Sources and Studiesin the History of Mathematics and

Physical Sciences

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J.Z. Buchwald J. Liitzen G.J. Toomer

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P.J. Davis T. HawkinsA.E. Shapiro D. Whiteside

SpringerNew YorkBerlinHeidelbergHong KongLondonMilanParisTokyo

Sources and Studies in theHistory of Mathematics and Physical Sciences

K. AndersenBrook Taylor's Work on Linear Perspective

H.J.M. BosRedefining Geometrical Exactness: Descartes'Transformation of the Early ModernConcept of Construction

J. Cannon/S. DostrovskyThe Evolution of Dynamics: Vibration Theory from 1687 to 1742

B. Chandler/W. MagnusThe History of Combinatorial Group Theory

A.I. DaleA History of Inverse Probability: From Thomas Bayes to Karl Pearson,Second Edition

A.I. DaleMost Honourable Remembrance: The Life and Work of Thomas Bayes

A.I. DalePierre-Simon Laplace, Philosophical Essay on Probabilities, Translated from the fifthFrench edition of 1825, with Notes by the Translator

P.J. FedericoDescartes on Polyhedra: A Study of the De Solidorum lilementis

B.R. GoldsteinThe Astronomy of Lcvi ben Gerson (1288-1344)

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H.H. GoldslineA History of the Calculus of Variations from the 17th Through the 19th Century

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T. HawkinsEmergence of the Theory of Lie Groups: An Essay in the History of Mathematics1869-1926

A. Hermann/K. von Mcyenn/V.F. Weisskopf (Eds.)Wolfgang 1'auli: Scientific Correspondence I: 1919-1929

Continued after Index

Andrew I. Dale

Most Honourable Remembrance

The Life and Work of Thomas Bayes

Wilh 29 Illustrations

Springer

Andrew [. DaleDepartmeni of Mathematical Statistics

University of NatalDurban 4001, Natal

South Africadale@nu.ac.za

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Library of Congress Catak>ging-in-Publication DataDale, Andrew I.

Most honourable remembrance : the life and work of Thomas Bayes / Andrew I. Dale,p. cm.—(Studies and sources in the history of mathematics and physical sciences)

Includes bibliographical references and index.ISBN 0-3K7-00499-8 (alk. paper)

I. Bayes, Thomas, d 1761. 2. Statisticians—Great Britain—Biography. I. Title. [I.Sources and studies in the hisu>ry of mathematics and physical sciences.QA276.157.B39D3S 2003519.5'092—dc21[B] 2003042474

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© 2003 Springer-Vcrlag New York, Inc.All rights reserved, This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springcr-Vurlag New York, Inc., 175 Fifth Avenue, New York, NY10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use inconnection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even ifthey are not identified as such, is not to be taken as an expression of opinion as to whether or notthey are subject to proprietary rights.

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To F.J.H.

PROLOCUTION

Si quelqu’vn me blasme de remarquer ces petites partic-ularitez, ie luy repondray, que Plutarque temoigne qu’ilauoit receu quelque sorte de deplaisir d’vne chose aussilegere: quand il se plaint de ceux qui n’auoient pas laissepar ecrit les noms des meres de Nicias, de Demosthene, deFormion, de Thrasybule, & de Theramines, renommez per-sonnages contemporains de Socrate: & au contraire il mon-stre scauoir bon gre a Platon & a Antisthene, parce quele premier auoit ecrit le nom du Precepteur d’Alcibiade,& l’autre, n’auoit pas dedaigne de nomer sa nourrice. Lanegligence des anciens Escriuains a este si grande, qu’elleest cause que sept villes ont este en dispute pour la nais-sance d’Homere: chacune se donnant cet honneur d’auoireste la mere nourrice du plus excellent Poete de la Grece;Ce qui m’a fait touiours penser, que c’est vn grand defaut aceux qui se meslent d’ecrire les Vies des Hommes illustres,de laisser en arriere quelques particularitez, qu’on n’eustpeut-estre pas considerees dans le temps qu’ils ont vescu,mais qui servient infalliblement estimees en vn autre ciecle.

Hilarion de Coste.La Vie du R. P. Marin Mersenne.

There are many passages in this book where I have been atsome pains to resist the temptation of troubling my readerswith my own deductions and conclusions; preferring thatthey should judge for themselves, from such premises as Ihave laid before them.

Charles Dickens.American Notes.

Preface

I find it impossible to write a preface tothis work, without discovering a little ofthe enthusiasm which I have contractedfrom an attention to it.

Joseph Priestley.The History and Present State of Electricity.

It is generally considered bad form in writing, unless on matters autobio-graphic, to make unbridled use of the perpendicular pronoun. The reader ofthe present book, however, may well wonder why one would want to studythe life and works of Thomas Bayes, ‘this strangely neglected topic’1, andit is only by a reluctant use of the first person singular on the part of theauthor that this legitimate question can be answered.

It was in the late 1960s that my interest in various aspects of subjectiveprobability was awakened by some of the papers of I.J. (‘Jack’) Good, andthis was followed by the reading of works such as Harold Jeffreys’s Theoryof Probability. In many of these the (apparently simple) result known asBayes’s Theorem played a pivotal role, and it struck me that it might beinteresting to find out a bit more about Thomas Bayes himself. In tryingto satisfy this curiosity in spasmodic periods over many years I discoveredthat little information seemed to be available. Writings by John D. Hollandin the 1960s shed some light on the matter, but it was only in periods ofsabbatical leave that I was able to undertake the intensive (and extensive)archival research needed to flesh out the shadowy figure after whom one ofthe major branches of modern statistics is named.

Where, for instance, was Bayes educated? The answer to this questionpresented itself by chance to me when I was visiting the University ofChicago. While waiting for some rare volumes to be retrieved from thedepths of the John Crerar Library I happened to pick up a catalogue ofmanuscripts in the Edinburgh University Library in which there appeareddetails of a certificate recording the admission of Thomas Bayes as a stu-dent. Visits to Edinburgh and the generous co-operation of the library staff

x Preface

resulted in the finding of a number of references to Bayes in the archivesof the University.

Similar serendipity led to my finding of the proof, written mainly inLatin, of one of the rules in Bayes’s Essay towards solving a Problem inthe Doctrine of Chances in a Notebook attributed to him and held in theEquitable Life Assurance Society in England. This not only reinforced theopinion that the Notebook was indeed written in Bayes’s hand, but alsoshed light on matters that Bayes had thought worth recording.

Once the question of Bayes’s education had been solved, or at least an-swered to the best of my ability, it seemed necessary to investigate hisfamily and life in Tunbridge Wells, where he spent many years as Presby-terian minister. Although some information on these matters was availablefrom (fairly) modern writings, it seemed expedient rather to consult writ-ers as near to Bayes as possible, and in doing so, of course, I was merelyemulating Edward Gibbon, who, in the preface to the fourth volume of thequarto edition of his Decline and Fall of the Roman Empire, stressed thathe had made it his earnest endeavour always to consult the original sources.Perhaps Gibbon, in his turn, was following the example set by Watson inhis edition of The Works of Horace:

Whoever attempts, at this distance of time, to write upon aClassic, ought not only to read over what the most approvedcommentators have written upon his works, but to compare thesame carefully with the text itself; otherwise he is in danger ofbeing misled: for many of them, instead of illustrating, onlydarken their author, and palpably mistake his meaning.[1792, p. v]

There are three eighteenth-century savants who are known to modernstatisticians and historians of statistics for single contributions2: ThomasBayes, Roger Boscovich3 and Thomas Simpson4. Bayes is known for hisTheorem, published in his posthumous Essay in 1764, Simpson for hiswork in 1755 (extended in 1757) on error distribution, and Boscovich forthe posing of a question in 1757 (and its solution in 1760) on the fittingof a straight line to observational data5. Boscovich and Simpson have bothbeen the subject of biographies6, and it seems not inappropriate that Bayesbe similarly commemorated, particularly as the tercentenary of his birth isupon us.

In drawing to a conclusion let me say a brief word about the presentationof material here. As regards the method of citation, works such as Bayes’sEssay towards solving a Problem in the Doctrine of Chances, published inthe Philosophical Transactions in 1764 in the volume for 1763, are citedas ‘Bayes [1763]’. In the reprinting of Bayes’s tracts, [n][n][n] indicates the endof page nnn in the original text. If a word in the original is broken across

Preface xi

two pages, [n][n][n] appears here at the end of the word, with broken formulaebeing similarly treated. Footnotes are given in the usual way, irrespectiveof how they may be positioned (e.g., broken over two pages) in the original.Things that appear in crotchets, [. . . ], are my corrections or additions, evenin exponents in formulae. In giving Bayes’s tracts I have allowed myselfthe liberty of a light editing. In the Essay, for example, what appear inthe original as ‘a remuch mistaken’ and ‘innature’ are given here as ‘aremuch mistaken’ and ‘in nature’, and ‘(a + b)b+q’ in the first paragraph ofProposition 7 is reprinted as ‘(a+ b)p+q’. On page [400][400][400] the last occurrenceof 2Eapbq appears in the original as 2Eapb

q, and the second Σ on page[416][416][416] appeared originally as E. No changes are more serious than these,and all would easily be made by the reader of the original texts.

Not being an expert in eighteenth-century history, I must bring to thisstudy a certain naıvete, a quality that I hope will be seen by the reader astouching rather than irritating and obtrusive. There are no doubt portionsof this work that could be better handled by a historian; but one of theresults of approaching a topic in ignorance is the pleasure of discovery7,and it is with the hope of communicating some of the pleasures I have metwith in this journey through Bayes and other coves that certain passages— for example the remarks on the Great Plague and the descriptions ofTunbridge Wells — are presented.

I hope, though, that the reader will not view this work in the light inwhich Nicodemus Boffin, ‘the minion of fortune and the worm of the hour’,viewed his new mansion, viz.8

He could not but feel that, like an eminently aristocratic familycheese, it was much too large for his wants, and bred an infiniteamount of parasites.

Rather, as George Berkely put it in Siris: a Chain of Philosophical Reflex-ions and Inquiries:

The displeasure of some readers may perhaps be incurred bysurprising them into certain reflections and inquiries for whichthey have no curiosity. But perhaps some others may be pleasedto find a dry subject varied by digressions, traced through re-mote inferences, and carried into ancient times. [¶350]

Durban, Natal, Andrew I. DaleSouth Africa

Acknowledgments

I am grateful to the following for granting permission for quotation fromthe works mentioned: The Mathematical Association, from the paper pub-lished in The Mathematical Gazette by G.J. Lidstone in 1941; The JohnsHopkins University Press, from A.W.F. Edwards’s Likelihood published in1992; the American Mathematical Society, from (i) the paper publishedin The Bulletin of the American Mathematical Society by F.H. Murray in1930, (ii) G. Boole’s A Treatise on Differential Equations (1959, 5th edi-tion) and (iii) F. Cajori’s A History of Mathematics (1991, 5th edition);the Royal Statistical Society, from the papers published in the Journal ofthe Royal Statistical Society, Series A (i) by S.M. Stigler in 1982 and (ii)by D.J. Bartholomew (comment by D.A. Gillies) in 1988; Springer-Verlag,Heidelberg, from (i) A.N. Kolmogoroff’s Grundbegriffe der Wahrschein-lichkeitsrechnung of 1933, (ii) O.B. Sheynin’s ‘R.J. Boscovich’s work onprobability’ in Archive for History of Exact Sciences (1973) 9: 306–324c©Springer-Verlag and (iii) my ‘A newly-discovered result of Thomas Bayes’in Archive for History of Exact Sciences (1986) 35: 101–113 c©Springer-Verlag; Barnes & Noble, and, separately, Palgrave Publishers Ltd, fromN. Page’s Dr Johnson. Interviews and Recollections (Macmillan) of 1987;Hodder and Stoughton, from R.W. Dale’s A History of English Congrega-tionalism of 1907; T. & T. Clark, from the Encyclopædia of Religion andEthics, edited by J. Hastings, of 1971; the Inter-Varsity Press, from TheNew Bible Commentary, edited by F. Davidson; C. Klincksieck et Cie, fromH. Laboucheix’s Richard Price: theoricien de la revolution americaine; lephilosophe et la sociologue; le pamphletaire et l’orateur ; the Wesley Histori-cal Society, from F.F. Bretherton’s paper published in 1916 in the Proceed-ings of the Wesley Historical Society ; the Associated University Presses(Golden Cockerel Press), from T. Hailperin’s Sentential Probability Logic.Origins, Development, Current Status, and Technical Applications; the In-ternational Statistical Institute, from G. Shafer’s 1985 paper in the Inter-national Statistical Review and F. Yates’s paper in the Revue de l’InstitutInternational de Statistique (1962); ITPS Ltd, for the passages (i) from

xiv Acknowledgments

A. Browning’s English Historical Documents 1660–1714, vol. VIII of En-glish Historical Documents (General Editor D.C. Douglas) published byEyre & Spottiswoode (Routledge) in 1953, (ii) from S. Korner’s The Phi-losophy of Mathematics: an introductory essay (1960, Routledge) and (iii)from G.H. von Wright’s A Treatise on Induction and Probability (Little-field, Adams & Co.) of 1960; the Epworth Press, from W.B. Whitaker’s TheEighteenth Century English Sunday. A Study of Sunday Observance from1677 to 1837 (1940); Faber and Faber, from M. Barton’s Tunbridge Wells(1937) and S. Spender’s World within World (1991); S.E. Black, from herBunhill Fields: the Great Dissenters’ Burial Ground (1990); Kegan Paul In-ternational, from M. Black’s The Nature of Mathematics: A Critical Survey(1933); Oxford University Press, from (i) M.G. Kendall’s ‘Ronald AylmerFisher, 1890–1962’, Biometrika (1963), 50: 1–15, (ii) R.L. Plackett’s ‘Stud-ies in the History of Probability and Statistics VII. The principle of thearithmetic mean’, Biometrika (1958), 45: 130–135 and (iii) J. Wishart’s‘On the approximate quadrature of certain skew curves, with an accountof the researches of Thomas Bayes’, Biometrika (1927), 19: 1–38 (the cita-tions are made by permission of the Biometrika Trustees). I regret that Ihave been unable to trace the authors or estates of these three items, andI thank them for their work; Edward Arnold, from A. O’Hagan’s Kendall’sAdvanced Theory of Statistics, Volume 2B. Bayesian Inference (CharlesGriffin); Academic Press, from my article in Historia Mathematica in 1991,and from the articles published in the same journal by G.C. Smith in 1980,vol. 7: 379–388 and R. Weinstock in 1998, vol. 25: 281–289; The Institute ofMathematical Statistics, from G. Shafer’s article in The Annals of Statistics(1982) 10: 1075–1089; Carus Publishing Company (Open Court PublishingCompany), for quotations from F. Cajori’s A History of the Conceptionsof Limits and Fluxions in Great Britain from Newton to Woodhouse of1919; the Societe Francaise d’Histoire des Sciences et des Techniques, fromJ.P. Clero’s Thomas Bayes: Essai en vue de resoudre un probleme de ladoctrine des chances, published in Cahiers d’Histoire et de Philosophie desSciences No. 18 (1988); Random House, Inc., from F.R.B. Godolphin’sThe Latin Poets (1949) and from O. Johns’s Asphalt and Other Poems(1917); Adolf M. Hakkert, from A.D. Leeman’s Orationis Ratio of 1986;Palgrave Publishers, from J.M. Keynes’s A Treatise on Probability of 1921(Macmillan & Co. Ltd); the University of California Press, from (i) AndrewMotte’s translation, Sir Isaac Newton’s Mathematical Principles of NaturalPhilosophy and his System of the World as revised by Florian Cajori, ofNewton’s Philosophiæ Naturalis Principia Mathematica Copyright 1934 byThe Regents of the University of California, Renewal copyright 1962 by The

Acknowledgments xv

Regents of the University of California and (ii) L.J. Savage’s ‘The founda-tions of statistics reconsidered’ in the Proceedings of the Fourth BerkeleySymposium on Mathematical Statistics and Probability, (ed. J. Neyman),vol. I, pp. 578–586, Copyright c©1961 by The Regents of the Universityof California; David Higham Associates Limited, from A.W.M. Bryant’sRestoration England published by Collins in 1960; P. Horwich, for the ma-terial from his Probability and Evidence (1982), published by CambridgeUniversity Press.

The quotations from A. Hald’s A History of Probability and Statisticsand Their Applications before 1750 c©1990 by John Wiley & Sons, Inc.and his A History of Mathematical Statistics from 1750 to 1930 c©1998by John Wiley & Sons, Inc. are reprinted by permission of John Wiley &Sons, Inc. The material from B. de Finetti’s Theory of Probability, vols 1& 2 (1974/5), c© John Wiley & Sons Ltd, is reproduced with permission.The material from Lorraine J. Daston’s Classical Probability in the En-lightenment, Copyright c©1988 by Princeton University Press, is reprintedby permission of Princeton University Press. The quotations from (i) R.C.Jeffrey’s The Logic of Decision, c©1965, 1983 by The University of Chicago,(ii) D.M. Jesseph’s Berkeley’s Philosophy of Mathematics, c©1993 by TheUniversity of Chicago and (iii) E. Nagel’s Principles of the Theory of Prob-ability, copyright by The University of Chicago and printed in the Inter-national Encyclopedia of Unified Science (Editor-in-chief Otto Neurath),Volume 1, Number 6, are all reprinted by permission of the publishers,The University of Chicago Press, as is (i) J.B. Bond’s paper published inIsis (1921), No. 11, vol. IV, 295–323 and (ii) P. Kitcher’s paper publishedin Isis (1973), 64: 33–49. The passage from Popper’s The Logic of Scien-tific Discovery (Hutchinson, 1968) is reprinted with the permission of MrsMelitta Mew, executrix of Sir Karl Popper’s estate. The material from HansReichenbach’s The Theory of Probability. An Inquiry into the Logical andMathematical Foundations of the Calculus of Probablity (1971), publishedby the University of California Press, is reprinted by permission of Dr M.Reichenbach.

Quotations of Samuel Pepys are reproduced from The Diary of SamuelPepys edited by Robert Latham and William Matthews (Copyright c© TheMaster, Fellows and Scholars of Magdalene College, Cambridge, RobertLatham and William Matthews 1983) by permission of PFD on behalf ofThe Master, Fellows and Scholars of Magdalene College, Cambridge, theEstates of Robert Latham and William Matthews. The numerous citationsfrom manuscripts relating to Thomas Bayes’s connexion with EdinburghUniversity are made with permission of Edinburgh University Library.

xvi Acknowledgments

Extracts from Stephen M. Stigler’s Statistics on the Table: The Historyof Statistical Concepts and Methods, Cambridge Mass., are reprinted bypermission of the publisher Harvard University Press, Copyright c©1999 bythe President and Fellows of Harvard College. The extracts from (i) J.E.Force’s William Whiston: Honest Newtonian (1985), (ii) N. Guicciardini’sThe development of Newtonian calculus in Britain 1700–1800 (1989), (iii)P. Walmsley’s The Rhetoric of Berkeley’s Philosophy (1990), (iv) E.G.R.Taylor’s The Mathematical Practitioners of Hanoverian England 1714–1840 (1966), (v) I. Hacking’s The Emergence of Probability (1975), (vi)I. Parker’s Dissenting Academies in England, Their Rise and Progress andtheir Place among the Educational Systems of the Country (1914) and (vii)H. Jeffreys’s Scientific Inference (1973) are reprinted with the permissionof Cambridge University Press. The extracts from T.L. Fine’s Theories ofProbability: an examination of foundations (1973) are reprinted by permis-sion of the publisher, Academic Press. The material from R.W. Home’spaper in Notes and Records of the Royal Society (1974–1975), 29: 81–90 isreprinted with the permission of the author and the Royal Society. The ex-tract from The Foundations of Scientific Inference, by Wesley C. Salmon,c©1967, is reprinted by permission of the University of Pittsburgh Press.The quotations from (i) volumes III, IX & XXII of the Dictionary of Na-tional Biography (1917), edited by L. Stephen & S. Lee, c© Oxford Univer-sity Press, (ii) A.C. Fraser’s The Works of George Berkeley, D.D. (1901,Clarendon Press), (iii) R.A. Fisher’s Statistical Methods, Experimental De-sign, and Scientific Inference (1995, edited by J.H. Bennett), a re-issue ofStatistical Methods for Research Workers, The Design of Experiments, andStatistical Methods and Scientific Inference c©1990 The University of Ade-laide, (iv) G.B. Hill’s edition of Boswell’s Life of Johnson (1934, ClarendonPress), (v) H. Jeffreys’s Theory of Probability c© Oxford University Press1961 (Clarendon Press), (vi) J.R. Lucas’s The Concept of Probability (1970,Clarendon Press) and (vii) D.O. Thomas’s The Honest Mind. The Thoughtand Work of Richard Price c© Oxford University Press (1977, ClarendonPress) are all made by permission of Oxford University Press.

Permission is granted by the Director of Dr Williams’s Library on behalfof the Trustees to quote (i) from a manuscript (1731–1732), copied by S.Palmer, entitled A View of the Dissenting Intrest in London of the Pres-byterian and Independent Denominations from the year 1695 to the 25 ofDecember 1731. With a postscript of the present state of the Baptists and(ii) from a note by W.T. Whiteley appearing in J.G. White’s The Churches& Chapels of Old London, with A Short Account of those who have Min-

Acknowledgments xvii

istered in Them (printed for private circulation in 1901), such permissionbeing granted subject to the following declaration: (a) that the Trusteeshave allowed access to the manuscript but are not responsible for the se-lection made, and (b) that the author, both for himself and his publisher,waives whatever copyright he may possess in the extracts made, so far asthe exercise of that right might debar other scholars from using and pub-lishing the same material and from working for that purpose on the samemanuscripts. Quotations from the 1974 edition of Richard Price’s A Reviewof the Principal Questions in Morals are made by permission of the editor,D.D. Raphael, the quotations from pp. xvi, xvii, xxiv, and xxxii being fromthe editor’s introduction and that from p. 74 being from the text of Price’sessay itself. The extracts from Bayes’s Notebook are given by permissionof The Equitable Life Assurance Society, in whose possession the Notebookis. Transcripts of Bayes’s letters to John Canton, and of other manuscriptsby him in the possession of the Royal Society, are reprinted by permissionof the President and Council of the Royal Society.

The Register of Burials in Bunhill Fields (RG4/3982) and the wills ofJoshua (PROB 11/746) and Thomas Bayes (PROB 11/865) are housed inthe Greater London Record Office in London, England. L. & B. Bailey’sHistory of Non-conformity in Tunbridge Wells (1970) is in the TunbridgeWells Library. The Stanhope of Chevening papers are housed in the Cen-tre for Kentish Studies, Maidstone, Kent: I am indebted to the ResearchArchivist, Mrs L. Richardson, for her help in accessing the manuscriptsand for permission to cite them here. W. Besant’s London in the Eigh-teenth Century and London City, published by Adam & Charles Black,are now out of copyright, as is Besant and G.E. Mitton’s The Fascinationof London. Holborn and Bloomsbury. The material quoted from R. Blan-chard’s paper in Restoration and Eighteenth-Century Literature. Essays inHonor of Alan Dugald McKillop (The University of Chicago Press), is notunder copyright. References to the list of subscribers to John Ward’s Livesof the Professors of Gresham College, the letters of Ward to Thomas Bayesand Skinner Smith, John Rippon’s lists of epitaphs in Bunhill Fields andTunbrigalia are from The British Library’s Add. Mss. 6207.f.1, 6222.c9,28513–28523 and 11631.e.73(1), and are made with acknowledgments tothe Library. Thomas Bayes’s Election Certificate is copyright by The RoyalSociety, and is reproduced with permission. The material c©1979 from TheScience of Music in Britain, 1714–1830 by J.C. Kassler is reproduced bypermission of Routledge, Inc., part of The Taylor & Francis Group. I regretthat, despite serious endeavours, I have not managed to contact the author,whose work is hereby acknowledged.

Four sketches are reprinted by permission of The Brtish Library (withshelfmarks), viz. (i) the portrait of Joshua Bayes, from W. Wilson’s The

xviii Acknowledgments

History and Antiquities of Dissenting Churches [489.b.12, 13], (ii) LeatherLane Chapel, from G.W. Thornbury’s Old and New London, vol. 2 [HLR942.1], (iii) the Parade at Tunbridge Wells, from J. Phippen’s Colbran’sNew Guide for Tunbridge Wells [796.e.14] and (iv) a map of TunbridgeWells, from J. Britton’s Descriptive Sketches of Tunbridge Wells [G3926].

Despite earnest endeavours, my attempts at tracing the current copy-right holders (to whom my sincere acknowledgements are due) of the follow-ing works have been unsuccessful: (i) R. Bayne’s edition of Joseph Butler’sThe Analogy of Religion, Natural and Revealed, published by J.M. Dent &Sons Ltd. in 1906; (ii) R. Nevill’s London Clubs their histories and treasures,published by Chatto & Windus in 1911; (iii) A. Armitage’s Edmond Halley,published by Thomas Nelson and Sons Limited in 1966; (iv) A.A. Luce &T.E. Jessop’s The Works of George Berkeley Bishop of Cloyne, publishedby Thomas Nelson and Sons Ltd in 1948; (v) P.F. Mottelay’s BibliographicalHistory of Electricity and Magnetism chronologically arranged (1922), pub-lished by Charles Griffin; (vi) D.D. Roller’s The De Magnete of WilliamGilbert (1959), published by Menno Hertzberger; (vii) W.G. Bell’s TheGreat Plague in London in 1665 (1951), published by The Bodley Head(Random House); (viii) R. Lynd’s The Pleasures of Ignorance (1928), pub-lished by Methuen & Co. Ltd; (ix) J.H. Wilson’s The Court Wits of theRestoration (1967), published by Frank Cass & Co. Ltd; (x) C.B. Boyer’sThe History of the Calculus and its Conceptual Development (1949), DoverPublications; (xi) R. von Mises’s Probability, Statistics and Truth (1981),Dover Publications; (xii) H. Poincare’s la Science et l’Hypothese (1903),published by Flammarion; (xiii) T. Carlyle’s Scottish and Other Miscella-nies (1915), W. Cobbett’s Rural Rides (1912), J. Stow’s A Survay of Lon-don (1912) and J. Butler’s The Analogy of Religion (1906), all publishedby J.M. Dent & Sons, Ltd (The Orion Publishing Group); (xiv) C. Eisen-hart’s ‘Boscovich and the combination of observations’, pp. 200–212 in L.L.Whyte’s Roger Joseph Boscovich, S.J., F.R.S., 1711–1787; Studies of hisLife and Work on the 250th Anniversary of his Birth (1961), L. Hogben’sStatistical Theory. The Relationship of Probability, Credibility and Error(1957), B. Lillywhite’s London Coffee Houses (1963), M.E. Ogborn’s Eq-uitable Assurances. The Story of Life Assurance in the Experience of theEquitable Life Assurance Society 1762–1962 (1962), B. Russell’s Historyof Western Philosophy (1961), all published by Allen & Unwin Ltd; (xv)F. Mosteller and D.L. Wallace’s Applied Bayesian and Classical Inference.The Case of The Federalist Papers (1984), published by Springer-Verlag;

Acknowledgments xix

(xvi) B. de Finetti’s ‘La Prevision: ses lois logiques, ses sources subjec-tives’, pp. 1–68 in Annales de l’Institut Henri Poincare (1937), publishedby Gauthier-Villars; (xvii) P. Walley’s Statistical Reasoning with ImpreciseProbabilities (1991), published by Chapman and Hall; (xviii) F.M. Clarke’sThomas Simpson and his Times (1929), published by Columbia UniversityPress; (xix) A.N. Whitehead’s Adventures of Ideas (1967), published byThe Free Press; (xx) D.T. Whiteside’s The Mathematical Works of IsaacNewton, vol. I (1964), published by the Johnson Reprint Company Ltd;(xxi) H.H. Horne’s The Philosophy of Education. Being the foundations ofeducation with the related natural and mental sciences (1916), publishedby Macmillan Publishing Company, Inc.; (xxii) I. Tweddle’s James Stirling‘This about series and such things’ (1988), Scottish Academic Press Ltd;(xxiii) C.L.S. Linnell’s The Diaries of Thomas Wilson, D.D. 1731–1737 and1750 (1964), published by the Society for Promoting Christian Knowledge;(xxiv) H. McLachlan’s English Education under the Test Acts (1931), pub-lished by Manchester University Press; (xxv) T.J.I’A. Bromwich’s An In-troduction to the Theory of Infinite Series (1931), published by Macmillan;(xxvi) T. Huxley’s Science and Education. Essays (1905) and Science andChristian Tradition. Essays (1909), published by Macmillan; (xxvii) H.E.Kyburg’s Probability and Inductive Logic (1970), published by Macmillan.

My thanks are due to the staff at many institutions where I have spentperiods of sabbatical leave during which much of this research was un-dertaken: in particular, the staff of University College, London; NicholasBingham and Rudiger Kiesel, then of Birbeck College; and Stephen Stigler,of the University of Chicago. I also thank H.W. Johnson, of the EquitableLife Assurance Society, for providing me with a copy of Bayes’s Notebook.

This tribute would be incomplete without mention of my indebtednessto Jo Currie, for her help in finding documents relating to Bayes in thearchives of the Edinburgh University Library, to Roger Farthing, for shar-ing his expert knowledge of Tunbridge Wells with me and for the JefferyFamily Tree reproduced here with his permission, and to Jackie Sylaides,for her typing of much of the original manuscript. I also acknowledge thehelp of my colleagues Hugh Murrell, for his checking of some numericalresults using Mathematica�, and Frank Sokolic and Brennan Walsh forthe professionally produced diagrams. Finally I thank the editorial staff ofSpringer-Verlag New York, Inc., for their assistance.

Contents

Preface ix

Acknowledgments xiii

1 Introduction 1

2 A Bayesian Genealogy 62.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 The early ancestors . . . . . . . . . . . . . . . . . . . . . . . 72.3 Joshua Bayes . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Appendix 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Thomas Bayes: A life 373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Thomas Bayes’s education . . . . . . . . . . . . . . . . . . . 383.3 Thomas Bayes and Tunbridge Wells . . . . . . . . . . . . . 583.4 The faded Bayes . . . . . . . . . . . . . . . . . . . . . . . . 963.5 Appendix 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4 Divine Benevolence 1014.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.2 The Tract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.3 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . 1404.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

5 An Introduction to the Doctrine of Fluxions 1825.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1825.2 The Tract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1925.3 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . 2195.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 2495.5 Appendix 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 250

xxii Contents

6 On a Semi-convergent Series 2536.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2536.2 The Tract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2546.3 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . 255

7 The Essay on Chances 2587.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2587.2 The Tract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2697.3 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . 2987.4 Editions of the Essay . . . . . . . . . . . . . . . . . . . . . . 334

8 The Supplement to the Essay 3368.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3368.2 The Tract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3388.3 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . 353

9 Letters from John Ward 3709.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3709.2 Letters to Bayes and Skinner Smith . . . . . . . . . . . . . 3719.3 Translation of Letters . . . . . . . . . . . . . . . . . . . . . 372

9.3.1 The letter to Bayes . . . . . . . . . . . . . . . . . . . 3729.3.2 The letter to Skinner Smith . . . . . . . . . . . . . . 373

9.4 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . 374

10 Miscellaneous Items 38410.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 38410.2 A Letter from Bayes to Canton . . . . . . . . . . . . . . . . 38510.3 Commentary on the letter to Canton . . . . . . . . . . . . . 38610.4 Item on Electricity . . . . . . . . . . . . . . . . . . . . . . . 39510.5 Commentary on the Item on Electricity . . . . . . . . . . . 39910.6 Papers in the Stanhope Collection . . . . . . . . . . . . . . 412

11 The Notebook 42011.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 42011.2 Attribution of the notebook . . . . . . . . . . . . . . . . . . 42011.3 The shorthand . . . . . . . . . . . . . . . . . . . . . . . . . 42111.4 General plan of the notebook . . . . . . . . . . . . . . . . . 42311.5 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . 423

11.5.1 Probability . . . . . . . . . . . . . . . . . . . . . . . 42311.5.2 Trigonometry . . . . . . . . . . . . . . . . . . . . . . 42911.5.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . . 43411.5.4 Solution of equations . . . . . . . . . . . . . . . . . . 43611.5.5 Series . . . . . . . . . . . . . . . . . . . . . . . . . . 44011.5.6 The differential method . . . . . . . . . . . . . . . . 45111.5.7 Numbers . . . . . . . . . . . . . . . . . . . . . . . . 45411.5.8 Miscellaneous mathematics . . . . . . . . . . . . . . 456

Contents xxiii

11.6 Natural philosophy . . . . . . . . . . . . . . . . . . . . . . . 45911.6.1 Electricity . . . . . . . . . . . . . . . . . . . . . . . . 45911.6.2 On the weight of a body . . . . . . . . . . . . . . . . 46411.6.3 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 46511.6.4 Harmony . . . . . . . . . . . . . . . . . . . . . . . . 467

11.7 Celestial mechanics . . . . . . . . . . . . . . . . . . . . . . . 47111.8 Miscellaneous matters . . . . . . . . . . . . . . . . . . . . . 480

11.8.1 Of the pyramid measured by Greaves . . . . . . . . 48011.8.2 Of weights and measures . . . . . . . . . . . . . . . . 48111.8.3 Extract from a dissertation upon cubits by Sir

Isaac Newton . . . . . . . . . . . . . . . . . . . . . . 48211.8.4 Aulay Macaulay’s shorthand . . . . . . . . . . . . . 48611.8.5 Estimate of the national debt upon 31 Dec. 1749 . . 48611.8.6 A shorthand verse . . . . . . . . . . . . . . . . . . . 48611.8.7 Warburton’s syllogism . . . . . . . . . . . . . . . . . 48711.8.8 Lettres Provinciales . . . . . . . . . . . . . . . . . . 48711.8.9 A Prescription . . . . . . . . . . . . . . . . . . . . . 490

11.9 Appendix 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . 49111.10 Appendix 11.2 . . . . . . . . . . . . . . . . . . . . . . . . . 49611.11 Appendix 11.3 . . . . . . . . . . . . . . . . . . . . . . . . . 49711.12 Appendix 11.4 . . . . . . . . . . . . . . . . . . . . . . . . . 498

12 Memento Mori 49912.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 49912.2 A small old book . . . . . . . . . . . . . . . . . . . . . . . . 51012.3 Register of Burials in Bunhill Fields . . . . . . . . . . . . . 51112.4 Inscriptions on vault . . . . . . . . . . . . . . . . . . . . . . 512

12.4.1 Inscriptions from Rippon . . . . . . . . . . . . . . . 51212.4.2 Further inscriptions from Rippon . . . . . . . . . . . 51312.4.3 Inscriptions on present vault . . . . . . . . . . . . . 51412.4.4 Inscriptions from Jones . . . . . . . . . . . . . . . . 516

12.5 Bayesian wills . . . . . . . . . . . . . . . . . . . . . . . . . . 51612.6 Speldhurst memorials . . . . . . . . . . . . . . . . . . . . . 524

Notes 526

Bibliography 595

Index 645