes: ‘Without the ingenious idea of providing a geometrical representation to the imaginary quantities, M. Cauchy’s theorem could not exist; and with regard to this, it is my duty to recall here a most curious little work, published by M. Mourey, in 1828’.398 The description of the work which follows is essentially the same as that given in Lec¸ons d’alge`bre (1835), except that here Lefe´bure notes that Mourey proves that an equation of degree m has m roots. 1845 Louis-Etienne Lefe´bure de Fourcy [175, pp216–219] [176, pp214–219] Lefe´bure considers Mourey’s mathematics in-depth for the first time; providing an exposition of the basics of his methods in the 5th edition of Lec¸ons d’alge`bre. The section on Mourey follows on from one detailing how to manipulate expressions of the form a + b √ −1; taking powers and nth roots, etc. Lefe´bure intends to provide the reader with a brief summary of the way Mourey proposed to free analysis from imaginary quantities; sufficient for the reader to grasp the meaning of √ −A2 in the new algebra. In Mourey’s system, he remarks, the expression √ −A2 represents perpendicularity rather than impossibility. It is significant—with algebraic reform in mind—that Lefe´bure credits Mourey with having reconstructed algebra in its en- tirety; by establishing the rules of algebraic calculus, then moving on to equations. Lefe´bure extends his exposition of Mourey’s methods in the 6th edition of Lec¸ons d’alge`bre (1850). Emphasis in this edition is on the agreement between the results of the new algebra and those of ordinary algebra—made possible by Mourey’s notion of 398‘Sans l’ide´e inge´nieuse de donner une repre´sentation ge´ome´trique aux quantite´s imaginaires, le the´ore`me de M. Cauchy ne pourrait point exister; et la justice exige que je rappelle ici a` ce sujet un petit ouvrage fort curieux, publie´ par M. Mourey, en 1828’. [174, ppv–vi] 171 verseurs. Lefe´bure hopes that his exposition of Mourey’s new doctrine is sufficient to show that imaginaries would have presented few difficulties to mathematicians had they had, from the outset, as clear and precise a meaning as Mourey has given them. It is interesting to query why a period of seventeen years elapsed before Lefe´bure wrote at length about Mourey. Perhaps the 1845 publication was the first opportu- nity that arose to include a longer discussion on Mourey. Perhaps he chose to wait until elements of the new doctrine became popular. I suggest the possibility that there may be a link with the position he assumed in 1843 as successor to Sylvestre Franc¸ois Lacroix (1765–1843) at the Faculte´ des Sciences. When Lefe´bure became an Examiner for Admissions at the E´cole Polytechnique in 1826, he was not able to continue teaching there because the roles of examiner and teacher were considered incompatible.399 As a result of this, the new position brought with it the luxury of much free time, which Lefe´bure spent in writing text- books for students of mathematics. His books proved remarkably popular: in effect, they provided the only means by which entrant students to the E´cole could familiar- ize themselves with their examiner’s mathematical interests and so they constituted invaluable revision material. They were, however, severely criticized for their sim- plicity and for the author’s lack of originality and innovation.400 In 1843 Lefe´bure succeeded Lacroix in the Chair of Differential and Integral Calculus at the Faculte´ des Sciences. I imagine that the role might have demanded improvements in the standard of Lefe´bure’s publications: including an exposition of Mourey’s methods would have been an effective way to add substance. 1845 Ambroise Faure [177, preface] Faure (1795–1871)—Professor of Mathematics and Physics at the Colle`ge de Gap and at the E´cole Normale de Gap—publishes a book on the theory and interpretation 399 [161] 400‘Mais, e´crira Larousse, ils sont “totalement de´pourvus d’ide´es originales et de gouˆt pour les innovations, meˆme les plus le´gitimes”.’ [162, p2] 172 of imaginary quantities. In the preface he cites Mourey, along with Bue´e, Argand and Franc¸ais, as one who attempted to find meaning in the abstract symbols of imaginaries. He states, for the sake of priority, that in 1828—before Mourey’s book was published—his manuscript was seen by Joseph Fourier (1768–1830), Secretary of the Acade´mie des Sciences.401 1846 Charles W. Hackley [178, pp244–246] Hackley (1809–1861)—Professor of Mathematics and Astronomy at Columbia Col- lege, New York—publishes the textbook, A Treatise on Algebra. It is based on the finest English, French and German sources and contains ‘all that is important in the higher parts of Algebra’.402 It includes a translation of the section on Mourey from Lefe´bure’s Lec¸ons d’alge`bre (1845). 1846 Nouvelles annales de mathe´matiques [147] An appeal is placed for biographical information on Mourey. 1852–1862 William Rowan Hamilton and Augustus De Morgan [179] Hamilton and De Morgan discuss Mourey at intervals in correspondence during this period. Hamilton’s copy of Mourey is borrowed from De Morgan. 1858 Augustus De Morgan [180] De Morgan publishes a paper on the existence of roots in which he provides a summary of Mourey’s proof, with improved notation, and an algebraic substitute for a geometric step in the proof. 401Knowledge of Faure’s reference to Mourey was acquired through [160, p67f(no.17)]. The same source [160, pp63–66] also provided additional insights into Le´febure (1835, 1840) and Liouville (1839, 1840). 402 [178, piii] 173 1861 C.-V. Mourey [145] Mourey (1828) is re-printed in Paris. The author’s address is not given in this sec- ond edition. There are no other revisions. 1868 Abel Transon [181, pp193f, 199, 202] Transon (1805–1876)—Examiner for Admissions at the E´cole Polytechnique—publi- shes a paper on the application of directed algebra to geometry in the Nouvelles an- nales de mathe´matiques. Therein, he makes a number of brief references to Mourey. He recognizes in Faure’s proof that every equation has at least one root—given in his 1845 work—the same issues that Liouville had called attention to regarding Mourey’s proof. He addresses his argument for the existence of imaginaries to ‘cet ami de l’E´vidence’, imitating Mourey.403 1870 Michel Chasles [182, p62] Chasles’ report on the progress of geometry is published. Therein, he refers to Mourey and Warren as re-inventors of the doctrine of Argand. He writes that Mourey has generalized the ideas of Argand and Bue´e.404 1886 P. G. Tait [146, p447] Tait’s article, ‘Quaternions’ is published in the 9th edition of the Encyclopaedia Bri- tannica. Therein, he refers to Mourey and Warren as independent re-inventors of the new doctrine of imaginaries. 403 [181, p199] 404Michel Chasles (1793–1880): student at the E´cole Polytechnique (1812–1815); Professor of Geodesy, Mechanics and Astronomy at the E´cole Polytechnique (1841–1851) and Chair of Higher Geometry at the Sorbonne (1846–1880). A member of the Acade´mie des Sciences (elected 1851); a Fellow of the Royal Society of London (elected in 1854, received the Copley Medal in 1865) and the first foreign member of the London Mathematical Society (elected 1867). His 1837 work, Aperc¸u historique (‘Historical View of the Origin and Development of Methods in Geometry’) established him both as a mathematician and an historian of mathematics. [183] 174 1887 Gino Loria [184] Loria publishes a paper on the F.T.A. in Acta Mathematica, in which he remarks on the similarity between Mourey’s proof of the F.T.A. and one published recently by a Norwegian mathematician named Holst.405 He argues that the proofs are very similar in substance.406 1904 Elie Cartan [186, pp337–338, 341f(no.58)] In Cartan’s article on complex numbers—published in the Encyclope´die des sciences mathe´matiques pures et applique´es—he refers to Mourey as one among a number in the early nineteenth century whose work justified and legitimized the calculus of imaginaries. He makes an important distinction between the geometrical repre- sentation of complex numbers—where we associate a + b √ −1 with a point (a, b) on the plane—and the geometrical theory of complex numbers—in which complex numbers are defined by vectors which are subject to defined operations. He sees Mourey’s work as a rigorous exposition of the vector theory and notes that Mourey is aware of the need to define operations. Making the comparison with Argand and Franc¸ais, Cartan notes that in Argand’s work the distinction between representation and theory is not clear and that Franc¸ais makes no reference to the properties of the operations such as commutativity, etc.407 1924 Julian Coolidge [188, pp26–27] In his book, The Geometry of the Complex Domain, Coolidge includes Mourey in his account of the history of the representation of the binary domain. He remarks that Mourey ‘writes with a notable exuberance [. . .] but [. . .] is by no means lack- ing in penetration and mathematical insight’.408 He notes that Mourey has: (i) 405For Holst’s paper see Acta, volume VIII. 406Gino Loria (1862–1954): Italian mathematician and historian of mathematics; Professor of Higher Geometry at Genoa University (1886–1935). [185] 407Knowledge of Cartan’s reference to Mourey came from [187, p724]. 408 [188, p26] 175 an awareness of the conditions for the equality of directed and non-directed lines; (ii) the notion of the directed angle and associated notation; (iii) the notion of a geometrical operator in the rule for multiplication; (iv) the nth roots of unity de- duced from the rule for taking powers and (v) a proof of the F.T.A. He also remarks that nowhere does Mourey stress that any directed line may be represented by a linear combination of 1 and 11. I believe this is implicit in Mourey’s remark that all directed numbers can be formed from the unit line by multiplication, division and version: recall that Mourey has for the unit line 1r = cos r + √ −1 sin r. 1929 G. Windred [189, pp538–539] Windred publishes an article, ‘History of the Theory of Imaginary and Complex Quantities’ in The Mathematical Gazette. Therein, he describes Mourey’s work as one of two ‘notable contributions’ to appear in 1828.409 The other is Warren’s. Like Coolidge, Windred notes that Mourey has the notion of a geometrical operator. In addition, he suggests that Mourey was perhaps the first to see the need for stating the conditions of equality for vectors and remarks that while Mourey’s notation is no longer used, some of the problems he treated remain of ‘distinct mathematical value’.410 1997 Gert Schubring [168, pp12–13] Schubring contributes a chapter to Le Nombre une hydre a` n visages. Therein, he refers to Mourey from the perspective of the history of the vector concept. Much of what he writes about Mourey I do not agree with, in particular: that Mourey persisted with the definition of subtraction in arithmetic as he moved into algebra and refused to accept a broader meaning of subtraction in algebra; and that, like Bue´e, he had success in geometrically (but not algebraically) bringing together the notions of length and direction. Still, two remarks of his are of interest: first, that he was unable to obtain any biographical information about Mourey and second, 409 [189, p538] 410 [189, p539] 176 his suggestion that Hermann Grassmann (1809–1877)—who worked on developing a general calculus of vectors from 1832—may have been influenced by Mourey’s 1828 work, as it was cited in the work of Moritz Wilhelm Drobisch (1802–1896) who was a popular German mathematician of the period. 1997 Christian Gilain [160, pp58–66, 67f(no.17)] The work done by Gilain in his chapter in Le Nombre une hydre a` n visages consti- tutes the most in-depth study of Mourey’s mathematics since the nineteenth century (apart from my own). He describes Mourey’s proof of the F.T.A. in detail, following Mourey’s text closely, and makes a brief comparison with Argand’s proof: both have the same level of generality but Mourey’s proof is longer and less straight-forward. He also makes some comments on the lack of rigour in Mourey’s proof. A survey of the reception of Mourey’s work in the nineteenth century is also provided. It covers: Warren, Lefe´bure, Liouville and Faure. A summary of Liouville’s papers on Mourey also appears. Gilain remarks that Liouville made fewer changes to Mourey’s proof than we would expect; notably, he retains the geometrical aspects of Mourey’s proof. Gilain suggests that Sturm’s proof of Cauchy’s theorem may have been inspired by Mourey’s work: Sturm may have acquired knowledge of Mourey through Lefe´bure’s reference to Mourey in the preface to Lec¸ons d’alge`bre (1835). 2005 Gert Schubring [190, pp569–570] In Conflicts Between Generalization, Rigor, and Intuition, Schubring gives his in- terpretation of Mourey: Mourey’s publication confirms that the epistemological orientation was to reject any generalizability of algebra, and in claiming geometry to be the authoritative instance of mathematics that guaranteed sense and meaning. Mourey admitted subtraction only in arithmetic, provided that the subtrahend was smaller than the minuend. [. . .] Mourey was so radical in this that he excluded subtraction from the operations of algebra altogether.411 411 [190, p569] 177 My interpretation of Mourey is quite different: I see him as a progressive, rather than as an opponent of change. 178 18 30 18 35 18 40 18 45 18 50 18 55 18 60 18 65 18 70 18 75 18 80 18 85 18 90 Mo ure y’s 1st edn (Pa ris) Wa rren (En glan d) Dro bisc h(G erm any ) Lef´ ebu re( Par is) Lef´ ebu reo nC auc hy’ sth m. (Pa ris) Lio uvi lle (Pa ris) Lio uvi lle, Lef´ ebu re( Par is) Lef´ ebu re, Fau re( Par is) Hac kley (US A), App eal for bio g.i nfo .(P aris ) H am ilto n& De Mo rga n(I rela nd, Eng lan d) De Mo rga n(E ngl and ) Mo ure y’s 2nd edn (Pa ris) Tra nso n(P aris ) Cha sles (Pa ris) Tai t(S cot lan d) Lor ia( Ital y) F ig ur e 5. 18 : T im el in e to sh ow di ss em in at io n of kn ow le dg e of M ou re y’ s co nt ri bu ti on , w it h so m e in di ca ti on of ge og ra ph y. 179 5.4.4 Hamilton and De Morgan’s correspondence on Mourey, 1852–1862 At intervals during the ten-year period between 1852 and 1862, Sir William Rowan Hamilton and Augustus De Morgan (1806–1871) corresponded on C.-V. Mourey’s 1828 work. The correspondence—which was published, in part, in Graves’ Life of Hamilton (1889)—provides valuable information on: (i) the availability of Mourey’s book and the distribution of copies; (ii) Hamilton’s initial interest in Mourey and (iii) Hamilton and De Morgan’s impression of Mourey’s mathematics and their as- sessment of the evidence which might have suggested that Mourey had anticipated Hamilton in the discovery of quaternions. The correspondence between the two men was initiated by De Morgan in 1841. In his first letter to Hamilton [179, p245] De Morgan reminded Hamilton of their first encounter, in London twelve years prior, and expressed an interest in Hamilton’s theory of triplets which had featured in Hamilton’s ‘Essay’ [191] in the Transactions of the Royal Irish Academy. At the time of the correspondence, Hamilton was Andrews Professor of Astron- omy at Trinity College, Dublin and De Morgan was Professor of Mathematics at University College, London. The correspondence between Hamilton and De Morgan relating to Mourey: abridged and with editorial additions to establish context Hamilton to De Morgan — 8 January 1852. Hamilton is still working towards publication of his Lectures on Quaternions. In this correspondence he confesses to De Morgan his fear of having been ‘too diffuse’ and lays out before him plans to remedy the fault: he proposes to produce ‘a copi- ous table of contents’ and ‘a concise and readable preface’ which might include some historical references.412 In closing, he asks of De Morgan: ‘Can you assist me to pro- 412H to DeM (8 Jan. 1852) in [179, p314]. 180 cure Mourey, Paris, 1828?’413 Clearly, Hamilton intends to include Mourey in his sketch of the historical developments which preceded his discovery of quaternions.414 De Morgan to Hamilton — 10 January 1852. De Morgan writes to Hamilton to inform him that he is able to furnish him with a copy of Mourey. He writes: I can lend you Mourey; as you ought to see, with this letter. You will take care of it I know; and you may return it at leisure. Mourey would have been very remarkable if Warren had not appeared in the same year. [. . .] By and bye, when the French—tardily—begin to cultivate algebra as a science, they will declare that Mourey did it all. So I would not on any account lose Mourey.415 Hamilton to De Morgan — 13 January 1852. In reply to De Morgan’s satirical remarks about French priority, Hamilton refers to Servois’ correspondence in Gergonne’s Annales in which Servois had suggested the trinomial form (p cosα + q cos β + r cos γ) × (p′ cosα + q′ cos β + r′ cos γ) = (cos2 α + cos2 β + cos2 γ) = 1 but was unable to advance any further, not knowing what form p, q, r, p′, q′, r′ would take.416 Hamilton writes: Thanks for your promise to lend me Mourey, of which I shall take every care when it arrives; and the post to this place appears to be safe, though slow. I heard of it a good while ago, and shall be very glad to see it, though I fancy the book to be little else than double algebra.417 If the French want an anticipation, though not a very 413H to DeM (8 Jan. 1852) in [179, p315]. 414See [141, pp(31–32f)] for Hamilton’s reference to Mourey in Lectures on Quaternions. 415DeM to H (10 Jan. 1852) in [179, p316]. 416See [192, p235] for Servois’ suggestion of the trinomial form. 417In using the term ‘double algebra’ to describe Mourey’s mathematics, Hamilton is crediting Mourey with having constructed an algebraic system, which is something quite apart from ordinary arithmetic. 181 complementary one! of the quaternions [see the word “absurdes”], I can point out what might with some plausibility be claimed by them as such.418 Hamilton then reproduces Servois’ trinomial form and writes: ‘You see that I solve his problem by p = i, q = j, r = k, p′ = −i, q′ = −j, r′ = −k.’419 14 January 1852. Continuing with the same letter, Hamilton acknowledges receipt of Mourey. He has already begun a careful consideration of its contents. He writes: P.S.—The Mourey has arrived, and shall be taken every care of. In a sense, I have already read it through, but must re-consider the proof of the existence of a root. I see that you or I—but I hope it will be you—must write, some time or other, a history of √ −1.420 De Morgan to Hamilton — 15 January 1852. De Morgan makes some remarks on Servois’ trinomial form. He closes the correspon- dence—in reply to Hamilton’s comment on 13 January—with confirmation that ‘Mourey is nothing else but a part of double algebra’.421 Hamilton to De Morgan — 2 June 1852. Hamilton updates De Morgan on his latest mathematical investigations and offers to lend to him Cauchy’s 1825 memoir on definite integrals with imaginary limits, which relates to his (Hamilton’s) current investigations. He suggests that he might return De Morgan’s copy of Mourey at the same time as sending the Cauchy memoir; but, for the time-being, he wishes to hold on to them both.422 418H to DeM (13 Jan. 1852) in [179, p316]. 419H to DeM (13 Jan. 1852) in [179, p317]. 420H to DeM (14 Jan. 1852) in [179, p317]. 421DeM to H (15 Jan. 1852) in [179, p318]. 422H to DeM (2 June 1852) in [179, p371]. 182 Hamilton to De Morgan — 14 July 1854. In this correspondence, Hamilton relates to De Morgan the progress he has made to date with Mourey. He has produced copious notes which constitute an abstract of the book, so that he might return Mourey but continue to discuss the work in correspondence with De Morgan. Hamilton is interested in Mourey’s definitions and fundamental principles, not his applications of the theory. He is keen to discuss Mourey’s concept of version and to address the issue of priority over geometrical operators. He writes: Having so long procrastinated about returning your Mourey, I wished to retain some notes of the work, which might assist my own memory, and enable me perhaps to write a little to you about it, when it shall be in your hands again, as I really hope it soon will be, for I have lately made nearly as full an abstract of the book, in one of my own manuscript volumes, as I wish to have at hand. In fact between extracts (copied in the French, for practice), abridgements, and comments of my own, I have already filled fifteen large pages of such a volume, and got as far as his solution of a cubic, so that not much more remains. I should like to consider, if I can spare some quiet hours for it, his proof (or alleged proof) that every algebraic equation has a root; but I care more for his conceptions, definitions, and notations, which I think that I now perfectly understand, than for such applications of his theory. What he says about version, versors, &c., might no doubt have set me on the track of the quaternions, if I had seen his book as early as I did Warren’s; but I had not only formed my own general views, but had published the name and sign (i.e. my sign U ) of the versor of a quaternion, at least as early as July, 1846, in the Philosophical Magazine, if not elsewhere before that date; and at a time when I had (certainly) not seen, and (I think) not heard of, the work by Mourey. The conception of the quaternion, as a geometrical operator, which at once turns (or verts) and stretches (or tends) a line, was familiar to me at least as early as 1844 [. . .] Warren, I think, was shy of putting rotation prominently forward, if at all (but I must look into his book again); with Mourey it is quite a key, for the plane, as with me for space . . .423 423H to DeM (14 July 1854) in [179, pp488–489]. 183 Hamilton’s notebook MS.1492/95 The manuscript volume which Hamilton refers to in the above letter is extant in the Manuscripts Library of Trinity College, Dublin. Within MS.1492/95, a 13”×8” leather-bound ledger with 287 pages, there are thirty-four pages on Mourey, written predominantly in 1854.424 Hamilton worked his way through De Morgan’s copy of Mourey (1828) during the summer of 1854; copying extracts from the book in French and writing in English [in square brackets] when he came across some aspect of Mourey’s work which he felt was worthy of remark. To highlight important sections, Hamilton drew a single or double vertical line in the margin of the notebook. Interspersed within the mathematics is the correspondence between Hamilton and De Morgan on Mourey copied into the notebook, usually by Hamilton from memory. Hamilton included the correspondence in the notebook so as to keep all his material on the same subject together, as was his practice. Hamilton seems to have devoted more attention to the consideration of Mourey’s definitions and principles, than to Mourey’s applications of his theory. However, there are two exceptions: (i) inspired by Mourey’s application of his theory to the solution of the cubic, Hamilton devised an alternative method using quaternions; and (ii) he discussed at length the merits and faults of Mourey’s proof of the F.T.A. and was inspired to write his own paper on the subject, which is transcribed in Appendix F.425 Hamilton never published the paper. The following letter, from Hamilton to De Morgan (17 July 1854), is a good example of the correspondence which passed between them concerning Mourey’s proof.426 424See MS.1492/95, pp. 241–274. 425For Hamilton’s solution of the cubic using quaternions, see MS.1492/95, pp. 273–274. 426H to DeM (17 July 1854), MS.1492/95, pp. 259–261. This letter was not published in Graves’ Life of Hamilton. I am grateful to the Keeper of the Manuscripts at Trinity College, Dublin for giving their permission to publish it here. 184 Obs July 17th 1854 My Dear De Morgan I have pretty well satisfied my curiosity about Mourey’s work, & a very few addi- tional memoranda will suffice for any purposes of mine. It (the book) has only been taken up by me at some rare moments, but it appears to me to deserve to be better known. I think that Mourey really does prove Ist, that every eqn of the form (A), page 104, xn + b xn−1 + bc xn−2 + ...+ (bcd...)x− g = 0; +c +cd +d +bd +... +... (A) has at least one root, x, namely a certain line AP , if b, c, d, . . . & g denote given lines: though I don’t admit the validity of his proposed proof, pages 113, 114, that this eqn (A) has as many as n roots (comp. p.266.) However it was quite unnecessary that he shd attempt to do so, at that stage. He ought to have been content with proving, as I think he has fairly done, that “toute proble`me de cette nature”, namely wh. conducts to an eqn of the form (A), “peut eˆtre re´solu, au moins d’une manie`re” (p.104); or in other words that an eqn of this form (A) has always at least one root x. Let that one root be x1; we may, as in all elementary books on Algebra, divide by x−x1, & so depress the degree by a unit in the exponent. Suppose it be otherwise & previously known, IInd, that an eqn of this lower degree n− 1, say his eqn of p.115, zn−1 − pzn−2 + qzn−3 − · · · ± s = 0, (C) has always n− 1 roots, b, c, d, . . . so that up to some given value of n− 1, we are sure of the existence of b, c, d, . . . in the n− 1 equations b+ c+ d+ · · · = p bc+ bd+ cd+ · · · = q, bcd · · · = s; 185 then the eqn of the next higher degree, p.114, xn + pxn−1 + qxn−2 + · · ·+ sx+ t = 0 (B) whatever may be the values of its coeffs p, q, . . . , s, t, takes the form (A), where g = −t, & ∴ by his geoml proof (which I have just now turned roughly into a more algebraic form), it has at least one root x1, as above. Applying then again the IInd premise, or admitting that there are always n− 1 roots of an eqn (C) of the (n− 1)th degree, we see that there are n roots in all, of the eqn (A) of the nth degree; & ∴ of every eqn of that degree. Thus (tho’ I know that you don’t want these illustrations), the eqn (C) is linear & has one root, b = p, when n = 4; ∴ the genl quadratic eqn x2 + px + q = 0, (B), is reducible to the form (A), x(x + b) − g = 0; ∴ it has one root x1; ∴ it can be depressed to the first degree, & ∴ it has one other root x2; the genl quadc eqn has ∴ 2 roots, x1, x2. Again, this being admitted, the genl cubic x3 + px2 + qx+ r = 0 may be put under the form x(x + b)(x + c) − q = 0; it has ∴ by Mourey’s principle, one root, & ∴ may be depressed by divn to a quadc wh. will supply two other roots. Thus all depends on (A) having one root; & Mourey has only (as I think) spoiled the statement of his argument, by seeking to show, in the latter part of p.113 & in the earlier part of p.114, by geoml considerations, that (A) itself has n roots. Nor does he (as I conceive) succeed in showing this. He shows indeed that in discussing the eqn AP ·BP ·CP ·DP ·EP = g, we may select any one, say A, of the given points A,B,C,D,E; describe a certain closed curve δ about it; & determine on that curve at least one position P , wh. satisfies the condns of the question: and then commence anew with any other of the given points, as B, & find a point P ′ on the new curve, wh. likewise satisfies the conditions proposed. But he gives no reason for P & P ′ being different. The n curves δ might be conceived to intersect in one common point P . Therefore altho’ “le proble`me peut eˆtre resolu de n manie`res” [c’est-a-dire par n me´thodes, ou avec n origines differents, employe´s comme autant de points auxiliares] it is not yet proved, in this way, that the unknown line AP admits of n distinct values, or that the eqn (A) has n roots. But one suffices. Yours very sincerely, Wm. Rn. Hamilton. 186 Correspondence in Life of Hamilton cont. Hamilton to De Morgan — 23 November 1857. In a letter to Hamilton on 20 November, De Morgan had claimed to have discovered a proof that every equation has a root; a proof which he described as ‘so elementary that it must be the proof in future’.427 Hamilton writes in reply: I am very glad to hear that you think you have discovered the proof of the general existence of a root of an algebraic equation. Notwithstanding all that has been done by others, it was (for perfect repose to the mind) a thing still to be sought for. Perhaps Cauchy’s proof came nearest to satisfying me, of any which I have tried to examine. But even Mourey’s method, with some correction of details, was not (I thought) a total failure. You understand, therefore, that I am not asserting (or admitting) the insufficiency of all known proofs, when I say that I look forward with much interest to the communication of some new and simple proof by you.428 De Morgan to Hamilton— 27 November 1857. Thus reminded of Mourey’s proof, De Morgan is encouraged to revisit it and makes the comparison with his own. He writes: ‘Your mention of Mourey’s proof—which I had forgotten the existence of—made me look at it again, and I find no affinity with mine, which is of the Cauchy family, and would have struck Cauchy, if he had lived, I suppose.’429 The same date, later. De Morgan writes: I have been looking at Mourey’s proof to-day, i.e. of the existence of the root of an equation. It is perfectly sound, and is, I feel sure, the double algebra proof. But it wants a little intelligiblization to suit it to modern notions. I never did more than 427DeM to H (20 Nov. 1857) in [179, p530]. 428H to DeM (23 Nov. 1857) in [Ibid.]. 429DeM to H (27 Nov. 1857) in [179, p532]. 187 glance at it; but I now see I wanted, what I have been doing lately, to get it glanced into me. I shall get it into my paper on the subject.430 De Morgan’s paper on the existence of roots Just over a week after this correspondence (on 7 December 1857) De Morgan pre- sented his paper on the existence of roots to the Cambridge Philosophical Society. From an examination of Sturm’s proof of Cauchy’s theorem on imaginary roots [171], De Morgan had been led to a demonstration of the existence of roots which he be- lieved to be ‘the natural prefix to Sturm’s demonstration’.431 He found, in addition, an extension of Cauchy’s theorem. Both are incorporated in his paper, which also includes a consideration of the proofs by Argand and Mourey on the existence of roots. De Morgan had issue with Mourey’s notation and with his reliance on geometry in his proof of the F.T.A. Unimpressed by Mourey’s original notation, he attempted to introduce improved notation in his summary of Mourey’s proof. He also felt that it was necessary to supply an algebraical substitute for the geometric step in Mourey’s proof; the step relating to the change in the prime-directeurs of the radii.432 De Morgan’s view with regard to mathematical proofs was that one should to be able to fill any gaps which appear in the proofs using algebra; otherwise, the proof is not algebraic, nor does it have the potential to be so. He believed that a proof ought to be rigorous and based on accepted principles that are closely related to the theorem in question. Mourey’s proof of the F.T.A. has gaps: it is intuitive, rather than rigorous, on issues of continuity. Had Mourey demonstrated far more rigour in analysis, then perhaps De Morgan would have been convinced that the geometrical arguments in the proof were worthy of being retained and might, in 430[Ibid.] 431 [180, p261] 432For details of De Morgan’s substitution see [180, pp264–265]. 188 fact, do just as well as algebra. It has already been remarked (page 162) that it is unreasonable to expect this. Peacock’s criticism of Mourey De Morgan’s views on Mourey were clearly influenced by the Revd. George Peacock (1791–1858) who had taught De Morgan at Trinity College, Cambridge. In his paper on the existence of roots, De Morgan cites Peacock’s criticism of Mourey from his 1834 report (quoted below). Note that Mourey’s name is misspelt and the year of publication is also incorrect. Peacock writes: It is not very difficult to establish this fundamental proposition [the existence of roots of equations] by reasonings derived from the geometrical representation of impossi- ble quantities. This was done, though imperfectly, by M. Argand [. . .] and has been since reconsidered by M. Murey [sic], in a very fanciful work upon the geometrical interpretation of imaginary quantities, which was published in 1827 [sic]. It seems to me, however, to be a violation of propriety to make such interpretations which are conventional merely, and not necessary, the foundation of a most important symbol- ical truth, which should be considered as a necessary result of the first principles of algebra, and which ought to admit of demonstration by the aid of those principles alone.433 To think that the F.T.A. can be proved by algebraic principles alone is unrealistic. We now understand that the theorem is really about complex numbers and so we expect proofs to involve concepts from analysis where complex numbers are defined. It is not surprising that Peacock is critical of Mourey. Like Mourey, Peacock was concerned with the fundamental principles of algebra and was determined to reform algebra in order to overcome the problems associated with negative and complex numbers; however, his understanding of how generalization ought be introduced into algebra—through his Principle of the Permanence of Equivalent Forms—was rather more limited than Mourey’s.434 433 [193, p305] 434Peacock’s Principle of Equivalent Forms allowed the operations of arithmetical algebra to be adopted into symbolic algebra. See [194] for a paper on Peacock’s development of symbolic 189 Correspondence in Life of Hamilton cont. Hamilton to De Morgan — 4 January 1858. In a letter to Hamilton on 1 January, De Morgan wrote of his recent interest in Argand and remarked on the similarity between Argand and Cauchy’s proof of the existence of a root. In reply, Hamilton writes: Your remark about Argand’s investigation respecting roots of equations, which I never had time to read, reminded me that I had said something hastily, which might seem depreciating as regarded Mourey’s analogous investigation. I thought the latter, which I did read (in 1854), quite satisfactory, so far as concerned the existence of one root; and that he ought to have stopped there, and referred the rest to algebra.435 But he went on to make the superfluous effort, in which I thought he failed, to prove all the n roots by geometry.436 Hamilton to De Morgan — 25 March 1862. Hamilton has returned Mourey to De Morgan, after borrowing it for a second time; though, apparently, he has not returned all of it! He writes: ‘I trust that you have received your “Mourey”, minus a paper wing, which may yet be found’.437 De Morgan to Hamilton — 1 April 1862. De Morgan acknowledges receipt of Mourey, which affects the end of their corre- spondence relating to him. He writes: ‘I have received the Mourey. Never mind the cover. Uno avulso non deficit alter.438 I am thoroughly embusinessed for a few algebra, with respect to his Principles of Algebra (1830) in particular. The paper also contains a short discussion on Hamilton’s initial criticism of Peacock’s symbolic algebra and his invention of quaternions as an exercise of algebraic freedom. 435By ‘referred the rest to algebra’ Hamilton meant that once the existence of one root is proved, the existence of all the other roots follows by polynomial induction. 436H to DeM (4 Jan. 1858) in [179, p541]. 437H to DeM (25 March 1862) in [179, p577]. 438This is a quotation from the 6th book of Virgil’s Aeneid. It translates as: ‘When one is removed, there is no shortage of another.’ I am grateful to my brother, Benjamin for the reference and 190 days—after which at you again.’439 Two years later Hamilton returned to the subject of Mourey, in correspondence with his personal friend, the mathematician, Dr. Andrew Hart (1811–1890). At the time of the correspondence, Dr. Hart was a Senior Fellow of Trinity College, Dublin. Hamilton to Hart — 30 June 1864. It appears that Hamilton has acquired a copy of the second edition of Mourey. Once again he is concerned over priority. He writes: In driving home yesterday I looked into the Second Edition (Paris, 1861) of Mourey’s very ingenious little work. It was lucky that I could at once supply Salmon with a reference to the page of my Preface to the Lectures, in which I had cited the First Edition (Paris, 1828). But it is foolish to consider any such work as an anticipation of the quaternions. This brilliant and patriotic notion occurred lately to a French correspondent of our friend Salmon, who was so good as to send me the letter to read. The relation is rather of contrast than of resemblance, as in this very note to you I partly show.440 Systems which interpret +1 differently cannot have much in common. I forget in what year it was that I first heard of Mourey; nor is it of the slightest importance. As long ago as 1829 my attention was called by John T. Graves to the work of Mr. Warren, published in Cambridge, in 1828, On the Square Roots of Negative Quantities. The systems of Warren and Mourey, both published in 1828, are substantially the same; though the Frenchman was livelier and smarter. What was best in both had been anticipated in France by Argand.441 translation. 439DeM to H (1 April 1862) in [179, p579]. 440See MS.1492/195, pp. 57–59, in the Manuscripts Library of Trinity College, Dublin, for Hamil- ton’s comparison of the systems: Graves chose not to include this section of the correspondence in Life of Hamilton. 441H to Hart (30 June 1864) in [179, pp189–190]. 191 Salmon’s French correspondent Hamilton became acquainted with the Revd. George Salmon (1819–1904) in 1841, when Salmon joined the staff in the mathematics department at Trinity College, Dublin.442 From the letters which passed between Hamilton and Salmon, we know that the ‘French correspondent’, referred to above, was a M. Kuntz who was a stu- dent at the Lyce´e Charlemagne in Paris. He had written to Salmon on 8 June 1864 with the intention of bringing Mourey’s work to Salmon’s attention. He had read Salmon’s textbook, A Treatise on the Analytic Geometry of Three Dimensions (1862), in which Salmon had included a section in the appendix on Hamilton’s quaternions. Kuntz believed that while Hamilton might be credited with having in- vented the terminology associated with quaternions—scalars, vectors and tensors—it was Mourey who deserved priority as the first to have the concept of directed lines in the plane. Salmon sent Kuntz’s letter on to Hamilton; telling Hamilton, in an accompanying letter, that he had never heard of Mourey before. Kuntz’s letter and the subsequent related correspondence between Hamilton and Salmon were copied into another of Hamilton’s notebooks, which again is preserved in the Manuscripts Library of Trin- ity College, Dublin.443 On 11 June Hamilton replied to Salmon: Mourey however was by no means the first to treat of directed lines, [at least within the plane] [. . .] Argand (1806–1813) was decidedly prior to him. [Your correspondent may take comfort from the thought that Argand also was a Frenchman.]444 [. . .] The thing (or thought), which I express by the word “vector”, was never claimed by me 442Salmon was appointed to a divinity lectureship at Trinity College, Dublin in 1845 and as Done- gall Lecturer in Mathematics in 1848. [195] 443See MS.1492/195, pp. 1–9 in the Manuscripts Library of Trinity College, Dublin. Kuntz’s letter was copied into the notebook by Hamilton’s wife, Helen. Again, I am grateful to the Keeper of the Manuscripts at Trinity College, Dublin for giving their permission to publish this correspondence. 444We now know that Argand, though resident in Paris, was of Swiss nationality. 192 as my own: nor can I, on the other hand, admit that Mourey, or his predecessor Argand, had in any degree anticipated the quaternions. Vectors might have been familiarly known for two or three thousand years, instead of merely fifty or sixty, and the world been as far off as ever from quaternions.445 Hamilton returned Kuntz’s letter to Salmon on 15 June. Initially, Hamilton had intended to respond publicly to the young Frenchman in the Philosophical Magazine. However, Salmon reassured him that such action was not necessary and explained that he had already dealt with the matter by writing to Kuntz personally: My Dear Sir Wm, I do not think it is necessary for you to take any public notice of my French Correspondent’s letter, who for all I know may be a young man of no mark. If his criticisms had appeared anywhere in public, the case would be different. I wrote telling him he ought not to judge of your system by the meagre sketch which was appended to my book on surfaces, but that he ought to consult your own Lectures. I told him of the reference you had given to Mourey’s work: & I availed myself of your hint & in telling [him] that Mourey had been anticipated made the communication more pleasant by speaking of his “Countryman” Argand. Finally I added that nothing that had been done by Mourey, Argand or anyone else for the Geometry of the Plane could be considered as an anticipation of Quaternions as I thought my correspondent himself would fully recognize if he made himself better acquainted with the latter system. I remain very sincerely yours Geo. Salmon.446 Hamilton thanked Salmon for his conduct but explained that he was not content to let the matter rest and would return to the issue in future correspondence with Salmon. 445H to S (11 June 1864), MS.1492/195, pp. 3–4. 446S to H (16 June 1864), MS.1492/195, pp. 8–9. It appears that Hamilton never published on the matter in Phil. Mag. 193 De Morgan and the history of mathematics De Morgan had a keen interest in the history of mathematics. Cajori writes: ‘Few contemporaries were as profoundly read in the history of mathematics as was De Morgan.’447 Smith writes: ‘He devoted considerable attention to the history of mathematics, but his articles are not only eccentric but unreliable.’448 De Morgan’s interest in the history of mathematics led to a number of publi- cations, in: Smith’s Dictionary of Greek and Roman Biography (1862–1864); the Penny Cyclopaedia (1833–1843), published by the Society for the Diffusion of Use- ful Knowledge; the Companion to the Almanac (various years) and his Arithmetical Books (1847). This last has been described as ‘probably the first significant work of scientific bibliography’.449 De Morgan’s collection of scientific books—which comprised around three thou- sand titles—was bought after his death by Lord Overstone who then presented the collection to the library of University College, London.450 This is the current lo- cation of De Morgan’s 1828 edition of Mourey. The University of London’s Senate House Library also has a copy of the first edition of Mourey: this particular copy once belonged to John T. Graves. It is peculiar that Hamilton had asked De Morgan to help him acquire a copy of Mourey when his friend, John Graves owned a copy. The British Library also has a copy of the first edition of Mourey. 5.5 Benefits of the algebraic perspective In studying Mourey from an algebraic perspective we have derived a number of ad- vantages. The true focus of Mourey’s researches has come to light and we have seen more readily: (i) evidence of a logical, rigorous and even axiomatic approach to his 447 [196, p316] 448 [197, p462] 449 [198] 450According to De Morgan’s entry in [13]. 194 development of the new algebra and (ii) implicit evidence of what Wussing terms ‘group-theoretic thought’.451 We also find ourselves in a better position to appre- ciate the relevance of Mourey’s work to today’s mathematicians and historians of mathematics: Mourey’s mathematics is about the fundamental principles of algebra which is the foundation of our reasoning in the subject; it is also a fresh example of the interaction of number theory, algebraic equations and geometry, which have been described by Wussing as the ‘three historical roots of abstract group theory’.452 I am not suggesting that Mourey ought to occupy a place in the history of the development of algebra or group theory. I believe his work to be, in some respects, an isolated incident that had little or no impact on contemporary developments: it attracted no real publicity; notably, it failed to achieve the level of publicity which Mourey himself predicted, in the form of opposition to his radical new system. What I do hope for is that the reader may now appreciate the following. (i) Mourey was concerned with establishing a rigorous foundation for algebra. (ii) He understood that deficiencies in contemporary algebra were inhibiting progress. (iii) He had a clear sense that arithmetical algebra had to be left behind and that rigour, generalization and a higher level of abstraction had to be embraced. He under- stood that this was key in reviving algebra and driving it forward. The fact that Mourey was an innovator, ahead of his time, is plain in his mathematics. With the benefit of hindsight, historians of mathematics can now recognize the quality and far-sightedness of some of Mourey’s mathematical ideas. 451 [199, p17] 452 [199, p16] 195 Overall final remarks To conclude this thesis, I will suggest some of the ways in which it might contribute to the history of mathematics and, more generally, to the history of science. — I have highlighted the existence of primary source material relating to Tait, facilitating fellow researchers in their efforts. — Through original research, I have supplemented the existing literature on Tait, providing new insights into: Tait’s family history; his unique contribution to the ever-relevant science-versus-religion debate; his involvement in the fields of probability and statistics; and, with respect to the history of the Argand diagram, Bishop Terrot’s work on the geometrical representation of complex numbers. — In addition, I have given an identity to the elusive Frenchman, C.-V. Mourey who, for the past 186 years, has remained an unknown to historians of math- ematics. I have re-examined his 1828 work and have brought to light an unpublished paper by Sir W. R. Hamilton on the F.T.A. which was inspired by Mourey. In covering a variety of aspects of Tait’s life and work, it is hoped that this thesis has given an indication of the vast scope of his expertise and interests, and has appealed to the varied interests of a diverse readership. The intention was to stimulate interest in Tait, who has been unfortunately overlooked in recent times. 196 Seldom in a generation does it fall to the lot of the scientific worker to become a pre-eminent leader in his subject; and not often in the course of the centuries is such a leader also pre-eminent as a clear and original writer, as an unusually effective lecturer, and as an exerter of powerful personal influence. Each of these qualifications by itself would be suffi- cient to make its possessor a man of distinction. He who possesses them all is truly great as a worker and teacher, great also as a man when his personal aims and influences are all in the direction which he deems to be highest and best. Other men gather the long record of his work which he leaves behind, and it is passed on to help the work of other ages. His works do not cease with his life, they follow him. His influence continues to spread too, if in diminished stream, unavoidably and unconsciously, by means of those who have come under the compulsion of his personal- ity. He exerts compulsion, not on his pupils merely, but on his compeers also; and through this his influence redoubles itself. From the record of his life and work, men who never knew him may come under the magic spell and be enrolled in the list of his disciples. Such a man was Peter Guthrie Tait, Natural Philosopher.453 453 [200, p1] This quotation is taken from an obituary tribute to Tait written by Professor William Peddie (1861–1946): assistant to Tait (1883) and lecturer in natural philosophy (1892–1907) at Edinburgh, later Professor of Physics at University College, Dundee (1907–1942) and President of the Edinburgh Mathematical Society (1896–1897). Peddie gives a good account of Tait’s contributions to mathematics and physics, with particular emphasis on Tait’s experimental work in his Edinburgh laboratories and his influence as a teacher. I found the seven-page booklet amongst the Papers of the R.S.E., in a folder containing documents relating to the Tait Memorial Fund: perhaps the booklet was circulated to gather support for the Tait Memorial Movement. A copy of the booklet is also preserved in Tait’s scrapbook. 197 Bibliographical essay In addition to the three primary unpublished sources on which this thesis is based— Tait’s scrapbook, the Tait–Maxwell school-book and Tait’s pocket notebook—I have made use of a variety of archival material and a number of secondary sources. For information on Tait’s family history, I consulted the digitised family history records for Scotland, which are available at The ScotlandsPeople Centre, Edin- burgh. This supplemented information available from ancestry.co.uk and from a genealogical table for the Ronaldson family (MS.36998/38, Special Collections, the University of St Andrews) which was presented to the University of St Andrews by Tait’s granddaughter, Miss. Margaret Tait on 26 April 1974. I was also privileged enough to have personal communication with Susan Rutherford (Tait), paternal great-granddaughter of P.G.T. At the Edinburgh Academy archive, I found a number of documents relating to Tait’s schooldays. The following proved especially useful. The Edinburgh Academy List, 1824–1974 is a record of the pupils and staff at the Academy during the pe- riod between 1824 and 1974. It was published by the Edinburgh Academical Club in 1974 for the 150th anniversary of the Academy. The Academy’s Prize List is a published list of pupils receiving prizes at the Academy’s Public Exhibition Day— the school’s annual prize-giving event in July, which was first held in 1825. The pupils’ prize-winning poetry is also published alongside the list of prize winners. The Annual Report of the Rector and Directors contains information on the general running of the school; finances, staff appointments, etc. The Edinburgh Academy Chronicle is the school’s newsletter. It was launched in 1893, with a number of editions published per year. It features: important school events (including guest lectures, sports matches, etc.), news on former pupils, literary articles written by current pupils, etc. At the National Library of Scotland in Edinburgh, I consulted the Papers of the R.S.E. (Acc. 10000), giving special consideration to: the minute books, letter books, bound volumes of visitors to the R.S.E. and two folders of documents relating to the Tait Memorial Fund (Acc.10000/166,167). Acc.10000/166 contains documents 198 relating to the setting up of the Tait Memorial Committee, decisions about a fitting memorial to Tait and the raising of funds. Acc.10000/167 contains correspondence between the R.S.E. and Tait’s son, William Archer Porter Tait, and between the R.S.E. and the University of Edinburgh. The correspondence relates to the invest- ment and distribution of donations. At the Manuscript Library of Trinity College, Dublin, I consulted Hamilton’s notebooks. Amongst the library’s collection of two-hundred Hamilton notebooks, MS.1492/95 and MS.1492/195 were of particular interest in relation to C.-V. Mourey. MS.1492/95 is a 13”×8” ledger with 287 pages. It contains: a copy of Bue´e’s 1806 paper, copied into the notebook by Hamilton’s sister, Eliza, together with some remarks by Hamilton; a consideration of the work of Carnot and Grassmann; and thirty-four pages relating to Mourey (pages 241–274). MS.1492/195 is another ledger notebook. It contains: correspondence between Hamilton and the Revd. George Salmon, Dr. Andrew Hart and others, including Tait. On the inside cover of the notebook (page iii) Hamilton recorded Tait’s address at Greenhill Gardens, Edin- burgh. The correspondence with Salmon and Hart is principally on Mourey and concerns priority in the discovery of quaternions. In search of biographical information on Mourey, I went to the Archives de Paris, at 18 boulevard Se´rurier 75019 Paris. I consulted the ledger volumes that contain: information on property owners in Paris (their name, occupation etc.); property valuations and a record of when properties changed hands and for what reason (e.g. through sale or succession). The addresses I had for Mourey were in volume nos. DQ18.206, DG18.323 and DQ18.364. I also consulted microfilmed copies of the re- constituted civil records for the period between the C16th and 1859. The principal secondary source relating to P. G. Tait is the scientific biography written by C. G. Knott (1856–1922), published in 1911: Cargill Gilston Knott, Life and Scientific Work of Peter Guthrie Tait (Cambridge : at the University Press, 1911). Knott was a former student of Tait’s at Edinburgh and his research assistant in the natural philosophy department between 1876 and 1883. His biography of Tait was described by Sir Edmund Taylor Whittaker (1873–1956), Royal Astronomer of 199 Ireland, as “‘one of the best scientific biographies ever written”’.454 Knott’s biography supplemented the two volumes of Tait’s scientific papers which were published in 1898 and 1900 by Cambridge University Press: Peter Guthrie Tait, Scientific Papers, 2 vols. (Cambridge : at the University Press, 1898 and 1900). A bibliography of Tait’s published works is given in Knott’s biography and in a paper compiled by Chris Pritchard for the Tait Centenary meeting at the R.S.E. in 2001:‘Provisional Bibliography of Peter Guthrie Tait’, ed. Chris Pritchard on be- half of the Royal Society of Edinburgh, a paper presented at Peter Guthrie Tait (1831–1901 ): Centenary Meeting, July 2001; The Royal Society of Edinburgh. . Chris Pritchard has also published a number of papers on Tait’s involvement in knot theory, golf science and the promotion of quaternions. For details consult the References section at the end of this thesis. 454 [33, p49] 200 Appendices A FAMILY TREE REPORT FOR THE TAIT FAMILY Information on the Tait family has come from a variety of sources: personal commu- nication with Susan Rutherford, paternal great-granddaughter of P.G.T., searches at The ScotlandsPeople Centre in Edinburgh; information available through an- cestry.co.uk and that contained in a genealogical table for the Ronaldson family (MS.36998/38, Special Collections, the University of St Andrews) which was pre- sented to the University of St Andrews by Tait’s granddaughter, Miss. Margaret Tait on 26 April 1974. Only sources other than these have been referenced in the report. Readers may find it helpful to consult the genealogical tables in Figures A.1 and A.2 (pages 218–219). Table A.1: Family group report for the Tait family. Peter Guthrie Tait Father John Tait (1785–1837) Private Secretary to the 5th Duke of Buccleuch, Walter Francis Montagu Dou- glas Scott (1806–1884). Son of Patrick Tait (c.1746–1819), a shoemaker who died in Selkirk, and Margaret Guthrie (b. c.1747), daughter of Robert Guthrie of Selkirk. Married Mary Ronaldson on 27 June 1829 in Dalkeith, Midlothian. Mother Mary Ronaldson (1795–1846) Daughter of John Ronaldson (1756–1820) (a tenant farmer in Sauchland, Crich- ton, Midlothian) and Ann(e) Turnbull (d.1837) who were both buried in Borth- wick cemetery, Midlothian. Birth 28 April 1831. Dalkeith, Midlothian. Baptism 10 June 1831. Dalkeith, Midlothian. Witnesses: William Lamb (Selkirk); John Ronaldson (maternal uncle; writer, Edinburgh). Marriage 13 October 1857 to Margaret Archer Porter. Shankill, Antrim, Ireland. 201 Death 4 July 1901. Age 70. At Challenger Lodge, Wardie, Leith. Usual residence: 38 George Square, Edinburgh. Occupation: Emeritus Professor, Edinburgh University. Cause of death: arteriosclerosis (stiffening of the arteries); weak heart for 7 months. Death certified by: Gibson. Death certificate signed by: J. G. Tait (son), resident at 38 George Square, Edinburgh. The funeral: 6 July 1901 at St John’s Episcopal Church, Edinburgh. Place of interment: St John’s Church Yard, to the east of the church.455 Challenger Lodge was the residence of Sir John Murray (1841–1914), Tait’s friend and former pupil, who had invited Tait to stay at the residence in order to recuperate, following a period of ill health which began when his son Freddie was killed in the South African War in February 1900.456 Will Written 15 January 1875. Updated 23 March 1896. Trustees: Margaret Archer Tait (Porter); Alexander Crum Brown [Brother- in-law, Doctor of Medicine, Prof. Chemistry and Chemical Pharmacy]; and William Ramsay Kermack [Writer to the Signet, Edinburgh]. Additional new trustees: Tait’s sons John Guthrie [Prof. in the Central College, Ban- galore, India], William [civil engineer, Edinburgh], Freddie [Lieutenant 2nd Battalion Black Watch, Royal Highlanders] and Alexander Guthrie [resid- ing at 38 George Square]; and Harry Cheyne [Writer to the Signet, Edinburgh]. The beneficiaries. To his wife: £500 for providing family mournings and household and other expenses; rents, dividend, interests, whole free annual income and profits of remainder of estate and effects. If she enters into a second marriage, provision immediately ceases and she will no longer be a trustee. If she dies or marries again, monies will be divided equally between the children: boys to receive at age 21; girls age 21 or on marriage, whichever is first. 455 [18, p41] 456 [18, p40] 202 Assets On 28 August 1901, assets included: cash £39.7; house furniture and effects (38 George Square Edinburgh) £175.18.6; current accounts in the National Bank of Scotland, £376.5.7, at Mackenzie and Kermack £215.1.1; savings in the National Bank of Scotland £1436.18.6; stocks and shares, William Younger and Co. Ltd. £465.12.6, City Property Investment Trust Corporation Ltd. £979.19, John Fraser and Sons Ltd. £205; bonds in the Royal and Ancient Golf Club St Andrews £25.10.2; rents, for stables in Meadow Lane, Edinburgh £910 for the half year to Martemmas 1901; fenduty payable at Whitsunday yearly, Duke of Buccleuch for subjects of Dalkeith annual sum of £11; ground annuals payable at Whitsunday for subjects at Taitshill, Selkirk £306 and payable at Martemmas for subjects at Taitshill, Selkirk £266; sum in bond and disposition in security for £2500 granted (May 1895) by son William Archer Porter Tait to P.G.T. reduced to £400; life assurance policies, The Scottish Provident Institution £5990.10, Scotland Annuable Life Assurance Society £3652.18.6; pension from the University of Edinburgh £953.19.2 per annum; fee due for examining a D.Sc. thesis for the University of Edinburgh £3.3; sum due from son John Guthrie Tait for advance on premium of insurance of his life £54.13.4.; interest in estate of his uncle, the late John Ronaldson of Somerset Cottage, Edinburgh £1935.16.3, and of his aunt, the late Margaret Ronaldson £774.18.9. Estate in Scotland: £24795.18.1. Estate in England (£143.7.4): balance owing to the publishers Macmillan and Co. £18.7.4; with Macmillan, interest in copyright for Heat, The Unseen Universe and Quaternions with Kelland £25; with A & C Black, interest in copyright for Light, Properties of Matter, Dynamics and Newton’s Laws of Motion £100. Testate. Total assets in the U.K.: £24,939.5.5. Duty paid: 4.5% £1,268.8.11. Residences—indicated by the 10 year census. 1841 23 Warriston Crescent, Edinburgh. P.G.T. age 10, living with mother Mary (45) [independent means]; sisters Anne (8) and Mary (5); and 1 servant. 1851 Peterhouse College, Cambridge. P.G.T. age 19 [undergraduate pensioner], living with (amongst others): William A. Porter (26); William J. Steele (19); Frederick Fuller (31); and Edward J. Routh (20). See Figures A.3 and A.4 (pages 220–221). Tait collaborated on a book with William Steele (1831–1855), Dynamics of a Particle (1856); married William Porter’s sister, Margaret (1857); and competed against Fuller and Routh for the Chair of Natural Philosophy at the University of Edinburgh (1860). 1861 13 Buccleuch Place, Edinburgh. P.G.T. age 29 [Professor of Natural Phi- losophy], living with: wife Margaret (21); child Edith (1); and 2 servants. Sometime between 1861 & 1871 6 Greenhill Gardens, Edinburgh.457 In William Thomson’s obituary trib- ute to Tait, Thomson recalls that Tait produced a list, etched in charcoal, on the bare plaster of his study wall at this residence. Tait had ordered the cur- rent scientific figures by merit: Hamilton, Faraday, Andrews, Stokes and Joule ranked first in the column. Clerk Maxwell was as yet too young to appear.458 457 [201, p193] 458 [28, p364–365] 203 1871 17 Drummond Place, Edinburgh. P.G.T. age 39 [Professor of Natural Philosophy, University of Edinburgh], living with: wife Margaret (31); children Edith (11) [scholar], John (9) [scholar], Mary (6) [scholar], William (5) [scholar], Freddie (1); and 4 servants. 1881 38 George Square, Edinburgh. P.G.T. age 49 [M.A. Professor of Natural Philosophy], living with: wife Margaret (41); children John (19) [student at Cambridge], William (15) [scholar], Freddie (11) [scholar], Alex (8) [scholar]; and 3 servants. 1891 38 George Square, Edinburgh. P.G.T. age 59, living with: wife Margaret (51); children Edith (31), Alex (18) [Student of Arts], William (25) [Civil En- gineer]; and 2 servants. 1901 38 George Square, Edinburgh. P.G.T. age 69, living with: wife Margaret (61); children Edith (41), John (39) [Professor of English], Alex (28) [Mer- chant’s Clerk]; and 3 servants. Education. Early education at Dalkeith Grammar School and Circus Place School, Edinburgh.459 1841–1847 The Edinburgh Academy. In Dr. Cumming’s class. [4, p120] Associations: schoolboy friendship with fellow student, James Clerk Maxwell (1831–1879). 1847—1848 The University of Edinburgh. Studied mathematics under Philip Kelland (1808–1879) and natural philosophy under James David Forbes (1809–1868). 1848—1852 The University of Cambridge. Peterhouse College: admitted 21 June 1848 (pensioner); matriculated in the Michaelmas term of 1848. Coached by pri- vate mathematics tutor, William Hopkins (1793–1866). B.A. (1852); Senior Wrangler (2nd Scot on record) and 1st Smith’s Prizeman; elected a Fellow of Peterhouse (1853); M.A. (1855).460 See Figure A.5 (page 222) for the Tripos Examination results. Associations: William Steele; and the Porter brothers, William Archer Porter (1824–1890) and James Porter (1827–1900). See Figure A.6 (page 223) for a photograph of Tait and Steele as graduates in 1852. Occupations: Professor of Mathematics; Professor of Natural Philosophy. 1854–1860 The Queen’s College, Belfast. Professor of Mathematics: elected 14 September 1854; also held voluntary classes to supplement honours lectures in natural philosophy. Associations: Thomas Andrews (1813–1885), James Thomson (1822–1892) (William Thomson’s brother), Charles Wyville Thom- son (1830–1882) and James McCosh. Andrews introduced Tait to experimental work and to Sir William Rowan Hamilton (1805–1865).461 459 [18, p3] 460 [13] 461 [18, p12] 204 1860–1901 The University of Edinburgh. Chair of Natural Philosophy: succeeded Forbes. Associations: William Robertson Smith (1846–1894), Tait’s lab as- sistant (1868–1870); and Balfour Stewart (1828–1887), Forbes’ former student and lab assistant. Memberships, fellowships, prizes, honours. Royal Society of Edinburgh. Elected Ordinary Fellow (07/01/1861); proposed by Philip Kelland. Served as: Councillor (1861–1864); Secretary to the Ordinary Meetings (1864– 1879) and General Secretary (1879–1901). Awarded the Society’s Keith Prize (1867–1869, 1871–1873) and Gunning Victoria Jubilee Prize (1887–1890).462 Associations: William Thomson (1824–1907). The Royal Society of London. Awarded their Royal Medal (1886); however, Tait chose never to become a Fellow of the Society.463 Honorary member of The Literary and Philosophical Society of Manchester (1868). Hon- orary degrees: The Catholic University of Ireland (Sc.D., 1875); The University of Glasgow (LL.D., 1901) and The University of Edinburgh (LL.D., 1901). See Figure A.7 (page 224) for Tait’s notification of his honorary degree from Edinburgh. Honorary fellowships: Societas Regia Hauniensis, Copenhagen (1876); The Edinburgh Mathematical Society (1883); The University of Cambridge, Peterhouse (1885); Socie´te´ Bavate de Philosophie Experimentale, Rotterdam (1890); Societas Regia Scientiarum, Upsala (1894) and The Royal Irish Academy (1900).464 John Ronaldson (maternal uncle) Father Only son of John Ronaldson (1756–1820), a tenant farmer from Sauchland, Crichton, Midlothian. Mother Ann(e) Turnbull (d.1837). Birth 17 October 1812. Sauchland, Crichton, Midlothian. Baptised 1 December 1812, Sauchland, Crichton, Midlothian. Marriage Single. Death 26 December 1864. Age 52. Somerset Cottage, Raeburn Place, Edinburgh. Cause: Disease of the aorta. Death certificate signed by: P.G.T. (nephew, living at 6 Greenhill Gardens). Occupation: writer. Single. Place of interment: Dean cemetery, Edinburgh. 462 [29] 463 [18, p49] 464 [18, p47] 205 Will Written 22 July 1862. Trustees: P.G.T. [Master of Arts, Professor of Natural Philosophy, Uni- versity of Edinburgh] and William Ramsay Kermack [Writer to the Signet]. Beneficiaries: entire estate to sister, Margaret Ronaldson, residing at Somerset Cottage; upon her death, the residue of the estate to go to P.G.T. (nephew), Anne Margaret Tait and Mary Tait (nieces, both resident at Somerset Cottage). Assets On 2 February 1865, assets: cash in the house £10, household furniture and silver plate £263.8, current account at the National Bank of Scotland £89.8.7, shares in the British Railway Company £807.2 and £1008.17.6, promissory note by John Macnab of the Oriental Bank Corporation £507.6.11, balance on current account Messers Mackenzie and Kermack £45.14.7, stocks in the National Bank of Scotland including dividends £2088, stocks in the Edinburgh and Glasgow railway £338.6. Total assets £5965.5. Residences—indicated by the 10 year census. 1841 Claremont Street, Midlothian. J.R. age 25 [writer], living with: sister Margaret Ronaldson (25); Ann Ronald- son (20); and 1 servant. 1851 Somerset Cottage, Raeburn Place, Edinburgh. J.R. age 37 [Clerk in the National Bank of Scotland], living with: sister Mar- garet Ronaldson (39) [Annuitant]; nieces Anne M. Tait (17), Mary Tait (15) [scholar]; and 2 servants. 1861 Somerset Cottage, Raeburn Place, Edinburgh. J.R. age 48 [Banks Clerk], living with: sister Margaret Ronaldson (50), nieces Anne M. Tait (27), Mary Tait (25); 1 servant; and a visitor Peter Melrose [a shoe maker]. Occupations: writer; clerk in the National Bank of Scotland. Influence on P.G.T. Following the death of both their parents (John in 1837 and Mary in 1846), P.G.T. and his two sisters went to live with Mary’s bachelor brother, John Ronaldson, and her maiden sister, Margaret Ronaldson, at Somerset Cottage. Although John was a banker by profession, he had a keen interest in scientific investigation and spent much time with P.G.T., enjoying scientific pursuit which must have influenced P.G.T.’s enthusiasm for science.465 Anne Margaret Tait (sister) Father & mother Same as P.G.T. 465 [18, p3] 206 Birth 14 April 1833. Dalkeith, Midlothian. Baptised 16 June 1833, Dalkeith, Midlothian, in the presence of the congrega- tion. Marriage Single. Death 26 June 1915. Age 83. 15 Fettes Row, Edinburgh, Midlothian. Cause: Pneumonia 6 days. Death certificate signed by Edith Tait (niece, re- siding at 19 Abercromby Place, Edinburgh). Single. Place of interment: Dean cemetery, Edinburgh. Mary Tait (sister) Father & mother Same as P.G.T. Birth 4 August 1835. Dalkeith, Midlothian. Baptised 15 September 1835, Dalkeith, Midlothian. Witnessed by John Patter- son, agent of the Leith Bank, Dalkeith, and John Ronaldson, writer, Edinburgh. Marriage Single. Death 26 April 1913. Age 77. Somerset Cottage, Raeburn Place, Edinburgh. Cause: Carcinoma (malignant tumour) of the intestine. Death certificate signed by Mary Cathcart (niece, residing at 44 Melville Street, Edinburgh). Single. Fundholder. Place of interment: Dean cemetery, Edinburgh. Mary was a pupil of the Scottish landscape and marine painter, Samuel Bough (1822–1878) at the Royal Scottish Academy. Residences—indicated by the 10 year census. Tait’s maternal aunt and uncle, Margaret Ronaldson (1809–1892) and John Ronaldson (1812–1864), lived in Somerset Cottage with Tait’s sisters, Anne Margaret Tait (1833–1915) and Mary Tait (1835–1913). It appears that John, Margaret, Anne and Mary were all unmarried, had no children of their own and remained resident at Somerset Cottage until their deaths. All four were buried in Dean cemetery in Edinburgh. Somerset cottage was built in 1832. It was one of three detached villas; two of which were replaced by tenements and no longer exist. The property was bought by the Edinburgh Academy, probably soon after Anne and Mary died. During the 1920s, it was converted into two villas, upper and lower, to provide accommodation for Academy staff. The Academy sold the property in the 1960s. It became a guest house and then the Raeburn House Hotel. Following recent redevelopment, the grade B listed Somerset Cottage is now operating as ‘The Raeburn’, a boutique hotel with bar and restaurant. The property is situated on the south east corner of the Edinburgh Academy sports fields. It is a fifteen minute walk from Somerset Cottage to the Edinburgh Academy. Margaret Archer Tait (Porter) (Spouse) Father James Porter (c.1788–1851) Presbyterian minister in Drumlee, Castlewellan, Co. Down, Ireland. Died in a fall from his horse. 207 Mother Eliza Archer ( d. c.1877) Margaret was a sister to the Porter brothers, William Archer and James, who Tait had known at Cambridge. Tait grew close to the family during his time in Belfast, from 1854. Margaret was one of twelve children. Birth 1 May 1839. Co. Down, Ireland. (Date from her grave stone in St John’s Church Yard, Edinburgh.) Marriage 13 October 1857 to Peter Guthrie Tait. Shankill, Antrim, Ireland. Death 27 October 1926. Age 87. The Rectory, Colinton, Midlothian. Cause: Cardiac failure due to old age. Death certificate signed: H. Reid of the Rectory, Colinton (son-in-law, resident at The Rectory, Colinton, Midlothian). Place of interment: Tait family grave, at St John’s, Edinburgh. Residences—indicated by the 10 year census. 1861–1901 See P.G.T. 1911 38 George Square, Edinburgh. M.A.P.T. age 71 [widow], living with: son William (45) [Civil Engineer, Em- ployer]; daughter-in-law Anne (34) [John Guthrie’s wife] and Anne’s children Margaret (3), Patrick (1); and 4 servants. Alexander Crum Brown (brother-in-law) Born 26 March 1838 in Edinburgh. Son of the Revd. John Brown (1784–1858) who was biblical scholar and minister of the United Presbyterian Church of Scotland, Broughton Place, Edinburgh. Alexander’s mother, Margaret Fisher Crum (1799–1841) was John’s second wife. Alexander had a step-brother and step-sister from his father’s first marriage. Alexander married Jane Bailie Porter (1836–1910), sister to Tait’s wife and the Porter brothers, in Belfast in 1866. He died on 28 October 1922, aged 84, at 8 Belgrave Crescent, Edinburgh. He studied medicine at the University of Edinburgh: M.A. (1858); M.D. (1861) and D.Sc. (1862, London). Between 1863 and 1869, he was Lecturer in Chemistry at the Edinburgh Extramural School and from 1869 to 1908, Professor of Chemistry at the University of Edinburgh.466 Alexander was a Fellow of the R.S.E. (07/12/1863), proposed by Sir Lyon Playfair. He served as: Councillor (1865–1868, 1869–1872, 1873–1875, 1876–1878, 1911–1913), Secretary to the Ordinary Meetings (1879–1905) and Vice-President (1905–1911). He was awarded the Society’s Keith Prize (1873–1875) and the Makdougall-Brisbane Prize (1866–1868) which was a joint award.467 He was President of the Edinburgh Medical Missionary Society between 1911 and 1916.468 466 [202, p895] 467 [29] 468 [202, p895] 208 Children of P.G.T. and Margaret Archer Porter (6) Edith Tait (c.1860–1948); John Guthrie Tait (1861–1945); Mary Guthrie Tait (1864–1946); William Archer Tait (1866–1929); Frederick Guthrie Tait (1870–1900); Alexander Guthrie Tait (1873–1934). Edith Reid (Tait) Birth c. 1860. Belfast, Ireland. Marriage 24 June 1902. St John’s Church, Edinburgh. To Harry Seymour Reid (1866–1943): born in Glasgow; son of Daniel Reid, a ship worker; later Rt. Revd. Bishop of Edinburgh (1929–1939). At time of marriage: Harry was a clerk in holy orders, residing at 5 Granville Terrace, Edinburgh; Edith was residing at 38 George Square, Edinburgh. Harry died at 3 Pilmour Place, St Andrews, aged 76. Harry Seymour Reid’s portrait (from 1930) is housed in the National Portrait Gallery, London, as part of the ‘Photographs of Anglican Bishops, 1860s–1940s’ set of portraits. Death 1 September 1948. Age 88. At 3 Pilmour Place, St Andrews. Cause: Cardiovascular degeneration. Death certificate signed: William Shaw Andrews (intimate friend; resident at The Eastory, St Andrews). Widow. Residences—indicated by the 10 year census. 1861 & 1871 See P.G.T. 1881 Cannot be found on the Scottish or English 1881 Censuses. 1891 & 1901 See P.G.T. 1911 27 Argyle Crescent, Portobello, Edinburgh. Edith age 51, living with husband Harry (44) [Episcopal Clergyman]; and 2 servants. John Guthrie Tait Birth 24 August 1861. 6 Greenhill Gardens, Edinburgh. 209 Marriage 7 January 1904. Bangalore, Madras, India. To Anne Smith Cook (1876–1951): born Forfar, Arbroath; daughter of John Cook (teacher in the mathematics department at Abroath High School; Principal of Doveton College, Madras (1882–1907); Principal of the Central College Bangalore (1882–1907); F.R.S.E. (1894, proposed by P.G.T.).469) Anne died at 38 George Square, Edinburgh, aged 75. She is buried in Morningside cemetery in Edinburgh, while John Guthrie is buried in the Tait family grave in St John’s Church Yard. Death 4 October 1945. Age 84. 38 George Square, Edinburgh. Cause: Enlargement of the prostrate, urinary bladder fistula, suppression of urine, rheumatoid arthritis. Death certificate signed: M. Tait (daughter). Oc- cupation: retired College Principal. Residences—indicated by the 10 year census. 1871 & 1881 See P.G.T. 1891 Cannot be found on the Scottish or English 1891 Censuses: he went to India in 1890.470 1901 38 George Square, Edinburgh. See P.G.T. Presumably he returned for a short period following his brother Freddie’s death. 1911 Cannot be found on the Scottish or English 1911 Censuses. We can presume he is still in India as The Edinburgh Academy Register has his address in 1914 as ‘Central Coll. House, Bangalore, S. India’.471 Education. 1871–1877 The Edinburgh Academy. Mr. Carmichael’s class.472 1880–1883 The University of Cambridge. Admitted Peterhouse [pensioner]; matricu- lated Michaelmas 1880; classical Tripos 1st class 1883; B.A. 1884; M.A. 1890.473 Sporting achievements. 469 [29] 470 [4, p336] 471[Ibid.] 472 [4, pp335–336] 473 [13] 210 Rugby: Played for A.F.C. (The Academical Football Club at the Edinburgh Academy) and London Scottish F.C.; blues for Cambridge (1880, 1882); played for Scotland, versus Ireland (1800, 1885). Golf: Semi-final of the Golf Amateur Championship, Hoylake (1887); London Scottish Golf Gold Medal, Hope Grant Medal, Bangalore Golf Club Gold medal (twice). Winner of prizes at Bangalore Rifle meetings and S. India Rifle Association meetings.474 Occupations: barrister-at-law, scholar of English Literature, College Principal. 1884 Barrister-at-law: admitted Lincoln’s Inn; called to the bar 25 April 1888.475 1890 Government education department, Mysore, India.476 1908 Principal of Mysore Government Central College, Bangalore.477 Previously Professor of Languages there and Vice-Principal.478 Examiner in English, the University of Madras.479 Military service. 1914–1919 Served in the Great War: Lieutenant Colonel, Bangalore Rifle Volunteers.480 Memberships, fellowships, prizes, honours. 1937 Royal Society of Edinburgh. Elected Fellow (01/03/1937), proposed by: Sir D’Arcy W. Thompson, William Peddie, A. Crichton Mitchell, Sir Edmund T. Whittaker.481 Fellow of Madras India University.482 Mary Guthrie Cathcart (Tait) Birth 22 July 1864. 6 Greenhill Gardens, Edinburgh. 474 [4, p336] 475 [13] 476 [4, p336] 477[Ibid.] 478 [13] 479[Ibid.] 480[Ibid.] 481 [29] 482 [13] 211 Marriage 10 September 1885. Episcopal Church, St Andrews. To Charles Walker Cathcart (1853–1932): born 42 Drummond Place, Edin- burgh; surgeon; son of James Cathcart, a wine merchant in Leith. At time of marriage: Charles resident at 44 Melville Street, Edinburgh; Mary resident at 4 Alexander Place, St Andrews. Witnesses: James Porter (Mary’s uncle, officiating minister), R. W. Irvine, J. G. Tait (Mary’s brother). Charles Cathcart was a Senior Lecturer in Clinical Surgery (1913–1917) at the University of Edinburgh; previously Lecturer in Anatomy at Surgeons’ Hall, Edinburgh (1881–1885). He was a Consultant Surgeon at the Edinburgh Royal Infirmary and Conservator of the Museum of the Royal College of Surgeons in Edinburgh (1887–1900). He was also a Lieutenant-Colonel in the Royal Army Medical Corps (Territorial Force), at the 2nd Scottish General Hospital, at Edinburgh War Hospital and at Edenhall Hostel for Limbless Sailors and Soldiers (1914–1919).483 He wrote a number of medical books, including A Surgical Handbook, jointly with Professor F. M. Caird, which ran to twenty editions.484 Charles died at 12 Randolph Crescent, Edinburgh, aged 78. His usual residence: 13 Newbattle Terrace, Edinburgh. Death 22 November 1946. Buxton, Derbyshire. Usual residence 16a Trinity Church- Square, London. Widow. It seems that Mary moved to London following her husband’s death. Residences—indicated by the 10 year census. 1871 See P.G.T. 1881 The School House, St Andrews. St Andrews School for Girls, St Andrews, Fife. Mary age 16 [scholar], living as a boarder. 1891 8 Randolph Crescent, Edinburgh. Mary age 26, living with: husband Charles (38) [surgeon]; and 2 servants. 1901 8 Randolph Crescent, Edinburgh. Mary age 36 [living by own means], living with: children Theodora (8) [scholar], Francis (6) [scholar], Helen (3); brother William (35) [civil engineer]; and 3 servants. Mary’s husband Charles does not appear on the Scottish or English 1901 Censuses. 1911 All the Cathcarts disappear from the Scottish and English 1911 Censuses, except for Theodora: she is a student, aged 18, a visitor at St Margaret’s School in Polmont, East Stirling, perhaps on teacher training. In 1913, however, the Cathcarts reappear; resident at 44 Melville Street, Edinburgh. William Archer Porter Tait Birth 25 March 1866. 6 Greenhill Gardens, Edinburgh. 483 [203] 484 [204] 212 Marriage Single. Death 23 June 1929. Age 63. 17 Greenhill Gardens, Edinburgh. Cause: Parkinson’s disease 5 years, cerebral haemorrhage. Death certificate signed: J. G. Tait (brother, residing at 38 George Square). Single. Occupation: Civil Engineer, retired. National Probate Calendar has William resident as 38 George Square, Edinburgh and 72 George Street, Edinburgh. The Edinburgh Academy Register has William resident (in 1914) at 38 George Square, Edinburgh.485 Place of interment: Tait family grave, at St John’s, Edinburgh. Residences—indicated by the 10 year census. 1871–1891 See P.G.T. 1901 8, Randolph Crescent, Edinburgh. See Mary Guthrie Cathcart (Tait). 1911 38 George Square, Edinburgh. See Margaret Archer Tait (Porter). Education. 1875–1881 The Edinburgh Academy. Mr. Carmichael’s class.486 1885 The University of Edinburgh. BSc Engineering, D.Sc.487 Occupations: civil engineer. 1887–1890 Training (Engineer): Sir J. Wolfe Barry and H. M. Brunel (son of I. K. Brunel).488 1891–1894 Assistant Engineer, Glasgow Central Railway.489 1894–?? Partner in J. & A. Leslie & Reid. Involved in the building of the Talla Reservoir, which was opened in 1899.490 485 [4, p360] 486[Ibid.] 487 [29] 488[Ibid.] 489[Ibid.] 490[Ibid.] 213 Engineer, Edinburgh and District Water Trust and several other authorities; Manager of Edinburgh Royal Infirmary.491 Memberships, fellowships, prizes, honours. 1891 Institute of Civil Engineers. Awarded Miller Prize (1891) and Telford Premium (1906) for contribution of professional papers. William was a member of the Inst. C.E.492 1898 Royal Society of Edinburgh. Elected Fellow (02/05/1898), proposed by: Sir W. Thomson, Charles Alan Stevenson, George Barclay, John Stugreon MacKay. Served as: Councillor (1914–1917, 1918–1921); Vice-President (1921–1924).493 Sporting achievements. Played for: London Scottish F.C.; West of Scotland F.C. and A.F.C. (The Academical Football Club at the Edinburgh Academy).494 Frederick “Freddie” Guthrie Tait Birth 11 January 1870. 17 Drummond Place, Edinburgh. Marriage Single. Death 7 February 1900. Age 30. Koodoosberg Drift. Killed in action, during reconnaissance under General Macdonald. He was shot through the heart while making an advance of 50 yards on the Boer position. His body was buried on the banks of the Riet River. P.G.T. received the news of Freddie’s death by telegram on 14 February; it affected him profoundly.495 There is a monument in memory of Freddie at the Tait family grave at St John’s, Edinburgh. Residences—indicated by the 10 year census. 1871–1881 See P.G.T. 1891 Aldershot Hampshire. Freddie age 21, living in barracks, 2nd Battalion Infantry. 491 [4, p360] 492[Ibid.] 493 [29] 494 [4, p360] 495 [53, pp220–226] 214 Education. 1879–1883 The Edinburgh Academy. Mr. Carmichael’s class.496 May 1883– Dec. 1886 Sedberg School. A boarding school for boys in Cumbria. Sent on the advice of Bishop Sandford, Rector of St John’s, Edinburgh and a family friend. Freddie excelled at mathematics and French.497 Military career.498 Sept. 1889 Enters Sandhurst. Promoted to Corporal at Christmas 1889. Passes out (July 1890) with special honours in riding and military administration. Oct. 1890 Gazetted to 2nd Battalion of the Leinster Regiment, the 109th foot. Spends time in Folkstone and Aldershot. 1890 Promoted to 2nd Lieutenant. Promoted to Lieutenant (1893). June 1894 Joins 2nd Battalion of the 42nd Royal Highlanders (The Black Watch). Sta- tioned at Edinburgh Castle for two years, until autumn 1896 when sent to Ballater (Scotland) as her Majesty’s Guard. A period at York follows, until the end of 1897. Aug. 1898 Appointed as Inspector for the Scottish District, following brief spell in Colch- ester, Eastern District. Resident in Glasgow from October, with weekends spent in and around Edinburgh. 18 Oct. 1899 Resignation from staff appointment sanctioned: Freddie had applied to rejoin the Black Watch in case of active service. On declaration of War in South Africa, he voyages to South Africa on ‘The Orient’ (24 Oct.). 11 Dec. 1899 Wounded in action: hit in thigh. Recuperated at Wynberg Hospital, then Claremont Sanatorium. 7 Feb. 1900 Killed in action: shot in the heart on first day of action following recuperation. Sporting associations and achievements. 496 [4, p379] 497 [53, p35] 498Source: [53, various places]. 215 Played rugby for A.F.C. (The Academical Football Club at the Edinburgh Academy).499 Colours for rugby football (1889–1890): Sandhurst XV, ‘one of the best forwards in the team’.500 Introduced golf to Sandhurst: ‘Before Tait’s time golf had been unknown at Sandhurst, so he laid out a short course for his fellow cadets, and instructed them in the ways and mysteries of the sport.’501 Member of the Royal and Ancient Club at St Andrews (joined Spring 1890).502 Scotland’s amateur golf champion (1896, 1898); runner-up (1899).503 Alexander Guthrie Tait Birth 2 February 1873. 17 Drummond Place, Edinburgh. Marriage Single. Death 11 May 1934. Age 61. Highfield Dreghorn Loan Colinton, Midlothian. Usual residence: Ladebraes Villa, St Andrews. Cause: Carcinoma (tumour) of the rectum 1.5 yrs, cachexia (wasting syndrome) 6 months. Death certificate signed: J. G. Tait (brother, resident at 38 George Square). Single. Occupation: Glass merchant, retired. Place of interment: Tait family grave, at St John’s, Edinburgh. Residences—indicated by the 10 year census. 1881–1901 See P.G.T. 1911 100 Upper Parliament Street, Liverpool. A.G.T. age 37 [glass merchant, a boarder]. The Edinburgh Academy Register has Alex at the same residence in 1914.504 Education. 1883–1885 The Edinburgh Academy. Mr. Shipton’s class, later Mr. McBean’s class.505 1891 The University of Edinburgh. M.A.506 499 [4, p379] 500 [53, p59] 501[Ibid.] 502 [53, p60] 503 [4, p379] 504 [4, p397] 505[Ibid.] 506[Ibid.] 216 Occupations: glass merchant. Grandchildren of P.G.T. and Margaret Archer Porter (7) By John Guthrie (3) Margaret Tait (1907–1996); Patrick Tait (1909–1935); Peter Guthrie Tait (1914–1984). All three children were born in Bangalore, India. Margaret was an aca- demic who lived in St Andrews. She died aged 88. Patrick was in the Mysore service. He had a B.A. from Cambridge. He died of tuberculosis, aged 26, at the family home, 38 George Square, Edinburgh. His father John signed the death certificate. Patrick’s body was interred in the family grave at St John’s, Edinburgh. Peter Guthrie was a District Commissioner in the Colonial Civil Service. He died in Selkirk, aged 70. He married Marjorie Agnes Hope Gillespie in Edinburgh on 19 February 1938. The couple had seven children. By Mary Guthrie (4) Theodora Cathcart (1892–1951); Francis John Cathcart (1894–1916); Helen M. Cathcart (1897–1992); Louisa Marion Cathcart (1902–1988). The children were born at 8 Randolph Crescent, Edinburgh. Theodora was a teacher. She never married. She died in London, aged 59. Francis was a 2nd Lieutenant in the Royal Field Artillery. He was educated at Loretto school in Edinburgh and studied engineering at the University of Edinburgh. He served at Gallipoli and at Mesopotamia during the advance to Bagdad (1916). He was killed in action on 3 June 1916, aged 21. He never married. Helen married Archibald Mcneilage, a mechanical engineer, in Edinburgh on 17 June 1924. She died in Edinburgh, aged 95. Louisa worked as a private secretary. She married Christopher F. Millett, an advertising artist from London, on 23 December 1930. Harry Seymour Reid, Bishop of Edinburgh, officiated at the marriage, which took place at Christ Church in Morningside, Edinburgh. Louisa died in Morningside, aged 85. 217 F ig ur e A .1 : A nc es to rs of P . G . T ai t (1 83 1– 19 01 ). 218 F ig ur e A .2 : D es ce nd an ts of P . G . T ai t (1 83 1– 19 01 ). 219 F ig ur e A .3 : T ai t [n o. 3] an d W ill ia m St ee le [n o. 4] re si de nt at P et er ho us e C ol le ge , C am br id ge , 18 51 . F ro m th e E ng la nd C en su s of 18 51 . [2 05 ] T ai t’ s bi rt h pl ac e is gi ve n as L am b et h, Su rr ey , E ng la nd , w hi ch is in co rr ec t. A s a p en si on er , T ai t w ou ld ha ve ha d to pa y fo r hi s ow n fo od an d tu it io n. T ai t w as ad m it te d to P et er ho us e C ol le ge on 21 Ju ne 18 48 ; he m at ri cu la te d in th e M ic ha el m as te rm th at ye ar an d in 18 52 ca m e ou t as Se ni or W ra ng le r an d F ir st Sm it h’ s P ri ze m an . 220 F ig ur e A .4 : O th er s re si de nt w it h T ai t at P et er ho us e: Fr ed er ic k Fu lle r [n o. 3] , W ill ia m A . P or te r [n o. 5] an d E dw ar d J. R ou th [n o. 12 ]. Fr om th e E ng la nd C en su s of 18 51 . [2 05 ] 221 Figure A.5: Tripos examination results, Cambridge, 1852. Tait is Senior Wran- gler. Sourced from Tait’s scrapbook. Reproduced with the kind permission of the J.C.M. Foundation. Tait had been coached by the private mathematics tu- tor, William Hopkins (1793–1866) who is referred to by Craik [14, p107] as the “wrangler maker”. Hopkins also tutored Maxwell, Thomson, Stokes and Cayley. 222 Figure A.6: Tait (first on the left) and William Steele (1831–1855) (third on left); graduates at Cambridge, 1852. [18, pfacing 11] Original from Tait’s scrapbook. While at Cambridge, Tait developed a close friendship with William Steele. Af- ter graduation, both became Fellows of Peterhouse and they began collaborating on a book, Dynamics of a Particle which was published in 1856. Sadly, Steele died with barely a few chapters written so Tait finished the work. 223 Figure A.7: Notification of Tait’s honorary degree from the University of Edin- burgh. Sourced from Tait’s scrapbook. Reproduced with the kind permission of the J.C.M. Foundation. The ceremony was scheduled for 27 July 1901. Tait died just a few weeks prior on 4 July. 224 B TAIT’S POEM ON THE FRANCO–PRUSSIAN WAR (1870) This is an unpublished poem by Tait which has been sourced from his pocket note- book. Written in October 1870, the untitled poem finds its context in the height of the Franco–Prussian War (July 1870 to May 1871). Tait writes that he has sent the poem on to ‘Russel’. This is, presumably, Alexander Russel (Russell) (1814–1876), editor of The Scotsman newspaper: his identity is suggested by evidence from Tait’s scrapbook.507 A search of the Scotsman Digital Archive (1817–1950) indicates that Tait’s poem was not published in the newspaper.508 Sent to Russel 11/11/70. Octr 31/70. A Who is this G fat but quick?509 The hound that crouched ’neath B’s stick510 What time the plunderers of the Dane Quarrelled about their shameless gain. Beery & fat and scant of wind 507Tait refers to Russel and The Scotsman in an annotation accompanying the original draft of a poem by William Robertson Smith in the scrapbook. Smith’s poem is adapted from John Milton’s poem, ‘On Shakespeare’ (1630). In his note, Tait writes that Russel had agreed to publish Smith’s poem in The Scotsman. 508Scotsman Digital Archive (1817–1950) is a digital resource provided by the National Library of Scotland . 509G : Tait’s shorthand for ‘German’. 510B : Otto von Bismarck (1815–1898); Prusso–German diplomat and statesman; chief architect of the German Empire; held responsible for having provoked the Franco–Prussian War. 225 He puffs along the battle plain For is not B’s “stick” behind? Who’s dead to honor, lives to pain. This is your G, fat yet quick Driven to war by B’s stick. — What is this G’s lawful prize? Whate’er finds favor in his eyes. The accursed one who hounds him on Knows well his self-respect is gone. He fears his reckless discontent, And so in devilish mood Delighted sees it find a vent In rapine, lust and blood. That is this German’s lawful prize Whate’er finds favor in his eyes. B What does his master hope to gain? That does not seem so very plain. To inscribe in each historic tome Another rush of Goths to Rome?511 Seeks he the immortality Of him who fired Diana’s shrine,512 Or with the ambition cursed is he With Caliph Omar’s fame to shine?513 511Goths: a Teutonic people originating from South Sweden (Gotland) who encroached on the Roman Empire during the fourth century. They split into two divisions: the Visigoths and the Ostrogoths. Under Alaric, the Visigoths devastated Greece and sacked Rome in 410 A.D. 512Diana: the Italian goddess of the woods, women, childbirth and the moon. 513Caliph Omar (581–644): adviser to Mohammed; succeeded Abu Bakr as 2nd Caliph. During his reign, Islam became an imperial power. He died at the hands of a foreign slave. 226 What then does B hope to gain? I give it up—my quest is vain— — But what then will this G gain? The answer is both full & plain— Contempt from every honest man The thief’s reward, the murderer’s ban, When Europe’s slow but sure police Are set upon his bloody track And all shall feel that lasting peace Requires he should be beaten back. These will the rabid Germain gain Fettered at length in Europe’s chain. C But are not Gs civilized? Is justice not among them prized? These statements which have long been made But yesterday were not gainsaid— But he who runneth now may read Unlikely as it may seem This quiet content, devoid of greed Is but an empty dream. For Germans are not civilised Say rather they are brutalized. — What should the wretched Fman feel,514 Downtrodden by the G’s heel? Glad that the veil is drawn aside Which did so long the monster hide That lust of Blood & Rapine rife 514Fman: Tait’s shorthand for ‘Frenchman’. 227 Are plainly now revealed Which secretly preparing strife Were but by Tartuffe’s cant concealed.515 This satisfaction he may feel Though crushed beneath that brutal heel. D Say what shall be the wretches fate Who finds this monster at his gate? Dares he to act the part of man And shoot the murderer if he can? Dares she her honor to defend Who[se] face has pleased some Gman boor,516 Or dare the starving peasant tend His little stock, his winter’s store? The gallows is the wretches fate Behold this monster at his gate. — Death and Dishonour, that is all. In vain for mercy do ye call. Hell is abroad—his hounds obscene Are loosed on every village green— The fairest spots on earth that smiled Are soiled by murderer’s tread The grey-beard and the sucking child Heighten the piles of dead. Pity has fled, & right is wrong, Nature aghast—Oh Lord how long? E But, Frenchman, though thou feel the curse, 515Tartuffe: a comedy by Molie´re. 516Gman: a variation on Tait’s shorthand for ‘German’. 228 Rejoice—thy foeman’s case is worse. When from his hordes thy land is free Thou shalt enjoy thy liberty— He, crushed beneath an iron hand,517 With none from “stick” to save, May yell in praise of Vaterland518 But is not less a Slave! Hurrah—each mangy skulking hound In Bismark’s leash is firmly bound. — All honor, Bismark, to thy stick519 Which makes thy beery slaves so quick— But act with caution—have a care— And dread the vigor of despair! Even Germans may at last feel shame The “stick” so long to bear— Syne play to thee this pleasant game For “turn about” is fair. And Frenchmen will pronounce it “chic” When Bismark’s slaves give him the “stick.” 517Iron hand : Bismarck was known as “The Iron Chancellor”. 518Vaterland : German homeland or fatherland. 519Bismark : a variation in the spelling of ‘Bismarck’. 229 C SCHOOLDAYS AT THE EDINBURGH ACADEMY Included in this Appendix is the following: — the syllabus in 1846–1847 for the 6th and 7th classes; — a list of prizes and medals won by Tait at the Academy; — a reproduction of the results of the 1846 Academical Club Prize competition; — and a transcription of the mathematics examination paper from the 1847 Aca- demical Club Prize competition. The Academical Club Prizes were instituted in 1831 by the “Accie Club”—an as- sociation of Academy old-boys founded in 1828.520 Initially, the prizes were awarded for Latin verses but in 1846, under the influence of the Rector, John Williams, they came to be awarded for the best performance in voluntary written examinations across all subjects: the competition was open to classes 5–7 and the examinations took place in school, over a period of three days.521 In the 1846 competition, Tait beat Maxwell in mathematics and overall; however, in 1847 the situation was re- versed as Maxwell placed first in mathematics that year and Tait came second. The 1847 mathematics exam paper was attempted by both Tait and Maxwell. It constituted the mathematical component of the Academical Club Prize competi- tion. I am grateful to Andrew McMillan, Honorary Archivist at the Academy, for his kind permission to publish the paper.522 In Section 4 on geometry, the questions are based on propositions found in Euclid’s Elements. They require either a proof or a ruler and compass construction. 520 [5, p162] 521 [5, pp162–163] 522The examination paper was published in the Academy’s Prize List for 1847 [206, pp15–17] which is preserved in the archives of the Edinburgh Academy. 230 6th class 7th class English Shakespeare, Campbell, Irv- ing’s Elements of Composition. Weekly Themes. (4 hrs) Composition and Elements of Physical Science. (2 hrs) Latin Portions of Horace, Virgil and Livy. Exercises, Verse and Prose. Elements of Physical Sci- ence. Antiquities. Portions of Cicero, Tacitus, Ju- venal, Horace, Catullus. Exer- cises, Prose and Verse. Greek Portions of New Testament, Homer, and Euripides, Sand- ford’s Exercises. Greek Prose Composition. Ancient Geogra- phy. Evidences of Christianity. (18 34 hrs, incl. Latin) Portions of New Testa- ment, Homer, Sophocles and Xenophon, Dunbar’s Exercises, Antiquities. Exercises, Prose and Verse. (16 12 hrs, incl. Latin) Arithmetic/ Mathematics Arithmetic: Cube Root. Geom- etry, four books. Algebra, to Quadratic Equations. (5 hrs) Mathematics : Six Books of Eu- clid, Trigonometry, Mensura- tion, Algebra, Quadratic Equa- tions, &c. (6 hrs) French Grammar, Phrases, Charles XII. (3 hrs) Grammar, Portions of Voltaire and Racine. (3 hrs) German Grammar, Extracts, Prose and Verse. (2 hrs) Table C.1: Syllabus in 1846–1847 for the 6th and 7th classes. Information abstracted from the Directors’ Report for 1847 [59, p7], preserved in the archives of the Edinburgh Academy. Tait was in the 6th class in 1846, Maxwell was in the 7th. There is a heavy emphasis on the classics because the Edinburgh Academy was founded in order to fulfil a need for a school in Edinburgh which could offer a high standard of classical education (higher than that offered by the Scottish seminaries). Edinburgh’s Royal High School provided a classical education but the founders of the Edinburgh Academy felt that greater provision was needed for the teaching of Greek, to compete with England’s public schools. 231 Prizes for scholarship and prizes for particular merits. Geits class 1841–1842 Dux, Best English Reader, 2nd Best English Scholar. Second class 1842–1843 Dux, 2nd Best English Scholar. Third class 1843–1844 Dux, 4th Best Arithmetician. Fourth class 1844–1845 Dux, Best Arithmetician, Best Latin Verse (‘Aeneas in tumulo cereris’). Fifth class 1845–1846 Dux, Silver Medal for Geometry, 2nd Best English Scholar, Best French Scholar. Sixth class 1846–1847 Dux, Mitchell Medal for Best Mathematician, 2nd Best French Scholar, 2nd Best Examination Papers on Physical Science, Best Latin Verse (‘Sit felix proventibus annus: Rusticus loquitur’). Academical Club Prizes. 1846 Tait’s results : 3rd overall; 8th in Latin; 1st in Mathematics; 5th in English and French; equal 3rd in History, Geography, and Scripture Biography. J. C. Maxwell’s results : equal 6th overall; 3rd in Mathe- matics; 6th in English and French; equal 3rd in History, Geography, and Scripture Biography. 1847 Tait’s results : 3rd overall; equal 3rd in Latin; equal 4th in English; 2nd in Mathematics. J. C. Maxwell’s results : 2nd overall; 2nd in Latin; equal 4th in Greek; 1st in English; 1st in Mathematics. (First overall in the 1847 competition was Lewis Campbell, later Professor of Greek at St Andrews and co-author of Life of James Clerk Maxwell (1882).) Table C.2: Prizes and medals won by Tait at the Edinburgh Academy (1841– 1847). Information collated from the prize lists for 1842–1847, issued on Exhi- bition Day which was the annual prize-giving event in July. Sourced from the archives of the Edinburgh Academy. 232 Figure C.1: Academical Club Prize results, 1846. Tait places third overall but is top in mathematics. Sourced from Tait’s scrapbook. Reproduced with the kind permission of the J.C.M. Foundation. 233 Mathematics Examination Paper, the Academical Club Prize, 1847 FOURTH DEPARTMENT. MATHEMATICAL PAPER. SECT. I.—ARITHMETIC. 1. What fraction of a pound is 79 of a guinea? 2. When .365 of a day have elapsed, what is the exact time? 3. Find to 5 places of decimals what fraction of the year had elapsed at 10 a.m. on the 1st of June. 4. Find the square roots of 11675889, 82369, 0.16, and .000625. 5. Find the fourth root of 28561. 6. Multiply 34.643 by 8.1068. SECT. II.—MENSURATION. 1. Find the area of a quadrilateral whose diagonal is 97 feet, and the perpendic- ulars 17 and 23 yards. 2. Find, approximately, in square feet, the area of a regular hexagon, the length of a side being 10 yards. 3. Find the area of the whole figure included in the construction of Euclid I. 47; if the base of the right angled triangle measures 20 feet, and the perpendicular from the vertex to the hypotenuse 8 feet. 234 SECT. III.—ALGEBRA. 1. Prove that am × an = am+n; also that a m an = a m−n. 2. Prove that if 2 be divided into any two parts, the difference of their squares is always equal to twice the difference of the parts themselves. 3. Multiply a+x+x2 +x3 +x4 by a−x; collecting the coefficients of like powers of x. 4. Find the value of (a− b)(a+ b− c) + (b− c)(b+ c− a) + (c− a)(a+ c− b). 5. Find the greatest common measure of 18x3 − 75x2 + 83x − 20, and 36x3 − 96x2 + 13x+ 5. 6. Find the product of the fractions ax(a−x)2 and a2−x2 ab . 7. Find the square root of 4x4+4x3−11x2−6x+9, and 4x4−12x3+11x2−3x+ 14 . 8. Solve the equation 3x+714 − 2x−7 21 + 2 3 4 = x−4 4 . 9. Divide 21 into two parts, such that 10 times one of them may exceed 9 times the other by 1. 10. Shew that the difference between the sum of the cubes of two quantities, and the cube of their sum is equal to three times their product multiplied by their sum. 11. Prove algebraically that the square of half a given straight line is greater than the rectangle contained by any two un-equal parts into which the line can be divided. 12. Solve the equations (1) x+ y = a x2 − y2 = b    (2) 3x− 4y = 5 5x− 8y = 7    235 SECT. IV.—GEOMETRY. 1. Define a point,—a plane superficies,—a circle,—parallel straight lines.523 2. The greater side of every triangle has the greater angle opposite to it. 3. If two triangles have two sides of one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them of the other; the base of that which has the greater angle shall be greater than the base of the other. 4. If the square described upon one of the sides of a triangle be equal to the squares described upon the other two sides of it; the angle contained by these two sides is a right angle. 5. Divide a straight line into two parts, such that the rectangle contained by the whole and one of the parts, shall be equal to the square of the other part. 6. If two sides of a quadrilateral be parallel; the triangle contained by either of the other sides, and the straight lines drawn from its extremities to the bisection of the opposite side, is half the quadrilateral. 7. Prove that the opposite angles of any quadrilateral figure, inscribed in a circle, are together equal to the two right angles. State and prove the converse of this. What mode of demonstration is usually adopted by Euclid in proving converse propositions? 8. On a given straight line describe a square, and, on the side opposite to the given line, describe equilateral triangles lying in opposite directions; circles described through the extremities of the given line, and through the vertices of these triangles are equal. 523superficies: in geometry, ‘a continuous extent having only two dimensions (length and breadth, without thickness); an entity such as forms the boundary or one of the boundaries of a solid object, or separates two adjacent portions of space; a surface’. [207] 236 9. If a straight line bisecting the vertical angle of a triangle also bisect the base, prove that the triangle is isosceles. 10. In an obtuse angled triangle, if perpendiculars be drawn from the points bi- secting the sides, they will all pass through the same point. May the same thing be proved in an acute-angled triangle? and if so, what difference occurs in the situation of the point of intersection? If the triangle were inscribed in a circle, prove that the point of intersection is the centre. 11. If the diameter of a circle be one of the equal sides of an isosceles triangle, the base will be bisected by the circumference. 12. Given the diagonals and perpendicular breadth of a parallelogram, to construct it. 13. Divide a circle into two parts, such that the angle contained in one segment shall be equal to twice the angle contained in the other. 14. If an equilateral triangle be described about a circle, the line joining any angle and the middle of the opposite side has its third part intercepted between the angle and the circumference. 15. Given one angle, the side opposite, and the sum of the other two sides, to construct the triangle. 237 D TRANSCRIPTION OF TAIT’S NOTES FROM TERROT’S PAPER Images to accompany the text—comprising photographic extracts of the Tait–Maxwell school-book—appear in Figure D.1 (pages 245–253). The few editorial corrections that I have made to the transcription are recorded in Table D.1 (page 254). On the Imaginary roots of Negative Quantities. By the Right Reverend Bishop Terrot. 1847 1. √ −1 is called impossible or imaginary ∵ no ordinary algebraic quantity which must be either + or − can give when squared a negative result. Considering however the common application of Algebra to Geometry we easily see, that the assumption that every line must be either + or − is inconsistent with the possibility of drawing a line in any direction. +1×a means a line whose length is a drawn in one direction, −1× a means the same length of line but drawn in a different direction, and to say that a line of the length of a cannot be drawn in any other direction than one of these is absurd. √ −1 ∴ is not impossible any more than − or +1 and shows only the direction of the line to which it is affixed. 2. If from C [Fig 1] we draw any number of lines such that they shall be in continued proportion and make at the same time ∠ACA1 = A1CA2 = A2CA3 &c., then calling CA = 1, CA1 = a, CA2 = a2 or the lines are in this series a0, a1, a2, a3 &c., while the angles which they make with the line CA are 0, ϑ, 2ϑ, 3ϑ &c., being the angle ACA1× exponent of that radius vector (CAa for example) from which to CA they are measured. Thus the line whose angle of inclination is on nϑ has its length = an & vice versaˆ. 238 3. If we now assume the several lines CA, CA1, CA2, &c., [Fig 2] all equal or radii of a circle the case will not be altered. Let n be a divisor of 2rpi or let ϑ = 2rpin . Thus the Radius an = a 2rpi ϑ is the same in length & position as CA ∴ a1 = 1 1 n = 1 ϑ 2rpi . We know from ordinary Algebraical principles that the several nth roots of unity may be expressed by the series a, a2, a3, &c. It therefore follows that we may take the successive Radii of a circle at equal angles for the several roots of unity & conversely. If R be the numerical length of radius that radius inclined to the first at ∠ϑ is = R × 1 ϑ 2rpi . We ∴ call 1 ϑ 2rpi the coefficient of direction because it refers only to the direction, never to the length of a line. Thus, a× 1+ √ −3 2 is a line = a simply. 4. Let us next suppose n = 2, AB will be a diameter & if CA = 1, CB = −1. But a2 = 1 ∴ a = ±1. But the radii being a, a2, a must evidently be = −1 & a2 = +1. Next let n = 4, CA, CD, CB, CE are the 4 roots of the equation a4 − 1 = 0. But the roots are ±1 & ± √ −1. Here CA & CB are symbolized by +1 & −1 respectively ∴ CD & CE must be symbolized by + √ −1 & − √ −1 respectively, it being however quite optional which direction from C we account positive or negative either in the horizontal or perpendicular lines. 5. It appears from the foregoing Props. that if a line is symbolised by = a · 1 ϑ 2rpi we know both its length & direction. a · 1 ϑ 2rpi ∴ represents the actual transference of the point in space by moving from A to C [Fig 3]. But it is also clear that its actual transference in space though not its distance travelled would be the same did it move from A to B & then from B to C. Thus ∴ (AC× its coefficient of direction) = (AB× its coefficient of direction) + (BC× its coefficient of direction). Therefore also the sum of any two lines making an angle with each other is = the diagonal of their parallelogram completed. Even in this startling form it is only the general assertion of a proposition particular cases of which we admit when we say AB1 +B1C = AC or that AC + CB1 = AB1. 1. As examples to elucidate this let ABC (Fig 4) be an isosceles right angled triangle described on the radius AD. If we call AB the radius or Hypotenuse a each of the sides will be in length a√ 2 & AB is symbolized by a×1 45 360 = a×1 1 8 = a× 1+ √ −1√ 2 . 239 But AC = a√ 2 . CB being perpendicular to original position is = a√ 2 × √ −1 (Prop. 4) ∴ AC + CB = a× [ 1√ 2 + √ −1√ 2 ] = a× 1+ √ −1√ 2 = AB. 2. Let BAC = 60◦, BCA = 90◦, then AB in length & direction is a · 1 60 360 = a · 1 1 6 = a · 1+ √ −3 2 , AC = a 2 , CB in length = a · √ 3 2 ∴ in length & direction jointly = a · √ 3 √ −1 2 = a · √ −3 2 ∴ AC + CB = a 2 + a · √ −3 2 = a · 1+ √ −3 2 = AB. 3. Let the triangle (Fig 5) be Equilateral & let AB be the original position. Let AB = a, AC = a · 1 1 6 , CB = a · 1 −1 6 ∴ AC +CB = a · [ 1 1 6 + 1 −1 6 ] = a · [ 1 1 6 + 1 1 1 6 ] = a · [ 1 1 3 +1 1 1 6 ] = a · [ −1+ √ −3 2 + 1 ] × 2 1+ √ −3 = a · [ 1+ √ −3 2 + 2 1+ √ −3 ] = a = AB 6. In the foregoing Props. & Examples it has been taken for granted that we know not only the several nth roots of unity but also their proper order; that is the order in which as coefficients they express the radii drawn to the extremities of the arcs ϑ, 2ϑ, 3ϑ, &c., with the original radius. But when we determine the roots of xn − 1 = 0 we obtain them in no fixed order. To discover this order we must observe that two roots are always of the form a± √ −b comparing which with (Fig 6) a is evidently the part symbolical of the cosine, + √ −b that of the sine because it is affected by √ −1 and is ∴ perpendicular to original radius. Thus ∴ in a ± √ −b, + refers to radii in the upper semicircle & − to those in the under; and the two radii whose symbols differ only in the sign of √ −b are at equal angles to the original radius on opposite sides of it. ∴ the root in which a is greatest is nearest to the original radius. Thus the roots of n6 − 1 arranged properly are 1, 1+ √ −3 2 , −1+ √ −3 2 ,−1, −1− √ −3 2 , 1− √ −3 2 symbolizing the radii drawn respectively to the ends of the arcs 0◦ or 360◦, 60◦, 120◦, 180◦, 240◦, 300◦. For if +1 be first −1 having no sinal part must be in the middle. Next 1+ √ −3 2 & −1+ √ −3 2 must be in the upper half of the circle and 1+ √ −3 2 must come first because its cosine is in CA. And so with the rest. 7. It appears from Props. 4, 5 that the radius drawn to the end of an arc ϑ is = 1 ϑ 2rpi and this again by a± √ −b where a is what is trigonometrically called the cosine & √ b the sine of ϑ. Now (Fig 6) let ∠ACA1 = ϑ, ∠ACA2 = 2ϑ, &c., ∠ACAp = pϑ, 240 then CA1 = CD + √ −1 ·DA1 = cosϑ+ √ −1 · sinϑ, CAp = cos pϑ+ √ −1 · sin pϑ. But by prop. 2, CAp = CA1 p = ( cosϑ+ √ −1 · sinϑ )p ∴ ( cosϑ+ √ −1 · sinϑ )p = cos pϑ+ √ −1 sin pϑ, which is Demoivre’s Theorem. cor. If pϑ = 2pi, cos pϑ+ √ −1 · sin pϑ = 1. Hence ( cosϑ+ √ −1 · sinϑ ) , ( cos 2ϑ+ √ −1 · sin 2ϑ ) &c., represent the several pth roots of unity. If we arrange the angles, instead of ϑ, 2ϑ, 3ϑ &c., in pairs thus ϑ & p− 1 · ϑ, 2ϑ & p− 2 · ϑ &c., the several expressions for x − the several pth roots of unity or the simple factors of xp−1 = 0 taken in pairs corresponding with the above will be ( x−cosϑ− √ −1·sinϑ ) & ( x − cos p− 1ϑ − √ −1 · sin p− 1ϑ ) which last is = ( x − cos pϑ− ϑ − √ −1 · sin pϑ− ϑ ) = ( x− cos 2pi − ϑ− √ −1 · sin 2pi − ϑ ) = ( x− cosϑ+ √ −1 · sinϑ ) . In the same way the next pair must be ( x − cos 2ϑ + √ −1 · sin 2ϑ ) & ( x − cos 2ϑ − √ −1 · sin 2ϑ ) . Multiplying these together for the quadratic factors of xp − 1, we obtain when p is even xp − 1 = (x2 − 1)(x2 − 2x cosϑ + 1) · (x2 − 2x cos 2ϑ + 1) to p 2 terms. But when p is odd x p − 1 = (x − 1)(x2 − 2x cosϑ + 1) &c., to p+12 terms, where ϑ it may be observed is = 2pip . 8. sinA+B = sinA · cosB + cosA · sinB cosA+B = cosA · cosB − sinA · sinB Let arc AB (Fig 7) = A, BD2 & AD1 each = B. Then by Prop. 3, CB = r · 1 A 2pi , CD1 = r · 1 B 2pi , CD2 = r · 1 A+B 2pi ∴ CD2 = r · 1 A 2pi · 1 B 2pi . But by Prop. 7, 1 A 2pi = cosA + √ −1 · sinA, 1 B 2pi = cosB + √ −1 · sinB ∴ 1 A+B 2pi = cosA × cosB − sinA × sinB+ √ −1 ( sinA · cosB+ cosA · sinB ) , but 1 A+B 2pi = cosA+ B+ √ −1 sinA+ B. Equating then the sinal & cosinal parts of these, we have, cosA·cosB−sinA·sinB = cosA+ B, sinA · cosB + cosA · sinB = sinA+ B. 241 Definition. It should be observed that in the following propositions a line expressed by letter simply as AB must be considered both as to length & direction while when in brackets thus (AB) its length alone is referred to. Thus (AB)1 ϑ 2pi = AB. 9. In any right angled triangle the sum of the squares of the sides is = square of hypotenuse. Let CA (Fig 6) = r, then CA1 = r · 1 ϑ 2pi , & CAn−1 = r · 1 −ϑ 2pi ∴ CA1×CAn−1 = r2×1 ϑ 2pi × 1 1 ϑ 2pi = r2. Also CA1 = (CD1)+ √ −1(D1A1), CAn−1 = (CD1)− √ −1(D1A1) for (D1A1) = (D1An−1) ∴ CA1×CAn−1 = (CD1)2 + (D1A1)2 which is ∴ = r2 = (CA)2 = (CA1)2 its equivalent in area. 10. Cotes’ Properties of the Circle. Let the circumference be divided into n equal parts and join OP1, OP2, OP3, &c., (Fig 8) and also join P1, P2, P3 with C any point in the Diameter. Then CP1 = OP1 − OC, CP2 = OP2 − OC &c., ∴ CP1 ·CP2 ·CP3 · · · CPn = Σn · (OA)n−Σn−1 · (OA)n−1 · · · ±OCn, where Σn is the product of all the coefficients of direction for OP1, OP2, &c., Σn−1 the sum of ∧ (the product sq? P.G. Tait) these coefficients taken n− 1 together & so on. But these coefficients are also the roots of the Equation xn − 1 = 0. Now the product of the roots of this Equation with their signs changed is −1 & Σn is = the product with their signs unchanged. Therefore if n be even Σn = −1 but if odd +1, and in either case Σn−1, Σn−2 &c., each = 0. Hence CP1 ·CP2 ·CP3 · · ·CPn = ±(OA)n± (OC)n; the upper signs to be used when n is even, the lower when odd. Here CP1, CP2 &c., consider the lines both as to length and direction, we must ∴ divide the first or multiply the second by the product of all their coefficients of direction. If n be even the several pairs as CP1, CPn−1 are evidently of the form (CP1) · 1 ϑ 2pi and (CPn−1)·1 −ϑ 2pi ∴ CP1×CPn−1 = (CP1)×(CPn−1) and this is true for every pair except CA = (CA)·+1 & CB = (CB)·−1 ∴ (CP1)·(CP2)···(CPn) = (−OAn+OCn)·−1 = OAn − OCn. But if n be odd the several pairs remain as before only no P falling on B, −1 is not a coefficient of direction ∴ (CP1) · (CP2)· &c., = OAn − OCn as before. 242 Cor.1. If C be on the opposite side of O from A, the other conditions remaining the same OC is negative. If n be even the deduction in the prop. remains unchanged. But if n be odd, (CP1) · (CP2) · · &c., = OAn +OCn. Here it may be remarked that when lines as OA are in the original direction, since the coefficient of direction in that case is unity it is immaterial whether we write OA or (OA). Ex. Let n = 3 & OC = 12 , then, (AC) = 3 2 , (CP1) = (CP2) = √ 3 2 ∴ (CA) · (CP1) · (CP2) = 32 · √ 3 2 · √ 3 2 = 9 8 = 1 + 1 8 = 1 3 + 12 3 = OA3 +OC3. Cor.2. If C be in OA produced the reasoning & result will be the same as in the prop., only, that now CA & CB being of the same affection −1 is not a divisor of the second member of the Equation, &, (CP1) · (CP2)· &c., = (OC)n − (OA)n. 11. If from A the extremity of the Diameter (Fig 8) the circumference be di- vided into n equal parts & if these several extremities be joined, then (AP1) · (AP2) · · · (APn−1) = nCAn−1. As in former prop. AP1 = CP1 − CA, AP2 = CP2 − CA & so on ∴ AP1 · AP2 · · ·APn−1 = CP1 − CA · CP2 − CA &c., to n− 1 factors = Rn−1 · {Sn−1 − Sn−2 · · · ± S1 ± 1} where S1, S2 &c., are the sum, sum of products two & two, &c., of all the values of 1 1 n except unity there being no line drawn from A to the circumference in the direction CA. S1, S2 &c., are ∴ the coef- ficients of the Equation x n−1 x−1 or of x n−1 + xn−2 + · · ·+ 1 = 0 with the signs changed for the products of odd numbers of roots, unchanged for even ones. If ∴ n− 1 be even Sn−1 = +1, Sn−2 = −1, & so on. If n− 1 be odd Sn−1 = −1, Sn−2 = +1 & so on. ∴ AP1 · AP2 · &c.,= Rn−1 × ±{1 + 1 + 1 to n terms} = ±nRn−1 according as n− 1 is even or odd. If n− 1 be even, AP1 · AP2· &c., = (AP1)(AP2)· &c., the several pairs of coefficients giving unity for their products. If n− 1 be odd, then the several pairs give as before their product unity but there remains the factor −AB which has for its coefficient −1. ∴ in either case (AP1)(AP2) &c., (APn−1) = nRn−1. 243 12. If by this method we undertake to prove that the angles at the base of an Isosceles triangle are = eachother we have (AC) = (BC) (Fig 5). But AC = (AC) · 1 A 2pi = (AC) · [a + √ −b], CB = AD = (AC) · 1 −B 2pi = (AC) · [a′ + √ −b′]. But AC + CB = AB. ∴ (AC) · (a + a′ + √ −b + √ −b′) = AB = a positive quantity consequently the sinal parts destroy one another or √ −b = − √ −b′ or b = −b′. Therefore the angles A & B have their sines of equal length but of different affections. The angles themselves ∴ being together less than pi are geometrically equal to each other. Cor. Much in the same way we might prove that in every triangle the greater side has the greater angle opposite to it & vice versaˆ that the greater angle has the greater side opposite to it. May 27th 1847. P. G. Tait. 244 Figure D.1: Photographic reproduction of Tait’s notes from Terrot’s paper. Sourced from the Tait–Maxwell school-book. Reproduced with the kind per- mission of the J.C.M. Foundation. 245 246 247 248 249 250 251 252 253 Editorial corrections made to the transcription of Tait’s notes from Terrot’s paper Reference Editorial correction §2 Tait has: ‘∠ACA1 = A1CA22 = A2CA3 &c., then calling CA = 1, CA1 = a, CA2 a2’. I cannot see a reason for the superscript 2 in CA22 so I have removed it. I have also added in an equals sign between CA2 and a2. Ex.1 Tait has: ‘∴ AC +CB = a× [ 1√ 2 + √ −1√ 2 ] = a = a× 1+ √ −1√ 2 = AB’. I have removed the = a as it appears only because there is a break in the line. §7 Tait has: ‘∴ ( cosϑ+ √ −1 · sinϑ )p = cos pϑ+ √ +1 sin pϑ’. This is incorrect: it should be −1 under the square-root sign on the right hand side of the equation. The error also appears in Terrot’s paper (p.350). Looking carefully at Tait’s original notes, it seems that Tait had spotted Terrot’s mistake and attempted to correct it. §9 Tait has: ‘which is ∴ = r2 = (CA2) = (CA1)2’. I have repositioned the superscript 2 outside the bracket (CA). §10 Tait has: ‘and this is true for every pair except CA = (CA) ·+1 & CB = (CB) · −1 ∴ (CP1) · (CP2) · · ·CPn = (−OAnn +OC n) · −1 = OAn−OCn’. I have added in the bracket around CPn which Terrot (p.353) and consequently Tait have omitted. Also there is no reason for Tait to have the subscript n in −OAnn so I have removed it. §11 Tait has: ‘if these several extremities be joined, then (AP1) · (AP2)(APn−1) = nCAn−1’. I have added in · · · on the left hand side between (AP2) and (APn−1) to show that all the APi up to i = n− 1 are being multiplied together. Table D.1: Editorial corrections made to Tait’s notes from Terrot’s paper. 254 E BUE´E’S 1806 PAPER AND GERGONNE’S TWO-DIMENSIONAL TABLE In 1806 Adrien-Quentin Bue´e’s paper, ‘Me´moire sur les quantite´s imaginaires’ [144] was published in French in the Philosophical Transactions of the Royal Society of London. It came to feature in many accounts of the history of the geometrical representation of complex numbers; no doubt because the significant and pioneering contributions of Wessel (1799) and Argand (1806) went unnoticed by the mathemat- ical community.524 The question of priority was raised by Sylvestre Franc¸ois Lacroix (1765–1843) in a note [210] published in volume IV of Gergonne’s Annales. Lacroix was writing to draw attention to Bue´e’s contribution following the re-discovery of Argand by Jacques Franc¸ais (1775–1833) in 1813. Argand responded by insisting that he had worked independently of Bue´e and without knowledge of his contribu- tion.525 Reviews of Bue´e’s paper were largely unfavourable. It was even feared that the publication had tainted the reputation of the Royal Society, in whose Transactions it had appeared.526 Nevertheless, the paper remains of historical interest for the fol- lowing reasons. (i) It was the first non-English-language publication in a nineteenth 524Fortunately, Argand’s contribution was brought to light in 1813, by Jacques Franc¸ais (1775– 1833) in his paper [208] published in volume IV of the Annales de mathe´matiques. For an account of the rediscovery of Argand see [209]: it contains an English translation of Argand’s 1806 pamphlet and of some of the subsequent related correspondence in the Annales between Franc¸ais, Gergonne (the editor) and Servois (another contributor); it also provides a good account of the historical context in the section entitled, ‘Notes’ (pages 85–135). 525See [211, p209] for Argand’s response to the issue of priority raised by Lacroix. 526Examples of reviews of Bue´e’s 1806 paper: [212], [213] and [214]. According to [215, pxix] this last anonymous review [214] was written by ‘Playfair’. I assume this is John Playfair (1748– 1819): Professor of Mathematics (1785–1805), then Natural Philosophy (1805–1819), at the University of Edinburgh; Fellow of the Royal Society (1807) and one of the founding members of the R.S.E. [216] 255 century British journal; although the contribution is not considered foreign as Bue´e was living in England at the time as a political refugee.527 (ii) It was this paper which first drew George Peacock’s attention to the subject.528 (iii) New evidence has come to light which reveals Bue´e’s initial motivation. According to two sources, Bue´e’s paper was written in response to William Frend’s The Principles of Algebra (1796). Both sources refer to a letter Bue´e had written to Frend on 21 June 1801. From the Annual Report of the Royal Astronom- ical Society (1842): ‘Among his [Frend’s] papers is preserved a letter to him from M. Bue´e, a Frenchman residing in England, dated June 21, 1801, containing the form in which the perusal of Mr. Frend’s work made the writer put together his own views of the subject’.529 And in an historical note in The Algebra of Coplanar Vectors and Trigonometry the author, Hayward writes: I have a letter in my possession from M. Bue´e to Mr. Frend, dated June 21, 1801, by which it appears that the former was desired by a gentleman in whose house he was living (as tutor, perhaps) to write a private reply to Mr. Frend’s objections [to negative and imaginary quantities].530 Strangely, Bue´e makes only a single insignificant reference to Frend. One would assume from the paper that Bue´e’s principal influence was Carnot’s Ge´ome´trie de position (1803). Adrien-Quentin Bue´e (1748–1826): a biographical sketch Snippets of biography that I have been able to obtain record Bue´e as a French Catholic priest who was forced to flee Paris during the Revolution. The Nouvelle biographie universelle describes Bue´e as a French writer and math- ematician. He was born in Paris in 1748 and died there on 11 October 1826. His 527 [217, p76f(no.74)] 528 [218, pxxvii] 529 [219, p151] William Frend (1757–1841) was a member of the Council of the R.A.S. 530 [215, ppxviii–xixf] 256 first ecclesiastical role was as organist at the Basilica of Saint-Martin in Tours. He fled Paris in 1792 and sought refuge in England, where he remained for a period of twenty-one years. On his return to Paris, he became an Honorary Canon of the city.531 His leisure time was devoted to music and the exact sciences. He left a number of manuscripts in which he considered various mathematical problems.532 From La France litte´raire we learn that Bue´e had a number of publications through a periodical pamphlet published during the Revolution entitled, La Me`re Duchesne.533 La Me`re Duchesne—a variant of Le Pe`re Duchesne—allowed contribu- tors to express their views on the current political situation through the character of la me`re Duchesne. Bue´e wrote to voice his opposition to the Civil Constitution of the Clergy Law of 1790. All Catholic clergy were required to swear an oath of allegiance to the Civil Constitution which made the Roman Catholic Church in France subor- dinate to the French government. Those who refused—the “non-jurors”—suffered restricted freedom to carry out their ministry, imprisonment and persecution. According to [222], when Bue´e escaped Paris and settled in England he estab- lished himself in Bath and supported himself financially by publishing pamphlets, articles and leaflets on various aspects of mathematics and science. More on Bue´e’s 1806 paper Much of Bue´e’s paper is unintelligible and hardly resembles mathematics at all. Some of what he says in relation to the sign √ −1 appears insightful (e.g. that it is as a sign of perpendicularity, or a mean proportional between +1 and −1); however, credibility is lost when the document is considered as a whole. Bue´e has a variety of interpretations of the sign √ −1. It is: a sign of impossibility in the context of the solution of a problem involving impossible conditions; a mean proportional between +1 and −1; a descriptive sign of perpendicularity and a mean 531Honorary Canon of Paris: an honorary title granted in France, at one time, by the Bishop to other members of the clergy. 532 [220, p730] 533 [221, p555] 257 ‘quality’ between two opposite qualities + −. Bue´e has two species of algebra: (i) universal arithmetic and (ii) a mathematical language. In the first, the signs + − denote addition and subtraction. In the second, they are taken as signs of ‘quality’; and, according to Bue´e, any quantity affected by a sign of quality is capable of meaning or interpretation. Thus, when √ −1 is considered as a mean quality between the qualities + −, quantities affected by the sign √ −1 take on a variety of fantastic interpretations. A few examples suffice: (i) as an amount of books, neither in the possession of a man (+), nor owed by him (−); (ii) with +t signifying the month approaching and −t the month past, −t √ −1 2 denotes the first half of the current month and +t √ −1 2 the second half, with their sum (equal to zero) expressing the current month; and (iii) placed before the expression for a cube or parallelepiped, √ −1 indicates that the cube or parallelepiped is, in fact, a void. Making a proper assessment of Bue´e is made difficult by the fact that he does not attempt to prove established results but instead works through a variety of bogus problems. Intermittently, however, we encounter something which we recognize as being definitely incorrect. Consider, for instance, his application of Pythagoras’ theorem to the lines AB = 1 and AD = √ −1 (Figure E.1, page 259). He has BD 2 = AB 2 + AD 2 = 12 + ( √ −1)2 = 12 + (−1) = 1− 1 = 0 He recognizes that the result is absurd but fails to realize that the apparent paradox arises over his confusion between the concepts of length and vector : the length of AD is 1, not √ −1. His explanation for what has happened is bizarre and not worth reproducing. Amidst all this confusion, however, there appears—out of nowhere—something of substance: Bue´e gives the direction of a line as eiθ, in which he surely takes inspiration from Euler.534 He writes: let √ −1 = 1 × e pi 2 √ −1 (e being the base of hyperbolic logarithms and pi half of the 534Introductio in analysin infinitorum (1748), a work which is cited by Bue´e elsewhere in the paper in relation to another matter. See [144, p25]. 258 circumference of a circle of which the radius is 1); 1 × e pi 2 √ −1 signifies the line AD [Figure E.1] of which the direction is e pi 2 √ −1.535 Figure E.1: Bue´e’s application of Pythagoras’ theorem. [144, pfacing 88] Gergonne’s two-dimensional table of real and imaginary magnitudes In 1811 Joseph Gergonne (1771–1859), editor of the Annales de mathe´matiques, conceived of a two-dimensional table of real and imaginary magnitudes. He revealed his conception in an editorial footnote to Jacques Franc¸ais’ paper [208], published in volume IV of the Annales, in which Franc¸ais famously communicated the ideas 535‘soit √ −1 = 1×e pi 2 √ −1 (e e´tant la base des logarithmes hyperboliques et pi la demi-circonfe´rence d’un cercle dont le rayon est 1); 1×e pi 2 √ −1 signifie la ligne AD dont la direction est e pi 2 √ −1’. [144, pp40–41] 259 contained in Argand’s 1806 pamphlet.536,537 In revealing his conception it was not Gergonne’s intention to strip Franc¸ais of priority, merely to show that their shared conception was very much “in the air”. Joseph Diaz Gergonne (1771–1859): a biographical sketch The life of Joseph Gergonne was a life of military service and mathematics. Between 1791 and 1795 the Frenchman saw action as Captain of the National Guard working towards stability for post-revolution France; for the French army defending Paris against the Prussians and as secretary to the general staff of the Moselle army. In 1794 he attended the Chaˆlons artillery school and was commissioned as a lieutenant. The same year he saw action in Spain. From the time of his arrival at Nimes in 1795, however, Gergonne was able to devote himself to mathematics. He was appointed to the Chair of Transcendental Mathematics at the E´cole Centrale, which had re- cently been established in the city. In 1810 he founded the first scientific journal devoted purely to mathematics, the Annales de mathe´matiques pures et applique´es, after having experienced difficulties in getting his own work published. The journal ran until 1832 and was succeeded by Liouville’s Journal de mathe´matiques pures et applique´es in 1836. In 1816 Gergonne moved to Montpellier to take up the Chair of Astronomy at the university there. In 1830 he became Rector at the university. He retired from his position in 1844 and died in Montpellier in May 1859, aged eighty-seven. 536See [208, pp71–72f]. The inspiration for Franc¸ais’ paper had come from a letter found amongst his late brother’s papers. The letter had come from Adrien-Marie Legendre (1752–1833), who had acquired his knowledge of the ideas from an unnamed source (Argand). Franc¸ais acknowl- edged his source and wrote of his hopes that the publicity generated by the publication of the paper might bring to light the original author of these ideas (Argand). Argand subsequently wrote to the editors of the Annales, declaring himself to be the said author and explaining that in 1806 he had published a pamphlet on the subject which Legendre had looked at in manuscript form. Argand provided the editors of the Annales with a summary of his pamphlet to support his authorship claim. 537Gergonne’s table is referred to in [209, pp95–96, 116–117] but my knowledge of it came directly from the Annales. 260 Gergonne’s primary mathematical interest was geometry. In 1813 he was awarded a prize by Bordeaux’s Socie´te´ des Sciences, Lettres et Arts for an essay on the rival methodologies of analytic and synthetic geometry, along with another entrant, M. Armand de Maizie`re.538 Gergonne was an advocate for the merits of analytic geom- etry and it was his position on the rival methodologies which brought him into his first conflict with Jean-Victor Poncelet (1788–1867), a proponent of synthetic geom- etry; later, in 1826, they entered a serious dispute over priority in the discovery of the principle of duality.539 Both Gergonne and Poncelet were students of Gaspard Monge (1746–1818), the founder of descriptive geometry.540 More on Gergonne’s two-dimensional table The impetus for Gergonne’s table (Figure E.2, page 262) was a paper published in volume I of the Annales entitled, ‘The´ore`me ge´ne´ral sur l’invariabilite´ de la forme des fonctions’ [228]. The paper was written by M. Armand de Maizie`re, winner of the essay prize, along with Gergonne, in 1813.541 Maizie`re considered the relationship between the independent and dependent variables, x and y, of a function ϕ. He had to ensure that x and y were subject to the law of continuity; that is, if we take two neighbouring states, xa and xa+1, of the independent variable x, then the difference between the two corresponding y states, ya and ya+1, of the dependent variable y should fall below a given limit, however small. Gergonne considered how one would measure the proximity of two neighbouring terms in cases where at least one term is non-real: he proposed that the difference between the two terms be written in the form p+ q √ −1, which tends 538 [223, p459] Gergonne’s essay was never published, however, he does give a summary of it in [224]. Maizie`re’s essay, according to [223, p459]: Me´moire qui a partage´ le prix en 1813 (Paris, 1814), p. 28. 539 [225, pp577–578] 540Uncited sources of biographical information on Gergonne: [226] and [227]. 541According to [228, p368], Armand de Maizie`re was a mathematics teacher at the lyce´e (high school) of Versailles. No further biographical information is available. 261 to zero when p and q tend to zero.542 ..., −2 + 2 √ −1, −1 + 2 √ −1, +2 √ −1, +1 + 2 √ −1, +2 + 2 √ −1, ... ..., −2 + √ −1, −1 + √ −1, + √ −1, +1 + √ −1, +2 + √ −1, ... ..., −2, −1, ±0, +1, +2, ... ..., −2− √ −1, −1− √ −1, − √ −1, +1− √ −1, +2− √ −1, ... ..., −2− 2 √ −1, −1− 2 √ −1, −2 √ −1, +1− 2 √ −1, +2− 2 √ −1, ... Figure E.2: Gergonne’s two-dimensional table of real and imaginary magnitudes for integer values, conceived of in 1811. Reproduced from [208, p71f]. In private correspondence on the paper, Gergonne had proposed the idea of a double-entry table: About two years ago, I wrote to M. de Maizie`re about his paper inserted at page 368 in the first volume of this series and informed him that we were probably wrong to expect to include all numerical values in a simple series; and that by their nature, they seemed to form a table with double entries, which, limited to integers, could be represented as follows: [Figure E.2] so that already, like M. Franc¸ais, I assumed numbers in the form n √ −1 on a line perpendicular to the line which contains numbers of the form n; and that, like him again, I represented the numbers not belonging to those two lines by the sum of their projections on each of them.543 542‘Pour cela nous remarquerons que la diffe´rence de deux pareils termes peut toujours, en ge´ne´ral, eˆtre suppose´e imaginaire et de la forme p+q √ −1; or il n’y a pas de doute qu’une telle expression ne puisse tendre vers ze´ro, puisqu’il suffit pour cela que p et q tendent eux-meˆmes vers cette limite commune.’ [228, pp369–370f] 543‘Il y a environ deux ans qu’e´crivant a` M. de Maizie`re, au sujet de son me´moire inse´re´ a` la page 368 du 1.er volume de ce recueil, je lui mandais qu’on avait peut-eˆtre tort de vouloir comprendre toutes les grandeurs nume´riques dans une simple se´rie; et que, par leur nature, elles semblaient devoir former une table a` double entre´e qui, borne´e aux seuls nombres entiers, pourrait eˆtre figure´e comme il suit: [Figure E.2] en sorte que de´ja`, comme M. Franc¸ais, je supposais les 262 Following publication of Franc¸ais’ paper in the Annales, another contributor, Franc¸ois Joseph Servois (1768–1847) proposed a way of extending Gergonne’s table into three dimensions. Gergonne’s response to Servois was to disclaim credit for originally having designed the table with the intention of extending it into three dimensions: a triple argument table had occurred to Gergonne but only after he had read the memoirs of Argand and Franc¸ais.544 So while Gergonne had hit upon a way of representing complex numbers as co- ordinates in the plane, his conception lacked a geometrical definition of addition and multiplication for complex numbers as directed lines. However, the table was devised before the rediscovery of Argand and without knowledge of Bue´e, which serves to elevate Gergonne’s achievement and make his table all the more interesting.545 nombres de la forme n √ −1 situe´s dans une ligne perpendiculaire a` celle qui renferme les nombres de la forme n; et que, comme lui encore, je repre´sentais les nombres e´trangers a` ces deux lignes par la somme de leurs projections sur l’une et sur l’autre.’ [208, pp71–72f] 544See [192, pp234–235, 235f] for Servois’ suggestion and Gergonne’s response. 545Gergonne had the opportunity to declare any knowledge of Bue´e in his editorial comments on Lacroix’s note [210] in which Lacroix called attention to Bue´e’s contribution. 263 F HAMILTON’S UNPUBLISHED PAPER ON THE FUNDAMENTAL THEOREM OF ALGEBRA: INSPIRED BY MOUREY I discovered the following unpublished paper by Sir William Rowan Hamilton in his notebook MS.1492/95 (pages 263–265) in the Manuscripts Library of Trinity Col- lege, Dublin, during a research trip to the library in February 2014. I am grateful to the Keeper of the Manuscripts at Trinity for kindly giving their permission to publish here this transcription of the original. To make the transcription easier to read, I have italicized the text which Hamil- ton had underlined (except for titles) and rendered bold the text which Hamilton had double underlined. I have made no other changes; notably, I have chosen to keep Hamilton’s contractions just as they appear in the original, e.g. ‘imagy’ (imag- inary) and ‘wd’ (would), as I feel that they help to establish a stronger impression of Hamilton for the reader. I understand, however, that these contractions would be replaced by full words by an editor or publisher who was preparing Hamilton’s paper for publication. Hamilton’s notation — Where cossin appears, as in equations (a) (6) (9), read along the top first, then the bottom, to give two separate equations. E.g. the first equation (a) gives: r′ cos v′ = r cos v−a cosα from the top and r′ sin v′ = r sin v−a sinα from the bottom. — Similarly for >< in equation (2). — v(n+1) is v with n+ 1 dashes. E.g. v(3) = v ′′′ . Similarly for α(n). — The ∫ sign, as in equation (11), denotes integers and is not to be confused with the integrals of integral calculus. 264 Theorem of Mourey. (not stated precisely in this way by him; he writes (p.104), xn + b xn−1 + bc xn−2 + ...+ (bcd...)x− g = 0 +c +cd +d +bd +... +... ) “Every equation of the form x(x− a)(x− a′) · · · = b (A) has at least one (real or imagy) root, whatever given (real or imagy) values the consts a, a′, . . . and b may have.” Modified statement of the theorem. If a, a′, . . . and b be any n given positives, and α, α′,. . .β any n given reals; it is possible to satisfy the system of the 2n eqns, r′ cos sin v′ = r cos sin v − a cos sin α, r′′ cos sin v′′ = r cos sin v − a′ cos sin α′, . . . (a) rr′r′′ · · · = b, v + v′ + v′′ + · · · = β + 2mpi, (b) (where m is some whole no) by at least one system of n positives r, r′, r′′, . . . & of n reals, v, v′, v′′, . . . Modified Demonstration. Let rv be the least positive root of the equation,546 546Hamilton has substituted x = reiv, x−a = r′eiv ′ = reiv − aeiα, x−a′ = r′′eiv ′′ = reiv − a′eiα ′ , etc. from equations (a) and b = beiβ into equation (A). He has then multiplied the resulting equation together with its complex conjugate to give b2 = r2 { r2 − 2ar cos (v − α) + a2 }{ r2 − 265 b2 = r2v { r2v − 2arv cos (v − α) + a 2 }{ r2v − 2a ′rv cos (v − α ′) + a′2 } · · · (1) in wh. it is supposed that b > 0; so that rv > 0, rα > < a, rα′ > < a′, . . . (2) and rv+2pi = rv [Because all the coeff s of the √ (1) remain unchanged.] (3) Let r′v, r ′′ v , . . . be the other positive & periodic functions of v, wh. are obtained by substg for r in the eqn (a) its value rv; & in order to render determinate the correspondg functions v′v, v ′′ v , . . . let us suppose that they receive no sudden changes (of the form 2mpi), & also (as is allowed by the eqn) that each v(n+1) becomes = v, when v = α(n) + pi; so that if we introduce these n− 1 other continuous functions of v, wv = v ′ v − v, w ′ v = v ′′ v − v, . . . (4) we shall have wα+pi = w ′ α′+pi = · · · = 0 (5) To render wv = ∓pi, it wd be necessary to suppose v′v = v ∓ pi, (rv + r ′ v) cos sin v = a cos sin α, rα + r ′ α = a > rα; (6) If then we have, on the contrary, rα > a, we must conclude that the continuous function wv (which may become = 0) can never attain either of the limits, ∓pi, but is always included between them, or that wv > −pi, wv < +pi, if rα > a (7) 2a′r cos (v − α′) + a′2 } . . ., call this (†). Then for each angle v we look for the smallest positive r which satisfies (†). Hamilton calls this rv. Substituting r = rv into (†) gives equation (1). 266 The function wv is ∴ in this case periodic, (like rv, &c) because it is determined without ambiguity by means of its cosine & sine; thus wv+2pi = wv, v ′ v+2pi = v ′ v + 2pi, if rα > a (8) On the other hand, if v′v − α = 0 or = 2pi, then rv cos sin v = (a+ r′v) cos sin α, rα = a+ r ′ α > a; (9) If then the inequality rα > a be not satisfied, we must regard the continuous func- tion v′v − α as never attaining either of the limits 0 & 2pi, although it receives the intermediate value pi, when v = α + pi; hence v′v > α, v ′ v < α + 2pi, v ′ v+2pi = v ′ v if rα < a (10) In like manner v′′v+2pi − v ′′ v = 2pi or 0, all ∫ as rα′ > or < a ′, (11) and similarly for v′′′, . . . While ∴ the quantity v is continuously increased by 2pi, the continuous function v + v′v + v ′′ v + . . . (12) is increased at least by 2pi, & may be increased by 4pi, or by 6pi, . . . All ∫ as we have the system of the n− 1 inequalities rα < a, rα′ < a ′, . . . (13) or have one or more of these n− 1 opposite inequalities, rα > a, rα′ > a ′, . . . (14) In every case ∴ , during this contins increase of v, the function (12) passes at least once through the stage β + 2mpi, whatever assigned constant β may be: & the 2nd eqn (b) is satisfied, under the form, 267 v + v′v + v ′′ v + · · · = β + 2mpi But the first eqn (b) is also satisfied, by (1), under the form, rvr ′ vr ′′ v · · · = b and the equation (a) has been satisfied, by the choice of the functions of v. The theorem of Mourey is ∴ proved to be true. (Obsy July 18th 1854. W.R.H.) [Presumably, Hamilton meant the following to be an addition to the paper . . . ] Admitting the theorem that an eqn of the form (A) has at least one root, say x1, so that x1(x1−a)(x1−a′) · · · = b, where a, a′, . . . b are any real or imagy consts, & x1 is some real or imagy quantity, whose existence is proved as above, we may substi- tute this last value for b, transpose, develope, and divide, (without remainder), by x = x1. The quott will be a polynome in x of the degree n− 1, (A) having been of the degree n. Suppose it known, then, that every eqn of this depressed degree n− 1 has n−1 roots, or may be decomposed into n−1 binomial factors of the form x−a; it will follow on the one hand that the genl eqn of the nth degree may be put under the form (A), & ∴ that it has at least one root; & on the other had, after depression by divn, that it has also n− 1 other roots. On the whole then, Mourey’s theorem proves that “the genl eqn of the nth degree has n roots, if the general eqn of the next lower degree have the next lower number of roots”; that is, in the last analysis, if the genl eqn of the first degree have one root: but this is manifestly true. Therefore, &c. 268 Mourey connects the proof of his own theorem (for such I think it may be properly called) with the probm to find a point P , in the same plane with any n given points A,B,C, & such that the product of the n paths or ways (chemins), to it from them, may be equal to a given path; AP · BP · CP = a given directed line LM . His reasoning appears to me to be correct in the main, but he seems somewhat to spoil his argument by seeking to show (p. 114 at the top), that the eqn (A) itself has as many as n roots, whereas it was enough to prove (what alone he does prove, in my opinion) that it has at least one. The combination of it with eqns of lower degrees is required, & is sufft to prove finally, as above, the known theorem, that “the general eqn of the nth degree has n roots”. But the proof given in this paper has been entirely suggested by Mourey’s work. W.R.H. 269 G MOUREY’S TERMINOLOGY AND NOTATION: A LOOK-UP TABLE The following table provides a summary of Mourey’s original terminology and no- tation. Its entries are given in the order in which they appear in Mourey (1861) so that it might be used as a companion-guide to the text. In itself the table provides a good summary of Mourey’s approach. Table G.1: Mourey’s terminology and notation: a look-up table. Term, notion, notation (en Franc¸ais) Definition in English Example of usage or a sketch to aid understanding ligne directive ou chemin [n.] directed line or path; a straight line leading in a given direction origine [n.] origin; the initial point of a directed line terme [n.] terminus; the terminal point of a di- rected line 270 (chemins) de suite in succession; the terminal point of one path is the initial point of the other (chemins) concurrents concurrent; (paths) leading in the same direction (chemins) oppose´s opposed; (paths) leading in opposite directions (chemins) inverses inverse; (paths) of the same length, leading in opposite directions unite´ relative relative unit; unit of measurement See nombre relatif. nombre relatif relative number; quantity measured in terms of the unite´ relative E.g. Unit of measurement: 1 metre taken from left to right. The numbers 1, 2, 3, . . . represent 1m to the right, 2m to the right etc. The numbers −1,−2,−3, . . . represent 1m to the left, 2m to the left etc. concret [adj.] concrete; describes the nombre re- latif when the unit of measurement is specified 271 abstrait [adj.] abstract; describes the nombre re- latif when the unit of measurement is not specified Note. However, these abstract numbers can be thought of as quantities relative to an abstract unit 1, for example. nombre directif directed number In the context of paths, an abstract number may be regarded as a quantity relative to an abstract geometrical unit, i.e. a path of arbitrary length and di- rection. E.g. the directed number 2 can be thought of as a path in the same direction as the unit path 1 (the line of length 1 in the direction of the pos- itive real axis) but of twice the length. Non-real quantities are generated by a rotation of the unit path. (nombre) positif [adj.] positive (number); having the same direction as the abstract unit E.g. 1, 2, 3, . . . with the unit +1 (nombre) ne´gatif [adj.] negative (number); having a direc- tion opposite to that of the abstract unit E.g. −1,−2,−3, . . . with the unit +1 verser [v.] to make turn or rotate ‘verser AB de r’: to rotate the directed line AB through an angle r about its origin, expressed by ABr = AC. angle directif directed angle; an angle leading in a particular direction Convention. The unit of the directed angle is an anti- clockwise rotation of 90◦, denoted by a subscript 1, i.e. AB90◦ = AB1. Similarly, AB180◦ = AB2, AB−90◦ = AB−1 and AB60◦ = AB 2 3 . 272 verseur (de AB) [n.] the angle through which a directed line is rotated ‘L’angle r est le verseur de AB’: r is the angle through which the directed line AB is rotated. rapport directeur (de AC a` AB) the angle leading from AB to AC, expressed by AC ∵ AB (read ‘AC directeur a` AB’ or ‘AC recteur AB’) ‘L’angle r est le rapport directeur de AC a` AB’: r is the angle leading from AB to AC. coˆte´ dirigeant leading side; the side from which the directed angle originates coˆte´ dirige´ facing side; the side towards which the directed angle approaches See coˆte´ dirigeant. How Mourey indicates a directed angle. A directed angle is expressed by a series of letters: the first letter corresponds to the vertex(1), the second to the coˆte´ dirigeant (2), and the last to the coˆte´ dirige´(3). If no convention is established as to the sense in which the angle turns then it is necessary to place one or several letters between (2) and (3). E.g. the directed angle leading from AB to AC via x is expressed by ABxC. 273 e´galite´ spe´ciale des angles directifs =´ special equality of directed angles; equality of directed angles modulo 4 right angles (equivalently modulo 2pi) r =´ r + 4 =´ r + 8 =´ · · · =´ r + 4n (where n is an integer) dige`ne [n.] from the Greek meaning ‘two ori- gins’; a geometric figure composed of two paths (not necessarily with the same origin), which we com- pare in order to find the rapport di- recteur between them. See rapport directeur. The value of the dige`ne, QC ∵ AB′, is the directed angle AB′C ′, which has the same coˆte´ dirigeant AB′, and whose coˆte´ dirige´ AC ′ is concurrent with QC. e´qui-dige`ne [n.] an equation expressing the equality of two expressions of the form a ∵ b, where a and b are paths, i.e. equal- ity of two dige`nes. See dige`ne and rapport directeur. E.g. SE ∵ RD = QC ∵ PB version [n.] the act of rotation; the operation by which we rotate a path. See verser. e´qui-quotient [n.] an equation expressing the equality of two ratios of directed quantities E.g. a : b = c : d AB+ notation expressing a positive path of the same length as AB, no matter the sign of the path AB. Read ‘AB positive’. E.g. (42)+ = 4 de´verseur + [n.] the sign + used in the manner con- sistent with the notation AB+. See AB+. 274 amr notation signifying a directed line of length am rotated through an angle r, i.e. (am)r and not (ar)m σ notation representing the set of val- ues 0, 4q, 8q, . . . i.e. multiples of four right angles r =´ r + 4 =´ r + 8 =´ · · · =´ r + σ See e´galite´ spe´ciale des angles di- rectifs. (signe) super- e´gal := super-equals (sign); symbol used to indicate that two quantities are equal under the radical sign E.g. −1×−1 := 14 not : = 1 as √ −1×−1 = √ 12 × 12 = √ 14 = 1 4 2 = 12 and not = √ 1 = 1. prime-directeur [n.] the rapport directeur of a path in the positive direction (chemins) super-e´gaux [adj.] super-equal; (paths) of the same length, the same direction and with the same prime-directeur a= notation expressing a path parallel to unity and of the same length as a. Read ‘a paralle`le’. a=1 notation expressing a path perpen- dicular to unity and of the same length as a mi-de´verseur = [n.] the sign = used in the manner con- sistent with the notation a=. See a=. 275 References [1] John J. O’Connor, Edmund F. Robertson, ‘A Visit to James Clerk Maxwell’s House’, in The MacTutor History of Mathematics Archive (November 1997) accessed 1 Jan. 2014. [2] John J. O’Connor, Edmund F. 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