Never at Rest: A Biography of Isaac Newton 9781107392793, 1107392799

This richly detailed 1981 biography captures both the personal life and the scientific career of Isaac Newton, presentin

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Never at Rest: A Biography of Isaac Newton
 9781107392793, 1107392799

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Table of contents :
Cover
Title
Copyright
Contents
Preface to the paperback edition
Preface
Acknowledgments
A note about dates
Abbreviations used in footnotes
1 The discovery ef a new world
2 A sober, silent, thinking lad
3 The solitary scholar
4 Resolving problems by motion
5 Anni mirabiles
6 Lucasian professor
7 Publication and crisis
8 Rebellion
9 Years of silence
10 Principia
11 Revolution
12 The Mint
13 President of the Royal Society
14 The priority dispute
15 Years of decline
Bibliographical essay
List of illustrations
Index of Newton's life and works
General index

Citation preview

Never at Rest

A Vulga r Mechanick can pra ctice what he has been taugh t or seen done, bu t if he is in an erro r h e kn ows not how to find it out and co rrect it, and if y o u p ut him out of his road, he is at a stand; Whereas he that is able to reason nimbly and judi­ cious ly about figure, force and motion, is never at rest till he gets over ev ery ru b . Isaac Newton t o Nathaniel Hawes 25 May 1694

NEVER AT REST A Biography

of Isaac Newton

RICHARD S. WESTFALL

Professor of History of Science Indiana University

CAMBRIDGE UNIVERSITY PRESS

to

GLORIA

CAMBRIDGE UNIVERSITY PRESS

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Information on this title: www.cambridge.org/9780521231435

© Cambridge University Press 1980 T his publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1980 Reprinted 1981, 1982 First paperback edition 1983 20th printing 2010

A

catalog record for this publication is available from the British Library. ISBN 978-0-521-23143-5 Hardback

ISBN 978-0-521-27435-7 Paperback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

Contents Preface x xiii Acknowledgments xvi A note about dates Abbreviations used in footnotes

xvii

The discovery of a new world 1 2 A sober, silent , thinking lad 40 3 The solitary scholar 66 105 4 Resolving problems by motion 140 5 Anni mirabiles 6 Lu casian p rofessor 176 7 Publication and crisis 238 281 8 Rebellion 9 Years of silence 335 10 P rincipia 402 469 11 Revolution 1 2 The Mint 551 13 President of the Royal Society 627 14 The priority dispute 698 15 Years of decline 781 1

Bibliograph ical essay List of illustrations 887 Index

875 885

Preface to the paperback edition

OT a great deal of time has p assed s ince I wrote a preface for the

Nfirs t printing of this book, and with that preface brought more

than twenty years of work to a conclusion. After s uch a period I was, I confes s , ready to move on to o ther m atters , and with the exception of one Newtonian topic, I have done so. As a result I am not now prepared o r inclined to revise the work in any fundamental way. Beyond correcting a few typo graphical and factual errors , I confine myself to indicating here some pass ages that reviews and recent publications h ave convinced n1e ought to b e altered , and to mention­ ing the one topic on which my o wn research since co n1pleting the biography has led me to a deeper understanding . In Chap ter 2 , pp. 57-58, I as sert that Newton's secondary edu ca­ tion did not include any s ignificant m athemati cs . Recen tly, D. T. Whiteside has uncovered in Grantham a po cketbook dated 1 654 that contains extensive " Notes for the Mathem aticks. " [See " New ton the Mathematici an , " in Z. Bechler, ed. , Contemporary Newtonian Re­ search (D o rdre cht , 1 98 2 ) , pp . 1 1 0- 1 1 . ] I t appears to Whiteside to have b een written in the hand of Henr y S tokes , mas ter of the gr am­ m ar s chool of Grantha m , whi ch Newton began to attend during the year following the notes. Although most of the p ass age is devoted to elementar y calculations , such as rules for determining the areas of fields , that a countr y gentlem an might have found useful, not all of it remains at thi s level . S ixty-five p ages on "The Measureing of Tri an­ gles [ & ] Circl es" include ins tru ctions on how to cal culate a table of sines (a task that Newton undertook, albeit bri efly, at Cambridge) , and a method for ins cribing an equilateral septu agon in a circle (an­ other topic that Newton encountered anew in his reading at Cam­ bri dge) . S tokes even included the limits that Archi medes establis hed for the value of pi, more than 310/71 and les s than 31h. As Whites ide does not fail to rem ark, this passage of the " Notes " inevitably re­ m i n d s o n e o f t he g e o m e t r i c fi g u r e s i n c i s e d in t h e p l a s t e r a t Wo olsthorpe . The pocketbook s trongly implies that m y words on Newton's gram mar school education need b asic revision. I t is highl y probable that he was far from a novice in mathematics when he ar­ rived at Ca mbridge, and the sudden i m m ersion in m athem a tical study that began sometime during his undergraduate years as well as the burst of cre ativity that acco mpanied it, though not rendered v 11

Vlll

Preface to the p ap erback editio n

one whit less s tunning b y this new information, do become consid­ erably more comprehensible . In his review of this b ook and in private communications with me, White side has argued that in Chapter 4 , the m aj or dis cus sion of Newton's mathematics (and I woul d add, in pp . 23-38 of Ch apter 1, the dis cus sion of seventeenth-century m athematics before Newton) , I have neglected the distinction between indivisibles and infinitesi­ m als , and that, as a result , the clarity of my exposition and s ome of the conclusions I draw s uffer. I do not consider myself to be an his to­ rian of mathem atics , and I am not now ready to plunge b ack into the demanding world of seventeenth-century mathematics in order to m ore fully clarify the m atter in my own mind. Consequently, I am not able here to offer the revisions that are probably in order. Suffice it to s ay that in my experience, when Whiteside speaks on sev en­ teenth-centu r y m athem atics , the wise attend. Hence I ca ution read­ ers that in those pass ages , where something of importance t o them inv olves indivisibles o r infinitesimals , they would do well to cons ult the work of others , s uch as Whiteside 's edition of the Ma th ematical Pap ers) his other writings , or the b ooks I cite in the foo tnotes and bibliography. Cha pter 9 contains a dis cussion (pp . 35 1 -6) of a theological work, Th eologia e gen ti lis origines p h ilosophicae J which Newto n began to compose in the 1 680s . The m ore I thought about it and ab out the fact that Newton continued to refer to it the rest of his life, the more im portant it seemed to m e . About the time this biography firs t ap­ peared , I became convinced that I had not sufficiently understood this m anus cript; it cons titutes the one Newtonian topic that I have e xplored further since publication of the b ook. The study confirmed m y gro wing belief that it was the most important theological treatise Newton ever composed, even though he never put it in a finished form. One must not be misled by the earnest narratives , which ring so quaintly in our twentieth-century ears , of the lives of N o ah and his offs pring. Thes e were the familiar s cholarly themes o f the day, and Newton could no m ore leap out of his own age than any of us can. He could, however, bend this accepted material to radical new purposes . I t appears to me that the Origines can be adequately de­ s cribed only as the firs t of the deist tracts . I have dis cussed it at length and compared it to superficially similar works upon which he drew in an essay-" Isaac Newton's Theologiae Gen tilis Origines Ph ilosophi­ cae J " in W. W. Wagar, ed. , The Secular Mind (New York, 1982) , p p . 1 5-34 . To the best of my knowledge, this e s s a y i s the only dis cussion in print , beyond the brief pass age in the present biograph y, of this important Newtonian manus cript, and I urge anyone interes ted in Newto n 's theological views to consult my ess ay.

Preface to the paperback edition

IX

I devote pp. 7 40-4 of Chapter 1 4 to the correction of the error in Proposition X , Book I I , for the second edition of the Principia) re­ marking at one point that "One would like to know n1orc ab out the circun1 s tan ces ." Volun1e 8 of Whiteside's edition of the Matlzematical Papers has appeared s ince I wro te tha t line, and, as a result, we do know more about the circums tances . The treatment of that matter (pp . 3 1 2-424) is one of the tours de force of Whites ide 's final volurne . He assemb les the papers on which Newton initially confirmed the validity of Bernoulli's obj ection, and he identifies , as Bern oulli was not able to do , the precise nature of the error. There follow the man­ uscript s of six s ucces s ive attacks on the problem, together with seven other draft passages , leading up to N ewton 's s ucces sful l o cation of the erro r and his correction of i t . Indeed , as the m anuscri p ts reveal, N ewton even pro ceeded to an alternative demonstration of the cor­ rect res ult . Mos t interes ting of all , Whi teside argues persuas ively that only a con1putational error prevented Newton from realizing that he could h ave co rrected the proposition within the fran1ework of the o riginal demons tra tion . Bernoulli seized on the radically amended p ropos ition - for whi ch he claimed credit, as the newl y printed pages m ade its l ast-minute ins ertion clear - as evidence that Newton had not understoo d s econd derivatives when he composed the first edi­ tion of the Principia. That charge , endles sly repeated, figured promi­ nently, much to Newton's em b arrass ment, in the priority dispute. This b rief paragraph cannot begin to do j us tice to the richnes s of Whiteside's presentation , to which I refer readers wishing to kn ow more ab out the incident. In his review of the present work in the American Historical Review, 87 (1982) , 1353, I . Bern ard Cohen points out that the interpretations in the book are mine and are not necess arily shared by other Newto­ nian s cholars. I never intended i t otherwise . When I signed my name to the b ook, I unders tood that I was taking s ole resp onsibility for its content , and I rather ass umed tha t readers would unders tand the same. I have no privileged acces s to final truth, of cours e , and P ro­ fessor Cohen may prove to be right about the interpre tations he chal­ lenges in his review . Suffice i t to s ay that I rem ain unrep en tant as of this moment, and aside from the four pass ages above, the b ook con­ tinues to rep resent my view of Isaac N ewton .

Preface

HE utility

T

of bio g raphy , D r . Johnson argued, rest s on the fact that we can enter by sympathy into situations in which o thers have found them selves . P arallel circu m stances to which we can conform our minds shape every life. Even the g reat are not re­ m oved fro m the facto rs co mmon to all: " We are all prompted b y the same motives , all deceived b y the s a m e fallacies , all animated b y hope, o b structed b y danger , ent angled b y desire , and s educed b y pleasure . ' ' I must confess that twenty years devoted t o the biog ra­ phy of Newton have not in my case confi rmed D r . John son ' s dic­ tum . The m o re I have s tudied him , the m o re Newton ha s receded fro m me. It has been my privileg e a t various times to know a nu n1ber of b rilliant men , men who m I a cknowledge without hesita­ tion to be m y intellect u al sup eriors . I have never , however , met one against who m I was unwilling to measu re myself, s o that it seemed reasona ble to say that I was half a s a ble as the per son in question , or a third o r a fou rth, but in every case a finite fraction . The end result of my study of Newton has served to convince me that with him there is no measure . He has become for me wholly other, one of the tiny handful o f supreme geniuses who have shaped the catego­ ries of the hu man intellect , a man not finally redu cible to the criteria by which we comp rehend ou r fellow beings , tho se parall el circum­ stances o f D r . Johnson . Why then , one might ask, am I attempting to write Newton's biography? M y second prefatory confes sion is that increa singly I have asked the same question myself. Had I know n , when in youth­ ful self- confidence I com mitted m yself to the task, that I would end up in similar self-dou bt , su rely I would never have set out . I did perceive that it would be a long and arduou s t ask, though I was willing to undertake the labo r . I thought at the time it would take ten years, not far sho rt o f eternity a t that point in my life, though now , in its brevity , an indication o f the chasm between expectation and realit y . Perhaps even the prospect of twenty years wou ld not have tu rned me back, but the o ther chasm, the unexpected gulf opening between me and my subj e ct , would have been another m atter . As I face the situation now , not in pro s pect but in retro­ spect , the lingering influ ence of the Puritan ethic m akes the pro s­ pect of discarding the fruits o f so m u ch earnest t oil abhorren t . For that matter , could any other potential biographer of Newton escape the s am e dilem m a? Only another Newton could hope full y to enter x

Preface

XI

into hi s being , and the econo my of the human enterp ri se is such that a second Newton would not devote hi mself to the biography of the fi rs t . If hi s tory has a functi on - and my doubts have never extended to questioning that it has - perfo rce it must deal wi th the Newtons. Everyone who is info rmed agrees on the need fo r a new bi ograp hy to repl ace Sir David B rews ter ' s masterpi ece , which is now one hundred and twenty-five yea rs old . Others have aspi red. It is not fo r me to decide whether I have s ucceeded where they failed. Wi th all of the hesitations the paragraph above implie s , I place the result of long years on the altar of hi s tory wi th the hope tha t it may add its bi t to the understanding of the past to which the modern age has already contributed so much. In writing Newton's bi ography , I have attemp ted , in accordance wi th my unders tanding of bi ography a s a literary fo rm , to avoid compo sing an essay on Newtonian science . At the same ti me I have sought to make Newton the s cientist the central character of my drama. While he devoted hi msel f extensively to other acti vities which a biography cannot igno re , fro m theology on one hand to administration of the Mint on the other, Newton holds our atten­ tion only because he was a s cientist of transcendent importance. Hence I tend to think of my wo rk as a scientific bi ography , that is , a bi ography in which Newton' s s cientifi c career fu rnishes the cen­ tral theme . M y goal has been to present his science , not as the finished p roduct which has done so much to shape the whole of the modern intellect , but as the developing endeavo r of a living man confronting it as p roblems still to be s olved. Scientists and philoso­ phers can probe the finished p roduct. My interes t in this biography centers exactly on what was not yet complete , the obj ect of New­ ton' s o wn activity , the subs tance of a life devoted to p robing the unknown. I ha ve tried to p resent his scientifi c endeavo rs in the context of his life, fi rst in Woolsthorp e and Grantham, then in Ca mbridge , and finally in London. To an extent few others have equaled , however, Newton was a man of learning . Never fully at ease wi th others , he held his distance and lived largely in the setting of hi s own s tudy . His books furnished the context of hi s life more than Cambridge or London did . A bi ographer igno res thi s truth about Newton at his own peril . I have done my bes t to keep it in mind and to p resent a pi cture of Newton in which the pursuit of truth , most imp o rtantly though not exclusively s cientifi c truth , formed the essence of his life. To the extent that I have suc ceeded in thi s , the biography as a whole will al so succeed . The fi rst volume of the Royal S o ciety ' s edition of Newton' s Corresp ondence appeared about the ti me I began serious work on the bi ography. Now all seven volumes are in p rint , and the footnotes in my book bear testimony to thei r indispensable aid . They are only

x 11

Preface

part of the flood of Newtonian p ublications du ring the last two decades , on all of which my own wo rk rests directly , to all of which I want to acknowledge a debt which is in fact bey ond ac­ knowledgment . The Correspondence has been necessary , as has the White side edition of the Mathema tical Pap ers, the final vol ume of which (to my loss I am su re) ha s yet to be published as I write this P reface . One of the features of my biograp hy , which s ets it apart from earlier ones , i s the chronolo gical account of Newton ' s m athe­ m atical activity . The account is my own , and I take full resp onsi bil­ ity for it . It would ha ve been imp ossibl e , ho wever , without White­ side's monument of scholarship . Bey ond these general publications a re a nu mber of others m o re restricted to single a spects of Newton's life: I . Bernard Cohen's Papers & Letters plus his edition of the Princip ia with v ariant read­ ings u ndertaken j oint ly with the late Alexandre Koyre; A . R . and M. B . Hall ' s Unpublished Scientific Papers; John Herivel 's Backgrou nd to Newton 's 'Princip ia '; and (though it differs fro m the above in being fo rm ally a monograph) B . J . T . Dob b s ' s Fo undations of New­ ton 's A lchemy . The p ublication of these wo rk s has signifi cantly ex­ panded the oppo rtunities of Newtonian scholarship . I am not the fi rst to benefit fro n1 them , no r will I be the last . I regret that my wo rk i s done too soon to derive further benefit from the p ublica­ tion of optical p apers that Alan S hapiro has undertaken . During the time I ha ve been at work on Newton , I have received as sistance of m any kind s from m any sources . Grants fro m the N a­ tional S cience Foundation, the Geo rge A . and Eliza Gardner How­ ard Foundation, the American Council of Learned S ocietie s , a nd the National End owm ent for the Hu manities; and s abb atical le aves fro m Indiana University ha ve provided most of the tim e for study and writing , much of it in England , where the great bulk o f Newton's pap ers exi st . One of tho se years I had the p rivilege and advantage to be a Visiting Fellow of Clare H all , Cambridge. The National Science Foundation and Indiana Uni versity ha ve also he lped to fi­ nance the acquisition of photocopies of Newton' s p apers . The staffs of many li braries have outdone thems elves in kind a ssistance , most prominently (in propo rtion to my dem ands) the C ambridge Uni­ versity Library , the Trinity College Library , the Widener Library at Harvard , the B abson College Library , the Indiana University Li­ b rary , and the Public Reco rd Office. Mo st of the typing I owe to a succession of secretaries over the years in the Dep artment of His­ to ry and Philosophy of Science at Indiana University , b ut a mong them e specially Karen Blaisdell . The help of Anita Guerrini in proofreading has been invaluable . I cannot sufficiently express my grJtitude to those I ha ve mentioned and to many others who ha ve help ed in less central way s . I can at least try to exp ress it, and I do.

Preface

x 111

N o r can any autho r omit hi s famil y . By the ti me my chil dren reached consciousnes s , I had emb arked on the biography . Now I finis h i t as they co mplete thei r educations and set out on thei r own . The whole of thei r inti mate experience of me has been flavored by the addi ti onal presence of Newton . I do no t know if I would have been a more sati sfactory father wi thou t the bio graphy . Suffi ce it fo r them t o realize that wh at they put u p wi th o v e r the years d i d in the end achi eve some sort of conclu sion, incarnation as a b ook . My wi fe of course endu red more , and I thank her for enduring it wi th grace and un ders tandin g . In the end she discovered the only ade­ quate defense - s he is wri ting a book he rsel f and may , if I i mprove , acknowledge my encouragement and support in her own preface . R.S.W.

Acknowledgments

WISH

to ackno wledge permi s sion granted me by the B abson Col­ lege Lib rary to reproduce an alchemi cal diagra m , a plan of the Jewi sh temple , and a scheme of the twelve gods of the ancient peoples , all found among the Grace K . B abson Collecti on; by the Trustees of the B ritish Museu m to rep ro du ce a pi cture of the i vory bust by Le Marchand; by the University of Californi a Pres s to rep roduce six di agrams fro m thei r editi on of the English translati on of the Principia; by the Syndics of Cambri dge University Library to reproduce eleven sket ches and p ass ages fro m th e P o r t s m o u th Papers ; by C ambridge University Pres s to rep rodu ce the pi cture of Croker 's medal of Newton fro m John Crai g , Th e Mint (C am­ bri dge , 1 953) ; by the Joseph Halle S chaffner C ollection, University of Chicago Li brary , to rep roduce a drawing of che mi cal furnaces ; by Colu mbi a Uni versity to rep roduce an engraving of the Ri chter miniatu re and a lithograph of the Gandy po rtrait fro m the David Eugene S mi th Collection , Rare B ook and Manuscript Li brary; by the President and Fell ows of C o rpus Christi College , Oxfo rd , to repro duce a drawing of the co met of 1 680- 1 ; by Lo rd Egremont and the Petworth Es tate to rep ro duce the Kneller portrait of 1 720; by the Bibliotheque Publique et Universi taire de Geneve to rep ro­ duce the portrait of Nicolas Fati o de Duillier; by W. Heffer & Sons Ltd . of C ambri dge , Englan d , to repro duce the p o rtrait of Newton in thei r possession; by the Jewish National and Uni versity Li b rary

I

XIV

Acknowledgments

to rep rodu ce a scheme of the Revelation of S t . John the Divine fro m the Y ahuda Papers ; by the Provost and Fellows of King' s College , C ambri dge, t o reproduce a set .- o f alchemical symbols from the Keynes Co llection and an ivo ry plaque by Le Marchand ; by the Tru stees of the N ational Po rtrait Gallery to reproduce the Kneller p o rtrait of 1 702 and a portrai t , artist unknown , of 1 726 ; by Neale Watson A cademic Publications , In c . , to rep ro duce fou r diagrams fro m R i chard S . Westfall , Force in Newton 's Physics (London , 1 97 1 ) ; by the Warden and Fell ows o f New College , Oxfo rd , to rep rodu ce a drawing of Woolstho rpe , a pi cture of the house on S t . Martin ' s Street , a drawing o f the experimen tum crucis, and a scheme o f chro­ nology fro m the New College M S S ; by Lo rd Po rtsmou th and the Tru stees of the Portsmouth Estates to reproduce two Kneller por­ traits as well as the Tho rnhill po rtrait of 1 7 1 0; by the Royal S o ciety to reproduce their drawing of the reflecting telescope , the sketch by S tukele y , an ivo ry plaque by Le Marchand , the Jervas portrait of 1 703 , the V anderbank p o rtrait of 1 725 , and the V anderbank p o rtrait of 1 726 ; by S otheby Parke B ernet & Co . to rep roduce pi ctu res of a bust an d a plaque sculp ted by Le Marchand ; by the Uni versity of Texas to rep roduce the Newton family tree ; by the M aster and Fellows of Trinity C ollege , Camb ri dge , to rep roduce the Tho rnhill portrait of 1 7 1 0, the M urray p o rtrait of 1 7 1 8, the V anderbank por­ trait of 1 725 , and the S eeman portrait o f 1 726 ; and by the Y ale Medi cal Lib rary to rep roduce the drawing of Jupiter enthroned from their alchemical p ap er. I wis h further to acknowledge the permis sion and courtesy given me by the American Philosophical S o ciety ; B abson C ollege (fo r the Grace K. B abson Colle ction) ; the B odleian Li brary; the S yndi cs o f the C am bridge University Library (fo r the Portsmouth P apers and other M S S) ; the University of Chi cag o Library (for the Josep h H all e S chaffner C olle ction) ; the William Andrews Clark Memo rial Library (of U CLA , Los An geles , C alifo rnia) ; the Rare Book and Manuscript Li brary of Columbia University (for the David Eugene S mith Colle ction) ; the Fran cis A. C ountway Library of Medicine (Bo ston , Massachu setts) ; the Edinbu rgh University Library; the Emmanuel College, Cambridge , Library ; the Syndics of the Fitz­ william Mu seum , C ambridge; the Huntington Library ( S an Ma­ rino , C alifo rnia) ; the Jewish National and Uni versity Lib rary (for the Yahuda MS S) ; the Provost and Fellows o f King's College , Cambri dge (for the Keynes M S S) ; the P ierpont Mo rgan Library ; the Warden and Fellows of New Col lege , Oxford; the Royal Soci­ ety; the S mithsonian In stitution Libraries (fo r the Dibner Collec­ tion) ; the Department o f Sp eci al C olle ctions of the Green Library , S tan fo rd Uni versity (for the Newton Colle ction) ; the Controller o f H . M. S tationery Offi ce (for C rown-copyright re cords i n the Publi c

Acknowledgments

xv

Record Office) ; the Uni versity o f Texas (fo r manu scripts in the Humanities Research Center, Au stin) ; and the Master and Fellows of Trini ty College, Cambri dge , t o cite manuscripts . The Uni versity of Cali fornia Press has all owed me to q u ote from the Caj ori edition of Newton ' s Principia: Cambridge Uni versity Press t o q u ote from I . Bernard C ohen and Alexandre K oyre , eds . , Isaac New ton 's Ph ilosoph iae Natura lis Princip ia Math ematica; from B . J . T . D obbs, Th e Foundations of Newton 's Alch emy; from A . R . and M . B . H all , eds . , Unp ub lished Scientific Pap ers of Isaac New ton; from H . W. Turnbull et al. , eds . , The Corresp ondence of Isaac New­ ton; and from D. T . Whiteside , ed . , The Mathematical Papers of Isaac Newton: D over Publi cati ons , Inc . , to quo te from their edition of Newton ' s Op ticks: A . E. Gunther to quote from R . W . T. Gunther, Early Science in Oxford: H arvard Uni versity Press t o quote from I . Bernard C ohen , ed . , Isaac Newton 's Papers & Letters o n Natural Ph i­ losophy: History of Science to q u ote from Karen Figala, " N ewton as Alchemist" : Oxford University Press to q u ote from Mark Curti s , Oxford and Cambridge in Transition; from John Heri vel , Th e Ba ckground to New ton 's 'Principia '; and from Frank Manuel , Th e Religion of Isaac Newton: Th e No tes and Records of th e Royal Society to q uote from J . E . McGuire and P . M . Rattansi , " Newton and the ' Pipes of Pan' " and from R . S. Westfall , "Short-writing and the State of Newton ' s Conscience , 1 662 " : and Yale University Press to quote from Marjorie Hope Nicolson , Con way Letters. I gratefully ac­ knowledge all of their kindnesses .

A note about dates

ECAUSE England

had not yet adopted the Gregorian calendar (whi ch it treated as a piece of popish superstition) , it was ten days out o f phase with the Continent befo re 1 700 , whi ch England observed as a leap year, and eleven days out of phase after 28 February 1 700 . That is , 1 March in England was 1 1 March on the Continent befo re 1 700 and 1 2 March beginning with 1 700. I have not seen any advantage to thi s work in adopting the cumbersome notation 1 / 1 1 March , etc. Everywhere I have given dates a s they were to the people involved , that i s , English dates for Engli shmen in England and C ontinental dates for men on the C ontinent , with­ out any attempt t o reduce the ones to the others . In the small number of cases where confu sion mi ght arise , I have in cluded in parentheses O. S . (Old S tyle) fo r the J ulian calendar and N . S . (New Style) for the Gregorian . In England the new year began legally on 25 M arch . S ome men adhered faithfully to leg al practi ce; many w rote double years (e . g . , 1 67 1 /2) during the perio d from 1 January to 25 March . Everywhere , except in quotation s , I have given the year as though the new year began on 1 January .

B

XVI

Abbreviations used in footnotes _,

Add MS

Ba bson MS Ba ily Bumdy MS CM Coh en Comm ep ist Corres CSPD C TB C TP Edleston Halls Herivel Hiscock ]B ]BC Keynes MS

Additional MS in the Camb ri dge Unive rsi ty Library (fo r thi s bo ok , that pa rt of the Addi­ tional M S S cons ti tuting the P o rtsmouth Papers) Newton manus cript in the l ibrary of Babson College , B abson P a rk , Mass . Franci s Baily, A n A ccount of th e Revd John Flamsteed, th e First Astronomer Roya l (Lon­ don , 1 835-7) Newton manus cript in the Dibner Collec­ tion , S mi thsonian Insti tuti on Libra ri es Council Minutes of the Royal Soci ety Isaac Newton 's Pap ers & Letters on Na tural Ph i­ losophy, ed . I . Bernard Cohen (Cambri dge , Mas s . , 1 958) Commercium ep istolicu m D . Johannis Collins, et aliorum de analysi promota (London , 1 7 1 3) Th e Co rrespondence of Isaac Newton, ed. H . W . Turnbull , J . F . S co tt , A . R . Hall , and Laura Tilling , 7 vol s . (Cambridge , 1 959-77) Calenda r of S ta te Papers D omestic Calendar of Treasury B ooks Calendar of Treasury Papers Correspondence of Sir Isaac Newton and Profes­ sor Cotes, ed. J . Edleston (London , 1 850) Unp u blished Scientifi c Papers of Isaac New ton , ed . A . R . and Marie B oas Hall (Cambri dge , 1 962) J. W. Heri vel , Th e Background to Newton 's 'Princip ia ' (Oxfo rd , 1 965) W. G. Hi scock , ed . , Dav id Gregory , Isaac Newton and Th eir Circle (Oxford , 1 937) Journal Bo ok of the Royal Society Journal Bo ok (C opy) of the Royal Society Newton manuscript in the Keynes Collec­ tion in the library o f Ki n g ' s College , Cambri dge

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xv 111

Math Mint Op ticks Prin Stukeley Var Prin

Villam il Yah uda MS

Abbrevia tions used in footnotes

The Mathematical Papers of Isaac Newton, ed. D . T . Whi tes i de , 8 vols . (C ambri dge , 1 967-80) Mint Papers in the Public Record - Offi ce Op ticks, based on the 4th ed . (New Yo rk , 1 952) Math ematical Princip les of Natural Ph ilosophy, tran s . Andrew M otte , rev . Florian Caj ori (Berkeley , 1 934) William S tukeley , Mem oirs of Sir Isaa c New­ ton 's Life, ed. A . Hastings White (London , 1 936) Isaac Newton 's Ph ilosophia e Natura l is Princip ia Math ematica , 3rd ed . wi th v ariant readings , 2 vols . , eds . Alexandre Koyre and I . Bernard C ohen (C ambridge , 1 972) Ri chard de Villamil , New ton: The Man (Lon­ don , 1 93 1 ) Newton manu script in Yah uda MS Va r. 1 in the Jewi sh N ational and University Library , Jerusalem

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The discovery ef a new world Convictus primus.

Seamdus.

Quadrantarij .

Mr. Johes Smith .

Ric: Smith

Jobes Bigge. Isaac Newton.

Ed: Lowrey. Coll: Trin:

EWTON'S

Jobes Nowell Tho: Ferrar. Barhamus Olyver. Jobes Doud . Jobes Hawkins. Jobes Rowland. Ed: Jolly .

Josua Scargell. Georg: Crosland. Hen: Wright. Jobes Tenant Eras: Sturton. 1

name in the matri culation book of Cambridge Uni­ versi ty on 8 July 1 66 1 , together with tho se of si xteen o ther s tu­ dents recently admi tted to Trinity Co llege, bears witness to an event so obvi ously si gnifi cant in his life (as i t must have been fo r the other si xteen , and as si miliar ev en ts have been fo r countles s y oung men through eight centuries of Western history) that one fli rts with banal­ i ty even to menti on it. He had left his home in the hamlet of Wools­ tho rpe in Lincolnshi re some five week s ea rlier, a raw provincial youth ventu ring mo re than ten miles fro m the pl ace of his birth p rob ably fo r the firs t ti me. H e had been admitted to Trinity College on 5 June. As it turned out , the change in scene inv olved much mo re than the inevi table shattering of rural p rovincialis m . In Cambridg e , Newton discovered a new world . In one sense , of cou rse, every youth who truly enters a university discovers a new world; such is the p rocess of educati on , the opening of fresh horizon s . Newton discovered a new world in a more con crete sense of the ph rase , however. B y 1 66 1 , the radi cal restru ctu ring of natural philosophy that is called the scienti fic revolution was well advanced . B ehind the fa mili ar facade of natu re , philosophers - we would call them scientists - had indeed discovered a new world, a quantitative world instead of the qualitativ e world of daily exp erience , mechani s tic instead of o rgani c, indefinite in extent instead of fini te, an alien world frightening to many but in its chal­ lenge thrilling to s o me. In Cambridge, Newton discovered this dis­ covery . I t was by no mean s inevitable or even probable that he do so , fo r Cambridge University did not thru s t the new world of scientific thought before i ts students . In all likeliho o d , the o ther sixteen young men who matriculated from Trini ty that day in July never susp ected

N

1 Cambridge University Library , Matriculations 1 6 1 3- 1702, 8 July 1661.

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its exi s tence. The obscurity o f their youth p assed imp erceptibly into the obscurity of their manhoo d , and no one today writes thei r biog­ raphies . Cambridge was a place where books were sold, however, and where libraries collected them. One who cho se could encounte r le arnin g which th e university i tself did not fo s ter� Newton cho s e , a n d with th e cho i ce de termined h i s pl ace in histo ry. Th ere were many facets to Newton' s life that were not concerned with natu ral science , and a bio graphy wo rthy of the name mu s t present th em . Nevertheless , the s ole reason one undertakes to write a biography of Newton is the relation in which he s to o d to the new world o f the sci­ entific revolution. As the heavens embrace our globe , so astronomy surrounded the scien tific revolution . To date the beginning of an intellectual mo ve­ ment is always an arbitrary choice , but the succession o f develop­ ments that followed the publication of Nicholas Copernicus's De revo lutionibus orbium coelestium (On th e Revolutions of th e Heavenly Sph eres) in 1 543 has led historians almos t unanimously to assign the birth of modern science to that year. Copernicus p ropo sed a new solution to the central problem that had occupied astronomy for two tho usand years , to account for the irregular motions of the planets agains t the unchanging background of the fixed stars. 2 P re­ vious astronomy had s tarted from the assumption , dictated by com mon sense and daily experience , that the earth is at res t . All the phenomen a observed in the heavens , then , are real motions . C oper­ nicu s p roposed ins tead that heavenly pheno mena are in p art mere appearance s which arise from the motion of the earth. To the earth Copernicus assigned two motion s , a daily rotation on its axis and an annual revolution about the sun. 3 The daily rotation from west to east accounted of course for the app arent daily rotation of the heavens from east to west . More than four hundred years after Copernicus , we still find it convenient to speak of the rising and setting of the sun and moon; befo re Copernicus , they were held t o do s o literally. M o re important for astron omy , Copernicus boldly wrenched the earth from its moorings in the center of the universe , labeled it as one planet among o thers , and set it in mo tion with the o thers around the sun. He p ro moted the sun fro m the rank of planet and placed it at the center of the system. The annual o rbit of the earth explained the mo s t dramatic of the planetary phenomena, thei r periodic retrogressions in their normal progression from west to east among the fixed s tars . To explain 2

I use the phrase "new solution" in contrast to the p revailing astronomy. It is well known that Copernicus was not the first to propose a heliocentric system . 3 Cop ernicus himself ass igned a third motion to the earth, an annu al conical motion of its axis . Kep ler later pointed out that it w as illuso ry .

Th e discovery of a new world

3

p

E

Figure 1 . 1 . A maj or defe rent on an epic ycle where by Ptole m aic as­ tronomy accounted for the apparent retro gressions of the planets .

these phenomen a , Greek astronomers had develop ed the device of the epicycle (Figure 1 . 1 ) . The p lanets did not travel in ci rcles about the earth; rather they traveled in circles (epicycles) the centers of which traveled i n circles (deferents) about the earth . Fro m our point of view, this solu tion to the major p roblem of p lanetary orbits had the effect of proj ecting the earth' s annual orbit onto each p lanet' s motion - as epicycle i n the cases o f the superior planets (Mars , Jupi ter , and Saturn) , as deferent in the cases of the inferior planets (Mercury and V enu s) . The essence of Copernicus ' s revi sion of as­ trono my lay in hi s inauguration of our point of view and in his demonstration that planetary retrogressions may be not real mo­ tions , but apparent motions deriving fro m the real mo tion of earth . Unfortunately , the matter could not rest on this plane of simplic­ ity . As ancient astronomers had established, one uniform ci rcular mo tion cente red in the earth was unable to account for the posi tion s of the sun during a year, and two uniform ci rcular motions , defer­ ent and epicycle , were unable to account for the positions of the planets . In all cases there were small deviations from the positions predicted by the circles , to account for which astrono mers had reso rted to o ther devices such as tiny epicycles and eccentric circles (Figure 1 . 2) . So also Copernicus found that a single circle could not account for the mo tion of the earth or of any p lanet around the sun . With the ancients he shared the conviction that the immutability and perfection of the heaven s require astronomy to confine i tself to co mbinations of the p erfect fi gure , the circle. Hence he too had recourse to tiny epicycles an d eccentrics to account for the same small d eviations . Mo re than half a century later, Copernicu s ' s greatest disciple , Johannes Kepler, co mpleted the structure of heliocentric astronomy

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E

E

(b)

(a) p



E

0

(c)

(d)

Figure 1 . 2 . Various devices in Ptolemaic astrono my. A shows the ef­ fect of a minor epicycle with the same period as the defe r­ ent; B, an epicycle on a m aj o r epic ycle; C, an eccentri c; and D, an eccentric on a defe rent . The size of the minor epi­ cycles and the amount of eccentricity in relation to the size of the deferent circle a re exaggerated considerably.

by abandoning the feature closest to Copernicus's heart, the perfect ci rcle . Employing the immense body of observations co mpiled by the Danish astronomer, Tycho B rahe , Kepler concluded that Mars travels about the sun in an ellipse , and he p romptly generalized the conclusion to all the planets . Published in his Astronomia nova (A New Astronomy , 1 609) , the ellipti cal shape of the orbits i s known as Kepler's first law. In one s troke he swept away all the machinery of epicycles and eccentrics . The minor deviations that had given rise to them merged in the elegance of a single curved line . To the first law h e add ed a second , that the area of the ellipse swept o ut by the radius vector of the moving p lanet is proportional to the time (Fig­ ure 1 . 3). With the second law he fulfilled a necessary demand of every working astronomer in supplying a means by whi c h to com­ pute the location of a planet at any time . T en years later, in the Harmonices mundi (Harmonies of th e World) , Kepler added a thi rd law

Th e discovery of a new world

5

Figure 1.3. Kepler's second law, that the area swept out by the radius vector is proportional to time. The eccentricity of the el­ lipse is greatly exa ggerated .

which tied the planets together into an organized system by relating the mean radii and periods of their orbits: T 2 ex: R3 . The same yea r that saw the publication of Kepler' s Astrono mia nova also witnes sed the entry into astronomy of the instrument des tined to become its principal tool . In 1609 Galileo Galilei fi rst tu rned a telescope on the heavens, and the following year he began to publish his observation s . Galileo was an ardent C opernican and attempted to turn his observa tions to the support of the C opernican system . He observed the rugged surface of the moon , which con­ tradicted earlier notions of the crystalline perfection of the heavens . So did the spots on the sun , which formed and dissolved in rela­ tively s ho rt periods of time, and which moved acro s s its face, indi­ cating that the sun turned on its axis . He discove red s atellites o f Jupiter, so that the earth ceased t o b e unique i n i t s accompanying satellite . He observe d the phases of Venus , which were incompat­ ible with the geocentric sys tem since they demonstrated that Venu s circles the sun . All of this and especially the l a s t lent support to heliocentric a strono my, though there was no way in which Galileo or anyone else could observe the motion of the earth or the central­ ity of the sun through a telescope. 4 4

The literature on the development of heliocentric astronomy is immense. A brief account can be found in Thomas Kuhn, The Copernican Revolution (Cambridge, Mass., 1957), a longer account in Alexandre K oyre , The Astronomical Revolution, trans. R. E. W. Maddison (London, 1973) .

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When Newton arrived in Cambridge , the learned world had had a fu rther half century to digest the work of Copernicus , Keple r , and Galileo . From the point of view of evidence , little had changed . Not until the nineteenth century did the Foucault pendulu m furni sh something like a direct demonstration that the earth turns on i ts axi s , and the observation o f stellar parallax directly confirm the annual motion . Nevertheless , by 1 661 the debate on the heliocentri c universe h a d been settled; those who mattered h a d surrendered to the irresistible elegance of Kepler' s unencumbered ellipse s , sup­ ported by the striking testi mony o f the telescope , whateve r i ts am­ biguities might be . For Newton , the heliocentri c uni verse was never a matter in question. But the heliocentri c universe , especially in Kepler' s formulation , offered as many challenges as conclusions. There had always been a tension within the geocentric system between physical and mathe­ matical accounts . For co smologi cal purposes , A ri s totle's homocen­ tric spheres , which had soli dified over the years into crys talline spheres , provided the physical picture of the universe . Working astronomers had meanwhile constructed p lanetary theory out of the various devices of deferents , epicycles , eccentrics , and equan ts . A final reconciliation between the two had been impossible , but as­ tronomers had learned simply to tolerate the dispari ty . Heliocentric astrono my quickly abolished the crystalline sphere s . In Kepler's vie w , Tycho ' s observations of comets had shattered the spheres; Kepler' s own ellipses confirmed thei r demi se . With the spheres went the very s tructure of the heavens , so that the new astronomy offered the i mprobable assertion that planets , without any means of support or guidance , whether vi sible or imagine d , trace and retrace precisely defined ellipses in the immen sity of space . To thi s unavoidable problem Kepler offered a solution in terms of forces centered in the sun , one pushing the planets along in thei r path s , another controlling their distances from the sun . Thi s celestial dynamics fu rni shed the th read of Ariadne th at Kepler' s in vesti gation su ccessfully followed . Mathematically , it was consistent with his laws . It is one of the anomalies of seventeenth-cen tury science , how­ ever, that a new science of mechanics rendered Kepler' s dynamics obsolete within a generation , although we continue to accept as valid the laws that ori ginally emerged from the dynamics . Descartes pro­ posed a d ifferent physical theory of the cos mos . Hi s vortex explained the gross phenomena of a heliocentric system; i t carried all the plan­ ets in the same p lane in the s ame direction around the sun . What the vortex could not p roduce were the exact mathemati cal relations of Kepler' s laws - indeed not any one of the three . Hence the old tension between the physical structure o f the cos­ mos and the mathematical theory of astronomy reappeared in a new

The discovery of a new world

7

setting . One difference rendered the new version of the tension impossible to tolerate . In the ancient world, physics took prece­ dence ove r as tronomy . Astronomers might save the phenomena with ingenious constructions , but no one asserted their physical reality. Keple r did assert the reality of his ellipses as paths that planets follow through space; and in the end , the seventeenth cen­ tury believed him . One way or ano ther, the disparity between the mathematics of the heavens and their physics would have to be resolved; until it was , a gaping hole leered at natural philosophers from the very heart of their new sci ence . Newton was one of those who q uickly p erceived thi s fact . Astronomy, the science o f the heavens , supplied the cosmic setting of the scienti fi c revolu tion . The most importan t ski rmi shes of the revolution took p lace on ea rth . Early in the seventeenth century , Galileo lai d the foundations of a new science of mechanics . It was impossible for there to be a radical restructuring of natural philoso­ phy in which mechanics , the science of motion , would remain un­ moved , for motion plays a central role in every conception of na­ ture . Like astronom y , mechanics had a tradition stretching back to the an cient world; unlike astronomy, it had been a topic of exten­ sive and p roductive discus sion during the Middle Ages . In addition , it was inextricably entwined in the new astronomy . On the one hand, the received mechanics of Ari stotle, even with i ts medieval modifications , refused to be reconciled to the assertion that the earth is in motion . On the other hand, in the climate of scienti fic thougnt as i t exi s ted in 1 66 1 , only the science of mechanics could resolve the cosmic problem po sed by the new astronomy . If astron­ omy p rovided the setting of the scientific revolution , mechanics supplie d i ts solid core . More than any o ther man, Galileo created the new mechanics . A convinced Copernican , he devoted his energy not to the sort of technical details whereby Kep ler brought as tronomical theory into agreemen t with the observed posi tions of p lanets , but to the ques­ tion of credence raised by the affront to common sense inherent in the proposition of a moving earth . The central argument in Ga­ lileo' s Dia logo sopra i due massimi sistem i del mondo (Dialogue on th e Two Ch ief World Systems) hinged on thi s point . As astro nomy , the Dialogue was a fraud. It expounded a heliocentric system based on circular orbits which could not even app roxi mate observed plane­ tary p o si tions . As mechanics , however, i t showed that a moving earth can be compatible both with untutored daily expe rience and with ca refully observed pheno mena of motion on the earth . Central to Galileo' s a rgument was the princip le of inertia . Galileo did no t in fact use the word " inertia" ; fo r that matter, he did not enunciate the

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Never a t rest

principle in the form we accept today . Nevertheless , he redefined the con cept of motion in such a manner that we recognize in i t the essential a spect of ou r principle of inerti a . To A ri s totle , t o mo ve w a s t o b e moved. The motion o f any body requi red a moving agen t . M o tion i mplied ontological change as well . The growth of an acorn , whereby i t realize s its potential to be an oak , was motion . The edu cation of a youth , whe reby he realizes his potential to be a man of reason , was motion . Manifestly , both processes requi re a cause , that i s , an agent or mover. Equally the mo­ tion of a heavy body fall ing , whereby i t realizes i ts po tential a s a heavy body to be as close to the center of the universe as p ossible , ap­ peared to require a moving agent. In the case of the heavy body , i ts fall was a natu ral motion caused by its nature as a heavy body once impediments to fall were removed. B o dies were also subject to vio­ lent motions in which external agents forced them to go where they had no inclination of th ei r own; th e moving agents were obvious enough in violent motions . Galileo set mechani cs on a new course by redefining motion to eliminate most of its A ri s totelian connotations . From the discussion of motion he cut away sprouting aco rns and youths alearning . Lo cal motion , whi ch had been fo r A ristotle the simplest case and hence the example most suited to analysis, became for Galileo the sum total of the meaning of motion. And to motion in thi s sense , he insi sted, a body i s indifferen t . A heavy body does not realize any potential when it falls; nothing it in changes . So also no violence is done to its nature when it is flung upward . Although he frequently used the phrase " natural motion , " any real distinction be­ tween natural and violent motions disappeared from his mechanic s . All motion i s one a n d the same . It i s n o t a process whereby potential is brou ght into realization . Motion i s simply a state in whi ch a body finds itself, a state to which it i s indifferent. P rojectile motfon had furnished the classic diffi culty to A ri stote­ lian mechanics , for a projectile continues to move after i t has sepa­ rated from its overt mover. Aristotle had solved the difficulty by arguing tha t the medium through which a projectile moves func­ tions as mover and su stains the motion . Medieval p hilosophers had transferred the mover from the medium to the body i tself. When it i s placed in motion, they argued, a body acqui res an i m petus , an internal motive force , which sustains the motion . Galileo reduced projectile motion to its simplest case , a ball set rolling on a ho rizon­ tal plane , and in his mind ' s eye he observed i ts motion. Since he requi red the frictionless p lane of rational mechanics and a perfectly round ball , he had to observe it in his mind' s eye . He concluded that under such i deal conditions a ball will continue to roll forever, as far a s the plane continues . On real p lanes , of course , real balls come to re st, but the smoother the plane and the rounder the ball the longer it will roll . Motion is a state to which a body is indiffer-

The discovery of a new world

9

en t . A ball on a horizontal plane can neither set i tself in motion nor bring i tself to rest. As Descartes, who shared this conception of motion, stated the case , philosophers had been asking the wrong que stion . They had asked what keeps a body in motion; one ou ght to ask instead what ever stops i t . With the n e w conception of motion , the p rinciple of inertia , if I may speak loosely , the diffi culties imagined to follow from the earth' s mo tion dissolved away . C annons fired east and west would carry e qual ranges; cannons fi red north and south would strike tar­ gets (or to speak more exactly perhaps of seventeenth-cen tury gun­ nery, would miss them with no rmal facility) . Obj ects dropped from towers would appear to observers to fall straight down since the observe rs are , of necessity, passengers on the same moving ea rth who participate in the common diurnal rotation . Galileo had more in mind than the question posed by astronomy, however. He propo sed nothing less than a complete recon s tru ction of the s ci ence of motion - into a new science , as he p roudly labeled i t in his fi nal book - and to reconstruct the science of motion completely was , in his view , to make it mathematical . Perhaps astrono my , which had always been a mathematical science of celesti al motion, suggested the model . Galileo was accustomed to say that in the Copernican system the ea rth became a heavenly body . If the immutable heavens alone offer a subj ect proper to mathematic s , the earth had been promoted into that clas s . One aspect of mechanics also offered a model of mathemati cal science . The balance, the lever, the inclined plane , the science of sta tics (as we would summarize them all) , had invited mathematical treatment both in the ancient world and in medieval Europe . A rchimedes especially , "divine Archimedes " as Galileo called hi m , had shown how the rigor of geometry could be app lied even to a mundane science . Not the least of Galileo' s self-esteem swelled from the realization that he had done what even divine Ar­ chimedes had not: To the mathematical science of bodies in equilib­ rium he had added a mathematical science of bodies in motion . Galileo' s new conception of motion supplied the cornerstone for the new structure . Defined mathematically , the motion of a perfect ball rolling on a frictionless horizontal plane i s uniform motion . In equal p eriods of time i t traverse s equal distances . Bodies also move vertically, of cou rse , and i t had long been observed that when they fall , they move with increasing velocity . Medieval p hilo sophers had even defined " unifurmly difform motion, " but they had analyzed i t in the ab stract without applying the defini tion t o real motions . C hanging the name to "uniformly accelerated motion , " Galileo identified it with the motion of heavy bodies fall ing . 5 A body in 5 As in the case of Copernicus, Galileo had a predecessor. Domingo de Soto, a Spanish scho­ lastic of the mid-sixteenth century, had identified free fall as uniformly difform motion . He had not elaborated a full system of kinematics on the basis of his insight, however.

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uniformly accelerated motion gains (or loses) equal increments of velocity in equal increments of time . Since he held , again in opposi­ tion to Aristo tle , that all bodies are composed of the same matter, which i s always heavy , he rea soned that bodies everywhere on the earth, whatever their size and whatever their substance , fall with a rate o f acceleration common to them all . As in the case of uniform motion , the assertion supposed ideal conditions analogous to his frictionless planes . Since a medium such as air i s always p resent , actual fall never realizes uniformly accelerated motion . A s with horizontal motions again , the more one approxi mates i deal condi­ tions , the more one approaches the defined motion . The speed with which ordinary bodies fall , combined with the crudity of the instruments available to measure time, made it im­ possible to check this theory directly . Galileo recognized , however, that he could ob serve the identical phenomenon, slowed down to a measu rable rate , on inclined p lanes . The established analysis of in­ clined planes even permitted him to calculate how much i t i s slowed down . U sing a water clock t o measure time, h e corrobo­ rated the fact of uniformly accelerated motion in nature . From the definition of uniformly accelerated motion Galileo pro­ ceeded to deduce the basic relations of kinematics that students still learn on thei r first introduction to mechanics , that a body falling from rest traverses distances proportional to the square of the time of the fall , that its velocity is proportional to the time of the fall and to the square root of the distance fallen . As a final tour de force , he dem­ onstrated that a projectile , which (again under ideal conditions) moves with a motion compounde d of uniform horizontal and uniformly ac­ celerated vertical elements , mu st follow a parabolic trajectory . 6 By 1 66 1 , the science of mechanics had assumed an anomalous stance . Though Galileo had not written in the Latin of the learned world but in Italian , a number of publici sts had made his re sults available to the European scientific communi ty . His relations of accelera tion , velocity , distance and time both in uniform and in uniformly accelerated motion had become the common p roperty of the science , accepted by all and questioned by none . Nevertheless, 6

The literature on Galileo is at least as immense as that on the new astronomy. The most influential work has been Alexandre Koyre, Etudes galileennes (Paris, 1939). Maurice Clave­ lin, La Philosophie nature lie de Galilee (Paris 1968), offers a more recent, excellent account. Stillman Drake has treated Galileo's mechanics in a series of articles too numerous to quote entirely here; among the most important are "The Concept of Inertia," Saggi su Galileo Galilei (Florence, 1967), pp. 3-14; "Galileo and the Law of Inertia," Ame rican Journal of Physics, 32 (1964), 601-8; "Uniform Acceleration, Space, and Time," British Journal for thf History of Science, 5 (1970), 21-43. He has recently summarized his work on Galileo iu Galileo at Work: His Scientific Biography (Chicago, 1978). Galileo, Man of Science, ed. Ernar McMullin (New York, 1967), contains articles which deal with every aspect and the majc interpretations of Galileo's science.

Th e discovery of a new world

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the reconstruction of mechanic s that Galileo had dared to promise had scarcely begun . The foundation on which his kinematics rested, the new conception of motion , presented diffi culties of comprehen­ sion not easily surmounted. Pierre Gassendi had given a wonder­ fully clear s tatement of it; later in the same work , he had also stated the concept of impetus and apparently had (ailed to recognize that the two were incompatible . John Wallis would repeat that perfor­ mance ten years hence . In fact , in 166 1 , only two signifi cant figures had embraced the principle of inertia, Rene Descartes and Christi­ aan Huygens . Indeed , it was Descartes and not Galileo who stated the principle of inertia in the form we accept today , insisting on the rectilinear character of inertial motion . His writings were the pri­ mary vehicle by which the con cept spread . In 166 1 , Huygens ' work on mechanics still remained locked in his manuscripts . Other stu­ dents of mechanics had derived Galileo' s kinematic relations from concepts of impetus . 7 In the meantime , two further i s sues of mechanics h ad been rai sed, both by Descartes . Partly for rea sons connected with his philosophy as a whole , which allowed a body to influence only those other bodies which it touched, D escartes undertook to define the laws o f impact . Almost no one accepted the laws a s he gave them in his Principia ph ilosoph iae (Princip les of Ph ilosophy, 1644) , but by stating inadequate laws , he bequeathed a problem to the science of mechanics . In the same work , he attempted to define the me­ chanical elements of circular motion , which had only become a problem with the principle of rectilinear inertia. Once again, Des­ cartes mangled the analysis and in so doing left a second problem . Both problems were attacked successfully by Christiaan Huygens in the late 1650s; but in 166 1 , he had not yet published eithe r result . To one attracted to it, the s cience of mechanics at that time pre­ sented more challenges than conclusions . Indeed no one , not even Huygens , had yet dreamed what impli­ cations lay hidden in those allu ring kinematic equations of horizon­ tal and vertical motion . In 166 1 , they remained as Galileo had pre­ sented them, brute facts of nature , enigmas to be probed, riddles to be explained . As yet, no causal dynamics made the uniformly accel­ erated motion of falling bodies appear as a natural result . When at length one was supplied, when the enigma was probed, not merely mechanics , but the entire body of natural philosophy shook with the con sequences . One can s carcely speak of this a s an unresolved problem in 1661 , since even its possibility had yet to be recognized . 7 See, for example, the discussion of Marcus Marci and of Evangelista Torricelli in Ri chard S. Westfall , Force in Newton 's Physics (London, 1971) , pp . 117-38 . Later Claude-Fran�ois Milliet de Chales , Giovanni Alfonso Borelli , and Edme Mariotte would carry through essentially identical analy ses (ibid. , pp . 200-3 , 213-30, 243--56).

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Implicit it might be , but the challenge was there nevertheles s , the greatest challenge of them all . Like mechanics , the science of optics had its own tradition , one more medieval than ancient , and one which also received a power­ ful s timulus from astronomy in the seventeenth centu ry . To ob­ serve the heavens was to employ the science of optics, whether one was conscious of it or not . The rectilinear p ropagation of light , which every observation assumed, was known to hold p recisely only for celestial bodies directly overhead . Atmospheric refraction distorted the observed positions of every other body , though no one knew exactly how much. In observing eclipses , moreover, as­ tronomers employed pinhole device s whi ch introdu ced fu rther p roblems in optics . Indeed, it was a problem introdu ced by the pinhole device which stimulated Kepler' s initial intere s t in optics , and the su btitle of his great work of 1 604, Astronomia e pa rs optica ( Th e Optical Part of Astronomy) , s tands as a permanent monument to the seminal role of astronomy in optics . Kepler' s p roblem was the app arent shrinkage of the moon as it passed acros s the face of the sun . Having shown that the apparent shrinkage was a pu rely optical phenomenon generated by the pinhole device and quantitatively related to the size of the pinhole , Kepler extended his insight to the explanation of vision . The pupil of the eye became the pinhole opening of the astronomical device , but equipped with a focusing len s . The retina became the screen on which the image fell . The concept of the retinal image, enunciated by Kepler, offered a new answer to the central question of vision on which the entire history of optics had turned before the seventeenth century . When the tele­ s cope burst upon an unsuspecting world six years late r, and the microscope shortly thereafter, optics assured itself an important role in seventeenth-century s cience . Foremost among the achievements of optics after Kepler' s semi­ nal work was the discovery of the law of refraction , a subj ect explored by Kepler even before the telescope, but one m ade crucial by the instrument. Published by D es cartes in 1 637, the sine law of refraction comp lemented the law of reflection known since ancient times . Not long thereafter Frances co Maria Grimaldi discovered the phenomenon of diffraction , although his work was not published until 1 665 . Together, the two discoveries sugges t the focus of sev­ enteenth-century optical s cience , not the p roblem of sight which had occupied optics before , but the nature of light. By the seven­ teenth centu ry , light was universally agreed to be an obj ective real­ ity and not a power emitted by a seeing subj ect. It was a constituent aspect of nature which could be s tu died on the s ame term s with the res t of natu re. The sine law of refraction added a new regularity to its behavior; the phenomenon of diffraction added mostly a puzzle .

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Neither seemed to entail a definitive ans wer to the basic question now at the center of s tudy: What is the nature of light? Descartes answered that it is a p ressure trans mi tted instantaneously through a transparent medium . Gassendi, the reviver of atomism , replied al­ ternatively that it is a stream of tiny particles moving with unimag­ inable speed . There were obvious p roblems wi th Gas s en di' s answer. Not only did it defy common sense, but it also app eared to contradict such p henomena as two men looking each other in the eye . On the o ther hand , very few liked D escartes ' s theory, which, in order to account for the geometrical regulari ties of light such as the very sine law he published, had resorted to the implausi ble argument that pressure, as a tendency to motion , obeys the same laws as motion itself. Nevertheless , D escartes ' s theory was capable of elaboration . His steady pressure might be converted to pulses of p ressu re, sometime called waves , and pulses or waves might over­ come the diffi culties presented by pressure alone. By 1 66 1 , at least two men were beginning to think in such term s : Robert Hooke and C hristiaan Huygens . Neither had published a theory of light by that time , nor was it evident that such a theory would solve all the p roblems . Moreov er, the terms in whi ch s eventeenth-cen tu ry s ci ence viewed nature as a whole had presented yet another problem to optics, the problem of color. Except in a small number of excep­ tional cases such as the rainbow (in which colors seemed to be associated more with the observer than with physical obj ects) , colors had been p laced among the properties of bodies . The new philosophy of nature asserted that all phenomena of colors are identical to the so-called exceptional cases . Size and shape alone are the real p rop erties of bodies; colors exi s t only insofa r as there are sentien t subj ects to observe them . From this point of view , colors were a ssociated with light . Colors became a problem in optics . No one asserted that light itself is colored . Rather, as light itself, in falling on a retina , arouses sensations of sight, so also some asp ect of light arou ses sensations of color . Once again , D es­ cartes was the man who posed the question; and once again , his proposed solution rou sed few enthusiasts . It remained to be seen what better answer could be offered . Like astronomy and mechan­ ics , optics in 1 66 1 was anything but a closed science. 8 8 Until recently , the history of optics has been studied much less extensively than the history of astronomy and mechanics . The established authority, Vasco Ronchi, Storia della luce (Bologna, 1939), only recently translated into English as The Nature of Light: An Historical Survey, trans . V. Barocas (London, 1970), has been superseded by several more recent works . For the history of optics through Kepler see David Lindberg, Theories of Vision from A l-kindi to Kepler (Chicago, 1976) . The best works on optics in the seventeenth century are A . I. Sab ra, Theories of Light from Descartes to Newton (London, 1967) and Alan E . Shapiro, "Kinematic Optics: A Study of the Wave Theory of Light in the Seventeenth Century , " 1 1 (1973), 134-266 .

A rch ive for History of Exact Sciences,

14

Never a t rest

A common thread which insistently catches the twentieth-century eye run s through the astronomy , mechanics , and optics of the six­ teenth and seventeenth centuries . Their achievements were ex­ pressed in mathematical te rms , and what we s eem to see i s the e stablishment of the pattern that has increasingly dominated natural science ever since , the mathematization of nature . Perhap s that thread did not s tand out so s trikingly to an observer in 1 66 1 . Al­ though seventeenth-century natural philosophers were as convinced as we are that a revolution was taking place in the s tudy of nature , it was not the mathematical description of a handful of phenomena or the ultimate implications that might lie behind the possibility of such descriptions which they considered to be the revolu tion . They thought instead of the new philosophy of natu re that had over­ thrown the Ari stotelian categories that had dominated natural phi­ losophy in the West for two thousand years . As far as men active in the s tudy of nature were concerned , the word "overthrown" i s not too strong . For them , Ari stotelian philosophy was dead beyond resurrection . In its place s tood a new philosophy for which the machine , not the organism , was the dominant analogy . Rene Descarte s , who contributed both to mechanics and to op­ tic s , contributed far more to the mechanical philosophy of nature . Although he was neither the firs t nor the sole p hilosopher in the seventeenth century to app roach nature in mechanical term s , and although he and others could draw upon the inspiration of ancient atomism, Descartes was nevertheless the principal architect of the s eventeenth centu ry ' s mechanical philosophy . Hinted at in his Dis­ cours de la m ethode (Discourse on Meth od) and its attendant essay s (1 637) , given metaphysical underpinning i n h i s Meditationes d e prima philosophia (Meditations on First Ph ilosophy, 1 641) , Descartes' s me­ chanical p hilosophy of nature was spelled out in full in his Principia ph ilosophiae (Principles of Ph ilosophy, 1 644) . Basic to it was the Cartesian dualism which attempted rigorously to separate mind or spirit from the operations of physical nature . The e ssence of mind (res cogitans) is the activity of thinking . In using the active participle to designate mind , Descartes deliberately specified that mind is the only locus of spontaneous activity in the univ�rse . In making its activity thought alone, however, he equally indicated that mind or spiri t does not play the role of an activating principle in physical nature . To be sure , the Creator summoned nature into be­ ing and continues to sustain it by His general concourse . He also decreed the laws by which nature operates, however, and He does not inte rvene to alter or obstruct their necessary operation . In addi­ tion , human souls can will motions for their bodies . Nevertheless, mo s t of the fun ctions of the human body , from digestion and growth on the one hand to reflex actions on the other, proceed

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independently of the human will . And the overwhelming majori ty of natu re' s phenomena , from the panorama of the heavenly vortices above u s , through the realm of animal and vegetable life about us , to the motions of the particles of bodies below the threshold of percep­ tion , are never affected by an act of a human will . Physical nature , composed of extended matter (res extensa) , i s a machine which .oper­ ates as machines must, according to the laws of mechanics . In the case of matter, Descartes consciously chose the passive participle , extensa , to expres s its nature . Wholly inert , shorn of any source of activity whereby it may initiate any change of motion , matter, to­ gether with the physical universe it composes , became the realm of pure phy sical necessity . Matter is divided into discrete particles which impinge on each other in their motion s , produ cing the pheno­ mena of nature . Though single particles may alte r their motions in impact, co ming to res t , speeding up , or changing direction , the total quantity of motion in the universe remains constant . This was the basic law of Descartes's mechanical universe; it insured that nonme­ chanical agents need not be introdu ced . Although Descartes did not employ the analogy of the clock , later mechanical philosophers would make i t the image of their conception of natu re . " Clock " to them summoned up the in1age of the great cathedral clocks . Their descending weights powered the hands turning on their axles and a great deal more . Mechanical men rang bells. Saints appeared through the doors . Cocks cro wed to tell the hour . One complex mechanism generated a multiplicity of op­ erations that suggested the infinite phenomena of the infini tely complex machine we call nature , except that the cosmi c clock m ade by the divine watchmaker required no winding up . The conse rva­ tion of the total quantity of motion, and the principle of inertia on which it rested, insured its eternal operation . If D escartes's mechanical philosophy of nature was the most pro minent, it was not the only one . In England, Thomas Hobbes produced a similar, though less elaborately formulated philosophy . In France , the p rincipal agent in the revival of atomr�m , Pierre Gassen di , offered another alternative . From a philosophical point of view concerned with the rigor of arguments , their differences were important. From the point of view of many practi cing s cientists , �uch a s Robert Boyle , the extent of their agreement obscured seri­ ous attention to their differences . All agreed on some form of dual­ i sm which excluded from natu re the possibility of what they called pejoratively " occult agents" and which presented natural phenom­ ena a s the necessary products of inexorable phy sical processes . All agreed that physical nature is composed of one common matter, qualitatively neutral and differentiated solely by the size , shape , and motion of the p articles into which it is divided. All agreed that the

16

Never at rest

p rogram of natural philo sophy lay in demonstrating that the phe­ no mena of nature are p rodu ced by the mutual interplay of material particles which act on each other by direct contact alone . Thus the two competing conceptions of light in the later seventeenth century were the two possible conceptions allowed by the mech a nical phi­ losophy; light was either particles of matter in motion or pulses of motion transmitted through a material medium . Descartes had defined the program of the new philosophy of nature in his sixth and final meditation . The first meditation , as he began the systematic search that led to a new foundation of cer­ tainty , had called the existence of the external world into doubt. We believe in it primarily on the evidence of our senses; but since our senses sometimes err, the existence of the external world cannot on their evidence claim the metaphysical certainty Descarte s was seek­ ing . By the ti me he reached the sixth meditation , Descartes was ready to replace the exi stence of the external world in the structure of certainty built on the rock of the cogito . The assertion of its existence now rested on a different foundation , not on the facile assumption of common sense buttressed by the unexamined per­ ceptions of the senses , but on the evidence of necessary argu ments fro m first principles . There i s , he added - and the statement was the most important assertion in natural philosophy during the entire seventeenth century - no corresponding necessity that the external world be in any way similar to the one our senses depict . Our senses reveal a wo rld of qualities . Reason tell s us that quantity alone exists , particles of matter differentiated solely by their size , shape, and motion , which pro duce sensations of qualities when they im­ pinge on our nerves . The p rogram of the mechanical p hilo sophy lay in de monstrating that assertion . Aspects of the mechanical philo sophy suggest an inherent har­ mony with the develop ments I have surveyed in astronomy, me­ chanics , and optics . The very word " mechanical" seems to incorpo­ rate the science of mechanics , and the program of �he mechanical philosophy , to trace all phenomena to particles in motion , seems to demand the same . The assertions that quantity alone is real and that qualities are only sensa tions recall the mathematically fo rmulated laws o f physics . The harmony may be more appa rent than real , however. The formalism of mathe matical laws never satisfied the mechanical philosophy' s demand that phenomena be explained in terms of particles in motion . Its explanations in turn refu sed to yield the mathematical l aws . As we have seen , Descartes' s vortices we re incompatible with Kepler' s laws , not just the ellipses, but all three laws . His explanation of gravity , like rival mechanical expla­ nations , was incompatible with Galileo' s kinematics of free fall . In order to derive the sine law of refraction, Descartes had to intro-

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duce arbitrary conditions which no one found believable; but when Pierre Fermat derived the law from the principle of least ti me , mechanical philosophers refused to have truck with what they con­ sidered occult notions . The words " quantitative'� and " mechanical " should not mislead us into seeing the two dominant t rends in seventeenth-centu ry science as different facets of one program . In 1 66 1 , to a young man attracted by the excitement of scientific discovery, the mechanical philosophy of nature possibly o ffered the more exciting pro spect. Here no limiting horizons re stric ted one' s attention t o confined problems such as the shape o f an orbit or the angle of refraction . If a new world had indeed been discovered, that world was found in the mechanical philosophy which took the world , the whole world , as its province . The aura of freshness still clung to i t . The classic exposition was le ss than twenty years old; Gassendi' s rival version , in its final statement, less than five . I ts promise had ravished young Christiaan Huygens in the Nether­ lands , and in England , young Robert B oyle . Dissatisfied with the arid formulations of a tradition that seemed capable only of repeat­ ing insights now two thousand years old , they grasped eagerly at the promise of a completely new program which offered progress instead of rep etition , true understanding of the depths of nature instead of superficial knowledge . It is a fact that no figure of impor­ tance in European science in the second half of the seventeenth century stood outside the p recincts of the mechanical philosophy of nature . 9 In 1 66 1 , nevertheless, to embrace the mechanical philo sophy was also to face a series of p roble ms which could not be avoided. For the philosophically serious , the rival mechanical systern s demanded choices - the p lenum or the void , the continuum or discrete parti­ cles . For the religiously serious , the exclusion of spirit from physi­ cal natu re required that the full rigor of the system be co mpromised in the name of Chri stian sensibility. For the scientifically serious , the app arent incompatibility of mechanical explanations with rigor­ ously fo rmulated mathematical laws p osed an equally serious prob­ lem . The mechanical philosophy excited the best minds of several generation s , but its problems were acute . Thei r very acuteness sug­ gests the possibility that disillusionment and revision might succeed the enthusiasm of youth . 9 Of the vast literature on Descartes, the great majority is devoted to issues that fall outside his natural philosophy . The best expo sitions of the mechanical philosophy of nature are found elsewhere: in R. G. Collingwood, The Idea of Nature (Oxfo rd, 1945); R . Harre, Matter and Method (London, 1964); Marie Boas [Hall], "The Establishment of the Mechani­ cal Philosophy, " Osiris, 10 (1952), 412--541 ; E. J . Dijks terhuis, The Mechanization of the World Picture , trans . C. Dikshoorn (Oxford , 1961); R. Lenoble, Mersenne ou la naissance du mecanisme (Paris, 1943) .

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Never at rest

S tanding largely outside the traditional boundaries of natural phi­ losophy was another area of investigation , chemistry, which gradu­ ally a sserted its rol e in an enterp rise .. which was itself gradually changing from natural philosophy to natural science . Although the Aristo telian doctrine of four elements offered a possible foundation for chemical theory, an extensive Aristotelian chemistry had never been elabo rated , and vi rtually no chemist in the seventeenth century was an A ri stotelian . In the sixteenth century, Paracel sus had pro­ posed a rival theory in which the four elements were supplemented by three p rinciples - salt, sulfur, and mercury - which functioned a s t h e p rimary agents i n chemical explanation. Perhaps he merely ad­ umbrated the theory and his followers elabora ted it. B y the dawn of the seventeenth centu ry , in any case, P a racelsian chemi stry dominated the field. Chemistry was far more experimental than natural philosophy a s a whole , with its background in the medieval universities and the ancient s chool s . The relative isolation of chemistry a s a field of stu dy stemmed partly fro m this fact . A student of mechanics worked at a desk with paper and p en . Until Tycho and the tele­ s cope brought about a new order, the same was largely true of an astronomer . The chemist, on the o ther hand, labored with cruci­ bles and alembics at a hot furnace: " sooty empiric" the g entlemen scholars scornfully called him . One consequence of chemistry' s experimental emphasis was an enormous expansion of its body of empirical knowledge . By the middle of the cen tury , chemi sts con­ trolled a corpus of experimental results whereby they were able to compound with a ssurance a range of substances wider by far than thei r predecessors had kno wn. But chemistry lacked an adequate stru ctu re of theory which could give coherence and direction to the rapid growth of information . Perhaps the single-minded ex­ perimentation in chemistry stemmed partly from the failure of theory . Paracel sus ' s three principles were more a doctrine of sub­ stance than a chemical theory . Salt, sulphur, and mercury, a s bo dy , soul , a n d spirit, were the necessary constituents o f a n y exis­ tent bo dy . One might philosophize in this vein , but one could scarcely organize a body of experimental information e ffectively. The major chemical generalization of the century , the recognition that alkalis and acids n eutralize each o ther, was announced by Johannes van Helmont, who was p owerfully influenced by Para­ celsian though t . The generalization can hardly be said to h ave grown naturally from the s tock of Paracelsian theory. Quite the contrary , it could only be grafted onto that theory by b rute force. Paracelsian chemists arbitrarily divided the p rinciple s alt to in­ clude, among o thers , acid salts and alkaline (or lixi viate) salt s . Van Helmont' s insight did not expand the explanatory power of Para-

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celsian theory; Paracelsian theory cast no illu mination on van Hel­ mont' s insight . In 1 66 1 , however, chemistry was undergoing what promised to be a momentous change . In that year Robert Boyle , who is some­ times called the father of chemistry , published his most famous, if not his most important book , The Sceptical Chymist . Boy le' s work consisted of a sus tained polemic against the Aristotelian concept of an element and the Paracelsian concept of a principle . He t reated the two as bro thers to be slain with the same sword . What he offered in their place was mechanical philosophy , that i s , chemistry stated in terms of the mechanical philosophy of nature . Boy le' s most famous sentence , his definition of an element, appeared in Th e Sceptical Chymist . It has been quoted and requoted by authors who have failed to notice that it merely repeated the traditional definition and that the entire work devoted itself to denying the pos sibility of elements in this sense , as Boyle explicitly stated in the final clause of the very sentence that contains the definition . Elements and princi­ ple s do not exist . What does exist is the qualitatively neutral matter of the mechanical philosop hy , divided into particles diffe rentiated only by size , shape , and motion . From their various combinations arise all the appearances of the substances with which chemists deal . To chemists Boyle offered full participation in the fraternity of natural p hilosop hers . By mechanizing chemistry , he effectively obliterated the barriers that had separated their enterprise from the rest of natural philosoph y . To the mechanical philosophy of nature he offe red what chemistry alone could p rovide , an articulated sci­ ence of matter, which surely had to be essential to a p hilosophy asserting that all phenomena result from particles of matter in mo­ tion . In 1 66 1 , Boyle ' s program , to which he and others would devote life times of labor , was only a p romise . He had shown that chemical phenomena could be expressed in mechanistic language . He had not yet shown that fuller understanding necessarily fol­ lowed . If his program was still only a p romise, nevertheless it was an exciting p romise, especially to those who found the new me­ chanical approach to nature exciting . 1 0 Even in the case of Boyle , chemistry at that time consisted of more than his program to mechanize it. Alchemy was still a living 10

On Paracelsus and Paracelsian chemistry, see Walter Pagel, Paracelsus . An Introdu ction to Philosophical Medicine in the Era of the Renaissance , (Basel and New York, 1958), and The Religious and Philosophical Aspects of van Helmont's Science and Medicine (Supplements to the Bulletin of the History of Medicine , No. 2, Baltimore, 1944) and Allen Debus, The English Paracelsians (London, 1965) . The best general expositions of seventeenth-century chemis­ try are Helene Metzger, Les doctrines ch imiques en France du debut du X VIr a la fin du X V/Ile siecle (Paris, 1923) and Marie Boas [Hall], Robert Boy le and Seventeenth- Century Ch emistry (Cam bridge, 1958) .

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enterprise and the sixteenth and seventeenth centuries witnessed its culmination in Wes tern Europe. One of the greatest of English alchemists , who wro te under the pseudonym of Eirenaeu s Phila­ lethes (probably George S tarkey) , was still alive in 1 66 1 , and Elias A shmole' s collection of B ritish alchemical writings , Thea trnm chemi­ cum Britann icu m , had appeared only nine years earlier. Boyle himself had been introduced to chemistry by the alchemical ci rcle in Lon­ don surrounding S amuel Hartlib , and through a long career de­ vo ted to integrating chemistry into the mechanical philosophy h e never ceased t o search into the A r t . Alchemy w a s alive i n Cam­ bridge as well , where Ezekiel Foxcroft , fellow of King' s C ollege , maintained connections with Hartlib ' s ci rcle in London. Undoubt­ edly there were also o ther alchemis ts in Cambridge. Alchemy was no one thing. By the seventeenth century it had been p racticed in the West in a series of different cultures, from Hellenistic through I slamic and Latin-medieval to the Eu ropean cul­ ture of the sixteenth and seventeenth centuries . Alchemy had had to adjust itself continually in order that different ages might compre­ hend i t . Medieval alchemists expounded the Art in terms of Aris­ totle' s four elements; alchemists of the late sixteenth century used Paracelsian salt, sulfur, and mercury . Hartlib' s ci rcle, influenced by the Pla toni st school of English philosophy , pursued a N eop latonic alchemy . The va rying philo sophi c costu m es , however, a lways clothed an animistic philosophy of nature which appears contradic­ tory in its very essence to the mechanical philosophy. Alchemists believed that life rather than mechanism stands a t the very heart of nature . All things are generated by the conjunction of male and fem ale; metals differ in no wise from the rest of nature . Like every­ thing else, metals grow in the womb of the earth - rather, metal grows , for if we speak in strict terms, alchemy did no t reco gnize more than one metal . That one, of course, was gold, the product that nature realizes when nothing interrupts her normal gestation . The o ther " metals , " abortions of nature , are potential gold that has failed to reach maturity . Alchemists were trying to comp lete what natu re had left incomplete . They were growing gold. Composed of living things , natu re to the alchemist abounded with centers of spontaneous activity, vital or active principles . These principles differed irreducibly from the inert matter of the mechanical philosophy which was able only to respond p assively to external actions . Alchemists sou ght to extract the active principles of nature fron� the feculent dross which weighted them down . (The scatological imagery of alchemy was quite as explicit as the sexual . ) They also sought t o obtain pure matter o r soil in which t o plant the seed s . We must recall tha t every theory of generation until the late seventeenth century reco gnized a female semen . Conception and

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generation were held to occur when the male semen fermented with it . For the alchemist , the philosophic sulfu r, inevitably the male principle in that age, the ultimate active agent, required the philo­ sophic mercury, the female p rinciple , the soil in which a s seed it could be planted. Alchemists were forever separating and purifying . Purification - here was the basic need . Before the soul co uld shine the feces that fouled it had to be scrubbed away . B efore it could fly , its bu rden of dross had to be remove d . Before it could truly live , its encumbering flesh had to be purged. But once it was free d , it was " the subj ect of wonders , " " the miracle of the world, " "a most puissant & invincible king . " 1 1 Towa rd alchemy the seventeenth century was ambivalent . 12 To cite but one example , B en Jonson's play The A lch emist testifies that some regarded alchemists as charlatans and rogues . By p retending to knowledge of how to make gol d , they earned their way by gulling the gullible . Al though there is no way to mea sure general opinion , it seems probable that a large number shared Jonson' s a tti tude . Before w e are temp ted t o see the beneficent influence of scientific skep ticism al ready at work , we ought to recall that Geof­ frey C hau cer exp ressed substantially the same opinion more than two centu ries earlier. After all , gold has been a desi rable commod­ ity in all ages , and there are inherent dangers in repeatedly p romis­ ing to make it unless one actually p roduces. The existence of skep­ tics is hardly surp rising . The continued appeal of alchemy in all 11

12

I have taken these particular express ions from Newton 's aichemical papers: Keynes MS 40 , f. 1 9v; Keynes MS 4 1 , f. 1 5 v; Keynes MS 40, f. 20. On alchemy in general, see M ircea Eliade, Th e Forge and the Cmcible , trans . Stephen Corrin (New York , 1 97 1 ) ; A rthur J . Hopki ns , Alchemy , Ch ild of Greek Ph ilosophy (New York, 1 934) ; John Read , Prelude to Ch em istry (London, 1 936) ; F. Sherwood Taylor , The Alchemists, Founders of Modern Ch em­ istry ( New York , 1 949) ; Robert Multhauf, Th e Origins of Ch em istry (London , 1 966) . A very small handful excep ted , the twentieth century is not , and even to bring up alchemy is to court misunderstanding . Herbert Butterfield asserted , without much deli­ cacy , that those who study alchemy in the twentieth century are " fabulous creatures themselves " who "seem sometimes to be under the wrath of God themselves . . . [and] seem to become tinctured with the kind of lunacy they set out to describe" ( The Origins of Modern Science , rev . ed . , New York , 1 965 , p . 1 4 1 ) . S ince I shall devote quite a few p ages to Newton's alchemical interests , I feel the need to m ake a personal declaration. Even though I am sure that no fabulous creature recognizes the lunacy with which he is tinctured , I can only state my own percep tion of the situat ion . I am not myself an alchem ist , nor do I believe in its premises . My modes of thought are so far removed fro m those of alchemy that I am constantly uneasy in w riting on the subject , feeling that I have no t fully penetrated an alien world o f thought . Nevertheless , I have undertaken to w rite a biography of New ton, and my personal preferences cannot m ake more than a million word s he wrote in the study o f alchemy disappear. It is not inconceiv able to most historians that twentieth-century criteria of rationality may not have p revailed in every age. Whether we like it or not , we have to conclude that anyone who devoted much of his time for nearly thirty years to alchemical study must have taken it very seriously ­ especially if he was Newton.

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ages, but especially in the middle of the seventeenth century , when the movement of thought seemed set so decidedly a gainst it, i s what require s explanation . I cannot p retend to give the explanation, either in general or in the p a rticular case that concern s me here: I can only suggest sev­ eral related speculations . Alchemy offered exactly what the me­ chanical philosophy of nature offered, though , to be sure , in a rather different package . That i s , alchemy also p romised to reveal the ultimate secrets of natu re . Vulgar multipliers were one thing; there were Ben Jansons aplenty to unmask them . Meanwhile seri­ ous men beyond suspicion of p etty fraud found in alchemy some­ thing more p recious than recipes to make gold, an all-embracing p hilosophy of i rresistible attraction . In the seventeenth century alone , to name only a few , Michael Maier, Jean d'Espagnet, Jo­ hann Gra s shoff, George Starkey , Alexandre Toussaint de Limojon de Saint Disdier, and E dmund Dickinson, who span the century from beginning to end, testify to that attraction . Are we to equate these men with Jonson ' s cheat? I s it impossible to imagine a grave young man, philosophically minde d, desirous of understanding not limited p roblem s but all that is - and perhaps convinced that he can - is it impossible to imagine the alchemical p hilosophers excit­ ing him even a s the mechanical philosophers did? In both philoso­ p hies, one had the pleasure all youths of spirit feel in rebelling against the e stablished order. In 1 66 1 , the achievements of the mechanical philosophy could sca rcely preclude the consideration of alternative philosophie s . If it resembled the mechanical philosophy i n i t s s cope , alchemy parted fro m it decidedly on a question of importance to most men of the seventeenth century . Without denying God as Crea­ tor, the mechanical p hilosophy denied the participation of spirit in the continuing opera tion of nature . Alchemy did not merely assert the p articipation of spirit; it asserted the p rimacy of spirit. All that h appens in natu re i s the work of active principles , which passive matter serves as a mere vehicle . To the religiou sly sensi­ tive , alchemy might have special attractions . Moreover , alchemy was pervaded with Chri stian symbolism . The delivery of active p rinciple from its body of death , its renewal and exaltation , showed fo rth the rebirth of the soul in Chris t in language that could not be mis taken . Even the resurrection of the body found its image in the purified terra alba p repared for the resurrected seed . Nor was it evident to the seventeenth-century mind that the two philosophies of nature were mu tually exclusive . The m echanical philosophy was a p rotean idiom . Those who spoke it were able to translate any theory into the language of particles in motion . On

The discovery of a new world

23

clo se examination, Descarte s ' s three elements appear to be cousins german to the Paracel sian tria prima . His fi rst element, the cease­ lessly agitated subtle matter, might double very well for the philo­ sophi c s u l fu r , the u l ti m a te a ctive p rincipl e in the a l chemical philosophy . Lines of demarcation that appear absolute to the twen­ tieth-centu ry mind were less clearly drawn in the seventeenth . It was perfectly possible to respond at once to the promise and excite­ ment of both , perhaps to attempt to supply the defects of each from the achievements of the other . Alchemy h ad the possible further attraction o f the deliberately arcane . Where the mechanical philosophy sought to illu minate ob­ scu rity by the light of reason , alchemy employed obscuri ty to p ro­ tect tru th from contamination . Frequently alchemists themselves concealed their identity behind pseudonyms , such as Eirenaeu s Philalethes . Alchemy did not speak in the ordinary language of ordinary mortals; it spoke in tongues , concealing its message in outlan dish imagery comprehensible only to the initiate d . Instead of a new p rogram to correct the errors of centuries, it offered an old p rogram , an ancient wi sdom h anded down generation by genera­ tion to a select few . An alchemi st was not made; he was chosen . He was one of the elect, a member of the invisible fraternity of the sage s of all generations . There is no need to contrast the twenteith centu ry with the seventeenth in this case; the lure of the arcane i s eternal . N o r is it impossible t o imagine h o w a young man con­ vinced of hi s geniu s might feel himself a membe r by right in the timeles s ci rcle of the elect . The antipodes of alchemy with its eternal and exaspera ting secre­ tivenes s was mathematics , the very claim of which to be called knowledge reste d on demonstra tions open to all . Where the one made its way deviously wi th allusion and symbolisrn , the o ther pro ceeded in the cold light of ri gorous logic . The diversity of the intellectual world of the seventeenth century has perhaps no better ill ustra tion than the coexistence of two such antithetical enquiries , both apparently i n flou rishing condition . Only t o later ages would it be clear that seventeenth-century al chemy was the las t blossom from a dying p lant and seventeenth-centu ry mathematics the first blooming of a hardy perennial . Whatever the state of alchemy, certainly it was manifest in 1 66 1 that mathematics was a flourishing enterprise . The very year 1 66 1 marked a tu rning point in seventeenth­ century mathematics . It witnessed the completion of the century ' s most influential mathematical publication, van Schooten' s second Latin edition of Descartes' s Geometrie with the addition of hi s own lengthy com menta ry and a number of ancillary treatise s . No other

24

Never at rest

work su mmed up the achievements of early seventeenth-century analysi s so well . A s a result , no work provided a more solid fo un­ dation fo r subsequent b uilding . The name " analysis , " which seventeenth-century mathematicians gave to their work , both attaches it to and distingui shes it from ancient geometry . Part of the common lore of seventeenth-century mathematics concerned a method, now lost, by which the ancient geomete rs had made discove ries . The geometrical works that sur­ vived fro m the ancient world demonstrated the discoveries in the familiar language of synthetic geometry , s tarting from self-evident axioms and proceeding by ineluctable steps of geometrical logic . Seventeenth-century m a th e m atici an s set o u t to re su rrect the method of discovery , or analysis , by which they were convinced the discoveries had originally been made . We still call what they created " analytic geometry . " Part of the p rogram was to show that the method provided a ready means of attack on knotty p roblems , such as drawing tangents to cu rve s , and most importantly on the climactic p roblem of ancient geometry , the three- or fou r-line locus (or locus solidus ) . In actual fact, seventeenth-century analysis called upon a form of mathematics , algebra , which had scarcely been tou ched by the ancient world and was largely the creation of West­ ern E u rope during the p revious century . The fu rther it p roceeded, the more analysis distinguished itself from classical geometry . As they began to sense the power of their new instrument, the analysts concerned themselve s less and less with the question of demonstra­ tive rigor. Rigor had p rovided the foundation of Greek geometry . Because o f its rigor, geometry h a d offered the living example of the ideal of science in the original sense of the word , that which is truly known , in contrast to mere opinion , which cannot clai m a similar degree of certainty . In the flush of excitement, analysts rushed for­ ward into another newly discovered world and left to s ucceeding ages the delicate problem of demonstrating , in the full sense of the word , that the world they had disco vered did indeed exist and was not me re rumor. The p roblem of the three- and four-line locus had been posed some two thousand years before in the following terms: Having three or four line s given in position, find the locus of points from which the same number of lines can be drawn to the gi ven lines , each making a given angle with each , and such that the p roduct of the lengths of two of the lines shall bear a constant propo rtion to the s q uare of the third ( the three-line lo cus) or a constant p ropor­ tion to the product of the o ther two (the four-line locus) (Figure 1 . 4) . Ancient geomete rs had known that conic sections furnish the sol ution, but, at least according to Pappus's account, on which seventeenth-century analysts had to rely since the t reati ses them-

The discovery of a new world

25

c \

\

\

--- --- T

\ \

/ / / / / /

I

/ /

I /

I

\

\ \

R

/

s

/ /

I

A

I I

Q

B

F igure 1 . 4 . Th e four-lin e locu s . Th e lines AB, BD, CD, and A C are giv en in p osition . Find the locu s of p oints P from whi ch lines PQ, PR , PS, and P T can be drawn to the four lines , each always making the same angle with th e li n e it meets , su ch th at PQ PR is always in a giv en rati o to PS PT. Th e locu s i s a conic th at passes through th e four int ersec­ ti ons (A , B , C, D, ) of the fo u r giv en lines . ·

·

selves h ad been lost, they had been dissati sfied with the generality of thei r solution . In hi s Geom etrie ( 1 637) , D es cartes applied algebrai c techniqu es to the p roble m . Descartes ' s work s tarted with the assertion that any p roblem in geomet ry can be redu ced to terms in whi ch knowledge of the lengths of certain lines suffi ces for i ts constru ction . The "length s of cert ain lines " referred to what we now call the coordi­ nates of a p oint . D escartes ' s coo rdinates look strange to anyone, familiar with " C artesi an coordinates , " who exp ects to find curves graphed on x and y axes . Instead of starting with a s ystem of coordinates and an equation , D escartes started with a di agram of the p roblem set. He chose two lines on the construction as his coo rdinat es and exp ressed the other lines in terms of tho se two , the two v ariables , for whi ch h e employed l etters from the end of the

Never at rest

26 \

\

\

T ' \

\

\

\

\

\

\s \

\

\

\

' R \

E

A

\

\

, 9 \

G

\ \

F

\

\

\

\

\

\

\

H \ \

- -- - - - - - - - - - - - - - - - - - ::-��-cD

Figure 1 . 5 . Descartes's diagram for the fo ur-line locu s . He set A B = x and BC = y . In effect , A is the o rigin, and the coordi­ nates of C, which is any p oint on the locus sought, are x, y . Manife st! y the two coordinates are not perpendicular to each other as the y are in what we today call C a rtesian coo rdinate s .

alphabet (Figu re 1 . 5) . O nly a later generation would reco gnize fully the simplifying advantages of perpendicular coordinate s . Mean­ while, Descartes was able to reduce the three- or four-line locus to a quadra tic equation in two variables . He recognized that such an equation represents a coni c , and he learned to distinguish the conics by the cha racteristics of their equations . The successful solution of Pappus ' s p roblem , as it was called , convinced Descartes o f the uni­ versal u tility of hi s method . A t much the same time, another French mathematici an, Pierre Fermat, was attacking the same p roblem in much the s ame way . Fermat went beyond D escartes in his recognition that any equa­ tion in two unknowns defines a curve . He realized as well the desi rability of perpendicular coordinates . Nevertheless , Fermat did not exert as much influence on the development of analytic ge-

The discovery of a new world

27

ometry as Descartes . Most of hi s works remained unpublished, although they did circulate in manuscript. In contrast , Desca rtes had a disciple and publicist in Franz van Schooten, who transl ated hi s Geometrie into Latin , the international language of lea rning, and embellished it in successive editions with commentaries of his own and of a circle of students educated in the tradi tion of Cartesian geometry . The full implications of the introduction of algebraic techni ques into geometry appea red only with thi s second generation . The ini­ tial connection with the outlook of Greek geometry and absorp tion in i ts p roblem s gradually receded, and the analytic consideration of algeb rai c equations in two vari ables which define curves gradually moved to the fo re . Jan DeWitt' s Elementa curvarum (Elements of Cu rves) , which was published in the second edition of S chooten' s translation , has been called the fi rst treati se in analytic geometry . The same claim has been made for John Wallis ' s De section ibus con icis (On Con ic Sections) , which appea red slightly earlier, in 1 655 . For my purpo ses the question i s not worth resolving . As the title of Wallis ' s work and the con.,.tent of DeWi tt' s testify , early analy ti c geometry w a s possible only because the Greek geo mete rs h a d dis­ covered and explored the conics . O n the conics the first generation of analytic geometers concentrated al mos t exclu sively , learning to exp ress t hei r characteri s ti cs in analy ti c form and extending knowledge of thei r p roperties . Fewer than five curves of the third degree had been s tudied by 1 66 1 , and in mos t cases thei r analyses had been defective . Nevertheless, in 1 66 1 , European mathematics po ssessed a new tool of immense power. The full extent of i ts power had yet to be explored . D rawing tangents to curves p resented one of the lead ing p rob­ lems to which analysis addressed itself. Drawing a tangent i s , of course , equivalent to finding the slope of a cu rve at any point, or what we now call differentiation . In his Geometry , Descartes had app roached the p roblem indirectly by devising a method to find the normal to a curve at any point; since the tangent is perpendicular to the normal at the point of tangency , the position of the no rmal iinmediately defines the positi on of the tangent. To find the no rmal at any point C, Descartes constructed a ci rcle, with i ts center P on the axis of the cu rve , that cuts the curve in two points , C and E . The simultaneous solution o f the equations o f the two cu rves gives the points of intersecti on . Now let the second intersection E coin­ cide with C; that i s , let the ci rcle be tangent to the curve a t C . S ince two roots have coalesced, the equation for the intersection shoul d have a double roo t at that point . Descartes p roceeded then by purely al gebraic means to establish the coo rdinates of the double root . Rather, since C is given , he established the coordinate on the

Never at rest

28

E

F

A

M

p

Figure 1 . 6 . Descarte s ' s method of finding tangents . B y finding the double root when E and C merge into a single point , he found the length of the subnormal MP, in effect the loca­ tion of P, the center on the axis A M of a circle that is tangent to the p arabola AEC at point C. The tan gent to the parabola at C is, of course , perpendicular to the nor­ mal PC .

axi s of P , the center of the circle tangent to the curve at C (Figure 1 . 6) . In contrast to Descartes , Fermat attacked the problem of tangents directly by apply ing the techniques of analytic geometry to the ancient concept that a tangent is the longest line from a point to a convex cu rve . Given a curve and i ts equation - recall that with Fer­ mat the cu rve was always a conic, a curve of second degree , as i t nea rly alw ays w a s with Descartes - we wish t o determine the tan­ gent to any p oint B . Let E be the point , as yet undetermined, at whi ch the tangent to B cuts the extended axis of the parabola DB . DC and CB are the x and y coordinates of the point B . (Actually , Fermat rej ected Descartes's use of x and y as the symbols fo r vari­ able s , p referring Viete ' s earlier no tation . ) Let I be anothe r point on the x axi s a short distance removed from C . Because the cu rve i s convex up , O J i s longer than the ordinate t o the curve a t I . From the equation of the curve and the two similar tri angles CBE, JOE , Fermat set up an inequality between (OJ 2)/(BC 2) and EI/EC . He then allowed I to coinci de wi th C so that the ord inate to the curve at I coincided with OJ . His inequali ty became an equation with a double root , from the solution of which EC , the length of the subtangent, emerged (Figure 1 . 7) . Since he sought to dete rmine the

29

The disco very of a new wo rld

B 0

D

E

c

Figure 1 . 7 . Fer mat' s method of finding tangents .

tangent di rectly, and since B C/EC defines the slope of the curve at B , Fermat' s p rocedure looks more like differentiation than Des­ cartes ' s determination of the tangent via the normal . Schooten' s second Latin edition of Des cartes' s Geom etry con­ tained two papers on maxima and minima, by Johann Hudde, set­ ting forth a p rocedure for finding double roots , whi ch figured both in Descartes' s method and in F ermat' s , whi ch he had seen in manu­ scrip t . If the equation has a double root , Hudde demonstrated that the root also satisfi es a second equation axn + (a + b)a 1xn - I + (a + 2b )a 2xn -2





+ (a + n b)a

.

n

=

0

whi ch is obtained by multiplying the firs t equ ation by the terms of an arithm eti c p rog res sion . Since the p roposition is general , any arithmetic progression can be used , and in practice mathematicians usually chose a p rogression su ch that the most troubles ome term in the original equati on was multiplied by zero and eliminated . The impli cation of Hudde' s rule app ears most readily if the progres sion of exp onents , n , n 1, n 2, . . . 1 , 0, is used . Since the zero coefficient eliminates the constant term , an x can be canceled from each of the remaining terms , and the equati on becomes -

-

n xn - I + (n - l )a 1xn - 2 + (n - 2)a 2Xn -3

.



.

+

an - 1

=

0

what we call the first deriv ativ e of the original equation , set equ al to zero in order to find a maximum point, as in Fermat' s method of tangents . (The equ ation in question is not that of the curve but that of EC , which becomes a maximum when EB is tangent to the curve . )

30

Never at rest

The problem of tangents , much in the air by the middle of the seventeenth century , also received a kinematic solution which re­ veals another facet of seventeenth-century mathemati cs . Both To r­ ricelli and Roberval treated curves as the paths of points in motion. From thi s point of view , the instantaneous direction of the moving point at any posi tion on the curve defines the tangent . Torricelli employed thi s method to determine tangents for parabolas of all degrees and for spirals . Roberval ' s method was simila r enough to To rricelli' s that a charge of plagiari sm arose , one of the innumera­ ble wrangles in seventeenth-centu ry mathematics , when i dentical problems and i dentical considerations impelled mathem a ticians in identi cal directions . The concept of a curve as the line generated by the composition of two known motions did not spring newbo rn from the b rains of seventeenth-century mathematicians . Neverthe­ less , the kinematic app roach to mathematics most revealed i ts af­ finities with the develop ing science of mechanics. I t also offered one possible road by which the new analysis could lead on into still unexplo red territories . D u ring the early seventeenth century , mathematics concerned i t­ self extensively with another question not initially an aspect of analysis , the p roblem of calculating areas under curves and volumes enclosed by curved surfaces . The use of infinitesimals found i ts w ay into mathe matics p rimarily via thi s rou te . Kepler' s Steriometria dolio­ ru m (Measurement of th e Volume of Casks , 1 6 1 5) , which su mmed infinitesimal layers to calculate the volumes , ma rked the beginning of the assault on such problems along this front . Twenty years la ter, C avalieri ' s Geometria indivisibilibus (Geometry by Means of In­ div isibles , 1 635) attempted to place the use of indivisibles on a rigo r­ ous basis . By that time , the method was literally part of the atmo s­ phere that mathematicians breathed , and even though Cav alieri first published gene ral results for areas under the series of curves , x = y , x = y2, x = y3, . . . , x = y n , Torricelli , Roberval , Pascal , Fermat, Gregory of S t . Vincent , and Wallis either were al ready at work with similar methods and arriving a t simila r results , or soon would be . 1 3 13 Their ability in effect to integrate this series of equations does no t contradict my earlier statement that analysis was confined almost entirely to second-degree equations befo re

1661. The curves that correspond to these equations of higher order were not studied .

The results for the lower powers showed a consistent pattern and were simply generalized by induction . Even the curves that co rrespond to the equations of the lower powers , beyond the p arabola y = x2, were no t studied; the "integrations " were carried through in isolation from any analysis o f the curves. In the case o f the simple parabola , mathemati­ cians usually thought in terms of computing volu mes rather than finding the area under a curve, but there was, of course , no equivalent analogy for the equations of h igher order . E quations of more than second deg ree were also frequently used to expound Hudde's rule, but once ag ain the curves that the equations define were not examined .

The discovery of a new world A

B ___

c

3 1. F

M

D

Figure 1 . 8 . C avalieri ' s calculation of the area of a trian gle by means of infinitesim als. The diagram sho ws only one of the infinite co rresponding pai rs of line s in the two triangles .

C avalieri utilized a tri angle, the si mplest of plane figures . On the base of the triangle , he constructed a p a rallelogram such that its diagonal , one side of the tri angle , divided the parallelogram into two i dentical tri angles . He then constructed lines parallel to the base and separated by equal dis tances such that the lines , infinite in num­ ber, filled the two tri angles and , in effect , cons ti tuted their areas . Cavalieri was seeking a rigorous demonstra tion . He knew that he could neither add an infini te number of lines nor arrive at a two­ dimensional area from such an addition . Howeve r, he could dem­ onstrate the ratio between two i nfini te sums when ' every line (which he treated implicitly a s an infini tesimal area) in one sum has the same ratio to a corresponding line in the other. In the case of the two tri angles the ratio i s equality . Every line EH in triangle FDC corresponds to an equal line BM in triangle CAP. Hence the two sums a re equal; the area of tri angle FDC equals the area of tri angle CAP; and the area of each equals hal f the area of the p arallelogram AFDC composed of the two (Figure 1 . 8) . In modern notation, if we consider the diagonal as the line x = y that d ivides a square, Cavalieri had demonstrated

32

1

a

a

Never at rest X

dy

=

1

Y

dy

�2

=

Proceeding in an analogous way , he s ummed up ratios of squares on the infinitesi m als and arrived at the conclusion that the volu me of a pyramid (the sum of the squares of the infinitesi m als in a tri angle) i s to the volume of the parallelepiped of equal height on the same base as 1 to 3. In our notation, the area under the cu rve x =y 2 ,

la 0

x dy

=

la 0

y 2 dy

a3 =

-

3

In a later work , he extended thi s result to all integ ral powers , x

Yn

la

X

dy

=

la 0

y n dy

=

an + t n + 1

--

By other routes o ther mathematicians arrived at the same result , which w as an accepted common possession of the m athematical com muni ty by the mid-1 650s . I t w as not , of course , expressed i n our terminology , and the inverse relation o f "integ ration" t o " dif­ ferenti ation , " which leaps fro m the p age when the two are ex­ pressed in mo dern notation, had not been recognized . Obviously , the equilateral hyperbola , xy = 1 , or y = 1 /x , could not fit into the general expression for areas , but in 1 647 Gregory of St . Vincent successfully applied infinitesimals to the area under thi s curve as well (Figure 1 . 9) . Although Gregory him self did not see the full i mplications of hi s work , o thers soon did. In our notation,

lb

y dx

=

lba

dx X

-

=

log b

-

log a

In the seventeenth-century , when log ari thms were still a new tool , thi s result , for the hyperbola y = 1 / (1 + x ) , p rovi ded a useful meth­ od , not to evalu ate areas from logs , but to calculate Jogs fro m areas . Althou gh the method of infinitesimals led mathematic s tow ard the concep t of integration, i t was inherently clumsy and crude . By 1 66 1 , i ts fertility was app roaching exhaustion . O ne of the d i s tinctive new features in seventeenth-centu ry m athematics , the u se of infini te series , which began to appear p rominently only in the middle of the century , developed pri marily from attacks on p roblems of areas . Like mos t of seventeenth-cen­ tury mathematics , this approach developed out of i ts counterpart in ancient geometry , in thi s case the method of exhau stion. When the new analysis was grafted onto the method of exhaustion , a wholly new branch of mathematics , i nfinite series , sprang forth . Although precursors can be cited, such a s Viete' s expre ssion for

The discovery of a new world

33

A

D N

J

F

L H

E OK

B

G

M

c

Fig ure 1 . 9 . G regory co mpare d the a reas DEGF and FGCH by means of infinitesimal segments equal in a rea .

the value of 1T , the prominent role of infini te series began with Gregory of S t . Vincent' s investigation of areas under the equilateral hyperbola . Once mathematicians recognized the logari thmic nature of the area function, they searched fo r means to calculate i t by app roximation , either by adding inscribed quadrangles of ever­ decreasing size or by subtracting tri angles of ever-decreasing size from the circumscribed quadrangle (Figure 1 . 1 0) . In the fi rst case , the area ABCE i s app roached by BAFC + kFnd + mnp b + hklf + . . . When OA = AE = AB = 1 , and AB i s d ivided into equal segments , the area ( = log 2) equals 1 / (1 2) + 1 /(3 4) + 1 /(5 6) + 1 1(7 8) + . . . , a series that allows one to calculate the loga­ rithm to whatever deg ree of accuracy he choo ses by continuing the series . During the next quarter-century a whole series (no t infini te to be sure but sufficiently long nevertheless) of series expansions of the logari thmic function , based on the area of the hyperbola, were developed as tools to simplify the calculation of logari th m s . Grego ry 's Opus geometricum (Geometrical Work) also considered infinite series by demonstrating that the sum of an infinite nu mber of geometrically p roportional indivisibles can app roach a fini te value as i ts limit . John Wallis ' s Arith metica infinitoru m (A rithmetic of ·

·

·

·

a

B

A

0

A

E

0

A

B

B Figure 1 . 1 0 .

Two means of generating infi nite series t o comp ute the area of ABCE ( = log 2 , when AB = AE OA ) . In diagra m A, the area is app roached by add ing p rogres­ sively s maller rectangles un der the curve . In diag ram B, the area is ap p roached b y subtracting p rogressively s maller triangles fro m the rectangle ABDE. =

The disco very of a new world

35

Infinites) , p ublished nine yea rs after Gregory ' s work , rev ealed p o ssi­ bilities in the concept of infinite series that extended far beyond the logarithmic function . Casting caution to the winds and relying on his intuition of continuity in mathemati cal pattern s as a substitute fo r demonstrative rigo r, Wallis set ou t on a rather different p ath toward C avalieri ' s goal , to determine the ratio of the su ms of two infinite series of indivisibles as a means to calculate areas . Consider the parabola y == x 2 • Let the base line of the a rea under the cu rve fro m x == 0 to x == 1 be divided into an infini te number of equal segments . Consider the length of the first ordinate to be 0 2 , of the second 1 2 , of the thi rd 2 2 , and so on to n 2 • Now co mpare the sum of the o rdinates to the sum of the ordinates (all equal to n 2) that make up the rectangle enclosing the segment of the cu rve being evaluated . Wallis p roceeded by a rough induction . If the first two o rdinates alone a re consi dered , 02 + 1 2 1 1 1 1 1 + + l2 + l2 = 2 = 3 6 = 3 � If th ree are considered ,

02 + 1 2 + 22 5 1 1 + ----6 . 2 22 + 22 + 22 = 1 2 = 3 After a small nu mber of si milar calculations , all of which revealed the same p attern , Wallis was ready to generalize:

02 + 1 2 + 22 + 32 + . n2 + n2 + n2 + n2 + .

. . n2 . . n2

1 1 + = 3 6n

Since n is indefinitely large , the ratio of the area under the curve to the s quare enclo sing it approaches the limiting value of 1 /3 (Fi gure 1 . 1 1 ) . I n a si milar manner, he determined that, for the curve y == x 3 , the value o f the ratio approaches 1 /4 a s a limit. Ano ther pattern was now emerging , and Wallis did not hesi tate to grasp i t . For all posi­ tiv� integers k , the ratio of the area under the curve y == xk to the area of the enclosing square equals 1 1 (k + 1 ) . This was hardly a sta rtling result in 1 656 . Expressed in modern terminology, it was equivalent to the already established conclusion ,

11

x k dx

1 __ =k + 1

Undoub tedly the agreement with established results increased Wallis ' s confidence in his p rocedures . His fu rther extension of them did break new ground . He generalized the results to fra ctional val­ ues o f k , which he introduced into mathematical notation . He showed that the areas under binomials (such as y == (a + x ) 2 ) can be evaluated by multiplying the binomial out and evaluating the term s one by one . Wallis ' s ulti mate interest, however, lay in a much more

Never at rest

36

0

D

0

0

0 A

Fig ure 1 . 1 1 .

T

T

T

T

Walli s ' s de monstration of th e rati o between the square

A TOD and th e area AO T under th e parabola.

difficult problem, the a rea under the ci rcle. Wh at he sought was the ratio between the enclosing square and the quad rant of the ci rcle y = ( 1 - x 2) 11 2 , whi ch would be equiv alent to the v alue of 4/7T. H e began b y employing the method ab ove t o compose a table , for integ ral v al ues of p and n , of what we would now exp res s as

f'

(1

-

x ' 1v)" dx

that is, of the ratio of the enclosi ng s quare to the area under the cH rve between the y axis , and x = 1 , where it cuts the x axi s (Fi gure 1 . 1 2) . p

/n 0

2

3

10

0 1 2 3

1 2 3 4

1 3 6 10

4 10 20

11 66 286

10

11

66

286

1 84756

The discovery of a new world

Fig ure 1 . 1 2 .

37

Wallis ' s table exp ressed the rati o of the area of the s quare to the area under the curve y = ( 1 - x 11P) n for values o f p and n from 0 to 1 0 . The diagram rep resents p = 1/2 , n = 1/2 , the equation for a circle of radius 1 , from which W allis established an infinite sequence for the value of 'TT (or, more precisel y , 4111") by inte rpolating into his table for integral values of p and n .

As Wallis recognized , the symmetry of the table repeats the pattern of the P a scalian triangle . He now sought to expand the table and, relying ag ain on his faith in the continui ty of patterns , to interpolate the values for 1 /2 , 3/2 , 5/2 . . Specifically , the function p == 1 12 , n == 1 /2 would be the valu e he sought , .

f'

1 ( 1 - x 2) 1 1 2 dx

He designated the value of thi s ratio by the symbol D . F rom the pattern in the rows and columns in which D does not appear, Wallis w a s able to fill in the enti re table . The row p == 1 /2 contains the fallo w ing value s: 1,

D,

3 2 '

1,

D,

3 2 '

4

3°'

.

15 8

1 05 48

_?_±_ D 15

or 3 2

5 4 4 ' 3

6

3

5

7

SD, 2 · 4 · 6 · . .

Since the alte rnate terms, in which D does not appear, constantly inc rease , Wallis concluded tha t the terms with D a re also intermedi­ . a te in value to the ' pre ceding and succeeding te rms . B y extending

38

Never at rest

the row , he was able to establish ev er narrower limits a round the value of D . Hence D ( == 4/7T) app roaches the limit, as the terms of the sequence are extended indefini tely, 3 · 3 · 5 · 5 · 7 · 7 · 9 . . 2 · 4 · 4 · 6 · 6 · 8 · 8 . . .

.

Infini te series were still a recent innov ation i n 1 66 1 . The extent of the new possibilities they opened were as yet unprobed . As the example from Wallis illustrates , they had been used only to calcu­ late certain v alues . From such cases , the concep t of a limiting value whi ch can be app roached to any desi red degree had emerged . No one had yet perceived how more flexible series co mpo sed of powers of a variable instead merely of numbers would extend the range of the new device. Mathemati cs as a whole at that time consi sted of innov ati ons like infini te series not yet fully p robed , and not yet bound together into a method or system that would reveal and exploit thei r mutual relati ons . Nevertheless , the very number of innov ations was excit­ ing. The world of mathematics was in ferment . On ev ery side it was spilling out over its earlier boundaries . 1 4 Let us try to place ou rselves in the posi tion of a young man in 1 66 1 , eager for knowledge, though of wholly untested capacity, as the new world of lea rning unrolled i tself before his eyes . What an i ncredible challenge to the i magination -- a world undreamed of in rural Lincolnshi re, a world of many continents as extensive in thei r diversi ty as in thei r number. To the north lay the frigid lands of mathemati cs where one must breathe the bracing atmosphere of ri gor. To the south lay the fetid tropical j ungles of alchemy with thei r strange mythi cal fauna . Temperate lands for experi mental in­ vesti gation lay between . Mani festly , the very v astness of the new wo rld placed it beyond the cap acity of any one mind to grasp and to comp rehend , findi ng an ordered cosmos where only chaos ap­ peared . Perhap s . Perhap s not . Perhaps some rare individual, one of the intellectual supernovae who have burs t intermittently but mo s t i nfrequently into the visi ble heav ens o f a s tartled wo rld , might 14 As in the case of some of the earlier topics , there is a considerable body o f literature on mathematics in the seventeenth century which I shall not attempt to su mmarize here . Indispensable though diffi cult is D . T. Whitesi de , " Patterns of Mathemati cal Th ought in the later Seventeenth Century , " Archive for History of Exact Sciences , 1 ( 1 96 1 ) , 1 79-388 . See al so H . G . Zeuthen, Gesch ichte der Mathematick im 1 6 . und 1 7. jahrh undert (rep rint ed . , New York , 1 966) ; J . E . Hofmann , The History of Mathematics to 1 800 , trans . F . Gayn or and H. 0. Midoneck (Totowa, N .J . , 1 957) ; Carl Boyer, History of Analytical Geometry (New York, 1 956) , and The Concepts of the Calculus (New York, 1 939) ; Michael S . Mahoney , The Mathemat ical Career of Pierre de Fermat (Princeton , 1 973) ; and Margaret Baron , The Origins of the Infinitesimal Calculus (Oxford, 1 969) .

The discovery of a new world

39

grapple effectively even with such a task . Other worlds new to Newton also opened themselves to his gaze in Cambri dge , and his exploration of them played an important part in his life . Had he limi ted himself to them , however, his name would have passed long since into oblivion . As I said before, the only reason anyone wri tes a biography of Newton i s because he chose to enter a wo rld not only new to hi m as to all undergraduates , but new to man himself. In C ambridge , Newton discovered that a new world had been discove red. He discovered as well something s till more i mportant . The early adventu rers had only s couted i ts coasts . Vast continents remained to be explo red .

2

A sober, silent, thinking lad

SAAC New ton was born eighteen and a half years before he en­ I tered Cambridge , early on Christmas D ay 1 642 , in the manor

house of Wools thorpe near the village of Cols te rworth, seven miles s outh of Grantham in Li ncolnshi re . Since Galile o , on whose di scov­ eries much of Newton ' s ow n career in s cience would s quarely rest, had died that year , a si gni fi cance attaches i tself to 1 642 . I am fa r fr o m the fi rst to note it - and will b e undoubtedly fa r from the last. B o rn in 1 564 , Galileo had lived nearly to ei ghty . New ton would live nearly to ei gh ty-five. B etween them they vi rtu ally sp anned the enti re scienti fic revoluti on , the central core of whi ch thei r combined work constitu ted . In fact , only England ' s stiff-necked Protestanti sm pe rmi tted the chronol ogi cal liaison . B e cau se it consi dered that p ap­ ery had fatally conta mi nated the Gregorian calendar, Engl and was ten days out of phase w ith the Continent , where it was 4 Janu ary 1 643 the d ay Newton w as born . We can s acri fi ce the sy mbol with­ out losing anythi ng of substance . It matters only that h e was b o rn and at such a ti me that he could utilize Galileo ' s work. P rior to Isaac, the New ton family was wholly without di stincti on and wholly withou t learning . Since it knew steady economic ad­ vance du ring the centu ry p rior to Isaac' s birth , we may assu me that it was not without diligence and not without the i ntelli gence that can make di ligence fruitfu l . In his old age, New ton may h ave enter­ tai ned the idea that he was des cended from a gentle Scottish family of East Lothi an , one of whose members came to England w ith James I . 1 He settled in fact fo r an ancestry much more humble , however, and mu ch cl oser to reali ty . A ccording to his offi ci al pedi­ gree , the family des cended from a John New ton of Westby (a village about five miles southeast of Grantham) , who li ved in the mi d-si xteenth centu ry . Indeed there is a pedigree of another branch of the Newton family entered in the visi tation of Linc olnshi re of 1 634 whi ch traces the family to this same John Newton, who i s said to be de_s cended from the Newtons of Lancashi re . It app ears more probable that the family was indigenou s to the area, taki ng i ts name from one of the nu merous New tons (from " new town") found nearby as they were found over all of England , whi ch dated from 1

The story is based o n a conversation of James Gregory with New ton i n 1 725 , repeated more than fi fty years later via two intermedi aries in a letter of Dr. Th o mas Reid (see A ppendi x x x x 1 1 in David Brewster , Memoirs of the Life, Writings , and Disco veries of Sir Isaac Newton , 2nd ed . , 2 vols . [Edinbu rgh , 1 855] , 2, 537-4 5 . ) 40

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sober, s ilent, th inking lad

41

the great medieval expansion of population in the eleventh and twelfth centuries. 2 A Simon Newton, the fi rst of the family to rai se his head tenta­ tively above ru ral anonymity , lived in Westby in 1 524 . Along with twenty-two other inhabitants of Westby , he had achieved the status of a taxpayer in the subsidy granted that year. Fou rteen of the twenty-two , including Simon Newton , paid the minimum assess­ ment of 4d . Eight others p ai d asses sments ranging from 1 2d to 9s 6d , and one, Thomas Ellis , who was one of the richest men in Lincolnshire, paid over £ 1 6 . 3 If the Newtons had risen above com­ plete anonymity , clearly they did not rank very high in the social order , even in the village of West by . Since the average village in that p a rt of Lincolnshire consisted of about twenty-five or thi rty households , his assessment may indicate that he and thirteen others occupied the lowest rung on the Westby ladder. They were climb­ ing , ho wever , and rather rapidl y . When another subsidy was granted in 1 544 , only four men from Westby had the p rivilege to pay; two of them were Newtons . Simon Newton was gone , but John Newton , p resumably the son of Simon, and another John Newton (the John Newton of Westby mentioned above) , p resu m­ ably his son, were now , after a man named Cony , the mo st flour­ i shing inhabitants of Westby . 4 In hi s will of 1 562 , the younger John Newton s till styled himself " husbandman" ; twenty-one years later, his son , a third John , died a " yeoman , " a step up the social ladder, and a brother William of the same generation also claimed that standing . 5 Inevitably, Newton' s pedigree has been worked out in considera­ ble detail , first b y Newton himself, later by the antiquarians whose attention the great attract . A list of his uncles , great-uncles , and the like , and the relationships in which they stood to him is of less interes t than the i mpli cations wrapped up in the s h i ft fr o m husbandman t o yeoman . I n Lincolnshire, the sixteenth and seven­ teenth centuries witnessed a steady concentration of land and wealth with a consequent deepening of soci al and economic distinctions . The Newtons were among the minority who prospered . Westby is located on a limestone heath , the Kesteven plateau , a wedge of high ground thrust up toward Lincoln between the great fens to the east 2

C . W . Foster , " Sir Isaac Newton's Family , " Reports and Papers of the Architectural Societies of the Co unty of Lincoln, Co unty of York, Archdeaconries of Northampton and Oakham, and County of Leicester, 39, part I ( 1 928) , 4-5 . Newton 's own researches into the genealogy of his family , w hich also trace it back to the Newtons of Westby , are found in Keynes MS 1 1 2; Babson MSS 440, 441 ; and in an unnumbered m anuscript in the Humanities Research Center, University of Texas. 3 Foster , "Newton's Family , " p. 5 . 4 Ibid. , p . 5 . 5 Ib id. , p p . 29, 36- 7 .

Never at rest

42

and the fenny bo tto m lands of the Trent valley to the west . The plateau had always p resented i tself as a likely highway to the north . The Romans had built Ermine Street along i ts back , and the G reat North Road of mediev al and early modern England followed the same route as far as Grantham , where it vee red off to the west toward an easier passage over the Humber . Even today the main highway north near the ea s tern coast of England crosses the plateau along the same path . Wool sthorpe , where New ton was reared , lay less than a mile from a maj o r thoroughfare of his day . If the plateau was a natural highway , i t was not a natu ral granary. The soil was thin and poo r . M uch of the arable land could su stain only a two-field rotation, which allowed it to stand fallow half the ti me. Enclosure here p roceeded slowly , while large stretches of uncultivated waste were used in common as sheep walk s . Wool from sheep was the foundation of the plateau ' s agricultu ral econ­ o m y . In co mpensation for nigg ardly soil , the plateau bo re a rela­ tively sparse population. Those who would could p rosp er. The Newtons would. The tale is told in the details of successive wills . Fro m John Newton of Westby , who left a will when he died in 1 562 , each genera tion for a century left a consi derably au gmented estate . Rather , they left au gmented estates . The Newtons were also a prolific clan (Figure 2 . 1 ) . John Newton of Westby had eleven children , of who m ten survived . His son Richard, Isaac ' s great­ grandfa ther, had seven children of who m five survived . Isaac 's grandfather , Robert , had eleven , of who m six survived . There was no single inheritance which was augmented and passed on . The inheritance was continually being divided, but most of the seg ments took root and flourished. By the middle of the seven­ te en th century a considerable nu mber of subs tantial yeo men na med Newton were sp rinkled over the area around Grantham , all of them descendants of John Newton of Wes tby , husbandman . 6 No doubt the fact that this John Newton married very well - Mary Nixe , the daughter of a p rosperous yeoman - helped his p osition. He must also have known how to handle the dowry , however , for he was able to p rovide handso mely for three sons . The descen­ dants of one of them , Willia m , p rospered even more th an the rest; in 1 66 1 one of his descendants , yet another John , push ed his way into the squirea rchy as Sir John Newton, Bart. 7 In 1 705 , Isaac Newton anxiously pursued his son , also Sir John Newton , B art . , for cc rroboration of his pedigree . At about the ti me of his death , John I Jewton of Westby purchased an extensive farm of well over a h undred acres , including sixty acres of arable , in Woolsthorpe 6

Ibid . , passim .

7 Ibid . ,

p.

10.

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so ber, silent, thinking lad

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Figure 2 . 1 . The Newton fa mily tree as Ne wton himself dre w it up for his co at of arms in 1 705 . ( B y pe rmission of the Uni­ versity of Texa s . )

for ano the r son, Richard. 8 Woolsthorpe lay app roximately three miles southwest of Westby, and Richard Newton was I saac New­ ton ' s g reat-grandfather. To put the family ' s economic position in perspective , the ave rage estate in the 1 590s of p ea sants on the heath with p roperty , that is, of the wealthier peasants , was about £ 49 . The richest yeoman , a s measured by his will , who died in Lincolnshire in the 1 590s left personal p roperty of nearly £ 400 . Very few wills in that time left goods worth more than £ 1 00 . 9 Richard Newton , whose father established him on a farm pur­ chased for £40, left goods inventoried at £ 1 04; the inventory did 8 Ibid. , p . 6. 9 Joan Thi rsk , English Peasant Fanning (London , 1 957) , shire and the Fens (London , 1 952) , p . 1 03 .

pp .

55 , 1 03-4; M. W. Barley, Lincoln­

44

Never at rest

not include the land or the house . 10 It did include a flock of fifty sheep , well over the average number. 1 1 Sheep were the measure of wealth on the heath . Not only did John Newton of Westby endo w three sons magnifi­ cently by yeo man stan dards, but he also married a daugh ter to Henry A skew (or Ayscough) of Harlaxton . 12 The Ayscough s were a prominent Lincolnshi re family , though i t i s not clear what, if any , relation Henry A skew bore to the main stem of the family, whose seat lay well to the north . It was not the last alliance between the two families . Robert Newton, Isaac's grandfather, was born about 1 570. He inheri ted his father' s p roperty at Woolsthorpe to which he added the manor of W ool sthorpe by purchase in 1 623 . The mano r was not in p ro sperous condition . It had changed hands by sale four times during the p revious century . 1 3 Nevertheless i ts value was reckoned at £ 30 per year. 1 4 Added to the original estate, it gave the family a comfortable living indeed by yeoman standards of the day . So­ cially, i t may have elevated Robert s till further. He was now lord of a mano r , legally entitled to exe rci se the powers of local authori ty, such as conducting court baron and court leet, whi ch , as s till opera­ tive elements of local administration , had jurisdiction o ver minor breaches of the peace and could levy fines but not impri son. The lord o f a manor was no husbandman . 15 In December 1 639, he settled the enti re W ool sthorpe p roperty on hi s eldest surviving son, Isaac, and Hannah A yscough (or Askew) , to whom he was be­ tro thed . Isaac was h ardly a young man . He had been born on 2 1 September 1 606 . Although Hannah Ays cough' s age i t n o t known, i s seems likely that she was well beyond maidenhood herself; her p arents had been married i n 1 609 and her brother William was probably the William A skue who matriculated in Camb ridge from Trinity College early in 1 630 . Nevertheless, the couple did not marry at once , and the re is every suggestion that they waited to obtain the inheri tance fi rst. After all , Robert Newton was nearly 1° 12 13

14 15

1 1 Thirsk , Peasant Fanning, p. 7 1 . Foster, "Newton 's Family , " p p . 37-9. Foster, " Newton 's Family , " p . 1 1 . Tumor Papers , Lincolnshire Archives . I have not seen the Tumor Papers but I heard about them from P rofessor Gale Christianson of Indiana State University , who received his information from M r . K . A . Baird of Colsterwo rth. As far as I kno w , Mr. Baird is the only person who has looked at the papers since the family deposited them in the archives nearly forty years ago . I draw my information fro m his unpublished report , which bears the mark of reliability . Foster, "Newton's Family , " p . 1 3 . I n his notes intended fo r a life of Newton, John Conduitt, the husband o f Newton 's niece, Catherine Barton Conduitt , rather insisted on their position as lords of a manor . In the rolls , he said , the family styled themselves " Lords of the M annour of M o rtimer in the parishes o r p recincts of Wols trope & Costerwo rth in the soak of Grantham in the County of Lincoln" (Keynes MS 1 30 . 3 , sheet 2) .

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sober, silent, th inking lad

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seventy . He obliged in the autumn of 1 64 1 ; the follo wing Ap ril they were uni ted. 16 The A yscough match was another distinct s tep forward for the Newtons . Hannah was the daughter of James Ayscou gh , gentle­ man , of Market Overton , county Rutland. As marriage portion, she brough t with her a p roperty in Sews tern , Leicestershire , wo rth £ 50 per year. 17 It i s diffi cult to imagine the match without New­ ton' s recently purchased d ignity of mano ri al lord. Hannah brought mo re than addi tional wealth . For the first time , the Newtons made contact with formal learning . B efore 1 642 , no Newton in Isaac's branch of the family had been able to sign his own name . Their wills , d rawn up by scri veners or curates , bore only their mark s . Isaac Newton , the father of o u r subj ect, w a s unable t o sign his name , and so w as his brother who helped prepare the inventory of his possessions . 18 In contrast, one Ayscough at the very least w as educate d . William , Hannah' s bro ther, M . A . Cambridge , 1 63 7 , pursued the calling fo r which learning w a s e ssential . 1 9 O rdained to the clergy of the Anglican Church, he was instituted to the recto ry of Burton Coggle s , two miles east of Colsterworth , in January of the year in which his si ster married Isaac Newton . The nearly exclusively masculine orientation of genealogical re­ search in the p ast , buttressed by the analogou s practice of patro­ nymics, has focused attention enti rely on the Newton family . Can­ non Foster publi shed an impeccable investigation of " Si r Isaac Newton' s Family , " in which Hannah Ayscough , daughter ofJames Ayscou gh and possibly related to the A skews of Harlaxton with whom the Newtons had mated once two generations earlier, sud16

Foster, "Newton's Family , " p . 1 5 . A mista ken impression of distance and of farreaching connections may arise from the fact that three counties were involved. Since Woolsthorpe lies near the corner where the three meet , no more than five miles separated the three villages . Robert Newton had not reached any great distance to find a bride for his son . The match was distinctly a lo cal affair. 18 Foster , " Newton's F amily , " pp . 43-56 . Although Richard , the brother , could not sign the inventory in 1 642 , he was able to sign his own will in 1 659 . Neither his son nor his wife were able to sign their wills at a later tim e , however . 1 9 It is reasonable to question the extent of Hannah Ayscough Newton's literacy in the light both of her husband's illiteracy and of the general level of education among women at the time . S he did sign her own will in 1 672 (ibid. , p. 53) . Moreover, one letter to her son Isaac has survived and is reproduced here in chapter 5 (Corres 1, 2) . It is the letter of a woman for whom writing was clearly a heavy burden , two sentences long, or w hat would be two sentences if she had kno wn ho w to punctuate and capitalize . It did not communicate anything beyond the fact that she had received a letter from him. There must have been other letters of this sort, it is true; indeed this one refers to an earlier one that Isaac had m entioned receiving . It seems very unlikely to me that there were many , and I find it quite impossible to imagine a p rofuse and revealing correspondence which would deepen our understanding of their relationship if it had survived . 17

46

Never at rest

denly appears a s an adj unct to the Newtons . Perhaps we ought rather to view the match fro m the Ayscough perspective , but no dedicated genealogist has seen fit to investigate that family . We know only that James Ayscough , gentleman , of Market Overton had a daughter Hannah . The name raises the possibility of a blood relationship to the important Ayscough s of northern Lincolnshire , but n o available empi rical evidence either supports o r ca�ts doubt on the speculation. 20 Whether they were connected o r not, the genealogy of James Ays cough would undoubtedly be somewhat more impressive than that of Robert Newton , though his branch of the family would apparently show a recent decline in contrast to the Newtons' s teady rise . 2 1 The intellectual his tory of England i s not sprinkled with Ayscough s , however, and anyone seeking to explain Isaac Newton' s brilliance genetically i s not likely to find it forecast among them any more than i t is among the Newtons . The overriding importance of the Ayscough contribution lies elsewhere . As it turned out, Isaac was reared entirely by the Ays cough s . We can only speculate what would have happened had his father lived . The father was now the lo rd of a manor, as his own father had not been while he himself was being reared . Per­ haps he would have seen the edu cation of his son a s a natural consequence of his position. However, his brother Richard , who was only a yeoman to be sure and not the lord of a manor , did not see fit to educate his son , who died illiterate . 22 Being reared a s an Ayscough , Isaac met a different s e t o f expectations . The pre­ sence of the Reverend William A yscough only two mile s to the east may have been the cri tical factor. At a later time , his interven­ tion helped to direct Isaac toward the university . Whatever the individual roles , the Ayscoughs took it for granted that the boy would receive at least a basic education . We have some rea son to doubt that the Newtons would have done so . 20

21

22

William Stukeley , a young er contempo rary of Newton's who collected a great deal o f information about his life , took t h e connection fo r granted . H e calls t h e Ayscoughs " a very antient and wealthy family in Lincolns hire , from a hamlet of that name , near Bedal , Yorkshire" (Stukeley, p . 34) . I n his memoir t o Fontenelle, Conduitt probably drew upon Stukeley when he called them " an antient and hon b le family in the County of Lincoln . . . " (Keynes MS 1 29A , p . 1 ) . Conduitt said that the Ayscoughs were "formerly of great consideration in thos_e parts , " by w hich he meant the area about Colsterworth , since he connected them with Great Ponton, about four miles no rth. (Keynes MS 1 30. 2, p. 9) . The phrase suggests a gentle family fallen on hard times and fo rced into marriage with rising yeomen to recoup their fo rtunes . Foster , " Newton's Family, " p . 55 . Cf. n . 1 8 above. However, Isaac Newton pere also called himself a yeo man in his will (ibid. , p. 45) . Isaac Newton fi ls, the son o f Hannah Ayscough , was the first in his line to call hi mself a gentleman (initially , as far as we know , in the visitation o f Lincolnshire of 1 666; ibid . , p . 3) . Of course, he had com menced B . A . at Cambridge University , which autom atically conferred gentle status.

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sober, silent, thinking lad

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Six months after his marri age , I saac Newton died early in Oc­ tober, 1 642 . He left behind an estate and a pregnant widow but virtually no information about himself. We have only one brief description of him , from a century and a half after his death , by Tho m a s M aude , who clai med to h ave inqui red diligently into Newton' s ancestry among the descendants of his half brother and half sisters and a round the pari sh of C olsterworth . According to Maude , I saac Newton the father was " a wild, extravagant, and weak man . . . ' ' 23 Such may have been the case; since Maude did not even get his name right , however, calling him John , we are s carcely compelled to accept the description . About his e state we are directly informed by his will . Since it define s the economic position of Isaac Newton (the son) at the time of his birth , i t de­ serves some scrutiny . In addition to his extensive lands and the mano r house, Isaac Newton , senior, left goods and chattels valued at £ 459 1 2s 4d . His flock of sheep, the ultimate measure of wealth in those p a rt s , numbered 234 , which compare s with an average flock of about 35 . He owned apparently ' 46 head of cattle ( divided among three categories which are partly illegible on the document and hard to interpret in any case) , also several times the average . In his barns were malt , oats, corn (probably barley , the staple crop of the heath) , and hay valued at nearly £ 1 40. Since the inventory was made i n October, these i tems undoubtedly repre sent the harvest of 1 642 . By putting oats (£ 1 1 5 s) in a separate category , but coupling corn and hay (£ 1 30) in another, the men who drew up the inven­ tory made it diffi cult to interpret. The oats and hay would have been fodder for the winter; surely the corn would not have been . The cattle (worth £ 1 0 1 ) , and the s heep (worth £ 80) would have consumed the fodder du ring the app roaching winter, so that it does net constitute a final pro duct of the estate . Part of the final product was wool , and the inventory includes wool valued at £ 1 5 . It i s unlikely that the 1 642 clip , from June , would s till have been on hand; £ 1 5 is too small a sum in any case , since the annual clip averaged between a fourth and a thi rd of the value of the flock . The estate also included, of course , extensive agri cultural equipment and furnishings for the house . I t included a s well rights to graze sheep on the common . 24 The value of such rights is impossible to esti­ mate , but when wool is king , grazing rights are gold. Like fodder, of course, they would only be means to the annual pro duct, how­ ever. At this remove i t i s impossible to determine the total annual value of the estate . An estimate of at least £ 1 50 per year does not 23 24

Thomas Maude , Wensley-Dale; or, Rural Contemplations, 3 rd ed. (London , 1 780) , p . 29n . Newton still possessed these rights , plus some more attached to p roperty acquired later , in 1 7 1 2 when he w rote to Henry Ingle of Colsterwo rth about an agreement to enclo se the commons (Corres 5, 346-7) .

48

Never at rest

seem unreasonable . 25 We should add that the inventory may have been lower than the long-term average value of the estate . The 1 620s had been a hard decade, and p robated inventories throughout the 1 630s were lower in consequence . They did not fully recover their former level until about 1 660 . Newton' s mother reserved the income fro m the p aternal estate to Isaac when she remarried; the dowry lands in Sewstern appea r to have been included. In addition , her second husband settled a further piece of land on him . 26 Ulti­ mately Newton inherited the entire paternal estate together with the land from his s tepfather and some additional properties purchased by his mother. I have summarized the estate in financial terms because that was its only meaning in Newton' s life . At one ti me the family intended that h e manage it. This was not to be, ho wever, and the estate functioned in his life only as financial s ecu ri ty . What­ ever p roblems might awai t the child sti ll unborn when the mven­ tory was made, poverty was not likely to be among them . The only child of Isaac Newton w a s born three months after hi s father' s death in the manor house at Woolstho rpe early Chri stmas mo rning. The posthumous offspring, a son , was named after his father, I saac . Already fatherless , apparently premature , the baby was so tiny that no one expected it to survive . Over eighty years 25

26

The will and inventory are in Foster, "Newton's Family , " pp . 45- 7 . Stukeley stated that the m anor of Woolsthorpe, Newton' s paternal estate , was worth £ 30 per annum; in addition he had an estate at Sew stern from his mother that was worth £ 50 per annum (Stukeley, p . 36) . Stukeley app ears merely to have repeated the assessment of the manor , which was n o t the entire estate a t Woolsthorpe. Thomas M aude, whose sou rce was apparently the profligate Reverend Benj amin Smith , Newton's half-nephe w , stated that Newton's whole estate at Woolsthorpe was worth about £ 1 05 per annum at his death . (Maude , Wensley-Dale, p . 29n) . In 1 694 Newton told David Gregory that the estate he inherited was worth £ 1 00 a year . Among the Tumor Papers there is an undated and unsigned list entitled "A valuation of the lands of the general persons heare underwritten as followeth, " which appears to list the p roperty owners of the p arish. The valuation estimates the estate o f Newton's mother at £ 70 , a figure which would not include the dowry in Sewstern or the estate of the Reverend Barnabas Smith in N orth Witham . (E ven s o , she was the wealthiest person in the p arish by far , with more than one-third o f t h e total valuation . ) According t o the family tradition o f the Ays coughs , a s written do wn by William Ayscough's granddaughter, the estate was worth £ 1 20 per annum ( I . H . Uames Hutton] , " New Anecdotes of Sir Isaac Newton , " Annual Register, 1 9 (1 776] , 24) . Only if we take seriously the limitation to Woolsthorpe, which would exclude both Hannah Newton's dowry in Sewstern (the p roduce o f which was p robably in the inven­ tory) , the p roperty from Barnabas Smith , and later additions in Buckminster, do I find it possible to reconcile these figures with the facts of the inventory of his father's will . The estate could have declined in value, but there was no general decline in agri cultural values in Lincolns hire during that period . In attempting to ass ess the estate, I have relied heavily on comparative information in Thirsk , Peasant Fanning, passim . In 1 733 the estate �hat Newton inherited, given as about 270 acres, sold for £ 1 600 (Tumor Papers) . Foster , " Newton's Family , " p . 1 7 .

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later, Newton told John Conduitt , the husband of his m ece, the family legend about his birth. s r I . N . told m e that he had been told that when he w a s born he was so little they could put him into a qu ar t pot & so we � kly that he wa s forced to have a bolster all round his neck to keep it on his shoulders & so little lik ely to live that wh en two women were s ent to Lady Pakenh am at No rth Witham for s omething for him they s ate down on a s tile by the way & said there was no o ccasion fo r making haste for they wer e sure the child would be dead before they could get back. 2 7

Apparently hi s life hung in the balance at least a week . H e was not bapti zed until 1 J anu ary 1 643 . We exp ect li ttle informati on on the following years , and we are not di sappointed . The great Civil War raged through Eng land , but it is hi ghly unlikely that i ts anxieties p rinted them selv es on the infant' s psyche . B efore he was a year old the parliamentary forces had secured Lincolnshi re; t roop s may hav e passed up and down the Great North Road within sight of the manor hou se . They would have been bodies of men without meaning to the tiny boy; the only man over wh om the household could have worried w as already dead . An event of overwhel ming i mportance sh attered the secu rity of his chi ldhood immedi ately foll owing his thi rd birthday . Conduitt obtained an account of it from a Mrs . H atton, nee Ayscough: Mr S mith a neighbouring Clergy man , who had a very good Estate , h ad lived a B atchelo r till he was pretty old , & one of his parishioners advis eing him to marry He said he did not know where to meet with a good wife: the man ans wered , the widow Newton is an extraordi­ nary good woman: but saith Mr S mith , how do I know she will have me . & I don' t care to ask & be denyed . B ut if you will go & ask her , I will pay you for your day's work . He went accordingly . Her answer was , She would be advised by her B ro : Ay s cough . Upon which Mr S mith s ent the same person to Mr Ay s cough on ye same errand , wh o , u p on con su lting with his Sister, treated with Mr Smith: who gave her son I s aac a parcell of Land , being one of the terms insisted upon by the widow if she married him . 2 8

Barnabas Smith was the rector of North Witham, the next village south along the Witham, a mile and a half away . Born in 1 582 , he had matri culated at O xford in 1 597 , commencing B . A . (as grad ua­ ti on was called at the ti me) in 1 60 1 and proceeding M . A . in 1 604 . 27

Condu itt' s memorandu m of a conversati on with New ton , 31 A u g . 1 726 (Keynes MS 1 30. 1 0) . As he worked the memorandum into his p rojected life of New ton, he amended it in two successi ve drafts , each ti me for the worse (Keynes MS 1 30 . 3 , sheets 5 and 6; Keynes MS 1 30. 2 pp . 1 5- 1 6) . The progressive deterioration of the story makes a depressing 28 Keynes MS 1 25 . warning of the dangers of attempted literary elegance .

$.

Figure 2 . 2 . A map from the early nineteenth century of the corner of Lincolns hire where Newton was b o rn and rea re d . O nly two places connected with Newton and his family are not included on the map . B uckminste r , the locati on of two fields that his mother wi lled to him , ought to appear near the left-h and edge of the map , about a mile no rthwest of Sews tern and directly west of Stainb y. If the map e xtended 1 1/4 inche s fa rthe r s o u t h , Market O v e rton , the o riginal h o m e of his mother, w o u ld app e a r , 3 1/4 mil e s direc tl y south of Sewstern. (From E dm und Tumor, Collectio n s for the History of the Tow n a n d Soke of Gra n tham, London , 1 806 . )

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"P retty old , " as M rs . Hatton' s account has it, rather understates the matter; he was sixty-three years old when he added " Smith" to Hannah Ayscough Newton' s lengthening string of names . Nor had he lived a bachelor . He had buried a wife the previous June , and he had not allowed much grass to grow over her grave before he mended his single estate . 2 9 We do not know a great deal about the rector of North Witham . To start with the best, he owned books . Newton's room a t Wools­ thorpe contained , on the shelves that Newton had built for them , two or three hund red books , mostly editions of the Fathers and theological treati ses , which had belonged to his stepfather. 3 0 Pur­ chasing books with intent to study i s , of course , not the only w ay to obtain them . One might inheri t a theological library , for ex­ ample , if one' s father was a clergyman as Barnabas Smith' s fa ther was . At any rate he had the books . He may even have read a bit in them . In a huge notebook, which he began in 1 61 2 , Smi th entered a grandly conceived set of theological head ings , and under the head­ ings a few pertinent passages culled from hi s reading . If these notes rep resent the sum total of his lifetime assault on his library, i t is not surpri sing that he left no reputation for learning . Such an expanse of blank p aper was not to be discarded in the seventeenth centu ry . Newton called i t his " Waste B ook , " and what Barnabas Smith had once intended as a theolo gical commonpla ce book witnessed the bi rth of the calculus and Newton's first steps in mechanics . 3 1 Possi­ bly the library started Newton' s theological voyage to lands his stepfather would not have recognized . Smith must have been vigorous , not to say lu sty; though already sixty-three when he married Hannah Newton , he fa thered three children before he died at seventy-one . No surviving story suggests that he conce rned himself much with the likelihood that the three child ren would soon be left without a fa ther, even as another boy had been . Beyond the books and the vigor, little else about him sounds attractive . He occupied the rectory in North Witham be­ cau se his fa ther, the rector of South Witham, had bought i t for him in 1 6 1 0 by purchasing the next presentation from Sir Henry Paken­ ham , who controlled i t . In the following year, a visitation by the Bishop of Lincoln reported that the Reverend Mr. Smith was of good behavior, was nonresident, and was not hospitable . 3 2 In ef­ fect , Barnabas Smi th ' s father had purchased a comfortable annuity for his son . He received the income from North Witham for over forty years . For the fi rst thirty, as far as we know, he conformed without p rotest to the ever more Arminian policies of the estab3 0 Stukeley, p . 1 6 . 2 9 Foster, " Newton's Family , " p. 1 7 . 32 Add M S 4004. Foster, "Newton's Family" p . 1 7 .

31

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lished chu rch . With the Civil War came the Puri tans and t h e Covenant. T h e Reverend Mr. Smi th remained undisturbed in his living . The second Civil War brought the Independent s and the Engagement . By no w, large numbers of s teadfast Anglican clergy had p referred ej ection to conformity , and many were suffering real d ep rivation, but not the Reverend M r . Smi th . When he died in 1 653 , he had grasped his living firmly through all the uph eavals - a pliable man , obviously, more concerned with the benefice than with p rinciples . Althou gh they had never met, John Milton knew him well . An ow of such as for their b ellies sake , Creep and intrude, and climb into the fold? Of other care they little reck 'ning mak e , Than h o w to s cramble to t h e shearers feast, And shove away the worthy bidden guest; Blind mouthes ! 33

Not that the living of North Witham was Barnabas S mi th ' s p ri­ mary means of support . He had an independent income of about £ 500 per annum - " wch in thos e days was a plenti ful estate . . . , " said Conduitt, in his single essay at unders tatement . 34 For N ewton , his s tep father' s wealth meant in the end a si gnificant increment to his own possessions . A s M rs . Hatton' s account s tates , part of the mar­ riage s ettlement was a parcel of land for him , increasing his paternal estate. Years later, Newton inheri ted from his mother additional land s that she had purcha sed for him , undoubtedly from the estate of her second husband . 3 5 The will of N ewton ' s uncle, Richard New­ ton , suggests an economic s tatu s similar to that of N ewton' s fa ther. The will of Hannah Ayscough Newton Smi th bespeaks a wholly different level . The Ays cough marriage had been a s tep upward fo r the Newtons , more in s tatus than in wealth . The Smith marriage brought a maj o r increase in wealth . In return , it dep rived N ewton of a mother. H i s s tepfather had no intention of taking the three-year33 34

35

"Lycidas , " in The Works of John Milton, ed . Frank A . Patterson (1 8 vols . ; New York, 1 93 1 -8) , 1 , Pt . 1 , 76- 83 . Keynes MS 1 30 . 2 , p . 1 1 . Acco rding to the Ayscough family tradition, Smith was "a rich old batchelor , who had a good estate . " In the context this meant a much better estate than the N ewton estate of £ 1 20 per annum (I. H. Uames Hutton], " New Anecdotes , " p. 25) . The two wills of Barnabas and H annah Smith are in Foster, " Newton's Family , " pp. 504 . Beside some specific bequests (which include four silver vessels and a house in Lin­ coln) , Barnabas Smith left all his lands to his son Benj amin , £ 500 to each daughter , and all the rest to his wife. All the rest must have been considerable , since Hannah S mith, when s he died , w as able to leave an additional £ 50 to Benj amin (in addition to a property in Sewstern and an annuity of £ 40 per year that she had already given him) , an additional £ 80 to her m arried daughter Mary and her family , an additional £ 300 to her unmarried daughter H annah, and to Isaac two fields in neighboring Buckminister and a house in Woolsthorpe that she had purchased , plus the uncatalogued goods and chattels , most of w hich went to Isaac .

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old boy with the mother. Isaac was left in Woolsthorpe with his grandmother Ayscough . The Reverend Mr. Smith did have the house rebuilt for them; he could afford i t . The l o s s o f his mother must h ave been a traumatic event in the life of the fatherless boy of three . There w as a grandmother to replace he r, to be sure; but significantly , Newton never recorded any affectionate recollection of her whatever. Even her death went unnoticed . As we shall see , Newton was a tortured man , an ex­ tremely neuro ti c personality who teetered always , at least through middle age , on the verge of breakdown . No one has to stretch his credulity excessi vely to believe that the second marriage and dep ar­ ture of hi s mother could have contributed enormously to the inner torment of the boy already perh aps bewildered by the realization that he , unlike o thers , had no father. 3 6 Moreover, there is reason to think that Isaac Newton and Barnabas Smith never learned to love each other. Nine years after his stepfather' s death , when Newton was move d to draw up a list of his sins , he included , "Threatning my father and mother Smi th to burne them and the house over them . " 37 Probably every boy has angry confrontations with his parents , when puerile threats are screamed in fru stration . Neverthe­ less , the scene must have etched i tself deep ly on Newton's con­ sciousness if he recalled i t nine years later. For B arnabas Smith ' s part, h i s actions speak clearly enough . For more than seven and a h alf years , until he died , while the child of three grew to be a boy of ten , he did not take him to live in the rectory in North With am . The manor ho use of W oolsthorpe stands on the west side of the small valley of the river With am, a s tring down the K esteven pla36

37

In his psy choanaly sis of Newton (A Portrait of Isaac Newton [Cambridge, Mass . , 1 968)) , P rofessor Manuel makes the remarriage o f his mother the critical episode in New ton' s entire life. Already lacking a father, whom h e h a d never known, he was now bereft of the mother he had possessed exclusively . According to Manuel' s interpretation, the sense o f deprivation dominated his life . H e had been robbed of h i s m o s t p recious possession . He spent the rest of his life finding surrogates on whom to vent the rage that he had no t been able, in his infantile weakness , to pour out on its real obj ect, Barnabas S mith, the man who had ravished his mother. Robert Hooke , John Flamsteed , Go ttfried Wilhelm Leib­ niz , the unfortunate counterfeiters at the turn of the century - all of them suffered for the crime o f Barnabas S mith . Manuel's analysis is subtle, complex, and ingenious . Is it also true? It appears to me that we lack entirely a ny means o f knowing . It is plausible; it is equally plausible that it is miguided . I am unable to see how empirical evidence can be used to decide on it , one way or the other . I trus t it is clear that I am no t offering an alternative analysis . It would confront exactly the same problem of confirmation, and in any case, I have no qualifications whatever to o ffer one . Manuel's portrai t , as distinct from his analy sis, is a vivid and insightful account of established facts of Newton ' s life. No biog raphe r o f New ton can afford to ignore it. It does not attempt to give an account of his scientific career. Richard S . Westfall , " Short-writing and the State of New ton' s Conscience, 1 662 , " Notes and Records of the Royal Society, 18 (1 963) , 1 3 .

54

.

f�.'

Never at rest

:-

·· . . ' . .· . �j·.·..··: ·.�.·��..

.

'

Fi gure 2 . 3. An eighteenth-century drawing of Newton's ho me at W oolsthorpe . (Courtesy of the Warden and Fellows of New College , O xford . )

teau beaded with villages , leading toward the town o f Grantham . Built o f the gray limestone which also builds the plateau , the house forms a s quat letter T , with the ki tchen in the stem , and the main hall and a parlor in the cross-stroke (Figure 2 . 3) . The entrance, somewhat off-center between the hall and the parlor , faces the s tair­ way which leads ups tairs to two bedrooms . Here Newton was born , and here was the room he occupied while he grew to adoles­ cence . Beyond the fact that he attended day s chools in the neigh­ boring villages of Skillington and Stoke , we know little about his youth . The area was liberally sprinkled with aunts , uncle s , and

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cou sins of varying degrees . Wills tell us of two uncles Newton , one living in Colsterworth and one in Counthorpe three miles away , both with children app arently not far removed fro m Isaac in age . Three married aunts , all with children , lived in neighbo ring Skil­ lington . There were also Dentons , Vincents , and Welbys who were more distantly related on the Newton side of the family . At least some connection with them was maintained; in the affidavit that acco mpanied and justi fied his pedi gree in 1 705 , he stated that his grand mother Ayscough " frequently conversed with [his] g reat Uncle , Richard Newton" at Woolsthorpe . 3 8 There were Ayscoughs as well . His g randmother had grown up in the area , and besides her daughter Hannah there was another married daughter Sarah , not to mention the Reverend William Ayscough two mile s away. Nev­ ertheles s Isaac ' s boyhood appears to have been lonely . He formed no bond with any of his nu merous relations that can be traced in hi s later life . The lonely boyhood w a s the first chapter i n a long career of i solation . In August 1 65 3 , the Reverend Barnabas Smith died , and New­ ton ' s mother retu rned to Woolsthorpe to live . Perhaps the period that followed was a joyful interlude for the boy of ten to whom a mother had been restored. Perhaps some bitterness tinged his joy as a half brother and two half si sters shared her attention , one an infant no t yet a year old and another, j ust two , possibly dominating it. The fact is we do not kno w . We know only that the interlude was short . In less than two years , Isaac was sent off to grammar school in Grantham . Despite i ts name , the Free Grammar S chool of King Edward VI of Grantham was an insti tution some three hundred years old when Newton enrolled in it. The upheavals of the Reformation had led to its reestablish1nent under a new charter as a Grammar S chool of King Edward V I in the mid-sixteenth century . Undoubtedly the reformation of learning associ ated with the Renai ssance introdu ced more p rofound changes , but about them as they applied specifically to the grammar school in Grantham we know almost no thing . The school had an honorable reputation. More than a century earlier, William Cecil had studied there , more recently the C ambridge Platonist, Henry More , whom Newton would know at the univer­ sity . The current master, Mr. Stokes, was considered to be a good schoolmaster, we are told by Newtonian lore , which begins now to accu mulate. Pre sumably thi s means that the Reverend William Ayscough investigated him and app roved . By Newton ' s own testi mony, he entered the school in Grantham 38

Foster , " N ewton 's Family , "

p.

60 .

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Never at rest

v1hen he was twelve . 39 The considerable number of anecdotes about this period of hi s life concentrate on his academic prog res s and on extracurricular recreations . By telling us nothing at all about the nature of his studies , they leave us to assume that he studied what every boy in grammar school at that time stu died as a matter of cou rse . Latin and more Latin , with a bit of Greek toward the end and no a ri thmetic or mathematics worth mentioning - such was the standard curriculum of the English grammar school of the day, and such we mu st assume Newton confronted in the school at Gran­ tham which was respected, and which Mr. S tokes , reputed to be a good schoolmaster, ran . 40 A p ro duct of Renai ssance classici sm , the 39

I am relying on Conduitt's memorandum of a conversation of 2 1 Aug. 1 726 (Keynes MS 1 30. 1 0 , p. 2) . There has been considerable disagreement among biographers over the chronology of Newton's period in Grantham . Since this memorandum appears to be the most authoritative source, and since it is consistent with other established dates , I accept its testimony . It is necessary to note, however, and all the more so since I shall be citing this memorandum frequently , that one cannot accept it without reserv ations . In 1 726 , Newton was a very old man talking about ev ents that had happened more than sixty years before. In another memorandum of a conversation a year and a half earlier, on 7 March 1 725 , Condu­ itt remarked that Newton's head was clearer and his memory stronger than he had known him fo r some time . To say the least, this casts some clo ud over the memorandum of 3 1 August 1 726 . Nevertheless , we have n o better evidence. According to it, Newton went to the grammar school in Grantham at the age of twelve - i . e . , sometime in 1 65 5 . Possibly he arrived j ust in time to meet one of the illustrious alumni of the gram mar school in which he was enrolling , Henry More, who had established himself as a pro minent philosopher among the school of Platonists in Cambridg e. More was in Grantham in the latter part of April that year . When he visited Grantham a year earlier, he stayed in the home of M r . Clark where Newton lodged ; he m a y have done so also i n 1 655 (see M o r e t o Anne Conway, 8 May 1 654 and 1 8 April 1 655 in Marjorie Hope Nicholson, ed . , Conway Letters. The Correspondence of Anne, Viscountess Conway, Henry More, and Their Friends [New Haven , 1 930] , pp. 98, 1 07-8) . Whether or not they met now , the two would have occasion to meet in the future. Newton w as in Grantham initially four and a half years ; thus he would have been there in 1 658, the day of Cromwell' s death and the great storm which figures in one anecdote. He returned to Woolsthorpe for three-quarters of a year , and then came b ack to Grantham for three-quarters of a year, after which he enrolled in Cambridge . Projecting backwards from h i s ad mission t o Trinity , he came back t o Grantham about the beginning of September 1 660; he returned to Woolsthorpe earlier, about the beginning of December 1 659; and he went originally to Grantham in the late sp ring of 1 65 5 . Perhaps all of thes e dates should be set back by some indeterminate but necessarily fairly short interval to allow for a pause in Woolsthorpe between gram mar school and the univers ity , although there is no mention of such . There is evidence that we must set the dates back a bit over a month at the least. On 28 O ct. 1 65 9 , Newton was fined by the manor court of Colster­ worth and must therefore have been called ho me fro m school before the beginning of December . The exact dates are of no great significance; we would not know anything more even if they could be established . 4° Foster Watson , The English Grammar Schools to 1 660: Th eir Curriculum and Practice (Cam­ bridge , 1 908) , passim . Villamil contains two catalogues of Newton's library which indi cate that he had a considerable collection of the classics , including most of the works that Watson indicates were basic to the curriculum of the grammar schools . Many of them were p urchased after his grammar-s chool day s , however, since they w ere post-1 66 1 editions . Moreover , some of the p re- 1 66 1 editions came into Newton's hands after he

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grammar s chool of the seventeenth century had as i ts p ri mary goal the development of proficiency in Latin, no t merely the ability to read it, but al so to wri te and to speak i t . Latin was the sole p ath that led on to higher learning . Still the lingua franca of European schol ar­ ship, Latin was the means of comn1unicating with the learned world beyond the borders of Englan d . A generation earlier and some hundred miles to the south , John Wallis ' s education, as he remembered it late in his life , left him above all "well grounded in the Technical part of Grammar . . . " W allis made it clear that he meant Latin grammar. Of arithmetic and mathematics Wallis got none at all ; he learned of their very existence from hi s brother, who was p reparing fo r a trade . Moreover, of the more than two hun­ dred s tudents with him in Emmanuel Colle ge , C ambridge , he did not know of two , and perh aps there were none at all , who had more m athematics than he . 4 1 John M ilton' s tract, OJ Education, written in the 1 640s , tol d a similar s tory. Strangely, for one of the leading classical scholars of England , he summarized the grammar school s as " those G ramma­ tick flats and shallows where [the s tudents] s tuck unreasonably to learn a few word s with lamentable cons truction . . . "4 2 To be sure , Milton d i d not want less Latin i n the school s; h e merely assumed that everyone could rush through i t at his p ace . The very silence on such a vital question by the collectors of anecdotes s trongly sug­ gests that Newton' s education differed in no way from the ordinary one and some of the earliest surviving fragments of Newtoniana confi rm this . In 1 659 he purchased a small pocketbook (or note­ book as we would say) , dating his signature on the first p age below a Latin couplet with "Martij 1 9 , 1 659 . " If one assumes this means 1 659/60 , it belongs to the period when he was at Woolsthorpe . He devoted most of the notebook to " Util issimum prosodiae supple­ mentu m . " 43 Further, in the Keynes Collection in King's College there i s an edition of Pindar with Newton's signature and the date 1 659, and the B abson Collection has hi s copy of Ovid's Meta morphoses d ated that year. 44

41 43 44

arrived in Cambridge. For examp le , his copy of Cicero's Ep isto lae ad familiares, a basic work in the g rammar-school curriculum, contains the name of a m an from Pembroke Hall , and his Op era of Macrobius has on its fly leaf the inscription "J ohannes Laughton, 1 66 7 . " Both o f these works are in the partial reassembly of Newton 's library in Trinity Co llege, C ambridge. I have not bothered to look through every book in the collection for evidence of when he obtained it. Wallis to Thomas Smith , 29 Jan . 1 697 ; Thomas Hearne , Works, 4 vols . (London, 1 8 1 0) , 3, 4 2 Of Education , in Works, 4, 279 . cxliv , cxlvii-cxlviii . Trinity Colleg e , Cambridge, MS R. 4 : 48c . A . N . L . Munby , "The Keynes Collection of the Works of Sir Isaac Newton at King 's Colleg e, Cambridge, " Notes and Records of the Royal Society, 10, ( 1 952) , 49; A Descriptive Catalogue of th e Grace K. Babson Collection of th e Works of Sir Isaac Newton, comp . Henry P . Macomber (New York , 1 950) , p . 1 88 .

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The reader in the twentieth century , su rrounded by the achieve­ ments of modern mathematics and the mate rial cultu re i t has gener­ ated, can s carcely believe that the man who would d isco ver the calculus four yea rs after he left grammar school was probably not even i ntroduced there to the already thriving mathematical culture out of whi ch the calculus would come . Neither i s there any sugges­ tion tha t he s tudied natural philosophy . Neve rtheles s , the grammar school in Grantham served Newton well . Without exception, the mathemati cal works on which he fed a few years hence were wri t­ ten in Latin, as were most of his sources in natural p hilosophy . Late r s till , he could co mmunicate with European science because he wrote Latin a s readily a s English . A little ari thmeti c , which he could have absorbed in a day ' s time anyway , would scarcely have compensated fo r a deficiency in Latin. One other i mportant fea ture of the grammar school in the seven­ teenth century was the Bible . Studied in the classical tongue s , i t both supported the basic cu rriculum a n d reinforced the Pro testant faith of England . In Newton ' s case , biblical s tudy may have j oined with the Reverend S mi th ' s library to launch hi s vo yage o ve r strange theological waters . In Grantham , Newton lodged with the apothecary Mr. Clark , whose house s tood o n the High Street next to t h e George Inn . Also living in the house were th ree stepchildren of Mr. Cla rk , named Storer from his wife' s fi rst husband, a girl whose first name has been lost, and two boys , Edward and A rthu r . 45 It seems clear that Newton did not get along with the boys . Among the incidents that he remembered uncomfortably in 1 662 were " S tealing cherry cobs from Eduard S torer" and " Denying that I did so . " He also recalled "Peevishnes s at Master Clarks for a piece of bread and butte r. "4 6 A s fa r a s we know, Newton had grown up i n relative i solation with his grandmo ther. He was different from other boys , and i t i s not surp ri sing if he was unable to get along with them easily . As they came to recognize his intellectual superiori ty , the boy s in the s chool app arently hated him . Years late r there wa s only one , C hrichloe , whom he remembered with pleasure . Stukeley gathere d that the 45

46

his list of sins of 1 662, Newton mentioned Edward and Arthur Storer . Years later , Arthur Storer communicated observations of a comet to him from Mary land , and Ed­ w ard Storer was his tenant at Wools thorpe. The tw o brothers were nep hew s o f Humph rey Babington , fellow of Trinity College and later rector of Boo thby Pagnell , between Colsterworth and Grantham and somewhat to the east. Stukeley 's anecdotes about Newton in Grantham include a " Miss Storey , " the stepdaughter of M r . Clark and the niece of Humphrey Babington . I assume that Stukeley ' s Miss Storey was really Miss Storer , and that Edward and Arthur were her brothers and hence also residents of the house. Westfall , " S hort-w riting , " pp . 1 3- 1 4 . In

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others found him too cunning , able to get the better of them with his greater quickness . Perhaps i t was one such incident, ha rdly cal­ culated to endear him to boys already hostile, that Newton re­ corded in 1 662 . "Putting a pin in John Keys hat on Thy day to pick . h Im . " 47 The stories that Stukeley collected in . Grantham in the 1 720s stressed the fact that Newton preferred the company of girls . For Miss S torer, who was several years his junior, and her friend s he made doll furni ture, delighting in hi s skill with tool s . Indeed, as the two grew ol der, something of a ro mance apparently developed between Newton and Miss Storer. It was the first and last romantic connection with a woman in hi s life . The romance of an adolescent boy w ho prefers the comp any of girls is not l ikely to endu re. This one did not . Though Newton remembered M rs . Vincent (her mar­ ried name) as one of his two friends i n Granth am , i t was only Mrs . Vincent who tol d of the romance . For the most part , he kep t hi s own co mpany . He was always "a sober, silent , thinking lad, " Mrs . Vincent recalled, [and] " never was known scarce to play with the boys abroad. " 4 8 Earl y in Newton ' s sta y in Grantham , a cri si s occurred whi ch burned deeply into his memory . He had not even had time to assert hi s intellectual p rowess . Whe ther because he was ill-p repared by the village school s , or because he was alone again and frightened, he had been p laced in the lowest fo rm , and even there he stood next to the bottom. On the way to school one morning , the boy next above him ki cked him in the belly, hard. It must have been Arthur Storer. 49 Boys will be boy s , but even among boys a vicious ki ck in the s tomach requi res some provocation . Al ready there may have been one too many peevish scenes over bread and butter and cherry cobs and all the re st one can imagine . Thou gh he played with the gi rl s , Newton knew what he had to do : as s oo n as the s chool was over he challenged the boy to fight , & they went out together into the Church yard , the schoolmaster' s son came to them whilst they were fighting & clapped one on the back & winked at the o ther to encourage them both. Tho s r Isaac was not so lusty as his ant agonist he had so much more spirit & resolution that he beat him till he declared he would fight no more, upon wc h the schoolmaster' s son bad him use him like a Coward , & rub his nose

47

4s 49

Ibid. , p . 1 3 . John Keys attended the school in Grantham. If Newton put the incident on the list in 1 662, it cannot have passed for a joke though it may have been an attempt at one . Stukeley, p p . 45-6 . Cf. pp. 23 , 46 . One o f the sins that Newton listed in 1 662 was " Beating Arthur Storer" (Westfall, " Short-writing , " p. 1 4) . It seems p robable that the fight he remembered s eventy years later was the same one he remembered seven years later - the only fight in the list.

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Never at rest

against the wall & accordingly sr Isaac pulled him along by the ears & thrust his fa ce against the side of the Church . s o

Not content with beating him physically , h e insisted on worsting him academically as well; once on his way, he rose to be first in the school . As he rose, he left his trail behind him , hi s name carved on every bench he occupied . The benches do not su rvive , but a s tone windowsill still bea rs one of hi s signatures . 5 1 B y the time Stukeley was collecting anecdotes , Newton' s ge­ niu s was taken for granted . What everyone in Grantham remem­ bered about him were " hi s s trange inventions and extraordinary inclination for mechanical works . " He filled his room in the gar­ ret of Clark ' s house with tools , spending all the money hi s mother gave him on them . While the other boys p layed thei r games , he made things fro m wood, not just doll furniture for the girls , but also and especially models . A windmill was built north of Grantham while he was there . Althou gh water wheels were common in the area , windmills were not , and the inhabitants of Grantham used to walk out to watch i ts construction for diver­ sion . Only the schoolboy Newton inspected i t so closely that h e could build a model of it, as good a piece of workmanship as the original and one which worked when h e set i t on the roof. He went the ori ginal one better. He equipped his model with a treadmill run by a m�u se which was urged on either by tugs on a string tied to its tail or by corn placed above it to the front . Newton called the mou se his miller. He made a little vehicle for himself, a four-wheeled cart run by a crank which h e tu rned as he sat in it. He made a lantern of " c rimpled paper" to light his way to school on dark winter mo rnings , which h e could simply fold up and put in his pocket for the day . The lantern had other possibilities; attached to the tail of a ki te at night, it " wonder­ fully affrighted all the neighboring inhabitants for some tim e , and caus' d not a little discou rse on market days , among the country p eople , when over thei r mugs of ale . " By good fortune , Gran­ tham was not burned to the ground . 52 5° Keynes MS 1 30 . 2 , pp. 1 7- 1 8 . I have quoted Conduitt's original version (co rresponding to Keynes MS 1 30 . 3 , sheet 6) ; he later crossed out much of the account of the fight, including the role of the schoolmaster's son. Manuel makes a good deal of this incident . Arthur Storer is the first surrogate Barnabas Smith, and his final humiliation is the image of the fate o f future antagonists . As it appears to me, he rather slurs over the role of the schoolmaster' s son. If I am correct in identifying this fight with the beating Newton listed in 1 662, his remorse does not seem to me to be compatible with Manuel's interp retation. 5 1 It is reproduced in ] . A . Holden , " Newton and his Homeland - the Haunts o f his Youth , " i n Isaac Newton , 1 642- 1 727, a Memorial Volume, ed . W . J . Greenstreet (Londo n , 1 927) , opp . p . 1 42 . 5 2 Stukeley, p p . 38-42; Keynes MS 1 30 . 2 , pp . 22- 3 , 28-9. Conduitt's account derives from Stukeley' s .

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Newton spent so mu ch ti me at building that he frequently ne­ glected his school work and fell behind , whereupon he tu rned to his book s and qui ckly leaped ahead once more . Stokes remonstrated gently , but nothing could make hi m giv e up his mech ani cal contri­ v ances . H e could not leave them alone ev en on the Sabbath , al­ thou gh it filled hi m with remorse . 53 We know now that N ew ton found many of these contriv ances in a book called Th e Mysteries of Nature and Art by John B ate. In another notebook from Grantham, with the informati on that he purchased it for 2 1 /2d in 1 659, N ew­ ton took down extensiv e notes from B ate , on d rawing , catching bi rds , making v ari ous-col ored inks, and the like . Although they do not appear in his notes , most of his devices remembered in Gran­ tha m , including a windmill , were des cribed in the book . 54 Perhaps Newton' s ad oles cent geniu s shrinks a li ttle in the light of B ate' s book . H i s genius is scarcely in doubt, however, and the fact is that he found a book whi ch fed his natural interests . The re i s a tou ch of whi msy in some of these stories, wholly unexpected because wholly ab sent from the res t of his life. At thi s distance, there appears as well a patheti c attempt t o ingrati ate him­ self with his schoolfellows by such means . H e made lanterns for 53 54

" Making a mousetrap on Thy day , " " Contriving of the chi mes on Thy day , " "Twisting a cord on Sunday morning" (Westfall , " Short-w riting , " pp . 1 3- 1 4) . The notebook is now in the Pierpont Morgan Library in New York. On the flyleaf is the ins cription "lsacus New ton hunc librum possidet. teste Eduardo Secker. pret . 2 d o b . 1 659 . " Professor Andrade identified the notes with Bate's work (E . N . d a C . And rade , "Newton's Early Notebook , " Natu re, 135 [ 1 935] , 360) . Its connection with Bate's book is discus sed more fully in G . L. Huxley , " Newton's Boyhood Interests , " Harvard Library Bu lletin, 1 3 ( 1 959) , 348-54 . As with the other notebook , now in Trinity College , not everything in this one dates from the Grantham-Woolsthorpe period . After entering notes from Bate at one end , he entered extensive alphabetical lists of words under various headings such as " A rtes , Trades , & Sciences , " " Birdes , " " Cloathes , " and so on from the other . These lists , based on Francis Gregory's Nomenclatura brevis anglo-latino, were the example of a prominent characteristic of Newton , his desire , perhaps even compulsion , to organize and categorize information. There were to be many analogous exercises in the future , ranging from his " Quaesti ones quaedam Philosophicae" based on his reading of the mechanical philosophers beginning in 1 664 , to his Index chemicus, begun in the early 1 680s , which organized the results of his extensive reading in alchem ical literature . Like the notes from Bate , these lists appear to date from the Grantham-Woolsthorpe period . Manuel makes extensive use of New ton's additions to the lists he found in Gregory. On the unused pages in the center of the notebook, New ton later entered fairly extensive astronomi cal tables , on such things as the rising and setting of the sun , its al titude during the year , and eclipses of the moon, and an ecclesiastical calendar which begins with 1 662 . There is a description of how to make a sundial . Three pages describe technical details of the Copernican system , and several pages are devoted to elementary plane and spherical trigonometry . He also outlined a phonetic system . All of these notes in the center of the notebook are in a later hand, some as late as 1 664 , I think . Dates of the Cambridge terms , which are found with the astronomical tables , and the calendar beginning with 1 662 tend also to place them in his undergradu ate period .

62

Never at rest

them also , and who can doub t that they participated in the artificial meteor? When they flew kites, Newton investigated thei r properties to determine their ideal proportions and the best points to attach the strings . Apparently his efforts were in vain; he only convinced them of his greater ingenuity and completed thei r alienation . As Conduitt s ays , even when he played with the boys , he was always exe rci sing his mind . Ordinary boys mus t have found him discon­ certing . He told the Earl of Pembroke that the firs t expe riment he ever made was on the day of Cromwell ' s death, when a g reat s torm swept over England . By j umping fi rs t with the wind and then against it, and comparing his leaps with those of a calm day , he measured " the vis of the s torm . " When the boys were puzzled by hi s saying that the s torm was a foo t s tronger than any he had known before , he showed them the marks that mea su red hi s leap s . 55 According to one version of the s tory , he craftily u sed the wind to win a jumping contest - again the superior cunning whi ch made him suspect . Similar s tories of mechanical models are told of Robert Hooke' s boyhood . I n both cases, manual skill served them well i n constru ct­ ing equipment for experiments . Far more important, h owever, is the testimony of such stories to the pervasive image of the machine in the s eventeenth-century mind. _ Already that image had reshaped the conception of nature . The pu rsuits of his boyhood p repared Newton to embrace the mechanical philosophy as s oon as he met it. There were also o ther recreations in Grantham . Among them were sundial s . Apparently dials had attracted his attention even earlier; there is one mounted in the Colsterworth church suppos­ edly cut by Newton when he was nine . 5 6 Sundials involved much more than skill with tool s . They pre sented an intellectual challenge. He filled poor Clark ' s house with dials - hi s own roo m , o ther rooms, the entry , wherever the sun came . He drove pegs into the walls to mark the hours , half-hours , and even quarter-hou rs , and tied s trings with running balls to them to measure the shadows on successive days . By keeping a sort of almanac , he learned to dis tin­ guish the periods o f the sun so that he could tell the equinoxes and solstice s and even the days of the month . In the end the family and the neighbors came to consult " Isaac ' s dials . " 57 Thus did the maj esty of the heavens and the uniformity of nature spread themselves unforgettably before him . According to Conduitt, he was still watching the sun at the end of his life . He observed the shadows in 55 Keynes MS 1 30.2 , pp. 2 1 - 2. In the little book in which he collected anecdotes , Conduitt referred this one to Pembroke (Keynes MS 1 30 . 6 , book 2) . 56 Foster, " Newton' s Family , " p . 2 1 . 5 7 Stukeley, p . 43 ; Keynes MS 1 30 . 2 , p . 24. Again Conduitt's account derived from Stukeley ' s .

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every room he frequented , and if asked , would look at the shadows instead of the clock to give the ti me . 5 8 Living in an apothecary shop , he also interested hi mself in the composition of medicines . It was his first introdu ction to chemis­ try , whi ch would o ccupy more of his ti me than the heavens ._5 9 H e b ecame p rofi cient in drawing as well , and once more Cla rk' s house bore the brunt of his enthusias m . A later o ccupant of the garret room testified that the walls were covered with charcoal drawing s of birds , beasts , men, ships , and plants . He also d rew port raits of Charles I , John D onne, and the s choolmas ter S tokes . A few ci rcles and triangles also appeared on the walls - more of a forecast of the Newton we know than all of the portraits and birds and ship s together. And on nearly ev ery board , testifying to hi s i dentity like the desks in the s chool , stood the name " I s aac New­ ton , " carv ed and therefore ind elible . 6 0 What with carving s and d rawings and sundials , pokings about in the shop , and peevish s cenes over bread , the apothecary Clark n1ay hav e looked forward to the departu re of his p reco cious guest . That came late in 1 659. Newton was tu rning seventeen . It was time that he face the realities of life and learn to manage his estate . With that end in view , his mother called him home to Woolsthorpe . From the beginni ng the attempt was a disaster. As the hero-wo rshipp ing Conduitt has it, his mind could not brook s u ch " low employ­ ments . " H is mother appointed a tru sty serv ant to teach him about the farm . Set to watch the sheep , he would build model water­ wheels in a brook , both overshot and undershot, with p roper dams and slui ces . The sheep meanwhile would s tray into the neighbors ' corn, and his mother would have t o p a y damages . T h e record s on the mano r court of C ols terworth show that on 28 O ctober 1 659 Newton was fined 3s 4d " for suffering his sheep to break ye stubbs on 23 ouf loes [loose? i . e. , u nenclosed] Fu rlongs , " as well as l s each on two other counts , " for suffering his swine to trespass in ye corn fields , " and " for suffering his fence belonging to his yards to be out of rep ai r. " 61 On market days , when he and the servaµt went to town to sell the p rodu ce of the farm and to p urchase supplies , Newton would bribe the serv ant to drop him off beyond the firs t corner; he would spend the d a y building gadgets or with a book until the servant pi cked hi m up on the way home . If perhap s he 59 Keynes MS 1 30 . 2 , p . 2 1 . 5 8 Keynes MS 1 30 . 2 , p . 24 . 60 Stukeley, p . 43; Keynes MS 1 30 . 2 , p . 20 . Conduitt's account derives from Stukeley' s . Newton also made marki ngs o n the walls and windows o f the Manor House at Wools­ thorpe , many of them intersecting circles (H . W. R obinson, " Note on Some Recently Discovered Geometrical Drawings in the Stonework of Wools thorpe Manor House , " Notes and Reco rds of the Royal So ciety, 5 [ 1 94 7], 35-6) . 61 Tumor Paper s , Lincolnshire Archives .

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Never at rest

went to town , he would run directly to hi s old room at Clark's whe re a s tock of book s awaited, and again the servant had to con­ duct the business . Going home to Woolsthorpe from Grantham , one had to mount Spittlegate hill immediately south 9f town . I t w a s cu sto mary t o dismount and lead one's horse u p the steep hill . On one occasion , Newton became so lost in thought that he forget to remount at the top and led the horse all the way home; on another occasion (or perhaps in ano ther version of the same story) , the ho rse slipped hi s bri dle and went home while Isaac walked on, bri dle in hand, unaware that the horse was gone . Apparently the servant sto mached all of thi s . When Newton even fo rgot hi s meals , however, he despaired of ever teaching him . 62 Meanwhile , two o ther men were viewing M rs . Smith' s efforts from a different perspective . Her brother, the Reverend William Ayscough , had taken the young man' s measure , and he u rged his sister to send him back to school to p rep are for the university. The schoolmaster, Mr. Stoke s , was · if anything more insisten t . He re­ monstrated with Newton ' s mother on what a loss i t was to bury such talent in rural pursuits , all the more so since the attempt was bound to fail . He even offered to remit the forty-shilling fee p ai d by boys not re sidents of Grantham, and he took Newton to board in his own ho me . App arently Clark had had enough . In the autumn of 1 660 , a s C harles I I was learning to accustom himself to the per­ qui sites of the throne , a more momentous event took p lace to the north . Isaac Newton returned to grammar school in Grantham , with the university in p ro spect beyond . 6 3 The evidence available indicates that the nine months at home were a nightmare . 6 4 The list of sins in 1 662 suggests constant ten­ sion: " Refusing to go to the clo se at my mothe rs command. " 62 63 64

Stukeley, p p . 48-50; Keynes MS 1 30 . 2 , pp . 29-3 1 . Conduitt' s account in this case appears to have some additional material beyond Stukeley ' s . Stukeley, p . 5 1 ; Keynes M S 1 30 . 2 , p . 32 . In Stukeley 's account, Stokes was the key figure; Conduitt introduced William Ays cough as well . M anuel does not deal w ith this episode explicitly . For myself, I find it hard to reconcile w ith Manuel' s insistence on Newton ' s fixation on his mother and her central importance in his creative life. Manuel suggests that the return to Woolstho rpe in 1 665 , at the tim e of the plague, p rovided the psychic stimulus for the g reat discoveries of the annus mirabilis . H e even speaks of Newton's being tied t o h i s mother's umbilical cord . Newton could have spent his life with his mother in Woo lsthorpe; in fact, such would have been his easiest course, and he had to p rovoke a domestic crisis in order to escape . L ater , he could undoubtedly have returned from Cambridge to a vicarage or rectory in the vicinity , since a career in the church was the goal and actual practice of the overwhelming m ajority of university undergraduates in the seventeenth century . Instead , the excitement of learning seems to have swept every other factor in his life before it. Let me make it clear that the excitement of learning (under different rubrics) plays a major role in Manuel' s analysis also . N evertheless , I find it hard , as I say, to reconcile the episode of 1 660 with his analysis .

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" Striking many. " "Peevishness with my mother. " " Wi th my sis­ ter. " " P unching m y si ster. " " Falling out with the servan ts . " " Calling Dero thy Rose a j ade. " 65 He mu st have been insufferable . In Grantham, he had begun to sample how delicious learning could be . His inescapably intellectual nature had set - hi m apart from the other boys , but h e had no more been able to deny his natu re to win thei r favor than a lion can give up hi s mane . Just as he had begun to commit himself to learning, however, he had been called back to the farm to spend his life herding sheep and shoveling dung . Every­ thing with in him rebelled against hi s fate , and fortune was on his side . B y the intervention of Stokes and William Ayscough he was to feast on lea rning after all . His excitement still permeates Condu­ i tt' s account, blu rred neither by sixty-five years nor by C ond1-litt' s attemp t at grandiloquence . His genius now began to mount upwards ap ace & to s hine o ut with more strength, & as he told me himself, he excelled particularly in making verses . . . In everything he undertook he discovered an ap­ p lication equal to the p regnancy of his parts & exceeded the most sanguine exp ect ations his master had conceived o f him . 66

When Newton w as ready finally to leave , S tokes set hi s favorite disciple before the school , and with tears in hi s eyes made a speech in his p rai se , urging the others to follow his example . According to Stukeley , from who m Conduitt got the story, there were tears in the eyes of the other boys as well . 67 We can imagine! The school boys at Grantham were not the only group to whom Newton was a stranger and an enigma . To the servants at Wools­ thorpe he was simply beyond comprehension . Su rly on the one hand, inattentive on the o ther, not able even to remember his meal s , he app eared both fooli sh an d lazy in their eyes . They " rej oic' d at parting with h im, declaring, he was fit for nothing but the ' Versity . " 6 8 65 66

Westfall, " Short-writing , " p p . 1 3- 1 4 . 67 Stukeley , Keynes M S 130 . 2, p . 32-3 .

p.

51 .

68

Stukeley,

p.

51.

3

The solitary scholar

EWTON set out for Cambridge early i n June. There was no N greater watershed in his life . Although he would return to

W oolsthorpe infrequently during the next eighteen years , with two extended visits during the p lague, spiri tually he now left it, and what a later commentator has called the i diocy of rural life , once and fo r all . Th ree short years would put him beyond any pos sibility of return, though three more years , perhaps somewhat longer, had to pass before a p erm anent stay in Cambridge was assured . His accounts show that he stopped at S ewstern , presumably to check on hi s property there; and after spending a second night at S ti lton a s he ski rted the Great Fens, he arrived at Cambri dge on the fourth of June and p resented him self at Trinity College the following day . 1 If the pro cedures set fo rth in the statutes were followed, the senior dean and the head lecturer of the college examined him to deter­ mine i f he was fit to hear lectures . He was admitted - although there i s no reco rd whatever o f anything but the verdict , one feels con­ strained to add " fo rthwith . " He purchased a lock for his desk , a quart b ottle and ink to fill it, a notebook, a pound of candles , and a chamber pot , and was ready for whatever Cambri dge might offer. 2 Fifty years later , the German traveler Zacharius von Uffenbach found the city of Cambridge ' ' no better than a village . . . one of the so rriest places in the world . "3 Uffenbach was not viewing Cambridge against the ba ckground of Colsterworth and Grantham , however . To the young man venturing forth from rural Lincoln­ shire , the town of five or six thou sand inhabitants must have seemed a metropoli s , especially when Sturbri dge Fair, the greatest fair in England , located where coastal vessels could penetrate no farther inland th rough the waterways of the Fens, descended on the neighbo ring common in the late summer. In addition to the town, there was of course the university with i ts imp o sing colleges drawn up in ranks along the Cam . Even the worldly Uffenbach exempted them from hi s j udgment of Cambridge. To young Isaac Newton , they must have appeared grand beyond compare . They appear so still to older and more experienced visitors i n the twentieth century. 1 Tri nity College, Cambridge , MS R . 4. 48c . 2 Ibid. J. E. B. Mayo r , Cambridge under Queen Anne (Camb ridge , 1 870) , pp. 1 23 , 1 98 . Uffenbach's account of Cambridge is one of three pieces that compose Mayor's volume. In his notes to Uffenbach, Mayor qu otes very si milar judgments of the town by John Evelyn in 1 654 and by Edward Ward about 1 700 (p . 4 1 0) .

3

66

The so litary schola r

67

Admission to a college w a s not tantamount to a dmis sion to the universi ty . Many delayed m atriculation in the universi ty; a consid­ erable number who had no interest in a deg ree, to whi ch alone matri culation was relevant , managed to avoid i t altogether . New­ ton did intend to take a degree . On 8 July , together with a number of stud ents recently admitted to Trinity and to other colleges , he duly swore that he would pre serve the p ri vileges of the university a s much as i n him lay , 4 that he would save harmless its state, honor, and dignity a s long a s h e lived , and that he would defend the sam e by h i s vote and counsel; and to testify to the same he paid his fee and saw his name entered in the university ' s matri culation book . He was now a full-fledged member of the universi t y . M o s t of thos e who took t h e matriculation oath added their own silent glos s to the duties they swore to so solemnly . They would p re serve and defend the pri vilege s and honor of the universi ty to the extent that they might thereby exploit the venerable institution to their o wn p rofit . In 1 66 1 , C ambridge was mo re than four hundred years old . Organized i ni tially by a migration from Oxfo rd on the occasion of a cri si s between the universi ty and the town , C ambri dge had lived i n the shadow of the s enior university until the great exp an­ sion of higher education under Elizabeth and James . Not only had C ambri dge multiplied four- o r fivefold in size at that time , until it su rpassed Oxford and reached a maximum of more than three thou­ sand s ouls in the early 1 620s , but C ambridge out stri pped Oxford intellectually a s well . The heart of English Puri tanism , i t was the point of ferment in English intellectual life in the early s eventeenth century . Expansion was not without i t s perils , to be sure . The p ri­ mary p ressure behind it was the p erception that university e ducation and a university degree were means toward preferment by the state, especi ally in i ts department of ecclesiastical affairs , the Anglican chu rch . As both the monarchy and influential p eers qui ckly realized , the two universi ties collected together one of the largest reservoirs of patronage in the realm . No church universal now stood between the university and secular powers a s a buffer to p rotect it from their pressure s . As Crown and p eers on the one hand sought to control the university and th e patronage i t embodied , so on the other lesser figures flocked to recei ve thei r larges s . The established curri culum , dating from the medi eval university, was increasingly i rrelevant to the new function of the university , but piecemeal change s had not added up to a coherent alternative . By the second half of the century , C ambridge University , like Oxford , drifted toward the status o f a degree mill exploited without conscience by those fo rtunate enough 4 Keynes MS 1 1 6 . This intere sting phrase later made its way into Definition I I I o f the Principia , where Newton defined what he meant by the vis insita or vis inettiae of matter.

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to gain access to i t . Such an atmosphere might impinge in various ways on a youth who came in quest, not of a pla ce , but of knowledge . In 1 660 , there were additional perils for an insti tution recognized by all as the home of English Puri tani sm . Even Cambri dge had not passed untouched through the upheavals of the great Civil War and the Interregnum . In co mparison with Oxford, however, the blows it sustained had been mild. With the Restoration , all the conditions reversed themselves . Puritanism had los t more than political power. It had also lost confi dence in i ts mi ssion , and Cambri dg e scrambled to effa ce the memory of the sins of yore . Town and university together were ostentatious in proclaiming Charles II, to the extent that the Pa rliamenta ry In telligencer singled their celebration out as " very remarkable both for the manner and continuance . " On 1 0 May , the heads o f the colleges were summoned with their scholars to the public schools , where university lectures and exercises were held - all in academic gowns , the vice-chancellor and doctors in scarlet , the regents , non-regents , and bachelors with thei r hoods tu rned , the scholars in caps . A pro cession with music advanced to the cro ss on Market Hill where the proclamation was read, and on to · the Rose Tavern where it was read again . A band pla yed a great while from the roof of King ' s College Chapel , and bonfi res lit the night . On 1 1 May , the mayo r accompanied by the recorder and aldermen , all in their scarlet gowns , and by the freemen of the town mounted on horseback p roclaimed C harles in the name of the town no les s than seven time s , and bonfi res burned again . O n 1 2 May, soldiers fi red a volley from the roof of King ' s College C hapel. An effigy of C romwell hung from a gibbet in the market. The round underg raduate cap favored by the Puritans disappeared in favor of the square cap , which they had considered Romish; James Duport, a royal i st , congratulated the Puritans on squaring the ci rcle . 5 More diffi cult matters remained to be settled in private . The Common­ wealth had sold the royal fee-farm rents , part of the C rown' s estate, to various corporation s , and C ambri dge had been among the buyers . Although other corporations su rrendered their purchases to the Cro wn , the university dallied with the notion of requesting that the king graciously permit it to keep the rents . A more p rudent counsel urged that the university court royal favor by returning the fee-farm rents at once without waiting to be commanded. The 5 The account of the pro clam ation o f Char les II is taken from James B ass Mulling er, The Un iversity of Cambridge, 3 vols . (Cambridge, 1 873- 1 91 1 ) , 3, 554-5 , and ] . E . Foster, ed. , The Diary of Samuel Newton , Alderman of Cambridge ( 1 662- 1 71 7) (Cambridg e , 1 890) , p . 1 . The anecdote about Duport is found in Mullinger, The University of Cambridge, 3, 555; since Duport had retained his fellowship and his perquisites throughout the C ivil War and the Interregnum, his sarcasm strikes a singularly sou r note .

The solitary scholar

69

more p ru dent counsel, which prevailed , belonged to the master of Corpus Christi College, Dr. Love , who found him self shortly thereafter elevated to the deanery of Ely . 6 His advancement was a practical homily, the lesson of which was lost on few . A t least in the immediate aftermath of the Restoration , C ambridge pro spered without apparent inj u ry . Nearly all the heads of colleges were re­ placed; a mastership was too j uicy a rnorsel of preferment to be left untouched . Fellows ejected during the Civil War and the Common­ wealth were restored, but the years had taken their toll so that the changes were few . With the Act of Uniformity in 1 662 , an addi­ tional small number departed . Meanwhile Cambridge expanded rapidly during the 1 660s , admitting an average of more than three hundred new students each y ear and approaching again its maxi­ mum size of the 1 620s. The changeover emphasized anew the posi­ tion of the university in the network of royal p atronage and prefer­ ment. When Samuel Pepys , who had matriculated in C ambridge in 1 650 , visited it in 1 660 , he found the " old preciseness" had almost ceased to exist . 7 During the century since 1 560 , while Cambridge University , from o n e point of view , flou rished a s i t had never flou ri sh ed be­ fore, from another point of view i t almost ceased to exist. To i mpose discipline on the uni versities , and to control the religious groups on either extreme, Puri tans on one side, Catholics on the other, governmental policy under the Tudors had forced all the students into colleges . Long before 1 600 , the medieval hostels h ad dis app eared . Inc reasingly the colleges u surped the edu cati o nal duties of the university, so that by 1 660 the university h ad little function beyond the conferment of degrees . Among the colleges at C ambridge, none was more i mportant than the College of the Holy and Undivided Trinity , founded by Henry V I II in 1 546 , which Newton entered in 1 66 1 . Along with King' s College Chapel , i ts great court , virtually the entire college when Newton arrived, fur­ nished one of the two imposing spectacles of the university, " the fairest sigh t in C ambridge, " in the words of one observer. 8 To­ gether with its neighbor of equal size, St. John ' s College , i t domi­ nated the university; the two consti tuted a good third of the enti re insti tution . Intellectually , it dominated the university by itself. Three of the five Regius Professorships were attached to i t . During " Mullinger, The University of Cambridge, 3, 5 6 1 -2 . The Diary of Samuel Pepys, ed . Robert Latham and William Matthew s , 8 vols . continuing (Berkeley and Los Angeles, 1 970- ) , 1, 67. Note that Pepys's observation p receded the Restoration, although it was in the air, and his account of his visit to Cambridge makes it clear that everyone there was already adjusting to it. 8 Cited in W . W. Rouse Ball , Notes on the History of Trinity College Cambridge (London , 1 899) ' p . 75 . 7

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the rei gns of Elizabeth and James , Trinity had furnished a greater nu mbe r of bishops to the Anglican church than any othe r college in Oxford or Cambridge . In 1 600 , both archbishops and seven bi shops were Trinity men . Six of the translators of the Authorized Version came from the college . Like the university as a whole , Trinity ex­ panded anew and seemed to flourish in the decade fallowing the Restoration - a communi ty of more than four hundred men, fel­ lows , scholars , students , clerk s , choristers , servants, and twenty almsmen supported by the college under the terms of its charter . There i s nothing surprising i n the fa c t that Newton chos e t o enter Trinity , " the famou sest C ollege in the University , " in the opinion of John S trype, the future ecclesiastical hi storian , who was an under­ graduate in Jesus College at the time . 9 A s i t happens , pers onal factors probably influenced Newton ' s choice more than the reputation of the college . The Reverend William Ayscough , his uncle , was a Trin­ i ty man , and according to the account that Conduitt later got from Mrs . H atton , nee Ayscough, the Reverend Mr. A yscough persuaded Newton' s mother to send him to Trinity . 1 0 S tukeley heard in Gran­ tham that Humphrey Babington , the brother of Mrs . Clark and fellow of Trinity , was responsible . The doctor, S tukeley wrote , "is said to have had a particular kindness for him , which probably was owing to hi s own ingenuity . " 1 1 There i s some evidence to suggest a connec tion be tween N e wton and B abington . " Mr B abingtons Woman , " one of the bedmakers and chambermaids allowed to work in the college , appeared twice in the accounts Newton kept as a student , and he later indicated that he spent some of hi s ti me when he was home during the plague at neighboring B oothby Pagnell, where Babington held the rectory . 12 As a fellow with considerable seniority such that in 1 667 he became one o f the eight senior fellows who , along with the master, controlled the college (and reaped its ripest reward s) , and furthermore as a man who had de monstrated hi s ac­ ces s to roy al favor with two letters mandate immediately after the Restoration , B abington would be a powerful ally for a y oung man o therwise without connections . Both the nature of the college and the nature of Newton' s s tudies made a powerful ally desi rable at the least, and perhaps indispensable . For whatever rea son , on 5 June 1 66 1 , the famousest college in the university , quite unawa re , admit­ ted i ts famousest student. 9 11 12

In a let ter to his mother in 1 662; p rinted in Charles Henry Cooper , Annals of Cambridge, 5 1 ° Keynes MS 1 25 . vols . ( C am bridge, 1 842- 1 908) , 3, 504. S tukeley to Conduitt, 1 July 1 727 ; Keynes MS 1 36 , p . 7 . Trinity College, M S R . 4 . 48c; Fitzwilliam notebook (Fitzwilliam Museum , Cambridge) . In the university setting, "Mr. " was the contraction for "Magister , " M aster of Arts . Babington was created Doctor of Divinity in 1 669; hence the title "Doctor" by which S tukeley referred to him .

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Newton entered Trinity a s a subsiza r, a poor s tudent who ea rned his keep by p erforming meni al tasks for the fellows , fellow com­ moners (very wealthy s tu dents who paid for privileges such as eating at high table) , and p ensioners (the merely affluent) . S iza r and subsiz a r were terms p eculiar to C ambri dge; the corre spond ing O xonian word, servitor, expressed their posi tion unambiguou sly . So did the s tatu te s of Trinity C ollege , which called them " schola re s paupere s , qui nominentur Sizatores" a n d introduced the defini tion o f their statu s b y reference t o the requirement lai d on C hris tians to support paupers . 1 3 The statute s allowed for thirteen sizars supported by the college , three to serve the master and ten for the ten fellows most s enio r; they also defined subsizars as s tudents admitted in the same manner and subj ect to the same rules as s izars , but p aying to hear lecture s (at a rate lower than p ensioners) and paying for their own foo d . That i s , subsizars app arently were to be servants like sizars but not supported by the college - servants of fellows , of fel­ low commone rs , and of pensioners , according to whatever arrange­ ments they might make . 1 4 Essentially i dentical in s tatu s , sizar and subsizar stood at the bottom of the Cambri dge soci al structure , which repeated the distinctions o f English society . Like tho se of servants everywhere , the duties of the sizars were meni al. They functioned a s valets to fellow s and to o ther s tudents , rousing them for morning chapel , cleaning their boots, dressing their hai r, and carrying their orders from the buttery . In the hall , they waited on table s . Half a century earlier, an O xford s tudent , in jus tifying his expenditu re on a poor s tudent to his skeptical father, chose to p ose a rhetori cal question: " should I have ca rrie d wood, and dust, and emptied chamber pots . . . ? " 15 A Trinity C onclusion Book illustrated their dutie s indirectly , a s i t attempted to grapple with a s omewhat more p ressing p roblem in 1 66 1 . " O rdered that ye Woemen of ye C olledge dismiss their young maid servants , with ye 13 14

15

The Statutes of Trinity College from the year 1 560 are found in Appendix B of the Fourth Rep ort from the Select Committee on Education (London, 1 8 1 8) . I cite from p . 375 . I am unable to determine that there w as ever any consistently maintained distinction between sizars and subsizars in Trinity . Although the statutes made provision for subsiz­ ars , the first student admitted explicitly as a subsizar w as Joseph Halsey in 1 645 (W. W. Rouse Ball and J . A . Venn, ed. , Admissions to Trin ity Coll�ge, 5 vols . [London, 191 1 - 1 6]) . From that time , subsizars outnumbered sizars in the admissions book. Nevertheless , it is quite impossible to reconcile the number admitted exp licitly as sizars w ith the thirteen supported by the college. Perhap s the distinction, once it began to be made , w as between servants of fellows and servants of w ealthy students , but the Trinity records of admis­ sions , unlike those of St . John's College (Admissions to the College of St. john the Evangelist in the Un iversity of Cambridge [Cambridge, 1 882]) , did not list sizars as the servants of specific students. Brian Twyne to his father, 1 597; cited in Mark H. Curtis , Oxford and Cambridge in Transition, 1 558- 1 642 (Oxford , 1 959) , p . 56 .

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soonest, sometimes thi s quarter . That they carry about no burning tu rfe , that they go not for commons to ye ki tchen or for bear & bread to ye Buttri e for Fellowes , Scholars , or o thers , but that these be done by Sizars . " The sizars who waited on table dined after the hall was cleared on whatever the fellows left . Though the statutes specified that subsizars were to pay for their commons , they also specified that they were not to enter the hall with the other students . An order of 1 699 i mplied that they were even expected to subsi st on less . 16 Tht y sat in thei r separate pla ce in the college chapel . In so me colleges , special gowns visibly set them apart; Queen ' s and Corpus C hristi had separate sections for them in thei r regi sters so that even thei r names might not mingle with the others . 1 7 The language in which they were discu ssed almost su mmons up the image of chattel slav­ ery . " Let the Master have three, " said the Trinity statutes o f the thi rteen siza rs . In 1 707 , the fellows of Emmanuel College disputed the s ervi ces o f the siz ars ' ' b elon ging to the e i ght s enio r fellows . . . " 18 Small wonder that the historian of Balliol College states that a servitor (the Oxford equivalent of the sizar) was "a social pariah with whom men of ordinary good sense and good feeling hardly cared to be seen walking and conversing in public . " 19 If aU thi s was true , why was N ewton a sizar? Only one possible answer p resents i tself. His mother, who had begrudged him fu rther education in the first place, and (by one account) had sent him back to grammar school only when the forty-shilling fee was remi tted , now begrudged him an allowance at the university that she could have afforded easily . Though her income probably exceeded £ 700 per annum , Newton' s accounts seem to indicate that he received at 16

17

18

19

1 4 O ct. 1 66 1 (Master's Old Conclusion Book , 1 607- 1 673 , p . 272) . Fourth Rep ort, p . 375 . Point 2 of the order of 1 699 specified that the weekly quantum of every s cholar and pensioner in the Lo wer Butteries was not to exceed 4s, and that of every sizar and subsizar not to exceed 2s 6d (Con clusion Book , 1 646- 1 8 1 1 , p. 202) . David Arthur Cressy, "Education and Literacy in London and East Anglia, 1 580- 1 700" (dissertation , Cambridge University , 1 972) , p p . 242-3 . In his account of his life as a student in Cambridge in 1 667- 8 , Roger North , who was a fello w comm oner , indi cated that he envied the common scholars for their games and their freedom to ram ble, but he was "tied up by quality from mixing with them . . . " (Roger North, The L ives of the R ight Hon . Francis North, Ba ron Gu ilford; the Hon . Sir Dudley North; and the Hon . and Rev . Dr. john North . Together with the Autobiography of the Author, ed . Augustus J essop p , 3 vols . [London, 1 890] , 3, 1 4) . Fourth Rep ort, p . 375 . The example from Emm anuel is cited in E . S . Shuckburgh , Emman­ uel College (London , 1 904) , p p . 1 2 1 -2. Thomas Baker, an early eighteenth-century anti­ quarian of Cambridge, indicated that the admission records of St. John's contained subsiz­ ars as early as 1 572 along with the fello w or master "to who m the sizar belongs . . . " (History of the College of St. john the Evangelist, Cambridge, ed . J . E . B . Mayo r, 2 vols . [Cambridge, 1 869], 1 , 55 1 ) . H . W . Carless Davis , A History of Balliol College, rev . R. H . C . Davis and Richard Hunt (Oxford , 1 963) , p. 1 35 .

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most £ 1 0 per annum . 20 There i s a further possibility not inconsis­ tent with the above . Newton may have gone to Trinity sp ecifically as Humphrey Babington' s sizar, perhaps to attend to the interests of Babington , who at that time was resident in Trinity only about four or five week s a year. The payments mentioned above to " Mr Babingtons Woman" would fit into such a hypothesis . In the eigh­ teenth century , the Ayscough family tradition recorded the story 20

See especially the accounts in the Fitzwilliam no tebook (Fitzwilliam Museum) for the years 1 665 , 1 666 , and 1 667 . They app ear to sum his receipts from his mother (about £ 1 0 per annum) and his total expenditures over about three years . To be sure , these were highly unusual years when he was home much o f the time. However, the years include two payments of £5 each to his tutor . Since they are the only such payments recorded (though one must add that we do not have all of Newton's accounts as a student) , they may have been for the B . A . (paid in M ay 1 665) and the M . A . (paid apparently in M arch 1 666) . Such a rate seems reasonable. His tutor took fifty-three pup ils during the four years 1 66 1 - 5 , when Newton was an undergraduate. At £5 per sizar, and therefore at least at £ 1 0 per pensioner, Pulleyn would have made out quite handsomely during that p eriod from undergraduates alone. Newton's total expenditures between May 1 665 , when he received £ 1 0 , and March 1 667 , when he received £ 1 0 again , were £5 / 1 s/2d p lus £ 10 to his tutor. The period coincides p retty clo sely with the plague years; at most, Newton w as in Cambridge for six months between May 1 665 and March 1 667 . His earlier accounts in the Trinity notebook (MS R . 4 . 48c) are consistent with this level of expenditure ( cf. esp ecially the sum m ations) . In this notebook Newton recorded his expenses for the trip to Cam­ bridge and for settling in at the college. After a num ber of other item s , including a payment to Agatha, a chambermaid who appears to have been paid at most on a quarterly basis , he summed his expenditures to £3 /5s/6d : "Habui 4. 0 0 Habeo 0. 1 4. 6 . " Unfortu­ nately there is no date, but the clear indication is that he set out from ho me with £ 4 and that he stret ched it to last for a considerable p eriod . Inside the front cover of the notebook are two lists o f expenditures that appear to be dated: " 1 2 . 7 . " and " 8 . 1 1 . " The second list, co vering the four months July- November if that is what the otherwise unintelligible numbers mean , totals 9s/ 1 0d . In considering Newton's status a s a sizar , one must keep his mother's financial status in perspective. In his Natural and Political Observations and Conclusions up on the State and Condition of England written in 1 696 , Gregory King put the average annual income of baronets, the second highest category in his breakdown of the English social hierarchy , at £ 880 , and he put the average annual income of knights, the third category , at £ 650 . He also estimated that there were 1 60 households of temporal lords , the highest category , 800 households of baronets , and 600 of knights. If King 's estimates were near the mark, and if the rep orts of the Reverend Smith's estate were accurate , Hannah Smith's household thirty-five years earlier must have been among the fifteen hund red wealthiest in all of Eng land (Gregory King , Two Tracts, ed . George E. Barnett [Baltimore, 1 936] , p. 3 1 ) . Even if we assume that both were badly in error , she was still a wealthy woman by the standard s of the age. A century after Newton's student day s , James Boswell gave an account of Peregrine Langton , who lived in a village in Lincolnshire on an annuity of £ 200 p er annum (plu s , perhap s , some small additional rents) . Langton was able to m ain­ fain a household consisting of his sister and his niece as well as himself with four serv ants . He had a postchaise and three horses. He entertained frequently and well. To be sure , the thrust of Boswell's account was meant to emphasize Langton's excellent m anagement . Nevertheles s , N ewton's mother apparently had at least three times his income, and p rices had certainly not declined du ring the intervening century . Despite her wealth she forced her son to be a sizar .

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that " the pecuniary aid of some nei ghbo ring gentlemen" enabled Newton to study at Trinity . 2 1 As the recto r of B oothby Pagnell , B abing ton might fit that description . A t a later time it does appear that B abington' s support (that i s , his influence not his money) may have been crucial to Newton . We cannot avoid a further question . What impact, if any , did hi s status as s iza r have on Newton? He w as , after all , heir to the lord­ ship o f a manor. If the manor i tself was not grand, his family' s econo mic status , thanks t o the fortune o f Barnabas Smith , ranked above that of most gentry . Newton was used to being served, not to serving . His own record , drawn up in 1 662 , indicates that he had used the servants at Woolsthorpe harshly , and they , fo r their p art, had rej oi ced to see him leave . It i s hard to imagine that he did not find menial status galling . 22 His status p robably reinforced his nat­ ural p ropensi ty to i sol ation . Al ready in Grantham Newton had found it i mpossible to get al ong with his fellow students . If he thought he was escaping them to study with a superior breed in C ambridge, he was mistaken . The same boys were there; the names we re all that differed. Only now he was their servant, ca rrying their bread and beer from the buttery and emptying their cha mber pots . The one surviving anecdo te concerned with hi s relations with other students suggests that the i solation and alienation of Grantham had travele d with Newton to Cambridge , intensified perhaps by his menial status . Well over half a century later, Nicholas Wickins , the son of Newton' s chamber-fellow , John Wickins , repeated what hi s father had tol d him about their meeting . My Father ' s Intimacy w i th Hi m c a m e by meer accident My Father' s first Chamber-fellow b eing very disagreeable to him , he retired one day into ye Walks , where he found Mr Newton s o litary and dej ected; Upon entering into discourse they found their cause o f Retirem 1 ye same , & thereupon agreed to shake o ff their p resent disorderly Companions & Chum together, wc h they did as s o on as conveniently they could , & so continued as l ong as my F ather staid at College . 23

Since Wiclcins entered Trinity in January 1 663 , the encounter above occurred at least eighteen months after Newton' s admi ssion . I am inclined to think that the walks of Trinity had frequently known a 21

22

23

I . H. [ James Hutton] , " New Anecdotes , " p . 24. Hutton, who did not appear to know that N ewton had been a sizar, was sufficiently offended by the story that he eliminated it from his mother's account and referred to it only in a footnote. P rofessor Manuel does not consider this issue in his idy Ilic account of Newton's relations w ith his mother. Is it possible that he did not hold her responsible, by her stinginess , for his position? Wickins to Robert Smith , 1 6 Jan. 1 728; Keynes MS 1 37. Newton's accounts show a p ayment o f 1 s "To a po rter when I removed to another chamber" (Trinity College, MS R .4 .48c) .

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solitary figure du ring those eighteen months , as they would for thi rty-five years mo re. With the exception of Wickins , Newton fo rmed no single friendship that played a p ercep tible role in his life from among his fellow students , though he would live on in Trin­ ity with some of them until 1 696 , and even h i s relation with Wick­ ins was ambiguous . 24 Correspondingly , when Newton became En­ gland ' s rnost famous philosopher, none of his fellow s tudents left any recorded mention that they had once known him . The sober, silent , thinking lad of Grantham had become the solitary and dej ected scholar of Cambri dge. Significantly, I am inclined to think , Wickins was a p ensioner. Trinity was less rigid in segregating sizars than some of the col­ leges . It did not pre scribe separate gowns fo r them , and the possi­ bility of a sizar " chumming" (that i s , sharing a chamber) with a p ensioner exi s ted . A t first blush , it mi ght appear that Newton was more ap t to find congenial co mpanions among the other sizars . By and large, they were the serious s tudents . Whereas only 30 percent of the gentlemen who entered C ambridge continued to the degree, roughly four out of five sizars commenced B . A . 2 5 On the whole, however, they were a plodding g roup , narrowly vocational in out­ look , lower-class youths grimly intent on ecclesiasti cal p referment as the m eans to advancement . 26 Since he had entered Trinity at 24 To one of them , Francis Aston, he did w rite a letter in 1 669 (Corres 1 , 9- 1 1 ) . 25 I have conflated two sets o f statistics from Cressy in this statement. One shows that sizars were the most reliable in matriculating . Between 1 635 and 1 700 , 82 percent o f sizars matriculated in the university , whereas 71 percent of pensioners did so and only 49 percent of fellow commoners . Two-thirds of matriculated students pro ceeded to the B. A . a s opposed t o about 5 0 percent of a l l students admitted . The second table, based o n a systematic sample of one in five undergraduates who entered Caius in the seventeenth century , correlates the social origin of students with the percentage who g raduated . Whereas only 30 percent of the sons of gentlemen graduated , 68 percent o f the sons of clergy and professional men did , 79 percent of the sons o f tradesmen, and 82 percent o f t h e sons of yeomen and husbandmen . Most sizars come from the l a s t t w o categories (Cress y , "Education and Li teracy , " pp. 223 , 238) . 26 Cf. brief summaries of the careers of a random samp ling of sizars in Trinity at New­ ton' s time . Without exception, they pursued careers in the church . It is also worth noting that only one out of ten got a scholarship from the college and that none at all got a fellowship . James Paston , admitted subsizar 30 Jan. 1 660. B . A . 1 665 , M . A . 1 668. Ordained deacon 1 665 . Held a rectory in Suffolk from 1 667 to 1 722 , to which he added a second in 1 68 1 . His son , James Pas ton, attended Caius as a pensioner . Richard Howard , admitted subsizar 5 M arch 1 660 . B . A . 1 664 . Ordained deacon 1 665 . Held a vicarag e in Kent fro m 1 672 to 1 682 . Thomas Perkins , admitted sizar 7 M ay 1 660 . B . A . 1 664 . Ordained deacon and priest 1 664 . Held various chu rch living s in Hertfordshire, Surrey , and Essex before his death in 1 686 . Nicholas Pollard , admitted sizar 1 2 June 1 660 . B . A . 1 664. M . A . 1 668 . Rector of Barton St. Mary , Norfolk , in 1 667 and subsequently held two other livings in succession. Robert Grace , ad mitted sizar 1 3 June 1 660. B . A . 1 664. M . A . 1 668 . Ordained deacon in 1 664. Vicar o f Shenstone, Staffordshire , in 1 665 .

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eighteen , Newton was at least one year older than the average and perhaps two , another factor which separated him from them . Ge­ nius of Newton' s o rder does not readily find companionship in any society in any age. He was perhaps even less apt to fi nd it among the siza rs of Restoration Cambridge� _ As in Grantham , he was unable to conceal hi s brilliance. "When he was young & fi rst at university , " his niece Catherine Conduitt told her husband , "he played at drafts & if any gave hi m first move sure to beat them . " 2 7 The society of pensioners would at least mitigate the stigma of menial statu s . Wiclcins i s not the only evidence that he tried to assimilate himself with the p ensioners . He made loans to a number of them , Henry Jermin , Barnham Oliver, and Francis Wilford , though t o be sure most of hi s extensive bu siness in usury was conducted among his fell ow sizars . 28 His list of offenses in 1 662 included " Using Wilfords towel to spare my own" and " Helping Pettit to make his water watch at 12 of the clock on S atu rday night . " 2 9 Like Wilford , Pettit was a p ensioner; though the two instances differ in tone, they both testify to Newton's being in the company of pensioners . Hi s accounts contain the heading " Super­ sedens " with several entries of 6d following it. The only meaning I can attach to i t i s a payment to sit above his station in the hall . His Job Grace, ad mitted sizar 1 3 June 1 660 . Scholarship in 1 664 . B . A . 1 664. M . A . 1 668 . Held two church livings arid w as treasurer o f the Cathedral of Lichfield . Valentine Booth, admitted sizar 1 Feb. 1 661 . B . A . 1 665 . P robably the vicar of Stan­ ford, Leicestershire, 1 668 . Robert Bond , admitted sizar 2 1 May 1 66 1 . B . A . 1 665 . M . A . 1 668 . O rdained p riest 1 667 . Rector at Layer B reton, Essex, from 1 677 to 1 688 , w hen he died . Hump hrey Pagett , admitted subsizar 1 9 M arch 1 662 . B . A . 1 666 . M. A . 1 669 . Rector o f Peckleton, Leicestershire, 1 67 1 . Buried there 1 708 . John Baldocke, admitted subsizar 1 1 Ap ril 1 662 . B . A . 1 666 . M. A . 1 669 . Vicar o f Littlebury , Essex, 1 669- 72 , fo llowed b y a succession o f other church livings, until his 27 Keynes MS 1 30. 6, Book 2 . death in 1 709 . 28 Newton kept close track o f his loans . An X through the line reco rding o n e signified that it had been p aid . The Trinity notebook (MS R . 4 . 48c) , fro m his early years at C ambridge, has a loan o f 1 0s to Guy w ithout an X . When Newton toted up his financial status in March 1 667, at least three years later, he carried over an unpaid loan o f 1 0s to Guy (now Mr Guy) . There were no subsequent loans to Guy . The reco rds give no indication that Newton took interest. The sheer extent o f his lending activity suggests that he did , however, as does also his self-condemnation in 1 662 for setting his heart too much on money ( Richard S . Westfall, "S ho rt-writing and the State of Newton's Cons cience, 1 662 , " Notes and Records of the Royal Society, 18 [ 1 963] , 1 3) . Perhaps his activity in lending served further to isolate him . His supply o f money told the other sizars in unmistakable terms that he was not one of them. In any case, the student usurer has never been a popular figure, no matter how essential he may be. J ames Duport, the p rominent Trinity tutor, would not have app roved of Newton's p ractice under any ci rcums tances. One of his rules of conduct was directly relevant: "62 Doe you neither lend , no r borrow any things of any S choller or other " (" Rules to be obs erved by . . . schollers in the University" ; Trinity College, MS 0. l OA . 33, p. 8) . 29 Westfall , " Sho rt-writing , " p p . 1 4- 1 5 .

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accounts regularly showed payments to bedmakers and chamber­ mai ds , whi ch suggests that he tried to i mi tate the style of the pen­ sioners . So also his expenditure for clothe s was out of keeping with the stories of hand-me-downs and rags in whi ch sizars supposedly dressed. As far as ou r evidence reveals , Newton found no mo re companionship among the pen sioners than among the sizars - wi th the exception of Wickins , of cou rse. The · others may well have regarded him a s a climber, a strange fi gure fo r other rea sons as well , tolerated only to the extent that he always had money to loan . Hannah S mith ' s niggardly allowance, not enou gh to make him a p ensioner but apparently more than the sizars had , gave the initial impulse to Newton' s i solation in Cambridge . It was p robably in­ evitable in any case . In the summer of 1 662 , Newton underwent some sort of reli­ giou s cri si s . At least he felt i mpelled to examine the state of his consci ence at Whitsunday , to draw up a list of his sins before that date, and to start a list of tho se commi tted thereafter. His earnest­ ness did not su rvi ve long enough to extend the second list very far. Lest it fall under the wrong eyes , he recorded hi s sins in cipher , u sing Shelton ' s system of shortwriting just a s Samuel Pep ys was u sing i t at the same time for a livelier and more revealing record . 3 0 Many of the incidents that Newton remembered with shame be­ longed to Grantham and to W ool s thorpe, but some of them be­ longed to C ambridge : "Having uncleane thoughts words and ac­ tions and dreamese . " He had not kept the Lo rd' s day a s he ought: "Making pies on Sunday night" ; " Squirting water on Thy day" ; " Swimming i n a kimnel [ a tub] o n Thy day " ; 3 1 " I dle di scourse on 3 0 Newton seems to have met S helton's system while he was in Grantha m . The notebook in the Morgan Library has "A remedy for a Ague" in sho rt-w riting, apparently an exerci se in using the system . It follows the notes on Bate's book, and its title and first few words, written in Engli sh, are in his g ra mmar-school hand. The first p age o f the notebook in Trinity College (MS R . 4 . 48c) , which also dates to hi s grammar-school days , has a sen­ tence about the blood relati onshi p of his grandmother Ayscough and Mr. Beaumont's father - again apparently practice since it contains nothi ng to be concealed , and again suggesti ve o f Lincolnshire days . His interest in short-writing probably led hi m on to hi s interest in a phoneti c alphabet (Cf. the Morgan notebook and Ralph W . V . Elliott , " Isaac Newton as Phoneti cian , " Modern Language Review, 49 [ 1 954] , 5- 1 2) . 3 1 Water was regarded with grave suspicion in sixteenth- and seventeenth-century Cam­ bri dge . During the Elizabethan age , Whitgift issued a decree when he was vi ce-chancellor that any student who went i nto a river or pool to wash or to swi m should, on hi s first offense , be whipped publicly twi ce , before hi s college and before the whole university , and , on second o ffense, be expelled (Ball , Trinity, p . 68) . James Duport's " Rules" of about Newton's ti me repeated the injuncti on without the penalty . "Goe not into the water at all or very wareily once or twice in a Summer at most , but better it were I thinke if you could quite fo rbeare. " Duport had other reservati ons a bout water as well: "I am no great friend to goi ng downe the water, because I have observed oftenti mes it hath occasi oned the goi ng downe of the wind too much, and some under colour of goi ng a fishing , drop into a blind house and there drink like fishes" (Trinity College, MS O . l OA . 33 , p . 1 5) .

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Thy day and at o ther times"; " Carles sly hearing and committing many sermons . " He had not loved the Lord his God with all his heart and with all his soul and with all his mind: " Setting my heart on money learning p leasu re mor_e than Thee" ; " No t tu rning nearer to Thee for my affections" ; " Not living according to my belief ' ; " Not loving Thee for Thy self '; " Not desiring Thy o rdinances" ; " Not fearing Thee so a s n o t t o offend Thee " ; " Fearing man above Thee" ; " Neglecting to p ray . " 3 2 Relying upon this confe ssion and upon his interpretation of the lists of words in the Morgan note­ book , Professor Manuel concludes that Newton was borne down "by a sense of guilt and by doubt and self-denigration . The scrupu­ losity , punitiveness , austerity , discipline, and indu striousness of a morality that may be called puri tanical for lack of a better word were early s tamped upon his character. He had a built-in cen sor and lived ever under the Taskmaster' s eye . " 33 Newton' s undergraduate expenditure s appear to bear out Manuel' s j u dgment. If he treated himself now and then to cherries , " marmolet, " cus tards , and even a little wine on occasion, he felt obliged to enter them under Otiosi et frustra exp ensa as opp o s ed to Imp ensa prop ria , whi c h i n cluded clothes , book s , and academic supplie s . He even considered beer and ale a s otiosi, thou gh we might judge them prop ria as we reflect on the water available . 34 The Puritanical s tyle of Newton ' s life would have set him apart from the ordinary p ensioners even if his s tatus of sizar had not. His conduct largely repeated the rules and ideals that had domin ated the university du ring its heyday as the Puritan institution . The " Rules to be observed by . . . schollers in the University" composed by James Duport, a p rominent tutor in Trinity s till active when New­ ton a rrived, sound a tone remarkably like Newton' s confession . " Be diligent & cons tant at Chappell every morning . " " Ri se earlier on the Lords day . . . & be more carefull to trimme your soules then bodye s . " " If you be with company on the Lords day , let your discourse be of the sermon or of some other point of Religion . ' ' " Think every day to be your last, and spend it accordingly . " 35 Such was the public s tance of the university . Chapel twice a day , a t seven in the morning and at five in the evening , continued to be p re­ scribed for all under forty . Even when Puritanism had been in its p rime , such rules and i deals had been hard to impose on hig h-­ spirited youths . With the Restoration it became quite i mpossible . Exhortations by Duport and others were one thing; the orders an_d admonitions by university and college officials make it abundantly clear that reality contrasted brutally with the ideal . In 1 663 , Richard 32 33 34

Westfall , " S hort-writing, " pp. 1 3- 1 4. Frank E . M anuel , The Religion of Isaac Newton (Oxford , 1 974) , .PP · 1 5- 1 6. 35 Trinity College , M S O. l OA . 33 , pp . 1 - 3 . Trinity College, MS R . 4 . 48c .

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Smi th was su spended from the college " in regard of hi s very great & heinous mi s carri ages , & misdemeanors (wherei n he hath long continued disordering hi mself by intemp erate drinking & lying out of ye College . . . ) " ; in January 1 665 , Young the scholar was ex­ pelled " for his foule & scandalou s offence" ; in 1 66 7 , the master and senior fellows ag reed that the wi dow P owell "be not any furth er permi tted to attend at any Chamber of any Fellow or scholar or any o ther within thi s College" ; in 1 676 , Harry L up pincott confessed his "heinou s crimes" and craved the mercy of God that "in a booth at Sturbridge fai r [h e] did sweare wi cked oath s , and sing obscene songs . . . "3 6 While the college p uni shed specifi c offenses , the university dealt with general problem s . Coffeeh ouses had begun to appear . In 1 664 , the vi ce-chancellor and heads ordered that all in p up illari sta tu who frequented coffeeh ou ses without their tuto rs should be p uni shed " according to the statute for haunters of taverns and aleh ou ses . " 3 7 Since even Newton recorded several vi sits to a tave rn , though only after he had attained the statu s of bachelor, we can imagine how well both orders were enfarced . There were other dangers still more attractive , hou ses in B arnwell and even in Cambri dge "infa­ mous for harbouring lewd Women , drawing loose schollars to re­ sort thither . " Scholars were forbidden to enter such hou se s , whi ch were i dentified very expli citly , thereby adverti sing thei r location to those who had lacked the enterp ri se to di scover them on their own . 38 On 1 2 July 1 675 , the chancellor of the university , the Duke of Monmouth , not otherwise well known for his encou ragement of moral up li ft , inquired of the vi ce-chancellor and heads about the enforcement of the vari ou s statutes of the uni versi ty . They rep lied in so rrow that desp ite thei r every endeavor the statute forbi dding students to frequent taverns was " too frequ ently transgres sed , " that chapel servi ces were duly celebrated b ut attendance at them was poor , and that coffeehou ses were frequented by large numbers of all sorts - heads of hou ses ex cep ted , of cou rse . 39 Cambridge, that i s , was the scene o f those activi ties i n whi ch young men have engaged since th e world beg an . When we compare the reality of life in C ambri dge with Newton ' s examination of hi s conscience , we are perhaps not wholly surpri sed that Wi ckins found its a uthor in the Trinity walks solitary and dejected . Th ou gh a semblance of the old i deal re mained , i ncreasingly 3 6 Conclusion Book, 1 646- 1 8 1 1 , pp. 78 , 94 , 1 07� Master's Ol d Conclu sion Book , 1 6071 673, new pagination at rear of book, p . 6 . By and large, enforcement of discipline fell upon the sizars , who copied the style of the pensioners but did not have the connections 37 Cooper , Annals, 3, 5 1 5 . to escape the consequences . 3 8 Cambridge University Library , MS Mm 1 . 53, f. 98 . Printed in Cooper, Annals, 3, 57 1 . 39 Ibid. , 3, 568-9.

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emp t y i n the context o f the Resto ration , the uni versi ty itself adapted to the new order and encou raged activities fo rbidden a generati on earlier . The stage returned to Cambridge . Like other colleges, Trinity appropriated funds in 1 662 " toward ye charges of a S tage , & properties & a supper & for encou ragement of ye Actors of a C omedy out of ye Commencemt money s . "4 0 The foll owing March, they found occasion to use it again when the Duke of Monmouth , Charle s ' s illegitimate s on , then all of fou rteen years , recei ved a Master of Arts degree from the university , along with thirty-fou r cou rtiers accompanying hi m , and was entertained by a banqu et and comedy in the great hall at Trini ty . 4 1 The city fu r­ ni shed its share of excitement of another order. In a period of five days in 1 664 , a man who stood mute and refused to answer to a charge of robbery was sentenced to be p ressed to death and took an hour to die while he confessed wildly to the robbery and to all the other crimes he could recall , one Nelson was hanged for cutting his wife ' s throat , and a local attorney stood in the pillory on Peas Hill , all less than a quarter of a mile from Trinity . 4 2 N o record su rvives of Newton 's attendance at plays in C ambridge . It i s hard to im agine him mis sing the spectacle p rovided by the Duke of Monmouth ' s ceremonial visit . I t i s equ ally hard t o imagine him among the crowds who witne ssed the executions and jeered the unlucky man in the pillory , though his fellow students undoubtedly made up a large p roportion of them . Meanwhile, along with the tavern s and bawdy hou ses and comedies and executions there were also studies . By 1 66 1 the official cu rricu­ lum of Camb ridge, p rescribed by statute nearly a century before , w a s in a n advanced state o f decomposition . The system o f tutors within the colleges , whi ch had largely rep laced university lectures , had fallowed its own peculiar develop ment . Only a small minority of the fellows now en gaged actively in tutoring , largely to augment their income. The others , who accepted thei r stipends and divi­ dends in retu rn for nothing at all , referred derisi vely to those who tutored as "pupil mongers . " Whe reas sto ries of affectionate and enduring relations between pupils and tuto rs , who were expected literally to stand in loco parentis to the boys in thei r charge, were common du ring the fi rst half of the century , they became rare in the second half. 43 The re cords of Trinity C ollege make it clear that 4° Conclusion Book , 1 646- 1 8 1 1 , p . 6 1 . 41 Cooper, Annals, 3, 509 . 42 Foste r , ed. , Samuel Newton 's Diary, p . 1 0 . 4 3 C f. M atthew Robi nson' s relations with hi s tutor , Zachary Cawdre y , i n the 1 640s; " Auto­ bi ography of Matthew Robi nson" in ] . E . B. Mayo r , Cambridge in the Seventeenth Century, 3 vols . (Cambridge , 1 855-7 1 ) , 2, 1 6, 22 , 67. Ni cholas Ferrar's tut or exercised an immense influence on his pupil. Benj amin Whi chcote carefully supervised his pupils ' reading in order to raise up nobler thoughts in the m . Henry More read a chapter of the Bible to hi s

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the tutors , who looked after their own interests as well , functioned prinurily a s financial agents for the college; they dispersed pay­ ments to exhibiti oners and scholars , and they oversaw p ayments due to the college . 44 Increasingly , as the entire s tructure of required studies and exercises los t i ts claim to legiti m acy , tutors allowed students to go their own way . 45 Newton's tutor � Benj amin Pul­ leyn , was the champion pupil n1onger of Trinity during the period Newton was an undergraduate . In the five years , 1 660 to 1 664 , which embraced Newton ' s undergraduate career, he took on a total of fifty-s even pupils . Pulleyn , who was then a young fellow , re­ mained on at the college twenty-five more years . No s ingle piece of evidence suggests even friendship , much less intimacy , between the two . The surviving record of Newton� s reading indicates that Pul­ leyn guided him initially down the accu stomed p ath . When New­ ton found a new road for himself, he was able to follow it without perceptible restraint . The effect of the successive upheavals the university had been through in the two previous decades , first the Puri tan revolution and then the Restoration , had been limited to changes of personnel almost exclusively . The curriculum of the university had remained untouched o Influences dating back to the sixteenth century had been altering a spects of the medieval course of s tudy , howeve r . Under the guise of rhetori c , extensive reading i n the litterae hu maniores, which continued the thrust of grammar-school education including the mastery of Latin and Greek , had been inserted in the programs recom mended by tutors though the legally p rescribed curriculum remained untouched . 46 Even with these innovations � s tudy at C ampupils in his chamber every night and lectured them on piety . Joseph M ede heard a report from each of his pupils in his chamber every night and commended them to God by p rayer befo re he dismissed them . Alexander Akehurst watched carefu lly over his pupils' lives and friends and prayed with them every night in his chamber (James Bass Mullinger, Cambridge Characteristics in the Seventeenth Century [London, 1 867] , pp. 49-50, 89, 1 8 1 ) . 44 See various entries in Bailie's & Chamberlayne's Day Book, 1 664- 1 673, where tutors sign for the " wages " of exhibitioners , scholars j choristers , library keepers, chapel clerks and the like under their tutelage (e. g . , p p . 1 8, 34, 35, 80) . On 25 April 1 664 , the steward was ordered to take special care to call on tutors for the dues that their pupils owed the college (Conclusion Book, 1 646- 1 8 1 1 , p . 83) . 45 In the eariy 1 660s, John N orth's tutor gave him no direction, and he "soon fell to shift for himself, as a bird that had learned to pick alone . . . " (North, Lives, 2, 283) . A few years later, the same John North, acting as tutor to his younger brother, did not want to be bothered , so that Roger North read whatever he w anted to (ibid. , 3, 1 5) . The second case, with two b rothers involved , was hardly a normal tutor-pupil relation , of course. 4 6 Richard Holdsworth, "Directions fo r a Student in the University , " Emmanuel Co llege, MS 1 . 2 . 27 ; Du port, " Rules . " As Holds worth's "Directions " m ake clear, some of the classics could pose p roblems in a Puritan setting . " In Martial take those Epigrams wc h are marked fo r good ones by Farnaby , passing over the obscene & s currilous . " Newton' s reading no tes show that h e studied Gerard Vossius ' s Rhetorices contractae sive , pattitionum

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bridge had not broken the mold in which it had been cast four centu ries before , with its focus centered on Aristotle . It began with a heavy dose of logic (Aris�otelian logic) which together with ethics (Aristotelian ethics) and rhetoric provided the founda­ ti on for the study of Aristotelian philosophy , and it reached its culmination in the fo rmal dispu tations , conducted wi th A ristote­ lian s yllogism s , which were the standard academic exercises and examination s . Bo th H o lds worth' s " Directions" and D upo rt' s " Rules , " the most important su rviving records of tutorial practice in the seventeenth century , take the traditional curriculu m as an unquestioned assumption . Student notebooks , which su rvive in considerable numbers , confirm their evidence . 47 "The reading of Aristo tle will not only condu ce much to you r study of contro­ versy, being read with a Com mentator, but allso help you in Greeke , & indeed crown all your other learning , " Holdsworth admonished his students , "for he can hardly deserve the name of a S cholar , t hat is n o t in some mesu re ac qu ainted with hi s works . " " If at any time in your disputation you use t he Author­ ity of Aristotle , " Duport exhorted in a similar vein , "be sure you bring his owne word s , & in his owne language . In your answer­ ing reject not lightly the authority of Aristotle , if hi s owne words will permit of a favorable , and a sure interpretation . "4 8 Certainly Duport' s and probably Holdsworth' s advice were writ­ ten after t he publication of Descartes ' s philosophy had inaugu­ rated a new chapter in European thou ght . Duport inclu ded ex­ plicit instructions on di sputations; H oldsworth assumed that the student would spend much of his last two years in disputations , preparing for the pu blic Acts that would climax and conclude his cou rse of study , and he also j ustified the stu dy of rhetoric , under which reading in the classics was gathered , by its contribution to effective disputation . 49 If this account suggests that the unive rsi­ ties we re conservative , perhaps reactionary , Newton's first expe­ rience at C ambridge confirmed as much . Before he left ho me, his uncle William Ayscou gh, calling upon his own experience at Trinity thi rty years before, gave him a copy of Sanderson' s logic and told him it was the first book his tutor would read to him . W hen Newton got to Trinity , his uncle ' s prediction was proved

47 48 49

oratoriarum libri V, one of the standard university texts . There are no undergradu ate notes from the litterae humaniores , but he did purchase at least two history books , Hall ' s Ch roni­ cles and Sleidan 's Fo ur Monarch ies (as he called t hem) , perhaps t he beginning of his long study of chronolo gy . William T. Costello , The Sch olastic Curricu lum at Early Seventeenth-Cen tury Cambridge (Cambridge , Mass . , 1 958) , passim . Holdsworth , "D irections" ; Duport , "Rules , " p . 1 1 . Duport , "Rules , " pp. 8-1 1 ; Holdsworth, "Directions . "

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perfectly correct . s o D uring those thi rty years , the intellectual life of Europe had been turned inside out . As far as C amb ridge was con­ ce rned, nothing had happened . The story ends with a typically Newtonian flourish. He read the logic before he got to Cambridge and found that he knew more abou t it than the tutor- which m ay help to explain why no affection developed between the m . 5 1 One of Newton' s first purchases in C ambri dge was a no tebook , and probably it was in thi s one that he entered the fruits of his reading in the established cu rriculum . 5 2 As was his custom , he entered notes from b oth ends , starting at one end wi th Aristotle' s logi c and at the other end wi th Aristotle ' s Nichomachean ethics . In s o The story appears n o t t o b e entirely accurate since Newton ' s copy o f Sanderson h a s an ins crip tion indicating that he bought it in Cam bridge in 1 661 . The purchase at that tim e does tend to confirm the essence of the anecd ote . There is a nice personal touch connected with it . Robert Sanderson , the author of the logic, had been the rector of Boothby Pagnell , not far from Co lsterworth , when Newton was a boy . Hum phrey B abington, the bro ther of Mrs . Clark and possibly Newton's patron in Trinity , h ad succeeded him when he was elev ated to the bishopric of Lincoln with the Restoration. Newton spent part of his time at Boothby Pagnell during the plague years . It was there that he calculated an area under a hyperbola to fifty-two p laces , crowning his discovery of the binomial expansion . Seven other books with the inscription " Isaac Newton . Trin: Coll : C ant : 1 661 ' ' sur­ vive, four of them theologi cal works . I do not know of any notes that Newton took from them at this time Uohn Harrison , The Library of Isaac Newton [Cambridge , 1 978]) . 51 Conduitt 's memoran dum of a conversation wi th Newton on 3 1 Aug. 1 726; Keynes MS 1 30 . 1 0 , f. 2 . The story did not end with San derson' s logic. When Newton's tutor found him so forward, he told him that he was going to read Kepler 's optics to some gentlemen com m oners and that he should come to those lectures . Newton immediately bought the book , and when his tutor sent for him for the lectures he told him that he h ad read the book alread y . Frankly , this part of the story is unbelievable . What we kno w about the state of mathematics and natural philosophy in Cambridge makes it impos sible to con­ ceive that ordinary students there , not to mention ordinary tutors, coul d handle Ke pler's optics , whether this refers to the Paralipomena or to the Diop trice . Gentlemen comm oners were the least likely to be able to comprehend it, and the least likely to want to. As for Newton , he never mentioned Kepler ' s opti cs elsewhere. 52 Holdsworth's "Directions " had recommended that students not waste their time with the huge clumsy commonp lace books of an earlier age but that they get some " paperb ooks " of portable size; although he was not exp li cit, he seemed to suggest a separate one for each subject . Newton inheri ted the huge , predominantly blank comm on place boo k that his step father had once started . He renamed it his "Waste Book" (in reflection on his s tepfather? ) , and he used its nearly unlimited supply of paper for important es s ays in mechani cs and m athem atics . He also brought along the two tiny notebooks from his gram m ar-s chool days (the Morgan Library and the Trinity College no tebooks) . Mean­ while, he had several separate octavo notebooks for different subjects . Add MS 3996 was his bas i c notebook in whi ch he originally set down notes from his reading in the es tab­ lished curriculum and later notes from his reading in the mechanical philosophy . I call it his philos ophy notebook . Add MS 3975 started as an extension of the "Q uaestiones" in 3996 but gradually converted itself into a chemistry noteb ook . Add MS 4000 was his mathematics noteb ook , begun probably in 1 664 . The notebook in the Fitzwilliam Mu­ seum never took on a distinct character but it received m ore mathem ati cs than an ything else . He also had a large quarto notebook that he devoted to theology ; in the 1 670s he collected a great deal of material in it, but he used it very little during his s tudent days .

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both cases the notes were taken in Greek , mere transcriptions of sentences , u su ally the first sentences of chapters and thei r subdivi­ sions; they give the ap pearance of being as mu ch exercises in Greek as reading notes . 53 In neither case did he proceed all the way through the work in qu estion . Somewhat later, if we may j udge by the hand , he followed the notes from Ari stotle's ethics with others from the ethics of Eu stachius of St. Paul , a popular seventeenth­ century textbook which suitably Christianized Aristotle . Fairly early in his underg raduate career, he was introduced to D aniel Stahl ' s Regulae phi losop hicae, a recently publi shed compendium of Ari stotelian phi losophy laid out in the fo rm of di sputations with obj ections and replies handily provided . Stahl must have been a popular textbook since it quickly went th rou gh seven editions . Newton also failed to fini sh this wo rk , thou gh he proceeded farther than he did with any of the others . For rheto ric, he read Gerard Vo ssiu s ' s Rh etorices contractae, p roceeding only part way through the second of five book s . 54 He was introduced as well to Ari stotelian physics via the seven­ teenth-century peripateti c , Johannes Magiru s (Physiologia e peripateti­ ca e) . Here he fo�nd an exposition of the A ri stotelian cos mology , on which he took rather full notes . Something of the future Newton revealed itself when he departed entirely from M agiru s ' s order of present ation to collect together information on the periods of the celestial sphere s . ' � Sphaera nona & octava 4 900 anni s: O rbi s Q , 30: 1(. , 1 2: o , 2 annis . � , 348 � , 33 9 . J) , 27 diebu s & 8 fere horis circum ferter . ' ' From Magirus he also learned of v arious conceptions of light , and he copied down the argument against the corpo reality of light , namely that the sun would be exhau sted . Ag ain Newton stopped before he finished Magirus; later he retu rned to him , read two more chapters , then stopped fo r good and drew a line under hi s notes . He had been reading about phenomena such a s the rain­ bow , which were classified as " Apparent Meteors" in the Arist ote­ li an system . Under the line he entered a note which is symbolic of hi s di scovery of a di fferent school of natural philosophy : " Gali laeus sayth yt ye apparent diameter of starrs of t fi rst magnitude i s 5' ' = 300" ' = 0 , 083333d egr: &c . & of ye sixt m agnitude to bee 50" ' = O' , 83333 &c = octegr: , 01 38888 & c . Mounsieur Auzout esteems ye great dogs diameter to bee no more y11 2' ' & those of ye 6t m agni­ tude to bee 20' ' ' . ' ' 55 He never returned to Magiru s . 53

54 55

Add MS 3996 , ff. 3- 1 0 , 34-6 . Cambridge Universi ty Library has started foliati on from the end with the notes on logic; notes , not all on logic, extend through f. 30. Foliati on then s hi fts to the other end so that f. 34 starts the notes on ethi cs. Add MS 3 996, ff. 38-40 , 43-7 1 , 77-8 1 . Again the hand indicates that the notes on Vossius were made at a later ti me , after those on Stahl and Magirus . Add MS 3996 , ff. 1 6-26, 20, 20v , 26-6v , 26v .

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It may be signifi cant that N ewton did not finish any of the books from the established curri culum that he started . Nevertheless we should be mistaken to underestimate the importance of Aristotelian philosophy in his life . He came to C ambridge in 1 66 1 a p rovincial young man well grounded in Latin but enti rely innocent of syste­ matic natural philosophy . One must not confuse hi s interest in things mechanical with natural philosophy; while it may conceiva­ bly have predisposed him to favor the mechanical philosophy of nature once he met it, it had nothing whatever to do with natural philosophy as an attempt to account rigorously fo r the phenomena of nature . Although the Aristotelian philo sophy was passing out of favor, it was anything but nonsense . It was the fi rst sop histicated system N ewton met , and there is no reason to think that it failed to impress him initially as he emerged fro m hi s intellectual provincial­ ity , even though it was no longer presented with conviction . From it he learned the canons of rigorous thought , and it provided him with a system that organized the overwhelming diversity of nature into a coherent p attern . A s it is impossible to imagine the scientific revoluti on wi thout the background of medieval philosophy , so it is impos sible to imagine Newton ' s achievement without hi s prior ex­ posure to Aristoteliani sm . Nevertheless , the scientifi c revolution was under way. As the note on Galileo informs us , Newton dis­ covered it before he completed his undergraduate career. Because the university started his career as a natural philo sopher wi thin the A ristotelian s ystem , he had to recapitulate the prior history of the scientifi c revolution and have hi s own private rebellion against the orthodoxy established around him . And if he was neve r able to isolate limited problems from the total context of nature , if his concern with nature as an organized system never left him , part of the reason was hi s recollection of a completely different system to whi ch initially he had owed allegiance . It never ceased to be neces­ sary fo r him to j u stify the new system he had embraced . Although N ewton ' s note s make it clear that hi s initial studies at the unive rsity were di rected toward the traditional curriculum , a variety of evidence from elsewhere . make s it equally clear that the traditional curriculum was itsel f rapidly · approaching a state of cri­ si s . When it had been fo rmulated initially , it had embodied the mo st advanced position of European philosophy . By 1 66 1 , Eu ro­ pean p hilo sophy had moved on , and academic Ari stotelianism rep­ resented an intellectual backwater maintained in part by the legal mandate of a curriculum enacted as law and in part by men who had a vested interest in continuing a system to whi ch they had bound their live s . Intellectual vigor had departed long since. It was becoming an exercise performed by rote without enthusi asm . In the latte r half of the seventeenth century , nearly all of the universi ty

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lect u re ships be ca me s inecure s . The di sputati ons i n the p u bli c schools , the cli mactic exercises of the cu rriculu m , gradually became mechani cal perfo rmances without meaning . 56 According to the statut es , the student who had completed fou r years of study was required then to devote the whole of the Lent term to di sputations in the publi c schools, where university as opposed to colleg� exer­ ci ses took place . After a series of oral examinations and ceremonies , he was admitted as a determiner to "stand in quadragesima , " appear­ ing in the p ubli c schools every afternoon from one to five from A sh Wednesday to the Thu rsday before P alm Sunday to underg o ex­ amination by bachelors from other colleges and to defend a nu mber of theses in di sputations conducted according to the canons of Aris­ totelian syllogi sms .- -57 By Newton' s day , the attendance require­ ment s of the university had been relaxed to the point that anyone who was admitted before the end of the Easter term was allowed to commence B . A . in the Lent term four years hence. Thu s Newton was admitted to Trinity on 5 June 1 66 1 and commenced B . A . in 1 665 . The exercise of standing in quadragesima , together with the examinations preceding it , was already apparently a mere fo rmality . The fo rmal decision (or "grace ") of the uni versity senate to g rant Bachelor of Art s deg ree s in 1 665 , whi ch merely accep ted the lists submitted by the colleges , was passed on 14 January , before any of the examinations or disputations ha d taken place. 58 In the su mmer 56

57

58

Di sputations had not disappeared in Newton's time , however. Roger North reported that he observed them both in his college Uesus) and in the public schools in 1 66 7-8 . In the early 1 670s , disputations bulked rather large in the undergraduate career of William Taswell in O xford (A utobiography and Anecdotes, ed. George P . Elliott , in Camden Miscel­ lany , 2 [Camden Society, 1 852] , 1 7- 1 8; see p. 28 for an Act in 1 679, when he was a Master of Arts) . George Peacock , Observatio ns on the Statutes of the Uni versity of Cambridge (London, 1 84 1 ) , pr . 8 (the exerci ses as prescribed by the statutes of 1 570 , which remained in effect until the middle of the ni neteenth century) , v-xiii (the exercises as described by Mathew Stockys Esquire Bedel his Book in the second half of the si xteenth century) , lxv-lxxii (the exercises as described by Beadle J ohn Buck's Book of 1 665) . Cambridge U ni versi ty Li brary, Grace Book H , p . 339. Certainly the College con­ sidered its acti on decisi ve . Two days earlier Trinity had entered the list to whom Bachelor's degrees were granted in its Conclusion Book (Conclusion Book, 1 6461 8 1 1 , p . 93) . Cf. the letter of William Sancr oft , newly elected Master of Emmanuel, to hi s old tutor Ezechiel Wri ght in 1 663 . Sancroft told Wri ght that it wo uld grieve hi m to hear a public examination in t he college. In the whole universi ty, learning in Greek and Hebrew was out of fashion . The old philosophy was not studied and the ne w was not followed syste matically and seriously . " In fine, though I must do the present society right and say that divers of the m are very good scholars and orthodox (I believe) and du tiful both to king and church , yet , methinks I find not that old genius and spirit of learning generally in the college that made it once so deservedly famous . . . " (cited in Curtis, Oxford and Camb ridge, pp. 278-9) . M uch of the letter, of course , is about Emmanuel College , the old Puri tan stronghold, whi ch fell on desper­ ate times in the Rest oration .

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of 1 665 , the universi ty dispersed for most of two years because of the plague; it i s impossible to discern from the stati stics for degrees granted that the p lague ever occurred . 59 With the colleges co mpleting thei r conquest of the university , one might perhaps look fo r the prosecution of seriou s exerci ses and examinations in the colleges , with the university ratifying their de­ ci sions . What we know about the colleges hardly supports such a conclusion . Most of the fellows treated thei r fellowships as free­ holds to be enj oyed withou t corresponding duties . Only a s m all nu mber tu tored , and they did so pri marily for the extra inco me they g ained . In 1 659 , John Wilkins , the master of Trinity, drew up new regulations because of laxi ties in the examinations . 60 There were still exa minations of some sort in Trinity , at least in connec­ tion with scholarships and fellowships , when Newton was there . Neverthele ss, the general outlines of the pi cture are clear. The es­ tabli shed curri culu m had ceased to co mmand the allegiance even of tho se who maintained i t . 61 A serious and perceptive student could not have faile d to observe as much . Small wonde r that Newton never finished the Aristotelian texts he was assigned. S mall wonder also that he found other re ading. Perhaps one should not re gard hi sto ry as alternative reading; it figured strongly in Holdsworth's program . At any rate , two hi story book s , Hall 's 59

J ames Bass Mulling er, A History of the University of Cambridge (London , 1 888) , bet ween pp. 2 1 2 and 2 1 3 . Masters of Arts degrees , which were given in J uly , did fall in 1 666; the univer sity dispersed in June of that year . In February 1 667 , when the university had been vacated , with one intermission of about three months , for a year and a half, it got royal sanction to enact ordinances that such questionists as might be prevented by the p lague from being present in C ambridge on Ash Wednesday to receive degrees should not forfeit their seniority in their colleges (Mullinger, Th e University of Cambridge, 3, 62 1) . This action by the university demonstrates in two ways that the exercises had become formali­ ties , first by dispensing with them, and second by designating Ash Wednesday , when by statute they were supposed to begin , as the date on which degrees were received . Cf. the account of Thom as Fuller of the earlier p lague of 1 630 , which also forced the university to disperse: " But this corrup tion of the aire proved the generation of many Doctours , gradu­ ated in a clandes tine way, without keeping any Acts, to the great disgust of those who had fairly gotten their degrees with publick pains and expence . Yea, Dr. Collins , being afterwards to admit an able man Doctour, did (according to the p leasantnesse of his fancy) dis tinguish inter Cathedram pestilentiae, & Cathedram em inentiae, leaving it to his Auditours easily to app rehend his meaning therein " (The History of the University of Cambridge and of Wa ltham A bbey , new ed . , notes by James Nichols [London , 1 840] , p . 23 1 . ) N o one would have go tten the j oke i n 1 667 . Indeed , there would have been n o j oke . His story is about doctors , not bachelor s , but part of the change in Cambridge was the decline almost to disappearance of advanced degrees . 60 Mullinger , The University of Cambridge, 3, 547 . 61 Roger N orth indicated that his brother J ohn's account of Aristotelian logic was " with more freedom than the humour of the university , among the seniors at least at that time . would have allowed . " John N orth thought that it was not only useless but also pernicious (Lives, 2, 3 1 3) .

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Ch ronicles and Sleidan' s Fo ur Mona rch ies, were among his early pur­ chases in C ambridge . 62 Although he left nothing among his under­ graduate notes fro m his reading in these works , chronology re­ mained, in close association wi th his study of the prophe cies , one of hi s abiding interests . For a brief ti me , a bout 1 663 , he examined j u dici al astrology , according to a conversation he had wi th Conduitt ne ar the end of his life . 6 3 A strology was never part of the curriculu m . Phonetics and a universal philosophical language also had nothing to do wi t h established studies , although they , or at least t!-ie idea of a universal language , were live centers of intellectual interest at the ti me . There had been a nu mber of schemes fo r a universal language based, as Newton expressed it, on " ye natu res of things t hem selves wch is ye same to all Nations . . . " 6 4 So metime du ring his under­ graduate career, Newton came ac ross this literature ; he drew espe­ ci ally upon George D algarno ' s A rs signo mm ( 1 66 1 ) . To it he attached an inte rest in phonetics which may have derived from his study of Shelton ' s system of short-writing . 65 The basic featu re of such a lan­ gu age was a classifi cation of things and concepts into categories . All words wi thin one category would begin wi th the s ame letter; sub se­ quent lette rs (and phonetic symbols) would indicate the su bdivisions wi thin the category . Be cau se it would express t he nature of things themselves , such a langu age would transcend the barrie rs of n ation­ alities and be universal . Newton proposed a number of modifications to Dalgarno' s scheme . 66 Other interests soon pu shed the universal language aside , and he never returned to it . F requently , as in the case of John Wilkins ' s Essay To wa rd a Real Chara cter and a Ph ilosop hic Language (published in 1 668 , after New­ ton ' s ventu re in this field) , the concept of a universal language was coupled with critici sm of A ristotelian philosophy , which was held not to express the " real" natu re of thing s . Newton ' s you thful exer­ cise was not s o . Couched in Aristotelian terms , it reflected the sole philosophy to which he had been int roduced . Such was not long the case , however. In the notebook in which he had entered the fruits of his study advancing from each end , about a hundred pages remained empty in the center. Two pages devoted to Descartes ' s 62

63 64

65

Trinity College, M S R . 4 . 48c . The two books referred to were Edward Halle, The Union of the Two Noble and Illustrate Famelies of Lancastre & Yorke . . . (London, 1 548) and J oannes Philipp son (Sleidanus) , The Key of Histo rie . Or, A Most Meth odicall A bridgement of the Four Ch iefe Monarch ies . . . (London , 1 63 1 ) , a translation of Sleidanus ' s De quatu or summis imperiis . . . (Geneva, 1 559) . Conduitt ' s mem orandum of the convers ation of 3 1 Aug. 1 726; Keynes MS 130. 10, f. 2 . Ralph W . V . Elliott , " Isaac Newton ' s ' O f An U niversall Language ' , " Modem Language Rev iew , 52 (1 957) , 9. Elliott's article prints most of the manuscript, which is located now in the University of Chicago Library, MS 1 073 . Ib id . , p . 2. Cf. the phonetics in the Morgan Library notebook , which belon g to the sam e 66 Ib id. , p . 5 . period .

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metaphysics bluntly interrupted the Aristotelianism of the texts he had been reading . 6 7 A few p ages further on he entered the title, " Q uestiones quaedam Philo sophcae , and laid out a set of headings under which to colle ct the notes from a new course of readings . 68 Somewhat later, he wrote a slogan over the title , " A micus Plato ami cu s Aristoteles m agis ami ca verita s . " ·w hatever there m ay be of truth in the pages that follow , certainly there is nothing from Plato or Aristotle. Notes from Descartes , who se works Newton thor­ oughly digested in a way that he never had Arist otle , appear throughou t the " Qu aestiones . " Nor had he confined himself to Des­ cartes . He had also read Walter Charleton 's English epitome and translation of Gassendi , and perhaps some of Gassendi as well . He had read Galileo ' s Dialogue, though app arently not hi s Discourses . He had read Rob ert Boyle , Tho mas Hobbes , K enelm Digby , Joseph Glanville , Henry More, and no doubt others as well . Veritas , New­ ton ' s new friend , was none other than philosophia mecha nica . 6 9 There i s no way conclusively to date the beginning of the " Qu aestiones " althou gh v ariou s considerations suggest some time not too late in 1 664 . 7 0 Equ ally there i s no way to state with assur'1

67 68

69

70

Add MS 3996, ff. 83-Y . Add MS 3996, ff. 88- 1 35 . I shall amend Newton ' s title , the first word of which is archai c in spelling and the third word, in haste , simply incorrect , and refer to the passa ge as "Quaesti ones quaedam Philosophicae , " " Certain Phil osophical Questi ons . " Fifty yea rs later Newton told Conti that originally-by whi ch I take hi m to refer to the peri od I am now discussing-he had been a Cartesi an (Antonio-Schinella Conti , Prose e poesi, 2 vols . [Venice, 1 739- 56] , 2, 2 6) . Cf. A . R . Hall , " Sir I saac Newt on's Note-b ook, 1 66 1 -65 , " Cambridge Historical jo urnal, 9 ( 1 948) , 239-50, and Richard S . Westfall , "The Foundations of Newton' s Philosophy of Nature , " British Jo urnal for the History of Science, 1 ( 1 962) , 1 7 1 -82. It had to be before 9 Dec . 1 664 . On that day and on the following one he made and entered observations of a comet (Add MS 3996, ff. 1 1 5, 93v ) . Many years later he men­ ti oned hi s extended observations of this comet in a conversation with Conduitt on 3 1 Aug . 1 726 (Keynes MS 1 30. 1 0 , f. 4 "' ) . Already hi s hand had altered fr o m that of the initial entries and had ass umed the tiny perpendicular form that characterized it in 1 665 and 1 666. The earliest entries in the " Quaestiones" were made in a hand transiti onal between his gram mar-school hand, in which he took hi s early undergraduate notes , and the hand of 1 665- 6 . I know of no way to date the transiti onal hand with ass urance . On f. 93v there is a passage with a reference to Descartes in i t . Unfo rtunately , the passage to which it apparently refe rs does not fall on that page in any edition o f Descartes . It comes close in two editions of 1 664, however: the Weyerstaeten edition (from the Netherlands) and the Hart editi on (from London) . Since very little writing in the transitional hand survives, it is reasonable to ass ume that the transition went rather quickly from the grammar-school hand to that of 1 665- 6 ; the latter furnished the permanent model of hi s mature hand , being m odified only sl owly over the years ahead. Intellectual continuity suggests the same date , and so do accounts of hi s intr oduction to mathemati cs. Once the new program of study seized hi m , it seized hi m completely . Although entries were made in the " Q uaes­ ti ones" over a period of ti me perhaps as long as two years , the main body was not composed of note s made at leisure . The i nitial "Quaesti ones" were put together in hot haste by a ma n who had fo und hi s calling.

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ance the agency involved , but everything we know about Cam­ bridge suggests it had little to do as an institution with leading Newton to the new philosophy . One piece of testimony indicates that D escartes was very much in the air at the ti me , so that the advice of a tutor would scarcely have been requi red . Roger North , an unde rgrad u ate in C a m bridge in 1 667- 8 , whose tut o r , his brother, did not wish to be bothered and left him to follow hi s own inclinations , "found such a stir about D escartes , some railing at hi m and fo rbidding the reading him as if he had i mpugned the very Gospel . And yet there was a general inclination , especially of the brisk part of the University , to use him . . . " 7 1 Newton ' s notes imply that he also found a stir about Descartes and decided to investigate him . Beyond Descartes , we are left wholly to sp ecula­ tion , but it is not hard to imagine the process whereby Newton was led on from one author to another into a totally new world of thought . At last he had found what he came seeking in Cambridge. Without hesitation , he embraced it as his own . The very laxi ty of the universi ty now worked to hi s advantage . Pulleyn was probably happy enough not to be bothered , and Newton could pu rsue hi s interest unhindered . He set down fo rty-five headings under whi ch to organize the fruits of hi s reading , beginning with general topics on the nature of matter, place , time , and motion , proceeding to the cos mic order, then to a large number of tactile qualities (such as rarity , fluidity , softness) , followed by questions on violent motion , occult qualities , light , colors , vision , sensation in general, and finally concluding with a set of mi scellaneou s topics not all of whi ch appear to have been in the initial list . Under some of the headings he never entered anything; under others he found so much that he had to continue the entries elsewhere . The title "Quaes ti ones " adequately describes the whole in that the tone was one of constant questi oning . The questions were posed wi thin certain limits , however. They p robed details of the mechanical philosophy; they did not questi on the philosophy as a whole . Newton had left the world of Ari stotle forever. One product of his new world view was a temporary interest in perpetual motion . The mechani cal philosophy pictured a wo rld in constant flux. Newton , the tinkerer fro m Grantham , thought of various devices , in effect windmills and water wheels , to tap the currents of invisible matter. For example, he adopted the view that 71 North , Lives, 3, 1 5 . Hugh Kearney has found five student no tebooks fro m roughly this time with no tes from Descartes . Henry More's letters to Lady Conway indicate that he was introducing students to Descartes early in the 1 670s; from what we kno w of More 's in tellectual life, he was probably doing so earlier as well ( Hugh Kearney , Scholars and Gen tlemen . Un iversities and Society in Pre- industrial Britain [London , 1 970] , p . 1 5 1 ) .

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Figure 3 . 1 . Ne wton ' s sketch es o f perp etu al- motion machines powered by th e flu x of the gravi tational stream . (Cour­ tesy of the Syndi cs of Cambridge University Li brary . )

gravity (heaviness) i s cau sed by the descent of a su btle invi si ble m atter whi ch strikes all bodies and carries them down . "Whither ye rays of gravi ty m ay bee stopped by reflecting or refracting y111 , if so a perpetu all motion may bee made one of these two ways . " (Fi­ gures 3 . 1 and 3 . 2) . Under the heading of m agneti sm , he prop osed analogou s devices . 7 2 Most of the entri es in the " Qu aestiones" were derivative, notes from Newton' s reading . Nevertheless, the whole carried the unmi s­ takable i mprint of its autho r. To a remarkable deg ree the "Qu aes­ tiones" fo reshadowed the problems on whi ch hi s career in science would focus and the method by whi ch he would attack them. As to the latter , the title " Quaestiones , " whi ch descri bes not only the set of headings but their content , suggest s the active qu estioning that lay behind Newton ' s procedure of experimental enquiry . M any of the qu estions were di rected to the authors he was reading, whose opinions he did not merely regi ster passi vely . Des cartes' s theo ry of light rai sed a nu mber of obj ections . Li ght cannot be by pression & [c] for y11 wee should see in the night a wel or better y11 in ye day we s hould se a bright light above us becaus 72 Add MS 3996 , ff 1 2 P , 1 02 .

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Figure 3 . 2

Newton ' s sketches o f p e rp etu al- m otion m a chines powered by th e flux of the magneti c stream . (Courtesy of the Syndi cs of Cambridge University Li brary . )

w e are pressed downewards . . . there could b e n o refraction since ye same matter cannot presse 2 ways . a little body interposed cou ld not hinder us from seing pres sion could not render shapes so distinct . ye sun could not be quite eclipsed ye Moone & planetts would shine like sunns . A man goeing or running would see in ye night. When a fire or candle is extinguish we lo okeing another way should see a light. The whole East would shine in ye day time & ye wes t in -f night by reason of ye flood wc h carrys or Vortex a li ght would shine from ye Earth since ye subtill matter tends from i center . There is ye greatest pression on y1 side of ye earth from ye 8 or else it would not move about in equilibria but from ye 8 , therefo re i nights should be li ghtest . 7 3 73 Add MS 3996 , 1 03v .

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These were very searching quest ions indeed directed to the Carte­ sian explanation of li ght . Under the heading ' ' Of t Celestiall m at­ ter & orbes " he added a few more, pointing out that eclip ses wou ld be i mpossible according to the Cartesian theo ry since solid bodi es could transmit the pressu re in the vortex as well as the fluid matter of the heavens . 74 Every st atement in these passages was an implic it experim ent , an ob servati on of a criti cal phenomenon that ought to appear if the theory were tru e. When he considered theo ries of colo rs , he proceeded in the same way . Do colors ari se from mix­ tures of darkness and light? If they do , a printed page, black letters on a white sheet , ought to appear colored at a dist ance-another implicit experiment . Some of the experin1ents were posed expli c­ itly . Descartes had referred the tides to the p ressure of the moon on the fluid m att er of the tiny vortex su rrounding the earth . In a work by Boyle , Newton found a proposal to test the theo ry by co rrelat­ ing tides with the readings of barometers , whi ch ou ght to register the same p ressure . I mmediately he beg an to think of other conse­ quences the theory should entai l . Observe i f ye sea water ris e not i n days & fall a t nights b y reason of l earth pres sing from 8 uppon ye night water &c. Try also whither ye water is hi gher in mornings or evenings to know whither e or its vortex p ress forward most in its annuall motion . . . Try whither l seas flux & reflux bee greater in Spring or Autume in winter or Sommer by reason of ye 8 s Ap helion & p �rihelion . Whither ye Earth moved out of its Vortexes center bye Moones p ression cause not a monethly P arallax in Mars . &c. 7 5

There was no suggestion that Newton had made any of these ob­ servations . Nevertheless , if the essence of experiment al procedure i s active questioning whereby consequences that ought to follow from a theo ry are put to the test , Newton the experimental scientist was born with the " Quaestiones . " In 1 664 , su ch a method of inquiry had been little u sed . Newton ' s example was to be a powerful factor in helping experimental procedure convert natu ral philosophy into natural science. 7 6 As he became interested in light and vision , fo r which some form s of experimentati on requi red no equipment beyond his own eyes, Newton plunged fo rward with little thought of the conse­ quences . To test the power of fantasy , he looked at the sun with 74 76

7 5 Add MS 3996, ff 1 1 1 - 1 2 . Add MS 3996 , f. 93 . Two com ments from later periods suggest something of the extent o f Newton' s eager embrace of experimentation at this time. In the kitchen at Trinity, he remembered, he had cut t he heart of an eel into t hree pieces and observed how they went on beating in unison ( 1 3 Nov. 1 7 1 2 ; ]BC 1 0, 428) . For Dr. William B ri ggs , who was an undergraduate at Corpus Christi while Newton was at Trinity, he recalled an occa sion when t hey had dissected an eye togethe r (Newton to Briggs , 25 April 1 685; Corres 2, 4 1 7- 1 8) .

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one eye until all pale bodies seen with it appeared red and dark ones blu e. After "t motion of t spirits in my eye were almost de­ cayed " so that things were beginning to appear norm al , he closed hi s eye and " heightned [hi s] . fantasie" of seeing the sun . Spots of variou s hues appeared to his eye , and when he opened it ag ain pale bodies appeared red and dark ones blu e as thou gh he had been looking at the sun . He concluded that his fantasy was able· to excite the spirits in his optic nerve quite as well as the sun . 77 He also came close to ru ining his eyes , and had to shut himself up in the dark fo r several days before he could rid himself of the fantasies of color. Newton left the sun alone after that but not his eyes. A year or so later when he was developing his theory of colors he slipped a bodkin " betwixt my eye & t bone as neare to ye backside of my eye as I cou ld " in order to alter the curvatu re of the retina and to observe the colored circle s that ap peared as he pres sed (Figure 3 . 3) . 78 How did he fail to blind himself? In the grip of di scovery , Newton did not pause to reckon the cost . The content of the " Qu aestiones " i s equ ally redolent of the fu­ tu re Newton . The p assages " Of Motion " and esp ecially " Of vio­ lent Motion " m ark his introduction to the science of mechani cs. The latter p assage, really an essay , att acked the Aristot elian expla­ nation of proj ectile motion and concluded that the continu ed mo­ tion of a p roj ectile after it separates from the p roj ector i s due to its "natu rall g ravity . " 79 B y " gravity" he referred in thi s instance to an atomistic doctrine that every ato m has an inherent motility , called gravity , by whi ch it moves. The doctrine was similar , thou gh by no means identical , to the medi eval theo ry of impetu s , which stru g­ gled with the principle of inertia fo r Newton ' s allegiance fo r twenty years . He considered the cosmic order and Descartes ' s system of vortices . 8 0 Elsewhere in the not ebook , in a hand that corresponds with the later entries in the " Qu aestiones , " Newton also took notes from Tho mas Streete , Astronom ia ca rolina, whi ch effecti vely intro­ du ced him to Keplerian astronomy . He pondered the cau se of grav­ ity (that i s , heaviness) and pointed out that the " matter" that cau ses bodi es to fall must act on their innermost parti cles and not merely on their su rfaces . As I have already indi cated , light and colors occu­ pied a con siderable portion of the " Quaestiones " ; in their ·pages Newton reco rded the central insight to the demonst ration of which hi s entire work in opti cs was directed , that ordinary light from the sun is heterogeneou s , and that pheno mena of colors arise , not fro m 77

78 80

Add MS 3996, ff. 1 09 , 1 25-5v . The temp orary impairment of hi s si ght must have i m­ pressed hi m . He mentioned the experi ment in his Lectio nes op ticae about five years later (Add MS 4002 , p. 2 1 ) and as late as 1 691 in a letter to Locke (Co rres 3, 1 52-4) . 7 9 Add MS 3996, ff. 98-8v , 1 1 3- 1 4. A dd MS 3975 , p . 1 5 . A dd MS 3996 , in several places but especially ff. 93-3v .



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Fi gure 3 . 3

Newton ' s sketch o f an exp eriment t o pro du ce sensations of color by disto rting his eyebal l . (Co urtesy of the Syn­ dics of C amb ridg e Univ ersity Library . )

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the modifi cation of homogeneou s light as prevailing theory had it , but from the separation or analysis of the het erogeneou s mixture int o its component s . 8 1 The topics above ultimately expanded into the content of the two gre at work s , the Princip ia and the Op ticks, on whi ch Newton's en­ during reputation in science re sts . He was never merely an empi rical scienti st , however . In his own eyes , he was a philosopher, intent on understanding the natu re of things in the fullest sense of the phrase. From the perspective of natural philosophy , the " Quaestiones" were the first of the seri es of speculations that form the warp on whi ch he wove the fabric of hi s scientific career . Throughout his life , hi s speculations turned on a limited set of cru cial phenomena which seem to have functioned in his eyes as keys to the under standing of nat ure. Nearly all of them appear in the " Quaestiones . " Chemical phenomena do not , it is true. Even though he was reading Boyle , Newton did not di scover chemi stry - and alchemy - until sometime after the comp ositi on of the " Quaestiones " ; in the end , they fur­ nished the most criti cal inform ati on of all. Beyond chemi cal pheno­ mena , there were a num ber of others that showed up in e very specu­ lati ve scheme he composed from the Hyp othesis of Light of 1 675 (and it s precu rsors) to the final version of the Queries in the second and third Engli sh editions of the Opticks near the end of hi s life . The cohesion of bodi es , capillary action , su rface tension of fluids , the expansion and pressu re of air - these fou r (though not neces sari ly under these ru brics) all seized his attenti on in his initial reading in the new philo sophy and never released it. 8 2 Although the st ance of questioning remained predominant in the "Quaestiones , " an inchoate natural philosophy beginning to take shape can be dim ly perceived . If Descartes was cited mo st fre­ quently , his influ ence did not in the end do minate the " Qu aes­ ti ones . " Two other systems challenged hi s authority . On the one hand , Gassendi ' s atomisti c philosophy , known to Newton at thi s time primari ly through Charleton ' s Physiologia, offered a ri val me­ chanical system . Mo re than anything else , the " Qu aestiones" were a dialogue in whi ch Newton weighed the virtues of the two sys­ tem s . A lthough he appeared to reach no final verdi ct , it is clear that he inclined already toward atomi sm . After dep loying the st andard arguments against a plenu m , Newton opted for atoms , though not , or at least not initially , Gassendi ' s atoms . 8 3 I have already quoted 81 82

83

A dd MS 3996, ff. 27-30, 97, 1 22v . Add MS 3996, " Conjuncti on of bodys , " ff. 90v - 1 ; " Attraction Electricall & Fil tration [capill a ry action] , " f. 1 03 ; surface tension appears under " Of Water & Salt , " ff. 1 00v , 1 1 1 . " Mr . Boyle' s receiver" was mentioned several ti mes and the "utmost naturall" expansion of air was discussed (very briefly) under " Of Aer, " f. 1 00 . Add MS 3996, "Off ye first mater , " ff. 88-8v , 89v ; " Of Attomes , " ff. 89, 1 1 9-20.

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Newton ' s obj ections to Descartes ' s conception of li ght and his ex­ planation of the tides , and I have indi cated that he accepted a di ffer­ ent view of the cause of gravity (heaviness) . M atter and light were the most important ; to reject D escartes ' s opinion s on these two issues was to shatter the cohesion of hi s natural philosophy bey ond hope of repair . In hi s discu ssions of li ght and color, Newton left no dou bt that he held the corpuscular conception . 8 4 Desc artes may have introduced him to the mechanical philosophy , but Newton quickly transferred hi s alleg iance to atomism . There is also the possibility that the writings of Henry More guided Newton into the mechanical philosophy . 8 5 Descartes ' s name appeared so frequently in them that he could not have failed to notice i t . Whi chever he came to first , More represented the other current of thought that tempered Newton ' s enthu sias m for D es­ cartes . More's vi ews exerted a st rong influence on the orig inal essay on atom s that Newton wrote in the " Quaestiones . " Newton later crossed the essay out , however , and it was not here that More's position was vital. Like the other Cambridge Platonist s , Henry More was concerned by the mechanical philosophy' s possible ex­ clusion of God and spirit from the operation of physi cal nature. Whereas initially he welcomed Descartes as an ally of religion , the more he contemplated his sytem of nature, the more its impli ca­ tions alarmed hi m . In Hobbes he s aw the dangers spelled out ex­ pli citly. More w as concerned to reinstall spirit in the continu ing operation of nature , all of natu re . Especially in the last four entries of the " Qu aestiones , " " Of God , " " O f t Creation , " " Of ye sou le, " and " Of Sleepe and D reams &c, " whi ch appear by their po sition to be later additions to the original set of heading s , similar concerns made a tentative appearance in the " Qu aestiones . " 8 6 Their role in Newton ' s thought was destined to grow , diluting and modifying hi s initial me chanistic views . 84 85

86

" Of light , " ff. 1 03v , 1 28v ; " Of Species visible , " f. 1 04v ; " Of Colours , " ff. 1 05v , 1 22-4. Although Henry More had attended the same grammar school , and although the two men would later become acquainted, I think it is out of the questi on that More , OI)e of t he foremost intellectual figures in Cambridge , perhaps the fo remost, and a fellow of a different colle ge, woul d have known Newton t he undergraduate . Given More's emi­ nence , wi th the added indu cement of his connection with Grantha m , there is no problem at all in imagining how N ewton could have picked up hi s works . The enduring influence of Henry More and Cambridge Platonism on Newton has been the central t heme of J. E . McGuire i n a number o f articles: - see especially "Force, Active Principles , and Newton's Invisible Realm , " Ambix, 1 5 ( 1 968) , 1 54-208; " Atoms and the ' Analogy of Nature ' : Newton's Third Rule o f Philos ophizing, " Studies i n History and Philosophy of Science, 1 (1 970) , 3-58; "Body and Void and Newt on's De Mundi Systemate: Some New Sources , " Archive for History of Exact Sciences, 3 ( 1 966) , 206-48; and " Neoplatoni sm and Active Principles: Newton and the Co rpus Hermeticu m, " in Robert S. Westman and J. E . McGuire , Hermeticism and the Scientific R evolution, (Los Angeles , 1 977) . Add MS 3996 , ff. 1 28, 1 29, 1 30-30v , 1 32 .

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Meanwhile , natural philosophy was not the only new study Newton di scovered . He found mathemati cs a s well . A s with nat­ u ral p hilosophy, we have Newton ' s original notes that chart hi s cou rse . We also have a nu mber of account s , several in Newton ' s own words o f which one from 1 699 i s the most i mport ant , one in Conduitt 's memo randu m of a conversation with Newton on 3 1 August 1 726 , and another i n a memorandum o f Nove mber 1 727 , soon after Newton ' s death , by Abraham DeMoivre . The earliest of them dated from thirty-fiv e years after the event s it desc ri bed . Nev­ ertheles s , a reasonably consistent account , whi ch i s also reasonably consistent with Newton 's reading notes , emerg es from them . Ju ly 4th 1 699 . By consu lting an accompt of my exp enses at Cam­ bridge in the years 1 663 and 1 664 [Newton wrote as he lo oked over some early notes] I find that in ye year 1 664 a little before Christmas I being then senior Sophis ter , I bought S chooten' s Mis cellanies & Cartes 's Geometry (having read this Geometry & Ou ghtre d's Clavis above half a year before) & borrowed Wallis 's works & by conse­ quen ce made these Annotations out of S chooten & Wallis in winter between the years 1 664 & 1 665 . At wc h time I found the m ethod of Infinite series . And in su mmer 1 665 being forced from C ambridge by the Plague I compu ted i area of ye Hyperbola at Boothby in Lincolnshire to two & fifty figures by the same method . Is . Newton 8 7

In Conduitt ' s memo randum , it all beg an when Newton lit upon some book s on j u dicial astrology (an event whi ch DeMoivre pla ced at Stu rbridg e Fair in 1 663) . Being unable to cast a figure , he bought a copy of Eu cli d and u sed the index to locate the tw o or th ree theo rems he needed; when he found them obviou s , " he despised that as a trifling book . . . " DeMoivre' s account ag reed with Con­ duitt ' s except that he had Newton /g o on in Eu clid to mo re difficult propositions su ch as the Pythag orean theo rem , whereup on he changed his opinion , and read all of Eu clid through t wice . Su ch early study of Eu clid does not ag ree either with Newton ' s notes or with other part s of what he told Conduitt . Pemberton als o re co rded Newton ' s reg ret that he had not given more attention to Eu clid before he applied him self to Descartes . 8 8 He b ought Des cartes 's Geometry & read it by himself [Conduitt continued in language very similar to that in the DeMoivre account] when he was got over 2 or 3 p a ges he could understand no farther 8 7 Add MS 4000 , f. 1 4v . See a similar chronol ogy in Newton to Wallis, late 1 692 (Corres 7, 394) . Newton composed a number of other resumes o f his mathematical development some years later at the height of the controversy with Leibniz . They all agree essentially with the earlier one qu oted here . Cf. Add MS 3968 . 5 , f. 2 1 ; Add MS 3968 . 4 1 , ff. 76, 8 5 , 86\' . 88 Henry Pem berton , A View of Sir Isaac Newton 's Ph ilosophy (Lo ndon , 1 728) , preface .

Th e so litary scholar

99

than he began again & got 3 o r 4 pages farther till he came to another difficult place, than he began again & advanced farther & continued so doing till he made himself Master of the whole without having the least light or instructi on from any body . 8 9

Both account s ag ree in making Newton an autodi dact in m athemat­ ics , as he was in natural philosophy . Nearly twenty years later , when he was recommending Edward P aget fo r the position of mathemati cal master at Christ ' s Ho spital, Newton probably had his own experi ence in mind as he specifi ed Paget ' s qu alifi cations . Paget understood the several branch es of mathemati cs , he sai d , "& wch i s t surest character o f a true M athemati call Geniu s , learned these of his owne inclinati on , & by hi s owne indu stry without a Teacher . " 9 0 There had been even less mathemati cs in the uni versity than natu ral philosophy; not su rp ri singly , no stories su rvive of u ndergradu ates being stirred by Descartes ' s Geometry . Nevertheless , there is a curi­ ou s coincidence of time that has generally been igno red . The Luca­ sian ch air of mathemati cs , whi ch Newton himself would soon oc­ cupy, was establi shed in 1 663 , and the fi rst professor, I saac B arrow, delivered his inaugu ral series of lectures in 1 664 , beginning on 14 M arch . Cont rary to frequ ent assertions , Barrow was not Newton ' s tutor, and there i s n o evidence o f any fa mi liarity between them at thi s time. At least twice , Newton i mp lied that he had attended the lecture s , however; and thou gh they would probably not have di­ rected him to Descartes , given B arrow' s mathemati cal predilec­ tions , and though B arrow was not a maj o r influence on him, they could have stimu lated his interest in mathemati cs . 91 One wonders as well who in C ambridge cou ld have loaned him a copy of W alli s if it was not B arrow . In any event , the coincidence in time i s so close th at it strains credulity to deny any connection bet ween the lectures and Newton's sudden int erest . Newton ' s own notes ag ree with the account s of Conduitt and 8 9 Keynes MS 1 30 . 1 0 , f. 2v . " Mem orandums relating to sr Isaac Newton given me by Mr A braham Demoivre in Nov 1 727" ; University of Chicago Library, M S 1 075- 7 . The account in the DeMoivre mem orandu m of Newton's struggle with Descartes is so si milar to Conduitt's mem orandum of the conversation of 31 Aug . 1 726 that it is hard to believe they were wholly independent. Since the DeMoi vre memorandum is in Conduitt' s hand, it is entirely possible that it was Conduitt's account of a conversati on with DeM oivre instead of a copy of a paper written by DeMoivre . Conduitt di d not make copies of other items he received as he collected in formation about Newton . In the period 1 7 1 5- 1 6 Newton g ave a very similar account o f his early mathematical studies t o Abbe C onti , who was then in England (Conti , Prose e poesi, 2, 24) . 9 0 Newton to the Governors of Christ's Hospital, 3 A p ril 1 682; Corres 2, 3 7 5 . I wish to thank Professor Gale Christianson for pointing out the signifiGttl.Ce of this pa ssage to me . 9 1 Both references from around 1 7 1 5 , in statements connected wi\h the controversy with Leibniz (Add MS 3968 . 5 , f. 21 Add MS 3968 . 4 1 , f. 86v ) . He a1"0 told Conti that he attended Barrow's lectu res (Prose e poesi, 2, 25) .

1 00

Never at rest

DeMoivre that he plunged straight into mo dern analysis with no app reciable background in classical geometry . They agree as well with the centrality given Descartes . S chooten' s second Latin edition of the Geometry , with i ts wealth of additional commentaries , was his basic text , supp lemented by S chooten ' s Miscellan ies , Viete's works , Oughtred ' s algebra (the Cla vis Newton mentioned) , and Wallis' s A rith metica infinitorum . In roughly a year, without benefit of instruction , he mastered the enti re achievement of seventeenth-cen­ tury analysis and began to break new ground. Newton ' s surrender to his new studies was not without danger . In order to pursue them to a fruitful conclusion , he had to win a permanent position in Cambridg e , but rewards in Cambridge were not being p assed out for excellence in mathematics and mechanical philos ophy . Fello wships in Trinity went only to those who had first been elected as an undergraduate to one of the sixty-two schol­ ars hips supported by the college . In hi s first three years , Newton had not distinguished himself in any way . Trini ty had twenty-one exhibitions which carried annual stipends of about four pounds each . The college records give no indication of the criteria of selec­ tion . It is diffi cult to imagine that academic pro mise did no t figure , though need may have been the decisive factor . Suffice it to s ay that Newton did not appear among the ten , nearly all Pulleyn' s pupils , who received exhibiti ons in 1 662 and 1 663 . 9 2 Many featu res of the college worked to lessen his chances of a s cholarship . Statistics indicate that sizars had less chance than pen­ sioners , especially when enrollment was up and demand high , as they were in the 1 660s . Influence and connections were e ssential characteri stics of the system of p atronage that impinged on the whole university to the inj u ry of those , sizars above all , who lacked sponsors in high p laces . 93 Newton ' s chances were further dimin9 2 A com plete list of the twenty-one exhibitions and their stipends can be found in the Senior Bursar ' s Accounts of Trinity College for 1 664 . Five of them endowed by Mr. (i . e. , Magister) Hy lord , were added only in 1 663. Awards of exhibitions in 1 662 and 1 663 are listed in the Conclusion Book, 1 646- 1 8 1 1 , p p . 72 , 76, 77 . 93 Perhap s it is a mistake to project the system of patronage b ack to the level of scholarships, which were pretty small pickings although they were essential steppingstones to fellow­ ships. Fellowships were significant; they provided a perm anent income on which a man (such as a younger son or the son of a client) could live . Royal mandates , which were prom pted by the intercession of one courtier or another and were evidence then of a courtier's access to the king's ear, were fairly common for fellowships , although the number I have been able to lo cate is not proportionate to the indignation against them in some quarters of the university . (Mandates for masterships , important preferments, were nearly the invariable rule . ) Trinity received a letter mandate dated 7 Sept . 1 664 ordering the college to elect John Howarth to a fello wship "in consideracion of his owne merits , & of his relation to one of Our principall Servants . . . " (Trinity College, Box 29 . D) . He was elected . On 23 Nov. 1 666 , there was a similar one for Henry Cary , on a ccount cf his

The solitary scholar

1 01

ished b y the p rivileged g roup of West minster scholars who auto­ mati cally recei ved at least a third of the scholarships year after year , and with the scholarships the top ru ngs on the ladder of seniority fo r their year . During the entire centu ry , a good half of Trinity's fellows came fro m West minster School , and roug hly that propor­ tion held fo r the large group elected s cholars in 1 664 . 94 Indeed , with 1 664 Newton faced a cri si s . Trinity held elections to scholar­ ships only eve ry three or fou r years . The election in 1 664 was the only one during his career as a student . If Newton were not elected then , all hope of permanent residence in Cambridg e wou ld vanish fo rever . He chose exactly that time to th row over the recognized studies and pu rsu e a cou rse whi ch had no standing whatever in the colleg e' s scheme of values . P erhaps the app roaching elections , to be held in Ap ril, with th eir attendant examinations explain an otherwise anomalou s feature of Newton ' s notes on the establi shed cu rriculu m . H aving dropped Magiru s ' s peripateti c Physics, he took it up again and p lowed hi s way th rough two more chapters . Likewi se, he started Vossi u s ' s Rhetoric and the Ethics of Eu stacius of St . P a u l about this time - and likewise failed to finish both works . In the th ree cases , the notes su ggest last-minute boning fo r an examination . 95 Newton 's own own merit and the service and suffering of hi s father in the royal cause in the Civi l War (CSPD: C harles I I , 6, 280) . For whatever reason, this mandate was not obeyed. On 24 July and 30 Oct . 1 667, there were mandates for t he election of Valentine Pettit and John Goodwin to fellowships CSPD, 7, 322 , 553) . Both o f these mandates were obeyed . There were also petiti ons to the Crown in 1 665 and in 1 668 by two Westminster scholars , who had been disappoi nted in their expectation of fellowships, aski ng for letters mandate CSPD, 5, 1 41 , 8, 597) . They were further disa ppointed in t heir expectati on of letters manda te , and neither man ever obtained a fellowshi p . 94 Trini ty held elections to scholarships in 1 66 1 , 1 664, and 1 668. Both in 1 662 and in 1 663 , the Westminster group admitted the previous year (eight in 1 662 , seven in 1 663) took up scholarships by ri ght without election. The Trinity records also s how t hat one other student became a scholar in 1 662 . (No explanation is given; it may be simply a typo­ gra phical error .) The election held in 1 664 also pre-elected for 1 665; in all , forty-four students were elected to scholarships at that ti me , inclu ding the Westminster group (five) admitted in 1 66 3 . In 1 665, five more Westminster students took up scholarships; no other students became scholars in t hat yea r . The total number elected (or, in the case of the Westminster scholars , admitted) to scho i arships over the four-year span was sixty-five, of whi ch twenty-five were from Westminster . Of the sixty-five, twenty-five went on to fellowships; ni ne of them were fro m Westminster . In addition, another West minster scholar became a chaplai n in the college. Only five sizars and subsizars admitted in 1 66 1 , Newton's year , were elected to scholarships; two of them went on to fellowshi ps (Ball and Venn , Tri nity Admissions) . 95 There is reason to think that there was an examinati on. There is record of t he preceding one . " 1 1 April 1 66 1 . Ordered that t here be an election of scholars on April 22 & tha t they stand fo r places & be examined on the Friday & Saturday before" (Master' s Old Conclu­ sion Book , 1 607- 1 673 , p. 268) . The method o f exami nation in Caius was an extension o f t he practice o f disputations; the candidates i n Caius sat i n the chapel fo r three days , being exami ned by scholars on the first day and by the Dean and fellows on the second and

1 02

Never a t rest

account , a s related to Conduitt , implies that hi s tuto r P ulleyn may ha ve recognized his pupil ' s brilliance and tried to help him by en­ li sting Isaac B arro w , the one man in Trinity fit to j udge his compe­ tence in the unorthodox studies he had undertaken . The gesture nearly capped the deba cle , since Newton had been unorthodox even in his unorthodoxy. When he stood to be scholar of the house his tut our sent him to Dr Barrow then Mathemati cal p rofessor to be examined, the Dr exam­ ined him in Euclid wc h sr I. had neglected & kne w little or nothing of, & never asked him about Des cartes ' s Geometry wc h he was master o f sr I . was too modest to menti on it hi mself & Dr Barrow could not i magine that any one could have read that book without being first master of Eucli d , so that Dr Barrow conceived then but an indifferent opinion of him but howe ver he was made scholar of the house . 9 6

The final clause i s true; on 28 April 1 664 Newton was elected to a scholarshi p . It also poses a quandary : What can explain the deci­ sion? Perhaps the explanation is the obviou s one that springs imme­ diately to mind . Newton ' s genius readily outshone the mediocrity around him even in studies he had abandoned . Such an explanation seems to conflict , however , with Newton ' s account of the impres­ sion he made on Barrow , the leading intellect of the college. Mo re­ over, the realiti es of Cambridg e in 1 664 suggest another explana­ tion , that Newton had a powerful advocate within the college. There is good reason to think he had such an advocate. In 1 669 , as a new fellow , he was appointed tutor of a fellow commoner . Tutor­ ing fellow commoners was lucrative bu sine ss u sually reserved for important fellows . Two candidates fo r Newton 's patron present themselves . One is Barrow him self despite the story . It is not i m­ possible that Newton was misled as to the impression he made . Indeed it may be that B arrow told Pulleyn to send him ; if atten­ dance at his lectures paralleled that of Newton ' s later ones , he may have been curious about a fam iliar face. That i s all speculation , however. It i s not speculation that in 1 668-9 , Barrow was familiar enough with Newton ' s wo rk to send him Mercato r' s Logarithmo­ technia when he saw that it seemed to fo restall some of his work . In 1 669 , he obtained the Lucasian chair for Newton when he himself

9h

thi rd (John Venn, Caius Co llege, [London , 1 90 1 ] , p . 20 1 ) . In 1 692 , William Lynnet , a senior fellow , described the procedure then used in Tri nity for the election of fellows and scholars . He said that the y sat three days in the chapel from seven to ten and from one to four . The account assumes that fellows , or at least seni or fellows , examine d them orally . In the case of candidates for fellowshi ps, at least , they wrote an essay on the fourth day. (Edleston, p p . xlii-xliii) . Keynes MS 1 30 . 1 0, f. 2v . Newton also tol d Conti a version of this story (Prose e poesi, 2, 24) .

The solitary scholar

1 03

resigned , and in 1 675 , Barrow appears to have been deci sive in obtaining a royal dispensation for Newton . The other and more likely candidate i s Hu mphrey Babington . Recall Ne wton ' s ac­ counts which show that he employed " Mr B abingtons Woman . " Recall his statement that during the plague he was , at least part of the time , at Boothby , not far from Woolsthorpe, where the same Mr. Babington was rector . M r . Babing ton was also the brother of Mrs . Clark , with whom Newton had lodged in Grantham . Most important of all , he was approaching the statu s of a senior fello w , one o f the eight fellows a t the top of the ladder o f seniority , who ran the college in conj unction with the master. Moreover, the college would not have fo rgo tten that he stood well with the ki ng; twice he had obtained letters mandate in his favor in recent years . When B abington was later bu rsar of the college , Newton drew up tables to aid him in renewing college leases , and the two con ti­ nued to be associated in various acade mic affairs until Babington ' s death . Since B abington w a s resi dent only four o r five weeks o f the yea r at thi s ti me , however, his opportunity to influence the elec­ tion may have been small . Fou r years earlie r, the Reverend Wil­ liam Ayscough and M r . Stoke s had re scued Newton from ru ral oblivion . Someone pe rfonned that service again in Ap ril 1 664 , and on the whole Hu mphrey Babington appears most likely to h ave been the one . With his election , Newton ceased to be a sizar. He now recei ved commons fro m the college , a livery allowance of 1 3 s/ 4d per year, and a stipend of the same amount . 97 Far more important, he re­ ceived the assurance of at least four more years of unconstrained study , until 1 668 when he would incept M . A . , with the possibility of indefini te extension should he obtain a fellowship . The threat had lifted . He could abandon hi mself co mpletely to the studies he had fo und . The capacity Newton had shown as a schoolboy for ecstasy , total su rrender to a co mmanding interest , now found in his early manhood its matu re intellectual manifestation . 98 The tenta­ tiveness suggested by the earlier unfini shed notes vani shed, to be replaced b y the passionate study of a man possessed . Such was the characteristic that his chamber-fellow Wi cki ns remembered, having observed it no doubt at the ti me with the total inco mprehension of the Woolsthorpe servants . Once at work on a problem , he would forget his meals . Hi s cat grew very fat on the food he left stand ing 97 Junior Bursar ' s Accounts and Senior Bursar ' s A ccounts . 98 Recall his confession of 1 662: " S etting my heart on . . . learning . . . more than Thee . " Newton learned to rationalize his pursuit of learning , the co mmanding p assion of his life, in religious terms . He was studying God ' s word and God's work , and it seems clear enough where his heart was set . It had nothing to do with rational decision - he was helpless in the hands of his own genius .

1 04

Never at rest

on hi s tray . 99 (No peculiarity of Newton's amazed hi s contempo­ raries more consistently; clearly food was not so mething they tri­ fled with . 100) He would fo rg et to sleep , and Wickins wou ld find him the next m o rning , sati sfied with having di scovered s o me propo sition and wholly unconcerned with the night 's sleep he had lost . " He sate up so often long in the year 1 664 to observe a co met that appeared then , " he told Conduitt , " that he found hi mself much di sordered and learned fro m thence to go to bed betimes . " Part of the sto ry is true; he ent ered hi s ob servations of the comet into the "Quaestiones. " 1 01 The rest of it is patently false as Conduitt knew from personal experi ence . Newton never learned to go to bed be­ times once a problem seized hi m . Even when he was an old man the servants had to call him to dinner half an hou r before it was ready , and when he cam e down , if he chanced to see a book or a pap er , he would let hi s dinner stand fo r hou rs . He ate the gru el or milk with eggs prepared fo r his supper cold fo r breakfa st . 1 02 Con­ duitt observed Newton long after his years of creativity . The ten­ sion of the qu est that consu med him in 1 664 and the years that followed st retched whatever neu ro ses he had brought from Wools­ thorpe to thei r utmost limits . He was " much disordered " more than once , and not only fro m observing comet s . H i s di scovery o f the new analysis and the new natu ral philo sophy in 1 664 marked the beginning of Newton ' s scientific career . He considered the "Quaestiones" important enough that he later co m­ posed an index to them to su pplement thei r initi al organization under topics . 103 Sailing away fro m the old world of academic Ari s­ totelianism , N ewton launched his voyage toward the new . The passage was swift . 99

100

101 102

103

Apparently Newton told thi s to his niece . Conduitt credited the story to "C. C . " It was specifically his cat at the uni versity . (Keynes MS 1 30. 6, Book 2) . In addition to those already cited, Humphrey Newton mentioned it in both of his letters about his period with Newton in the 1 680s . (Keynes MS 1 35) . Stukeley heard stories about it in Cam bridge (Stukeley, p. 48) and Conduitt himself observed it in Newton' s old age (Keynes MS 1 30 . 6, Book 1 ) . As late as 1 742, Dr. George Cheyne reported hearing a similar story . (Natural Method of Curing Diseases of the Body and Disorders of Mind [London, 1 742] , p. 8 1 ) . Ni cholas Wickins to Robert S mith, 1 6 Jan. 1 728; Keynes MS 1 37 . Conduitt's mem oran­ du m of 3 1 Aug . 1 726; Keynes MS 1 30. 1 0 , f. 4v _ Add MS 3996, ff. 93, 1 1 5- 1 6. Keynes MS 1 30 . 6, Book 1 ; Keynes MS 1 30 . 5 , sheet 1 . Since Conduitt assigned this set of ane cdotes to "C. C . " (Catherine Conduitt) , they may refer to the early years of Newton' s resi dence in London. Newton may have become sensitive about the stories of his eccen­ tricities . They were apparently widespread; Nicholas Wickins spoke of them as m uch repeated. At any rate , in addition to telling Conduitt that he learned to go to bed betimes , he said the same thing to Stukeley as though he were trying to deny the stories (Stukeley to Conduitt , 1 5 July 1 727; Keynes MS 1 36, p. 1 1 ) . Add MS 3996, ff. 87-T

4

Resolving problems by motion

N his age of celebrity , Newton was asked how he had di scovered

I the law of universal gravitation . "By thinking on it continually , "

was the reply . 1 No better characterization of the man can be given , not only in its delineation of a li fe whose central advent ure lay in the world of thought rather than action , but also in its description of his mode of work . Seen fro m afar, Newton ' s intel­ lectual life appears uni m aginably rich . He embraced nothing less than th e whole of natu ral philosophy, whi ch he explored from several vantage point s which ranged all the way from mathemati­ cal physics to alchemy . Within natural philosophy, he g ave new di rect ion to optics , mech ani cs , and celestial dynami cs, and he in­ vented the mathematical tool whi ch has enabled modern science fu rther to explore the paths he first blazed . He sought as well to plumb the mind of God and His eternal plan fo r the world and mankind as it was presented in the bibli cal p rophecies . When we examine Newton ' s grandiose adventure minutely , it t:urns out to be a mixture of discrete p ieces rather than a homogeneou s me­ lange . His career was epi sodi c . What he thought on , he thou ght on continu ally , which is to say exclu sively , or nearly exclusively . What seized his attention in 1 664 , to the virtual exclu sion of everything els e , was mathematics . John Conduitt , the hu sband of Newton ' s niece and his intended biographer , almost invariably s mothered whatever insight he had in a froth of grandiloquence . One of his figures , applied to Newton's early career, bears repetition , however: " he began with the most crabbed studies (like a high sp irited horse who must be first broke in plowed grounds & the rou ghest & steepest ways or cou ld other1 The anecdote rests on some what shaky authority . The fi rst instance of it that I have fo und is in a note to Voltai re 's El ements de la ph ilosophie de Newton, Part III , chapter 3 (in French, "en y pensant sans cesse") . It did not appear in any of the editions published during Voltaire' s life , but first in the so-called Kehl edition of the Oeu vres completes de Voltaire, 70 vol s . (Paris, 1 785-9) , 3 1 , 1 75 . Pres umably , such notes were based on annotations that Voltaire made in an earlier editi on , b ut the relevant volume of the annotated edition has been lost. There is reason then to think that t he anecdote rests on Voltaire' s authority and derives from hi s stay in England , where he knew Newton's niece , but its authentici t y can hardly be called secure . It may be a cor rupti on of Conduitt ' s note: "Bentley sai d - Sr I . told hi m all hi s merit was patient thought" (Keynes MS 1 30. 5 , sheet 1 ) . Conduitt's note , in turn, may have derived , through Bentley , from Newton's letter of 10 Dec . 1 692: "But if l have done ye publick any se rvice this way 'tis due to nothing but industry & a patient thought" (Corres 3, 233) . 1 05

1 06

Never at rest

wise be kept within no bounds . " 2 Newton was to voyage o' er m any strange seas of thought, speculative adventures from whi_ch more than one explo rer of the seventeenth century never returr:�d . The discipline that m athematics i mposed on hi s fertile imagination marked the difference between wild fli ghts of fancy and fruitful discovery . It was supremely imp ortant that , almost fi rs t , m athe­ m atics commanded hi s attention . The surviving notes o f hi s initial studies i n m athematics bear out the various anecdotes that he plunged straightforward into Des­ cartes's Geometry and modern analysi s . The time was almost cer­ tainly 1 664 , probably in the spring or sum mer. His primary vehicle w as Schooten ' s pivotal second Latin edition of Descartes ' s Geometry with its wealth of additional co mmenta ries , suppo rted by reading in algebra , esp ecially from the works of Viete . He also made early contact with the m athem atics of infinitesimals as rep resented by John Wallis . It is quite impossible to dete rmine from the notes which came fi rst . It is equally impossible to see that anything im­ portant hinges on thei r chronological order. What matters is the voraci ty with whi ch he devoured whatever mathematics he found. Whi ston later remarked that in m athem atics Newton " could some­ times see almost by Intuition , even without Demonstration . . . " 3 Whiston had a proposition in the Principia in mind , but an examina­ tion of Newton ' s self-education in mathem ati cs compels one to a similar j udgment . Within six months of hi s initi ation into mathe­ matics , some of hi s reading notes were changing impercepti bly into original investigations . Within a year, he had digested the achieve­ ment of seventeenth-century analysis and had begun to pursue his own indep endent course into higher analysi s . 4 2 Keynes MS 1 30 . 4 , p . 4 . 3 William Whiston , Memoirs of the Life and Writings of Mr. William Wh iston (Lon don , 1 749) , p . 39. 4 D . T . Whites ide, " Introduction " , Math 1 , 1 4 . I revised this chap ter after spending much tim e with two students, George Anastaplo and Richar d Ferrier, on volume 1 of White­ side's edition. I owe many points in what follows to their insight s , which they worked out with extraordinary tenacity . In addition to the editorial apparatus in volume 1 of Math , Whiteside has written a number of excellent articles on Newton 's discov eries in mathematics: " Isaac Newton : Birth of a Mathem atician, " Notes and Records of the Royal Society , 19 ( 1 964) , 5 3-62; " New­ ton's Discovery of the General Binomial Theorem , " Mathematical Gazette, 45 ( 1 96 1 ) , 1 7580; " Sources and Strengths of Newton's Early Mathematical Thought, " in Robert Palter , ed . , The 'A nnus Mirabilis ' of Sir Isaac Newton (Cambridge, Mass . , 1 970) , p p . 69-85 ; and " Newton 's Mathematical Method , " Bulletin of the Institute of Mathematics and Its Applica­ tions, 8 ( 1 972) , 1 73- 8 . See also H . W. Turnbull , The Mathematical Discoveries of Newton (Londo n, 1 945) , and " Newton : the Algebraist and Geometer, " in The Royal Society Newton Tercentenary Celebrations (Cambridge, 1 947) , pp. 62-72; Joseph E. Hofmann , " S tu­ dien zur Vorgeschichte des Prioritatstreites zwischen Leibniz und Newton um die Entdeck­ ung der hoheren Analysis, " A bhandlungen der Preussischen Akademie der Wissenschaften,

Resolving p roblems by motion

1 07

Newton ' s early notes on analysis inves tigated the relation of particula r axes to the equation of a curve . He tried various trans­ formations of axes in order to simplify equations , seeking a regu­ larized p rocedure whereby one of the new axes would also be an axi s of the cu rve . 5 He was not yet beyond fai rly simple mi stakes. Like the ea rly analysts , he did not at fi rst comp rehend the signifi­ cance of negative ro ots . He drew the cubi c parabola , x 3 = a 2y , as though it were symmetrical with respect to the y axis , and some­ what later he confined his diagram of Descartes's foliu m , the curve x 3 - axy +y 3 = 0, to the first quadrant , tacitly assuming that its shape is identi cal in all fou r q uadrants . 6 He learned quickly , however. By October, he comp rehended negative roots clea rly enough to set down the rule that when the x axi s is the diameter of a cu rve such that it bisects all the ordinates , then y cannot appear in the equation in odd powers because there i s a negative root of equal absolute value corresponding to every p osi­ tive one . 7 A bove all , he seized and made hi s own the central insight of analyti c geometry . Already in September he proposed a problem to hi mself in the following tenns: "Haveing ye nature of a crooked line expressed in Algebr: termes to find its axes , to determin it & describe it geometrically &c . " 8 Perhaps because he came to it without a background in classical geometry , Newton had co mp rehended the thrust of analysis more clearly than its originators . The algebraic equation i s not merely a device to aid geometric constructions . The equation is more basic than the curve; the equation defines , or as Newton put it , expresses the natu re of the cu rve . Hence the equation itself should become the focus of attention . 9 Not that he was uninterested in the construction of curves . Very early , he collected from Schooten ' s work s various ways in which the coni cs can be cons tructed . He found nine constructions of the ellipse, six of the p arabola , and ten of the hyperbola. For each of the three curves , one construction was the basi c section through a cone . Most of the others treated the curve as the path of a moving point and described some device , such as the tra m mel fo r the ellipse , whi ch constrains the mo ving point such that it traces the curve (Figure 4 . 1 ) . It is mi sleading , however, merely to say that he found Mathematisch-naturwissens chaftliche Klasse, 1 943; Carl B . Boyer, The Concepts of th e Ca lculus. A Critical and Historical Discussion of the Derivative and the Integral (New York, 1 939) ; and J . 0 . Fleckenstein, Der Prioritatstreit zwisch en Leibniz und Newton (Basel and 5 Math 1 , 1 55-65 . Stuttgart, 1 956) . 8 Math 1 , 236. 6 Math 1 , 1 60, 1 84. 7 Math 1, 1 69 . 9 Timothy LeNoir , "The Social and Intellectual Roots of Discovery in 1 7th-Century Mathe­ matics" (dissertation, Indiana University , p p . 396-525) . It is impossible wholly to docu­ ment the extent to which I have profited from Mr. LeN oir's work.

Never

1 08

at

rest

e

g

a

�- -�����--��-----+-�

Figure 4 . 1

c

The t rammel construction of the ellipse. The tram mel dab moves so that a follows track gc and d follows t rack ef The length bd of the t ra mmel equ als half the maj or axi s of the ellip se , the length a b half the mino r axis.

the constructions in Schooten , since he ventured to discover some constructions of his own and to effect improvements in some of S chooten ' s . 10 He was also interested in a device he found in Viete , the mesolabum , by which one could lay off a given distance between two intersecting lines or between a circle and a line intersecting it (Figure 4 . 2) . 1 1 Since the instrument op­ ened the way to the cons truction of two mean proportionals between given lines , if its use were granted , that is � in effect , to the construction of cube roots , the mesolabum was Newton' s introduction to cubics . The collection of constructions of conics compiled from various locations in Schooten was a typical Newtonian exerci se , the ana­ logue in m athematics to the " Quaestiones quaedam Philosophicae" put together a few months earlier from the mechanical p hilosophers he had been reading . During the autu mn , Newton sought further to order his growing knowledge of the conics and of analyti c ge­ ometry in a p assage that went through five successive s tages as it gradually transformed itself in November into his own original essay: "To find ye A xi s or Diameter of any Crooked Line suppose­ ing it hath ym . " In its final version , it p resents a general procedure by which to determine axes o r , a s in cubics , the nonexistence of 10

Math 1, 29-39 .

11

Math 1 , 72-7 .

Resolving problems by motion

1 09

a

b

Figu re 4 . 2

The mesolabum is a physical instrument with arm gh of fixed length which slides along the beam agh while the beam in turn rotates around a . With it one can lay off along a line through a a given length gh between two inte rse cting lines, bh and gb , or a distance equal to the radius of a ci rcle between the ci rcle and its extended di­ amete r bh .

axes . 12 He al so investigated vertices and asymptotes . S carcely six months into his career as a mathematician , he was already consis­ tently concerned to develop general procedu res which would apply to all curves and the equations that define them , and not merely to solve particular problems . At the same time , he was discovering the simplifying advantage of u sually reducing equations and cu rves to standard coordinates, x and y , set perpendicular to each other, w hat we call Cartesian coordinate s . Inevitably , one of the problems h e undertook as h e explored the world of analytic geometry was the problem of drawing tangents to curves . He began with Descartes' s procedu re of determining the normal . Qui ckly he discovered Hudde' s rule in Schooten ' s edition , which he was u sing , and as qui ckly he mastered it. His innate urge to syste matize and to generalize asserted itself again . Once he had 12

Math 1, 1 67-20 1 . Further notes relevant to the same general topic continue to p . 2 1 2 .

Never at rest

110

b

d

c

Fi gure 4 . 3 . Newton's meth o d to determine the length of the subnor­ mal de .

grasped the method in individual problem s , he extended it into an equ ati on fo r the length of the subno rmal (which would determine the normal by locating the point at which it cuts the axi s) to any curve (Fig u re 4 . 3) . In finding de = v observe this rule . Multiply each terme of ye equat : by so many units as x hath di mensi ons in yt terme, divide it by x & multi ply it by y for a Nu merator . Againe multiply each terme of the equation by soe many units as y hath di mensi ons in each terme & divide it by -y for a denom : in the valor of v . 1 3

A mong other things , he p roceeded through ordered seri es of equ a­ tions , especially the series y

=

2

a ; x

y

a4 . . . = x3

in whi ch he found a pattern . In su ch patterns , Hudde' s rule began to reveal deeper levels of meaning . Newton did not invariably u se the exponent s of the variable as Hu ddeian multipliers . Hudde ' s rule was a method of simp lifying equ ations with double root s; one chose the sequence of multipliers in order to "blot out " the term m ultiplied by zero . 1 4 Thu s , in one problem , Newton faced an equa­ tion with three term s in x . Using multipli ers one le ss than the exponents of x , he blotted x out and was left with an equation in which v (the su bno rmal plus x ) appeared in only one term and hence cou ld be readily exp ressed . 15 Su ch problems rem ained par1 3 Math 1 , 236 . In hi s editorial note s , Whi tesi de summarizes this in modern notation as follows: the subnormal = y (dy ldx) = - y (fx !fy ) , where fx -and jy are the i mplicit deriva­ tives , dyldx and dxldy . Cf. earlier work on the same problem , p p . 21 6-233 . 1 4 Math 1 , 258 . 15 Math 1 , 2 1 9.

Resolv ing problems by motion

111

ticula r cases; when exponents were u sed as multipliers , allowing the equ ation to be reduced by one degree since the constant was blotted ou t , gene ral patterns began to emerge . Newton did not forget them . O ther new horizons also began to open before Newton . Why not carry Descartes' s procedu re a step farther? Descartes had deter­ mined the norm al to a cu rve at a given point by locating on its axis the center of a circle tangent to the curve at that point. His method depended on the realization that in the circle tangent to the cu rve the radii to the two points where larger circles on the same center cut the curve merge into one . Newton im agined two normals to the curve at neighboring points . He began to u se the symbol o fo r a sm all increment, so that the x coordinates of the two points were x and x + o . 16 Now let the two norm als merge into one . Newton reasoned that the two coinc iden t norm als m u st lie along the radius of the circle tangent to the cu rve and equ al to its cu rvatu re , or "crookednesse , " at that point . Hence an extension of Descartes ' s method one step farther would enable him to locate the center of cu rvatu re , usu ally not on the axis , for a given point and to deter­ mine the curvatu re as the reciprocal of the radiu s of curvature (Fi­ gure 4 . 4) . 1 7 In his usual fashion , he imm ediately set about ordering and generalizing his procedu re to all the conics . 18 He extended the procedu re yet another step by developing a general equ ation for the circle of cu rvatu re from which , via Hudde ' s rule , he could deter­ mine points of greatest and least curvatu re . 19 The investigation of the crookedness of lines carried hi m beyond anything he found in his readings . He achi eved success by December 1 664 , probably be­ fore his twenty-s econd birthday . By the following May , he was 16

17

Math 1, 246-8 . He had employed the notation earlier , in September, when he was working on Descartes 's ovals, the curve of the surface that would refract rays emanating from a point to a second point or focus (Math 1, 557) . Math 1 , 245 -5 1 . For the parabola rx = y 2 , ab = x, cb = y , bd = v, af = c, andfe d . The problem is to determ ine c and d, the coordinates of e, the point where the two norm als, ce and me, intersect . First Newton determ ined the su bnormal v = r/2 , a constant for the parabol a . By si milar triangles , he expressed c and d in terms of x, y, and v, an � arrived at the equ ation x + v + cvly d, or --dy + cv + vy + xy = 0. He removed v by substituting its value, r/2 . He removed x by substituting its value, y 2/r, from the original equ ation . Thu s 2y :� + r2y - 2dry + cr2 0. " N ow tis ev ident yt when t' lines em & ce are coincident yt ce is yt:' radius of a circle wc h hath ye same quantity of crookednesse w c h t' Parabola mca hath at ye point c . Wherefore I suppose ch & nm , 2 of t' motes of ye equ ation 2i� + rry 2.dry + err 0, to be equ all to one another . " App lying Hudde' s rule, he got d (6y 2 + r2)/2r and c 4y 3/r2 • Cf. a later reworking of the same problem which solv ed for c and d in terms of x (Math 1, 256) , and a brilliant extens ion of the analysis in which he clarified the geometric signifi cance of the steps in his procedure in terms of various possible circles cutting the parabola and tangent to it 1 8 Math 1 , 252-9 . (Math 1 , 259-6 1 ) . Math 1 , 265-7 1 . Whiteside points out an error in Newton 's solution; the p rogram itself was not m istaken , however . =

=

=

-

=

19

=

=

Never at rest

112

b

a

c

d

g

0

e

h i

Fi gure 4 . 4 . Newton ' s method to determine the location of the center of cu rvatu re f.

using his metho d of determining crookedness to rectify the evo­ lute s , the loci of centers of curvature , of certain curves . 20 Other aspects of seventeenth-centu ry mathematics also caught hi s interest . I have mentioned his notes on Viete ' s mesolabum , a me­ chanical device by which cube roots and trisections of angles can be constructed . An extension of thi s p assage introduced methods to develop tables of trigono metric functions from propositions on the functions of multiples of angles . 21 In fact , he set out to construct a complete table of sines to every minute . In the Fitzwilliam note­ book he assigned one page to each degree fro m 0 to 90 . Using methods from Viete , he computed a total of nine sines to fifteen places , before the sheer drudgery of the proj ect exhausted his pa­ tience . 22 In the computations , however, and in , othe r notes on methods of extracting roots and on Viete ' s calculation of 7T , New­ ton imbibed the concep t of continued decimals as approxim ations whi ch can never reach the exact value sought but can app roach it as closely as one chooses through the calculation of additional places . Thus he computed sines to fifteen p laces and found devices by whi ch he could compute roots and the value of 1T to fifteen places , or to fifty or five hundred if he wished . 20

22

Specifi cally the semicubical parabola, ky 2 x3, which is the evolute of the simple parab­ ola. The length of a segment of the evolute is equal to the difference in the two radii of 21 Math 1 , 78-88 . cu rvature drawn from its ends (Math 1 , 263-4) . Math 1 , 486- 8 . =

Resolving problems by motion m

113 s

b

q R

n

a

d

e

g

h

c

Figure 4 . 5 . The use of infinitesi mals and infinite series to find areas .

In Walli s ' s Arithmetica infinitorum, Newton also met the meth od of indi visibles . According to his own memorandum written in 1 699 beside his early notes , thi s occu rred in the winter of 1 664-5 . He mastered Walli s ' s method of squ aring (or, as we now s ay , integ rat­ ing) su rfaces under second-deg ree equ ations by comp aring the in­ finitesi m als that exhau st them with the squared infinitesimals of a triangle which are stacked into a pyramid . From the p arabola ry = bx - x 2 ' which he squ ared s atisfactorily , he tried to extend the method to the deceptively similar hyperbola, x 2 = dy + xy . 2 3 Inevi­ tably he failed , and in the process he made the elementary erro r of treating a curve without a term in y 2 as though it were symmetrical about the x axi s . Further not es from Walli s took u p the method o f infinite series with which Walli s had pu shed back some of the limits that confined the method of infinitesimals . If the line ac is divided into an infinite nu mber of equ al part s , ad, de, . . . , from each of which the parallel lines dn , ep, . . . , which increase in p roportion to one of the defined series , are drawn , all the lines m ay be taken fo r the su rface bqnac (Figure 4 . 5) . The proportion of the square acbm to the surface under the cu rve is as the ratio of the index increased by one to 2 3 Math 1 , 93-6 .

114

Never at rest

one . 2 4 If the cu rve i s the p arabola y 2 = rx, the lines inc rease acco rd­ ing to the sub secundanary series (in Walli s ' s terminol ogy) , the in­ dex of which is 1/2 . Hence the area of the squ are is to the area under the cu rve as ( 1/2 + 1 ) / 1 = 3/2 . Still following Walli s , he reco rded Walli s ' s sequ ence of comp ound series , y = a 2 - x2 , y = (a 2 - x 2 ) 2 , y = (a 2 - x 2 ) 3 , , and his interpolation of the series y = 1 (a 2 - x 2 ) 1 2 as a means of app roxirnating TT . The approximation he also reco rded; 4/TT i s g reater than 3/2 X 3/4 X 5/4 X 5/6 X 7/6 x 7/8 x 9/8 X 9/ 1 0 and les s than 3/2 x 3/4 X 5/4 x 5/6 X 7/6 X 7/8 x 9/8 X 9/1 0 X 1 1 /1 0 , a result analogou s to ro ot s and sines com­ puted to however many figu res one chooses . In the winter of 1 664-5 , or someti me near then , Newton ' s urge continually to o rganize his learning led him to draw up a li st of ' 'Problem s . " Initially he put down twelve , one of whi ch he later canceled . He added fu rther problems on several occasions , a s differ­ ent inks show , until he had li sted twenty-two in five di stinct group s . The first g roup included mo st of the problems in analyti c geometry to which he had addressed himself-to find the axes , di­ ameter s , centers , a symptotes , and vertices of lines , to co mpare their crookednes s with that of a circle , to find thei r greatest and least crookednes s , to find the tangents to crooked lines , and so on . The third group looked mostly toward the problems of quadratures (another seventeenth-centu ry term fo r what we now call integra­ ti on) to whi ch Walli s had introduced him-to find su ch lines whose areas , lengths , and centers of gravity may be found , to comp are the area s , lengths and gravities of lines when it can be done , to do the same with the areas , volu mes , and graviti es of solids , and so on . Several of the problem s were frankly mechani cal , and one treated a cu rve a s the path traced by the end of the line y , perpendicular to x , a s the line moves along x . I n both respects , the proble m s looked fo rward toward di stinctive features of his mathemati cs and of his mechanics . In all , the "Problems" laid out mu ch of the program that would occupy Newton during 1 665 . 2 5 •





His first important step beyond his mento rs , which he dated on several occasions to the winter of 1 664- 5 , was hi s extension of Walli s ' s u se of series to evaluate areas . Walli s ' s quadratu res had all been computed within fixed limit s; his series had been series of nu mbers whi ch he evaluated by co mparing them with other series equal to thei r largest term s , and by inducing the li mit toward whi ch the ratio converges from a few instances . As he studied i t , Newton realized that Walli s ' s method was more flexi ble than Wallis himself 24 Math 1 , 96- 8 .

2 5 Math 1 , 453- 5 .

Resolving problems by motion

1 15

had realized . It is not necessary always to compare the area under a curve with the area of the same fixed square . In the case of the ) , for example, any simple power functions (y == x , x 2 , x 3, value of x provides a base line that can be divided into an infini te number of segments , and with the corresponding valu e of y it implicitly defines a rectangle with which the area under the cu rve can be compared . In the case of the curve y == x 3, for any positive value of x the re i s a rectangle with sides x and x 3( == y ) . Newton accepted Wallis ' s demonstration that the area under the curve be­ tween the origin and x equals one-fourth the area of the rectangle , o r x 4/4 , a n expression not confined t o one fixed numerical value but valid fo r every value of x . Since a ratio was implied in the result , it did not have to be expressed explicitly . From Walli s he had also learned that the areas under curves defi ned by polynomial equations , such as y == x + 3x 2 + 2x 3 , can be treated as the sums of the areas under the individual term s . It followed that the quadrature of the polynomial above would be x 2/2 +x 3 + x 4/2 , another po lynomial stated in terms of the variable x and subject to evaluation for any value of x . The quadrature of each term in the polynomial su bsumed Wallis' s infinite series . Hence , by accepting Wallis ' s result s , Newton could extend Wallis' s method to construct series of a wholly different typ e . B y making the upper limit of quad­ rature a variable , he could co mpose series made up of powers of the variable . As Wallis had taught, polynomials are simple; finite series in powers of x exp ress their areas . Could he use the same term s as elements in infinite series to calculate areas that could not be ex­ pressed as finite po1 yno mials? To be specific , could he u s e them to evalu ate Wallis ' s problem , the area under y = (1 - x 2) 1 1 2 , the ci rcle? Newton had learned more than infinite series from Wallis . He had also learned an inductive method . With no more regard for rigor than Walli s had shown , he set off confi dently down the same road . Consider the set of curves y == ( 1 - x 2 ) 0, ( 1 - x 2) 1 1 2 , ( 1 - x 2) 1 , (1 - x 2) 3 1 2 , (Figure 4 . 6) . Squaring the curves with integ ral coeffici ents presents no problems . For any valu e of x (i mplicitly confined by the diagram to the range between zero and one, with the y a xi s , or x = 0, as the other boundary of the segment) , the area under y == ( 1 - x 2) 0 equals x, under y = (1 - x 2) 1 equals x - x 3/3 , under y == (1 - x 2) 2 equals x - 2x 3/3 + x 5/5 , . . . Newton drew up the results in a table as Wallis had done . Opposite each row he placed a power of x with a fractional coefficient; each colu mn co rre­ sponds to the index of the term ( 1 - x 2 ) . The table , manifestly a Pascalian array, gives the coefficients by which the term for that row must be multiplied in the quadrature of the cu rve . •











Never at rest

1 16 a

d

b

e

g h

v

c

q

p

Figure 4 . 6 . Newton' s quadrature of the ci rcle by interpolation , using a fa mily of relate d curves .

[OJ x - x :�/3 x 5/5 -x '/7

x

x x

x

1 0 0 0

[1 J

[ 2J

[3J

1 0 0

1 2 1 0

1 3 3 1 0

[4J

[S J

4 6 4 1 0

1 5 10 10 5 1 0

Now the proble m was to interp olate the colu mns fo r 1 12 , 312 , 5/2 , . . . , exactly Wallis ' s problem , though couched now in differ­ ent terms . The first two rows were evident: 1 , 1 , 1 , . . . and 1 /2 , 3/2 , 5/2 , . . . both filling in the obvious sequences . How could he proceed farther down the t able to the re maining rows? To inte rcal ate the rest [he expl ained l ater] I began to reflect that the denominators 1 , 3, 5, 7, etc. we re in arithmetical p rogression, so that the numerical coefficients of the numerato rs only were still in need of investigation . But in the alternately given areas [the columns under integ ral powers] these were the figu res of powers of the number 1 1 ,

117

Resolv ing prob lems by motion

namely of thes e , 1 1° , 1 1 1 , 1 1 2 , 1 1 3 , 1 1 4 , that is , first 1 ; then 1 , 1 ; thirdly 1 , 2, 1 ; fourthly 1 , 3 , 3 , 1 ; fifthly 1 , 4, 6, 4, 1 , etc . And so I began to inquire how the remaining figures in these series could be derived from the first two given figures , and I found that when the second figure m was given , the res t would be produced by continual multipli cation of the terms of this series , m - 0 1

x

m - 1 2

x

m - 2 3

---

m - 3 4

x --- x

m - 4

etc .

5

For example, let m == 4, and 4 X 1/2 (m - 1 ) , that is 6 will be the third ter m , and 6 x % (m 2) , that is 4 the fourth , and 4 x 1/4 (m - 3) , that is 1 the fifth , and 1 x 1/s(m - 4) , that is 0 the sixth , at which term in this case the series stops . Accordingly , I applied this rule for interposing series among series , and since , for the cir cle , the second term was ( 1/2x3)/3, I put m == 1/2 , and the ter ms arising were: -

1 12

x

1 12 - 1 2

1 -8 '

or

1 16

-

x

1 12 - 3 4

--

1 8

x

or

1 12 - 2 3

or

+

1 16 '

-

5 1 28 '

and so to infinity . Whence I knew that the area of the circular seg­ ment which I wanted was x -

( 1 /2)x 3 3

( 1 /8)x5 5

( 1 / 1 6) x7 7

(5/ 1 28)x9 9

etc .

And by the same reasoning the areas of the remaining cur ves , which were to be inserted , were likewise obtained: as also the area of the hyp erbola and of the other alternate curv es in this series

( l + x2 ) 012 , ( l + x2 ) 1 12 , ( l + x2 ) 212 ,

( l + x2 ) 3 12 etc .

. This was my first entry into thes e consi derati ons .

. . 26

His papers show that the initial insight did not immediately appear with the clarity this statement implies , and that the process by which he arrived at it reflected the influence of Walli s , which the statement obscures . 2 7 Nevertheless , he did perceive the ru le by which the column that co rre sponds to the circle is generated . Whereas integral powers expand into finite series when mu ltiplied out , fractional power s expand into infinite series in as cending 2 6 Newton to Oldenburg , 2 4 Oct. 1 676 (the Ep istola p osterior intended for Leibniz) ; Corres 2, 1 30- 1 . The o riginal Latin is on pp. 1 1 1 - 1 2 . 2 7 Thus hi s original statement of the continued product by which successive coefficients are found repeated a similar se quence that Walli s had used in his quadrature of the ci rcle (and which Newton had copied into his notes) , except that in Newton's sequence the figures in the numerator are all di splaced two positions to the left (Math 1, 1 0 1 , 1 08) .

118

Never at rest

powers of x . There i s no indi cation that Newton was unnerved by the p rospect . He must have anticip ated it. A s he knew from Wal­ li s , these columns inv olved 7T , and 7T can only be exp res sed b y an infinite extended decimal. You shou ld treat su ch p roblem s , he later exp lained , " as if you were resolving ye equ ation in Deci mall numbers either by divi sion or extraction of rootes or Vieta's A na­ lyticall resolution of powers; Thi s operation m ay bee cont inu ed at pleasu re , ye farther the better . & from each terme ariseing from this operation m ay bee dedu ced a p arte of ye valo r of y . " 28 With­ out this additional insight , the bino mial theorem wou ld have been impossible . With the p attern now clear , he filled in all the intercalated co lumns of his initial table , co rre sp onding to the fracti onal powers 1 /2 , 3/2 , 5/2 , . . . of the binomial (1 x2 ) . 2 9 Cou ld a more Walli sian procedu re be i m agined? Perhap s he mi ght wish to check it empirically by computing the value of 7T that his series would yield?3 0 Not at all! He was perfect ly confident . Calculating the well-known valu e of 7T was no challenge. It did not seriou sly occu r to him that the p rinciple of continu ity on whi ch he relied might betray him . L ater he wou ld recognize how fli m sy the foundation was , and he would p lace the binomial expansion on a firmer footing . Fo r the moment , untrou bled by doubts and filled with the thrill of di scovery , he ru shed fo rward to exp loit hi s powerful new tool . Now he cou ld attack the cu rve that had thwarted his effort to s quare it with the method of indi visible s . A s he note d , y (1 + x 2 ) 1 1 2 was virtually the same p roblem as the circle. The rectan­ gular hyperbola , xy == 1 , was more int eresting fo r several reasons . " In ye Hyperbola , " h e had written o n an early p age o f the Waste Bo ok, referring implicitly to the hyperbola xy == 1 , " ye area of it beares ye same respect to its A sy mptote whi ch a logarithme doth its nu mber. "31 The s ame relation applied to p roblems of compound -

==

28 The state ment comes from his tract of O ctober 1 666 , about eighteen m onths after the ori gi nal di scovery (Math 1 , 4 1 3) . 2 9 He noted that with one column given , one could construct the other colum ns by the same rule used in the basic Pascalian array - any number (i n an intercalated column) plus the number above it equals the number (in the intercalated column) to its right . The specific numbers in one of his early tables, in which the denominators under the 1/2 colu mn correspond to the series , whereas the other columns have been reduced, suggest that he filled in the other columns by this procedure (Math 1 , 1 23 ) . 3° For example , i f we set x = 1/2 , the area under the section of the circle minus the area of the ri ght triangle with legs .5 and V. 75 equals 7T/ 1 2 . Indeed Newton developed a series for the segment of a circle (Math 1 , 1 08) , but he did not calculate any valu e . 3 1 Math 1 , 457.

Reso lving problems by motion

1 19

I

h

g

a

b

d

c

e

Figure 4 . 7 . Ne wton' s quadrature of the hyperbola by extrapolation, using a fa mily of related curves .

interest . 32 Since the area under y = 1 /x is infinite when it is com­ puted fro m the y axis , Newton used the identical hyperbola to a different axi s , (defined by the equ ation y = 1 / ( 1 + x) (Figure 4 . 7) . A s before , he u sed a sequence of curves y = ( 1 + x ) 0 , y = ( 1 + x) 1 , , and again he constructed a table of coefficients y = ( 1 + x) 2 , for succe ssive terms . With a different bino mial , the seq uence of powers that correspond to each row is different: x , x 2/2 , x 3/3 , x 4/4 , . . . ; the Pascalian array of coefficients is the s ame . His problem in this case was not to interpolate fractional powers but to extrapolate back to the colu mn that co rre sponds to (1 + x) - 1 • This he could do by the pattern of the array even without the new rule for expanding bino mials . Since the coefficient in the top row must be 1 , the coeffi•



.

3 2 Math 1, 46 1 . About ten years later, when Humph rey Babington was bursar of Trinity, the finances of the college benefited from a table Newton drew up to aid in the renewal of leases (Edleston , p . lvi) .

Never at rest

120

ci ent b elow it must be - 1 to yield the 0 in the colu mn to the right . Hence the extended table appeared as shown here . [- 1] x

x x2 /2 x3/3 x4/4

x

-1

x

-1

x

[ 0]

[1]

[2]

[ 3]

1 0 0 0

1 0 0

1 2 1 0

1 3 3

[4] 4 6 4

The series for a valu� of x - in effect , the logarithm for ( 1 + x) - is expressed by x - x2 12 + x 3 13 - x414 . . . 3 3 Computing TT once more had been no challenge. A new method qui ckly to co mpute logarithms , the new device recently bestowed on seventeenth­ century calculators , was another matter . Filled with enthu si asm , he determined the values of x = 1 1 1 0 and 1 1 1 00 (logs 1 . 1 and 1 . 0 1 ) to 46 places . 34 He was fascinated with his new devi ce to the extent that he later recalculated the same logs and a few m o re to 55 places35 (Figure 4 . 8) . He had not yet graduated B . A . Scarcely a year had passed since he beg an to study m athem ati cs under his own untuto red tutelag e . Somewhat later , possibly i n the su mmer or autumn o f 1 665 , Newton systematized his work on qu adratures in fou r propo sitions . He had not been able to find anything general in quadratu res , he later told Walli s , " till he had reduc' d the Bu siness to the sole Con­ siderati on of Ordinates . " 3 6 Such had been the meaning of his two sequences of curves , y = ( 1 - x 2 r tn and y = ( 1 + xr ' and the diagrams whi ch accomp anied them , whi ch had furnished the keys to unlock the binomial expansion . In the paper on quadratu res , he m ade this approach systematic and general . Let ab = x and bd = y so that the equation of y as a function of x defines the curve addc (Fi gure 4 . 9) . When y i s expressed as some power of x , positive or neg ative , to find the area abd multiply the value of y by x and divide by the number whi ch expresses the power of x (after the multipli ca­ tion) . To illustrate , he went through two sequ ences: y = 1 , x , x 2 , x 3 . . , an d y = alb , (a lb )x , (a lb )x 2 , (alb )x 3 , . The areas are 2 3 exp ressed by x, x 12 , x 13 , x414 , . . . , and (alb) x, (al2b) x 2 , (al3b) x 3 , (al4b)x4 , , regular patterns whi ch did not escape his perception . •









35 Math 1 , 1 34-4 1 . 3 4 Math 1 , 1 1 3- 1 4. 33 Math 1 , 1 1 2 . 3 6 Joseph Raphson , Th e History of Fluxions (London, 1 7 1 5) , p . 3 3 .

Fi gure 4 . 8 .

The calcu lation of several log arith ms to 5 5 places , ca. 1 665 . ( Courtesy of th e Syndi cs of C ambri dge Univer­ sity Library . )

Never at rest

1 22

d

d

b

b

a

Fi gure 4 . 9. Newton's general ap proach to qu adratu res .

He noted that the fo rmu la applies to negative exponent s a s well as to positive ones , and that if the equ ation defining the cu rve ex­ presses y as a polynomial in x , the area is the su m of the areas for each term . Terms with x to the first power in the denominator obviou sly wou ld not fit into the pattern . For the general equ ation y == a 2 /(b + x) he gave the expansion a 2 /b - (a 2 /b 2 )x + (a 2 /b 3 )x 2 (a 2 /b4)x 3 , which can be s qu ared term by term according to the stated fo rmu la . Finally , fo r binomials rai sed to fractional powers , he stated his rule to expand the binomial into an infinite series in a more elegant fo rm than he had devised before . For any binomial of the fo rm (b + xr fn ' the coefficients of successive terms can be found by t he prog ression •



1

1



x

x

m n

x

x

(m

-

2n

n)

x

x

(m

-

3n

2n)

x

x

(m

-

4n

3n)

x

x

(m

-

4n) . . .

5n

Again each term is squ ared (or integrated) by the general fo rmu la . 37 Fifty years later , in his anonymou s Account of the Commercium Ep istolicum, Newton said that his method of quadratu res rested on three rules : first, the quadrature of xm fn n xm +n) fn l(m + n); seco nd , the addition and su btraction o f areas when the equ ation fo r y has two or more terms; and third , the reduction of fractions and radi cals and affected equ ations int o infinite series . 38 A lthou gh he had not yet applied his method to affected equations (that is , equ ations in whi ch x and y appear in the same term su ch that it is i mpossi ble to state one a s an expli cit function of the other) , the three rules had been implicit from the time of his initial steps with infinite series . They became expli cit in the systemati c pap er of 1 665 . In varying deg ree s , ==

37 Math 1 , 1 26-33. 3 8 " An Account of the Book entituled Co mmercium Ep istolicum . . . , " Ph ilosophica l Transac­ tions of the Royal Society, 29 ( 1 7 14- 1 6) , 1 76-7 .

Resolving problems by motion

1 23

all three owed a deb t to Wallis . The first was Newton ' s acceptance of the resu lts of Wallis ' s cu mbersome procedu re with infinite series . In Newton ' s hands , the first rule became more ftex] ble and usable , but he did not initially supply it with a foundation that was inde­ pendent of Wallis . Though later he would do so by treating quadra­ tu re (or integ rati on) as the inverse of differentiation , it was Wallis ' s result which allowed h i m t o arrive a t thi s insi ght . The second rule also came directly from Wallis . The evalu ation of the terms to be added and sub tracted depended , of course, on the first rule . The thi rd rule was Newton ' s own discovery inspired by the suggestion in Wallis ' s method . The thi rd rule was also cri ti cal . By giving the quadrature of vi rtually all the algebraic equ ations then known to mathematicians that could not be squ ared by the first two rules , it made the method general . As was u sually the case with his most signal insp irations , Newton appeared to resent hi s dep endence on Wallis . He was not overly fo rward in acknowledging hi s debt . His operations continu ally displayed patterns . The quadrature of y == xn i s [ 1 /(n + 1 ) ]x n + 1 • Could one not then use thi s pattern to " shewe ye natu re of another crooked line yt may be squ ared? " 39 In the sp ring of 1 665 , Newton began seriously to explore the possibili­ ties to which thi s avenue could lead . By now su rely he had become a stranger to hi s bed . More than one morning Wickins must have discovered a tau t fi gure bent over his incomprehensible symbol s , unaware that a night h a d passed and for that matter unconcerned. He was repai d by the discovery of the fundamental theorem of the calculus . 4 0 Consider a parabola defined by the equ ation rx == y 2 (Figure 4 . 1 0) : ac == r/4 == ad/2 . ap dt == a . Now define a new variable z , whi ch traces the cu rve p o . The subnormal appeared to be involved in the variable z , but by itself it was not the exp ression which , ==

39 Math 1 , 225 . 4 0 D . T . W hiteside, the editor o f the Mathematica l Papers , has sep ar ated a few t heorems from ff. 92 v - 1 1 6 of Add MS 4000 (the m athem atical notebook) and dated them to the sum m er of 1 665 (Math 1, 299-302) , whi le he h as dated most of the passage to autumn 1 664 (Math 1, 22 1 -33) . His notes indicate that the sep aration is based on the writing alone , and he agrees that the d ates are uncer tain . What he places in summer 1 665 comes at and near the beginning of the p assage and is not cons ecutive, though most o f it app ears to have been written (on verso sides and in blank spaces) after the material near it. I cannot find any adequ ate reason to suppose there was a long interruption , and I find his division misleading for two reasons . First, the entire p ass age assumes Newton ' s work on qu adra­ tures . Second , it seems incredible to me that he wou ld have set aside what was obviously an i m portant insight for nine months , as W hiteside's arrangement demands . Finally , in its relian ce on Descartes ' s method of tangents , it appears to me to precede an investigation of tangents e xplicitly dated to May 1 665 whi ch broke free of Des cartes 's method . Moreover , the new app roach to tangents contains strong suggestions that it emerged from the exp anded investig ation of qu adratures .

1 24

Never at rest s

_d

L____ ___ ____

n

a

c

e

g

0

p

Figure 4 . 1 0 .

The funda mental th eore m o f the cal cul us .

upon the application of the formula for quadratu res , would yield the original equation . If it were divided by y , however, then ge/ef = r/2 (rx) 1 1 2 = zla . z = (a /2) (rlx) 1 1 2 , " wch shews ye nature of ye crooked line po . now if dt ap , yn drs t = eoap . for supposeing eo moves uniformely from ap , & rs moves from dt wth motion decreaseing in ye same p roportion yt ye line eo doth sho rten . " [si c]4 1 Quickly he went through a number of other examples , of which the hype rbola y = a 2/x is perhap s the most revealing (Figure 4 . 1 1 ) . ak a -:- kh . oa == x . o d y . ca = v . og = z . By his usual method , Newton found v a 4/x 3 . He defined z by the and then the subnormal oc ( = x - v ) ratio ==

==

==

==

od oc

kh ( = a) og ( = z)

a 2 /x - a --

z = a 3/x 2 , " wch equation conteines ye nature of ye crooked line gh . Now supposeing ye line og always moves over ye same superficies in ye s a me ti me, it wi ll increase in motion from kh in ye s ame proportion yt it decrea seth in lenght & ye line ne wi ll move uni­ formely fro m (mq) , so yt ye space mqen == gokh . "4 2 In the operation of defi ni ng the new curve (gh in this case) , Newton ' s method of tangents via subnormals began to alter itself into differentiati on . 4 1 Math 1 , 299 .

4 2 Math 1 , 228 -9.

Resolving problems by motion

e

d

n

k

a

1 25

c

0

g

h

Figure 4. 1 1 .

The fundamental theorem o f the cal culus .

Since de i s normal to the curve , the ratio oclod used to define z , i s exactly equi valent t o the derivative o f y . The slope o f the parent cu rve in turn controls the motions by which the two areas are described . Two equal lines begin to move and in the first instant describe equal areas . In the case of the parabola , the line ap moves uniformly with a decreasing length while the constant length dt moves at a decreasing rate . For any value of x between a and e , the slope of the parent cu rve determines the length of ap and the rate at which dt moves . In the example with the hyperbola, uniform mo­ tion is attributed to the line mq . The ordinate of the cu rve gh 1noves with a velocity whi ch varies inversely as its length , that is with a velocity inversely as the slope of the p arent cu rve dm . Hence the area under the derivative curve is proportional in each case to the difference in the two corresponding ordinates of the original cu rve . Always systematizing , he ran throu gh the sequence of cu rves y == a 2/x, a 3/x 2 , a 4/x 3 , a 5/x 4 defining z in the same manner and finding a

Never at rest

12 6 n

m -------� a

h

Fi gure 4 . 1 2 .

The kineti c app ro ach to qu a d r a tu r es .

pattern , which he tabulated , fo r the quadratures of the newly de­ fined curves between the ordinates a and b . 43 A whole new app roach to quadratures was opening itself to Newton ' s advance . B efore , with Wallis , he had considered areas as static summati ons of infinitesi mals . Now he was treating them ki­ neti cally , the areas swept out by a moving line . In a sep arate pas­ sage , he had explored this idea in terms , not of areas alone, but of areas in equili brium about an axi s . In the diagram , gc == ac y . The area ghlc generated by the motion of gh , is in equilibrium about the axi s a cf with the area cdej generated by the constant de , whi ch moves along cd ( = x) as it increases (Figure 4 . 1 2) : "ck : ca : : motion of ye point a from c: motion of ye point a from m . ' '44 In this conte�t the problems of tangents and of quadratures suddenly were seen to have an inverse relation to each other . If the calculus had not been ==

43 Math 1, 228-33. 44 Math 1 , 240. Cf. the entire passage, pp. 238-41 . As Whiteside notes , i t appears i n the mathematics notebook shortly 'before the passage I have been discussing, separated only by an earlier set o f notes .

Resolv ing problems by motion

1 27

born , cert ainly it had been conceived . Newton had received his Bachelo r of A rts deg ree , if he had yet recei ved it at all , less than a month before . In mathematics , he had passed fu rther beyond the statu s of student than a single month can concei vably i mply . He had a b sorbed by now what books could teach him . Henceforth he would be an independent investigato r exploring realms never be­ fo re seen by hu man eye . The signifi cance o f the new insight was not lost o n him . H e set out at once to systematize it and to set it in the cont ext of his growing mastery in a paper, which went th rou gh two draft s , called "A Method whereby to squ are those crooked lines wch may bee squared . "45 Whereas initially his insight rested on the p atterns of coeffi cient s and exponents which revealed the inverse nat ure of the two operations separately grounded in procedu res he had adopted from others , he now offered a direct demonstration of it . For the two cu rves z = x3/a (above the x axi s in this case , the roles of y and z being re versed) and y = 3x 2 /a , he showed as before that the area under y (the deri vati ve cu rve) is proportional to the difference be­ tween co rresponding ordinates of z . Draw tangents to the cu rve of z at points f, m , and TT , and fro m the points of tangency and th e intersections of tangent s draw two perpendi cu lar sets of parallel lines (Figure 4 . 1 3) . kl bg

-

nb

bm

{3m

bg

-

a

-

y

When !!{3 and {3m are infinitely s mall , which he explicitly allowed , they are equi valent to the infinitesimal inc rements dx and dz . F rom the proportions , !!{3 bg = {3 m kl , and klvµ bpsg . We can extend the same reasoning to similar rectang les , " so yt t rectangle pa-hd is equ all to any nu mber of su ch-like squ ares inscribed twixt ye line nt/J & ye point d , wch squ ares if they bee infinite in nu mber , they will be equ all to t superficies dnt/Jwgg . " 4 6 Together with a second draft whi ch included another example , thi s was the only de mon­ stration of the fundamental theorem of the calculus that Newton offered . 47 It was confined to the two related cu rves z = x 3 /a and y ·

=

·

45 First draft , Math 1 , 302- 1 3 ; second d raft , with a slightly diffe rent title , Math 1 , 3 1 8-2 1 . The sec ond draft does not contai n a long table of integrals , which makes the fi rst version 4 6 Math 1, 304. much longer. 47 Math 1, 3 1 5 . In a third draft , he radically altered the dem onstration. P roposition 1 gives a gene ral equati on for the subnormal v. When z = a? fn /b, then v = mazxtn/nb . Proposi­ tion 2 states that when r is a constant , if rv zy , the area akhi [ = rz] = abej [ = f ydx ] . P ropositi on 3 states that i f y a? fn /b , the area abej [nl(m + n) ](alb)xtn . " Demon­ s t ra con . For Su pp ose a kh i i s a p a ra l le l o g ra m & equall to nax tn /(nb + mb) . Y1 is na x m +n )fn /(nbr + mbr) = ai = z . & (prop 1 ) az2 x( m +n) fn /brxz = azxm fn /br = v. & (prop ?1) rv = zy; y1 is a?tn lb = y " (Math 1, 3 1 9) . That is , the demonstrati on now rests entirely on t he inverse nature of the two operations . =

=

=

Never at rest

1 28 P

a

k

m

s

g

w

Figure 4 . 1 3 .

The fun damental th eorem of the calcu lu s .

== 3x 2 /a , (and , in the second draft ,

z

== a 3 /x and y == - a 3 /x 2 ) though

the procedu re as su ch was general. For the most part , he preferred to define the integ ral as the inverse of the derivative, resting his assu rance on hi s earlier work with qu ad ratures . Using hi s fresh insight , he included an extended table of integrals and their co rre­ sponding derivatives (to u se ou r rubrics) in the p aper. 4 8 N ow the problem of tangents had t aken on fresh interest , and in May 1 665 Newton retu rned to it . Perhap s hi s exposure to infinitesi­ mals in Walli s ' s method of qu adratures encou rag ed him to apply a simi lar device to Descartes ' s method of tangents . At any rate, he now chose two point s , e and f, on the cu rve aej that were removed from each other by an infinitesim al di stance (Figu re 4. 1 4) . He em­ ployed the symbol o , with which he had experi mented before, fo r 4 8 Later in 1 66 5 , he drew up two more tables of integrals (Math 1 , 348-54, 354-63) .

Resolv ing problems by motion

1 29

/ / / I I

I I I '

g

a

I I

I

I

I

I

b c

d I I

I

I

I \

\

I

I

I

I I

\ '

F igure 4 . 1 4 .

'

'

A new app ro ach t o tangents .

the increment of x . Hence if ab x , ac x + o , and cf == z == f(x + o ) . He ended up with v == o + a/2 + x . When the two points e andf conj oin , " wch will hapen when be == o , vanisheth into nothing , " the terms with o are " blotted out" and (in this example , ax + x 2 == y 2 ) v == x + a /2 . 49 The individual example interested him less than the procedure which , in typical fa shion, he im mediately set about gen­ eralizing . He noticed the fa miliar pattern of binomial coefficients . When y is expressed in seve ral terms containing x , z has the same terms and al so, since z == (x + o ) n , the te rms " multiplied by so many uni ts as x hath dimensions in i terme & againe multiplied by o & div ided by x . "5 0 He had already indicated th at terms with o to higher powers can be dropped since they will in the end be " blo tted out" when o vani she s . " Blotting out" was al so an operation famil­ iar fro m Hudde ' s rule . We may assu me that obviou s analo gies prompted his u se of the term here . He set up general expressions for v in equations in y 2 , y 3 , and y 4 , which he promptly general ized fu rther to what he called " An universall theorem for tangents to crooke d lines . . . " 5 1 With the device of infinitesimal increments in hand , he set about revising Descartes ' s method . The method m==

==

49 Math 1, 273. 5 0 Math 1, 273-4 . That is, if y has the term x\ z has x4 + 4x3o (or 4x 4o/x)) . 5 1 Math 1 , 276 . In fact, he had arrived at the identical expression the previous autumn by a different method (Math 1 , 236; cf. n. 1 3) .

Never at rest

1 30

g

Figure 4 . 1 5 .

a

b

c

d

Finding tangents fro m infinitesi mal in crements .

volved the use of two radii to get an equ ation , z 2 = y 2 + 2yo , whi ch he used to sub stitute fo r z in terms without an o . When the equ ation had a term in z to an odd power , the radi cal comp licated the algebra ho rrendou sly, yielding an equ ation whi ch even a seven­ teenth-century mathematician mu st have faced with a sinking heart . Why s hou ld he bother with radii? The infinitesimal increment o suggested an "infinitely little" triangle ejr (Figu re 4 . 1 5) . The inc re­ ment of y i s voly , and z == y + voly , whi ch introduced a ray of hope into the bleak algebrai c scene. The simplifi cation was pu rchased at the price of a qu estionable de vice, however . These operations cannot be allowed , he remark ed "unlesse infinite littlenesse may bee considered geometrically . " 5 2 Nevertheless, he persevered . If he cou ld di spense with Descartes ' s circle , w h y not dispense with the subnormal as well , and deal with th � tangent di rectly b y finding the subtangent bg ( == t) which deter­ mines the point where the tang ent cuts the axis? After all , his method of finding derivative curves that cou ld be squared had re­ vealed the cru cial signifi cance of the slope whi ch the tangent ex­ pressed directly . By similar triangles , tly == (t + o) lz , z == y + yolt . Where before h e solved fo r v , the subnormal, now h e s olved fo r t and got an equ ation exactly the inverse of the other ex cep t fo r a factor y . 53 He rewrote his earlier conclusions on center s of curva­ ture in term s of his new method of infinitesimals . The entire inves­ tigation had striven consistently , in typical Newtonian style , fo r generality. Now he propo sed a new notation su itable to " a general The oreme whereby the crookedness of any line may be readily determined . " Let X represent the equation that exp resses the natu re of the line set equ al to zero . Then ·X rep resents the s ame terms ordered according to the power of x and multip lied by any ari th­ metical progression . Interestingly , in striving fo r generality he re­ turned to Hudde ' s mu ltip liers as more general than exponents . X· 5 2 Math 1, 282- 3 .

53 Math 1 , 279-80.

Resolving problems by motion

131

represent s the resu lts of the same operation on the equation orde red by y , and · · . x , X- "., and ·X the equation when the operations are repeated a second ti m e . With these symbols he constru cted indi­ gestible general equations , which only eyes dazzled by the ex cite­ ment of discovery could have contemplated with equanimity , to determine the location of the center of cu rvature . s 4 Apparently Newton was uncomfort able with the infinitesimal basis on which his method of tangents now rested . In the fall of 1 665 , he began to extend hi s kinemati c app roach to areas to the generation of curves as well and to treat them as the locu s of a point moving under defined conditions . Years lat er , during the contro­ versy with Lei bniz , Newton said that B arrow ' s lectures m ay have led him to consider the generation of figu res by motion . ss It is necessary to remark that the idea was not unique to B arrow; it was part of the mathematical cultu re of the day . What he now propo sed was a new approach to tangents whi ch cast away once and fo r all the Cartesian scaffolding which had supported him during his mathem ati cal apprenticeship . 1 . If two bodys c , d describe ye strei ght lines ac, bd , i n ye same ti me, (calling ac x , bd y, p moti on of c , q moti on of d) & if I have an e quation exp ressing i relati on of ac x & bd y whose termes =

=

=

=

=

==

are all put equall to nothing . I multiply each terme of yt equati on by so many ti mes py or pix as x hath dimensions in it . & also by soe many ti mes qx or qly as y hath dimensi ons in i t . the sume of these produ cts is an equ ation expresing ye relation of ye motions of c & d . E xample if ax3 + a2yx - y3x + y 4 = 0 . y n 3ap xx + a2py - py 3 + aaqx - 3qyyx + 4qy 3 = 0. 5 6 54

x x X yy + x X· X xx c = x x x.· . x + 2 ·X X· ·X x d =

x

x :.X x

x x x yy + ·X x X- xx x x x · . x - 2 X x x: x - x X- · X x

c and d are the coordinates of the center of curvature . In modern notati on , as Whiteside show s , X x & , the hom ogenized first-order partial derivative with respect to x; X = y � ; : .X x 2 lc .r ; X · . = y 2 �y ; and ·X xy (cy (Math 1 , 289-90) . 55 About 1 7 1 4 , in connection with the controversy with Leibniz , Newton wrote that "its probable that or Barrows Lectures might put me upon considering the generation of figure s by moti on , tho I not now remember it . " (Add MS 3968 . 4 1 , f. 86"' ) . Newton made a si milar com ment about Barrow' s influence to Conti (Prose e poesi, 2, 25) ; see als o Newton to [? ] Sloane , ca. February 1 71 2 (Corres 5, 2 1 3) . In fact , there are not many references to Barrow among Newton's papers . I have found one very fav orable mention of Barrow ' s w ork in a draft of a paper connected with the calculus controversy . He suggested t hat the reader compare hi s letters from the early 1 670s with De a nalysi to find if his method o f fluxions is not there . " And then let hi m compare the differential method of Tangents of or Barrow published 1 670 & that of Mr Leibnits in his Letter of 21 June 1 677 & see if they be not the same , & if Mr Leibnitz hath added any thing more to the differentia of Mr Barrow then what Mr Newton in his Letters above menti oned gave him notice of. " This passage lends support to Barr ow's influence on Newton's mathe matical 5 6 Math 1 , 344 . development (Mint 1 9. 2 , f. 353\') . =

=

=

Never at rest

132 k ...---_

p

h

!-----

a

Fi gure 4 . 1 6 .

b

m

The qu adratrix kbf gene rated by the rotation of the ra­ dius ap and the motion of the line hb .

In our language , which beco mes increasing ly applicable with the progress of Newton ' s investigation , qlp is the deri vative of y with respect to x . A bout the same time , he began to apply analogous considerati ons to the construction of t angents to mechani cal curves . He examined the spi ral (generated by the uniform motion of a point along a line that is rotating uniformly around one end) , the quadratrix (the point of intersecti on between a radius ap that rotates through 90 deg rees and a line hb , perpendicular to the ini ti al posi tion of the radius , ak , that moves uniformly from k to a) (Figure 4 . 1 6) , the ellipse (treated as the point of intersection of the two lines rotating around the two foci such that the increase in one equals the decrease in the other) , and the cycloi d (traced by a point on a rolling wheel) . 5 7 The method had its perils , which Newton ini ti ally failed to avoi d enti rely . 58 Excited by hi s new approach , he im mediately undertook its revision and perfection . In doing so , he located and co rrected hi s own errors . By now , no doubt , Newton was neglecting hi s meals . Working in feveri sh haste , he w as ready by 1 3 November to system atize the 57 The initial paper on this topic probably dates from 30 O ct . 1 665 (Ma th 1, 369-76) . Already on 8 Nov . , Newton revised it (Ma th 1 , 377-80) . 5 8 Math 1 , 37 1 -2 . Whiteside's notes explicate Newton's errors .

Resolving problems by motion

133

new method in a paper entitled " To find ye velocitys of bodys b y t lines they describe. "5 9 When an equ ation i s given expressing the relation of tw o or more lines x , y , and z des cribed in the same time by two or more bodies A, B, and C, we seek the relation of their velocities p, q, and r at any point . Set the equ ation equ al to zero . Mu ltiply each term by pix times the exponent of x in that term , then each by qly times the exponent of y, finally by rlz times the exponent of z . Add all the terms together and set the su m equal to zero . The new equ ation expres ses the relation of p , q, and r. In a lemma, Newton offered what he expli citly labeled a demonstration . A s before, the lack of a generaliz ed symboli sm fo rced him to cou ch it in terms of a specific problem . Lemma. If two bodys A IB move uniformely ye one/other from alb to cld , e!f, glh , & c in ye same ti me . y11 are ye lines aclbd & ce/df & eg!fh &c as their velo citys plq . And thou gh they move not uniformly yet are

ye infinitely little lines wc h each moment they describe as their veloci­ tys are wc h they have while they describe them . As if ye body A wt h ye velocity p des cribe ye infinitely little line o in one moment . In yt moment the body B wt h ye velocity q will des cribe the line oqlp . For p :q : :o :oqlp . Soe yt if ye described lines be x & y in one moment , they wi ll bee x + o and y + oqlp in ye next .

Let the equ ation expres sing the line be rx + x2 - y 2 = 0. Newton su bstituted x + o and y + oqlp fo r x and y, getting the equ ation , rx + ro + x2 + 2ox + o 2 - y 2 - 2qoylp - q 2 o 2 /p 2 = 0. By the original equation , rx + x2 - y 2 = 0 . Hence ro + 2ox + o 2 - 2qoylp - q 2 o 2 /p 2 = 0 . Divide by o: r + 2x + o - 2qylp - oq 2 /p 2 = 0 . Since those terms with o are infinitely smaller than those without o, they can be blotted out . r + 2x - 2qylp 0, or pr + 2px = 2qy . =

Hence may bee observed: Fi rst , yt those termes ever vanish in wc h o is not because they are ye p ropounded equati on [ == 0 ] . Secondly ye remaini ng E quation being di vided by o those termes also vanish in wc h o still remaines because they are infi nitely little . Thirdly yt l still remai ning termes will ever have yt fo rme wc h by ye first preceding rule [the alg orithm gi ven above , wi th which the paper began] they should have . 60

Clearly , Newton thought the concept of generating curves by motion provided a new , more solid foundation fo r the resu lts he had already obtained . Later in the p aper, after he had stated and demonst rated its basic propo sitions , he added , "Hence may bee pronounced those theorem s in Fol . 4 7 [ of the Waste Book, the paper of May] . " 6 1 The new procedure did not produce new results . Rather i t offered a new basis on whi ch t o establish old results mo re firmly . From the idea of motion he derived the term " fluxional, " 59 Math 1 , 382-9.

60 Math 1, 385-6.

6 1 Math 1, 387.

1 34

Never at rest

which became his permanent descriptive word for his method . In fact , he had not escaped from infinitesimal s , however. His defini­ tions of p and q assume a third, invisible variable , time . The concept of absolute time entered inextricably into his mathematics at this time and found here its permanent rationale in his thought . If we represent a moment by t , then p oft and q oqlpt . 6 2 In the intuitive idea of continuous motion or. flow, infinites imals have been replaced by instantaneous velocities . The only device that Newton had avail­ able to express instantaneous velocity , however, was the ratio o f distance traveled t o the infinites imal unit of time i n whi ch i t took place . The idea of motion concealed its infinitesimal ingredient by transfe rring it to the unexpressed variable time . For the unit of time he had only one term , ·· a ' ' moment . ' ' The ' ' infinitely l ittle l ines' ' which bodies describe in each moment are the velo cities w ith which they describe the m . 6 3 His equations treated the increments as alge­ braic entities to be handled like other entities . He divided the equa­ tions b y o and then "blotted out" terms in which o still appeared because they were infinitely l ittle in relation to the others . The con­ cept of contiGuously varying motion , which appears intuitively to overco me the discontinuity of indivisible s , never ceased to appeal to Newto n ' s i magination . Nevertheless , he would in the end seek for another , more rigorous foundation for his calculu s . =

=

The autumn of 1 665 p assed i n incande scent intensity . Then , with the completion of the paper of 1 3 November , the l ight went out , as suddenly and totally as if Newton had extinguished a candle . Six 6 2 W hiteside offers a rather different interpretation of this method i n h i s notes t o the Mathe­ ma tical Papers, in which he develops a concept of what he calls " limit-motions , " and in his "Patterns of Mathematical Though t , " p. 36 1 . See Philip Kitcher, " Fluxions , Limits, and Infinite Littlenesse . A Study of Newton's Presentation of the Calculus , " Isis, 64 ( 1 973) , 33-49; and Tyrone Lai, "Did Newton Renounce Infinitesimals? " Historia mathematica, 2 (1 975) ' 1 27-36 . 6 3 Hence I find Whiteside's phrase " limit-motions, " and his rephrasing of Newton 's process as the modern definition of differentiation when o [dx ] approaches zero , misleading . The language of limits does not appear at all in Newton's papers from this time . o is an infinitesimal increment of x in the moment t . In the final step , terms containing o are "blotted out" because they are " infinitely less" than those without o . Fifty years later , in his anonymous " Account of the Commercium Ep istolicum, " Newton said that he used o for a finite moment of time or of any quantity that flows uniformly; he performed the whole calculation in finite figures without any approxim ation , and when the calculation was completed , he supposed the moment o to decrease in infinitum and to vanish. W hen he was not demonst rating , but only i nvestigating, he suppo sed o to be infinitely little in or der to proceed more swiftly (Ph ilosoph ical Transactions, 29 [ 1 7 1 4- 1 6] , 1 79) . Whatever the accuracy of the description for later periods, only the last statement appears to apply to the early documents . I agree that the concept of motion implies an intuitive approach to a limit, but in 1 665-6 Newton (and the mathematics of the age) did not have the tools with which to express it.

Resolving problems by motion

135

months passed in which , if we can trust the surviving record , he did not tu rn a finger toward mathematics . In May, so mething sti rred hi s interest anew, and he devoted three days to further elaboration of the idea of mo tion in two separate p apers compo sed on 1 4 and 1 6 May . Again the light went out , and once more some­ thing s ti rred him in October, when he drew his thoughts together in a more definitive essay . A thi rd ti me the light went out . It was as though the successfu l re solution of the problems that had been set fo r him had exhau sted his interest in mathematics . The re was no lack of other enthralling enqui ries to command his attention . As far as we can tell , he scarcely looked at mathem atics fo r the following two years . The three p apers of 1 666 all explore the method based on mo­ tion . The second two carry si milar title s , finally phrased " To re­ solve P roblem s b y Motion these following Propositions are suffi­ cient . ' ' 6 4 The first , theoretical , section of the paper is su rpri sing in its clo se app ro xi mation to mechanics . The initial propositi ons choose to speak of bodies in mo tion , rather than points , and they lay down rules fo r the analysis of rnotion into its components . To rectilinear motion Newton added ro tation , and noted that all mo­ tion s in a plane can be redu ced to one of the two or a compound of them . The principle would figure in hi s mechanics . He also defined what was meant by the center of gravity of a body and devoted the final section of the paper to locating centers of gravity of planes and to determining figures that are in equilibrium with each other ab out a given axi s . The solutions to such problems are exercises in mathe­ matics , of course . The content of the problems suggests some of the other interests which, in 1 666 , began to distract his attention from mathematics . Much the greatest part of the paper, however, cannot be mis­ taken for anything but what it i s , the exposition of a sophisticated mathemati cal metho d which Newton would later call hi s metho d of ftuxi ons . With Proposition 6 , bodies are fo rgotten , and the instanta­ neou s motions of a point in reference to two intersecting and mov­ ing lines are sho wn as ve ctors , a geo metrical foundation fo r hi s method of tangents . Proposition 7 states the method , that i s , the algorithm for finding the velocities p and q of bodies A and B when an equation in x and y , which defines the relation between the lines they describe in equal ti mes , i s given . Unlike the p aper of Novem­ ber 1 665 , the tract of October 1 666 uses o to represent a moment of 6 4 14 May: " To resolve these & such like P roblems these following propositions may bee very usefull" (Math 1, 390-2) . 16 May: " To resolve Problems by motion ye 6 follo wing p rop : a re necess ary & sufficient" (in fact , he added a seventh before he was done) . (Math 1 , 392-9) . "October 1 666 . To resolve Problems by Motion these following P ropositions are sufficient" (Math 1, 400-48) .

1 36

Never at rest

ti me, so that op and oq represent the infinitesimal lines described in a mo ment . Interestingly , in com posing what was meant as a gen­ eral method , Newton returned to Huddeian multi pliers . First , he called for multi plicati on by exponents . Then he added that when the equ ation is ordered first by powers of x and then by po wers of y , any arith metic progression , 3 , 2, 1 , 0 , - 1 , - 2 , will do . " O r more Gene rally ye Equ ation may bee multiplyed by ye termes of these progressions (ap + 4hp )Ix . (ap + 3bp )Ix . (ap + 2hp )Ix . (ap + hp )Ix . ap lx . (ap - hp )Ix . (ap - 2hp )Ix , &c . And (aq + 2.b q )ly . (aq + hq)ly . aqly . (a q - hq)ly & c . (a and h signifying any two nu mbers whithe r rationall or irrationall) . " 65 Important les sons learned early were not easily su rrendered even in the face of de monstrations that revealed th e peculiar relevance of the series of exponents . Proposition 8 proceeded to the inverse. Given the same two bodies A and B with velocities p and q and an equation expressing the relation between one of the lines x a nd the ratio qp , find the line y . "Could this ever [i . e . , always] bee done all proble ms whatever might bee resolved" , he stated . " B ut by ye following rules it may bee very often done . " 66 Newton proceeded then to p resent his 1nethod of qu adratu res and offered su ggestions for s implifying equ ati ons into integrable forms . " B u t this eighth Proposition may bee ever thu s resolved mechanichally , " he added , and he introdu ced an exp ansion of a l (h + ex ) by continued division and of (a 2 - x 2 ) 1 1 2 by continued root extraction . 6 7 With such proc edures , which im­ plied generality even though they were confined to specific ex­ amples , he placed the binomial expansion , which had rested here­ tofore on nothing more substantial than interpolations in patterns of exponents , on a new and more sol id foundation . By expansion into infinite series , those problems which cannot be resolved can be approxi mated to whatever accuracy one chooses . Section 1 conclu des with demonstrations of Propositions 7 and 8 , the two crucial elements o f the method . The de monstration of Proposition 7 re peats clo sely the s teps of his de monstration of the previo u s November, differing only in the equ ation it employs . " P rop gth is ye Converse of this 7th Prop . , " he asserted succinctly , "& may bee the refore Analytically de monstrated by it. " 68 Such was Newton ' s final concept of quadrature; the operation of squ aring an area under a cu rve i s the inverse of the ope ration of finding the ratios of velocities . By the latter method , he su ggested , " m ay bee gathered a Catalogue of all those lines wch can bee squ ared. ' ' 6 9 The int rodu c tory , theoretical sec ti on of th e tract com pri ses rou ghly a third of it. The rest is given over to the solu tion of 6 5 Math 1 , 402 . Of course t h e factors p ix and qly were in t h e other series als o . 6 9 Math 1 , 428 . 6 7 Math 1 , 4 1 3 . 6 8 Math 1 , 41 5 . 66 Math 1 , 403 .

Resolving problems by motion

1 37

problems . Newton found tangents to curves . He reworked his ear­ lier investigation of curvature to express it in terms of the new method , and went on to establish points of maxi mum and mini­ mum curvature and points of inflection , where the radius of cu rva­ ture beco mes infinite . As he had done before , he rectified certain curves , which are evolutes of others , by means of hi s solution of curvatu res . Wi th the equation of one curve given , he showed how to find other curves the areas of which have a given relation to that of the given curve . What appears as an arcane problem i s a method to transform equations into integrable fo rm . At the end , as I have mentioned , the p aper deals with centers of gravity and the equilib­ rium of planes about gi ven axes . Along the way , Newton dropped a number of clues that provide insights into the nature of his mathematical genius . He had a hi ghly developed sense of continuity which helped hi m to understand the relations of figures even wi thout demonstrations . When he devel­ oped equations fo r the center curvature in terms of perpendicular coordinates , he re marked that with oblique coordinates the circle of curvature would transform into an ellipse . 7 0 He understood the generality of procedures applied to parti cular cases . Hi s algori thm fo r differentiation was never demonstrated fo r a gene ral case , for which he had no adequate symbolism; rather he illu strated its valid­ ity fo r individual curves and intui ted the general statement frorn the gene rality of his procedu res . So also with hi s gene ral equations for the center of curvature . After finding the center of curvature for one cu rve , he used the result to induce equations valid fo r all cu rves . 7 1 Wallis ' s metho d of induction found in Newton a higher applicati on . Always interested in general solu tions , he was i mpa­ tient of mere co mputation . Thus he started to compute the points of maxi mum and mini mum curvatu re of a conchoid , x 2y 2 - (c 2 x 2 ) (b + x ) 2 == 0 . " The computation is too tedious , " he decided , as he gave up a task which no rational human being could contemplate quietly . 7 2 Taken all in all , the tract of October 1 666 on resolving proble ms by mo tion w as a virtuo so performance that would have left the mathematicians of Europe breathless in ad mi ration , envy, and awe. As it happened , only one other mathematician in Europe , I s aac B arrow , even knew that Newton exi sted , and it is unlikely that in 1 666 Barrow had any inkling of hi s acco mplishment . The fact that he was unknown does not alter the other fact that the young man not yet twenty-four, without benefit of formal instruction, had become the leading mathematician of Europe. And the only one who really mattered, Newton hi mself, understood hi s posi tion 7 0 Math 1, 424.

7 1 Math 1, 42 1 -3.

7 2 Math 1 , 426n.

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Never at rest

cle arly enough . He had studied the acknowledged masters . He knew the limits they could not surpass . He had outstripped the m all , and b y far . Ten years later, Newton wrote to John Collins that there is no cu rve line exprest by any aequation of three term s , thou gh t h e unknown quantities affect one another in it, or ye indices of their dignities be surd quantities . . . but I can in less then half a quarter of an hower tell whether it may be squ ared or what are ye simp lest figures it may be compared w t h , be those figures Conic sections or others . And then by a direct & short way ( I dare say ye shortest ye natu re of ye thing admits of for a gen eral one) I can compare them . . . . This may seem a bold assertion becau se it' s hard to s ay a figu re may or may not be squared or compared w t h another, but i t ' s p lain to me by ye fountain I draw it fro m . . . 7 3

Though the fountain had acqui red a few additional j ets by 1 676 , it remained essentially the instru ment of 1 666 . The instru ment of 1 666 derived in turn from the insights of 1 665 . As I unde rstand hi m , the year 1 665 was cru cial to Newton ' s self­ awarenes s . Al most fro m hi s first dawning of consciou sness , he had experienced hi s difference fro m othe rs . Neither in Grantham nor in C ambridg e had he been able to mingle successfully wi th hi s fellow students . The servants at Wool sthorpe had despi sed hi m . Always his ins atiable lu st to know had set hi m ap art . Now , finally , he had obj ecti ve p roof that hi s quest fo r learning was not a delu sion . In 1 665 , a s he realized the full extent of his achievement in mathemat­ ics , Newton mu st have felt the burden of geni us settle upon hi m , the terrible bu rden which he would have to carry in the isolation it i mposed fo r more than sixty years . From thi s time on , there is li ttle evidence of the futile efforts to ingratiate hi mself with hi s peers that app eared inte rmi ttently during hi s gram mar .... school and under­ graduate days . Accepting as sufficient hi s one close relationship with hi s chamber-fellow Wickins , he abandoned himself, as he had always longed to do , to the i mperious demands of Truth . Though he tu rned aside fro m mathematics at the end of 1 666 , Newton was not done with the method he had created by any means . Signifi cantly , he never tried to publish the tract of October 1 666 . 74 As he retu rned to it intermi ttently in the years ahead , he paid primary attention to improving the foundation of the method; his des criptions of hi s method at the ti me of the controversy with Lei bniz indicate how far he moved in that respect du ring about forty y ears of periodic revi sion . What he wished to be known for 73 Newton to Collins , 8 Nov. 1 676; Corres 2, 1 79- 80 . 74 Nor did he allow manuscript copies of it to circulate nearly as much as copies of later versions of the method did . Some copies did get abroad , apparently toward the end of his life and in connection with the calculus controversy; see Whiteside's discussion, Math 1 , 400n .

Resolving problems by motion

1 39

was not what he wrote in 1 666 , though its inspi ration deri ved directly fro m the early tract . He also extended the method to recal­ citrant p roblems , such as affected equations , that he could not handle in 1 666 , and he tackled other areas of mathematics as well . Neverthele ss , as he said , he never minded mathe matics so intensely again . His great peri od of mathematical creativity had come to a close . For the most part , his future activities as a mathemati cian would draw upon the insi ghts of 1 665 . Years later he told Whi ston " that no old Men (excepting D r . Wallis) love Mathematicks . . " 7 5 True, he was not yet an old man . O ther fascinating subj ects clam­ ored for the attention of the genius in which he was now confident , howeve r. .

75 Whiston, Memoirs, p p . 3 1 5- 1 6 .

5

Anni mirabiles

T O O K I NG back from the beginning of 1 666 , one finds it diffi cu lt L to believe that Newton tou ched anything bu t mathem atics du r­

ing the preceding eight een months . Clearly m athematics did domi­ nate hi s attention during the period , but it did not co�pletely oblit­ erate other int erest s . So metime du ring thi s period he also found time to compose the " Qu aestiones , " in which he digested current natural philosophy as efficiently as he did m athemati cs . Other nat­ ural philosophers were as igno rant of hi s existence as mathemati­ cians were . To those who knew of him , his fellow students in Trinity , he wa s an enig ma . The first blos soms of hi s geniu s flowered i n private, observed silently b y hi s own eyes a lone i n the years 1 664 to 1 666 , his anni mira biles . In addition to m athemati cs and natu ral p hilosophy , the university also m ade certain demands on his ti me and attention . He was scheduled to commence B achelo r of Art s in 1 665, and regulations demanded that he devote the Lent term to the practice of standing in quadragesima . Pictu red in ou r i maginati on , the scene has a su rreal­ i stic qu ality, medieval disputations j uxtaposed with the birth pains of the calcu lus . An investig ation of cu rvature was dated 20 Febru­ ary 1 665 , in the middle of the quadragesimal exercises , and in his variou s account s of his m athemati cal development he a ssigned the binomial expansion to the winter between 1 664 and 1 665 . 1 While Stukeley was a student in Cambridge over thirty years later , he heard t hat when Newton stood fo r his B . A . deg ree , " he was put to second posing , or lost his groats as they term it , whi ch i s look ' d ' upon as di sgraceful. " 2 The sto ry rai ses several problems . The senate had already passed the grace granting his deg ree before the exerci ses were held , and Newton signed fo r thi s deg ree with the other candi­ dates . 3 If the sto ry has any su bst ance, it wou ld have to apply to prior examinations in the colleg e. Nevertheless , as Stukeley re1 Math 1 , 259- 63 . Cf. Newton's memorandum quoted in chapter 3, (Add MS 4000, f. 1 4v) and various statements composed at the time o f the calculus controversy (Add MS 3968 . 5 , f. 2 1 , and Add MS 3968 . 41 , ff. 76, 8 5 , 86v) . 2 Stukeley, p . 53. " GROATS . To save his groats; to come o ff handsomely: at the universities , ni ne groats are deposited in the hands of an academi c office r, by every person standing for a degree : which if the depositor obtains with hono ur , the groats are ret urned to hi m" (Francis Grose , A Classical Dictionary of the Vulgar Tongue, ed. Eric Partridge [London , 1 963] , p. 1 72) . See Allan Ferguson , " A Note on a Passage in S tukeley' s ' Memoi rs of Sir Isaac Newton ' s Li fe' , Ph ilosoph ical Magazine, 7th ser . , 34 ( 1 943) , 7 1 . 3 Cambridge University Library, Subs criptiones II , p . 1 63. "

1 40

Anni mirab iles

141

marked , it does not seem st range since Newton was not mu ch concerned with the standard cu rriculu m . Once mo re the laxity of the uni versity wo rked to hi s advantage. Newton commenced B . A . largely because the university n o longer beli eved i n i t s own cu rri cu­ lum with enou gh conviction to enforce it. The one su rvi ving letter to Newton fro m his mother arrived in Cambridge at thi s time. Becau se the sheet has been to rn , a few words are missing . I sack received your leter and I p erceive you letter from mee with your cloth but none to you your si sters pres ent thai love to you with my motherly lov you and prayers to god for you I your loving mother hanah wollstru p may the 6. 1 6654

Obviously Hannah Ay scough Newton Smith was a barely literate woman; there is no reason to think that there was ever an extensive and revealing co rrespondence between them . Years lat er Stukeley heard in Grantham that Newton had written a nu mber of letters to a friend near Colsterwo rth when he was a student . 5 As far as we know , nothing of them su rvi ves . In the summer of 1 665 , a di saster descended on many p art s of England including Cambridge. It had " p leased Almighty God in his j ust severity, " as Emmanuel College put it , "to vi sit thi s towne of Camb ridge with the plague of pestilence . ' ' 6 A lthou gh Cambridge could not know it and did little in the following years to appease divine se verity, the two-year vi sitati on was the last time God would choose to chasti se them in this m anner . On 1 September, the city government canceled Stu rb ridge Fair and prohibited all public meeting s . On 10 October , the senate of the uni versity di scontinu ed sermons at Great St. M ary ' s and exercises in the public schools . 7 In fact , the colleges had p acked up and di spersed long before. Trinity re co rded a conclusion on 7 A ugust that " all Fellows & Scholars which now go into the Country upon occasion of the Pestilence shall be allowed ye u su all Rates fo r thei r Commons for t space of 4 Corres 1, 2 . 5 Stukeley t o Conduitt , 2 9 Feb . 1 728 ; Keynes M S 1 36 . Among the notes and anecdotes he collected , Conduitt noted that Ralp h Clark, an apotheca ry in Grantham , had seen several letters o f Newton when he was a student at the university. (Keynes MS 1 30. 6 , Book 4) . The note appears to derive from Stukeley's letter. 6 E. S. S huckburgh, Emmanuel Colleg e (London , 1 904) p . 1 1 4. 7 Cha rles Hen ry Cooper, Annals of Cambridge, 5 vol s . (Cambridge, 1 842- 1 908) , 3, 5 1 7 .

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Never at rest

ye month following . " 8 The records of the steward m ake it clear that the college, though ahead of the universi ty , was behind many of its residents who had fled already and therefore collected no allowance for the last month of the summer quarter. For eight months the university was nearly deserted . At Corpus Christi College, only one fello w , two scholars , and a few servants inhabite d the entire structure. They took a "pre serv ative powder" in thei r wine , burned ch arcoal, pitch , and brimstone in the gatehouse , and so mehow m anaged to survive both the plague and their own precautions . 9 To contro l p ani c , the vice-chancellor of the university j oined with the m ayor of the town in issuing bulletins of mortality every fortni ght , sep arating deaths due to the p lague from othe rs . E ach i s sue carried the announcement, " All the Colledges (God be prai sed) are and have continued without any Infection of the Plague . " 1 0 Perhaps the fli ght of thei r residents contributed as much as the me rcy of G od; the inhabitants of the town , who appear at the least equally worthy of mercy but had nowhere to go , fared less well . In the middle of March when no deaths had been reported for six weeks , the univer­ sity invited its fellows and students to return . By June it was evi­ dent that the visitation was not concluded . A second exodus oc­ curred , and the university was able to resume in earnest only in the sp ring of 1 667 . Many o f the s tudents attempted to continue organized study by moving with thei r tutors to some neighboring village. 1 1 Since Newton w as enti rely independent in his studies and had had his independence confi rmed with a recent B . A . , he found no o ccasion to follow Benj amin Pulleyn. He returned instead to W oolsthorpe. He must have left before 7 August in 1 665 because he did not receive the extra allowance granted on that date . 12 H i s accounts show that he returned on 20 March 1 666 . He received the standard ext ra commons in 1 666 and hence probably left for ho me in June . His accounts show again that he returned in 1 667 late in April . 13 8

10 11

12

13

9 Cooper , Annals, 3, 51 8 . Conclusion Book, 1 646- 1 8 1 1 , p . 97 . James Bass Mullinger , The Un iversity of Cambridge, 3 vols . (Camb ridge , 1 873- 1 91 1 L 3, 620. For example, John S harp, who was later an archbishop, removed to Sawston, a few miles south of C amb ridge, where he boarded with John Covell of his own college and other tutors who stayed there with their pupils (J . E . B. Mayor, Cambridge under Queeti Anne [Cam bridge, 1 870] , p . 470) . Steward's Book , 1 665 . New ton missed a payment of £ 1 1 s 8d . He began to draw com­ mons for fiscal year 1 666, which began with the Michaelmas term in 1 66 5 , receiving fu ll commons for the three terms wh en the college was not in session, in all £ 6 6s 8d . In fiscal year 1 667 he received the standard allo wance for the full Michaelmas term and five weeks o f the Lent term , £ 2 1 6s 8d in all , although he him self did not return until considerably after the co llege had officially resu med . His ac counts in the Fitzwilliam notebook list his receipt of£ 1 0 on 22 April 1 66 7 follo wed by his exp enses for the return to Cambridge.

Anni m irabiles

1 43

Much has been m ade of the p lague years in Newton' s life . He menti oned them in his account of his mathematics . The sto ry of the app le , set in the country , implies the stay in Woolsthorpe. In another much-quoted statement written in connection with the cal­ culus controve rsy about fifty years later , Newton mentioned the p lague years again . In the beginning of the year 1 665 I found the Method of ap p roximat­ ing series & the Rule for reducing any dignity of any Bino mial into su ch a series . The same year in May I found the method of Tangents of G regory & Slusius, & in November had the direct method of ftu xions & the next year in January had the Theo ry of Colou rs & in May fo llowing I had entrance into ye inverse metho d of ftuxions . And the same year I began to think of gravity extending to y e orb of the M oon & (having found out how to esti mate the fo rce with wc h [a] globe revolving within a sphere p re sses the su rface of the sphere) fro m Keplers rule of the periodical ti mes of the Planets being in sesquialterate proportion of thei r distances .fro m the center of thei r O rb s , I deduced that the fo rce s wc h keep the Planets in thei r Orbs mu st [be] re cip rocally as the squ ares of thei r distance s fro m the cente rs about wc h they revolve: & thereby compared the force requi­ site to keep the Moon in her O rb with the fo rce of g ravi t y at the su rface of the earth, & found the m answer p retty nearly . All this was in the two p lague years of 1 665- 1 666 . For in those days I wa s in the p ri me of my age for invention & minded Mathe mati ck s & P hiloso­ phy more then at any ti me since . 14

From thi s statement , combined with the other statements about hi s mathematics and the story of the app le , has come the myth of an annus m irabilis as sociated with Wool stho rpe . From one point of view , the lei sure of his fo rced vacation from academic requi rements gave him time to reflect . From another point of view , his return to the maternal bosom provided a crucial psychological stimulu s . 15 Either theory is impossible to prove or to disprove . We may be moderately skeptical of the second as we recall the less than total bliss of hi s year at home in 1 660 . It may be relevant as well that hi s last act before he retu rned to C ambridge was to pry an extra £ 1 0 from the tight fi st of his mother. I n any event , exclusive attenti on to the p lague years and W oolsthorpe disregards the continuity of hi s development . 16 Intellectually , he departed from Cambridge 1 4 Add MS 3968 . 4 1 , f. 85 . 1 5 " If 1 666 is the annus m irabilis, most of it was spent in the protective bosom of his mo ther while the plague raged without . . . " (Frank E. Manuel , A Portrait of Isaac Newton [C ambridge , Mass . , 1 968] , p. 80) . 16 Cf. D . T . Whiteside, " Newton 's Marvellous Year: 1 666 and All That, " Notes and Records of the Royal Society , 21 ( 1 966) , 32-41 . See also ] . W . Herivel , "Newton at Cambridge; the Develo pment of a Genius , " Times Educational Supplement, 9 June 1 96 1 , p. 1 1 94 .

1 44

Never at rest

more than a year before the p lague drove hi m away physically . He took important steps toward the calculus in the spring of 1 665 , before the plague struck , and he wrote two important papers dur­ ing M ay in 1 666 while he was back . Similarly , hi s development as a physicist flowed without break fro m the " Quaestiones quaedam Philosophicae . " If we focus our attention on the record of hi s studies , the p lague and Woolsthorpe fa de in importance in compari­ son to the continuity of his growth . 1 666 was no more mirabilis than 1 665 and 1 664 . The miracle lay in the incredible progra m of study undertaken in private and prosecuted alone by a young man who thereby assi milated the achievement of a century and p laced hi mself at the forefront of European mathemati cs and science. About the beginning of 1 666 , Newton ' s nearly exclusive concentra­ tion on mathemati cs stopped as suddenly as it had begun a year and a half before . 17 As we have seen , he retu rned to it briefly in May to write two papers which carried earlier work forward, and again in October to compose the i mportant tract on fluxions . During the following two years he app arently did not touch mathematics at all . Indeed, as he said, he never minded mathemati cs again with the same exclusive concentration. Newton was not a man of halfhearted pursuits . When he thought on something , he thought on it continually . By thinking continu­ ally on mathematics for a year and a half, he arrived at a new method that allo wed hi m to solve the initial problems , set for him by earlier mathematicians , with which he began . Now othe r inter­ ests represented by the " Quaestiones " could claim his attention . Once they had clai med i t , he thought on them as hard as he had on mathematics . One of these was the science of mechanics . The essay " On vio­ lent Motion" in the " Quaestiones" had introdu ced him to mechan­ ics . There he espoused the doctrine that a force internal to bo dies keeps the m in mo tion . In Descartes ' s Principles and in Galileo ' s Dialogue, he _confronted the radically different conception o f motion that we call today, using language whi ch Newton himself later made co mmon , the principle of inertia . In Descartes he also found two p roblems posed and i mperfectly answered , the mechani cs of i mpact and of ci rcular mo tion . They became the focus of hi s inves­ ti gation . It is incorrect to i mply that Newton turned hi s attention seriously to mechani cs only after hi s initial absorption in mathemat­ ics was b roken . To some extent, the two overlapped. H e dated hi s fi rst i mportant investigation in mechanics 20 January 1 665 , about 1 7 In this paragraph I am relying on the dating of papers in Whiteside' s edition of the Mathematical Papers, 1 and 2.

Anni mirabiles

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the ti me he began to stand in quadragesima . It does appear that his concern with mechanics was intermittent at first and that it con­ tinued into 1 666 more vigorously than his work in mathemati cs . The investigation of January 1 665 , re corded in · the Waste Book, carri ed the ti tle , " Of Reflections , " by which Newton meant im­ pact . 18 A tone of confidenc e not p resent in the " Quaestiones " in­ fused the passage . No longer the questioning student , he began to propound alternative solu tions . To be sure , he based hi s t reatment of impact squarely on Descartes ' s conception of motion . Ax: 1 00 Every thin g doth naturally persevere in y t state in wc h it is u n lesse it bee interrupted by some externall cau se, hen ce . . . [a ] body once moved will always keepe ye same celerity, quantity & determination of its motion . 1 9

Of Descartes ' s law of impact , which completed his discussion of motion , however, Newton did not say a word . He did not even bother to refu te i t . Instead he launched directly into his own analy­ sis of impact . Descartes ' s law of impact had led to conclusions which contradicted common experience . Newton saw that only a different approach could correct its faults . Like other ea rly students of impact , Descartes had been mes merized by the force with which a body s trikes , what he called the "force of a body ' s motion . " 20 The concept appeared obviou s . Everyone had experienced the fo rce of a moving body; eve ryone knew that it increased both with the size of the body and with its speed . Behind the obvious , however, lay unexpected difficulties on which attempts to analyze i mpact had repeatedly suffered shipwreck . Only Christiaan Huygens had un­ derstood like Newton the necessity of a new app roach; he had avoided the difficulties by converting impact into a problem in kinematics . In 1 665 , his work re mained unpublished . Newton' s solu tion differed as radically from Huygens ' s a s from De scartes' s . Instead of eliminating the treacherous concept of force , h e set ou t to modify it into a usable form . As far as i ts cause was concerned , he did not question Descartes . " Force is ye pressure or crouding of one Add MS 4004 ( Waste Book) , ff. 1 0- 1 5 , 38v. E xcept for canceled p assages , the entire investigation has been published in Herivel, pp . 1 32-82 . See J. W. Herivel, " S ur les premieres recherches de Newton en dynamique, " Revue d'histoire des sciences , 15 ( 1 962) , 1 05-40 , and " Newton on Rotating Bodies , " Isis, 53 ( 1 962) , 2 1 2- 1 8 . D . T . Whiteside , "Before the Prin cipia: The Maturing of Newton's Tho ughts on Dyn amical Astrono my , 1 664- 84 , " Journal for the History of Astronomy, 1 (1 970) , 5- 1 9; Ole Knudsen, " Newton ' s Earliest Formulation o f the Laws of Motion , " A ctes d u XF congres internationales d'histoire des sciences, 3, 344-8; Philip E . B . Jo urdain , " The Principles of Mechanics with Newton from 1 666 to 1 679 , from 1 679 to 1 687, " Th e Mon ist, 24 ( 1 9 1 4) , 1 88-224, 5 1 5-64 . 1 9 Herivel, p . 1 53. 2° Cf. Richard S . Westfall , Force in Newton 's Physics (London, 1 97 1 ) , especially chaps. 2, 3 , and 5, and Alan G abbey, "Force and Inertia i n Seventeenth-Century Dynamics, " Studies in Histo ry and Ph ilos ophy of Scien ce, 2 ( 1 97 1 ) , 1 -67 . 18

Never at rest

1 46

b ody upon another . " 2 1 This was a metaphysical proposition , how­ ever . How was one to shape it to the u ses of a quantit ative dynam­ ics? Newton set out toward that goal by -examining the implications of the very conception of motion proposed by Descartes . If a body perseveres in its state unless some exte rnal cau se acts u pon it, there must be a rigorous correlation between the external cause and the chang e it produces . " S oe much fo rce as is requ i red to destroy any quantity of motion in a body soe much is required to generate it; & soe m uch as is required to generate it soe much is alsoe required to de stroy it . " 22 " Tis knowne by ye light of natu re , " he asserted later as he revised hi s initial effort , "yt equall forces shall effect an equall chang e in equ all bodys . . . . For in loosing or to [sic] getting ye same quanty of motion a body suffers ye same qu antity of mu taion in its state , & in ye same body equ all forces will effect a equall chang e . " 2 3 Here was a new defini tion of force in which a body was treated as the passive su bject of external forces impressed upon it instead of the active vehicle of force impinging on others . More than twenty years of patient if intermittent thou ght would in the end elicit hi s whole dyna mics from thi s ini ti al insight . All of the possibilities inherent in the insi ght did not appear im­ mediately to the young man who w as being introduced to the science of mechanics and was grappling with the new conception of motion for the first time . Earlier, in the " Q uaestiones , " he had accepted the idea of a fo rce internal to bodies which keeps them in motion . It is clear to us that the earlier notion was incompatible with the principle of ine rtia and his new conception of force . It was not at first equally clear to Newton . Instead of rejecting the idea of inte rnal force ou trigh t , he attempted to reconcile it with the new concept he was developing . The force wc h ye body (a ) hath to preserve it selfe in its state shall bee equall to the force wc h [pu]t it into y t state; not greater for there can be nothing in ye effect wc h was not in ye cause nor lesse for since ye cause only looseth its force onely by co mmunicating it to its effect there is no reason why it should not be in ye effect wn tis lost in ye cause. 24

The more he wo rked on impact, however , the more the incompat­ ibility of internal force and the principle of inerti a revealed itself. In reworking his initial analy sis , he insisted more and more on changes of motion rather than mo tion itself. " A B ody is saide to have more or les se motion as it is moved wth more or les se fo rce , yt i s as there is more or lesse force requ ired to generate or destroy its whole mo tion . " 2 5 In such statements , the force with which a body 22 Herivel, p . 141 . 2 1 Herivel, p . 1 38 . 23 Herivel, pp . 1 57- 8 . 24 Add MS 4004 , f. 1 2' . N ewton canceled the passage . .

25 Herivel, p . 1 57 .

Anni mirabiles

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Figure 5 . 1 . O blique i mpact of two pe rfectl y elastic bodie s . (From Westfall, Force in Newton 's Physics. )

moves app roached our concept of momentu m . Its quantity was relative to the inerti al frame of reference . Force was measured by the generation (or destruction) of a given quantity of motion in any inertial frame of reference . The climax of the investig ation of i mpact came in Newton' s recogni tion that any two bodies isolated from external influences consti tu te a single sys tem whose common center of gravity moves inertially whether or not they i mping e on each other . H uygens ' s analysis o f i mpact had arrived a t the s ame conclusion b y exploring the implications of the principle of inerti a . 26 Newton arrived there by a dynamic analysis which thereby revealed the i mplications of the principle of inertia to him more fully . E xamining what he called " ye mutuall fo rce in reflected bodys , " he concluded that each one acts equally on the other and produ ces an equal change of motion in it. 2 7 To those who had tried to understand i mpact through the force of a body ' s motion , thi s conclu sion had appeared p atently false except fo r the special case of equal bodies with equal moti ons . Newton appears to have reached it by realizi ng that ev ery impact can be viewed from the special frame of reference of the common center of gravity of the two bodies . Fi rst he demonstrated that two bodies in uniform motion have equal motions in relation to thei r co m mon center of gravity and that thei r common center of gravity is ei ther at rest or in uniform motion , whether the two bodies are in the same plane or in different planes . No w let the two strike each other and rebound (Figure 5 . 1 ) . Fro m the earlier propositions it 26

Cf. Westfall , Force, chap . 4.

27

Herivel, p . 1 59 .

1 48

Never at rest

follows that b and c are in equal motion toward the line kp , the line of motion of thei r common center of gravity . When they meet at q, " so much as (c) p resseth (b ) from ye line kp; so much (b ) presseth (c) fro m it . . . " Therefore , when the two bodies are in e and g some time after the impact , they will have equal mo tions away fro m thei r co mmon center of gravi ty, which remains in uniform motion on the line kp . 2 8 Newton never forgot thi s conclu sion . It contained the first adumb ration of hi s thi rd law of motion , and the conclu sion itself app eared as Corollaries I II and IV to the laws , where it was treated as a necessary consequence of Law 1 (the princip le of ine rtia) and Law 3 . I n 1 665 , the Principia was more than twenty years away . The path tow ard it p roved not to be direct . Impressive as were the strides Newton had m ade , comp lexities remained that tended to reinforce hi s original idea of a fo rce internal to bodies which keeps them in motion . These complexities were associated with the me­ chanics of circular motion , the second p roblem posed by Descartes. Following both Descartes and com mon experience , Newton ag reed that a body in circular motion strives constantly to recede from the center, like a stone pulling on its string as it is whi rled about . 2 9 The endeavor to recede appeared to be a tendency internal to a moving body , the manifestation in circular motion of the internal fo rce which keeps a body in motion . Unlike the internal force of rectilin­ ear motion , it could not be made to disappear by shifting inertial frames of reference . Every indication from Newton' s p apers sug­ gests that the problems of ci rcular motion , together with considera­ tions external to mechanics , soon led him to reject the p rinciple of inertia . Twenty years later, the same problems of circular motion viewed from a new perspective would be decisive in hi s final con­ version to inertia . In seizing on circular motion , Newton again p aralleled the p ath taken earlier by Christiaan Huygens , who had also been dissatisfied with D escartes ' s treatment of it . Although thei r approaches dif­ fered , the discipline that mathemati cs exe rted on both w as evident; by instinct , as it were , they sought to reduce the mechanics of circular motion to quantitative term s . In Newton' s case , the analy­ sis grew out of his treatment of imp act . When he first considered it, he decided that the " whole fo rce" by which a body endeavors to recede from the center in half a revolution is double the fo rce able to generate or destroy its motion . 3 0 That i s , half a revolution is like a perfectly elastic rebound from an immovable obstacle . In both Herive l, pp . 1 68-9 . Newton himself used this illu stration; Herivel, p . 1 47 . See J . W . Heriv el, " Newton 's Discovery of the Law of Centrifugal Force, " Isis, 51 (1 960) , 546-53. 30 Herive l, p . 1 47 . 28

29

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1 49

d

c

Figure 5 . 2 . The force of a body moving impact .

m

a circle derived from

cases the original motion is exactly reversed , which requires a force twice that necessary to generate an equal motion . Later, after fu r­ ther consideration revealed the differences in the two cases , he ad­ ded the words " more y n " before the phra s e " double to the force . . . " Why did he speak of " whole force" ? Undoub tedly be­ cau se he saw a difference between the force from the center exerted continuou sly through an entire revolu tion and the force of an im­ pact which acts in an instant . It did not occur to him to distinguish the two by employing different words for them . To correct his mistaken measu re of the force from the center, Newton employed the model of. i mpact with more sophi stication . He imagined that a squ are circu mscribes the circular path and that a body follows a s qu are path inside the circle rebounding at the four points where the circle touches the outer s quare (Figure 5 . 2) . From the geo metry of the square he was able to compare the force of one impact, in which the component of the body ' s motion perpendicu­ lar to the side it strikes is reversed , to the force of the body ' s motion . "�a : a b: :ab.fa : : force o r p ression o f b upon jg at i t s reflect­ ing: force of b ' s motion . " In one co mplete circuit of fou r reflec­ tions , the total force is to the force of the body ' s motion as 4ab.fa , that is , a s the length o f the p ath to the radius of the circle . Newton

1 50

Never at rest

was mathematician enough to see without pausing to demonstrate that the s a me relation of path to radius would hold if the number of sides and i mpacts were doubled, then doubled again and again and agam . And soe if body we re reflected by the sides of an e qu ilaterall circum_­ scri bed polygon of an infinite number of sides (i . e. by ye circle it selfe) ye force of all ye refle ctions are to ye force of ye bod ys motion as all tho se sides (i . e . ye perimeter) to ye radiu s . 3 1 For one complete revolution the total force F is to the body ' s mo­ tion mv as 2TTrlr . Or F = 2TTmv . To see the s ignificance of the result ,

convert the total force of the body in one revolution to the " force by wc h it endeavours - fro m ye center" at each instant by dividing each s ide of the equation by the time of one revolution , 2TTrlv . The division y ields f = mv2 Ir, the formula we still use in the mechanics of circula r motion . 3 2 The formula for a body ' s endeavor to recede from the center, for which Huygens coined the name "centrifugal force, " gave Newton the means to attack a problem that he found in Galileo' s Dia logu e . It was an effort to answer one argument against the Copernican sys­ tem by s howing that the earth' s ro tation does not fling bodies into the air because the force of gravity , measured by the acceleration of falling bo dies , is greater than the centrifugal force arising from the ro tation . Newton ' s solution, chaotically recorded on a piece of parchment, the front side of which had been used by his mother for a lease, was closely associated with the investigations of mechanics in the Waste Book . In his first attempt, he used his first mistaken idea of the total force in half a revolution; and when he retu rned for a second attempt, he employed his later, co rrect form ula . All he needed in addition to his new formula were the size of the earth and the accele ration of gravity . For both , he used the figures he found with Galileo ' s solution of the problem in the Salus bury translation of the Dia logue, which appeared in 1 665 . He arrived at the conclu­ sion " y t ye force of ye Earth from its center is to ye force of G ravity as one to 1 44 or there abouts . " But why accept Galileo ' s figure for the acceleration of gravity? He suddenly realized that his measure of centrifugal force opened a further possibility; he could use it to measure g indirectly via a conical pendulum . The measurement , with a conical pendulum 8 1 inches long inclined at an angle of 45 degree s , revealed that a body starting from rest falls 200 inches in a second , a figure very close to the one we accept but roughly twice as large as the one he had found in Galileo ' s Dia logue . Hence he 3 1 Herive l, pp . 1 29-30 . 3 2 The use of mass here is anachronistic. Newton did not clearly define mass until early in 1 685 . I have been willing to insert it since he did realize the significance of the body 's size o r quantity .

A nn i mirabiles

151

A B ..----=--- D

c

/ E

Figure 5 . 3 . The force of a body moving in a ci rcle derived from its de viation from the tangent.

retu rned to his calculation and doubled the ratio of gravity to cen­ trifugal force . 3 3 Somewhat later, in a p aper which appears to date from the years immediately following his undergradu ate career, Newton retu rned to the same problem s . On thi s occasion , he calculated centrifu g al conatus more elegantly by utilizing the geometry of the circle instead of imp act. When a body moves in uniform circular motion , ti n1e is proportional to length of arc . Since a body will move in a strai ght line if it is not constrained to move in a ci rcle , Newton set the centrifugal tendency fo r the motion AD equal to BD, the distance the t angent diverges from the ci rcle at D (Figure 5 . 3) . When the arc AD i s "very small , " Newton could then apply the known ratio of BD to AD to calculate the instantaneou s force , and with the fo rce he could calculate the distance it would i mpel a body in a strai ght line , starting fro m rest , during the time of one revolution . He employed Galileo ' s conclu sion that distances traversed in uniformly accele rated moti on from rest vary as the squ ares of the ti mes , im_ ­ plicitly interp reting Galileo ' s kinematics in dynamic term s . His answer , that in the time of one revolu tion the centrifugal fo rce would move a body through a distance equal to 27T 2r , is m athemati­ cally equivalent to the earlier formula derived fro m impact . 3 4 Again he compared centrifu g al fo rce at the earth ' s surface to gravity; and since he did not round off his more accu rate measu rernent of g , he arrived this time at a slightly higher rati o , 1 : 350 . 33 Add MS 3958 . 2 , f. 45 . The manuscrip t is published and an alyzed both in He rivcl, pp . 1 8391 , and in Co rres 3, 46-54. 34 In the Waste Book, he conclu ded that in the time of uniform motion through one radian (or a distance along the circle equal to its radius) , the centrifugal force would generate a motion equal to the body ' s motion . Hence the distance traversed by an equal body s tarting from rest would be 1/2r. To compare the resu lts of the preseP.t paper , divide the distance, 21T2r, by the squared ratio of times , 47T 2·

1 52

Never at rest

So far he had come before . He was ready now to take a fu rther step . He compared the " endeavour of the Moon to recede from the centre of the Earth " with the force of gravity at the surfa ce of the earth . He found that g ravity is somewhat more than 4, 000 times as great . He also su bstituted Kepler ' s third law (that the cubes of the mean radii of the planets vary as the squares of their periods) into his fo rmu la fo r centrifugal fo rce: " the endeavou rs of re ceding from the Sun [he di scovered] will be reciprocally as the square s of the di stances from the Sun . "35 Here was the inverse-squ are relation re sting squ arely on Kepler ' s third law and the mechani cs of circular motion. To catch the full signifi cance of the statement one must reflect on the earlier ratio of gravity to the moon 's tendency to recede from the earth . He had found a ratio of abou t 4 , 000: 1 . Since he was u sing 60 earth radii as the moon ' s distance , the exact ratio according to the inverse squ are relation should have been 3 , 600: 1 . It is difficult to believe that this paper was not what Newton refer red to when he said that he found the compari son of the fo rce holding the moon in its orbit to gravity to " answer pretty nearly . " 3 6 Newton attempted to su mmarize his early work in mech ani cs in a paper from about the same time which he called "The lawes of Motion . "3 7 It is not an unfamiliar title to students of Newton or to student s of physi cs . The laws of motion as Newton understood them in the 1 660s differed sharply , however , from the laws of motion he pronounced in the Princip ia . In the early pap er he was concerned primarily with impact . The paper attempted to arrive at a general solution of the impact of two bodies with any arbitrary linear and rotational motions. The inclu sion of the latter marked an advance over the treatment of imp act in the Waste Book . In "The lawes of Motion , " Newton p ropo sed a definition of quantity of circular motion-the product of a body ' s bu lk and the velocity of a point on what he called its "Equ ator of circulation . " 38 He asserted 35 A dd MS 3958 . 5 , f. 87 . The paper is also printed and anal yzed in Heri vel, pp. 1 95-7 , in Co rres 1 , 297-303 , and in A . R . Hall, "Newton on the Calculation of Central Forces, " Annals of Science, 13 (1 957) , 62-7 1 . See also Leon Rosenfeld, "Newton and the Law of Gravitation , " A rch ive fo r Histo ry of Exact Sciences, 2 ( 1 965) , 365-86; and Ole Knudsen , " A Note o n Newton's Concept o f Force, " Centa urus , 9 ( 1 963-4) , 266-7 1 . 6 3 In 1 694, about twenty years before he wrote this famous line , Newton showed David Gregory a paper w ritten before 1 669 that contained "all the foundations o f his philoso­ phy . . . : namely the gravity of the Moon to the Earth, and of the planets to the Sun: and in fact all this even then is subjected to calculation" (Co rres 1, 30 1) . It is also difficult to believe that t he paper in question here was not the one Gregory saw . 3 7 Add M S 3958 . 5 , ff 8 1 -3v (the first versi on , called " The laws of Reflection") , 85-6v . The final versi on is printed in Heri vel, p p . 208- 1 8 . 3 8 Herivel has shown that Newton's "radius o f ci rculation" (which describes the equator .of ci rculati on) is equivalent to what mechanics now calls the radius of g yratio n (k) . He has also shown that Newton's quantity of circular motion (mkw) differs from what we call angul ar momentum (mk2w ) by a factor k.

Anni mirab iles

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the principle of the conservation of angular momentu m fo r the fi rst time in the hi story of mechanics: " Every body keep es the same reall quantity of circular motion and velocity so long as ti s not opposed by other bodys . " 3 9 Using what he called the " s mallnesse of resis­ tance" to change of motion , he developed a formula whi ch par­ celled out the tot al change of velocity of the two point s that come into contact among fou r factors , the progressive motions of the two bodies and their rotational motions . 4 0 For all its sophisti cation in the treatment of impact , " The lawes of Motion " contained internal problems whi ch help to illuminate the limits of Newton ' s early achievement in mechanic s . On the one hand , the paper employed the principle of inertia . On the other hand , it began with an assertion of absolute sp ace whi ch conflicted with the relativity of motion , a co rollary of inerti a , but which conforn1ed to the earlier conception of motion he had held . 4 1 Prob­ a bly fa ct o rs ex ternal to m e chanics influen ced Newt on most strongly , but within mechani cs the questions a ssociated with rota­ tion were the leading advocates of absolute sp ace, as his u se of the adjective " reall" with " quantity of circular motion" implies . He noted that a perfectly balanced rotating body will mai ntain the absolute inclination of its axi s , but if it is not perfectly balanced the "endeavou rs fro m the axis" will cau se it to wobble with a motion that is not merely relative but absolute. 4 2 The most basi c issue in his mechani cs had not been settled after all . Newton had not adopted the new conception of motion with finality . Within three years hi s hesitation would end fo r the time being with a p assionate rejection of relativism and inertia and a return to the concept of an internal fo rce generating absolute motion in absolute space . The fa ct that a paper entitled " The lawes o f Motion " could have as its goal a general solution of impact indi cates how far from completed Newton's dynamics was in the 1 660s . H i s inv estigation of impact had led to a concept of fo rce fully sui ted to imp act alone . By good fortune, the concept of force worked equ ally well with either conception of motion , as long as rectilinear mot ions alone were involved . When he applied it to proble1n s other than impact , such as circular motion , he immediately confronted a m biguities that he had only begun to learn to control. Newton had ca ught sight of the dynamics that would crown and complete Gali leo ' s kinemati cs; h e had scarcely begun t o examine i t s depth s . Above all , h e had only begun t o peer into the mysteries o f ci rcu­ lar motion . The man who would coin the phrase " centrip etal fo rce " had found the quantitative fo rmula fo r its illu sory mirror image , 39 Heri vel, 41 Heri vel,

p. p.

21 1 . 208 .

4 0 Heri vel, 4 2 Heri vel,

2 1 2- 1 3 . 21 1 .

pp . p.

Never at rest

1 54

the endeavor to recede from the center, or centrifugal force . Since he believ ed that bodies continue to move in a straight line unless so mething diverts th�m , it followed that any body moving in a circle must be constraine d to do so by so mething else . It follo wed also that Newton in the 1 660s , like De scartes before him , treated circular moti on as a s tate of equilibri u m between oppo sing fo rces . Thus , in his original calculation of thei r quantity , the ball in circular motion presses against the confining cylindrical shell , and the shell confine s the ball to its ci rcular p ath by pressing back . The circular motion manifests the equilibrium of the two fo rces . As long as he conceived of ci rcular motion in such term s , the internal endeavor of a body to recede from the center would appear to him as an argu­ ment against the principle of inertia . What then is one to make o f the story o f the apple? I t is too well attested to be thrown out of court . In Conduitt' s version, one of fou r independent ones, it ran as follows : In the year 1 666 he retire d again fro m Cambridge . . . to hi s mother in Lincolnshi re & whilst he was mu sing in a garden it came into hi s thought that the power of gravity (wc h brought an apple from the tree to the ground) was not limited to a certain distance from the earth but that this power must extend much farthe r than was usually thought . Why not as hi gh as the moon said he to him self & if so that must influence her motion & perhap s retain her in her orbit , where­ upon he fell a calcul ating what would be the effect of that suppo si­ tion but being ab sent from book s & taking the common esti mate in use among Geographers & our seamen before Norwood had mea­ sured the earth , that 60 English miles were contained in one degree of latitude on the su rface of the Earth his computation did not ag ree with his theory & inclined him then to ente rtain a notion that to­ gether with the force of gravity the re might be a mixture of that fo r ce wc h the moon w o u l d have if it was ca rried along in a 43 v o rtex . .

.

43 Keynes MS 1 30 . 4 , pp. 1 0- 1 2 . The account in the finished memorandum on Newton that he sent to Fontenelle differed slightly . Conduitt' s account is in many respects almost identical to that in the DeMoivre memorandum on Newton Uoseph Halle S chaffner Collection , University of Chicago Library, MS 1 075 . 7) except for the apple , which does not app ear in DeMoivre 's version . However, DeMoivre does place him in a garden . Perhaps Conduitt heard that detail from his wife , for it was his wife whom Voltaire cited as his au thority for the apple (Lettres ph ilosoph iqu es, Lettre XV ; Oeu vres de Voltaire, ed . M . Beuchot, 7 2 vols . [P aris , 1 829-40] , 37, 1 98-9) . William Stukeley also wrote that Newton told him the story on 1 5 April 1 726 (Stukeley , p. 20) . Robert Greene rep�ated substantially the same story in 1 727 on the authority of Martin Folkes , vice-president of the Royal Society while Newton was president, and later president himself ( The Principles of the Ph ilosophy of the Expansive and Contractive Fo rces [C ambridge, 1 727], p. 972) . It is surely significant that all of these accounts surfaced at about the same time and appear to date from the final year of Newton 's life. In my opinion, the date does not seriously compro­ mise accep tance of the incident itself, a con crete event that would have readily been recalled . On the other hand, Newton ' s age does not generate much confidence in his

Anni mirabiles

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Small wonder that such an anecdote , redolent of the Judaeo­ Chri sti an association of the app le with knowledge, continues to be repeated . Together with the m yth of the a nnus mira bilis and with Newton ' s memorandu m that said he found the calculation to answer pretty nearly , it has contribu ted to the noti on that univer­ sal gravitation appeared to Newton in a flash of insi ght in 1 666 and that he carried the Princip ia ab out with him essenti ally com­ plete for twenty years until H alley p ried it loose and g ave it to the world . Put in thi s form , the story does not survive comparison with the reco rd of his early work in mechanic s . The story vulga­ rizes universal gravitation by treating it as a bright idea . A bright idea cannot shape a scientific tradition . Lagrange did not call New­ ton the most fortunate man in history becau se he had a flash of insight . Universal gravitation did not yield to Newton at his first effort . He hesitated and floundered , baffled for the moment by overwhelming complexi tie s , which were great enough in mechan­ ics alone and were multiplied sevenfold by the total context . What after all was in the p aper that revealed the inverse-square relation? Certainly not the idea of unive rsal gravitation . The paper spoke only of tendencies to recede , and to Newton the mechanical phi­ losopher an attraction at a distance was inadmissible in any case. It was no accident that he placed impact at the center of the laws of motion . Revealingly , Conduitt (or DeMoivre) brought in the vor­ tex . 44 Nevertheless , Newton must have had so mething in mind when he compared the moon ' s centrifugal fo rce with gravity , and there is ev ery reason to believe that the fall of an apple gave ri se to it. Though he did not name the force explicitly , something had to press back on the moon if it remained in orbit . Something had to press back on the planets . Moreover, Newton rememb ered both the occasion and the calculation , so that fifty and more years late r they seemed to constitute an i mportant event in hi s development . Some idea floated at the border of hi s consciousness , not yet fully formulated , not perfectly focused , but solid enough not to disap­ pear. He was a young man . He had ti me to think on it as matters of great moment requi re . recollection of the conclusions he drew at that time, especially when his own papers tell a rather different story. See Jean Pelseneer , " La Pomme de Newton , " Ciel et te rre, 53 ( 1 937) , 1 90-3; and Douglas McKie and G . R. de Beer, " Newton ' s Apple, " No tes and Records of the Royal Society, 9 (1 95 1 -2) , 46-54, 333-5 . 44 So did W illiam Whiston in what m ay or may not be an independent account (Memoirs, p . 37) . Henry Pemberton, whose independence from DeMoivre and Whiston again can not be demonstrated , did not mention vortices specifically but did say that Newton long suspected the operation of some other cause beyond gravity ( View of Newton 's Ph ilosophy, [London , 1 728] preface) . Newton 's notes on the endpapers of his copy of V incent Wing, Astronomia Britannica (London , 1 669) , now in the Trinity College L ibrary, support the story that he believed some action of the vortex on the moon upset the perfect inverse­ square relation in its case.

1 56

Never a t rest

Motion and 1nechanics were not the only topics in natu ral philoso­ phy that com manded Newton ' s interest . A s i mportant in his own eyes were what he later called the " celebrated P haenomena of Col­ ou rs . "45 The phenomena of colors had beco me a celebrated topic in optics fo r at kast two reason s . What We call ch romatic aberration appeared in every tele scopic observation , coloring the i m ages and confu sing thei r focu s . In contrast , colors sharply focu sed the differ­ ent stances of the A ri stotelian and the mechanical philosophies of nature. It is scarcely su rp ri sing that colo rs were among the " Qu aes­ tiones qu aedam Philosophicae' ' compiled by the young mechanical philosopher in C ambridge. 4 6 He had found the i ssue in Descartes , in Boyle's Experiments and Considerations Touching Colors ( 1 664) , and in Hooke ' s Micrographia ( 1 665) . Dissatisfied with their explanations of colo rs , as his notes show , he turned his hand to hi s o wn . Like the chronology o f h i s early mechanics , the chronology o f his optical research is con fu sed . In the memo randu m on the plague years , he stated bluntly that he " had the Theory of C olou rs" in Janu ary 1 666 . He had said su b stantially the same thing not long after the discovery when he sent his first paper to the Royal Society early in 1 672: "in the beginning of the Year 1 666 (at which time I applyed my self to the grinding of Optick glasses of ot her figu res than Sp herica l, ) I procured me a Triangular glass-P risme, to try therewith the celebrated Pha enomena of Colours . "47 He proceeded then to descri be hi s theo ry before adding that the plague int er­ vened , dri ving him from Cambridg e , and it was more than two 45 Newton to Oldenburg , 6 Feb . 1 672; Corres 1 , 92. 4 6 Johannes Lohne , "Newton's 'Proof of the Sine Law and His Mathematical Principles of Colours , " A rchive for History of Exact Sciences, 1 , ( 1 96 1 ) , 389-405 , has suggested that we tra ce Newton's interest in colors to his grammar-school days and the reci pes for mixing colors that he copied from Bate's book . Frank Manuel, in contrast, sees all of Newton's op­ ti cal experi mentation as the manifestation of his unc ons ci ous desi re for inti m ate vi sual ex­ change with his mother (Po rtrait, p p . 78-9) . My presentati on is based on the premise that his wo rk in optics is best understood in terms of the current context of natu ral phil osophy. 4 7 Newton to Oldenbu rg , 6 Feb . 1 672; Co rres 1 , 92 . Descartes had made the prism the primary instrument of experimentation with col ors (Les Meteores, Dis cours 8; Oeu vres de Descartes, ed . Charles Adam and Paul Tannery , 1 2 vol s . [P ari s , 1 897- 1 9 1 0] , 6, 330) . In his Experiments and Considerations Tou ching Colou rs ( 1 664) , which was a maj or source of information to Newton, Boyle said the prism was the " usefullest instrument" men had ever em ployed for the investigation of colors ( The Wo rks of the Honourable Robert Boyle, ed . Thomas Bi rch , new ed. , 6 vol s . [London , 1 772] , 1, 738) . Newton's notes on the grinding of lenses are found in his mathematical notebook, A dd MS 4000 , ff. 26-33v . They have no date , but they follow his annotations on Wallis 's Arithmetica infinito rum, whi ch Newton's own memorandum written in 1 699 placed in the winter between 1 664 and 1 665 . See Lloyd W . Tayl or, "Newton 's Prism in the British Museum , " Nature, 138 ( 1 936) , 585; Rudolf Laem mel, "Die Prismen von 1 665 , " in Miszellen um Isaac Newton (Zii rich , 1 953) , p p . 2-4; I . Bernard Cohe n , " I prismi del Newton e i pri s m i dell' Algarotti , " Atti della Fondazione Giorgio Ranch i, 12 ( 1 957) , 2 1 3- 23; and Vasco Ronchi , "I ' prismi del Newton , ' del Museo Civico di Treviso , " ibid. , 12 (1 957) , 224-40 .

Anni mirab iles

1 57

years before he retu rned to the question . 4 8 There are at least th ree problem s in this narrative: Newton ' s location , the height of the sun , and the procu rement of the pri s m . If the beginning of 1 666 meant Janu ary whi ch the later memo randum explicitly asserted , Newton was not then in C ambridge as he implied . Moreover , in order to proje ct a spectrum more or less parallel to the floor , as he had to do if he was to view it on a wall , he needed a sun about 40 deg rees above the horizon , whereas the sun does not climb above 20 deg rees in Janu ary either in Cambridg e or in Woolsth orpe. Fi­ nally , in Janu ary he would have had to get the prism in Grantham , and prisms seem an unlikely item of co mmerce fo r a small market town . Newton' s conversation with Conduitt on 3 1 August 1 726 indi cated that they were hard enough to co me by in Cambridg e. I n Au gust 1 665 s r I . w h o w a s then n o t 24 bou ght a t Sturbri dge fair a prism to try some exp eriments u pon Des cartes ' s book of co lours & when he came home he made a hole in his shutter & darkened the room & pu t his prism bet ween that & the wall found instead of a ) with strai t sides & ci rcular ends &c . wc h circle the light made ( convinced him i m medi ately that Des cartes was wrong & he then found out his own Hyp othesis of colours thou he cou ld not demon­ strate it for want of another prism for wc h he staid till next Stu r­ bridg e fair & then proved what he had before found out. 49

Conduitt's ch ronology does agree with the height of the sun . Unfortunately , Newton had left Cambridge before the truncated Stu rbridge F air of 1 665 took place, and there was none at all in 1 666 . Conduitt seem s to have realized as mu ch , fo r he altered his original 1 665 to 1 663 . Perhaps we should not place much weight on the recollections of a very old man abou t events that occu rred sixty years earlier. The indi cated interruption does correspond to the paper of 1 672 , however, although the length of the int erruption differs in the two account s . s o No resolution of the di screpancies is entirely satisfactory . Newton ' s recollection of Sturbridge Fair may have been mist aken; there was also an annu al Midsu mmer F air, 4 8 Newton to Oldenburg , 6 Feb . 1 672; Co rres 1 , 95-6 . 49 Keynes MS 1 30 . 1 0 , ff. 2v -3. 50 Newton's accounts show that he purchased three prisms someti me after 12 Feb . 1 668 (Fitzwilliam noteb ook) . The chronology of his discovery is further muddied by a sentence in a letter to Oldenburg , 1 0 Jan . 1 676, replying to an intimati on by Gascoines that the directi ons for his basic p rism atic projection that he sent to Linus were different from those he pri nted. Newton denied they differed in any way from those he had followed " above these seven years" (Corres 1, 4 1 0) . This seems o place the crucial experi mentati on in 1 668, although the word " above" leaves it amt guous . His ini tial lecture said the p roj ec­ tion of the prism was the experiment that led ; 1 m to develop his theory of colors . If his later assertions about 1 666 have any meaning, the statement in 1 670 places the defini tive investi gati on , whi ch gave the projection central importance , at that time (Add MS 4002, p . 3) . .

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which managed to escape both plagues . If he pu rchased a prism there in 1 665 , he cou ld have taken it home with him and perfo rmed there basic , though perhaps cru de , experiment s connected with hi s initi al insight . His more elaborate experiments to confi rm it de­ m anded an assistant . 5 1 Perhaps he cou ld have dragooned a servant at Woolsthorpe, but one thinks instead of Cambridge and Wickins . The extra prism probably dem anded C ambridge, where the . M id­ su mmer F air of 1 666 mi ght have fu rnished it , and the sun was then high enou gh in the heavens . No more than two or three years hing es on the chronology in any case . He had the theory fully elaborated before Janu ary 1 670 when he lectured on it. Three years are not su ffici ent cau se to drown the ex citement of di scovery in a sink of eru dition . Suffi ce it then to say that a p assage on colo rs , in which Newton sharply qu estioned cu rrent theo ries , was among the earliest entries in the " Qu aestiones . " 5 2 Later , prob ably in 1 665 , he retu rned to colors again , utilizing empty p ages at the end of his original set of headings . 53 There is a good likelihood that the theo ry of colors in Robert Hooke ' s Micrograph ia ( 1 665) stimulated him . His immedi ate negative reaction to Hooke ' s account inaugu rated forty years of antipathy between two incomp atible men . As in mechanics , he was not cont ent any longer simp ly to qu estion . An alternative theo ry sp rang to mind . Hooke propo sed that " Blue is an impression on the Retina of an obli que and confu s 'd pulse of light , whose weakest 5 1 Few of us measure time by the movement of s hadows as Newton did. He would not have failed to realize what hi s fi rst attempt would have taught hi m in any case - that his spectrum moved rapidly across the wall. On a wall twenty-t wo feet away, it moved more than an inch every minute . Wi thout an assistant and some practice , no meas urements could have been taken and no experiments involving more than one prism set up . One needs to recall as well that on any one day he had less than two hours whe n the sun was properly located; and in Cambridge the sun does not shine every day. Lord Adrian has combined the measurements of rooms in Trinity with t he height o f the sun t o support the story that Newton lived at this time in a room on the north side of the great court (and hence with a window facing the noonday sun) between the master's lodge and t he chapel. He suggests t hat the experiments were performed there in early April, when the sun was at the ri ght elevation. The room in question is wide enough to all ow the twenty-two feet bet ween prism and spectrum that Newton mentions (The Lord Adrian , " Newton's Rooms in Trinity, " Notes and Records of the Royal Society, 18 ( 1 963) , 1 7-24) . As it hap­ pens , the dimensi ons of hi s room at Woolsthorpe, which measures exactly twen ty-two feet from the s hutter of the wi ndow facing south to the wall opp osite (if we ignore t he partiti on that now walls off a corne r but doe s not appea r to have been there in the seventeenth century) , fit with Newton's description of the experiment and make it poss_i­ ble that he performed it there . (I owe this information to M r . K . A . Baird of Colster­ worth, who recently measured the room, and to Professor Gale C hristianson, who com­ municated Baird's letter to me . ) 5 2 Add M S 3996, f 1 05v . 53 Add MS 3996, ff. 1 22-24v , 1 33- 5 . See A . R . Hall , " Sir Isaac Newton's Note-book , 1 66 1 65 , " Cambn'dge Historical Jo urnal, 9 ( 1 948) , 239-50.

Ann i m irabiles

1 59

part precedes , and whose strongest fellows . " Red i s the i mpres sion of "an oblique and confu s ' d pulse" of reversed order . 54 On the first page of his new set of notes , Newton contradicted the two funda­ mental as sertions of Hooke ' s theory , that light cons ists of pulses and that colo rs arise fro m confu sed i mpressions . " The more uni­ formely the globuli move ye optick nerves ye more bodys seme to be coloured red yellow blue greene &c but ye more variously t hey move them the more bodys appeare white black or Greys . " 55 If Hooke was the immediate target of the assertion , much more than Hooke ' s theory was involved. Like other mechanical philosophers , Hooke had merely provided a mechani sm for the exi sting theory of apparent colors , pheno mena such as the rainbo w and the colo red fring es seen through telescopes and pri s ms . The theory haci been fatally easy to mechanize . It had employed a scale of colors , which was also a scale of strength , running from brilliant red , considered to be pure white light with the least admixture of da rkness , to du ll blu e , the last step before black , which was the co mplete extinction of light by da rkness . The propo sal of the freshly minted Bachelor of Arts i mplied a co mpletely different relation of light and color. White light, ordi­ nary sunlight , i s a confu sed mixture . Individu al components of the mixture , which he cons idered to be corpu scles rather than pulses , cause sens ations of individual colo rs when they are sepa­ rated fro m the mixture and fall on the retina alone . Al ready he had drawn a picture of an eye looking through a pri s m at the colored fringes along a border between black and white . Fro m the two sides o f the border two rays were shown following different paths through the prism as they were refracted at differ­ ent angles and emerged along the same line incident on the eye (Figure 5 . 4) . " 1 Note yt slo wly moved rays are refracted more then swift ones. " 5 6 Though he would modify details as he clari­ fied hi s understanding of its i mplications , point one above con­ tains the insight on which Newton built hi s work in op tics. The insight fundamental to his dyna mics had happened less than a year befo re , both of them less than two years after he tu rned seriou sly to natu ral philosophy . He had a keen eye for the cri tical point at which to seize a proble m . He started with a n idea rather than a n observation . U nder the diagram of the prism and the eye was a table in whi ch he tried to 54 Robert Hooke , Micrographia (London , 1 665) , p . 64 . 55 Add MS 3 996, f. 1 22 . Newton ' s notes on Micrographia exp licitly included Hooke's theory of colors . The word " confused " did not appear in them. Newton did question Hooke's theory, which was inextricably boun d up with his con cep tion of light . If light is a p ulse, why does it not deflect fro m a straight line as sound does? How can the weaker pulse 5 6 Add MS 3996 , f. 1 22'·. move as fast as the stronger? (Ha lls, p. 403) .

1 60

Never at rest

Figure 5 . 4 . Newton ' s first suggestion of the diffe rential refrangi bility of light . (C ourtesy of the Syndics of Cambridge Univer­ sity Library . )

reason out the colors that would appear along the borde rs between various co mbinations other than black and white . Quickly the co m­ plexi ties of mentally sorting out slowly and swiftly moving rays refle cted from variou s bands along the borde r became more con­ fu sed than Hooke' s pulses. As suddenly as the original insight , a simplifying exp eri ment presented itself. An experi mental orienta­ tion pervaded the " Qu aestiones , " but until the investigation of col­ ors exp erimentation had been i mplicit rather than explicit, ques­ tions put but not experiments performed . At thi s point the period of adolescence was fulfilled , and Newton the experi mental scienti st reached maturity . That ye rays wc h make blew are refracted more yn ye rays wc h make , red appeares from this exp erimnt . I f one hafe [one end] of ye thred abc be blew & ye other red & a shade or black body be put behind it yn lookei ng on ye thred through a prism one halfe of ye thred shall appeare higher yn ye other . & not both in one direct line, by reason of unequall refractions in ye 2 differing colours . 57

The idea had been provi sionally confi rmed . Newton never fo rgot thi s experi ment; he continued to cite it as one of the basi c supp orts of his theory of color . A t the ti me he made the exp eriment , ho wever, the theory hardly exi sted . It was only a pro mi sing idea su pported by a single experi57 Add MS 3996, f. 1 22v.

Anni mirabiles

161

ment . Its im plications are obviou s to us who profit from three hundred years of digesting them . Newton had to grope his way forward as he denied a tradition two thousand years old which seemed to embody the dictates of co mmon sense . The concept of slow and swift rays was formulated within the context of a me­ chanical philo sophy , and it carried the usual connotations of weak and strong . It inclined hi m to think in term s of a two-color syste m , blue and red . I t inclined h i m as well t o im agine mechanisms b y which the "ela stick power" o f a body ' s particles determined how much of the motion of a ray was reflected ; " then yt body may be lighter or darke r colored according as ye elastick virtue of that bodys parts is more or lesse . " 5 8 Such ideas returned to the assu mp­ tion that colo rs arise from the mo dification of light , against which his central insight was directed . Perhaps it was here that his consideration of devices to grind ell iptical and hyperbolic lenses , which he mentioned in his paper of 1 672 , intervened . The background of the investigation was Des­ cartes' s announce ment of the sine law of refraction in his Dioptrique . A s use of the telescope had spread in the early seventeenth century , experience had shown that spherical lenses d o not refract parallel rays , such as tho se fro m celestial bodies , to a perfect focus . In La Dioptrique, De scartes had shown that hyperbolic and ellipti cal lenses would do so , given the sine law of refraction . Grinding them was anothe r question . Spherical su rfaces present no problem . Since they are symmetrical in all directions , constant tu rning and shifting of a lens adj u st the lens and the form against which it is being ground to each othe r so that a spherical surface i s bound to re sult . On the other hand, the grinding of an elliptical or hyperbolic surface is complicated indeed , exactly the proble m to challenge the mo del­ builder from Grantha m , who was now equipped with a thorough knowledg e of the conics . He sketched out several devices by which to produce them (Figure 5 . 5) . 5 9 And as he did so , he possibly reflected on the meaning of his earlier experi ment with the prism and the red and blue thread . Descartes ' s de mons tration had as­ su med the ho mogeneity of light . What if he did succeed in grinding elliptical and hyperbolic lenses? He still would not obtain a perfect focus because light is not homogeneous; the blue rays are refracted more than the red . At thi s mo ment, it appears to me , New ton began to realize the significance of his experi ment and of the idea behind i t . He stopped working on nonspherical lenses and never 5 8 Add MS 3996 , f. 1 23 . 59 Add MS 4000 , ff. 26-33v. A ll would have been subject t o the basic difficu lty , which does not arise w ith spherical lenses; the mutual grinding of lens and tool would gradually make the tool inexact.

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Figure 5 . 5 . Devices to produce hyperbolas . In drawing A , the appa ra­ tus turns ab out the axis a b . The arm cet, which is not in the same plane with ab, touches the hyperbola ch m . By moving the instrument to and fro , the ope ra tor can wear the plate into a h yperbola . O r , as Newton suggested , he can file the plate and use the instrument to check its accu­ ra c y . In drawing B, on the left , the strai ght edge cet o f a chisel, held at any angle so a s not to be in the same plane as the axis ab of the stock , cuts a conca ve hype rboloi d in the stock mounted on a lath e . In drawing B, on the ri g ht below , a si mil a r device is used to check the accuracy of the hyperbo loid whi ch the g rinding stone p, beneath the stock , shape s . On the right a b o ve , the concave hyperbo­ loid is used as a pattern to g rind a convex one , B, the tool to be used to g rind a concave hyperbolic lens . (C ourtesy of the Syndics of C a mbridge University Libra ry. )

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Anni mira biles

1 63

retu rned to them . 60 Later he showed that chromatic abe rration in­ troduces much la rger errors in lenses than spherical aberration . I n­ stead of lenses , he tu rned his attention to an experimental investiga­ tion of the heterogeneity of light and i ts role in the production of colors . I assume that this investigation came in 1 666 and that he referred to it both in 1 672 and in 1 726 . Only with it did he "have the Theory of Colours" in any legitimate sense of the p hrase . Newton recorded the investi gation as the essay " Of C olours " in a new notebook in which he extended seve ral of the top ics of the " Quaes tiones . " 61 Wi th his purpose now more clea rly focu sed, he marshaled known p heno mena of colors , which he had found in Boyle and Hooke , that exhibited the analysis of light into i ts com­ ponents . Thu s thin leaves of gold appear yellow from one side in reflected light but blue in trans mitted light from the other; with a solu tion of lignum neph riticum (nephritic wood , infusions of which were u sed medicinally at the time) the colors are the reverse . In both cases , the trans mi ssion of some rays and reflection of others 60

61

Newton did consider the pos sibility of an achrom atic lens composed of t wo different media on the t wo folios of the notebook immediately fo llowing the pass age on grinding lenses . These two folios were later torn out by an unknown person and are now in Shirburn C astle. Cf. Zev Bechler, " 'A less agreeable m atter' : The Dis agreeable Case of Newton and Achromatic Refr action , " British Journal for the History of Science , 8 ( 1 975) , 1 0 1 -26. Add MS 3975 , p p . 1 -20. Later he added another note on p . 22 . "Of Colours " is not a set of notes compiled from his reading and exp erimentation ; it is a consistent exposition of his theory of colors . With its general paucity of corrections , it may be the dis tillation of other papers now lost. It has no date, and it may have been entered into the notebook later t han 1 666 . If so, it should be seen as a polished statement of Newton 's investig ation of 1 666 . The assertion in the paper of 1 672 especially app ears decisive in placing the investigation in 1 666 . Other headings reminiscent of the " Quaestion es " that he set down initially in the new notebook included " Of Cold , & Heate, " and " Rarity, Density , Elasticity, Compression &c. " On N ewton ' s theory of colors , see A . R . Hall , "Further Optical Experiments of Isaac Newton , " Annals of Science, 1 1 ( 1 95 5) , 27-43; Michael Rob erts and E . R. Thomas , Newton and the Origin of Colours . A Study of One of the Earliest Examples of Scientific Method (London , 1 934) ; ] . A . Lohne , "Isaac N ewton : The Rise of a Scientist, 1 66 1 - 1 67 1 , " Notes and Records of the R oya l Society, 20 ( 1 965) , 1 25-39; J . A. Lohne and Bernard Sticker , Newtons Theorie der Prismenfarben (Munich, 1 969) ; Thom as S . K uhn , " N ewton's Optical Papers , ' ' in Cohen, pp . 27-45 ; Zev Bechler, " N ewton 's Search for a Mechanistic Model of Colour Dispersion: A Suggested Interpretation , " Archive for History of Exact Sciences, 1 1 ( 1 973) , 1 -37; Alan E . Shapiro, " The Evolving Structure of Newton's Theory of White Light and Color: 1 670- 1 704, " forthcoming in Isis; R. Furth, " N ewton ' s ' Opticks ' and Quantum Theory, " The Lodestone, 4 1 ( 1 948-9) , 23-30; Lord Rayleigh , "N ewton as an E xperim enter , " Nature, 1 50 ( 1 942) , 706-9; Leon Rosenfeld , " La Thforie des couleurs de N ewton et ses adversaires , " Isis, 9 ( 1 926) , 44-65 ; Vasco Ronchi, " L ' Ottica del Kep lero e quella di N ewton, " Atti de lla Fondazione Giorgio Ranchi, 1 1 ( 1 956) , 1 89-202 ; Augusto Guzzo , " O ttica e atomistica newtoniane , " Filosofia, 5 ( 1 954) , 383-4 1 9; and Maurizio Mamiani , Isaac Newton filosofo della natu ra . Le lezioni giovanili di ottica e la genesi del metodo newtoniano (Florence , 1 976) .

1 64

Never at rest

analyzes white light into its components . Newton was convinced that all solid bodies would behav e like gold if pieces thin enou gh could be obtained , and that the solu tion of lign um nephriticu m would appear blue from all sides if it were m ade thick enough so that no light could pass through . 62 If Newton turned available observations to the adv antag e of hi s theory , however, he relied primarily on his own experiments with the pri s m . Guided b y his ingenuity , the pri s m became a n instrument o f precision with which h e dis sected light into its ele mentary components . No o ther inves tigation of the seven­ teenth centu ry better reveals the power of experimental enquiry ani­ mated by a powerful imagination and controlled by rigorous logic . A mong the authors Newton had read, both Boyle and Hooke had employed variations of Descartes ' s proj ection of a prismatic spectrum to examine colors . Newton saw that he could bend the same experiment to test hi s own theory by i mposing carefu lly pre­ scribed conditions on i t . If in fact light is heterogeneou s and differ­ ent rays are refracted at different angles , a round beam should be proj ected by a prism into an elongated spectrum (Figure 5 . 6) . It would require enough distance to spread out , however . Rays are ideal entities; in actual experimentation a p hysical beam had to be used , and one big enough to give visible effects . If the sc reen were placed close to the prism as it had been in earlier experi ments , the expected elong ation would not appear. Descartes had received his spectru m on a screen only a few inches from the pris m . Hooke , who employed a deep beaker filled with water instead of a pri s m , had about two feet between refraction and screen . Boyle app arently used the floor, and hence had a distance of perhaps four feet . New­ ton proj ected hi s sp ectru m on a wall twenty-two feet aw ay . Where earlier investig ators saw a spot of light colored at its two edges , Newton saw a spectrum five times as long as it was wide . 63 His exposition in 1 672 contrived to su ggest an element of chance and surprise when he saw it; it was as accidental as the observati on of a barometer by Pascal 's brother-in-law on the summit of the Puy de Dome . Newton had cons tructed his experi ment to test what he wanted to test . Had the spectrum not been elongated , his p ro mising idea would have been refuted at its second step , and he could not have elaborated it into a theory . If he had not refuted himself, he was far fro m having proved any­ thing , as he knew very well . What he proposed was a radical re62 63

Add MS 3975 , p . 1 . Such was his report in 1 672 . In his first recorded perform ance of the experiment , he set down the dimensions of the spectrum as 2 1 /3 inches by 7 or 8 inches (Add MS 3975 , p . 2) . I t is true that Hooke's diagram o f a similar experiment indicated dispersion , but coloration was the object of Hooke's attention .

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Figure 5 . 6 . Ne wton ' s drawin g of the prismatic spectrum with t he bo ard xy used fu rthe r to narrow the pencil of light that enters the room through the hole k . (Courtesy of the S yn­ dics of Cambridge Unive rsity Library . )

1 66

Never at rest

ordering of the relati on of light and color. Whereas the received op inion considered white light as simple and colors as modifications of it, Newton asserted th at the light which provokes the sensations of indivi dual colors i s simple and white light a comple x mixtu re . Long-established views are not easily su rrendered . Possible obj ec­ tions were many; they would need to be answered . Perhaps the elongation was cau sed in some manner by limiting the beam inci­ dent on the prism - "the termination with shadow or darknes s , " which had long played a role in accounts of color. 64 When he placed the prism outside the shutter so th at unconfined light fell on it , however, it proj ected an identical spectru m th rough the hole . Perhap s the globuli of light acquired a spinning moti on and fol­ lowed a curved trajectory . When he followed the spectrum with a movable screen , he found that it sp read rectilinearly . Perhaps it resu lted from imperfections in the glass . He proj ected the beam th rough various p arts of th e prism , always with the same result when the prism was set at the same angle . When a second prism was available , he placed it immediately behind the first with its vertex reversed to m ake it refract in the opposite direction . If the elongation were due to i mperfections in the prism , a second prism (presu mably with similar imperfections) ought to increase them . Instead it resto red a round beam . He was not dealing with contin­ gent phenomena; if two opposite refractions neu tralized each other , they must be governed by law . The most important obj ection of all still rem ained . B ecau se the sun fill s a visual angle of 3 1 minutes , the beam incident on the prism was not composed of parallel rays . By the sine law of refrac­ tion , rays incident at different angles are refracted at different ang les . Could the elongated spectrum be an unexpected product of the sine law? Newton employed variou s devices to g et a beam composed of rays more nearly p arallel. B y u sing a boa rd with a hole in it p laced some twelve feet from the shutter he narrowed th e incident beam to about 6 minutes; all the dimensions of the spec­ tru m decreased by an equal amount , fu rther accentu ating the elon­ gation . 6 5 Later he even managed to perform the experi ment with the light from Venu s , which he collected with a lens . When he placed the prism between the lens and its focus , he found the focus " drawn out into a long splendid line . . . " 66 Such exp eriments displayed hi s imagination and hi s dexterity , but Newton knew well 64 6s 66

I am citing from his account in 1 672 (Newton to Oldenburg , 6 Feb . 1 672; Corres 1 , 92-4) . Add MS 3975 , p . 2 . Newton t o Oldenburg , 1 3 A p r . 1 672; Corres 1, 1 37 . For the m o s t part , little esthetic response crept into Newton 's accounts of his experiments with colors , but it can hardly be igno red in this case . He cited the experiment with Venus in his Lectiones opticae (Add MS 4002, p p . 1 5- 1 6) .

Anni mirabiles

1 67

that only a theoretical demonstration could finally meet the obj ec­ tion . It was not a difficult exercise for a mathematician of his ac­ co mplish ment . When the central ray of an incident pencil of ho mo­ geneou s light contained within an ang le of 31 minutes i s refracted equally at both faces of a prism , it emerges as a pencil contained within an angle of 31 minutes . 6 7 As it happens , equal refraction at each face is also the condition for mini mal refraction , so that N ew­ ton had only to turn the pri s m until the spectru m reached its low est position on the wall to obtain i t . His first reco rded proj ection of a spectru m noted that the rays were equally refracted by both faces of the pri s m . 68 Along with the distance of proj ection , equal refrac tion at the two faces was a planned condition of the initial experiment . So far was the elongated spectru m fro m a chance observa tion . Although the mathematical de monstration supp lied necessary rig­ or to the evidence of the spectru m , Newton ultimately found a fu rther experi ment that seemed to confirm it no less strongly . Like the final theory , the experiment did not appear in one burst of inspiration . It evolved through several stages until , as he realized its fo rce , he called it the exp erim entum crucis . 6 9 In its initial form , it was ill-de fined and hardly compelling . He simply held a second prism in the spreading spectru m five or six yards fro m the firs t . The blue rays suffered a greater refraction than the red . In neither case did the second refraction produce fu rther colo ration; blu e re mained blue and red remained red . 7 0 In 1 666 he went no farther . Only later did he realize the de monstrative potential he could gain by re fining the experi ment . Once Newton had fu lly seized the concept of analysis , he was able readily to generate other experiments to illu strate it. He could analyze sunlight into its co mponents by tilting a prism to the criti­ cal angle , where blu e rays , which are the mo st refrangible , began to be re flected from the second face while red rays were still trans mit­ ted th rough it (Figure 5 . 7) . He obtained an analogous separation with a thin film of air trapped between two prisms bound to­ gether . 7 1 Realizi ng that it was necessary to demonstrate that he could reconstitute white , he cast the spectra of three prisms onto each other so that they overlapped without coinciding . In the center, where all the colors fell , the co mbined spectru m was white . He attached a paper to the face of a prism with several slits pa rallel to the edges . On a sc reen held near the pri s m a line of color ap­ peared for each slit. As he moved the screen away , the center of the spectru m became whi te , but the full spectru m appeared ag ain , 68 A dd MS 3975 , p 2 . 6 7 Add MS 4002 , p p . 4 7 . The phrase appeared only in 1 672 (Newton t o Oldenburg , 6 Feb . 1 672 ; Corres 1 , 94) . Actually it was Hooke' s phrase, an em broidery on B acon 's instantia cru cis . 0 7 Add MS 3975 , p . 1 2 . 7 1 Add MS 3975 , p . 6 . -

69

.

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a

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J'

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Figure 5 . 7 . Three means of anal yzing light with p risms. At the top , a beam falls at a on the base eJ of the p ri s m . At the critical

Anni mirabiles

1 69

withou t fu rthe r experimental manipulation , as he moved the screen still farther. 7 2 Some seven years later, after the publication of hi s first paper in the Ph ilosophical Tra nsactions, Newton repli ed to a criti que by Huygens with a methodological ho mily . It see m s to me that M . Hugens takes an i mproper way of exa mining the natu re of colours whilst he proceeds upon comp ounding those that a re alrea dy compounded, as he doth in the former part of his letter . Perhap s he would sooner satisfy him self by resolving light into colours as far as may be done by Art , and then by exa mining the p roperties of those colours apart , and afterwa rds by trying the effects of reconj oyning two or more or all of those, & lastly by sepa rating the m again to exa min w 1 changes that reconj unction had wrought in them . This will prove a tedious & difficult task to do it as it ought to be done but I co uld not be satisfied till I had gone thro ugh it . 73

No doubt Huygens , the doyen of European science , did not relish the le cture fro m an unknown professor in Cambridge . Nevertheless it is a reasonable description of Newton' s procedure as he unrav eled the implications of hi s central idea. When we follow his p apers , we realize what hard work it was to cast off the dic tates of com mon sense embodied in a tradition two thousand years old . His ulti mate theo ry did not reveal itself all at once . He started with a new idea, a different solution to the prob­ lem of colors . I t took Newton several years , however, to fully realize that in pu rsuing his idea , that colo rs appear from the analysis of light instead of its modification , he had demonstrated a new and 7 2 Add MS 3975 , pp. 1 2- 1 3 . 7 3 Newton t o Oldenbu rg, 3 April 1 673; Corres 1 , 264 .

angl e , the more refran gi ble rays at the purple end of the spe ctrum are refle cted to c, while the less re frangible rays at the re d end pass throu gh the base and are refracted to b . In the middle , an e ye at 0 (close to the prism) views the sky as it is re fle cte d from the bottom of the pris m , ef. Be cau se of the diffe ren t angles between the eye and the two halves of the base , qf and ep , light is re fle cted to the eye from qf, but less from ep , where much of it is trans­ mitted through the base and re fracted toward g . Newton re po rted that the sky seen at ep l ooke d shade d . The parti­ ti on between the t wo hal ves pq appears blue because the red rays , being less refran gi ble , are transmitted and re­ fracted to ward g . At the botto m , the thin film of ai r be­ tween two p risms effects si milar analyse s . In all three dia­ gra m s , Newton did not atte mpt to indicate any refraction at the other faces of the p rism , since only the face ef effects the analysi s . (Courtesy of the Syndics of Cam bri dge Uni­ versity Li brary . )

1 70

Never at rest

unexpected property of light , its heterogeneity . " Of Colou rs" ap­ pears to embody the re sults of his investigation of 1 666 as he first began seriou sly to explore his insight . Without stating them explic­ itly , i t assu mes many of the proposi tions of his ulti mate theory . Rays of light differ both in the colors they exhibit and in thei r refrangibility . Though he fre quently referred to blue rays and red rays , Newton never understood the rays th em selves to be colored . A mechanical philosopher would never have entertained that no­ tion . Rays exhibit colors . " I call those blew or red rays &c w c h make ye Phanto me of such colours . " 74 Becau se they differ in re­ frangibility as well , the heterogeneous rays in sunlight are separated by the prism . Allied to the concept of analys is was the further proposition that indiv idu al rays are immutable in their properties . Newton had not co mprehended this at the ti me of his original idea when he had toyed with mechanisms of reflection which attempted , in effect, to combine mo dification with analys i s . By 1 666 , im mu­ tability had beco me clear, and experi ments seemed to support it . He painted a red patch and a blu e patch on a sheet of paper . When he ca st "Prismaticall blew" on them , they both app eared blu e , though one w a s much fainter than the other . When h e cast "Pris­ maticall red" on them , both appeared red; the other one was now fainter . 75 Nothing he did, neither refraction nor reflection , could alter the inherent propertie s of a ray of light . As he finally realized , analysi s entails im mutability . Colors are never generated; they are only made apparent by some process that separates the m from the heterogeneou s mixture of white light . Heterogeneity , the new property of light on which the concept of analysis depended , challenged the assu mption of two thousand years of optical re search. It reversed the relation of colors and whiteness . Whereas white had been associated universally with si mplicity and pu rity , bo th within optics and beyond it , Newton' s theory made individual colors simple and pure and demoted white ' to the statu s of a secondary appearance , the sensation of no single ray but of a heterogeneou s mixture . " I perswade my selfe , " he later remarked, " that thi s assertion above the rest appeare s Pa ra­ doxicall, & is with most difficulty admitted . " 7 6 Apparently in 1 666 he did not yet understand how paradoxi cal the asse rtion was . Only a couple of minor experi ments attempted to illustrate the recomposition of white . This too would await further elaboration in the futu re . The implications of his insight , only dimly perceiv ed at first , began to emerge more clearly in his investi gation of 1 666 . Nev7 5 Add MS 3975 , p . 3 . 74 Add MS 3975 , p . 2 . 7 6 Newton to Oldenburg, 1 1 June 1 6 72; Corres 1 , 1 83 .

A nn i m ira bi les

171

ertheless , a co mpari son of the essay " Of Colours " with the Lec­ tiones opticae, the polished product of the investigation of 1 668 and 1 669 , indicate s that Newton groped forward a step at a ti me . In 1 666 , several i mpo rtant points remained obscure . The e ssay " Of Colou rs" assu med a two-color sys tem , red and blu e . Twice , early in the es say he started to wri te down the order of colors , first in the spectru m and then in the film of air between two prisms . B oth times he left a blank space that he did not bother to fill in , as though the order of colo rs were of little interest . 77 When finally he did list them fo r the colored circles in a film between a lens and a flat sheet of glass, he named five: re d, yellow , green , blu e , purple . 7 8 " As white was made by a mixture of all sorts of colours . . . [he explained] Greene is made by a mixtu re of blew & yellow , purple by a mixture of red & blew &c . " 79 He had indicated already that yellow is merely dilu te red . Perhaps no aspect of the essay of 1 666 better illus trates the gradu al unfolding of the full meaning of his idea . More significant was the continuing confusion between his the­ ory of colors and his corpuscular conception of light . Already with the " Qu aestiones" Newton accepted the co rpuscular conc ep­ tion of light as one aspect of his atomistic philosophy of nature . 8 0 When he met an embryonic wave conception of light in Hooke ' s Micrograph ia , h e reacted agains t it with one o f h i s endu ring obj ec­ tions - that light ou ght then to s tray fro m i ts rectilinea r path as sound , which was agreed to be a wave motion , does . 8 1 Moreo ver, i t seems clear that the corpuscular conception functioned as a propaedeu tic to his theory of colors . The notion of strong and weak rays in which he couched hi s original insight readily clothed i tself in the images of large and s mall corpuscles . His geo metriza­ tion of colors found its counterpart in the paths of single corpus­ cles , wherea s the wave theory of light had always to think of a physical beam of finite di mensions . As the immu tab ility of rays impressed itself on hi s understanding , it became a fu rther argu­ ment for the corpuscular conception , finding i ts physical basis in the definition of an ato m . Thus there were many filiations binding the corpuscular conception of light to the new theory of color . His initial experiments with thin films in 1 666 served to reinforce the m . He observed that the circles of colors between a lens and a flat piece of glass appeared to grow in diameter the more obliquely he observed them . He i mmediately interpreted their size 77 Add MS 3975 , p p . 3 , 7 . 7 8 Add MS 3975 , p . 1 0 . H e cited the same fi v e as the components of white under somewhat 79 Add MS 3975 . p . 1 3 . different circumstances on p. 1 2 . 8 1 Halls, p. 403 . 80 Add MS 3996, ff. 1 04 ", 1 22-3".

1 72

Never at rest

in term s of the streng th of the ray ' s blow on the fi l m . A stronger blow allo wed the rays to pass th rough more readily in a circle of sm aller diameter, while a weaker blow caused it to p ass throu gh less readily and hence to appe ar as a larger circle . He worked out a formula , which he later crossed out, in which the diameter of the circl_e s varied inversely as the com ponent of motion perpendicular to the film . 8 2 Newton never enti rely separated his theo ry of colors fro m his conception of light . When he wrote Query 28 in the Op ticks in 1 706 , he as serted that wave conceptions of light always embodied mo dification theories of color . 83 Neverthele s s , the two were sep arate issues . He had created powerful experi mental evi­ dence to support the heterogeneity of light, whereas the corpuscular conception rem ained a matter of speculation , as he acknowledged forty years later when he argued for it only in the Queries of the Op ticks . In 1 666 , the distinction between the two had not become clear in his mind . One o ther i ssue rem ained: the colors of solid bodies . Newton built his theory of colors fro m experi ments with prism s . The vast maj ori ty of the colors we see , ho wever, are associated with solid bodies . Unle ss he accounted for their colors , his theory would be extremely limited . From th e moment of his ini tial insight , of cou rse , he had a gene ral account of the colors of solid bodies . Reflection can also analyze white light into its components . A body i s dispo sed to re flect some rays more th an others and appears to be the color it reflects bes t . His account of colors never dev iated from this position . In the beginning, however, the statement exp ressed an idea withou t empirical foundation and withou t qu anti tative con­ ten t . The essay " Of Colours" provided some em pirical foundation . When he painted red and blue patches on a piece of paper and viewed them in " P rism aticall blew" and "Prismaticall red, " both patche s appeared to have the color of the incident light, b ut the blue patch was fainter in red light and the red fainter in blue. " N ote y 1 ye purer ye Red/Bl ew is ye lesse tis vi sible wth blew/Red rays . " 84 Later he would add further empirical evidence as the immutab il ity of rays became clearer to him . Quantitative content was a more difficult matter. It was abso82 In fact , the formula was more complicated in two ways . It employed the components of m otion (or , as he put it, the sines of the angle of obliqu ity) both in the film of air and in the glass that confined the film. It further multiplied both components by the motion of the ray in the medium; in effect this made the formula a comparison of the squares of the motions because the index of refraction between the two media was understood to be the ratio of their motions . Since Ne wton crossed the formula ou t, its details are not important fo r my purposes . Its impor tance for me here lies in its implicit conception of light , exp ressed in its assumption of corpuscles moving with greater or les ser strength in 8 4 Add MS 3975 , p . 3 . 8 3 Op ticks, p . 362 . relatio n to a surface .

Ann i mirabiles

1 73

lutely essential . After his experiments with pri smatic spectra , analy­ sis by refraction could be expressed in rigorous qu antitative term s . Color ceased to b e a wholly subjective p henomenon since it was attached immutably to a given deg ree of refrang ibility . 85 With re­ flected colors , in contrast, he had attained no similar qu antitative treatment , and reflected colors constitute the overwhelming bulk of color p henomena in the world . He had picked up a suggestion , however. In Hooke ' s Micrograph ia he found de scriptions of colors in a variety of thin transparent bodies - in Mu scovy glass (or mica) , in soap bubbles, in the scoria of metal s , in the air between two pieces of glass . Newton himself observed the colors in a film of air be­ tween two pri sms , both in the transmitted and in the reflected light . The " p late of air (ej) is a very reflecting body , " he noted , and later he indicated that the colors of solid bodies are related to the colors of thin transparent films . 86 He even realized a means of doing w hat Hooke had confe ssed hi mself unable to do , measu ring the thickness of the films in which colors appear . When a lens of known cu rva­ ture was pressed on a flat piece of glass , a thin film of air was constituted between them . Circles of color appeared around the point of contact . Using the geometry of the ci rcle , the same propo­ sition indeed which he used in calculating centrifugal force , he com­ puted the thickness of the film fro m the curvature of the lens and the measu red diameter of the circles . " Of Colou rs" recorded Newton ' s first observation of " Newton ' s rings . " Hooke had asserted that the colors i n thin films are perio dic , that is , that su ccessive additions of a given increment of thickness cause the same colors to reappea r. Newton observed that success ive circles around the center did not increase linea rly in diameter; with­ out measu ring , he inferred that their diameters corre sponded to a linear increase in thickness of the film . "Let cd == the radiu s of curvatu re of the glass; ejgh ik circle s of colors; & el == fm /2 == gn /3 == hp /4 == ig /5 == kr /6 == the thicknesses of air . " 8 7 With a lens 25 in ches in radiu s , he measured the diameter of the fifth circle and calculated that the thickness of el, the first circle , was 1 /64 ,000-inch , a figure which he alte red several years later to 1 /83 , 000-inch after more refined measu rements . Thu s he had the outlines of a theory of reflected colors and an experi mental method whereby to elabo rate it. It appears , ho wever , that he did not yet gra sp the full connection between the two . The circles were me rely "circles of colours , " not 8 5 Or so N ewton finally believed . At one point, early in his career , he at least en tertained the idea that dispersion might vary with media so that an achromatic lens was theoretically possible (cf. Bechler , " ' A less agreeable matter ' " ) . 86 Add MS 3975 , p p . 8, 1 3 . 8 7 Add MS 3975 , p . 9 . On his diagram , el, fm , gn , hp , iq, and kr are the thickn esses of the thin film where successive circles appear at e, f, g , h , i, and k on the flat surface.

1 74

Never at rest

ci rcles of particular colors . He did nothing to measure the different thicknesses for different colors . Thi s aspect of his theory was in 1 666 the least developed . On close exa mination , the anni m ira biles tu rn out to be less miracu­ lou s than the annus mira bilis of Newtonian m yth . When 1 666 closed, Newton was not in com mand of the results that have m ade his reputation deathless , not in mathemati cs , not in mechanics , not in op ti cs . What he had done in all three was to lay fo undations , some more extensive than others, on which he could build with assur­ ance, but nothing was complete at the end of 1 666 , and most were not ev en close to complete . Far from diminishing Newton' s statu re , such a j udgment enhances it by treating hi s achievement as a hum an drama of toil and struggle rather than a tale of divine revelation . " I keep the subj ect constantly before me, " he said, " and w ait ' till the first dawnings open slowly , by little and little , into a full and clear light. " 8 8 In 1 666 , by dint of keeping subj ects constantly befo re him , he saw the first dawning s open slowly . Years of thinki ng on them continuou sly had yet to p ass before he g azed on a fu ll and clear light . By any other standard than Newtonian myth , the accomplish­ ment of the a nni mirabiles was astoni shing . In 1 660 , a provincial boy ate hi s heart out for the world of lea rning which he was app arently being deni ed . By good fo rtune it had been spread before him . Six years later, with no help beyond the books he had found fo r him­ self, he had made him self the fo remost mathematician in Europe and the equal of the fo re most natu ral philosopher . What is equally important for Newton , he recognized his own capacity because he understood the si gni ficance of hi s achievements . He did not merely measu re himself against the standard of Restoration Cambridge; he measu red him self against the leaders of European science whose books he read . In fu ll confidence he could tell the Royal Society early in 1 672 that he had made " the oddest if not the most consid­ erable detection wch hath hi therto beene made in the op erations of Nature . " 8 9 The parallel between Newton and Huygens in natural philosophy is rem arkable . Working within the same tradition , they saw the same p roblems in many cases and pu rsued them to similar conclu­ sions . Beyond mechani c s , the re were also p arallel investigations in optics . At nearly the same ti me and stimulated by the s ame book , Hooke ' s Micrographia , they thought of i dentical methods to measu re 88 B iographia Britannica (London , 1 760) , 5, 324 1 . No source for the quotation is given. 8 9 Corres 1, 82-3 .

Anni mirabiles

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the thickness of thin colored films . 9 0 No other natural philosopher even app roached thei r level . In the very year 1 666 , Huygens w ith all hi s acclai m was being wooed by Louis XIV to confirm th e renown of his Academie royale des sciences . There was no occa�ion for a young 1nan recently elevated to the dignity of Bachelor of Arts and working in isolation to be ashamed of his achievement , even if the Sun King , in his presu mption , had not placed a crown of lau rel on his bro w . 90 Hu ygens dated his experiments November 1 665; Oeu vres comp letes , pub. So ciete ho llan­ daise des sciences, 22 vols. (The Hague, 1 888- 1 950) , 1 7, 34 1 -8 .

6

Lucasian professor

OSSIBLY the fact that the uni versity closed down fo r the better

P part of two years , freeing Newton fro m control and supervi sion ,

aided hi s unorthodox prog ram of study . Such was the p rogressive decomposition of the established cu rriculu m , however, that it is far from clear that his intellectual development would have differed in any respect had the plague not intervened . While the requirements prescri bed an i mp ressive array of lectu res , disputations , and acts over a three-year period before a Bachelo r of Art s might be admit­ ted as an inceptor fo r the Master of A rts deg ree , most of them had ceased to be operative by the Resto ration , as repeated o rder s from Westm inster that they be observed testify . Even res idence had ceased to be required . Though Newton him self was not frequently absent from Cambridge once he retu rned after the plague, the col­ lege E xit and Redit B ook reveals that other scholars of the hou se felt free to corne and go as they plea sed , even in term time. New­ ton ' s friend Wickins, fo r example, who was a scholar drawing a stipend , left the college on 1 5 October 1 667 and did not return until the middle of January . 1 Unorthodox study was not manifestly worse than no study at all . The one discipline exerted was financial; Newton recei ved his stipends throu gh hi s tutor. 2 Although his papers do not indi cate that he again made any gestures whatever toward the establi shed cu rriculu m , hi s academic career neve rtheless marched through the remaining stages to full member ship in the college and uni versity without he sitation . S hortly after his return fro m Woolsthorpe late in April 1 667, the magnifi cent funeral of M atthew Wren , bishop of Ely , e sco rted by the entire acade mic com munity in full regalia according to their ranks and deg rees , must have reminded him that he stood then only on the first step of the university hierarchy and that others loomed immedi ately ahead . 3 In only a few months he would fa ce t he first and by far the most important of these , the fellowship election . As with the schola rship three years earlier , Newton ' s whole future hung in the balance of this election . It would determine whether he would stay on at Cambridge and be free to pursue hi s st udies or whether he would return to Lincolnshire , p robably to the village 1

Exit and Redit Book , 1 667- 1 703. Junior Bursar 's Acco unts, and Bailie' s & Chamberlayne's Day Book , 1 664- 1 673 . 3 See the account o f the fu neral in J . E . Foster , ed . , Th e Diary of Samuel Newto n , Alderma n of Ca mbridge ( 1 662- 1 7 1 7) (Cambridge, 1 890) , p p . 1 8-20 .

2

1 76

Lucasian professor

1 77

vicarage th at his family connec tions could have supp lied , where he might well have withered and decayed in the ab sence of books and the distraction of petty obligations . On the fa�e of i t , his chances were sli m . Th ere had been no election s in Trinity for three years , and as it tu rned out, there were only nine places to fill . The phalanx of Westminster scholars exerci sed their usual advantage. The grow­ ing role of political influ ence , whereby those with access to the cou rt won letters mandate fro m the king co m manding their elec­ tion , w as notoriou s . 4 For the rest, all depended on the choice of the master and eight senior fellows , and storie s of influ ence peddling filled the ai r. 5 The candidates had to sit in the chapel fou r days in the last week of September to be examined viva voce by the senior fellows , th e dying embodiment of the curriculu m Newton had sys­ temically igno red for nearly four years . How could an erstwhile su bsizar of whatever cap acity hope to prevail against such odds? If he too had a p atron , he might do more than hop e . In 1 667 , Hu mph rey B abing ton j o ined the ranks of the senior fe llo w s . Neither in Newton ' s p apers nor i n the surviving anecdotes does a hint of tension over the outcome appear. His accounts present a picture of relaxation which almost belies our other evidence of unre mitting, introverted study . Soon after his return , he spent 1 7s 6d to celebrate his Bachelor' s Act and on subsequent occasions tos sed away another of the £ 1 0 he had pried loose fro m Hannah 4 There were two mandates at Newton 's election , one for John Goodwin, who was ele cted fellow , and one for V alentine Pettit (CSPD: C harles I I , 7, 322 , 553) . Since t he college was in that year establishing the practi ce of electing only third-year bachelors , and since Pettit was only in his second year, the college deferred his election until 1 668 . O n the role of mandates , see E. F. Churchill, "The Dispensing Power of the Crown in E cclesias tical Affairs , " Law Quarterly Rev iew , 38 ( 1 922) , 297-3 1 6 , 420-34; J ames Bass Mullinger , The University of Cam bridge, 3 vol s . (Cambridge , 1 873- 1 9 1 1 ) , 3, 626-9; John Venn , Caius College (London , 1 90 1 ) , p p . 1 05 , 1 1 1 , 1 44 . Contempor ary comments and accounts are found in Ro ger North, The Lives of th e Righ t Hon . Francis North , Baron Gu ilford; the Hon . Sir Dudley North ; and the Hon . and Rev . Dr. John North . Together with th e Autobiography of the Auth or, ed . Augustus Jessopp , 3 vol s . (London , 1 890) , 2, 324; and N athaniel John ston, The King 's Visitorial Power Asserted (London , 1 688) , pp. 274-9 . Johnston 's book was an apology for the intrusions of James II into the universities that made effective use of es tablished p ractices to defend James' s actions . Tho mas Baker , History of the College of St. John the Evangelist, Cambridge, ed . J. E. B. Mayor , 2 vols . (Cambridge, 1 869) , 1 , pp . 2989, 543 , contains a nearly contemporary account . 5 Such is a m ajor theme of North's life of his brother John , master of Trinity (Lives, 2) . Inevitably there is much about it in Johnston , Visitorial Power. Baker accused Gunnin g , the master of S t . John's at the time when Newton was a student, and a man whom Baker generally admired , of misusing his influence in the election of fe llows . (History of St. Joh n 's p . 237) ; Cf. J . Beeby of Oxford to Secretary Williamson, 1 668 , in which Beeby deplored the destruction of free elections by mandates, but nevertheless thought it better that the King should co m mand an election "than that any p arti cular fellow or head of a house may sell it for £ /flO or £ 300 contrary to law or reason" (quoted in C hurchill , "Dispensing Power , " p. J ,

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Fi gure 6 . 1 . The gowns of seventeenth-century Cambri dge as Da vid Loggan p resented them . 3. An undergraduate of Trinity College . 5. A B a chel or of Arts . 7 , 8, 9 , 1 0 . Mas ters of Arts . (Fro m David Loggan, Canta brigia illustrata, n. d . , late seventeenth centu ry . )

Smith and then some with " acqu aintances" at taverns . He cheer­ fully confessed to a loss of 1 5 s at cards , compensated perhaps by a pu rchase of oranges for his sister. The accounts radiate confi dence as well . He invested £ 1 1 0s in tools , real tools including a lathe , such as he must have longed fo r in Grantham - not the pu rchase of a man seriou sly expecting to move on a year hence . His manifest intent to convert their chamber into a factory loft may have encour­ aged W ickins in his extended absences . And Newton inves ted handso mely in cloth for a bachelor' s gown , which cou ld be con ­ verted later into a master' s , eight and a half yards of "Woosted Prunella " p lus four yards of lining , fo r which he paid nearly £ 2 in all . 6 On 1 October a bell tolled at eight in the morning to su mmon the seniors to the election . The bell tolled again the following day at one to call tho se chosen to be sworn in: It tolled for Newton . Now at last the way was clear . The election promi sed permanent membe rship in the academic community with freedom to continue 6 Fitzwilliam notebook.

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the studies so auspiciou sly , as he at least understood , begun. Tru e , two more step s remained for him t o mount . I n October 1 667 , he became only a mino r fellow of the college , but advan cement to the statu s of maj o r fellow would follow auto mati cally when he w as created M aster of Arts nine months hence. The exerci ses for the degree had become wholly pro forma; no one was known to have been rej ected . The final step could come at any time in the follow­ ing seven years . The incumbents of two sp ecific fellow ships ex­ cepted , the sixty fellows of the colleg e were required to take holy orders in the Anglican chu rch within seven years of incepting M . A . Shortly after one o ' clock o n 2 O ctober 1 667 , Newton became a fellow of the College of the Holy and Undi vided Trinity when he swore " that I will embrace the tru e religion of Chri st with all my soul . . . and also that I will either set Theology as the object of my studies and will t ake holy orders when the time prescribed by these statutes arrives , or I will resign fro m the college . "7 The final re­ quirem ent was not likely to pose more of an ob stacle to a pious and earnest young man than the M aster's degree . As a minor fellow , Newton received a stipend (or " wages" in the blunter language of the day) of £ 2 a year , a li very allowance of £ 1 6s 8d , and a di vidend of £ 1 0 . Not yet admitted to the high table , he continu ed to dine with the scholars . The college assigned a room to him . On a li st dated 5 October 1 667, his name appears on the last line: "to sr N ewton-Spirituall Chamber . " 8 Edleston has specu­ lated that this referred to the ground-floor room next to the chapel in the northeast corner of the great cou rt . Wherever the room, the mere fact that the college assigned it to Newton tells us nothing about where he lived . The room was one of his perqu i sites . He might rent it out if he chose and pocket the income, and his ac­ count s do show that he recei ved £ 1 1 1 s Od chamber rent in the summer of 1 668 . Stukeley later heard that he lived on the north side of the cou rt between the master ' s lodge and the chapel . In any event , he continu ed to share a chamber with Wickins , who would also have one at his di sposal in another year . Undoubtedly the two split the rent fro m the room they did not use . 9 Two puzzling accounts of p ayments fo r a chamber to or fo r Hu mphrey Babing ton also su rvi ve , further testimony of a connection between the two whatever it means about Newton' s residence . 10 For one room or 7 9

10

Trini ty Statute s , Fo urth Report from the Select Co mmittee on Education (London, 1 8 1 8) , p . 8 Edl eston , p . xliii . 373 . Fitzwilliam notebook . Stukeley, p . 5 6 . Nicholas Wickins to Robert S mith, 1 6 Jan . 1 728; Key nes MS 1 37 . Wickins's son reported that Newton collected room rent for his fa the r and fo rwarded it . Add MS 3970. 3 , f. 469v . Yahuda MS 34. Both appear to be related to Newton's final chamber next to the great gate .

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another, he spent 2s 6d during 1 667 to hire a glazier and to point the windows and fireplace. Newton spent Chri stmas of 1 667 at home, leaving the colleg e on 4 December and retu rning on 1 2 Febru ary . There mu st have been some blunt talk abou t money and the style a regent master of the university expected to keep . He retu rned thi s time with £ 30 of H annah S mith' s money in his pocket . He spent a signifi cant part of it on clothes . 1 1 On 1 April 1 668 , the senate passed the grace grant­ ing M . A . deg rees to 1 48 inceptors including Newton. He paid the P roct o r £ 2 and the college £ 2 1 0s and he paid out 1 5s fu rther , "Expenses cau sed by my degre, " whi ch undou btedly went into the pocket of a tavern keeper . 12 He was created M aster of Art s on 7 July and became therewith a majo r fellow of Trinity College. A s a m ajor fellow , Newton' s stip end increased to £ 2 1 3s 4d per year and his livery allowance to £ 1 1 3s 4d . On behalf of each fellow , the college paid its steward 3s 4d commons for every week he was in residence, thou gh in fact the high table expended twice that sum to feed the fellows . The Elizabethan statutes had set the official stip end and allowances . Inflat ion since then had rendered the su m s insignifi cant , but as income fro m endowment s kept pace with inflation, the college devised a new system to su pport the fellows , an annual di vision of the su rplu s , o r dividend . All the colleges of Oxford and Cambridge adopted thi s device du ring the middle years of the seventeenth centu ry . By the definitive plan of divi sion whi ch Trinity adopted in 1 66 1 , the ordinary fellow of low seniority re­ cei ved a di vidend cf £ 25 . Until the latter years of the centu ry , the college paid out a dividend annu ally with only one exception . In addition, fellows resident more than half the year , as Newton in­ variably was, shared the profits of the colleg e bakery and brewery . The pandoxato r' s dividend, as it was called , brou ght the ordinary fellow £ 5 per year. In su m , Newton found himself the recip ient of 11

12

In hi s accounts immediately after hi s retu rn in April 1 667 Fitz william noteb ook: Two paire of shoos 0 8 0 Later in 1 667: 0 4. S hoos 0. 0 Cloth 2 yards & buckles for a Vest . 2. 0. 0 13. 2. To the Taylor Octob 29. 1 667 . To the Taylor. June 1 0. 1 667 1 0. 1. 3. 4. 1 0. 0. S hoos & mending In 1 668 : 0. S hoemake r 8. 5. Maki ng & c o f m y last suit 1. 11. 9 In 1 669: 0 2. 1 6 yards of Stu ffe for a Suit 8. 1. 13. 0 F or making &c. 3. 3. 1. For turning a Cloth suit Ibid.

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approximately £ 60 per year , between £ 20 and £ 25 of which he recei ved in the fo rm of food and lodgings . He recei ved in addition perquisites which cannot be reduced to measu re, such as the silver tankard reserved fo r hi m in the great hall and access to the bowling green . Since he was assessed ten shillings for the latter , it m ay have appeared to hi m more as an expense than a perqu isite; twenty years later Humphrey Newton observed that he never u sed it. 1 3 Thou gh hi s income derived from the landed endowment of the college , it was by law ex empt from taxation . To put hi s financial po sition into perspective , re call that the skilled work men who labo red on the Trinity library a decade later received between 20 and 25d per day, which wou ld have amounted to £26 to £31 per year if they worked six days a week fifty-two weeks of the year; they more probably earned between £ 1 5 and £ 20 . 1 4 Unlike the enforced celibates of the colleg e, they supported fa mili es on their income. Unskilled wo rk­ men received half that su m . In 1 658 , Samuel Pepys lived with a wife in London on a salary of £ 50 as a clerk at the Exchequ er . 15 Though one would not call £ 60 per annu m handsome, certainly it was comfortable , especi ally fo r one who se aspirations were intellec­ tual . In addition , Newton owned a s mall property near Wools­ thorpe, though income from it never appears in his accounts , and in due tim e he wou ld inherit much more. Newton received an added dividend during his fi rst y ear as fel­ low . On 2 A pril 1 669 , the college as signed S t . Leger Scroop e , a fellow commoner , to his tuiti on . Since su bstantial fees accompanied fellow commoners , they were u su ally reserved fo r i mportant mem­ bers of the college; fo r a num ber of years before he mig rated to the master' s lodge in Magdalene , fo r example , James Duport had m ade it hi s busines s to eng ross Trinity' s supply of fellow commoners . The assignment of such a plum to a new fellow without seni ority or influ ence of his own can only be seen , I believe, as further evidence that he had a patron in high place in the college . No mention of the relationship with Scroope made its way into New­ ton' s p apers . S cro ope neither matri culated in the unive rsity nor graduated from it . He left no trace on Newton ' s life . He aiso left no plate to the college, whi ch exp�cted such from its fellow com­ moners . Until Newton resigned his fellowship , the j unior bu rsar' s accounts annu ally carried the entry under the heading ' ' Plate not Recei ved" : " From Mr Newton, Mr Scroope' s . " Thomas Fuller referred to Trinity College a s ' ' the stateliest and most uniform College in Chri stendom , out of whi ch may be carved 1 3 Humphrey Newton to Conduitt, 1 7 Jan . 1 728; Keynes MS 1 35 . Keynes MS 1 35 also contains a second letter from Humph rey, 1 4 Feb . 1 728 . 1 4 The accounts for the library are in Trinity College , 0 . 4 . 47 . 1 5 Arthur Bryan t , Samuel Pepys , 3 vol s . (London , 1 933-8) , 1 , 46 .

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three Dutch Universities . " 16 There i s reason to believe that Newton savored the dignity to which he had risen . He spent time and money appointing his chamber. After the college had it plastered early in 1 668 , Newton paid £ 1 6s Od for putty and the services of a j oiner and painter. He purchased a leather carpet and j oined with Wickins to purchase a cou ch . Ticking and feathers for his bed cost nearly £ 2 , and he acquired a tablecloth and six napkins . 17 Some­ what later he added another rug and refurbished his bedroom . 18 He took care at once to prep are himself fo r hi s role in the colorful university processions . He purchased no less than eighteen yard s of tammy (£ 1 1 3 s Od) for his master' s gown , and lining besides (3s 6d) , and he paid a tailor £ 1 Os 6d to turn his bachelor' s gown and to make his new one . A hat cost 1 9s Od and a hood £ 1 3s 6d . 1 9 To celebrate his new dignity , Newton treated himself to his first visit to London . His accounts reco rd that he left the college on 5 August and that he spent nearly £ 1 0 in the city . A year earlier he had purchased Sprat ' s History of the Royal Society and had begun to purchase and read the Ph ilosop hical Transactions . He had read as well the wo rk s of prominent fellows of the Royal Society such as Robert Boyle and Robert Hooke . He could not have failed to ap p reciate the affinity of the society with the studies in which he had im mersed himself. Nevertheless , while he was in London Newton did not attemp t to ap proach the society , with whi ch his future was to be inextri cably woven . Membership in it would have to wait for their overt u re. We do not know what he did in London during the month he was there . On his return , he detou red to W oolsthorpe, possibly to show his mother the new finery her money had pur­ chased . She must have been imp ressed . She came across with £4 6s to cov er part of his London expenses and later in the year sent £ 1 1 more. Newton returned to the college on 28 September according to his account , on 29 September acco rding to the colleg e ' s . 2 0 He did not have long to wait befo re Cambridge p rovi ded an occasion to di splay his new grandeur. On 1 May 1 669, Cosimo de ' Medici , P rince of Tuscany , vi sited the town and uni versity . As he was conducted from the Ro se Tavern to the publi c schools , all the student s - and masters in their robes lined the walk . T he univer­ sity entert ained the prince, whose uncle had organized the famous 16 Thomas Fuller , Histo ry of the University of Cambridge, e d . James Nichols (London, 1 840) , p . 1 74 . 1 7 Fitzwilliam notebook . The Junior Bu rsar's Accounts fo r 1 668 i n Trinity sh o w an expendi­ tu re in the second quarter for plastering Newton 's chamber , though it need not refer to the chamber in which he lived . 18 Yahuda MS 34, f. 1 . Thi s is an account from 1 0 March 1 6 70 in a nearly i ndecipherable hand . It definitely mentions a rug and a bed for which Newton paid in part with hi s old 2 0 Ibid. Edleston , p. l xxxv . 1 9 Fitzwilliam notebook . bed.

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Accaden1ia del C imen to ( A cademy of E xperi 1nent) in Florence , with a philosophi c a c t , " De methodi philosophandi in experimen ti s fundata , et contra systema Copernicanum . " We are n o t informed that Newton was invited to p arti ci pate . The party moved on to the show places of the university , King ' s C ollege cha pel , S t . John ' s , and Trinity , " where schollers , B achel ors and Ma s ter of A rts o f that Coll . were orderly placed all along the first w alke on b o th sides to ye Lodge . . . " In the evening s tuden ts performed a co medy in the hall . 2 1 The next yea r the P rince of Orange, the fu ture monarch William I I I , vi sited , and in 1 67 1 Charles hi1nsel f. As he w as con­ ducted in the usual way fro m the town to the university through the ordered rank s of scholars and masters , the ' ' Conduit run claret wine when his Maj es tie passed by w ho was well pleased with i t . " The K ing was en tertained at dinner in T rinity , where the Mas ters of A rts in their ro bes served; New ton probably did not j oin them since he had by then assumed the dignity of university p rofessor. The college pre sented the inevitable co medy with which his maj es ty " expressed himself to be well plea s ed , as also with the good O rder o f the whole Un iversity . . . " The only thing with which he did no t express himsel f well pleased was the B ible the university p resen ted to him . 22 The visit co s t the university £ 1 , 039 S s l d and the town £ 268 1 l s 2d , inclu ding 6d p aid to John Fox to s weep the butchers ' s tall s . Behind the po mp of public display , ho weve r , mortal illness ate at the life bo th of the university and of Trinity . On the surface they seemed to flo urish . After the upheavals and contraction of the P uri­ tan revolution , they were expanding again and approa ching the peak in numbers that both had reached in the 1 620s . The expans ion p roved de ceptive; a precipitous decline , which reduced the univer­ sity in the sp ace of two decades nearly to hal f its former s ize , was about to set in early in the 1 670s . 23 University ins titutions were increasingly shams . O s tensibly the senate s till functioned as the 21 22 23

Foster , Samuel Newton 's Diary , pp . 44-5 . Charles Henry Coop er, Annals of CambridRc, 5 vols . (Camb rid ge, 1 842- 1 908) , 3, 536 . Cooper, A nna ls 3, 548- 9 . Foster, Samuel Newton 's Diary , pp . 64- 6 . A cen sus of t h e uni versity w a s publi shed i n 1 622 , which w a s very n ear t h e peak i n the early p art of the cen tury . It showed a total popu lation of 3 , 050 resident in the university , a nu m ber which in cluded fellows , students, an d serv ant s. In that year about 285 students graduated B . A . Matriculations were running above 450 per year; with the normal dis­ crepancy between adm issions and matricu lation s, about 550 new students were probably adm itted . Another census was taken in 16 72, the peak of the Restoration su rge. It showed a total pop ulation of 2 , 522 . About 265 graduated B . A . that year , and about 350 matricu­ lated . No census was pu blished in the early 1 690s . B . A . s had dropp ed to about 1 45 , however, and matriculations below 200 . B y extrapolation , the size o f the university must have been about 1 , 500, of whom well less than 1 , 000 would have been undergraduates . A chart showing B . A . s is found in J a mes Bass Mullinger, A History of the Un iversity of

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supre me governing bo dy; in fa ct , a s ma ll o liga rchy of c olle ge masters (or heads) appointed directly or co vertly by the C ro wn domin a ted it . The university officers , once chosen by the reg en t masters fro m their ranks , were now imposed on the m fro m outs ide or selec ted by ro tation from senior fellows of the colleges . In the fall of 1 669 , scarcely a year after Ne wton became a member, the senate tried to retain its power to elect the Esquire B edell . It no t only fa iled but received for i ts p ains a rebuke fro m the king , who empo wered the vice-chancellor to suspend any regent maste r who challenged hi s au tho rity . 2 4 Four years later, the heads seized effec­ tive no mination of the university orato r . Since the act deprived Isaac C ravens , a fellow of T rini ty , of the position, members of Trinity , led b y senior fe llo ws in clud ing Humphrey B abing ton , lodged a p ro te s t . Newton signed it along with o thers . A s usu al, the heads won . 2 5 In case there were any dou b t , Charles info rmed the senate of its s tatus with ab undan t clarity in 1 674 . On 1 1 July , he wro te to in fo rm the m that he was re moving the Duke of B u cking­ ham from his service . To curry favor with the Crown , the senate had elected B uckingham chan cellor of the university only three years e arlie r . Charles no w declared his election void, o rdered the university to p roceed at once to a new election , and reco mmended to thei r consi de ration his b a s tard son , the D uke of Monm outh . Witho u t a pause , the s en a te ele cted Monmouth , unanimously . They also thanked Charle s fo r his goodness in giv ing them the libe rty of such a choice . 2 6 Only a maj o r crisis in which the hea ds and the teaching masters might see their interests coincide would suffice to reanim ate the sen a te . P a s s ing g ra ces , which g ranted degrees to tho se who m the colleges named , had bec01ne the senate ' s p ri mary function . Thu s were the teaching masters disinherited fro m the in­ s ti tu tion their p redecessors had created and governed . I t is perhaps not surprising that du ring the res to ration the teach­ ing ma s te rs for the mo s t p a rt cho se no t to teach . Acts an d dispu ta­ tions ceased to be perfo rmed . The s tatutes required all Masters of Arts of four yea rs ' standing to pe rform divinity acts , the mo st dignified exercise of a curriculu m which saw theology as i ts culmi­ nation . In fact , they ne glected divinity acts to the point that the

24 25

Cambrid,ge (London , 1 888) between pp . 212 and 2 1 3 . Matriculations are listed in the Historical Reg ister of the University of Cambridge, ed . J. R. Tanner (Cambridg e, 1 9 1 7) , pp . 988-9 . There are a num ber of revealing statistics in David A rthu r Cressy, "Education and Literacy in London and East Anglia , 1 580- 1 700" ( diss ertation , Camb ridge Univer­ sity , 1 972) , especially a table on p . 237 . Three different interpretations of the decline of Camb ridge can be found in Hugh Kearney, Sch olars and Gentlemen . Universities and Society in Pre-Industrial Britain (Lon don , 1 970) , pp . 1 4 1 - 73 ; Mark H . Cu rtis , Oxfo rd and Cam­ bridge in Transitio n , 1 558- 1 642 (Oxford , 1 9 39) , pp. 272-81 , and Richa rd S. Westfall, " I saac Newton in Cam bridge, " forthco ming . Coop er, Anna ls, 3 , 537-9 . Cambridge Uni versity Library , Baker M S S , Mm . 1 . 5 3 , ff. 62-3 . 26 Coop er, A nnals , 3, 559-60 . Edlesto n , pp . xlvii-viii .

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court admonished the university to see to thei r performance , with the usual null effect. D espite his s olemn oath at Trini ty , Newton , for one , never kep t a divinity act . Univers ity p rofes sors converted thei r positions into sinecures , even those fo unded during the Res to­ ration . As Luca sian professor of mathemati cs , New ton res is ted this trend only during the first half of his tenu re . To some exten t , the decline of the university p roceeded in s tep with the rise of th e colleges . The colleges were the younger ins ti tu­ tions , most of the m like Trinity creations of the si xteenth century which the Tudor government fo s tered as ins truments of discipline in the two universities . The app arent p rosperity of the colleges had to do w ith externals alone . Their inner malaise was no les s p ro­ found than that of the unive rsity , and Trini ty did no t differ fro m the res t . In government , the ty ranny of the heads over the senate repeated itself in microcosm in the ty ranny o f each head over his house . 2 7 The neglect of e xercises and obligations repeated itself as well . 28 The fo undation of T rinity p ro vided for eight college lectur­ ers; like university p rofes sorships , the positions became s inecures . The grea t maj ori ty of fell ows evaded tutoring , which concentra ted itself in the hands of a s mall number, like Newton' s tutor Pulleyn, who undertook the duty for the extra income it p rovided. The others called them derisively "p upil mongers . " 2 9 O ther facets of the old Camb ridge di scipl ine collapsed as well . It was s y mpto matic of the new atmo sphere that the master and seniors of Trinity ag reed on 1 1 A ugust 1 668 that " no o ther Saturdaies be fish-daies in ye Hall but tho se which in ye Rubrick are app ointed to be o bserved as daies of fa s ting and ab s tinence . " 3 0 2 7 Cf. a story about Lazarus Seaman , master of Peterhouse, in th e 1 650s (Mullinger, The Univ ersity of Cambridge, 3, 392-4 1 6) ; William Tas well ' s relations with Dr. Fell , dean of Christ Chu rch a generation later (William Tas well , Autobiography and An ecdotes , ed . George P . Elliott , in Camden Misce llany , 2 [Cam den So ciety , 1 852] pp . 24-5) ; Richard B entley ' s use of his powers as master of Trinity early in th e eigh teenth century (James Henry Monk , The Life of R ich ard Bentley , D. D. , 2nd ed . , 2 vols . [London , 1 833] , 1 , 23 1 -2) . 28 In Trinity a conclus ion recorded in 1 683 noted the " very great failure and neglect" of college dis putations and established healthy (indeed prohibitive) fines for nonperformance (Conclusion Book, 1 646- 1 8 1 1 , p . 1 62) . Signed by the mas ter, John North, alone without an y of the senior fellows , this incident reflected th e tens ion between North and the rest of th e college, bu t their bad relations do not call the facts ass erted into qu estion . Late in 1 684 , after North ' s death , the vice-m as ter an d seniors fined three fellows for " their great contemp t" in b eing p resent in th e college but not opposing in th eir turn (ibid . , p. 16 7) . In 1 693 , fines had to be imposed for nonattendance at chap el (ib id. , p. 1 87) . In 1 698 , fines were levied anew (at a much lo wer rate than North had tried to collect) for neglecting to oppose at divinity disputations (ib id . , p. 1 97) . 2 9 Cf. William Taswell ' s com m ents on his income from tutoring (Au tobiograph y , pp . 28-9) . 3 0 Master ' s Old Conclus ion Book , 1 607-1 673 , p . 276 . Cf. Taswell' s co m m ents on the observance of days of abstinence at Oxford: "These nights were so far fro m being kep t as they should be, that we com monly lived more sumptuously than usual , at inns or co ffee­ houses . " He spoke of " cram ming myself with meat and drink . . . " (A utob iography, pp. 3 1 -2) .

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No s n1all share of re sponsibility for the de teriora tion of tone belonged to the systen1 of seniority . Seniority had alw ays p layed a role in the life of the colle ge , but in the s tatutes it vied with another governing p rinciple , acade mic achievement . For exan1ple , the col­ lege bestowed s tipends according to degree; a Doctor of Divinity d rew more than twice as much as a Master of A rts . B y the Restora­ tion , ho wever, the dividend furnished the main s ubstance of a fel­ low' s supp ort , and acco rding to the definitive formula of division adop ted by the college in 1 66 1 , the year of Ne wton ' s admission , seniority alone determined the level of the d ividend . 3 1 S eniority also dete rmined virtually all the other subs tantial rewards of college life: a s si gnment of cha mbers , college offices , university offices in Trini ty' s turn in the ro tation , especially p resentation to benefice s in the college ' s gift . The S teward ' s B ook for 1 666 listed Newton (who was then a scholar) , below another s cholar , John Herring , who stood above hi m s olely because he entered Trini ty on 3 June 1 66 1 , two days before Newton. A s it happ ened, Herring did not attain a fellowship , and his n a me dropped out after 1 668 . Herring would o therwise have continued one step ahead of Newton in college preferment until death or re signation, and any a mount o f activity in tu toring , any signal serv ice to the colle ge , or twenty Prin cip ias could no t have moved Newton ahead of hi m . Nor could any record of slo th have dropped Herring back . If the dead hand of seniority lay heavily on Trini ty , a s it did on ev ery college , the thro ttling hand of pa tronage clu tched at its thro a t . I n 1 664 , as alarm o v e r the rising tide of n1anda ted fellows hips spread through the univ ersity , St. John ' s p rotested to the king that m an­ da tes wo uld inj u re the college by " causing deserving persons . . . to seek interest at court rather than proficiency in le arning . " 32 Already the system of p atronage had p erv aded the university beyond the po ssibility of el imination , however . And nowhere did it entrench itself m ore deep ly than at the p innacle of the university hierarchy . Almo s t by definition, to be a master of a college was to be the o bj ect of patronage . John North , master of Trini ty fro m 1 677 to 1 683 , inveighed against the corruption of p a tronage but accep ted that tendered to him as a matter of right . 33 Benj a min Laney , restored to the maste rship of Pembroke in 1 660 , was rewarded for his loyalty du ring the Interregnu m with fu rther p referments as dean of Ro­ cheste r , bishop of Peterb o rough , and canon of Westminster, all of which app ointments he held at the same time . The master ship of S t . John' s , the Regius p rofes sorship of divinity , a p rebend i n Canterbury , and two p a rsonages likewise rewarded Peter Gunning for his 31 32

Master ' s Old Conclusion Book, 1 607- 1 673 , p . 265 . Mullinger , The University of Cambridge, 3 , 626- 7 .

33

North , Lives , 2 , passi m .

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loyalty . 34 He was master when S t . John ' s dep lored the encourage­ ment that mandates gave to seek interest at cou rt . S mall wonder such p ro tests bore little fruit . In lesser amounts , pa tronage filtered down through the enti re university . In Trinity , in 1 661 , Na thaniel Willis o b tained a dispens ation to hold his fello wship along with a rectory , which exceeded the s tatutory limit . A year later, Hu mphrey Babing­ ton , who m a le tter mandate had recen tly restored to his fellows hip , had the s a me statute dispensed with so that he might hold the rectory in Boo thb y Pagnell . Two years more an d Robert B ore man received the s a me p rivilege by dispens ation . 35 Ca mbridge might dep lore let­ ters mandate for others , but men who wanted letters mandate fo r themselv es filled Camb ridg e . A g o o d three-qu arters of the s tudents p roceeded to careers in the chu rch; they knew fu ll well whence came livings , as they were frankly called . N o thing is more revealing of the university than the ritual vol­ u mes of Latin verse published on eve ry occasion tha t tou ched the C ro wn . To fi ll these p rodu ctions , eve ry ambitious man in C am­ bridge period ically tortured his muse for his qu o ta of lines in o rder that hi s name not be mis sed . Ma sters and p rofes sors see med p ar­ ticularly anxious to appea r . None labored more dil igen tl y in the p rodu ction of verse than Is aac B a rro w , recently Regius P ro fessor of Greek a n d p resen tly Lucasian Professor of M athe matics , b u t clearly hung ry for more . Lacrymae Can tabrig ienses p ro claimed the title of the 1 670 edition , which mourned the death of Charle s ' s s ister b y poison , a hideous co mpila tion of sycophancy un moistened by any suggestion of a real te a r . Andrew Marvell ske tched the whole unive rsity when h e cha rac­ terized Francis Turne r , master of S t . John ' s , as ' - thi s close y outh who treads always upon the heels of Ecclesia s tical Prefe rment . "36 To savor the full a mbiguity of the situation , real ize tha t New ton , who never contributed to a valedictory volu me , and stood reso­ lu tely a side fro m the scra mble for p la ce , would in less than ten yea rs o we hi s s u rvival in C a mbridge to a manipulation o f the s tat­ u tes b y royal dispensation . There was no one in Ca mbridge who did no t have a favor to seek . The colleges themselves held extensive p a tron age, which they dispensed, inevitably , according to seniori ty . Trinity alone con34

M ullinger , The Un iversity of Cambridge, 3, 566- 7. B aker , His tory of St. John 's, p. 236 . C hurchill , Dispensing Power, pp . 3 1 0- 1 1 . Conclu s ion Book , 1 646- 1 8 1 1 , p . 68 . Trinity Co llege, Box 29 , D . C f. D . A . Wins tanley , The University of Cambridge in the Eighteenth Centu ry (C ambrid ge , 1 922) , which focuses p rim arily on the role of p atron age in the university , exercised now , not b y the mon arch , but by the Du ke of Newcastle, a p arlia­ mentary magn ate who was chan cellor of the uni vers ity for twenty years in the mid dle o f t h e century. 36 Andrew Marvell , Mr. Smirke: Or the Div ine in Mode, in The Complete Works of A ndrew Marvell , ed . A lexander B. G rosart, 4 vols . (priv ate circul ation , 1 872-5) , 4, 1 1 . 35

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trolled nearly fi fty advo wsons and added more continually . 3 7 It took a fellow nearly twen ty years to achieve the senio ri ty requi site fo r a college living . Once he had wai t�"d i t o u t , he could if he chose resign his fell o wship , bela tedly marry , and with inco me secure rear a family . In aJdition , ma s ters bestirred themselves in recommend­ ing fellows as chaplains , tuto rs , and vicars . Or if, for whatever rea son , marriage held no cha rms , a fello w with seni ority co uld beco me a college p reacher, which besto wed the right to hold a living in conj unction with a fello wship . In ei ther case , the fellowship rapidly came to appear p rimarily as a mod e of support, an en d in itself rather than a means to an academic end . Everybody trea ted i t a s a freehold which i mposed no obligations in return fo r its benefits , which were considerable . Its inco me did not even require one' s p resence . The S teward 's Books of T rini ty indicate that yea r in and year out more than a quarter and often more than a thi rd of the fellows resided in Camb ridge less than half the year; more than 1 0 percen t never showed up at all . The fa te of the s ta tute de m o ra in E mmanuel and Sidney Sussex college s , the two p reeminent Puritan ins ti tution s , is rev ealin g . Sir Walter Mild may founded E mman uel to enrich the church with an educated clergy; hence the s tatute de m o ra , which limited ten ure to ten years beyond the M . A . degree . A fello wship , Mildmay in sisted, was n o t mean t to p rovide a permanent abode . Under Lau d, the statute was revoke d . The revolutionary P uritan government resur­ rected i t . With the res toration , it disappeared fo rever . By the late seventeen th cen tury , s to ries of the effects of these changes on the fellows the mselves filled the air both of Cambridge and of Oxford . Freed fro m obligations , shorn of useful functions , the fellows su rrendered meekly to a life of indolence and boredo m relieved primarily by the solace of the table and the tave rn . In the splendid phrase of Roger No rth , they became " wet epicu res , " suc­ cumbing to a co rpulen t letha rgy liberally fla vo red with alcohol . 38 William Whis ton told of a s tudent at Clare with hi m in the 1 690s who decided that a reputation as a heavy drinker was the surest p a th t o a fe ll o w ship . 3 9 A t mu ch the s a me t i me in O xfo rd , 37

They a re li s ted in the Trinity Register , volumes for 1 664- 72 , 1 673-80 , 1 680-8, and 1 688- 1 702 . Four grants o f advowsons to the college in 1 667 , 1 673 , 1 678 , an d 1 68 1 are in Miscellaneous Papers Relating to Trinity College, vol. 3, Nos. 2 1 , 27, and 39 . In a m an uscript fro m a bout 1 736 , Tho mas B aker listed the num ber o f livings held by the v arious colleges ; by that time Trinity held seventy ( more than twice as m any as any o ther co llege) , ten o f which h ad incomes above £ 1 00 p er ann um (Cam bridge Uni versity Library , Baker M S S 3 8 North, L ives , 2, 272 . M m . 1 48) . 3 9 W illiam Whi ston , Memoirs of the Life and Writings ef Mr. William Wh iston ( L ondo n , 1 749) p. 1 29 . In 1 727, Louis de J aucourt, a co llaborator of Di derot's in the Encyclopedia , vi sited Cambridge; " whoever is ignorant o f the art o f d rinking a lo t and smokin g a lot , " he wrote in a letter , "is very un welco me in this Uni versity . . . " (qu oted in A r thur M . W ilson , D iderot [New York , 1 972 ] , p . 4 8 1 ) . .

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Hu mphrey P rideaux, when asked to recommend a college for hi s nephew , replied th at most of the men in au thority in the colleges the re were " s u ch as I cou ld sca rce co mmi t a dog to thei r charge . " 4 0 In its o riginal concept , a fellowship was meant to support lea rn­ ing . " When any person is chosen fello w of a college , " Nicholas A mhers t decla red early in the eighteen th centu ry , " he i mmediately beco mes a freeholder, and is settled for life in ease and p len ty . . . He wa stes the re s t of his days in lu xury and idlenes s : he enj oys hi mself, and is dead to the world: for a seni or fellow of a college lives and mo ulders away in a supine and regular course of eating, drinking, sleeping , and cheating the j uniors . " 4 1 A mhers t was a sati ri s t whose words should not be mis taken for disinteres ted de­ s crip ti on , but D r . Johnson was not pla ying the satiri s t when he visited his old O xford college, Pembroke , in 1 754 and met the Reverend M r . Meeke , a s tu den t with Johnson who had envied hi s learning , and no w a fellow of the colle ge . " About the s ame time of life , " John son said to hi s friend Wharton as they left , " Meeke was left behind at Oxford to feed on a Fello wship, and I went to London to get my living: now , Sir, s ee the differen ce of our literary characters . " 4 2 Early in the seven teen th cen tu ry , Trini ty had been the leading academic ins ti tu tion in Englan d . It had nu rtu red six of the tran sla­ tors of the A u thorized Version , more than any o ther college in Oxford or Camb ridge; it had fu rni shed more bishops to the Chu rch of England than any o ther foundation . Likewi se, Camb ridge had functioned as the point of fermen t in English intellectual life . Tradi­ tions do not die at once . John Pearson , maste r of Trinity from 1 662 to 1 673 (du ring Newton ' s early years there) , was one of the most es teemed clergy men of hi s age , and hi s su ccessor, Isaac Barrow, was an ou tstanding scholar of many facets . Elsewhere in the uni ver­ sity the Cambridge Platoni sts s u rvived for a ti me fro m the earlier age . A s they died off, no one replaced them . Early in the eighteenth cen tu r y , the Ge rman t raveler Zachari u s v on Uffenbach visited C amb ridge . His reci tal of the wretched s tate of the unive rsity and colleges i s a dep ressing chronicle of the level to which they had fallen . Though he did no t meet the university librarian , D r. Lau gh4 0 Quoted in Cha rles Ed ward Mallet , A History of the Unversity of Oxfo rd, 3 vols . (London 1 924-7) , 3, 5 6 . Cf. the diatribe o f D r . John E d wards written a bout 1 7 1 5 , agains t the "laziness and debauchery " of the fello ws o f the v arious colleges (quoted in James Bass Mullinger , St. John 's College [London, 1 90 1 ] , pp . 203-5) . Edwards was then a very old man, but it is no t possible to pass off what he says entirely as the du ckings o f the old ag ainst the young. S ee also the descrip tions of life in Resto ration O xfo rd ( Mallet , Oxfo rd, 2, 422-3) , in Jacob ite O xfo rd (ibid . , 3, 1 - 55) , and in eigh teenth-century Cambrid ge (D . A . Wins tanley , Unrefo nned Cambridge [C ambridge, 1 935] , pp . 256- 67) . 4 1 Q uoted in A. D . Godley , Oxfo rd in the Eighteenth Centu ry (London, 1 908) , p . 77. 4 2 James Bo swell , The L ife of Samuel Johnson L . L . D . (New York, n. d . ) , p. 1 6 1 .

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ton , who was out of to wn , he heard Laughton extolled a s a man of great learning . " Ra ra a vis in his terri s , " Uffenbach rema rked ta rtly to hi s diarv . 43 To be sure , Uffen bach rather made a ca reer of denigrating all things English . A mple eviden ce elsewhere confirms his j udgmen t , however . C a mbridge was fast app ro aching the s tatus of an intellectual wasteland . Consider the fo rty-one men who became fellows of T rinity in the th ree elections of 1 664 , 1 667 , and 1 668 . One of the fo rty-one was Newton , of cou rse . Of the o thers , Ro bert Uvedale beca me a pro mi­ nen t edu cator and horti culturist; he p u rsued his career en ti rely o u t­ side C a mbridge tho ugh he held onto his fellowship for fifteen years . Edward Pelling , who resigned his fell o wship after one yea r , went on to become an Anglican polemicist of heroic p roportions though his writings a re , I believe , virtually unknown today . Samuel S ca tter­ go o d , who held his fell owship for si xteen years , publ ished many sermons; Henry Do ve , Geo rge Seignior , and William B aldwin all published a small number . John B atteley gained s ome pro minence as an antiqu a rian after his tenure of seven teen years in his fellowship . John Allen , a fellow fo r thirty years , mo stly in absentia , published one sermon with the intrig uing title , in view of the oaths he had taken , " Of Perj ury . " Newton as ide , they do not fo rm an imposing g roup of in tellec tuals by any s tan dard . Nor were they mo re impres­ siv e as tutors . Four chose the role of p upil mong er, in the pej o rative phrase of the da y . Of the o ther thirty-seven , only ten eve r tutored a pupil , and those ten tuto red a total of sixteen . Newton w ith three and Wickins w ith two accoun ted for five of the s ixteen . The average tenure of th e forty-one was seventeen and a half yea rs ; eleven stayed mo re than twenty yea rs ; and four fulfilled Mild may ' s fears by mak­ ing the college their permanen t abode . Geo rge Modd , Patrick Co ck, Willia m Mayo r , and Nicholas Spencer all stayed on at Trinity over fo rty years . After more than thirty years , Spencer did take a degree as D o c to r of D iv inity; the o ther three took no further deg rees . None of the fou r ever tuto red a pupil . None of the fo ur ever p ubl ished a word . All survived to beco me senio r fello ws of the college and to reap its ripest rewards . After his creation as Master of A rts , New ton l ived in Trinity fo r twen ty-eight years . Those years coin cided roughly with the mo st disa strous period in the hi story both of the college and of the uni­ versity . Whatever his ini tial expectations may have been , he d id not find a congenial circle of fellow s chola rs . A philo sopher in search of �

4 3 J . E . B . Mayo r , Cam bridge u nder Queen A nne ( C ambridg e, 1 870) p . 1 40 . A cco rding to Offen bach, J ean LeClerc , the H uguenot s cholar, shared his o p inion . LeClerc complained to him "of the great laziness of Englishmen , and j u stly too; enjoying such large ben�ficia and noble libraries , they p rodu ced very little in the way of learning; which is only too true, w ith a few bright exception s " (ibid. , p. 427) .

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truth , he found him sel f ainong p lacemen in search of a p la ce . Thi s funda mental fact col ored the s cene in which virtually the whole of his creative life w as set . Against thi s ba ckground , we can read the vario us anecdotes that have s u rvived about his life in the college . Three so urces furnished mo s t of the s tories: William S tukeley , a s tuden t at Cambridge early in the eighteenth century and later a friend of Newton , who made it hi s b u s iness to co llect info rmation about him after his death ; Humphrey Newton, w ho served as Newton ' s a manuensis i n Cam­ bridge fo r five years in the 1 680s and wro te two letters about the experience after Newton ' s de ath; and Nicholas Wickins , the son o f John Wickins , who wrote abou t h i s fa ther' s recol lections t o Robert S mith soon after the death of Newton and eight years afte r the death of his father. The firs t two were no t enti rely independent s in ce b o th men l ived in G rantham in 1 627-8 and conferred to­ gether. S o me of S tukeley' s ane cdo tes sound enough like H umphrey Newton ' s tha t they seem to derive fro m him instead of fro m Cam­ b ridge . Humphrey Ne wton lived with Isaac Newton during a unique perio d while he was co mpo sing the Prin cipia . Perhaps we sho uld exercise some caution in treating his recollections as typ ical , though Newton ' s capacity to be do minated by a p ro blem did not confine itself to the Prin cipia . S tukeley rep orted tha t stories of New­ ton ' s absentmindedness were rife in Camb ridge . A s w hen he has been in t he hall at dinner, he has quite neglected to help himself, and the cloth has been taken away before he has eaten anything . That so meti me , when on surplice day s , he wo uld goe towa rd S. Mary ' s church, insted of college chapel , or perhap s has gone in his surpl ice to dinner in the hall . That when he had friends to ente rtain at his chamber, if he stept in to his s tudy fo r a bottle o f wine , a n d a thought came into hi s head , he would sit down t o paper and fo rget his friends . 44

Hump hre y Newton ' s chao ti c strea m of conscio usness contained s imila r re collections . He alwayes kep t cl ose to hi s studye s , very rarely went a visiting , & had as few V isiters , excep ting 2 or 3 P ersons , Mr Ellis o f Keys, Mr L o ugham [called Laughton in his other letter] of Trinity , & Mr Vi44 Stu keley , p . 6 1 . Nichol as Wickins reported that his father also called the stories of N ewton forgetting meals " wh at i' world has so o ften heard of s r Is aac . . " (Keynes MS 1 37) . .

The s tory about the su rplice was h eard and reco rded by Thomas P arne, among p ap ers he ass em bled early in the eigh teenth centu ry for a history of T rinity . He attributed it to a M r . B u rw ell, who Edleston sp ecu lates m ay h ave been A l exander Bu rrell , eleven years P arne' s senior in T rinity and possibly rel ated to a chap lain o f that name in Trin ity fro m 1 673 to 1 681 . " N ew ton h ath come into the H all witho ut his B and , an d w ent tow ard S t . Maries in his su rplice" (Edleston , p . l xx x) . On college s u rplice days , fellows were requ ired to w ear their s u rplices to the co llege chap el , not to S t . Mary' s , which was the university chu rch , and ob viou sly not to dinner in the hall .

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gam , a Chymi s t , in who se Co mpany he took much Delight an d Pleas ure at an E vening , when he came to wait up on Hi m . I never knew hi m take any Recreation or Pasti m e , either in Ridin g o ut to take ye Air, Walking , Bowling , or any other E xercise w hatever, Thinking all Ho urs lost, y t was not spent in hi s stud yes , t o wc h he kep t so clos e , y 1 he sel d o m left his Chamber, unles s at Term Time , w hen he read in ye schools , as being Lucasianus P rofesso r . . . He very ra rel y went to Dine in ye Hall unless upon so me P ublick D ayes , & then , i f H e has not been minded , would g o very carelesly, w th S hooes down at Heel s , S tockin s unty' d , surplice on , & his Head scarcely co mb' d. 45

"He would with great a cu tness answer a Q uestion , " Humphrey added in his second lette r, " b ut would very seldom start one . " 46 During five yea rs , Humphrey saw Newton laugh only once . He had lo aned an acquaintance a copy of Euclid . The ac quaintan ce asked what use its s tudy wou ld be to him . " Up on which Sir I s aac was very merry . "47 It is not hard to re cognize in the se anecdotes the man who uncon­ s cio usly sketched hi s own po rtrai t in his papers , a man ravished by the des i re to kno w . Equally , it i s not hard to recognize his s tatus in Trinity - i s o lation , indeed alienation . True , S tukeley mentioned friends being en tertained in his chamber, and Humphrey New ton na med three of them . The refe ren ces sca rcely suffice to erase the impres s ion left by the res t . Newton seldo m leaves hi s ch amb e r. He p refers to eat the re alone . When he does dine in the hall , he is hardly a genial co n1panion; ra ther he sits silently , never ini tiating a conversation , as is olated in hi s p rivate wo rld as though he had not co m e . H e does not j oin the fellows on the bowling green . He rarely visits o thers . None of those who visit him are fello ws of T rini ty. Of the three , we know that Newton later broke with Vigani be­ cause he " told a loose s tory about a Nun . . . "48 Wh a tever his 4 5 Keynes MS 1 35 . John Lau ghton , ad mitted pensioner t o Trinity i n 1 665 , w a s elected s cholar but not fello w . He did beco me a chaplain in the college, howev er , and college librarian . Later he was u niversity librarian from 1 686 to 1 7 1 2 , the one referred to by Offenb ach . John Ellis , admitted sizar to C aius in 1 648 , was a fello w there from 1 659 to 1 703 and an eminent tutor. Elected master in 1 703 , he was serv ing as vice- cha ncello r of the uni versity in 1 705 when he was knighted on the s a m e d ay as Newton . John Francis Vigani came to Cambridge abou t 1 682 and taugh t chemis try there in form a lly fo r twenty years b efore the university con ferred the title o f professor of chemi stry on him; i t did not confer any stipend to go with the title. At least p art of the time, he exp erimented and taught in T rinity, wh ere his laborato r y was app a rently in the ro om ( s till c alled Vigani 's room) which Newton himself had u sed for that p u rpose ea rlier . 4 6 Keynes MS 1 35 . 47 Stu keley , p . 5 7 . Stukeley a s cribed the story to H u m ph rey , who a llu ded to it without giving the whole since Stukeley had already relayed it to Conduitt. 48 Keynes MS 1 30. 6 , B ook 2. Con duitt a ttributed thi s s tory to hi s wife , Catherine , New­ to n ' s niece.

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friendships with Laughton and Ellis , they were not clos e enough to elicit co rrespon dence fro m ei ther side after Newton left Cambridge. On 18 M ay 1 669 , N ewton wro te a le tter to Franci s As ton , a fellow of Trini ty who had received leave to travel ab road and was then departing . According to the le tter, As ton h ad asked his advice about trav eling , and Newton co mp lied rather fully . The b ulk of the letter was worldly advice cribbed from a discourse s till fo und among th e Newton papers: " An Abridgemen t of a Manuscript of sr Robert Southwell ' s concerning travellin g . "49 A s ton should adapt his behavior to the co mpany he is in . He should ask questions b ut not dispute . He should praise what he sees rather th an cri ti cize . He should realize that it is dangero us to take offen se too readily abroad . He should obs erve various things abo u t th e economy , society , and governmen t of the co un tries he v i sits . A final parag raph added a number o f particu lar enquiries that Newton wished Aston to make , mos tly about alchemy an d mo s tly based on Mich ael Maier, Symbola aurea e mensae duodecim nationum (F rankfort, 1 6 1 7) . s o The letter to As ton is among the mo s t elo quent in Newton ' s correspondence . Not fo r i ts conten t-with i ts b orro wed ai r of worldliness , the letter itself i s more lu dicrous th an eloquen t . I t is fo und today among Newton ' s own papers , which suggests th a t he recognized he was cutting a ridiculous figu re as he as sumed a wo rldl y pos ture on the basis of one month in London and an essay by South well, an d decided no t to send it. The eloquen ce of the letter lies in i ts unique­ ness . It i s the only personal letter to or fro m a peer in Cambridge in the whole co rpus of Newton ' s co rresp ondence . In i ts uni quenes s , it add s col or to the po rtrai t of isolation in S tukele y ' s and H umphrey Newton ' s anecdotes . So do surviving accounts of the coll ege th at were collected in the second decade of the eighte enth century . Thomas P a rne, B . A . 1 7 1 8 , collected materials fo r a hi s tory o f the college , including the recollections of elderly fello ws such as George Modd . He reco rded particulars about Ray, Pearson , Barro w , Tho rnd ike , and Dupo rt . Newton was a fa mous man when Parne was d rawing his materials together, far mo re famous than the men abo ve , but only three references to hi m appear in the collection: his name (w ithout an y co mmen t whatever) a t the head of the lis t of writers ; the dates o f h i s elections t o Parliament and of h i s later unsuccessful b i d fo r election; and one b rief anecdote abo u t his absence of mind . 5 1 A t much the s a m e time , J ames P aine , w h o w a s elec ted t o a fello wship in 1 72 1 , s et down a conversation with Robert Creighton , who had been a fellow fro m 1 659 to 1 672 . Creigh ton recalled Pearson , 49 Keynes MS 1 52 . Co rres 1 , 9- 1 1 . Newton's early notes o n M aier are i n Keynes MS 2 9 . 5 1 Edlesto n , p p . l xxix-l xxx . so

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D ryden , Gale , Wilkins , and B a rro w; he did not mention Newton . 52 Nei ther did S amuel Newton , who as registrar and auditor was emp loyed by the college du ring the whole of Newton ' s tenure as fellow , en ter his name in the diary he kep t until Newton ' s ele ction to Parlia ment in 1 689 . 53 To be sure , S amuel New ton filled his diary more with events than with references to individu als . Nevertheless , i t seems evident that Newton did no t loo m p ro minen tly on the coll ege s cene . Two s to rie s do suggest tha t for their p a rt the other fell ows , whatev er thei r amus emen t at his absen tmin dedness , rega rded him with awe. In 1 667 , he seemed to have prophetic powers when the D u tch fleet in vaded the Thames . Thei r guns were heard as far as Cambridg, and the cause was well known; but the event was only co gni sable to Si r I saac' s sa gacity , who bo ldly pronounc'd that they had beaten u s . The new s soon confi rm' d it, and the cu rious would no t be easy whilst Sir I saac sati s fy ' d the m of the mode of his intel ligence , which was this; by carefully attending to the so und , he found it grew louder and louder, consequently came nearer; fro m whence he rightly infer' d that the Dutch we re victors . 54

When he walked in the fellows ' ga rden , " if some new gravel hap­ pen ' d to be laid on the walks , it was sure to be drawn over and over with a bit of s tick , in Sir Isaac' s diagra ms; which the Fello ws w ould cau tiously sp a re by walking beside the m , and there they w ould so metime remain for a good while . " 5 5 As fa r as we know , Newton formed only three close connections in T rinity , all of them ra ther elusive . With John Wickins , the young pen s ioner he met on a solitary walk in the college , he continued to share a cha mber until Wi ckins resigned his fello wship in 1 683 for the vicarage of S toke Edith . Wickins was frequently a bsent for extended perio ds , and during his final five years he was ha rdly there at all . When Robert S mith , Plu mian p rofessor of natural philoso phy 5 2 Quoted in W . W. Rouse Ball , Notes on the History ef Trinity College Ca m b ridge ( Lon don , 1 899) , pp . 97-9 .

53 Foster, Samuel Newton 's Diary , pp . 97-8 . He merely gave his na me and title . Newton ap peared only one other time, when William I II p ropo sed to appoint him p rovost o f King' s l a t e in 1 689 (ibid . , p . 1 02) . Samuel Newton did n o t e v e n mention h i s knighting, though he did men tion that o f his kinsman, John Elli s , on the same day . App arently the two Newtons were acqu ainted since Newton loaned " M r Newton " 1 8s in 1 665 (Fitzwil­ liam notebook) . 54 Stu keley , p p . 58-9 . This must have been a common sto ry , becau se it survived in a second account as well , though pla ced on this o ccasion in 1 672 with the battle of South wo ld B ay . E dleston quotes it from Jo hn Ni chols, The History and An tiqu ities of Hinckley (London , 1 780) , p . 6 1 n (pp . xlvi-vi i) . 55 Stukeley , p. 6 1 . Stukeley referred this s tory to Humphrey Newton ; hen ce it belonged to the time when N ewton w as co mposing the Principia .

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wro te to Wickins ' s son N icholas after Newton ' s death , he could find only three short le tters to his fa ther, which he thought were no t even worth tran scribing , plu s four or five other very short letters in which Newton fo rw a rded rent and dividends . Although it i s no t clear when the first three le tters were wri tten , i t appears tha t Newton severed connections with his cha mber fellow of twen ty yea rs . Nicholas Wickins d id say tha t Newton p aid for Bib les to be distributed among the poor in the parish . 56 The relation w ith Wick­ in s was special and the break with him probably more s ignificant than the b reak w ith Vigani . In addition to Wickin s , there were relationships with Humphrey Babing ton and I s aac B a rro w about which we know even les s . The aliena tion fro m college s o ciety wo rked to Newton ' s adv an­ tage . As D r. Johnson ' s friend , the Reverend M r . Meeke , was to learn , the increasing triviality of the fellows ' lives could en tang le a pro mising man and des troy him . Passiona tely in clined to s tudy in any case , New ton tu rned away fro m his pee rs and in upon himself and su rrendered comple tely to the pursuit of kno wledge . The S tew­ a rd ' s Books of the college sho w tha t he seldom left . In 1 669 (which mean s , in the college books , the twelv e months th a t ended with Michael mas , 29 Septembe r 1 669) he was there all fifty-two weeks; in 1 670 , fo rty-nin e and a half; in 1 67 1 , fo rty-eight; in 1 672 , fo rty­ eight and a half. When he did leave , i t was usually fo r a t rip ho me . A decade later, Hu mphrey Newton found tha t he seldo m went to morning chapel since he s tudied un til two o r three ev ery morning . Fo r th a t matter , he seldom interrupted his s tudies to a ttend evening chapel either, though he did go to chu rch in S t . M a ry' s on Sun­ days . 57 " I believe he grudg ' d yt sho rt Time he spen t in eating & sleeping , " Hu mphrey New ton observed . 58 The Reverend John North , master of the college fro m 1 677 to 1 683 and resident in i t fo r a time befo re tha t , who rather fancied himself a s cholar, " be­ lieved if S ir Isaac Newton had no t wrou ght with his hands in making expe riments , he had killed hi mself with s tudy . " 59 The laxity of the sys tem , whi ch had helped him already as an undergraduate , continued to favor hi m . If it de manded no thing of the Geo rge Modds and P atrick Cocks , like wise it demanded no th­ ing of him . His u se of his leisure may have unsettled the others , but the essence of the sys te m was tole ran ce . Unrelenting s tudy on a fello wship intended to support s tudy was not demonstra bly more subve rsive than dra wing d ividends in ab sentia . Suppo rted co mfor­ tably , N ewton was free to devote himself wholly to w hateve r he chose . To remain on , he had only to avoid the three unfo rgiv able 5 6 Keynes MS 1 37 . 57 Stukeley , p . 60 . Stukeley referred th e inform ation to Humphrey Newton . 59 North , Lives , 2 , 284 . 5 8 Keynes MS 1 35 .

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sins : crime, heresy , and marriage. Safely ensconced with Wi ckins in the orthodox fastness of Trinity College, he was not likely to sa cri­ fi ce hi s secu rity for one of them . Beside s the three topi cs of the anni mirabiles a new subj e ct began now to engross him . His accounts show that in 1 669 he spent 1 4 s fo r "Glasses" in Cambridge and 1 5s more for the same in Lon­ don . 60 He made some other pu rchases in London as well . �

For A qua Forti s , subli mate , oyle perle [sic - per se? ] fine Silver, Antimony, vinegar Spirit of Wine , White 2. lea d , Allome Nitre, S alt of Tartar , � 0. 0. 8 . A Furnace A tin Furnace 0. 7 . Joyner 0. 6. Theatrum Chemicum 1. 8.

0 0 0 0 061

He also paid 2s to have the oil transported to Cambridg e. Theatrum Ch em icum referred to the hu ge comp ilation of alchemi cal treati ses publi shed by Lazaru s Zetzner in 1 602 and re cently expanded to six volum es . More than woodworking was going on in the chamber shared by the long-su ffering Wickins . Years later N ewton re­ marked to Conduitt that Wi ckins , who was stronger than he, u sed to help him with his kettle, "for he had several furnaces in his own cha mbers fo r ch y mical exp e ri ment s . " 62 W hen Newton ' s ha i r turned gray early in the 1 670s , Wi ckins told hi m it was the effect of hi s concentration . Newton , whom Hu mphrey Newton saw laugh only once , would j est that it was " ye E xperimts he made so often w1h Quick Silver, as if fro m Hence he took so soon that Colou r . " 6 3 Chemi cal exp erimentation had medical implications also , and Newton had more worries than g ray hair. He su spected he had tub erculosi s , for whi ch , Wi ckins recalled , he treated him self with Lucatello ' s Balsam . As it hap pens , Newton ' s formula for "Luca­ tello's Balsome" has su rvived, a heady mixture of turp entine, the be st damask rosewater , beeswax , olive oil , and sack , flavored with a pinch of red sandalwood and a dash of oil of St. John ' s wort . A cco rding t o Newton ' s recipe, i t w a s good fo r measles, plague, and smallpox, fo r all of whi ch one was to mix a quarter o f an ounce in a little b roth , take it warm , and sweat afterwards (which should have been no problem) . For the bite of a mad dog , one appli ed it to 60 The T ri ni t y Exit and -Redit Book shows that he was ab sent from 26 N ovember to 8 D ecem ber (Edleston , p . lxxxv) . This is the absence shown in the S teward's Book for 1 670. 61 Fitzwilliam notebook. 62 Conduitt's memorandu m of 31 Aug . 1 726; Keynes MS 1 30 . 1 0 , f. 3v . 6 3 Nicholas Wi ckins to S mith; Key nes MS 1 37 . Humph rey Newton also reported that New­ ton was g ray by thirty (Keynes MS 1 35) .

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the wound as well , while wind colic , g reen wounds , sore breasts , burns and bruises were treated by external application alone . 64 Al­ though " Lucatello 's Balso me" did not clai m to be effective again st consu mption , Newton apparently concluded that such a sovereign remedy would not likely fail in thi s , either . As Wickins reme m­ bered i t , " when he had compos' d Hi mself, He would now and then melt in Q uantity ab1 a qr o f a Pint & so drink i t . " 6 5 Meanwhile , chemis try was no t his only study . In 1 669 , he took up mathematics and optics again . Probably New ton devoted at mo st a bare modicum of attention to mathematics during 1 667 . Perhaps he co mposed a paper on analytic geo metry and translated a short flu xi onal essay into Latin at th at time; it is impossible to date the m with assu rance . 66 Two pieces related to analy tic geometry rather than the flu xional calculu s pos sibly stem fro m 1 668 . One , on the o rganic cons truction of conics , generalized individual meth ods of generating conics , which he had met in Scho oten ' s work , in to a device that could describe the general coni c with a con tinuous , unin terrup ted mo tion . Two angles fixed in size ro tate abo u t poles d and e ; their legs intersect at points b and p (Figure 6 . 2) . When interse ction p traces a given cu rv e , the directri x , in tersection b de­ scribes a new cu rve, the des cribend, whi ch is either of equ al degree o r , when the directrix embo dies certain conditions , of higher de­ gree . With certain rectilinear directri ces , b can describe a conic through d and e . Si milarly Newton argued that the device can trace a cubic and a quartic fro m a conic , and a quarti c , a quin tic a nd a sexti c fro m a cubic directrix o f suitable properties . 6 7 Twenty years la ter, he inco rpo rated the bu rden of the paper in to Section V , B ook I, o f the Prin cip ia . The other paper that may date fro m 1 668 , though it may also belong to 1 670 when John Collins was thru sting rela ted questions befo re Newton , is more significant. 6 8 En umeratio cu rva ru m triu m dimension um ( The Enumeration of Cubics ) was a sophisti cated exercise 6 5 Keynes MS 1 37 . 6 4 S tanford University MS 538 . 6 6 " P roblems o f Curves " ; Math 2 , 1 75-84 . D e solutione p roblematum p e r motu m ; Math 2 , 1 94200 .

6 7 T h e p aper exists in four succes sive forms; Math 2, 1 06-50 . T h e fi n a l v ersion (pp . 1 34-50) bears the title De modo describendi con icas sectiones et cu rvas trium dimensionu m quando sint p rim i g rad us. &c . Whiteside dates it p ro v i sionally to late 1 66 7 o r 1 66 8 . 68 D ating papers b y t h e h a n d in wh ich they are w ritten is chancy business at b e s t . O n that ev iden ce alone , an d w ith explicit reserv ations about it, Whiteside p laces this p aper in late 1 667 or 1 668 . F rom such a time, the p ap er springs b efo re us w ithout motive and without intimate connection to the main themes o f Newton ' s mathematical work . 1 670 at least p rovides a motiv e, the qu estions Collins persi sten tly rai sed about cubics . See W. W. Rouse B all , " O n Newton ' s Classification o f C u bic C u rves , " Proceedings of the L ondon Mathematical Society , 22 ( 1 890- 1 ) , 1 04-43; and H. Hilton , " N ewton on Plane Cubic Curves , " in W. J. Green s treet , ed . , Isaac Newton , 1 642- 1 727 (London , 1 927) , pp . 1 1 5- 1 6 .

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q

Fi gure 6 . 2 . N ewton ' s organi c construction of curv es . The two fi xed angles bdp an d hep pivot on the points d and e. A s th e inters ecti on p traces th e directrix pq , inters ecti on b traces th e descri b end be , de , a conic (in this cas e , a hyp erbola) through d an d e .

i n analyti c geo metry whi ch burst the swaddling clothes i n whi ch the infant s cience had been nou rished and extended it successfully to curves of higher o rder than the conics . To be s u re, analysis had confronted cubi cs before, but in 1 668 only five individ ual cubics had been described , and they imperfectly . When Newton had tried to describe two cubics in 1 664, he had failed to do so adequ ately . What he now undertook was a systemati c taxonomy of cubi cs whi ch classified them into cases , species , and (within species) forms and the grades they can take on . He began with a kinematic descrip tion of curves . The line BC (the describer) moves on the base AB , with whi ch it maintains a fixed angle , while the point C moving on BC traces the cu rve CE (Figure 6 . 3) . That i s , AB and BC a re the two coordinates of the curve.

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A

0(

B

E

Figure 6 . 3 . The re duction of the cubic with coordinates AB , BC to the different coordinates af3 , {3C .

And since any fixed line af3 may be the base and the describer {3C may be ordinate to it in any given angle , it is evident that the natu re of the same single cu rve may be expres sed in an infinity of ways . It i s o u r p u rpose to s how i n what manner one m a y arrive a t t h e simplest and thereby enu merate and determine the sp ecies of curves . 69

Newton then pro ceeded to show tha t by a suitable choi ce of coo rdi­ nates the gene ral cu bic ai3 + bxy 2 + cx2y + dx :3 + ey 2 + Jxy + gx2 + hy + kx + l

=

0

can always be reduced to eli minate fou r te rms . The conversion to ne w coo rdinates involved a he rculean exerci se in algeb ra which pa ssed throu gh an equation wi th eighty-fou r term s . He emerged su ccessfully with the canonical equation of the general cubic b xy 2 + dx:3 + g x2 + hy + kx + l

=

0

He went on to illu strate nine cases of cubi cs , and wi thin the cases sixteen species . Wi th the further divisions of species into forms and grades , he arri ved in all at fifty-eight distinct types of cu bics , all of which he plotted with some care (Figu re 6 . 4) . As an aid in classifying and pl o tting , Newton de fined the concept of the diametral hyper­ bola , a conic hyp erbola drawn to the axi s and one of the asymptotes of the cubic . Like the diameter of a coni c , the diame tral hyperbola bisects cho rds (including tho se between two b ranches of the cu bic) and cu ts the cu rve at extrema . A qua rti c equation , ob tained from the cubic and diame tral hyperbola solved simu ltaneou sly to yield the extrema , bore much of the b u rden of Newton' s taxonomy . 7 0 6 9 Math 2 , 1 1 . 70 The Enumeratio went thro ugh three version s : a fi rst incomplete one (Math 2 , pp . 1 0- 1 6) ; a second incomp lete one (ib id . , pp . 1 8-36) ; and the final one for that time (ibid . , pp . 36-84) .

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F i gure 6 . 4 . Newton ' s drawing of some of the cubic s . (Courtesy of the Syn dics of Cambridge University Library . )

In h i s treatment o f cubi cs , Newton amply displayed the power of his geo metric imagination . In analogy to the continuity of conics , whereby the hyperb ola p asses into the parabola, the ellipse, and finally the circle , so in th e cubics " the fi rst [case] passes into the second via the thi rd, that i s , as d p asses into - d via O; . . . the third pas ses into the fourth and fifth , the fourth into the fifth via the sixth ,

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�·

and the first three in to the seventh , the seventh into the eighth and the eighth into the nin th . " 7 1 A s eparate paper co mpo sed at much the same ti me , perhaps as an introdu ction to the enumeration , simila rly set out to general ize such characteristics and properties of coni cs as o rdinates , dia mete rs , axes , vertices , and asympto tes to make them app licab le to curves in general . 7 2 In all , New ton ' s work on cubics 7 1 Math 2, 99 . 7 2 E arly draft s , Math 2 , 90-2; fini shed v ersion , ibid . ,

pp .

94 - 1 04 .

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raised the study of ana lyti c geo metry to a new plane of generality . As with his meth od of fluxions , he thru st it back in his desk , and there it remained until he du sted it off twenty-five years later and finally published it after still another de cade . In 1 669, events focu sed Newton 's attention once more on his fluxional method and forced him to take it fro m his desk . Though he did not then publi sh i t , at least he made it known . Toward the end of 1 668 , Nicholas Mercator published a book, Logarithm otech­ nia , in whi ch he gave the series for log ( 1 + x ) , whi ch he had derived by simp ly dividing 1 by ( 1 + x ) and squaring the series term by term . As the title suggests , he realized that the series offered a simpli fied means to calcu late logarithms . S o me months later - the exact time is unknown , but it ap pears fro m the dates of follow ing events to have been in the early month s of 1 669 -John C ollins sent a copy of the book to Isaac Barro w in Cambridge . Collin s was a mathemati cal imp resario who had made it his busi­ ne ss to foster his favorite study . To that end , he fun ctioned as a clearinghouse for info rmation , attempting by his corresp onden ce to keep the growing mathemati cal co m munity o f England and Europe abreast of the latest develop ments . No doubt Collins was playing thi s role when he sent a copy of Mercator's work to th e Lucasian pro fes sor of mathematics . Late in July , Collins re ceived in reply a letter whi ch info rmed hi m that a fri end of Barro w ' s in Cambridge, " that hath a very excellent geni us to those things , brought me the oth er day s o me pap ers , wherein he hath sett downe meth ods of calculating the dimensions of magnitudes like th at of Mr Mercator concerning the hyperbola, but very generall . . . "73 Barro w was not mis taken in thinking the paper would please Collins , and he promised to send it with his next letter. About ten days later, Collins did re cei ve a paper with the title De ana ly si per a equ ationes numero term inorum infin itas (On Analysis by Infin ite Series) . Late in A ugust, he learned who the author was . "His name is Mr New,ton; a fellow of our College, & very young (b eing but the se cond yeest Master of A rts) but of an extraordinary genius & proficiency in these things . " 74 Among other things , the episode informs us that Barro w and Newton were now acquainted . Ap parently they had been for some time; C ollin s would later note that Newton contrived a general method of infinite series " above two yeares befo re Mercator pub73 B arro w to C ollins , 20 Jul y 1 669; Co rres 1, 1 3 . 7 4 B a rro w to Collins , 20 Au g . 1 669; Corres 1 , 1 4- 1 5 . The edi tor o f the Correspondence suggests that " yeest" is a dialect form of " yo u ngest . " In August 1 669, Newton was far from the second you ngest Master of Arts in Trinit y . I have not seen th e o rigi nal letter, b u t the phrase woul d make sense if " the second yees t" said, o r was meant to say , "a s econd year . "

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lished any thing, and co mmunicated the same to D r Barrow , who acco rdingly hath attested the same . "75 Hence when Barrow re­ cei ved Mercato r's book , he realized its implication fo r Newto n's work and sho wed it to hi m . The epis ode served also t o confront New ton with the enormou s anxi eties that p r o spe cti ve publica tion a ro u s e d . D es cr i bing the event s a few years later , he said that upon the appearance of Merca­ to r's b o ok , "I began to pay less attention to these things , suspecting that either he knew the extraction of roots as well as di vi sion of fra ctions , or at least that others upon the di scovery o f division would find out the rest [of the bino mial expansion] befo re I could reach a ripe age fo r w ritin g . "7 6 Fo rget the main and the final clauses; they imp o sed later reflections on his ini tial rea ction. What he found in Mercato r's book was half of the discovery that had set him on his way four yea rs earlier . If Mercator had done it for the hyperbola , would he not do it fo r the circle as well (i . e . , the series for (1 - x 2 ) 1 1 2 , "the extraction of roots ") ? Mo reover , Mercator had applied series exp ansion to quadratu res . To Newton at lea st , the whole of his p roud advance stretched out directly beyond the door Mercato r had opened . We know from Collins 's co rre spondence that other s ca ught the published hint . Lo rd Brouncker cla imed to have found a series for the area of a circle . James Gregory was working toward one . More than once , Mercato r him self clai med to have o ne . 77 It is unlikely that Newton heard of these claims , but his imagination would have filled them in , fo r he knew that infinite series were in the air and that other mathemati cians were at work . I n ha ste , h e compo sed a treatise, drawn from h i s earlier papers , whi ch by its generality (in contrast to Mercato r' s single series) would a ssert his pri o rity . Still in haste , he took it to Barrow , who 75 Collins t o James Gregory , 2 1 Jan . 1 67 1 ; Co rres 1 , 60. Nea rly fifty years later, during the pri ority dispute , Ne wton w rote a paper stating that Barrow's letter to Collins wa s grounded on what Newton " had com municated to him from time to time before the Logarithmotechnia came abroad . . . " (Add MS 3968 . 27 , f. 390'' ) . 76 Newton to Oldenburg, 24 Oct. 1 676 ( the Ep istola posterior for Leibniz) ; Corres 2 , 1 33 ; original La tin, p . 1 1 4 . If Willi a m Derha m's memory served h i m well, Newton gave hi m a somewhat differen t account of the effect of Mercator's b ook upon hi m . "That when he was a Junior [sic - J unior Sophi s ter?] at the Uni versity , he ha d th oughts o f these things , but b r ought the m to no perfection , by rea son he was fo rced to leave the U ni versity in the Pla gue-yea r 1 66 5 , & 1 666 . But at hi s return to Cambridge , Mercator' s Logarithmotechia , & what or Wallis p u blished ab out y1 ti me , revi ved his th o ughts of these matters , and then he b rought them to more perfection , & communicated them to or Barro w , a t y1 ti me of the same college; who acquainted Mr Collins & others w th them" (Derham to Con duitt, 1 8 J uly 1 733; Keynes MS 1 33) . 77 I am follo wing Whi teside ' s con vincing a rgument here; Math 2 , 1 65-8. In fact, othe rs did pick up Merca tor' s hint and proceed to the derivation o f other series . In 1 671 James Gre gory , and in 16 73 Leibniz used analogous methods to derive the in verse ta n series (Math 2 , 34n) .

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p ropo sed to do the o bvious thing and send i t to Collins . As he faced the implications of that move , Newton ' s haste melted sud­ denly aw ay . Hithe rto he had co mmuned with hi mself al one , aw are of hi s own achievemen t but secure fro m profane criticism . Lately he had communi cated something to Barro w , but Barrow was a memb er o f the closed so ciety of Trinity , the only one in it able even to rea d his pape r. What now opened before New ton was s o mething much more , an d app aren tly he shrank back in fright , all too aw are of his unripe age and much else besi des . When B arrow w ro te on 20 July , he had the paper in his possession but was not allo wed to send it. We can only imagine what went on during the foll owing days , tho ugh his le tte r of 3 1 July gives more th an a hin t . I send y o u the papers of m y friend I p ro mised . . . I p ray having peru sed them as much as you thinke good, remand the m to me; acco rding to his desire , when I asked hi m the liberty to impart the m to you . and I pray give me notice of your receiving them with yo ur soonest convenience; that I may be satis fyed of their recep tion; be­ cau se I am afrai d of the m; venturin g them by the post, t hat I may not longer del ay to corrispond with your desire . 78

Only when Collins' s en thu siastic response cal med hi s fears did Newton allow Barrow to divulge his name and g ive permission fo r B roun cker to see the p aper. In all , i t was a fai r show o f app rehen­ sion b y a man who knew himself to be the leading mathematician o f Europ e . As b o th its ti tle and the ci rcums tan ces o f i t s co mpositi on s uggest , De analy si concerned its el f primarily with infini te series in thei r app lication to quadratures , though it did contrive to ind icate some­ thing o f the scope of the general me thod of fluxions . D rawn from the October tract, it added a few features to the earlier treatment. It began with th ree rules : the quadrature of y = ax m fn , the quadratu re of cu rves co mposed of several such te rms , and the reduction of more co mpounded terms to simp ler ones like ax m fn by division and the extraction of roots . The thi rd rule , of course, bore the burden of the paper. In stead of stating the general fo rmula of the bino mial expansion , New ton cho se to develop series by division - of a 2/(b + x ) fo r exa mple - and ro ot extraction - o f (a 2 + x 2 ) 1 12 fo r example . Any equation , he asserted , " however complicated with ro o ts and denominators it may be, " can be reduced to an infinite series of simple terms , ea ch of whi ch can be squared . 79 What he now added to his method for the first ti me was an iterati ve p rocedure by which to reduce affected equation s , such as y 3 - a 2y - 2a 3 + axy - x 3 = 0, to an infini te series which exp res sed y as a function of in creasing or decreasing powers of x . " Indeed , " 7 8 Co rres 1 , 1 4 .

79 Math 2 , 2 1 9 .

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he concluded , "since every problem on the length of curves, the quantity and s u rface of solids and the centre of gravity may ulti­ mately be redu ced to an inquiry into the qu antity of a plane su rfa ce bounded by a cu rve line, there i s no necessity to adj oin anything about them here . " 8 0 Nevertheless , he did adj oin a problem in recti­ fication and a general expo sition of how the problems listed can be treated a s p roblems of area s . And by means of his iterative proce­ dure for affected equations , he developed fro m the series expressing the area of an hyperbola and the arc of a circle new series to exp ress their bases - in effect , the series fo r er ' sin x ' and cos x ' here ex­ pressed for the first time in European mathematics . With a touch of virtuosity , Newton went on to develop series for the areas of two mechanical curves , the cycloid and the quadratrix. Even the length of the qu adratrix ' s arc can be determined, he ad­ ded, though the computation i s difficult . Nor do I know anything of this kind to which this method does no t extend its elf, and then in various ways . . . . And whatever common an alysis p erfo rms by equati ons made up of a finite number of terms (whenever it may be possible) , t his method can always perform by infinite e quations : in consequence , I have never hesitated to bestow on it also the name of analysis . Indeed , dedu ctions in the latter are no less certain than in the other , nor its equations less exact . . 8 1 .

D e a na lysi did not confine itself t o the method o f calculating area s . It also expounded Newton' s concep t of the generation of areas by the motion of lines , whereby infinitesimal moments are continu ou sly added to the finite area already generated . At the end of the paper, to assert hi s priority , he b riefly expo sed hi s method of tangents as the inverse of the method of quadratures . A demonstra­ tion of Ru le 1 u sed the method of tangents to obtain , from an equation fo r the area z as a function of x , the equation fo r y a s a function of x , the cu rve of which z rep resents the area . 8 2 Thu s , however briefly , De ana lysi d i d indi cate the full extent and power of the fluxional method . With the transmission of De analysi to John Collins in Lond on, Newton' s anonymity began to di ssolve . Though a mediocre ma­ them atician at best , Collins could recognize genius when he saw it . He recei ved De analysi with the enthu siasm it deserved . Before he fulfilled Barrow' s requ est and retu rned the paper, he took a co py. He showed the copy to others , and he wrote about the contents of the tract to a nu mber of hi s correspondents : James Gregory in S cotland , Rene de Sluse in the low countries , Jean Bertet and the Engli shman F ranci s Vernon in France , G . A . B o relli in Italy , R i ch80 81

Math 2 , 233 . Math 2 , 24 1 -3 . Translation altered slightly .

82

Math 2 , 243- 5 .

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ard To wneley and Thomas Strode in England . Years later , when Newton went through Collins papers , he was surprised to learn j ust how widely the paper had circulated . " Mr . Collins was very free in communi cating to able Mathemati cians what he had receiv 'd from Mr. Newton . . . , " he wrote anon ymously in the suppo sedly im­ partial Comm ercium ep isto licum . 83 Meanwhile , Collins and Barrow wanted to publish it as an appendix to Barrow' s fo rthcoming lec­ tu res on opti cs . This was more than New ton could contemplate . He drew back . A letter of Collins indi cates they applied mo re sua­ sion than a mere suggestion in passing; he tho ught Newton would " give way " eventually . 84 He was mistaken . New ton wi thheld the pu bli cation of his method , the fi rst episode in a long hi story of si milar withdrawals . Thus quietly did Newton' s apprehensions sow the seeds of vicious conflicts . De ana lys i was no t, however, devoid of effect on Newton's life . At the very time he co mmuni cated it to Barrow , Barro w was con­ templating re signation fro m the Lu casian professorship of mathe­ matics . The professorship, established scarcely five years earlier by the bequest of Henry Lucas , was the fi rst new chair founded in Camb ridge since Henry VIII had created the five Regius professor­ ships in 1 540 . With the Adams professorship of Arabic, established in 1 666 , it b rought the number of similar positions in the university to eigh t . It was the only one concerned in any way with mathemat­ ics and natural philosophy , whi ch were otherwise hardly tou ched upon by the curriculum . By existing standards , Lucas endowed it magni fi cently; with its stipend of £ 1 00 , more or less , from the in come of lands purchased in Bed fo rdshire, it ranked behind the masterships of the great colleges and the two chairs in divinity (whi ch were usually occupied by college masters) as the ripest plum of patronage in an insti tution much concerned with patronage. On 29 October 1 669 , this plum fell into the lap of an obs cure young fellow of peculiar habits , apparen tly without conne ctions , in Trin­ ity College - to wi t, Isaac Newton . The re a re various stories about Barrow's resignation and New­ ton ' s appointment . One version holds that Barro w recognized his master in mathemati cs and resigned in his favor . 85 Frankly , it is quite i mpossible to square thi s account with the features of life in the Restoration universi ty as we kn ow them . Ano ther more recent 8 3 Comm epist , p . 1 20. 8 4 Colli n s to James Gregory , 25 Nov . 1 669 and 1 2 Feb . 1 670; Corres 1 , 15, 26 . 8 5 Newton himself was responsible for this sto ry. He told Conti that Barro w had wo rked out a rather long solution to a p roblem about the cycloid and was struck with amazement when Newton gave a solution in six lines . Barrow then resigned his ch air to N ew ton , confessing " that he was more learned than he" (Antonio-Schinella Conti , Prose e poesi, 2 vols . [Venice, 1 739- 56] , 2 , 25-6) .

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one , m o re in harmony with the times , suggests that Barrow was angling fo r a higher po sition . 86 It i s known , I think , beyond dou bt that B arrow was a man ambitious fo r p referment . One has only to recall his invariable contributions to the valedictory volumes p u b­ li shed by the universi ty - not to mention their length ! - to realize ·as much . Within a year of his resignation , he was appointed chaplain to the king , and within three years ma ster of the college. Neverth e­ les s , there was no rule that de manded he resign the Lucasian ch air befo re he cou rted further p referment , o r for that matter that he fo rgo the ubiquitou s royal dispensation to enj o y both at once, and it is hard to leave the third account of his resignation wholly out of the picture. In his own ey es, Barrow was a divine , not a mathenia­ tician; he resi gned to devote himself to his true calling . Seven­ teenth-century s o ciety b eing what is was (that i s , not wholly in­ commensu rable with twentieth-century so ciety) , this motive wa s in no way incompati ble with the other . Contemp orary co mments agreed in the assertion that Barrow effecti vely appointed Newton . Collin s understo od as much , and a generation later Conduitt did also . 8 7 There is every rea son fo r u s to accep t the account as well . By the Lu casian st atutes , Lucas ' s two exe cuto rs , Robert Rawo rth , an attorney , and Thomas B u ck , the university printer , appointed the professo r while they were alive , as they were in 1 669. The mere fact of Barro w ' s ap pointment as the fi rst Lucasian p rofessor suggest s that he stood in well with them o r had mean s to influence them . So d o the Lu casian statutes , whi ch they had drafted in 1 663 , and which em bodied modifications of the statutes fo r the Regiu s chairs of Greek and Hebrew that Barrow had obtained in 1 66 1 when he held the Greek professorship . The letter patent with the modifi cations specifi cally mentioned B arrow and thu s seems to have been inspired by him . It allo wed the two p ro fes­ sors , who were paid by Trinity College, to retain the incomes fro m their fello wships with the chairs; it also allo wed them to hold eccle­ siastical ap pointments that did not entail cu re o f souls . The Lucasian statutes extended the same rights and restri ctions to the new pro fes­ so r. B arro w ' s anticip ation of his own ap pointment may perhaps be evident in one omission from the statutes , whi ch did not fo rbid the incumbent to hold offi ce in hi s college. Others were not prep ared to grant Barrow a coup of this dimen sion , however, and the le tter patent of 1 664 , confirming the statutes , reinstalled the prohibition . In all , the two do cuments made the terms of the p ro fessorship 86 Math 3, xivn; and W hitesi de ' s life of Ba rrow i n Dictiona ry ef Scien tific Biography , 1 , 473 .

8 7 C ollins t o J ames Gregory , 25 Nov . 1 669; Co rres 1 , 1 5 . C oll ins to David Gregory pere ; 1 1 A u g . 1 676 � Co mm ep ist, p . 48 . Collins to J ohn Wallis , p robably 1 678; Corres 2, 241 . C onduitt's memoir; Keynes MS 1 29A , p . 1 1 .

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nearly iden tical to tho se of the Regiu s chairs . B arrow wrote his hand all over the statutes . It is not su rpri sing that he was able to effect the no mination of hi s successo r . The publication of Merca­ to r' s book had been almost an act of P rovidence . De analysi , which it p ro voked , rai sed Newton to the Lu cas ian chai r . Acco rding t o t h e statu tes , the Lucas ian professor w a s requ ired to read and expound " some part of Geo m etry , Astrono n1 y , Geog ra­ phy , Optics , Statics , or some o ther Ma thematical discipline , " each week du ring the three academic terms . For each lectu re that he missed , he would be fined fo rty shill ings . Each yea r , he had to deposit in the university lib rary cop ies of ten of the lectu res he had read; here too the statutes established a fine fo r failure to co mply . In addition to the lecture , the professor was to make himself available two hou rs each w eek to respond to questions and to cla rify difficul­ ties . He had to reside in the univ ersity continuou sly du ring term time . He could leave fo r more than six days only fo r serious reasons app rov ed by the vice-chancello r; and if he absented himself fo r as long as half a term , the vice-chancellor was instructed to appoin t a sub stitute who would receive all the p rofessorial income . The letter pa ten t that confirmed the statutes defined the aud ience o f the p ro­ fessor; it requ ired all undergradu ates beyond their second y ear and all Ba chelors of Arts until their thi rd year to attend under pain of the usual p enalties . Although they allo wed hi1n to hold his fellow­ ship , the statute s , as I have indicated , fo rb ade the professor to ac­ cep t any ecclesiastical p romotion that involved cu re of s ouls and requ ired residen ce el sewhere . To this restriction , the letter pa ten t added the prohibition fro m college o r univ ersity office. In co mpen­ sation , it confined his tuto rial activity to the wealthy fello w com­ n1oners . Besides the considerable inco me , the p rofessor also re­ ceived a considerable p romo tion in statu s ; his scarlet go wn set him off fro m the mere teaching masters . Though we do not hav e New­ ton ' s accoun ts , we may presume that he did not stint in assuming his new dignity . The statu tes did not fail to requi re that the profes­ sor be a man of learning and of good repute . He was subj ect to r e mo v a l fo r con v i c ti o n " of any s eriou s cri me ( su ch as Les e­ Maj esty , heresy , schism , voluntary ho micide , no table theft , adu l­ tery , fo rnication , perj u ry) , " or fo r being "intol erably negligent" in his duties . 8 8 Newton did not so immerse himself in his studies as to be indif­ ferent to his material welfare o r in co mpetent to attend to i t . In the mid- 1 670s , the tax commissioners of Bedfordshire tried to collect taxes on the lands from which his income came . He wrote one of 88

The Lu casian s tatutes a nd t h e letter patent confirming them a r e p rinted in fu ll i n lvfoth 3 , xx-xxvu.

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them indignan tly , explaining th at the properties were not college lands but were attached rather to his p rofessorshi p . Citing the words of the Act , he conclu ded that they plainly in cluded him, " & by consequence excuse m e expresly fro m paying. fo r any o f the profi ts of my P rofessorship . "8 9 I n Resto ration Camb ridg e , performance ten ded t o dive rge , often wildly , from statuto ry requ irements . As far as the stu den ts were concerned , the fresh b u rden of lectu res imposed upon them was only another item on a list now univ ersally igno red. By 1 660 , college tu to ring had vi rtually co mpleted i ts conquest of university instruction . B arrow had complained of the neglect of his lectu res when he was p rofessor of G reek . " S opho cles and I acted in an empty theatre; . . . there was no choru s , no t even of boys . . . " 90 Edmund Castell , the first Adams p rofessor of Arabic, met the sa1ne indifference . Withou t fu rther ado , he posted a sign on the door, " Tomorro w the P rofessor of A rabic goes into the wilderness , " and conve rted the position into a sinecu re , which it remained fo r a centu ry and a half. 91 While we have no information abo u t New­ ton' s early experience, we do know what Humphrey New ton found upon his arriv al fifteen years later. When New ton lectu red, he recalled , " s o few went to hear Him , & fewer yt unders tood him, yt oftimes he did in a manner, for want of Hearers , read to ye Walls . " He usually lectured for half an hour, th ough he retu rned in less than a qu arte r of an hour when he had no audience . Humphrey added that New ton had no pupils at that ti me . 92 In thi s he was mistaken ; the college records indicate that Newton took on his thi rd and last fellow co mmoner du ring Humphrey' s stay with him . Perhaps the discrepancy calls the account into question; more p rob­ ably it is a co mment on the closeness of Newton ' s relation with the pupils he tu tored . As for the two hours a week for consultations , no eviden ce whatev er of such survives . 8 9 Co rres 7 , 37 1 . The h and is a bout 1 675 , and the draft of the letter appears on a sheet with experiments on diffraction that confirm that tim e .

90 Quoted in J ames B ass Mullinger , Cam bridge Characteristics in the Seventeenth Century (L on­ don , 1 867) , p . 55 .

9 1 Quoted in Win s tanley , Un refo nned Cambridge , pp . 1 32-3 . In 1 7 1 0, Uffenbach w as a m azed to fin d that no lectures were being given in the uni versity . I t w as sum mertime; in the winter th ree o r four were given , " to the b are w alls , fo r no one comes in" ( M ayor, Cam bridge u nder Queen A nne , p . 1 24) . When Conyers Middleton did battle w ith Ri chard B entley in 1 727 , he employed Bentley ' s neglect o f the Regius p rofessorship o f divinity as one o f his weapon s . Earlier p rofessors o f divinity , he said, had pu t up notices o f lec tures , showed u p , " and actually read a theolo gical lecture whenever they found an aud ience ready to attend them , which w as som etimes the case" (qu oted in Winstanley , Unrefonned Cambridge , p . 98) . C f. Winstanley's and Godley' s descrip tio ns of the in creasing neglect of p ro fess orial du ties in both uni versities (ibid . , pp . 1 03-38 . Godley , Oxfo rd , pp . 43-7) . 9 2 Keynes MS 1 35 .

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One of the fundamen tal facts of Newton 's tenure as Lu casian p rofessor of mathemati cs is th e paucity of refe ren ce to his teaching . For fo rty years afte r 1 687 , he was the most famous in telle ctu al in England , and there was every incentive fo r fo rmer studen ts at the university to recall thei r conne ctions with hi m . Even Will iam Whis­ ton , who became his d isciple and su ccesso r , could barel y remember having heard him . 93 As far as we kno w , only two others ever claimed to have been ins tru cted by him . 94 The Lu casian chair was 11o t created as part of a cons ciously fo rmu lated revision of the cu r­ riculu m . It came in to being b e cause one man , Henry Lu cas , th ought Cambridge should have its equivalen t to the Savilian chairs in Oxfo rd . Mathemati cs was further removed fro m the in te rests of the average Re sto ration student than A rabi c , and New ton pitched his lectu res on a plane apt to make them incomprehensible . More­ over, no man , no matte r what his genius , could have reversed the decay of th e university lecture sys tem . The prohibition agains t college offi ce did operate , though p ro ba­ bly only because New ton chose that it should . During th e same period , Regius p rofess o rs of Greek and Hebrew held college of­ fi ces . 95 At a time when all the life in Cambridge was concentrating itself in th e colleges , exclusion fro m college office tended fu rther to isolate New ton within the university . 93 Whis ton , Memoirs , p . 36 . 94

Both of them u sed his fame to their fu ll adv antage, and in so doing emph as ized the lack of o ther s u ch recollections . Henry Wh arton , a d m itted to Caius in 1 680, was the pupil there o f John Ellis , whom H u m ph rey Newton men tioned as a friend of Newton . A ccord­ ing to the " L ife" of Wharton , p refaced to the s econd ed ition o f his Fourteen Sermons (London , 1 700) , he attained con siderable s kill in mathematics . " Which last was much encreased by the kindness of M r . Isaac Newton , Fellow of Trinity College , the inco m para­ ble Lucas-Professor of Mathematicks in the University, who was pleased to give him fu rther ins tru ction in that noble S cien ce , a m ongst a select Company in his own p rivate Cham­ ber" (Quoted in Math 4 , 1 1 ) . A mong Wharton's papers is a cop y in his hand of Newton 's " Epitome of Trigonometry" with the note tha t Newton g ave it to him (i . e . , to co py) in 1 683 . I cannot forebear to won der why none o f the others in tha t "s elect company" (which had dis persed b y the time H u m ph rey Newton a rrived shortly th ereafter) came forward similarly to pat himself on the b ack in public. Sir Tho m as Parkyn s , a d m itted to T rinity in 1 680 , and late r author of Progymnasmata . The Inn-Play: or Comish-Hugg Wrestler (London , 1 727) , asserted th at he owed h is application of m athe m a tics to w res tling to his tu tor , D r . B athu rst, a nd to S ir Is aac Newton , mathematics p ro fes s o r . "The latter , seeing m y In clinations that Wa y , inv ited me to his p u blick L e ctu res , for wh ich I thank him , tho ' I was Fellow Com moner , and seldom , if ev er, any such were call ' d to the m . . . " (p . 12) . Whatever we can make out of this b izarre reference , it does not appear from Parkyns's a ccou nt that he belonged to Wha rton ' s select grou p . Pos sibly the time was righ t; Parkyns left for G ra y ' s Inn in 1 682 . 95 Cf. " The Case of the Heb rew & G reek P rofess ors in regard to P rea cherships & c " ; T rinity College, Box 29, C . I I I , a . The paper, a plea fo r the righ t to hold church livings w ith the p rofess orships , ass erted (correctly) th at Benj amin Pul leyn , Greek p ro fessor fro m 1 674 to 1 686 , and Wo lfran S tubbe, Hebrew p rofessor from 1 688 to 1 699 , both held college o ffices w ith the chairs .

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Meanwhile, there were the weekly lectures . O r perhaps th ere were the weekly lectures . As far as the reco rd info rms u s , Barrow had already reduced the requirement to lectures during one of the three term s ; Newton acquiesced in that schedule . He delivered a course of lectu res in the Lent term of 1 670 , soon after hi s appoint­ ment . Thereafter he gave a series during the Michaelmas term (or at any rate he dep osited manu scripts with such dates on them) each year thro ugh 1 687. After 1 68 7 , he su ccu mbed to the prevail­ ing mode and held the position as a sinecure for fourteen years, during five of which he was not even resident in Cambridge. The reco rd of his abs ences from Trinity during the earlier period sup­ ports the conclu sion that he lectu red only one term per year . Although he did not leave frequently , when he did do s o , he was as apt to go during term ti me as during vacations . Indeed he left for t w o week s in London less than a month after his appoint­ ment . 9 6 N o r did he concern himself excessively with the requ ire­ ment that he dep osit copies of ten lectures each year. In all , he eventually dep osited fou r manu scripts that purported to contain annual cou rses of lectu res throu gh 1 687 . 97 Questions have been raised about all fou r , so that it is quite impo ssible to know for sure on what he lectured . It appears highly p robable that the lec­ tures through 1 683 co rresponded rou ghly to those in the deposi ted manu scripts . 9 8 Beginning in 1 684 , he may have lectured on the Principia) but the deposited manu scripts were merely drafts of that work whi ch he sent in as the easiest way to fulfill the requirement . As the topic for his first course of lectures , Newton cho se , not the subject of De ana lysi J indeed not mathemati cs at all , but optics . He had required external stimulu s to co mp o se De ana lysi . D u ring the following two years , he devoted a fair amount of time to mathe­ mati cs , but again under external stimulu s . What he turned to of his 96 Edlesto n , p . lxxxv . 97 Cam bridge Uni versity Library , D d . 4 . 1 8; D d . 9. 46; Dd. 9. 67; D d . 9. 68 . 9 8 Altho ugh D d . 9 . 67 p u rp o rts to be the optical lectures delivere d in fo ur course s , there is an earlier version among Newton ' s papers (Add MS 4002) tha t was o rganized d i ffe rently , did not give all o f the same date s , and presented a s two courses o f lecture s vi rtually all tha t the deposited lecture s p resented a s four. There is g o o d rea son t o think the depos ite d man uscript wa s orig inally a revi sion o f Add MS 4002 p repare d for publication a n d depos­ ited ( with suita ble dates in serte d) when he aban doned the p roj ect. D d . 9 . 68 cla ime d to be lectu re s on algebra delivered between 1 673 and 1 683 . New ton put the ma nuscripts to­ gether onl y in 1 683-4 . When John Flam steed vi sited Cambridge in July 1 674 , Newton gave him a paper o f notes for a lectu re supposedly delivered in midsummer o f tha t yea r. The deposited ma n uscri pt contains virtually the identical material s split between two lectu res in Octobe r 1 674 . The ma nus cript fu rther p retended tha t a course o f le ctu re s was given i n the autumn o f 1 679, when we know tha t Newton was in Wool sthorpe after the death of his m other (Math 5, v ii-xii, 3-6) .

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own accord was optics and the theo ry of colors . 99 His accounts recorded the purchase of three p risms some time after February 1 668 , probably du ring one of the summer fairs . His earliest su rviv­ ing letter, da ted 23 February 1 669 , described hi s fi rst reflecting telescope and referred obliqu ely to his theory of colors . Thu s there i s reason to think that Newton had resu med hi s investigation of colors before hi s appointment, and that he chose to lectu re on the topic then foremost in his mind . Two or three years earlier, he had sketched out his theory of colors . Now the p roblem seized hi m in ea rnest and would not release hi m until he had con quered it. Years later D r . Cheyne was " credibly infonned " that when New ton was carrying out the investigation , " to qu icken hi s faculties and fix his attention , [he] confined hi mself to a s1nall qu antity of bread, du ring all the ti me, with a little sack and wa ter, of which , without any regulation , he took as he found a craving or failu re of spirits . " 1 00 Wicki ns p robably had to contend with more than neglected meals and a roo m plunged into midnight at noon . The new experiments , which were more sophisticated, requi red a n as sistan t, and Wickins undoubtedly had to endu re forced labor. Newton besieged his friends to supplement hi s supply of equipmen t . A paper composed about this ti me , which gave a method to determine the cu rvature of len ses fro m a mea su re ment of focal di stance, employed the object glass of D r . Babington ' s telescope as a concrete example . 1 0 1 P rimar­ ily , he drew upon hi s own earlier work . Reaching back to his incomplete investigation of colors , he now worked out the full implication s of his central insight and b rought his theory of colors vi rtually to the form that he published more than thirty years la ter as his Opticks. Aside fro m specifically mathematical sections , Newton ' s Lectiones op ticae corresponded to Book I of the ulti mate Op ticks. Hence the lectures concerned themselves primarily with p rismatic p heno mena as they related to the heterog enei ty of light . The late Invention of Telescopes [he began, for the benefit of what­ ever audience was p resent] has so exercised most of t he Geo meters, that t hey seem to have left nothing unattempted in Opticks , no room for further Improvements . . . . But since I observe the Geometers hi therto mistaken in a particular Property of Light, that belongs to its Refra ction s , tacitl y founding thei r D emonstration s on a ce rtain P hysical Hyp othesis not well established ; I j udge it will not be unac­ ceptable if I bring the Principles of this Science to a m o re strict The Lectiones op ticae, p ublished in the eighteenth centu ry from his desposited manuscript , began with an e xtensive mathematical section . In the manus crip t that seems clearly t o represent t h e lectures given (Add MS 4002) , t h e e xp erimental inves tigation o f colors came fi rs t . 1 00 Quoted i n Edleston, p p . xli-xlii . 1 01 Math 3, 525 .

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Examination , and subjo in , what I have dis co vered in these Matters , and found to be true by manifo ld Experience , to what my reverend Predeces so r has last deli vered from this Place . 1 02

He launched forthwith into an exposition of hi s theo ry o f colors and the heterogeneity of light . The renewed research and thought on which the lectures rested clarified aspect s of hi s theory that had remained relatively crude in 1 666 . Earli er he had exp ressed his theo ry in terms of a traditional two-colo r system . Now the meaning of the continu ou s spectrum fo rced itself upon him . If there were only two colo rs , he should have seen two separate circles . Instead , of course, he always ob­ tained a continuou sly illuminated oblong . Though he frequently spoke o f seven colo rs , both in the lectures and the years ahead , and though he even co mpared their po sitions in the spectru m to the divi sions of the musi cal octave , he underst ood that such divisions were wholly arbitrary . Instead of two colo rs, o r seven , there were an infinite nu mber co rresponding to the infinite angles of refraction between the extremes for purple (whi ch he no w u sed , in place of blue) and red . As he clarified the theory in 1 669, so also he strengthened its experimental foundation . In 1 666 , he had begun , in a crude way , to employ a second p rism to refract sep arate part s of the spreading spectrum . Now he refined the experi ment into a fo rm whi ch could rigo rou sly refute the theory of modifi cation . He placed the seco nd prism half-way acro ss the roo m with its axi s perp endi cular to the fi rst so that the full spectru m fell upon it . If, as the theo ry of modification might argue, di sp ersi on as well as colo ration were a modification introduced by the prism, the second p rism ought to spread the spectrum into a squ are . Quite the contrary , it p roduced a spectru m inclined at an angle of 45 degrees . 1 0 3 Further i mprove­ ment s suggested them selves . With the second prism set parallel to the fi rst , he cov ered its face ex cept fo r a small hole whi ch admitted individual colo rs isolated from the rest of the sp ectru m , and he comp ared the qu antities of their refractions . Finally he realized the 1 02

Lectiones Opticae (London , 1 729) , p p . 1 -2; A dd MS 4 002 , p . 1 . The final clau se referred to

1 03

Lectiones, p . 3 2 ; A dd M S 4002 , p . 25 . J ohannes Lohne h a s p ointed o u t a n error i n New­

Barro w' s optical lectu re s that ha d been delivered the yea r befo re . ton' s de scription o f his results (" Newton ' s ' P roof o f the Sine Law and His Mathematical Princi ple s o f Colors , " A rch ive fo r Histo ry of Exact Sciences, 1 ( 1 96 1 ) , 389- 405) . What appea rs to me as a simple error ari sing from the geometry of the room in whi ch Newton ha d to perform the experiment appears to Lohne as a crisi s in the fou ndation o f Newton' s mathemati cal science o f light . I am u nable to accept thi s interp retation of the basic thru st of Newton ' s opti c s . See also Lohne , " Experi me ntu m cru cis , " Notes and Records of the Royal So ciety , 23 ( 1 968) , 1 6 9-99; and Ronald Laymon , " Newton ' s Experimentum crucis and the Logic of Idealization and Theory Re futation , " Studies in Histo ry and Ph ilosophy of Science, 9 ( 1 978) , 5 1 -7 7 .

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p

Figure 6 . 5 . The experimen tu m crucis .

i mportance of a demons trably fixed angle of incidence on the sec­ ond pris m . To achieve this , he used two boards with small holes , one placed immediately beyond the first pris m , the other i mmedi­ ately in front of the second (Figure 6 . 5) . Since the boards were fixed in position , the two holes defined the path of the beam which fell on the se cond p ris m , also fixed in position beyond the se cond hole . By turning the first p rism slightly on its axis , Newton could trans­ mit either end of the spectru m , pretty well if no t perfectly isolated fro m the rest, into the second prism . There , as he exp ected , the blue rays were refracted more than the red . Neither beam suffered fu rther dispersion . It was this experiment which Newton called , his exp erim entum cru cis in 1 672 , though he did no t employ that phrase either in his lectures or in his Op ticks. No doub t an experimen t that is crucial in the full sense of the word for the confirm ati on of a theory is impossible . If we consider only the two alterna tives under con sideration at the time , it does appear that Newton' s exp erim en­ tu m cru cis refuted his co mpetito r , the theory of mod ifi�ation . When the science of optics had fu lly digested the meaning of the experi­ men t, it never retu rned to the concept of modification ag ain . In 1 669 , Newton also g reatly expanded his experimen tal demon­ stration that white is me rely the sensation cau sed by a hete rogeneous mixtu re of rays . No single part of hi s investigation underwent g reater exp ansion , almost as though Newton found this conse quence of his theory difficult to accept and needed to convince hi mself. To the b rief experiments of 1 666 in which he cast overlapping spectra on each o ther , he added one in which a lens collected a diverging spec­ trum and res tored it to whi tenes s . If he intercepted the converging rays before the focus , he ob tained an elongated spectrum redu ced in size . At the focu s , the spectru m disappeared into a white spot. Be-

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Fig u re 6 . 6 . The reconstitution of whi te l i ght with

a

lens .

yond the focus , the spectrum reappeared with i ts o rder reve rsed (Figu re 6 . 6) . Beyond the lens , no operation was performe d on the light. When the spectrum merged into the fo cus , col ors merged into whi teness . Since the indi vidual rays retained their identity , colors reapp eared as they separated anew beyond the focu s . New ton was aware that impres sions on the retina endure for about a second . The fu rther th o ught occu rred th at all the elements of the hete rogenous mixture that p rodu ces the sens ation of w hite need no t be p resent at once . He mou nted a wheel beyond the lens s o that cogs inte rcep ted individu al colors of the conve rging spectru m . When he tu rne d the wheel slowly , a succession of colors appeared at the focu s . When he turned it fast enough that the eye could no longer distinguish the su ccession , white appea red once more . 1 04 One conse quen ce of the heterogeneity of light w as the distinc tion between colors on the one hand and the scale fro m white to black on the other. The scale remained as a scale of intensity alone . Its intermediate steps are the shades of gray . As far as color is con­ ce rned , the en ti re sc ale is identical , the sensation of a hetero gen eous mixtu re n1ore or less intense . 1 0 5 Newton ' s experimental imagina­ tion ro se to the challenge of illustrating this further paradox . When colored bodies reflected a beam of sunlight onto a white sheet of paper, the paper appea red in the color of the body . If his theory were co rrect , the paper ou ght to app ear white in the light refl e cted from a shiny black body . It did . 1 0 6 In a ddition to the theory of colors , the lectures also contained 1 04

Lection es , p p . 1 7 1 -2 1 3 ; Add MS 4002 , pp . 30- 57 . These p ages con tain the entirt.' discu s­

sion of white, not just the experimen ts desc ribed in my text . See Alan E . Sh apiro , "The Evolv ing S tru ctu re of Ne wton 's Theory o f White Light and Colo r: 1 670- 1 704 , " forth­ coming in Isis . 1 0 5 He associated white , bl ack , and the grays already at the tim e of hi s initial insi ght (A dd MS 3996 , f. 1 2 2 ) . 1 0 6 I take this exp eri ment from a s lightly later period . New ton to Oldenburg , 1 1 June 1 672 (his reply to Hooke's criticism of h i s p aper) ; Corres 1, 1 83-4 .

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extensive math e mati cal demonstrations whi ch fu rther revealed N ewto n ' s ma the matical powers (if that were needed) but did no t ma terially advance the argu ment central to his contribu tion to op­ tics . One demon stration did compare the errors in lenses arising fro m spherical and chro matic aberration . On the assu mpt; � : that the diameter of a lens is s mall in relation to its ci rcle of curva ture , he de1nonstrated that the error introdu ced by chro matic aberration is " " fa r greater" than that by spherical aberration. To b e specific, when the dia meter of the lens is 2 inches and its radius of curva ture 1 20 inches , the mini m u m circle at the focus within which all the differentially refracted rays of different col ors pass is 1 , 500 ti mes greater than the mini mum circle of ho mogeneous rays . 1 0 7 A s fa r as the theory of col ors is concerned , the Lection es op ticae, which Newton pro bably composed late in 1 669 and in 1 670 , con­ cen trated on pri s matic pheno mena . They did not elaborate the ex­ planation of the colors of solid bodies that Newton had sketched in his essay , " O f Colou rs . " I assu me , therefore , that the investigation which became the foun dation for such an elaboration , contained in a paper with the title , " O f ye coloured circles twixt two contiguous glasses , " dated fro m 1 670 at the ea rliest . 1 08 Newton e mployed the techn iqu e su ggested in 1 666 . He p laced a lens o f known curva tu re on a fla t piece of g lass , causing a series of col o red rings to appear to hi m as he looked down on the apparatu s . I n 1 666 , h e us ed a lens wi th a radius of curva ture of 2 5 in ches . N ow he emp loyed a len s with a radius of 50 feet, in creasing the diameters of the rings nea rly five times . B eyond the cu rvatu re of the len s , the paper g av e very little information abou t the technique he e mployed . He indicated that he had to press the glasses hard to b ring them togethe r , and he spoke of tying the m tog ether . Apparently he bu ilt a fra me o f some sort, which pressed the glasses together w hen he tied it tightly and hel d them in pos ition for measurements . Since one experi n1ent men tioned that the only force used in it was the weight of the uppermo st glass , he seems to have placed them fla t on a table and viewed them fro m above. In the Opticks he indicated that his eye 1 07

/\Jath 3, 5 1 1 , 5 1 On . An erro r by Newton , wh ereby h e fo rgot t o squ are on e ratio , m ade

1 08

A dd ,\;JS 3970 . 3 , ff. 350-53\ . T h e hand in which the paper w as w ritten also suggests a

his result roughly three-halves as large as it should have b een .

d ate a round 1 670 . Since Newton composed the first d raft o f the so-c alled " D is course of Observations , " whi ch he fin ally sent to the Royal So ciety in 1 675 , and which is m o s tly i denti cal to Book II o f the Op ticks, in 1 672 , the p ap er in qu estion he:- e cannot h ave b een composed l ater than 1 672 (see Richard S . Westfall , " N ewton 's Reply to Hooke and the Theo r y of C olors , " Is is 54 [ 1 963] , 82-96) . I have pu blis hed the p aper in its en tirety : " Isaac N e w ton 's Colou red C ircles twixt Two C ontiguous Glasses , " A rchivefo r His tory of Lxact Scien ces, 2 ( 1 965) , 1 8 1 - 96 . S ec also Vasco Ron chi , " U n G rande uomo fron te a un grandc m i stcro . N ew ton e l e interferenzc luminos e , " Bolletino dell 'Associazione Ottica Italiana occazionc , 3 ( 1 938) , 1 5 4-64 . ,

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was 8 to 9 inches abo ve the lens and that he used a pair of compasses to take the diame ters fro m the surface of the upper glas s . A s with hi s p rismati c observations , the powerfu l experimental imagination which conceived the means to re duce thin films to measu rement canno t fail to imp ress . Once again , the cou rse of the unknown y o ung man in C a m b ridge ran p a ra ll e l to th a t o f Huygens , the ackno wledged leader of physical science i n Eu rope . Stimu lated by Hooke 's o bserv ations , Huygens thought o f the sa me device to measure thin colored films at almo st exa ctly the sa me time as Newton 's original ob servations in 1 666 . 1 0 9 Newton 's skill in pe rfo nnance , the leg acy of his years in G rantha1n , ou tran th at of Huygens fro m the beginning , and in the more sophisticated exp eri­ ments of 1 670 he si mply eclipsed his unsuspected rival . 1 1 0 What he demanded of his measu re ments tells us mu ch about the man . Mea­ suring with a comp ass and the unaided nake d eye , he expe cte d accuracy o f le ss than one-hundredth of an inch . With no app aren t hesitation , he recorded one ci rcle at 23 1 12 hundredths in diameter and the next at 34 1 /3 . When a small divergence app eared in his resu lts , he re fused to ignore it but stalked it relentlessly until he found that the two faces of hi s lens differed in cu rvatu re . The diffe rence corresponded to a measurement of less than one-hun­ dredth of an inch in the diameter o f the inner ci rcle and abou t two-hundredths in the diameter of the sixth . " Y et many times they imposed upon mee , " he added grimly to his successful elimination of the e rro r. 1 1 1 N o o n e else in the seventeenth centu ry would hav e paused fo r an error twice that size . Newton was confident enough in his te chnique that he used hi s results to corre ct the radius of curvature of the lens; in the " Discou rse of Observa tions ' ' of 1 672 (and in the Op ticks) he put it at about fifty-one fee t . I n addition t o the error fro m the curvature , h e found that the diameter of the first circle was consi stently too large . Huygens had disco vered the same thing . For hi m , the fi rst circle app eared at a Huygens , Oeu vres comp letes , pub . S o ciete hollandaise d es S cien ces , 22 vols . (The H ague , 1 888- 1 950) , 1 7, 34 1 - 5 . Huygens da ted his inv estiga tion November 1 665 , a few months before the probable time o f Newton ' s first performan ce o f the exp eriment, early in 1 666 . Initia lly Huygens u s ed two lenses , both o f ten-foot radiu s , thou gh on a s e cond try he saw the advantage o f one lens w ith a greater radius (forty-five feet) pressed on a flat plate o f glass . Like New ton in his first experiment, he simply assu med the p eriodicity of the rings ; he mea s u red only the innermost and ou termost. 1 1 0 Huygens divided the differen ce between the calcu lated thi ck nes ses o f the innerm ost and ou termost cir cles by the number o f cir cles minus one to get the in cremen t o f thickness that causes a circle to appea r . His calcu lated thicknes s o f the innermost circle was three times too large, that o f the outermost two times . The combination o f the two erro rs redu ced his error on the in crement to only SO p ercent too large. In co m p arison , New­ ton ' s calcu la ted thi ckness of the innermost circle in 1 666 was about 50 per cent too large, and h e impro ved his a ccu racy greatly in 1 670 . 1 1 1 Add MS 3970 . 3 f. 35 2\' ; W es tfall , " N ewton 's Coloured Circles , " p . 1 95 , cf. n . 1 0 .

1 09

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Ne ver a t res t

calculated thickness of . 000034 inch; seven additional circles ap­ peared w ith increments of . 0000 1 4 inch , and no mo re appea red beyond th e thickness of . 000 1 34 inch. The bizarre nature of the re sult p ro bably contributed to Huygens ' s ab andonmen t of his o b­ servations . In contrast , Newton refused to accept the ano malous fi rst ci rcle . The fact that only a finite number of circles appeared never bothered hi m; th eir gradual fa d ing into white as successive circles overlapped follo wed directly from his th eory of colors . The thickness for the first ci rcle , howev er, should have equ aled the increm ent fo r s u cces sive ones . H e had been impres s ed by the amount of pressure necessary to b rin g the glasses into contact. Was it pos sible that he had distorted the shape of the lens b y his pres­ sure , thus artificially enlarging the first ci rcle? Various modifica­ tions o f th e experiment allowed him to conclude le giti mately that he had . A g ain , in his meas urements co mparing ci rcles in water with tho se in ai r , he modified his procedure un til he obtained resu lts that agreed with the accepted index of refra ction between the two media to two significant fi gure s . It i s impossible t o compare Newton's resu lts a gainst modern measurements with fin ality . For the most part he me as ured dark circles . Since the lower end of the third spectrum o v erlaps the upper end of the s econ d , there is only one circle tha t is truly dark; subsequent ones are no t really analogous . The figure Newton ob­ tained at this time for the thi ckness of the fi rst dark ci rcle is be­ tween 1 0 and 20 percent too large . Other co mparisons involve arbitrary choices of wavelengths; insofar as they can be don e , they indicate about the same degree of erro r , with the thickn ess always too hig h . 1 1 2 S igni ficantly , he reduced the sizes a bit when he com­ po sed the " Dis course of Observations " in 1 672 , and still mo re for the Op ticks . A s far as the comparison is meaningfu l , his final fi gures do not appear to diverge from 1nodern measuremen ts , an extrao rdi­ nary achievement fo r a pioneering investig ation . The paper on colo red circles fit in to many of Newton ' s inte rests . The difficulty of brin ging a convex lens into conta ct with a fla t sheet o f glass impressed him; it appea red in all of hi s s ub sequent speculations as one of the key pheno mena fo r unders tanding the nature of things . The experiment with water between the glasses gav e him first-hand experience with capillary action , tho u gh earlier he had made notes in the " Q uaestion es" about it fro m his reading . He made a fi lm of wa ter by le ttin g a drop " creepe" between the glas ses . Fro m the varying results of two experiments , in one of which the g lasses were pressed much harde r , he concluded that the creep mg in of the water altered the curva ture of the glass because 1 12

C f. Westfall , " N ewton ' s Colou red Circles , " pp . 1 85-6.

Lucas ia n p rofesso r

219

water has less " incongruity" with g lass than air. Congruity and incongruity were concepts he had met in Hooke's Micrographia; they too had a long history ahead of the m in Newton ' s specula­ tions . S o did the aether to whi ch he alluded sev eral ti mes , as he had done in the essay of 1 666 . A l ready at that ti m e he had i mplied the princip al features of his m echani cal account o f optical pheno mena . It is no t , he asserted , "ye superficies of Gl asse or any s moth pellu cid body y1 re flects l ight but rather ye cause is ye diversity of A ether in Gl asse & aire or in any contiguous bodys . " He spoke of pulses in the aeth e r in connection with thin films . 1 1 3 References to pulses which i mplied hi s mechanical explanation of the periodic rings , also filled the paper on colored circles . The pulses were no t l ight . Rather they we re vib rations in the aeth e r , set up by the blow of a co rpuscle of light on the first surface of a film , which determined whether or no t the co rpu scle would be able to penetrate the second surface and thus be trans rnitted, or would be reflected . Beyond its rev ela tion of Newton's experi m ental acumen and its inciden tal information on his sp eculative sys te m , the pap er tells us much ab out the progress of hi s theory of colors . A s late as 1 670 (i f my p roposed dating is correct) , Newton had still not sep arated hi s theory of colors fro m his conception of light . A t least a good hal f of hi s interest in the circles seemed to stem fro m his conviction that they supp orted the corp uscular conception of light . Thus he de­ voted much of his effort to measu rin g circles at various obliquities . The more ob liqu ely he viewed them , the larger they appeared . In one experiment, h e measured them fro m five different angles , in another fro m four, in four others from two . He concluded that the diameter of a ci rcl e is proportional to the co secan t of the ray ' s obliquity ( secant of t h e angle of in cidence i n our te rminolo gy) " or recip rocally as ye sines of its obl iquity; that i s , reciprocally as yt part of the inotion of ye ray in ye said fi l me of ai re wch is perpen­ dicular to it, or reciprocally as ye force it s trikes ye refracting surface wth all . " Hence the thi cknesses of the films vary as the s quares of the co secant " o r recip rocally as ye quares of ye sines , motion , or percus sion . " 1 1 4 A stronger blow allowed the co rpuscle to pa ss more easily an d co rresponded then to a thinner film . Similar considera­ tions stood behind his co mpari son of circles in wa ter and in air . The thi ckness here v a ried reciprocally as the subtlety of the mediu m . He was satisfied to see hi s measuremen ts of films in ai r and water yield a ratio equal to the index of refraction between the two media , since ==

113 114

Add M S 3975 , pp . 1 4 , 1 0 . Add MS 3970 . 3 , f. 350; Westfall , " N ew ton 's Colou red Circles , " p . 1 9 1 . I c annot e xplain his use o f the squ are here, It had app eared earlier in h i s att emp ted formula for the sa me pheno mena in the essa y " O f Colours . " In fact , his d ata showed that the thi ckness v aried inversely as the first po wer of the perpendicular component of motion , not as the squ ar e .

220

Never at rest

he considered that indices of refraction were ano ther exp ression of the res istances of media to light . In the end , ho wever, these specu­ lations led into contradictions . When he co mpared colors , he found that red rings were larger than blu e . That is red rays cause bigger pulses . He was already convinced frorn refractions that red rays are stronger, and he was n o t prepared to set a higher priority on his speculations about obliquity , in which the weaker blow caused the bigger pulse . By the time he drew up the " Discourse of Observa­ tions , " the critiques of his paper of 1 672 had taught him to separate the theory of colors fro m the conception of light . He allo wed the concern wi th obliquity to recede into the backgroun d , and the speculations about the streng th of blows to disappear. Sim ilarly , when he began to measu re rings , he had no t yet firmly fixed thei r relation to co lors in his min d . Only at the end of the paper did the observations co me to the aid of the theory of colors . Perhaps this was a s troke of good fortune , for it allowed Newton to concen­ trate hi s attention on an aspect of the ci rcles , thei r periodici ty , which had no necessary conne cti on to hi s theory of co lors . Hooke had asserted the perio dicity of colors in thin films . Ne wton and Huygens had bo th assu med the periodici ty of the circles they saw in 1 666 , but nei ther had demonstrated it. The p aper " Of coloured ci rcles " first established the periodicity of some optical pheno mena by careful measuremen t . Fro m the geo metry of the circle , the thickness of the fi l m be tween the lens and the flat sheet is proportional to the square of the d ia meter of the colored circles . Newton measu red dark circles between colored ci rcles , squared their diameters , and found a si mple arithmetic progres sion . If the thickness of the first dark ci rcle were set at 2 uni ts , successive circles appeared at thi cknesses of 4 , 6 , 8 , 1 0 and 1 2 uni ts; colored ci rcles appeared between them at thi cknes ses of 1 , 3, 5, 7, 9, and 1 1 units , beco ming less distinct until they merged co m pletely into whi tenes s . No matter what the obliquity , no matter what the medi um , air or water, the same progress ion held . 1 1 5 There could be no question about the periodicity of the ci rcles . Whereas Huygen s si mply forgot his measu remen ts when periodicity became an embarrass ment to hi s treatment of light , Newton's measurements etched its reality so deep ly on his cons ciousness that he cou ld not forget it even tho ugh it even tually became an even greater e mbar­ rass men t to hi s treatmen t . Neve rtheless , i n the " Discourse of O bservations , " he p ushed pe­ riod icity away from the cen ter of attenti on almost as far as the effects of obliquity , and he poin ted the whole investig ation firmly to ward the exp lication of colors in s olid bo dies , that is tow ard the analysis of heterogeneous light by re flection . As irony would have it, the details 1 15

A dd MS 3970 . 3 , ff. 350- 2 \ W estfall , " N ewton 's Coloured Circles , "

pp.

1 9 1 -4.

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L u casian p rofessor

of his exp lication have not survived , whereas the general statement made already in 1 666 , that bodies reflect so me rays mo re than others , has s u rvi ved . On the other hand , when periodicity was found to b e a p roperty o f light itself, it constituted as imp o rtant an addition to optics as heter ogeneity . To cap the irony , periodicity p layed the central role in the overthrow of Newton's co rpu scu lar conception of light in the nin eteenth centu ry , though he had seen hi s ob servati ons initially as a suppo rt for co rpu scu larity . On its final page, the paper on colo red circles turned directly to the theo ry of colo rs when Newton tried to measure the difference in the thicknesses of films in which red and purple ap peared . The fact of d!fferent thicknesses was evident , of cou rse , in the su ccession o f co l o rs in the inner rings . Measu ring them proved to be more difficult. Instead of attemp tin g to measu re the col o rs on the cir cles themselves , where only the innermost and least reliable cir cle was free from overlap , he tried to cast col o rs separated by the prism on his ap paratus . Lim itati ons of space dictated his method . There was no place where he could p roj ect a spectru m down verti cally far enough to let the col o rs sep arate . Hence he proj ected the col o rs onto a white sheet o f paper which reflected the light down on his lens . Apparent ly the intensity was not great enou gh to allow mea­ su rement s of circles , but Newton's in genuity rose to the challen ge. Simply la ying the lens on the sheet of glas s , he cast purp le on the paper. As he rotated the prism th rough the spectru m , circles ex­ panded and new ones app eared in the center . The nu mber of cir cles from purp le to red co rresponded to the nu mber of extra half-pu lses for pu rp le at that thickness . When he then p ressed the lens down on the glass , more circles ap peared in the center and expanded unti l the glasses came into contact . The nu mber of red circles co rresponded to the number of half-pu lses for red at that thicknes s . The rati o of the total number of circles to the nu mber of red circles gave the ratio between the half-pulses for purp le and red . Newton set it at 1 4: 9 or 20: 1 3 . 1 16 This ratio remained the empiri cal foundation of Newton' s quantitative treatment of col o rs in solid bodies . B odies are co mposed of transp arent particles the thickness of whi ch deter­ mines the colors they reflect . He had demonstrated with the prism that o rdinary sunlight is a heterogeneous n1ixture of rays , each with its own immutable degree of refran gibility . " And what is said of their refrangibility may be understood of their reflexi bi lity; that is, o f their di sp ositions to be reflected , so me at a greater , and others at a less thi ckness of thin plates o r bubbles , namely , that tho se disposi­ tions are also connate with the rays , and immutable . . . " 1 1 7 Hence 1 16 117

Add MS 3970 . 3 , f. 353v ; Westfall, " Newton ' s Coloured Ci rcles , " From the " Di s course o f O bservations " ; Co hen, p . 224 .

p.

1 96 .

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all the pheno mena of colors derive from pro cesses of analys is , whether refraction o r reflection , whi ch separate indi vidual rays from the mixture. In 1 666 , Newton laid out the program and car­ ried it through fo r refractions . Only about in 1 670 did he fully work out the details fo r the colors of solid bodies . With 1 670, Newton' s creative work in optics virtually came to an end . He had worked ou t the imp li cations of his initi al in sight , answering to his own satisfaction the questions he had set himself. Though he would devote considera ble time to the expo sition of his theo ry , first in 1 672 , later in the 1 690s , and carry out s o me ' m in or experimentation , he had effectively exhausted his interest in the su b­ ject . Never again was it able to co mmand his undivided attention . D u ring this same period , wi lly nilly , Newton als o did s ome wo rk on mathemati cs . Two enthu siastic and persu asive men , Isaac Bar­ row and John Collins , now knew his power and refu sed to let it rest . B arrow involved him in the pu bli cation of his two sets of lectures . To the Lectiones X VIII ( 1 669) on optics , New ton added tw o small improvement s for whi ch B arrow thanked him in the preface . He did not name him , ho wever , pro bably at Newton's requ est . At one point , Barro w ' s lectu res emp loyed the theo ry of colors that Newton had dispro ved . Its use suggests at lea st that Newton had not co mmuni cated this di scovery to hi s new patron . New ton did advise Barrow to include a cert ain passage in his Lec­ tiones geom etricae ( 1 670) . In July , Barrow presented an ins cribed copy of the work to hi m . 1 1 8 In his own opti cal lectures , Newton went out of his way several times to defer to his predecessor. 1 1 9 Nor was the relation one-sided . B arrow allowed him to us e his extensive mathematical library . Barrow als o set him mathem ati cal tasks . In the fall of 1 669 , he suggested that Newton revise and annotate the Algebra of Gerard Kinckhuy sen , which had re cently been trans lated from D utch into Latin . It was also Barro w who set him at work a year later on a revision and expansion of De analysi . The epi sodes fro m 1 669 to 1 67 1 constitute most of the known relationship between the two men . It was j u st as we ll for New ton 's career that he chose to please the older schola r . Already Barro w had shown himself to be a powerfu l patron . Newton was to need hi s aid one mo re time . John Collins proved a more pertinaciou s gadfly . He had been the ultimate sou rce of the Kinckhuysen Algebra , which he had had tran slated from D utch to supply the lack of a good introduction to the subj e ct . Late in November 1 669, when New ton made hi s sec1 18 11 9

Math 1 , x v ; Math 3, 70- l n , 440n , 479n , 490n . Math 3, 44 1 , 453, 455- 65 , 46 1 .

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ond trip to London , he met Co llins . Newton's accounts sug gest that alchemy rather than mathemati cs was his primary purpo se on the tri p . Either he or Barrow info rmed Co llins of the visit , how­ ever , and Collins could not let the opportunity pass . A year later , he described the meeting to James Greg o ry . I never saw Mr Is aac Newton ( who i s younger then yours elfe) but twice viz so mewhat late upon a Saturday night at hi s lnne , I then proposed to hi m the adding of a Musicall [i . e . , harmonic] Progres­ si on , the which he pro mis ed to consi der and send up . . . . And againe I saw hi m the next day having invited him to Dinner . 1 20

It was out of the question that Collins shou ld allow his n ew discov­ ery to es cape fro m his net of co mmuni cation . Newton 's exchange with C ollins effectively intro duces hi s survi ving co rrespondence. Collins had planted the question about the harmonic s eries art­ fully . F o r his part , Newton glo wed in the warmth of appreciation , whi ch Collins reinfo rced by the gift of a copy of Wallis's Mechanics sent vi a B arrow . In Janu ary , Newton initiated the co rrespondence with a lengthy and difficult letter , on whi ch he must have s pent much o f the intervening time, responding to the problems Collins had proposed at their meeting . He o ffered variou s devices to sum up a finite nu mber o f terms in any harmoni c series and a method to app roximate the answer . 1 2 1 Collins never did co mp rehend what Newton had written , but he s ent in return his own thoughts on the harmonic series and a new problem . H ow could one co mpute the rate of interest N on an annu ity of B pounds for thi rty-one years purchased fo r A pounds? On 6 February 1 670, Newton sent a fo r­ mu la by which to compute N when A and B are given , and he des cribed to Collins the sort of annotations he was preparing on the Kin ckhuysen Algebra . 1 22 By 1 8 February , Collins had added the questions of Mi chael Dary , a computer and gauger , and New ton en clo sed a let ter fo r Dary with a series fo r the area under a circle . 1 2 3 In July , Co llins sent the London bookseller Moses Pitts , who was planning to pu blish the Kin ckhuysen , to see Newton in Cam­ bridg e , altho ugh ultimately the two men never met . 1 24 Newton co mp leted his annotations to Kinckhuysen by the sum­ mer of 1 670 , though he later added to them at Collin s ' s behest . This paper does not approach his fluxional papers in significance . Nevertheless , i t w a s the work of a man who had probed the depths of his to pic, and it revealed his mastery of basic algebra . His com­ ments st rove to simplify Kinckhuysen's more cumberso me p roce­ dures and to pro pose general metho ds where Kin ckhuysen dealt in 1 2 ° C olli n s to James Gregory , 24 Dec . 1 970; Corres 1 , 53. 1 2 1 Corres 1 , 1 6-20 . 1 22 Co rres 1 , 23- 5 . 1 2 3 Co rres 1 , 2 7 . The letter t o Dary i s los t . 1 24 Newton t o C olli n s , 1 1 J u l y 1 670; Co rres 1, 3 1 . C ollins to Newton , 13 J ul y 1 670; Corres 1 , 32.

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particular one s . A t Collins 's requ est , he added significant portions on extracting the roots of cubic equa tions , inclu ding the identifica­ tion of imaginary roots , and he wrote a masterfu l exposition of ho w to redu ce pro blems to equations , in which he treated algeb ra as a lan guage akin to other langu ages and the constructio n o f equ a­ tions a s an exe rci se in translation . 1 2 5 Newton's " Obse rvations on Kinckhuysen" served further to increase his fa me within a limited circle o f mathematicians . John Wallis , to who m Collins had not shown De analysi because of his reputation fo r p la gia ry , did hear about the annotations; he o ffered the op inion that Newton could bring them out as an independent treati se of his own. 1 2 6 To wneley longed to see the Kinckhu ysen volu me " with th ose wonderfull ad­ ditions of Mr Newton . " 1 2 7 Ja mes Gregory, a mathematician who app roached N ewton' s stature , continued to co rre spond with Col­ l ins about Newton' s m ethod of exp and ing b inomials into infinite series. 1 2 8 Whether he understood the full extent of the publicity or not, Newton no sooner sensed the consequ ences of Collins ' s adu lation than hi s anxiety , lulled initially by the pleasu re of recog nition, b e ­ gan to mount anew . It was evi dent already in his letter of 1 8 F ebru­ ary 1 670 . Collins had asked to pu blish the formula for annuities . Newton agreed , " so e it b ee wthout my name to it . For I see no t what there is de sirable in pub lick esteeme , were I able to acqu ire & maintaine it. It would perhaps increa se my acqu aintance , ye thing wch I cheifly study to decline . " Al ready he had begun to fend o ff the su ggestions that he publish D e analysi . Now h e began t o with­ draw from Collins ' s overeager embrace as well . He info rmed Col­ lins in th e letter of 1 8 February that he had found a way to co mpu te the harn1onic series w ith logari thms , but he did no t inclu de it sinc e the calculations were " trou blesom . " 1 2 9 Collins did not hear fro m him ag ain until July . When he finally sent Collins the " O bs erv ations on Kinckhuysen " in July , Newton acco mpanied them with a letter filkd with defen­ sive diffidence. He hoped he had done what Collins wanted . He left it entirely to Collins whether to p rint any o r all of it . " F o r I assu re you I writ wt I send you not so much wth a designe yt th ey should bee printed as yl your desires should bee satisfied to have me revise " I n a lgcbram Gerardi Kinckhuysen observationes " ; Math 2 , 364-444 . See a l so the rela ted p aper , "P roblems fo r Construing Aequ ations , " Math 2, 450- 5 1 6 . See Christoph J . S criba , " M ercator' s K inckhuysen-T ranslation i n the Bod leian Lib rary a t O xford , " British Jo urnal j(n th e History of Science , 2 ( 1 964) , 1 45-58 . 1 26 W allis to C o llins, 25 Jan . 1 6 72 ; S teph en Peter R igaud , ed . , Cmesp ondence of Scien t�fic Men of the Seventeenth Centu ry , 2 vol s . (O xford , 1 84 1 ) , 2 , 529 . 1 2 7 Town elcy to Collin s , 4 Jan . 1 672 ; Co rres 1 , 78 . 1 28 James G regory to C ollins , 23 Nov . 1 6 70; Con-es 1 , 45 - 8 . 1 2 9 Con-es 1 , 27 .

1 25

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225

ye booke . And so soone as you hav e re ad ye papers I have my end of writing the m. " There re mains [he added] but one thing mo re & thats about the Titl e page if you print these alterations wch I have made in the Autho r: Fo r it may bee esteemed unhandsom & inj urious to Kinck huysen to father a booke wholly upon him wch is soe much alter' d from what hee h ad made it. But I think all will bee safe if after ye words [nunc e Belgico Latine versa] bee added [et ab alio Autho re locupletata . ] or some other s uch note . 1 3 0 Not " enri ched by Isaac Newton , " but "en riched by ano ther au­ tho r " ! O thers might long to see his wonderful additions . New ton himself was p rimarily concerned that his name not appea r . Enco uraged b y receiving the anno tations , Collins hastily wro te ba ck that he no ted Newton's agree ment with him on the ins uffi­ ciency of Kinckhuysen 's treatment of surds . He sent along th ree books and asked Newton to pick out the best discussion of surds to insert in the volume . Rather wearily , New ton asked to have the manuscript back . 13 1 It came at once with another letter filled with fu rther questi ons and the p ro mise of further publicity; " your paines herein , " Collins assured hi m , " will be acceptable to s o me ye ry eminen t G ran dees of the R Societie who must be made accquainted therewith . . . " 132 It was a clumsy thing to say to a man who h ad recently to ld hi m he studied chiefly to di minish his acquaintan ce . Over two months passed befo re Newton rep lied . On 27 September he infor med Collins that he had thought s o me of co mposing a co mpletely new introduction to algeb ra . But considering that by reason of severall divertise ments I sho uld bee so long in doing it as to tire you patience wth expectation , & also that the re being severall Introductions to Algebra al ready published I might thereby gain ye esteeme of one ambitious among ye croud to have my scribbles printed , I have chosen rather to let it passe wtho ut much altering what I sent you before . 1 33 C ollins never saw the manuscript again . He also heard no more fro m Newton for ten months . Although he had misj udged Newton initially , Collins now real­ ized that he was dealing with a man extraordinary in mo re ways th an mathematical genius . He responded to Newton' s silence with his own , and in December he described his relations with New ton to James Grego ry . G regory was eager to learn about Newton ' s general method of infini te series . Collins told h i m h o w Newton co mmuni cated indiv idual series but not the general method , tho ugh 1 3° Con-es 1 , 30- 1 . 1 3 1 Collins to N ewton, 1 3 July 1 670; Con-es 1 , 32-3 . Newton to Collins , 1 6 Ju ly 1 670; Con-es 1 , 34-5 . 1 3 2 Collins to Newton, 1 9 July 1 970; Corres 1 , 36 . 1 33 C orres 1 , 43-4 .

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he understo od that he had written a treatise about it. Collins had sent him the annuity problem , hoping thereby to learn the general method . Newton had sent back only the fo rmula ; " hence ob serving a warinesse in him to imp art , o r at least an unwillingness to be at the paines of so doing , I de sist , and doe not trouble him any more . . . " 1 34 But in the end Collins could not deny his self-impo sed mission and desist fo rever . In July 1 67 1 , he wrote a ch atty letter about mathematics and the Kinckhuysen edition , whi ch would have a better sale , he remarked , if it carried Newton ' s name. He also sent Newton a copy o f Bo relli ' s new book . Newton resp onded with deliberate incivility by suggesting that Collins not send him any more book s . It would be su fficient if he merely info rmed him what was publi shed . He did mention that he had intended to visit Collins on the occasion of the re cent induction o f the D uke of Buckingham as chancellor of the university, but a bout of sickness had preven ted hi s making the trip to London . So mewhat g rudgingly , it appears , he also added that he had reviewed his introduction to Kinckhuysen during the winter. And partly upon Dr Barrows instigation , I began to new metho diz ye discourse of infinite series , designing to illustrate it wth s uch prob­ lems as may (some of them perhaps) be more acceptable then ye invention it selfe of wo rking by such series . But being suddainly diverted by some buisinesse in the Country, I have not yet had leisure to return to tho se thoughts , & I feare I shall not before winter . But since you info rme me there needs no hast, I hope I may get into ye humour of compl eting them before ye impression of the introduction , because if I must helpe to fill up its title p age , I had rather annex something wch I may call my owne, & wch may bee acceptable to Artists as well as t other to Tyros . 1 35 The new meth odized disco u rse, known as the Tractatus de metho­ dis serierum et fiuxionu m (A Trea tise of the Methods of Series and Flux­ ions) , though Newton hi mself did not give it a title , was the mo st ambitious exposition of his fluxional calculus that Newton had yet undertaken . 136 D rawing on both De a nalysi and the tract o f O ctob er 1 666 , he produ ced an expo sition of hi s method di rected to that circle of mathemati cal art ists with who m he had communed so far only passively in the rea ding of thei r w o rk s . A lthou gh the treati se 134

Co rres 1 , 53- 5 . I am u ns ure h ow to reconcile C ollins ' s reception o f De analys i w i th thi s letter . It is true that De analysi gave methods o f expanding by divi sion and root extrac­

tion wi thout s tating the genera l binomial theo rem. Nevertheles s , it o u tl ined Newton ' s enti re fl u xi onal calculus as well as qua dratu res b y infi nite series . P robably Colli ns d i d not u n derstand what he had. 13 5 C ollins to Newton , 5 July 1 671 ; Corres 1 , 65-6 Newton to C ollins , 20 July 1 67 1 ; Co rres 13 6 Math 3 , 32-328 . 1 , 67-9. .

227

L u cas ian p refessor D

F

Figure 6 . 7 . A reas treated as fluent quantities with ordinates as th eir fluxions .

was i m mensely important , there is no need to rep eat here what I have described at some length earlier . Suffice it to say that in virtu­ ally every respect , Newton both deep ened and exp anded his treat­ ment and made his exp osition more systemati c. He gave sp ecial attention to the two basi c op erati ons , posed here as problems 1 and 2, to which all difficulties in the analy sis of cu rves reduce. 1 . Given the length of the space continuously (that is, at every time) , to find the speed of motion at any ti me proposed. 2. Given the speed of motion continuo usly , to find the length of the space described at any ti me pro posed . 1 37 D i ssatisfi ed with his exposition of integ ration , Newton later ad­ ded a new problem: To determine the area of any p roposed cu rv e . " The resolut ion o f the p roblem i s based o n that of establi shing the relation ship between fluent quant ities from one between thei r flux­ ions (by P roblem 2) . " If the line BD by the motion of which the area A FDB is described advances along AB , conceive that the rec­ tangle A CEB is des cri bed by part of the line, BE, one unit in length (Figure 6 . 7) . Tak e BE as the fluxion of the rectangle; then BD will be the fluxion of the area . A B x . Hence ABEC = 1 x = x . BE x ( = 1 ) . AFDB x. BD = z = zlx (since x = 1 ) . " Conse­ quently , by the equation defining BD is at once defined the fluxio­ nal ratio zlx, and from this (by P roblem 2 , Case 1 ) will be eli cited the relationship of the fluent quantities x and z . " 138 That is, from ==

==

13 7 13 8

·

==

Math 3 , 7 1 . Math 3 , 2 1 1 . I have followed Whi te side ' s use of x and i in his tra nslation. Actua lly Ne wton u sed m and r at thi s time and developed the dot notation only later .

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Never at rest

the fo rmula of the curve expressing y ( z) as a function of x ' we can , via p roblem 2 , the method of qu adratu res , arri ve at a new formula expressing the area z as a function of x . From thi s intro­ duction, Newton proceeded to the qu adrature of polynomials , of equ ati ons that can be exp anded into infinite seri es, and of affected equ ations , and he included an extensive table of integrals . ==

By means of the preceding catalo gues [he concluded} not merely the areas of curves but also other quantities of any kind generated at an analogo us rate of flo w may be derived from their fluxions , and that through the medium of this theorem: A quantity of any kind is to the unity of its own class as the area of a curve to the surface unity if only the fluxion generating that quantity shall be to the unity of its own kind as the fluxion generating the area to its own unity - that is , as the line , moving normally upon the base , by which that area is describ ed , to the linear unity . In consequ ence, if a fluxion of what­ ever kind be expressed by an ordinate line of this sort , the quantity generated by that fluxion will be expressed by the area described by that ordinate. Or if the fluxion be expressed by the same algebraic terms as the ordinate line , the generated quantity will be expressed by the same ones as the area described. 139

One of the not able features of De methodis is Newton ' s growing attention to the foundation of his fluxional calculu s . Fro m the be­ ginning of his u se of the concept of motion , its intuiti ve idea of continu ou s flow had cla shed with the i nfinitesi mal devices by which Newton expressed it . De ana ly si offered some tent ati ve steps toward the resolution of the confli ct . 1 4 0 De methodis extended them . Signifi cantly , the treati se began with an expo siti on of infi nite series in whi ch Newton enunciated a concept of convergence. As the qu otient s of affected equations are extended , they " e ver more closely app roach the root till finally they differ from it by less than any gi ven qu antity and so, when they are infinitely extended , differ from it not at all . . . " 1 4 1 Although De methodis continu ed for the most part to u se the langu age of infinitesi mals , the concept of li mit values whi ch can be app roached more closely than any defined di fference probably influenced Newton ' s understanding of h i s terms . 1 42 When h e dealt with problems su ch as the area of the ci ssoid , in which he appealed directly to comp arable quantities be­ ing generated in equal mo ment s , the ideas i mplicitly present in his concept of motion of flux emerged most clearly . In su ch demon­ st ration s , he stated , 139

Math 3 , 285 .

For e xample, in hi s derivati on of the algorithm for finding a fluxion; Math 2 , 243- 5 . Math 3 , 67 . C f. D e Analysi ; Math 2 , 245-7. 14 2 F o r e xamples o f usage that i mplies such , see Math 3 , 79-83 , 1 23, 1 33, 2 99, 305, 3 1 1 . 14 °

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I take quantities as equal whose ratio is one of equality . And a ratio of equality is to be regarded as one which differs les s from equality than any ratio of inequality which can possibly be assigned . . . . I have here used this method . . . since it has an affinity to the ones usually em­ plo yed in these cases . However , that bas ed on t he genesis of surfaces by their motion of flow appears a more natural approach. 1 43

S o als o Newton' s willingness to u se the fluxions of fluxions and his treatment of angles of conta ct indicate that his fluxional meth od rested on something far m o re su btle than a simpleminded appeal to infinitesimals . When Newton introduced the concept of the qu ality (or variation) of curvat ure, he had in effect to employ a third deriv­ ative . It is tru e that he tended to mask the issue by setting up a new equation of whi ch he found the fluxion. Thu s , in this case, he defined the radius of cu rvature as a variable (stated as a function of x ) , the fluxion of whi ch expres sed the variati on of curvature . Neverthele ss , he was handling the equi valent o f a third derivative, and he was not one to deceive himself about his own pro cedu res . 1 44 The p roblem of cu rvature rai sed the related issue of degrees o f infinity . His investiga­ tion of the cy cloid led to the conclusion that its radius of cu rvature at its end is zero . The cycloid, he asserted , is " mo re cu rved at its cu sp F than any cir cle , fo rming with its tangent ,BF p roduced a conta ct angle infinitely greater than a circle can with a straight line . There are also contact angles infinitely greater than cycloidal ones and others in turn infinitely g reater than these , and so on infinitely , but still the greatest are infinit ely less than rect ilinear ones . " 1 45 In a separate pap er asso ciated with De methodis and perhaps in­ tended as an addendum to it , Newton b roke through the li mitations o f available language and placed his method on a new foundation more adequate to its cent ral inspiration . It began with fou r axio ms . Axiom 1 . Magnitudes generated simultaneous ly by equal fluxions are equal. Axiom 2. Magnitudes generated simultaneous ly by fluxions in given ratio are in the ratio of the fluxions . Note: by simultaneous generation I understand that the wholes are generated in the same ti me. Axiom 3. The fluxion of a whole is equal to the fluxions of its parts taken together . Here note that increasing fluxions are to be set positive, decreasin g ones negative . Axiom 4 . Contemporaneous mo ments are as their fluxions . 1 4 6 1 4 3 Math 3 , 283 . Cf. his rectification of the Archim edean spira l by trans form ing it into a para b ola of equal length; Math 3 , 3 1 3 . 1 4 5 Math 3 , 1 65-7. 1 44 Math 3 , 1 87-9 1 . 1 4 6 Math 3, 331 . B y a moment, New ton meant the increm enta l cha nge of a fluent ma gnitude . A flu xi on was the velocity of change, the m oment d ivided by the " m oment" of time .

Never at rest

230 c

c

F

e

E

g

G

A

B b

Od

Figure 6 . 8 . Th e con cep t of nas cen t or u lti mate ratios .

Newton then p ropo sed a s theo rem 1 that when there are four perpetu ally proportional fluent quantities , the su m of th e prod ucts of each extreme with the other 's fluxion i s equal to the sum o f the products of each middle term with the other's fluxion. That i s , if AIB = CID (so that AD = B C) , then A flD + D flA = B flC + C fill . To p rove this theo rem , he set out with a purely analyti c approach which depended on eli minating terms in whi ch a mo ment M of "infinite smallness" appeared . He stopped in mid­ sentence and set out anew with a geo metri cal model in whi ch fou r line s , A B , A C , AD , and A E , became the flu ent quantities . AB/AD AEIA C ; hence the re ctangle A CFB equals the rectangle AEGD (Figure 6 . 8) . Let the lines "increa se fluxionally by their resp ective mo ments Bb , Dd , Ee , and Cc . . . " Since the re ctangles remain equal by assumption , their increments mu st also be equal . Ab Cc + A C Bb = Ad Ee + AE Dd . By axiom 4, moments are as thei r fluxions ; hence we can replace Cc , Bb . . . with fl(A C) , fl(AB) . . . Therefore ·

·

·

·

==

·

·

·

Ab

·

fl(A C) + A C

·

·

fl( A B) = Ad

·

fl(A E) + A E

·

fl(AD)

With his geo metri c model , Newton initially treated the mo ments once again as stati c, infinitesi mal incre ments . Such was not his goal , however. He cro ssed the passage out and pursued the intuitive concept of motion to a different end .

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L u casia n p rofessor

Now let the rectangles Af and Ag di minish till they go back into the primary rectangles AF and A G : Ab will then come to be AB while Ad becomes AD . Hence at the last moment of that infinitely decreas­ ing fluxion - that is , at the first moment of flux of the rectangles AF and AG when they start to increase or diminish - there will be AB

·

fl (A C) + A C

·

fl(A B)

=

AD

·

fl(A E) + A E · fl(AD) .

As was to be proved . 1 47 Thu s was born the concept of nascent or ultimate ratios whi ch brou ght the notion of limit v alues , present in hi s method o f infinite series , to his method of fluxions and ena bled Newton mo re ade­ quately to express the intuitions p resent in the idea of fluxional change. A s co rollaries to theo rem 1 , Newton derived the basi c rules fo r the fluxions of products , quotients , roots , and p owers . He ad ded eleven mo re theo rem s as "foundations fo r demonst rations " ; and he illu strated the po wer of the method he was propo sing by quickly doing the areas under three of the mo re intractable cu rves known to contempo rary mathemati cs , the Gutschoven qu artic (x 2 (a 2 y2) 4 y ) , the ci ssoid , and the conchoid. 148 F o r all its brilli ance , the most remarkable thing about De meth odis with its associated papers is the fact that Newton never comple ted it. From his letters , it appears that he st arted the treatise in the winter of 1 670- 1 . A trip ho me in the spring interrupted him , and when he wrote to Collins on 20 July 1 6 7 1 , he said he had not returned to the papers and di d not exp ect to before winter . "I hope I may get int o ye humour of completing them . . . " Since New ton was p rone to similar co mments , whi ch were defen sive maneuvers to ward off criti cism by pretending lack of interest , we should pau se befo re we take the co m ment seriou sly . The manus cript itself does seem to bear him out , however. It reveals an initial effo rt whi ch ca me to a halt , a renewed attempt which advan ced a bit further , and ultimate abandonment . 149 In May 1 672 , N ewton in­ fo rmed Collins that he had written the better half of the treatise the -

==

1 47 Math 3, 33 1 -5 . 1 4 8 The paper, whi ch ha s n o title, i s found i n Math 3 , 328- 52 . 1 4 9 Add MS 3960. 1 4 . N ewton folded fifty small sheets into a rough pa mphlet containin g two hun dred pages whi ch he sewed to gether a t some point (i ncluding a t least one other ha lf-sheet with an early emendation) . I find in the pa mphlet ' s size the suggestion that he expected to fill about two hun dred pages . He never did get to the second half, which remained empty except for a few e mendations entered there . When he returned to the treatise, he a dded a few more sheets whj ch he si mply fol ded in to the center w ithout sewin g , and (perha ps at th is ti me) a dded some corrections and ampl ificati ons to earlier parts . There is a list of p roblems from ca . 1 670 (Math 3 , 28-30) . wruch appea rs to be an outline o f the proj ected treatise. What he completed does not contain a number of the p roblem s on the list, though it contains well over half of them .

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Never at rest

p revious winter , but it had proved larger than he expected . It was not done; he might "possibly" co mplete it . By July he did not know " when I shall proceed to finish it . " 15 0 In fact , he never did . No doubt th e reluctan ce of London booksellers to publi sh mathe­ mati cal books , whi ch usu ally lost money , played a role in New­ ton 's di lat o rines s . 1 5 1 It is impos sible to as sign the determining role to thi s facto r, however . The Royal S ociety subsidized the publi ca­ tion of H o rrox ' s Op era in 1 672 . Ed1nund Gunter's Workes were republished the following year . In 1 674 , B arro w brought out a new edition of hi s lectu res , and in the years ahead p roceeded with publi­ cations of Euclid's Data and Elem ents and of Archimedes and Apol­ lonius . Other mathemati cal works , mos tly elementary ones to be sure, appeared continually . Had Collins ever gotten his hands on Newton 's De meth odis , he would have moved hea ven and earth to put it into print as Halley did with another treatise fi fteen years later . It was not the dep ression of the publishing trade, rather it was Newton who abo rted the publi cation of a work which would have transfo rmed mathemati cs . He never retu rned hi s annotations on Kinckhuysen to Collin s , and he finally killed the edition by buying out the interest of the book seller Pitts fo r fou r pounds . Collins never saw anything of the maj o r treatise beyond the barest of tan­ talizing hints in a couple of letters . The unresolved, unresolvable tension that pulled Newton to and fro , as he responded to the warmth of prai se , then fled in anxiety at the scent of criti cism , worked now to sup press his masterful treatise . F o r that matter , he was not terribly interested in it in any case . As he had told Collin s , he was not in the hu mo r to complete i t . Nearly all of Newton 's bu rst of mathemati cal activity in the pe­ riod 1 669- 7 1 can be traced to external stimuli , to Barrow (armed initially with Mercato r's w o rk) and to Collins . His own interests had moved on . By 1 675 , C ollin s , who con fes sed that he had not heard fro m him fo r nearly a year , repo rted to Grego ry that Newton w as "intent upon Chimi call Studies and practises , and b oth he and D r B arrow &c [were] beginning to think e mathcall Speculations to grow at least nice and dry , if not somewhat barren . . . " 15 2 Ho wever , it proved to be impos sible fo r Newton to reti re again to the anonymity of hi s s_a nctu ary . An irresisti ble cu rrent that would not let his gifts be hidden bore him fo rward . If not mathemati cs , then something else . Fittingly , it was a product of his hands rather than a child of his b rain whi ch brought the crafts man fro m Gran1 5 1 Cf Math 3 , 6- 7 n . 1 5 ° Corres 1 , 1 6 1 , 2 1 5 . 1 5 2 Colli ns t o J a mes Gregory , 1 9 Oct. 1 6 75 ; Corres 1, 3 5 6 .

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tham fully into the view of the European scientific co mmunity . Though , contrary to the account long accepted , we now know that Newton's the ory o f colors did not lead him wholly to desp air of refra cting t eles copes, 1 53 he did bu ild a reflecting telescope neverthe­ less . He cast and ground the mirro r fro m an alloy of his o wn invention . He built the tube and the mount . And he was proud of hi s handi w o rk . He was still proud when he recalled it for Conduitt nearly sixty years lat er: " I asked him , " Conduitt re co rded , " where he had it made, he said he made it hi mself, & when I a sked him where he got his tools said he made t hem him self & la ughing added if I had staid fo r other people to make my tools & things fo r me, I had never made anything of it . . . " 1 54 The telescope was about six inches long , but it magnified nearly fo rty times in dia meter, which , as Newton could b e b rought t o admit , w a s more than a six-foot refracto r could do . Later he made a second telescope. "When I made these, " he confessed in the Op ticks , " an Artist in London undertook to imi tate it; but using an other way to p oli shing them than I did, he fell mu ch short o f what I had attained to . . . " The Lucasian p ro fessor was unable to restrain himself from proceeding to lecture the artisans of London on the secret s of thei r craft. The P o lish I used was in this manner . I h ad two round Copper Plates , each six Inches in Diameter , the one convex, the oth er con­ cave , gr ound very true to one another . On the convex I ground the Object-Metal or Concave which was to be polish'd , 'till it had taken the Figure of the Convex and was ready for a P olish . Then I pitched over the convex very thinly , by dropping melted Pitch upon it, and warming it to keep the Pitch soft , whilst I ground it with the con­ cave Co pper wetted to make it spread eavenly all over the convex. Thus by wo rking it well I made it as thin as a Groat, and after th e convex was co ld I ground it again to give it as true a Fi gure as I could . Then I took Putty which I had made very fine by washing it from all its grosser P articles , and laying a little of this upon the Pitch , I ground it upon the Pitch with the concave Copper , till it h ad done making a Nois e; and then upon the Pitch I ground the Object-Metal with a brisk motion , for about two or th ree Minutes of ti me , leaning hard upon it . Then I put fres h Putty upon the Pitch , and ground it again till it had done making a noise, and afterwards ground the Object-Met al upon it as before . And this Work I repeated till the Metal was polis hed , grinding it the last ti me with all my strength for 153

1 54

Math 3 , 467-9 , 5 1 2- 1 3n . Cf. Zev Bechler, " ' A less agreeable matter' : The D i sagreea ble Case of Newt on and Achromatic Refraction , " British Jo urnal fo r the History of Science, 8 ( 1 975) ' 1 0 1 -26 . Condu itt ' s memorandu m of 3 1 Aug . 1 726; Key nes MS 1 30. 1 0 ,

ff. 3_3v .

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a good while together , and frequently breathing upon the Pitch, to keep it moist without laying on any more fres h Putty. 1 55 Whate ver the su ccess of the refle cto r, the telescope Humph rey Newton found fifteen years later , stationed at the head of the stairs down to the garden , where Newton used it to observe co mets and planet s , was a refracto r . 156 Meanwhile he found it quite impossi ble not to show off his cre­ ation . His earliest survi ving letter , of February 1 669 , i s a des crip tion of it to an unknown co rrespondent written as a result of a p romise to Mr. Ent , to whom he had presu mably shown or mentioned the telesco pe. 157 When he met Collins in London at the end of 1 669, he told him about the telescope, allowing it in his account to magnify 1 50 ti mes . 158 He must have been showing it off in Cam b ridge . In December 1 67 1 , Collins repeated to Francis Vernon what Mr. Gale (a fellow of T rinity) had written of it from Cambridge . 159 By Janu­ ary , Towneley was asking about it excitedly , and Fla msteed had heard of it both from London and from a relative who had re cently been in Cambridge . 16 0 Perhaps Collins had never info rmed the eminent grandees of the Royal Society about Newton 's mathema ti­ cal achievements , but they heard about the telescope all right and asked to see it late in 1 67 1 . At the very end of the year, Barrow delivered it to them (Figu re 6 . 9) . When it arri ved , the telescope cau sed a sensation . Early in Janu­ ary , Newton re ceived a letter fro m Henry Oldenbu rg , the se cretary of the society . sr

Your Ingenuity is the occasion of this addresse by a hand unknowne to you. You have been so generous , as to impart to the Philos ophers here, your Invention of contracting Teles copes . It having been con­ sidered, and examined here by some of ye most eminent in Opticall Science - and practise, and applauded by them, they think it n eces s ary to use some meanes to secure this Invention from ye Usurpation of fo rreiners; And therefo re have taken care to represent by a scheme that fi rst Specimen , s ent hither by yo u , and to describe all ye parts of ye Instrument , together wth its effect , compared wth an ordinary, but much larger , Glasse; an d to send this figure , and description by ye l55

Opticks , pp . 1 04-5 . Keynes MS 1 35 . See A . A . Mills and P . J . Turvey , " Newton 's Telescope. An examina­ tion of the Reflecting Telescope Attributed to Sir I saac Newton in the Pos session of the Royal Society, " Notes and Reco rds of the Royal So ciety , 33 ( 1 979) , 1 33-55. 1 57 Co rres 1 , 3-4 . 1 58 Collins t o James Gregory , 2 4 D e c . 1 970; Co rres 1 , 5 9 . 1 5 9 Co rres 1 , 5 . 16 0 Towneley to Collins , 4 Jan . 1 672; Co rres 1 , 78 . Fla msteed t o Collins, 3 1 Ja n . 1 672; Co rres 1 , 88. 1 56

A.

Figure 6 . 9 . The Royal Society ' s drawing of Newton's teles cope . Th e two crowns are the s ame object , an orn ament on a wea­ th ercock 300 feet away , as s een through Newton ' s tele­ s cope (A ) an d as seen through a refracting t elescope 25 inch es long (B) . (Co urtes y of the Royal Soci ety . )

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Secretary of t R. Soc. (where you were lately by t Ld Bp . of Sarum [ Seth Ward] proposed Candidat) in a solemne letter to Paris to M. H ugens , thereby to prevent the arrogation of such strangers , as may perhaps have seen it here , or even wt h you at Cambridge; it being too frequent , y1 new Invention · and contrivances are snatched away from their true Authors by pretending bystanders ; B ut yet it was not tho ught fit to send this away w1h out first giving you notice of it, and sending to you t very figure and description , as it was here dra wne up; y1 so you might adde , & alter , as you shall see cause; wch being done here wt h , I shall desire your favour of returning it to me w1h all convenient speed , together w1h such alterations , as you shall think fit to make therein . . . sr

your humble servant Oldenburg 161

Tru e to their w o rd, the Royal So ciety sent a descri ption of the instru ment to Hu ygens; they even sent a general account written on 1 Janu ary ahead, so concerned were they to secu re th e credit to Newton . 162 Huygens was no less plea sed than they; he called it the "marv ellou s telescope of Mr. Newton . . . " 16 3 The S ociety em­ ployed Chri stopher Cock , an instru ment maker in London, to con­ stru ct a rele cting tele scope fou r feet in length , and later one of six feet , though both failed fo r want of satisfactory mirrors . 16 4 Having nominated Newton, they proceeded to hi s full election to the soci­ ety on 1 1 Janu ary . The ritu al dance perfo rmed with Collins now co mmenced anew. Newton fairly beamed as the warm glow of p raise fell about him . A t the reading o f your letter [he replied t o Oldenburg] I was sur­ prised to see so much care taken about securing an invention to mee , of wc h I have hitherto had so little value. [sic! ] And therefore since the R . Society is pleased to think it worth the patronizing , I must acknowledg it deserves much more of them for that, then of mee , who , had not the co mmunicati on of it been desired , might h ave let it still remained in private as it hath already done some yeares . 16 5 Despite hi s p retense of indifference, Newton picked up their inten­ tion to send a des cription to Hu ygens and suggested that they be su re he realized the telescope eliminated colors from the image. He volunteered instru ctions about its maintenance, and in his next two letters he sent info rmation about alloys he had tried for mirro rs . He readily agreed to the publi cation of the descripti on without even suggesting that hi s name be withheld . 16 1 1 65

1 62 Co rres 1 , 72. 1 6 3 Corres 1 , 89. Co rres 1 , 73. Newton to Oldenburg , 6 Jan . 1 672; Co rres 1 , 79.

1 64

Co rres 1 , 83 , 1 04 .

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237

I am very sensible of the honour done me by ye BP of Sarum in p ro p o sing mee Candidate [he concluded his initial reply to Oldenburg] & wch I hope will bee further conferred u pon mee by my Election into the Society . And if so , I shall endeavour to tes tify my gratitude by communicating what my poore & solitary endeavours can effect towards ye promoting your Philoso phicall designes . 166

The Roy al Society could not have guessed that the final sentence contained a hidden p romise . Newton unveiled it on 1 8 Janu ary . He info rmed the society that " I am purpo sing them , to be conside red of & examine d , an acco mpt of a Philosophicall di scovery wch in­ duced mee to the making of the said Teles cope, & wch I doubt not but will prove mu ch mo re gratefull then the communi cation of that instru ment , being in my Judgment the oddest if not the most con­ sidera ble detection wch hath hitherto beene made in the operati ons of Nature . " 16 7 The ri tual dance had further figures , however . As he had found already with Collins , it proved to be not quite that simple to divulge hi s discovery . A week and a half later , he had not yet sent it, and he felt compelled to go through the remaining steps . He wrote that he hoped he could "get some spare bowers " to send off the account . 1 68 Wi ckins needed the sp are hou rs as much as Newton , who set him to work copying the p aper . The die had been cast , however . It was too late to withdraw . On 6 February 1 672 , Newton finally mailed an account of his theo ry of colors to Lon­ don . Fo r the mo ment the po sitive p ole p revailed . S wep t along by the su ccess of his telescope, New ton stepped publicly into the com­ munity o f natural philos ophers to whi ch he had hitherto belonged m se cret . 1 66

Co rres 1 , 80.

1 67

Co rres 1 , 82 .

1 68

Co rres 1 , 84.

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pap er on colo rs that Newton sent to the Ro yal Society early in 1 672 in the fo rm of a letter addres sed to Henry O lden­ burg did not contain anything new from Newton' s point of view . The o ccasion p rovided by the telescope had co me at an opportune time. At B arrow's behest , Newton had been revising his lectures fo r p ubli cation during the winter. 1 He had not found it a g reat cho re to p roduce a succinct statement of his theo ry bu ttre ssed by three p ri s mati c experiment s that he took to be most compelling . He thought it relevant to include a sp ecial discu ssion of how the dis­ covery had led him to devise the reflecting telescope. The continu­ ing co rrespondence p rovoked by the initial paper , whi ch intruded int ermittently on his time and consciou sness during the following six years , also involved only one addition to his optics , his intro­ duction to diffraction and brief investig ation of it . Aside from dif­ fra ction, the entire thru st o f his concern with optics during the period was the exposition of a theo ry already ela bo rated . The continuing di scu s sion fon:ed Newton to cla rify s ome issues . When he wrote in 1 672 , he had not yet fully sep arated the i ssu e of heterogeneity from his co rpu scular conception of light , and he al­ lowed himself to assert that, becau se o f his di scovery , it could " be no longer disputed . . . whether Light be a Body. " 2 He cou ld hardly have been more mi staken . Within a week of the paper's presentation, Robert Hooke p roduced a critique that mi sto ok co r­ puscularity fo r its central argument and p ro ceeded to disp ute it with some asperity . The le sson was not wasted . Though he continued to believe in the co rpu scular conception, Newton learned to insist that the essence of his theo ry of colo rs lay in heterogeneity alone . 3 Thi s

T

1

3

Cf. Collins to G regory , 23 Feb . 1 672 (H . W . Turnbull, ed . , James Grego ry Tercentenary Memo rial Vo lume [London , 1 939] , p . 2 1 8) ; Collins to Gregory , 1 4 March 1 672 (Co rres 1 , 1 1 9) ; C ollins to Newton , 3 0 April 1 672 (Co rres 1 , 1 46) . Add MS 4002, the o riginal ma nu­ script of the lectures , contained eighteen lecture s . Dd . 9 . 67, the manuscript tha t Newton deposi ted as his lectures , contained thirty-one . Although the numbers exa ggerate the ex­ tent of expansion , there was certainly a revision , and it appears to me that D d . 9 . 6 7 was the product of the winter of 1 6 71 -2 . See George Sarton , "D iscovery of the Di spersion of 2 Co rres 1 , 1 00. Li ght and of the Nature of Color (1 672) , " Isis, 14 (1 930) , 326-4 1 . Cf. hi s statement in February 16 76 against objections based on hypotheses about the nature of light. " That in any Hypothesis whence t rays may be supposed to ha ve any orig inall diversities, wh ether as to size or figu re or mo tio n or force or quality or any thing els imagina bl e wc h may suffice to difference those rays in colo ur & refra ngibility, there is no need to seek for other causes of these effects [colors and different re fracti ons] then those original diversities. Thi s rule being laid down, I arg ue thus. In any Hypothesis whatever , light as it comes from ye Sun mu st be 238

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was a matte r of clarifi cation and exposition , ho weve r, not an alte ra­ tion of his theo ry . The very fact that six years of discus sion effected no change , such that his Op ticks, finally publ ished in 1 704 , merely res tated conclu sions wo rked out in the late 1 660s , te stifies to the intensity and rigo r of the early investigation . The dis cussion that follo wed on the paper of 1 672 tells us less about optics th an ab out Newton . For eight years he had locked himself in a re mo rseless struggle with Truth . Genius of Newton ' s orde r exacts a toll . Eight years of unea ten meals and sleep less nights , eight years of contin ued ecstasy as he faced Truth dire ctly on gro unds hitherto unkno wn to the human spirit , took its further toll . And exasperation that du llness and stupidity should distract hi m fro m the fu rther ba ttles in which he was already engaged on new fields added the final straw . By 1 672 , Newton had lived with his theo ry fo r six years , and it now see med obvious to him . F o r everyone else , ho wever , it s till embodied a denial of co mmon sense that made it d ifficult to accept. Their inability to re cognize the fo rce of his de mons trations qu ickly drove New ton to dis traction . He was unp repared fo r anything excep t im mediate acceptance of his theo ry . The continuing need to defend and explain what he took to be settled plunged hi m into a personal crisis . T o be sure , the ini tial response gave no hin t of the crisis to follow . Al most befo re the ink had dried on his paper of 6 February , .. Newton received a le tter from Oldenburg . Filled with lavish praise , it info rmed him that his p aper had been read to the Royal Society, where it " mett both with a singular attention and an unco mmon applause . . . " The Society had ordered it to be printed fo rthwith in the Ph ilosophical Transa ctions if Newton would agree . 4 The tension caused by the de cision to send the paper can be heard in Newton ' s relief a s h e read Oldenburg' s letter. I before tho ught i t a great favour t o have beene made a member of that honourable body; but I am now more sensible of the advantage . For beleive me s r I doe not onely esteem it a duty to concurre w t h them in ye promotion of reall knowledg, but a great privelege that instead o f exposing dis co u rses to a prej udic't & cens orious multitude (by wch means many truths have been bafled & lost) I ma y w t h freedom apply my self to so j udicio us & impartiall an Assembly. 5 He as sented to the publication of the paper with only a slight - fo r him , compulsory - demur. supposed either homogeneal or heterogeneal . I f y e last, then i s that Hypothesis co mpre­ hended in this general rule & so cannot be against me: if the first then must refractions have a po wer to modify light so as to chan ge it's colorifick qualification & refrangibility; w c h is against experience. " Newton to Oldenburg , 1 5 Feb . 1 676; Corres 1 , 4 1 9-20 . Oldenbu rg t o N ewton , 8 Feb . 1 672; Co rres 1 , 1 07- 8 . New ton t o Oldenburg , 1 0 Feb . 1 672; Corrcs 1 , 1 08-9 .

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Accord ingly , the paper appeared in the Ph ilosoph ical Transa ctions fo r 1 9 Feb ruary 1 672 . Tog ether with the des cription of his tele­ scope, which the follo wing issue carried , i t established Newton ' s rep utation i n the wo rld of natural philosophy . Oldenburg took care to p ublicize both items in his extensive co rrespondence with natu ral philosophers thro ughout E urope. The replies he received ind icate that both were no ticed . 6 The telescope caught the eye of leading astronomers everywhere - Cas sini , A uzo u t , and D enis in Paris , and Hevel ius in Danzi g . In the summer, Ol denburg re­ ceived a report from Flo rence that a reflecto r built on Newton ' s plan h a d already been constructed there . A Frenchman , Casse­ grain , did New ton the ultimate honor of claiming prio rity fo r the in venti on. In the same letter that reported Cas seg rain ' s as serti on, Olden b u rg felt co mpelled to men tion that James G reg ory had publis hed a plan for a reflecting telescope in his Op tica p romo ta , 1 663 . 7 As fa r as Cassegrain was con cerned , no one seems to have taken hi s clai m , advanced afte r th� pub lication of New ton ' s tele­ scope, seriously . Grego ry never pressed his; and ever generous , he acknowledged the importance of Newton ' s paper on colors in es­ tab lishing the theoreti cal significance of the reflecting telescope. 8 Th e G regorian and the Cas seg rani an models differed from the Newtonian and from each other in the shape and p lacemen t of the secondary mirro r . Newton argued , not witho ut some heat , fo r the superio rity of his arran gemen t . 9 No one ch allenged the fact that Newton had been the first to produce a reflecting telescope that worked . The paper on colors also received i ts share of attenti o n . Ol den­ b u rg specifically calle d it to Huygens ' s atten tion when he mailed the Ph ilosoph ical Transactions to him . Huygens replied that " the new Th eory . . . appears very ingenious to me . " To be sure , Huygens later expres sed reserva tions about the theo ry; meanwhile , in April , Newton received what can only have appeared as praise fro m the recognized leader of E u ropean scien ce . 1 0 A yo ung English as trono­ me r , J ohn Flams teed , who would soon become the first As trono­ me r R oyal , commented on the paper, tho ugh witho ut m uch co m­ prehension . 1 1 A young German savant resident in Paris , Go ttfried Wilhel m Leibni z , then unknown but as determined to make his way in natural philosophy as he was des tined to , indica ted that he 6 7 8 9 10 11

A . Rupert and Marie Boas Hall, eds . , The Co rrespondence of Henry Olden burg , 1 2 vols . (Madison, 1 965- ) , 9, passim . O ldenburg to Newton , 2 May 1 672; Co rres 1 , 1 50 . G regory to Collins , 23 Sept . 1 672; Corres 1 , 240 . Newton to O ldenburg , 4 May 1 672; Corres 1 , 1 53- 5 . Oldenburg sent the comment i n a letter o n 9 Ap ril 1 672; Corres 1 , 1 3 5 . Flamsteed to Collins , 1 7 Ap ril 1 672; Corres 1 , 1 45 .

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had seen i t . 12 Towneley reported to Oldenb u rg that Sluse had as ked him to translate it in to F rench so that he might read i t . For himself, Towneley found the paper "so admirable" that he urged the pub li­ cation of a La tin tran slation fo r the benefit of philo·s ophers across Europe . 13 As a result of the telescope and the paper on col o rs , New ton s oon found hi mself the recipient o f p resen tation copies of books by Huygens and Boyle . Never again could he return to the an onymity of the early years in Cambridge . Once and fo r all , he had ins talled himself in the co mmunity of Europ ean na tural p hi­ los ophers , and among its leaders . Newton did no t see every co mmen t on his theo ry of colors that Olden b u rg an d o thers received . He did see enough that he sho uld have been gratified by its o verwhelmingly favo rable recep tion . The praise was no t unanimou s , ho wever. Newton had concluded the paper with a seeming invitation to co mmen t and criticism: "That, if any thing seem to be defective, or to thw art this relation, I may have an oppo rtunity of giving fu rther direction abo u t it, or of ackno wledging my erro rs , if I have co mmitted any . " 14 Alas , within two weeks he received a leng thy critique from Robert Hooke, the es tab lished master of the subject in England , a condescending com­ mentary that contrived to imply that Hooke had perfo rmed all of Newton ' s experi ments him self while i t den ied the con clu sions Newto n drew fr o m them . Ini ti ally , Newton ch o s e to ignore Hooke ' s tone . I received your Feb 1 9th. And having considered Mr Hooks observa­ tions on my discourse , am glad that so acute an objecter hath said no thing that can enervate any part of it . For I am still of the same j udgment & doubt not but that upon severer exa minations it will bee found as certain a truth as I have asserted it . You shall very suddenly have my answe r. 1 5 The cri tique must have rankled more than he let on , however . In­ stead of receiving the answer suddenly , Oldenburg had to wait th ree month s ; and when it arrived , i ts tone was rather less unruffled . Mean while other co mments and critiques arrived . Sir Ro bert M o ray , the fi rst president of the Royal Society , proposed four ex­ periments (which betrayed no unders tanding of the question) to test 12 13 15

Leibniz t o Oldenbu rg, 26 Feb . 1 673; Oldenburg Correspondence, 9 , 49 1 . 14 Corres 1 , 1 02 . Towneley to Oldenburg, 24 April 1 673; ibid. , 9, 622 . New ton to Oldenbu rg, 2 0 Feb . 1 672; Corres 1 , 1 1 6 . See Leon Rosenfeld , ' ·La Thfo rie des cou leurs de Newton et ses adversaires , " Isis , 9 ( 1 926) , 44-65 , and "La P remier conflit entre la thforie ondu latoire et la thforie corpusculaire de la lu miere, " Is is , 1 1 ( 1 928) , 1 1 1 22; A . R . Hall and Marie Boas , "Why Blame Oldenburg? " Isis , 53 (1 962) , 482-9 1 ; Zev Bechler, " New ton ' s 1 672 Optical Controversies: A S tudy in the Grammar of Scientific Dissent , " in Y. Elkana , ed . , The In teraction between Science and Ph ilos ophy ( Atlantic High­ lands, N .J . , 1 974) , pp . 1 1 5-42; and Alan E . Shapiro , " Newton ' s Definition of a Light Ray and the Diffusion Theo ries of Chromatic Dispersion , " Isis , 66 ( 1 975) , 1 94-2 1 0 .

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the theory . 16 More significant were th e obj ections of th e French Jesuit, Ignace Gaston Pard ies , p ro fessor at the College de Louis­ le-Grand and a respected member of the Parisian scienti fic com­ muni ty . He pointed out that for certain positions of the p rism the sine law of refraction could account for the diverging spectrum since all the sun ' s rays were no t incident on th e prism' s face at the same ang le , and he questioned the exp erimentu m cru cis on the same grounds of unequal incidence . 1 7 In fact , Newton's initial paper had adequ ately answered both objections . Nevertheless , Pardies ' s le tter was an intell igent c0 1nment by a man obviously knowledg eable in optics . It was also respectful in tone , though Pardies made the mistake of opening with a reference to Ne wton's " very ingenious Hypothesis . . . " Hooke had also called the theory of colors New­ ton ' s "hypothesis" several times . Now he began to bridle . I am content [he con cluded hi s rep ly t o P a rdie s , manifestl y discontent] that the Reve rend Father calls m y theo ry a n hypothesis if it has not yet been p roved to his satisfaction . B ut my design was quite different, and it seems to contain nothing else than certain p roperties of light which , no w discovered , I think are not difficult to p rove, and which if I did not know to be true , I should p refer to reject as vain and empty speculation, than ackno wledge them as my hypothesi s . 1 8 Pardies d i d not propose t o start a quarrel . H e apologized hand­ so mely and accep ted Newton's exp lan ation of why the unequal incidence of th e sun ' s rays on the p rism cou ld not explain the divergence of the spectru m . He rai sed a further question, ho w­ ever: Could not Grimaldi's recent discovery , diffraction, explain the divergency? In answer to this [Newton replied] , it is to be observed that the doctrine which I explained concerning refraction and colours, con­ sists only in certain p roperties of light, without regarding any hypo­ theses by which those p roperties might be explained. Fo r the best and safest method of p hilosophizing seems to be, first to enquire diligently into the properties of things , and to establish tho se p rop­ e rties by experiments and then to proceed more slowly to hypotheses for the explanati on of them . For hypotheses should be employed only in explaining the properties of things, but not assumed in deter­ mining them; unless so fa r as they may furnish experiments . For if the pos sibility of hypotheses is to be the test o f the truth and reality of things , I see not ho w certainty can be obtained in any science; since numerous hypotheses may be devised , which shall seem to 16 17 18

Cohen , pp . 75-6. Newton responded in his letter to Oldenbu rg of 1 3 April 1 672 ; Con-es 1 , 1 36-9 . Pardies to Oldenburg, 3 0 March 1 672; Con-es 1 , 1 30-3 . New ton t o O ldenbu rg , 1 3 Ap ril 1 672; Corres 1 , 1 44 . O riginal Latin , p . 1 42 .

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overco me new difficulties . Hence it has been he re tho ught necessary 19 to lay aside all hypo theses , as fo reign to the purpose . A s the rest of the co rrespondence would fu rther demons trate , the discu ssion of col ors provided Newton with his first serious oc ca­ sion to exp lo re qu estions of scien tific metho d . Pardies exp ressed himself satisfied with the add iti on al exp lanations that New ton offered , th ough th ere is no ev idence that he accepted the theo ry. During all this ti me , Hooke's critique of the Feb ru ary paper and the need to reply hung over New ton ' s head . Hooke and Newton were pro bably fated to clash . Newton had conceived his theory of col ors in reaction to Hooke's . For his part , Hooke cons idered hi m­ self the au tho rity on optics and res ented the appearance of an in ter­ loper . When Newton 's telescope set the Royal Society agog , he sub mitted a memorandu m abou t a disco very using refractions that would perfect op tical in struments of all s orts to the l imit any one could desi re , fa r beyond Newton's inven tion . Unfortunately , he concealed the d isco very itself in a cipher . 2 0 He ap proached the paper on colors in much the same way , with a magis terial tone of au thority whi ch would have been galling to a person less sens itive than Newton . Two scientists more differen t are hard to imagine . Th ough highly g ifted , Hooke was mo re plausible than brilliant. He had ideas on every subj ect and was ready to pu t them into print withou t much hesitation . Newton in con trast was o bses sed with the ideal of rigor and could hardly convince hims elf that anything was ready for pub lication . Hooke later confessed that he spent all of th ree or fou r h o u rs co mpo s ing hi s observ ati ons on N ewto n ' s paper . 2 1 H e had cause t o regret h i s haste . Newton spent th ree months on his respons e . It may be relevant as well that Hooke was sick eno u gh with cons umption that later in the year he was not expected to survive. 22 Hooke sub mitted his cri tique to the Royal So ciety on 1 5 Feb ru­ ary, one week afte r Newton ' s paper was read . Newton had a copy by 20 Feb rua ry. Hooke g ranted Newton 's experiments , " as having by many hundreds of tryalls found them s oe , " bu t no t the hypo the­ sis by which he exp lained the m . " Fo r all the exp ts & obss: I have hi the rto made , nay and even tho se very exp ts whi ch he alledged , doe seem to me to prove that light is nothing but a pulse o r mo tion .

.

Cohen , p . 1 06 . I have made some slight alterations in this eighteenth-century transla tion . O riginal Latin in Corres 1 , 1 64 . w Collins ' s memorandu m , w ritten in early 1 672 , on Newton ' s letter o f 23 Feb . 1 669 t o a friend; Corres 1 , 4 . Cf. the similar report in the minutes of the Royal Society for 1 8 Jan . 1 6 72; Tho mas Birch , Th e Histo ry of the Royal Society of London, 4 vols (Lon don, 1 756-7) , 3, 4 . Cf. also Olden burg to Huygen s , 1 2 Feb . 1 672; Corres 1 , 1 1 0. 2 1 In a pa per about Newton ' s reply; Co rres 1 , 1 98 . 2 2 Cf. Collin s to G regory, 2 6 Dec. 1 672; Corres 1 , 255 . 9

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p ropagated thro ugh an homogeneo us , uniform and transparent medium: A nd that Colour is nothing but the D isturbance of yt light . . . by the refraction thereof . . . " The b urden of Hooke's critique was the reassertion of his own version of the modification theory as he had p ublished it in Micrograph ia . He p rotested as well agains t Newton ' s abandonment of refra cting telescopes . " The truth i s , the Difficulty of Removing that inconvenience of the sp litting of the Ray and consequently of the effect of colours , is very g reat, b u t no t yet insuperable . " He had already overcome it in microscop es, he asserted , but had been too b u sy to apply his d i s co v ery to telescopes . Like a t rue mechanical p hi loso p her, Hooke kept returning to picturable images such as split rays to express hi s theory of colors . He saw Newton ' s theory in si milar terms , as primarily an exposition of the corpuscular hypothesis , and h e assured Newton that h e could solve the pheno mena of light and colors no t only by his own hypothesi s , but by two or three others as well , all different from Newton' s . He failed en­ tirely to come to grips with Newton ' s experimental de monstration of the fact of heterogenei ty . 2 3 Although Newton initi ally p romised to reply at once , he planned an ans wer that required more time . Whatev er else Hooke' s criti que contai ned , it reasserted the modification theory of colors without compro mi se. Newton decided to seize the opportunity it offered for a fully elaborated exposition of hi s own theory of analysis . He drew heavily on hi s Lectiones op ticae for experimental s u pport that he had omitted in his b rief initial paper. Nor did he stop there . He composed as well an exposition of the pheno mena of thin films as they pertained to the colors of bodies and the heterogeneity of light , more than a first draft of the " Disco urse of Observations " of 1 675 and Book II of the Op ticks, since extensive passages appeared verba­ ti m as they would be published thirty years hence . Wha t Newton drafted in the early months of 1 672 was a treati se on op tics which contained a sometimes b riefer exposition of all the elements of hi s ulti mate work excep t Book II, Part IV (the phenomena of thick plates) , Bo ok III (his brief expo sition of diffraction) , and the Q ue­ ries . Since it included the fi rst sketch of hi s "Hypothesis of Light" (1 675) , it did contain material analo gous to some of the Q ueries . 24 P ublished in 1 672 , the small t reatise would have advanced the sci­ ence of op tics by thi rty years . 23 24

Corres 1 , 1 1 0- 1 4 . For a more favorable assessment of Hooke ' s critique, see A . I . S ab ra, Theories of L ight from Descartes to Newton (London , 1 967) , p p . 25 1 -64 . Add M S 3970 . 3 , ff. 433-44 , 5 1 9-28. Although separated , these two batches g o together; f. 5 1 9 follows the introduction to it at the bottom of f. 442 \' . F or a fuller discussion see Richard S. Westfall , "Newton ' s Reply to Hooke and the Theory of Colors, " Isis, 54 ( 1 963) ' 82-96 .

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It was not published, however. In March , he told Oldenburg he had not yet co mpleted i t , and in April he delayed again . 2 5 Perhaps by n o w he was looking for an excu se not to send i t . Two years earlier, he had refused to have hi s name attached to a formula for annui ties lest i t increase hi s acquaintance , which he stu died chiefly to dj minish . The telescope and the paper on colors had shown how right he had been . By early May , four months after he sent the telescope to London , he had received twelve letters an d written eleven answers about the telescope and colors - hardly a crushing burden , but not a de crease in acquain tance either. 26 Discu ssing the events of the spring four years later, Newton told Olden burg th at " frequent interruptions that immediately arose fro m the letters of various persons (full of obj ections and of o ther matters) quite de­ terred me fro m the design [of publishing the Op tical Lectu res ] and caused me to accuse myself of imprudence , becau se , in hunting for a shadow hitherto , I had sacrificed my peace , a matter of real sub­ stance . " 27 If New ton was looking for an excu se , Oldenburg ' s le tter of 2 M ay offered one . Ol denburg u rged hi m to o mit Hooke's and Pardies ' s names fro m his ans wers an d to deal with their o bjections alone , " since tho se of the R . Society ought to aime at no thing , but the discovery of tru th , and ye improvem 1 of knowledg e , and not at the pro sti tuting of persons for their mi s-apprehensions or mis­ takes . " Another statemen t in the letter, that "some begin to lay more weight upon [your theory of light] now , than at first , " may have served to increase his unhapp iness with the suggestion . 28 Had Oldenburg misled him about the reception of his paper? Initi ally , New ton acquies ced in the reques t , though i t clearly i rritated hi m. He was more th an irri ta ted after he had brooded over it for two weeks . I understood not your desi re of leaving out Mr Hooks name, because the contents would discover thei r Author unlesse the greatest part of them should be omitted & the rest put into a new Method wthout having any respect to ye Hypothesis of colours des cribed in his Micrographia . An d then they would in effect become new objections & require another Answer then what I _have written . And I know not whether I s hould dissatis fy them that expect my answer to these that are already sent to me . He had decided, he continued , not to send all that he had prep ared , though he still inten ded to in clu de a dis course on "the Phaenomena of Plated Bodies , " in which he sho wed that rays differ in reflexi bil25 26 27

Newton to Oldenburg, 19 March 1 672 , 13 April 1 672; Con-es 1, 1 22 , 1 37 . I a m counting all o f the letters fo r which solid evidence exists , n o t all o f the letters that survive. Newton to Oldenburg , 24 Oct. 1 676 (the Ep istola posterior for Leibniz) ; Co n-es 2, 1 33 . 28 Corres 1, 1 5 1 . Original Latin , p . 1 1 4.

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ity as well as refrangibility and related the colors of bodies to the thickness of thei r particles . 2 9 A few days later, in a state of great agitation , he wrote to Collins , thanking him fo r hi s offer to undert ake the publi cation of hi s opti­ cal lectures . But I have now determined otherwise of them ; finding already by that little use I have made of the Presse, that I shall not enjoy my former serene liberty till I have done with it; wch I hope will be so soon as I have made good what is already extant on my account . He could not put the subject out of hi s mind , and after a paragraph about his mathemati cal work he returned to it . I take much s atisfaction in being a Member of that h o nourable body the R . So ciety; & could be glad of doing any thing wch might de­ serve it: Which makes me a little troubled to find my selfe cut s hort of that fredome of communicati on wch I hop ed to enjoy, but cannot any longer without giving offense to some persons whome I have ever resp ected . But tis no matter , sin ce it was not for my o wn sake or advantage l I should have used t hat fredome. 3 0 When he finally sent the reply on 1 1 June , Newton had eli mi­ nated the discourse on thin films along with most of the material fro m hi s Lectiones . What he di d send was an argument pointed to the issue of analysis versu s modifi cation. Though not so well known as the paper of February , the answer to Hooke supple­ mented it brilli antly in the u se of prismatic pheno mena to support the theo ry of colo rs . 3 1 Equ ally , it p resented an argument ad homi­ nem . Far fro m omitting Hooke's name, Newton inserted it in the fi rst sentence of the rep ly , in the last , and in more than twenty­ five others in between . He vi rtually composed a refrain on the name Hooke. Su cces sive drafts of variou s passages progressed through three and fo u r stages, each one more offensi ve than the la st . 3 2 I must confesse [he said in the final version of the opening paragraph] at ye first receipt of those Considerations I was a little tr oubled to find a person so much concerned for an Hyp othesis, from whome in particular I most expected an unconcerned & indifferent examinati on of what I p ropounded . . . . The first thing that offers itselfe is les se agreable to me, & I begin with it because it is s o . Mr Hook thinks himselfe concerned to reprehend me for laying aside the thoughts of improving Op tiques by Refractions . But he knows well yt it is not for Newton to Oldenburg , 2 1 May 1 672; Corres 1 , 1 59-60. 3 0 Newton to Collins , 25 May 1 672; Co rres 1, 1 6 1 . 3 1 Co rres 1 , 1 7 1 -88. The faithful Wickins made a copy of the letter that is now in Add MS 3970. 3 , ff. 489- 500. 32 Successive drafts in Add MS 3970 . 3 , ff. 433-4, 445 , 447. Cf. Westfall , " Reply to Hooke. " 29

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one man to pres cribe Rules to ye studies of another , es pecially not without understanding the grounds on wch he proceeds . After setting Ho oke straight on that sco re , Newton turned to H ooke ' s considerations of his theo ry . And those consist in as cribing an Hypothesis to me wch is not mine; in as s erting an H ypothesis wch as to ye principall p arts of it is not against me; in granting the greatest part of my dis course if explicated by that Hypothesi s ; & in denying some things the truth of wch would h ave appeared by an experimentall examination . Newton showed Hooke how to recon cile the wave hypothesis to his theory of colo rs , before he o ffered the opinion that Hooke' s theo ry was " not onely insufficient, but in some respects unintelligi­ ble . " He even instru cted Hooke , who had boasted that he could perfe ct optical in struments in general, how to improve mi cros copi­ cal o bservations , Hooke's special pro vince , by the use of mono­ chro matic light . So mu ch fo r H ooke's pretended perfection of re­ fracting in stru ment s! In a co vering letter of the same date to Oldenburg , New ton assu med that H ooke would find nothing o bje ctionable in the reply since he had avoided " obliqu e & glancing expressions . . . "33 That is , he emplo yed the b roadswo rd in stea d of the rapier . Whe re H ooke ' s observations had been irritatingly patronizing , Newton's reply was vi ciou sly insu lting - a p aper filled with hatred and rag e . It esta bli shed a pattern fo r his relations with Hooke that was never broken . The Royal Society fo rebore to print Hooke's critique lest it appear di srespectful to Newton . 34 It did allow H ooke to endure the hun1iliation, first of hearing the response read at a meetin g , then of seeing it in print in the Ph ilosophical Transactions . Hooke was not the sole cause of Newton's exasperation . The entire exchange, the need to amplify and explicate what seemed perfectly obviou s , annoyed him . On 1 9 June, he asked O ldenburg that he ' ' not yet print any thing more concerning the Theo ry of li ght befo re it hath been m o re fully weig hed . " 35 On 6 J uly , he requested again that Pardie s ' s second le tter not be publi shed , though he relented when Oldenburg told him it was already at the printer s . 3 6 In the letter of 6 July , he tried to restate the qu estion in a fo rm that would terminate discus sion . I cannot think it effectuall for determining truth to examin the sev er­ all ways by wch Phaeno mena may be explained , unles se there can be a perfect enu meration of all those ways . You know the proper 33 34 35 36

Con-es 1, 1 93 . Hooke w a s allowed t o read a brief re ply t o Newton a t the meeting of 1 9 June (Co rres 1 , 1 95- 7) . H e d i d n o t even get t o read his longer reply (Con-es 1 , 1 98- 203) . Con-es 1 , 1 94. Newton to Oldenbu rg , 6 , 1 3 Jul y 1 672; Co n-es 1, 2 1 0, 2 1 7.

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Method for inquiring after the properties of things is to deduce them from Experiments . And I told you that the Theo ry w c h I pro­ pounded was evinced to me , not by inferring tis thus because not other­ wise, that is not by deducing it onely from a confutation o f contrary suppositions , but by deriving it from Experiments concluding positively & directly . The way therefore to examin it is by considering whether the experiments wch I p ropound do prove those parts of the T heory to w c h they are applyed, or by prosecuting other experiments wc h the T heory may suggest fo r its examination . He pro ceeded then to redu ce his theory to eight qu eries whi ch cou ld be an swered by experiments . Let all objecti ons from hypo­ theses be withheld . Either show the insufficien cy of his experi men ts o r produ ce other experi men ts that contradict hi m . " For if the E x­ perimen ts , w c h I u rge be defective it cannot be difficult to show the defe cts , but if valid , then by proving the Theory they must render all other Obje ctions invalid . " 37 Newton obviously mean t that the experimen ts he had sent al­ rea dy an swered his eight qu eries . Alas , the Royal Society directed th at experiments be performed to test them , an d Oldenburg , with all the delicacy of an uncoordinated cow , asked Newton to suggest some . It was 21 September before he b rought hi mself to reply , an d then only to say he was busy with other things . Oldenburg heard no more that au tu mn . Neither did Newton ' s o ther corresponden t, Collins , until he received a lengthy co mmen tary on Gregory's re­ ma rks about teles copes in December. Newton exp lained to Collins tha t he had written s u ch a " long scribble . . . because Mr Gregory' s discours looks as if intended for the P ress . " 38 Finally , in January , Oldenburg su cceeded in elici ting a rep ly fro m Newton to a query on the improbable subje ct of cide r (" wch liqu o r I wish , wth you , p ropagated far an d near in England . . . ") 39 Indeed , cider later be­ came one of their stap les of correspondence , a subje ct free of emo­ tional inves tment whatever its conten t of spirit . Having not understood the silence , Oldenburg immediately fol­ lo wed Newton 's response by forwarding a new critique , from no less a fi gu re than Huygens . This was the fourth co mmen t Newton had received fro m Huygens , each one less en thus iastic than the one before . When the paper appeared , Hu ygens found it " very ingeniou s . " In the summe r , it still seemed " very pro b ab le" to hi m , though he dou b ted what Newton said abou t the magni tude of chro matic abe rration . Newton sent a b rief exp licati on . 4 0 By 3 8 Newton t o Collins, 1 0 Dec . 1 672; Co rres 1 , 252 . Co rres 1, 209- 1 0 . The initial exchange has been los t. I t is known from Oldenburg to New ton , 1 8 J an . 1 673; Co rres 1, 255 . 4 0 Oldenburg to N ewton (con taining Huygens ' s comment) , 2 July 1 672; Co tTes 1 , 206- 7 . Newton t o Oldenburg , 8 July 1 672; CotTes 1 , 2 1 2- 1 3 . 37

39

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autumn , Huygens tho ught things co uld be otherwise than the the­ o ry held and suggested that Newton be content to let i t pass as a very p robable hypothesi s . " Mo reover , if it were true that from thei r o rigin some rays of light are red , others blu e , etc . , there would remain th e great difficulty of exp laining by the mechanical philo sophy in what thi s diversity of colo rs consists . " 4 1 Oldenburg fo rwa rded the comment to Newton; Newton did no t answer . Of all the natural philosophers in E u rope, Huygens was s ubjecting Newton ' s theory to its most searching scrutiny. In January 1 673 , he sent his fourth and fulle st co m men t . I t was also his most critical . I have s een , how Mr. Newton endeavours to maintain his new The­ ory concerning Colours. Me thinks, that the most impo rtant Objec­ tion, which is made against hi m by way of Qu aere, is that, Whether there be mo re than two sorts of Colours. For my part, I believe, that an Hyp othesis , that should explain mechanically and by the nature of motion the Colors Yell ow and Blew, would be sufficient for all the rest, in regard that tho se others , being only mo re deeply charged (as appears by the Prismes of Mr. Hook) do p roduce the dark o r deep­ Red and Blew; and that of these four all the other colors may be compounded . Neither do I see , why Mr. Newton doth not content hi mself with the two Colors , Yell ow and Blew; for it will be mu ch more easy to find an Hypothesi s by Motion, that may explicate these two differences, than for so many diversities as there are of other Colors. And till he hath found this Hyp othesis , he hath not taught us, what it is wherein consists the nature and difference of Colours, but only this accident (whi ch certainly is very considerable,) of thei r different Refrangibility . 4 2 Once again , the mechanical philo sophy with its demand for pictur­ able explanatory i mages obstructed the understanding of Newton ' s discovery that light is heterogeneous . Newton waited two more months to respond , only to indicate then that Huygens ' s private letter to Oldenburg did not call fo r an answer from hi m . If Hu ygens expected an answer, ho wever, and intended "y1 they sho uld be made publick , " he would do so if he had Huygen s ' agreement " y 1 I may have lib erty to publish what passeth between u s , if occasion be . " In case that were not cu rt eno ugh , he added something else fo r Oldenburg . s r I desi re that you will procure that I may b e put out from being any longer fellow of ye R. Society . Fo r though I honour that body, yet since I see I shall neither profit them, nor (by reason of this 41 42

H uygens to Oldenbu rg, 17 Sept. 1 672; Corres 1, 235-6. Oldenbu rg to Newton (containing Huygens's comment) , Corres 1, 255-6. I quote Olden­ burg's translation ; Coh en , 1 36 . The "Prismes of Mr. Hook" referred to experiments in Micrographia with glass containers of prismatic shape filled with yellow and blue liquids .

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distance) can partake of the advantage of their Ass emblies , I desire to withdraw. 43

In regard to the threat of withd rawal , Oldenbu rg expostulated with Newton briefly and offered to have him excused fro m " ye trouble of sending hither his qterly payments & without any reflec­ tion . "44 Newton, who was seeking to avoid co mplications , not to multiply them , did not pursue the i ssue , and it simply passed . In Ap ril, he sent a reply to Huygens which returned again to the issue o f explanato ry hypotheses . He could not rest satisfied with two colors because experiments showed that other colo rs are equ ally primary and cannot be derived fro m red and blue. Nor was it easier to frame a hypothesis for only two " unless it be easier to s uppose that there are but two figu res sizes & degrees of velocity or fo rce of the aetherial corpu scles o r pulses rather then an indefini te variety , wc h certainly would be a very harsh supposition . " No one i s su r­ pri sed that the waves of the sea and the sand on the sho re reveal infinite variety . Why should the co rpu scles of shining bodies pro­ duce only two sort s of ray s? But to examin how colours may be thus explained Hypothetically is besides my purpose. I never intended to show wherein consists the nature and difference of co lours , but onely to show that de facto they are originall & immutable qualities of the rays wc h exhibit them , & to leave it to others to expli cate by Mechanicall Hypothes es the nature & difference of those qualities ; wch I take to be no very difficult matter . 45 He went on to discuss the other issues Huygens had raised; and though he avoided the deliberately insulting tone of his reply to Hooke, he could not conceal his vehemence. Certainly Oldenbu rg did not miss i t . " I can assu re you , " he wrote to Huygens , " that Mr. Newton i s a man of great cando r, as also one who does not lightly put fo rward the things he has to say . "4 6 Hu ygens , who was' not u sed to being addressed as a delinqu ent schoolboy, did not miss it ei the r; " seeing tha t he maintains his do ctrine with some warmth , " he rep lied , "I do not care to dispute . "47 He did permit himself a few pointed co mments in a tone of icy hauteur . With one mo re letter fro m a so mewhat chastened Newton , the exchange came to an end . Though Huygens had ainple excu se to take of­ fense, he recognized the qu ality of his opponent and chose not to . 43 44 45 46 47

Newton to Oldenburg , 8 March 1 673; Corres 1 , 262 . Oldenburg to Newton , 1 3 March 1 673 ; Co rres 1 , 263 . Newton to Oldenburg , 3 April 1 673; Corres 1 , 264 . Oldenburg to H uygens , 7 April 1 673; Oldenburg Correspondence, 9 , 5 7 1 . Origi nal French in Co rres 1 , 268 . Huygens to Oldenbu rg , 1 0 June 1 673; Corres 1 , 285 .

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The very letter that carried his response enclosed a li st of English scientist s to whom Oldenbu rg should present copies of his newly publi shed Horologium oscillatorium ; Newton was among them . What is more to the point , Huygens allowed himself to be convinced, even though the heterogeneity of light posed diffi cu lties , which he never su rmounted , to the specifi c form in which he cou ched hi s wave theory of light . 4 8 Oldenbu rg had mentioned Newton 's th reat to withdraw from the Royal Society to Collins , who in turn comm ented on it to Newton . I supp ose there hath been done me no unkindness [Newton wrote hi m in May] , for I met wth nothing in l kind besides my expecta­ ti ons . But I could wis h I had met with no rudeness in some other things . And therefo re I hope you will not think it strange if to prevent accidents of that nature for ye future I decline that conversa­ tion wch hath o ccasioned what is past. 49 When Collins showed him thi s , Ol denbu rg a sked Newton to "passe by the incongruities " committed against him by members of the Royal Society . After all , every assembly had members who la cked discretion . The incongruities you speak of, I pass by [Newton told hi m] . But I must , as formerly , signify to you, l I intend to be no further solli ci­ tous about matters of P hilosophy. And therefo re I hope you will not take it ill if you find me ever refusing doing any thing more in l kind , or rather l you will favour me in my determination by pre­ venting so far as you can conveniently any objections or other philo­ sophicall letters that may con cern me. s o Oldenbu rg did not receive another letter from Newton for eighteen months . Collins also found hi s co rrespondence interrupted . In the su mmer of 1 674, Newton a cknowledged the receipt of a book on gunnery and even comm ented on its content . " If you should have occasion to speak of this to ye Author, " he added , "I desire you would not ment ion me becaus I have no mind to concern my self further about 48

49 so

In para graph 2 1 , chap. 5 of the 1 678 version of hi s Treatise of L ight, which he read to the Academie , Huygens appeared to accept Newton ' s results though he still hoped to find a mechani cal explanation of them (Oeuvres completes , pub. Societe hollandaise des Science s , 22 vol s . (The Hague , 1 888- 1 950) , 1 9, 385-6) . In letters to Leibniz ( 1 1 Jan. 1 680) and to Fulleniu s ( 1 2 Dec . 1 683 an d 3 1 Aug . 1 684) , he accepted Newton ' s demonstration that chromatic aberration is much g reater than spherical (ibid. , 8, 257 , 478, 534) . He accepted tha t result also in his Dioptrica , which belonged to the period 1 685-92 (ibid. , 1 3, 4 83-7; cf. pp. 55 1 - 7, 621 -73) . He did not accept Newton ' s corpuscular conception of light, of course , and he remained dissatisfied that Newton o ffered no explanation in mechanica l terms of wha t color is (Huygens to Leibniz , 29 May 1 694; ibid. , 1 0, 6 1 0- 1 1 ) . Newton to Collin s , 20 May 1 673; Corres 1 , 282. Newton to Oldenburg , 23 June 1 673; Co rres 1, 294- 5 .

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it. " 5 1 Late in 1 675 , Collins told Gregory that he had nei ther seen nor w ritten to Newton for a year, "not troubling hi m as being intent upon Chimicall Studies and practices , and both he and D r Barro w & c beginning t o thinke mathcall Speculations t o grow at least nice and dry , if no t so mewhat barren . " 5 2 Collins' s correspon­ dence with Newton never rev ived. Oldenburg and Collins had functioned as Newton' s contact with the learned world outside Cambridge. Although ample opp ortuni­ ties to correspond directly with men of the caliber of Gregory and Huygens had presented them selv es, Newton had refused to grasp the m . He communicated with others through the two intermedi­ aries , who virtually monopolized hi s correspondence . In cutting their access to him , Newton attempted to regain his former soli­ tude . After the publications of 1 672 , however, that was i mpossible . Huygens ' s p resentation copy of the Ho ro logium demonstrated as much , and in September 1 673 , Boyle confirmed the point by p re­ senting hi m a copy of his book on effluvia . Nevertheles s , for the moment , a modicu m of criti ci sm had sufficed , first to incite hi m to rage, and then to drive him into isolation. Meanwhile , life in Cambridge continued its course . As fa r as we know, Newton completed his le ctu re s on optics in the autu mn of 1 672 and the following autumn began a series on algebra (ulti­ mately published as Arith metica universalis, 1 707) which continued for eleven years . In the summer of 1 672 , he appeared in p rint as the editor of Bernard Varenius , Geographia universa lis (originally pub­ lished in the Netherlands in 1 650) . Although we know nothing about the occasion of thi s publication , it is reasonable to see the hand of Isaac Barrow behind i t . Newton later confessed that all he did was supp ly the schemes which were referred to in the original edition but no t p resent . 53 Early in 1 673 , his erstwhile patron, who had been re sident in London for a period , returned to Trinity as master of the college . The news had p receded the fact; already in December, Newton wrote to Collins about it as common knowl­ edge, adding that no one rej oiced over the appointment more than he . 54 At that time, he could not know how i mportant it would be to him two years hence . Newton was absent fro m Cambridge for a month in the su mmer of 1 672 . After a vi sit to Bedfordshi re , undoubtedly in connection 51 52

53 54

N ewton to Collins , 20 June 1 674; Co rres 1, 309 . Collins to G regory , 1 9 O ct . 1 675 ; Con-es 1 , 356. Earlier, on 29 June 1 675 , Collins told G regory that Newton's main attention w as centered on chemical studies and experiments; Co rres 1, 345 . Keynes MS 1 30. 5 , sheet 1 , and Keynes MS 1 30 . 6 , Book 1 . N ewton to Collins , 1 0 D ec. 1 672; Corres 1 , 252 .

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with the lands that supported his p rofessorship , he spent a couple of week s at home . On hi s return , he p aid a mysteriou s call to Stoke Park , N ortha mptonshire , where he stayed with unidentified friends fo r nearly two week s . s s H i s three week s ' absence in the spring of 1 673 p robably indi cated another trip home. He did not return to Woolsthorpe again for at least two and a half years , when a ten-day absence in October 1 675 may have been spent there . Newton probably moved into his permanent chamber in Trinity on the first floor beside the great gate near the end of 1 673 (Figure 7 . 1 ) . An account headed ' 'The Income of that Chamber in wc h Mr Isaac Newton now inhabits" began with an entry for a payment to Thom as Coppinger attached to "Mr Thorndicks income" . s 6 Her­ bert Thorndike had inhabited the chamber until he re signed his fellowship in 1 667. His lease from the college of the tithes and parsonage of Tru mpington that year had forced his resignation , since the college statutes fo r bade a fellow to hold a lease from the 55 56

He wrote three letters to Oldenburg (6 , 8 , 13 July) and one to Collins ( 1 3 July) from there ; Co rres 1, 208- 1 1 , 2 1 2- 1 3 , 2 1 7- 1 8, 2 1 5- 1 6 . The I ncome of that Chamber in wc h Mr Isaac Newton no w inhabits . Imp rimis paid by the said Is aac Newton to Mr Tho Coppinger for 2 . 1 h 1 8 . s 6d M r Thorndicks income upon Dr Mr Thorndicks Babingtons account I ncome Paid more by or Babington to M r Tho . Coppinger upon the same 3. 0. 0 account Paid to Silk for a new door out of the Chamber 0. Portal into the Garden 8. 6 2. To the Porter for a Pump & setting of it down 6. 8 Add MS . 3970 . 3 , f.469v . Newton crossed the final item ou t . The account its elf dated from the 1 690s . On the same sheet, he referred to an observation Halley made abou t light and colors from a diving bell, an observation he included in Op ticks, Book I, Part II, Proposi­ tion X . Halle y did not begin to dive with a bell until 1 69 1 . I do not pretend to know why Newton should have been drawing up the account at th at time; poss ibly Babington' s death at the beginning of 1 692 provided the occasion . The terms o f the account baffle me. Babington may have appeared in it becau se of his office as bursar of the college in 1 674 . I do not unders tand what the "income" of the cham ber could refer to or in what sense a former resident of a cham ber cou ld co mmand its income. There is another account, appa rently related to the chamber, which I date by i ts hand alone to the late 1 670s . I t also included references to income and to Dr. Babington . Paid in part of Income due to or Babington 8 . 0 . 0 3. 8. 0 Paid more for Income to or Bab. Paid for making ye Oven mouthed Chimney in ye Cham ber 0. 7. 8 0. 1 4 . 6 Paid for ye fire irons there Paid for a stone roll in ye Garden besides 0. 1 7 . 0 ye frame Paid for making a new Cellar behind ye 3. 5. 2 Chappel 0. 1 0 . 0 For a door bolts & a lock to ye Cellar

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college . In his will of 1 672 , Tho rndike bequeathed the lea se to the bi shop of Rochester , the dean of Ch rist Church , and the master of Trinity , and to their su ccessors , and decreed that they should allow the profits of the lease to the resident vi car , Thomas Coppinger , M . A . 1 626 , a g raduate of Trinity and brother of Thorndike ' s sister­ in-law . 57 Coppinger died on 25 Ap ril 1 674 . The reco rds in the Junior B u rsar' s Account s of expenditures on fellows ' gardens indicate the possibility that Thomas Gale su cceeded Thorndike in the chamber. Gale resigned his fellowship s01neti me between Mi chael­ mas and Christmas 1 673 , freeing the chamber . 58 The same college Paid for altering & hanging ye Chamber To ye Joyner S mith Painter Uphol ster Gla sier

1 . 6. 0 0. 1 9. 0 1 . 0. 0 14. 13. 9 1 . 15. 0

Deduct Oven Chimney & Glasier

36. 1 6 . 1 2 . 02 . 8

Remains

34 . 1 3 . 5

23 . 2 . 3 Thirds deducted Two Chairs wt h arms 1 1 b 6s 8 _ 1b o . s o . d Ei ght Chairs wt h out arms 41b Ten C ushi ons 5 1 b 3. 0. 0 Scritore 3 1 b 6 Russia Leather chairs 0. 15. 0 A Che st 0. 2. 6 Two Spanish Tables wt h neats leathe r Carpets Ya huda MS 34 , f. 2 . I n accordance with hi s own instructions, Newton crossed out the entry for making the o ven-mouthed chimney , though he did not cross out the entry fo r t h e glazier . The s u m for the chai rs a n d cushions w a s originally (and correctly) £ 1 0 6s Od . I ass ume thi s was a joint account , and that he understood the special chimne y , clearly for hi s alchemical experi ments , to have been his p rivate expense. One possible explanation for the deduction of a t hi rd ins tead of a half is Wickins's nonresidence by the late 1 670s . I have no explanation for Babington's presence in the account beyond the speculation offered above for the other account . Herbert Thorndike , Theologica l Wo rks, 6 vols . (O xford, 1 844-56) , 6, 144-5 . The Senior Bursar's Book recorded a pay ment to Gale o f one quarter's s ti pend for the year that began with Michaelma s 1 673 .

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57 58

Figure 7 . 1 . Loggan 's engraving of Trinity Co llege in th e late s ev en­ teenth centu ry . The ch amb er in which Newton first lived as a fellow may have been on th e north side of the court (th e right-hand side in this p rint) , more or less opposite the figu res on th e path there. His final ch amber was on the first flo o r (i . e . , ab ove th e ground floor) immedia tely no rth of the great gate . The garden beside the chap el was attached to that ch amber . (Fro m David Loggan , Canta­ brigia illustrata, n . d . , late s ev enteenth century . )

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accounts do not show any expenditure on Newton' s garden before 1 683; the recorded expenditure in th at year has been taken as the earliest evidence of Newton' s habitation of the chamber. The entry in his account , with the pay ment to Copp inger, seems unequivocal, ho wev er. Newton continued to re side in the chamber, first with Wickins when he was resident and then alone , until he left T rinity in 1 696 . Newton traveled to London at the end of A ugust 1 674 to partici­ pate in th e installation of the D uke of Monmouth as chancellor of the university . The vi ce-chancellor and heads had deci ded that at least si x Masters of A rts from the three great colleges and three fro m all th e others should attend , and that most of them should be regents " who se o rnaments (as twas thought) would give the great­ est grace & beauty to the Procession . " The scarlet robe of a univer­ sity p rofes sor manifestly filled that category . On 3 September they gathered at Derby House, some 480 in all, including nobles and kni ghts g raduated fro m the university , and they set out at four o' clock when word arrived that Monmouth was ready to receive them . P receded by mounted members of the King ' s Life Guards to clea r a way th rough th e crowd , the j unior bedell led the way with the bedell ' s staff, follo wed by the regents , non-regents , univ ersity officers and doctors of the various faculties , " all habited in the O rnaments agreeable to thei r respective O rders & Degrees . " Mon­ mouth received th em at the door of Worcester House, where mus­ keteers kept the throng fro m entering with the proce ssion . The usual round of fulsome orations accompani ed the admini s tration of the oath before they all sat down to the obligatory feast . In "a few but very full & affectionate words , " Monmouth declared his plea­ sure at his election , which he owed of course to the explicit order of hi s father, Charles I I . For hi s part , Charles declared hi s plea sure at the univ ersity ' s docility by distributing £ 300 among its officers and granting the m what was called a " large Concession" to admit such as they thought fit to the degree of Master of A rts . They made la rge use of the large concession , undoubtedly with no s mall profit to them selv es to supplement the £ 300 . 59 Though he was ,in London for a week , Newton made no recorded effort to meet with mem­ bers of the Royal Society . Willy nilly , Newton devo ted some time to mathematics during these years . Every indication suggests that he did not do so sponta­ neou sly . He did not p ursue a coherent program of mathematical 59

Ch arles Henry Cooper, Annals of Cambridge, 5 vols. (Cambridge, 1 842- 1 908) . 3, 560-3 . The Fellows Exit & Redit Book in Trinity shows that Newton left the college on 28 August and returned on 5 September . Two other fello ws left earlier in August, one other on 28 August , and four on 1 September . I assume that most of them represent the Trinity delegation at the installation .

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investigation. He frequently prote sted hi s lack of interest . Nev­ ertheles s , his mathematical genius was known now , and mathe­ matical di scou rse was fo rced upon him whether he wanted it or not , much of it through the agency of John Collins . In December 1 672 , for example, Collin s wrote him about a method of drawing tangent s developed by Rene de Sluse , a mathemati cian in the low countries , which was later publi shed in the Ph ilosophica l Transa c­ tions . Newton rep lied that the method was ap parently identical to hi s own . A s though determined to tantalize Collins beyond end u r­ ance , he worked one specific problem without saying a word to explain his procedure. This s r [he concluded maddeningly ] is one particular , or rather a Corollary of a Generall Method wc h extends it selfe wth out any trou­ blesome calculati on , no t onely to the drawing tangents to all curve lines whether Geometrick or mechanick or how ever related to streight lines or to other curve lines but also to the resolving other abstruser kinds of Problems about the crookedness , areas , lengths , centers of gravity of curves &c. Nor is it (as Huddens method de maxi mis et minimis & consequently Slusius his new method of Tan­ gents as I presume) limited to aequations wc h are free from surd quantities . This method I have intervowen wth that other of wo rking in aequations by reducing them to infinite series . I remember I once occassionally to ld Dr Barrow when he was about to publis h hi s Lectures that I had such a metho d of drawing Tangents but som e divertisement or other hindered me from describing it to him . 6 0 So much fo r Collins . Six months later , after Oldenburg had re­ layed this hint of a hint to Slu se, who immediately wanted to learn more, Newton refused to di scu s s it in a manner calculated to make Collins realize he was not to repeat what Newton said . 6 1 By this time, i f not before, Newton' s exasp eration with the corre spondence about colors was interacting with his mathemati cal corre spondence further to enhance hi s already considerable reluctance to communi­ cate. The very next sentence of his letter to Collin s in May 1 673 took up hi s threat to resign from the Royal Soci ety , whi ch Collins had heard about from Oldenburg . Years later, when he cited the December letter to Leibniz as evidence of hi s early discovery of the calculus , Newton wi shed he had said more. Probably it was the receip t of Huygens ' s Horo logium oscillatorium in the su mmer of 1 673 which p rompted Newton to develop a dem­ onstration of the i soch ronou s p roperty of the cycloid . 62 However briefly , Huygens ' s work recalled Newton to the topics in mechan60 Newton to Collins , 10 Dec . 1 672; Co rres 1 , 247- 8 . 6 1 Newton to Collins , 2 0 M a y 1 673, a n d Collins t o Newton, 1 8 June 1 673; Co rres 1 , 282 , 62 Math 3, 420-30. 288 .

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ics he had pursued in the 1 660s . Huygens ' s investigation of cen­ trifugal fo rce , he wro te to Oldenburg , " may p ro ve of good use in naturall Philosophy & As tronomy as well as mechanicks . "63 He would find oc casion to re call thi s letter also , to cite against Hooke rather than Leibniz. Ba rrow continued to press him into service . Newton w en t over both Barro w ' s edition of Euclid and his edition of Archi medes to find co rrections for the list of errata . Tho mas Horne , a fellow of King ' s , app lied to him fo r help in unders tanding Descartes ' s Geome­ try . Fro m outside Cambridge, John Lacy , one of the King ' s survey­ o rs , put a question in mathematics to hi m . 64 A note dated 1 673 in the appendix to Edward Sherbu rne , Th e Sphere of Manilius ( 1 675) , stated that Newton had a treatise on dioptri cs ready fo r the p ress , "and divers Ast ronom ica l Exercises, which are to be subj oyned to M r . Nich olas Mercator's Ep itome of Astronomy , and to be printed at Cambridge . " He added that Newton also plann ed to publish a general analytic method based on infinite series fo r the quadrature of figures , centers of g ravity , v olu mes , surfaces , and recti fications . 65 W hatever the source of S herburne' s info rmation , we kno w no thing more abou t the as trono mi cal exer­ cises . They did no t appear in Mercato r ' s b ook when he published it in 1 676 , tho ugh a reference to Newton , who had shown the author a very elegan t hypothesis on the moon ' s libration , establishes that the two had met . 66 Tho ugh Newton was holding Collins at arm ' s length no w , Col­ lins was still the primary agent who broadcast the news of his mathemati cal ab ility . Late in 1 673 , he set Michael D ary , a co mp uter and gauger who , at the ripe age of sixty , had beco me Collin s ' s pro tege , o n Newton . F o r more th an a yea r, Dary forced a co rre­ sp ondence , mostly about algebra, on hi m . 67 Of m o re s i gni fi can ce was the corres p ond ence in 1 675 with another co mputer, John S mith , who m Collins also directed to New ton . Smith planned to prepare a table of square , cub e , and fou rth roo ts of all the nu mbers fro m 1 to 1 0 , 000 . He applied to Newton fo r help in redu cing the app alling burden of co mputati on . Newton adv ised a method )f in terpolation . In the beginn ing , Smith 63 64 65 66 67

Newton to Oldenbu rg , 23 June 1 673; Con-es 1 , 290 . Collins to Newton , ca. July 1 675 ; Co rres 1 , 346 . Horne to Newton , 22 Aug. [?1 676]; Con-es 2, 86- 7 . Newton to Lacy , n . d . but hand of early 1 670s; Co rres 7, 36 1 -2 . Ed ward Sherburne, The Sphere of Manilius (London , 1 675) , p . 1 1 6 . Cited i n Edleston, p . Ii . In hi s letter t o Oldenburg o f 2 3 June 1 673, commenting on Huygens ' s Ho rologiu m , New ton mentioned the moon ' s motion (Corres 1, 290) . Newton to Dary, 6 Oct. 1 674; Dary to Newton , 1 5 Oct. 1 674; Newton to D ary, 22 Jan . 1 675 ; Corres 1 , 3 1 9-20, 326 , 332-3 . There is evidence of quite a fe w other letters . I t i s worth noting that Newton corresponded directly with D ary , possibly becau se h e was only a computer and in no way a peer.

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wo uld need to extract a hundred roots to ten or ele ven places for each table , one for each hundredth nu mber. New ton sent him some instructions based on the binomial theorem on how to mini mize that task . Worki ng from the hundred roots , he could co mpute all the intervening ones correct to eight places by a method of interpo­ lation , first every tenth root, and fro m the m the nine in between . Once he computed the ini tial set of roots , nearly all the additional co mputation would consist only of addition and subtraction . 68 Un­ like the elementary p roble ms set hi m by Dary , the problem of interpolation presented a challenge whi ch sei zed Newton' s interest , and he devoted some ti me to i t . Nothing came of S mith' s proj ect , but Newton ' s interest produced the foundations of modern interpo­ lation theory . Late in 1 676 , he s tarted a brief treatise on interpola­ tion under the title Regu la differentia rum . 6 9 In typical fa shion , he strove to t ranscend parti cular methods in order to "embrace ev ery­ thing in one single general rule . . . " 7 0 He ab andoned the Regu la unfinished to start anew on what became a systematic exposition of inte rpolation by means of central differences . In the latter paper, the foundation of hi s Meth odus dijferentia lis published in 1 7 1 1 , Newton derived what are now referred to as the Newton-S ti rling and New­ ton-Bessel formu las . 7 1 Early in the twentieth century , a commenta­ tor on Newton ' s method of interpolation compared hi s work , an occasional piece flung off hastily in the midst of other concerns , with the acco unt of interpolation that was then considered authori­ tative. " Modern wo rkers , " he concluded , " have struggled up to the level reached by Newton . . " 7 2 In a letter for Leibniz written in October 1 676 , Newton referred to hi s work on interpolation and stated the basic theorem on which it rested, that interpolation i s equivalent t o finding , a t the desired point, the ordinate of a curve that passes through the given values between which one i s interpo­ lating . 73 He did not , however , include in the letter the method of interpolati on , which he may not yet have completed . Like the rest of hi s mathe mati cs , it remained a private possession shared as yet by no one . .

In 1 676 , a mathematical correspondence that Oldenburg carried on with the aid of Collins as part of hi s program of philosophical 6 8 Newton to Smith , 8 May , 24 July , 27 Aug . 1 675; Corres 1 , 342-4, 348-9 , 350- 1 . Note that N ew ton also correspon ded directly with S mith . 6 9 Math 4, 36-50 . See Dun can Fraser, " Newton a n d Interpolation , " i n Isaac Newton, 1 6421 72 7, a Mem o rial Volume, ed . W. J. G reenstreet (London , 1 927) , pp . 45-69 , and Newton ' s 71 7 0 Math 4, 47 . Math 4, 54-68 . Interp o lation Fo nnulas (London , n . d . ) . 72 Fras er, Newton 's Interpolation Fo nnu las , p . 7 1 . 73 New ton to Oldenburg , 24 Oct. 1 676 (the Ep is tola posterior for Leibniz) ; Co rres 2, 1 37 . Original Latin , p . 1 1 9 .

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co mmunication spilled over to include Newton . The correspon­ dence dated back beyond the early months of 1 673 , when a young German philosopher, Gottfried Wilhelm Leibniz , vi sited the Royal Society . Leibniz had arrived in Paris the previous year in the diplo­ matic service of th e Elector of Mainz . Already marked as a man of genius , he readily made hi mself at home in the intellectual circle of the French capital and among the members of the Academie des Sciences . He convinced him self that he must become p rofi cient in mathe mati cs , for which task he recruited the aid of no less a tutor than Chri stiaan Huygens . Pressing financial necessities also played a role in Leibniz' s activities . Without means of hi s own , he had only hi s intellect on which to live, though it was not an inconsiderable as s et . When h i s pa tron in M ainz died , leaving hi m v i rtu ally stranded, his practical need became acute . He hoped desp erately for an app ointment to th e Academie , or for some other position which would enable hi m to stay in a center of learning where he could thri ve amidst an intellectual communi ty not to be fo und in hi s native Germany , still shattered by the trauma of the Thirty Years ' W a r . Both for intellectual and for p ractical reasons , Leibniz culti­ vated learned circle s in western Europe . The Royal Society in Lon­ don rivaled the Academie in Pari s . Already in 1 67 1 , he had thrust hi mself before it with his dedication of an essay on motion. He visited London and the Royal Society in January and February 1 673 and was elected to membership; after he returned to Pari s , he took care to maintain a steady correspondence with Oldenburg, who needed little encouragement in any case . At the beginning of 1 673 , Leibniz was still very much a tyro in mathemati cs , but he was adv ancing with giant strides toward its leading ranks . He made mathemati cs the focus of the correspondence with Oldenburg . Ol­ denburg , who was not a mathemati cian , p ressed Collins into ser­ vice to support hi s end of the exchange . Collins , of course, had made it his business to stay in touch with the leaders of B riti sh mathe matics , especially Gregory and Newton . In response to Leibniz ' s questions , Oldenburg mailed hi m on 6 April 1 673 a long report , which Collins had drawn up fo r hi m , on the status of B riti sh mathe matics . Though Newton ' s name was by no mean s the only one mentioned , it did figure prominently; one paragraph especially gave an intriguing resume of the p roble ms he could solve with his method of infinite series , though it did not give any suggesti on of what the method was . 74 D uring the follow­ ing years , the corresp ondence continued while Leibniz' s own grasp of mathematics broadened rapidly . He tended to ask more than he revealed , but in thi s he did not differ from the established mode of 74

Oldenbu rg Correspondence, 9, 563-7 .

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the day . He had powerful p ractical motives not to reveal hi s o wn progress . His inventions consti tuted his sole cap ital . The possibi lity of a po siti on in Pari s , membership in the Acade mie or the Ramus chair o f mathematics (v acated by Roberval ' s death) , depen ded on hi s achievement . Besides , the rep lies fro m London did not commu­ nicate anything funda men tal . In April 1 675 , in response to speci fic questions , Lei bniz recei ved th e most info rmative letter yet , a resume (ulti mately fro m Collins) of p rogress with infini te series , which seemed to Leibni z to be the focus of B riti sh mathe matics . He was no t unacquainted with the topic hi mself. Grego ry and Newton do minated the report , which included a number of series exp an­ sions . 75 Abo ut the same ti me , in expectation of a visit to England and a discussion with C ollins , Leibniz used Ol denburg ' s letter o f 1 673 as th e basis of a memorandum i n which h e listed questions he wanted to ask . 76 The memorandum is of interest fo r its unselfcon­ scio us revelation of what Leibniz thought about Newton in 1 675 . Newton's name app eared in i t , of co urse. Leibniz wanted to learn more about hi s method of infini te series , the one thing he had heard about his mathemati cs . But Newton ' s was only one name among others . By no means did Leibni z see hi m in 1 675 as the command­ ing figure of English mathe matics . B uilding on hi s o wn earlier progress , Leibniz achieved the funda­ mental insights of hi s differential calculus , which was virtu ally identical to Newton's fluxi onal method , during the autumn of 1 675 . He developed hi s distinctive no tation, in which the calculu s still expre sses itself, at that time . All of this has been established, not fro m Leibni z's assertions , but fro m hi s manuscripts , just as Newton ' s invention of the fluxi on al method has been . 77 At the enJ of 1 675 , it is doub tful that Newton was aware of Leibniz's exi s­ tence , though he may have heard his name spoken at the R o yal Society earlier that year in connection with a bitter exchange be­ tween O ldenburg and Hooke ab o u t Huyg ens ' s s p ring-d ri ven watch . To the best of our knowledge , Newton was also unaware that reports on hi s mathematical achievements , with materials fro m hi s letters and fro m De analysi, were being sent to Leibni z . For thi s , he alone was t o blame. H e had consistently disco uraged communi­ cation and cut himself off when others were eager to discuss an d learn . Years later, after a bi tter p riority dispute had b ro ken o u t , when Newton learned what Collins h a d sent , h e drew his own sini ster conclusion s . What i s clear fro m the correspondence , how75 76

Oldenburg to Leibniz, 1 2 April 1 675 ; Der Briefwechsel van Gottfried Wi lhelm Leibniz m it Mathematikern , ed . C . I . Gerhardt (Berlin , 1 899) , pp. 1 1 3-22 . Co rres 2, 235-6. See also Joseph E . Hoffman , Leibniz in Pa ris , 1 672- 1 676. His Growth to Mathematical Matu rity (Cambridge, 1 974) , p. 291 . Hoffman was the first to identify thi s 77 memorandum . Ibid. , p p . 1 87- 201 .

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ever, i s th at by the end of 1 675 , the cri ti cal period in Leibniz's own develop men t, he had received only s o me of Newton 's resu lts with­ out demonstrations , and that these results had been confined to infinite series . To be sure , infini te series fo rmed an integral part of the flu xi on al method , but Leibniz had not heard of thei r b roader ramifications . In 1 676 , Newton both learned who Leibniz was and entered the co rrespondence hi mself. Leibniz wrote to Oldenburg in M ay about two series he had recently received fro m Collins via a Danish math­ ematician , Geo rg Mohr , the series expressing the sine of an angle given the arc , and the inverse series which expressed the arc or ang le given the sin e . Actually , the letter fro m Oldenburg a year ea rlier had also included both series ; only now did their elegance i mpress Leibni z . He asked fo r demons trations of them . 78 Both Ol­ denbu rg an d Collins u rged Newton to resp ond . The request came at an un welco me ti me . A new corresp onden ce that challenged hi s theo ry of colors had op ened , and Ne wton was allo wing it to agitate hi m exces sively . Nevertheles s , he acceded to the request , and on 1 3 June 1 676 , he co mpleted a letter fo r Leibni z and Ehrenfried von Ts chi rnhau s , another German mathematician then resi den t in Paris and a friend of Leibni z , who was also involved in the co rrespon­ den ce tangentially . Once again , Newton chose not to enter in to direct co mmunication . He addres sed the letter to Oldenburg , who fo rwarded a copy , together with another res ume of the work of Grego ry and others that C ollins prepared , to Leibniz an d Ts chirn­ haus on 26 July . Not wishing to tru st a le tter he recognized to b e in1po rtan t t o the mail , Oldenbu rg h a d wai ted until he found a ca rrier who would deliver it to Leibni z . Newton wro te t w o letters fo r Leibniz i n 1 676 . Nearly fo rty years later , he ci ted them as evidence against Leibniz in the priority dis­ pute and lab eled them the earlier le tter an d the later letter, the Ep istola prio r and the Ep istola posterio r. Replying in the first to Lei b­ niz's question about the foundation of the two series , he drew upon the co mbined reservoir of De analys i and De meth odis to present a general expo sition of series . The foundation of the redu ction of functions to infinite series lies in division and root extraction , he explained , carried out in sy mbols just as they are carried out with decimals , but the operations are shortened by the use of the bino­ mial theo rem . He stated the theo rem and illu s trated its use with nine exam ples . He p roceeded then to his method of extracting the roots (expressed as infinite series , of cou rse) of affected equ ati ons . Nine fu rther examples derived the two series about whi ch Leibniz 78

Leibn iz t o Oldenburg , 2 May 1 676; Corres 2 , 3-4 . See Christoph J . Scriba, "The Inverse Method of Tangents: A D ialogue between Leibniz and Newton ( 1 675- 1 677) , " Arch ive for History of Exact Sciences, 2 ( 1 962-6) , 1 1 3-37.

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had asked , suggested the use of series to determine areas , volumes , and so o n , and showed the reduction of mechanical cu rves , such as the quadratrix , to series . In these examples , Newton assu med and employed his algo rith ms fo r differentiation and integratio n with out expounding them . With all the letter offered , Newton could not refrain fro m tantalizing Leibniz as he had tantalized Collins by sug­ gesting that he held more in reserve. Fro m all this it i s to be seen h o w much t h e limits of analysis are enlarged by such infinite equations ; in fact by their help analysis reaches , I mi ght almost say, to all problems , the numerical problems of Diophantus and the like excepted. Yet the result is not alto gether univers al unless rendered so by certain further metho ds of develop­ ing infinite series . For there are some problems in which one canno t arrive at infinite series by division or by the extraction of roots either simple or affected. But how to proceed in those cas es there is now no time to explain . . . 79 If Lei bniz had hitherto considered Newton merely as one am ong a nu mber of Engli sh mathem ati cians , the Ep istola p rior di sabu sed him . N o r was he relu ctant to express his ad miratio n . " Yo u r letter , " he wrote to Oldenburg immedi ately upon its receip t , " contains more numerous and more remarkable ideas about analysis th an many thick volumes published on these matters . . . New ton' s dis­ coveries are w o rthy of his genius , whi ch is so abundantly made manifest b y his opti cal experi ment s and by hi s catadi optrical tube [the reflecting telescope] . " He went on to sho w Newton that he knew a thing o r two about infinite series himself, to expound his general method of trans fo rmations , as he called it (withholding s omewhat as New ton had done) , and to put some specific questions to New to n . 80 Tschirnhaus als o responded enthu siastically , but with his dep arture fro m Paris he cea sed to parti cip ate in the exchange. Lei bniz ' s new questions provided the occasion of the Ep istola posterior. Befo re Newton could write it, however , Leibniz himself visited London fo r ten days in O ctober . Despite all his hopes , his search fo r a position in Paris had failed , and he had finally accep ted an ap pointment at the cou rt of the D uke of B runswi ck-Lunebu rg . He stopped i n London o n h i s way t o Hanover. While he was there , he conversed with Collins . It was hardly a meeting of equ als , Col­ lins scarcely more than a co mputer , Leibniz a mathem ati cal genius of the first order , who swep t Collins qu ite off his feet and left him convinced that Engli sh mathemati cians lagged far behind . 8 1 D az79 80 81

Co rres 2, 39. Original Lati n , p. 29. The whole Ep istola prior is found on pp. 20-32 , and the English translation on p p . 32-4 1 . Leibniz to Oldenburg , 26 July 1 676; Co rres 2, 65-7 1 . Original Latin, p p . 57-64. Collins to S trode , 24 Oct. 1 676; Corres 2, 1 09 .

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zled b y the visito r , Collins opened his files to hi m . Leibniz read De ana lysi and a fulle r exp osition of G reg o ry' s work than that sent to hi m , a piece called the Histo rio la whi ch in cluded Newton ' s letter on tangen ts . Though he took no tes on the las t , he did not take notes on Newton 's flu xi onal p ropo sitions at the end of De analysi no r on G regory's method of maxi ma and mini m a . His no tes concentrated on infinite series , whi ch he saw as the subje ct in which B ritish mathemati cs could ins truct him . The ab sence of no tes on the flux­ ional calculus i mplies that he saw no thing there he did no t know alread y . 8 2 After Collins reg ained hi s equilib ri u m following Leib­ niz 's departure , he realized the extent of his indis cretion . He did not tell Newton what he had shown to Leibni z . Apparently , fro m the con ten t of the Co mmerciu m episto licum, Newton learned only later that Leibniz saw De analysi . For his part , Leibniz chose no t to menti on i t . Even before Leibniz's visi t , Collins had been impressed eno ugh by the reply to the Ep isto la prio r to urge New ton anew to publish hi s metho d . O bsessed with the latest exchange on colo rs , Newton tho ught o therwise . I look upon your advice as an act of singul ar friendship [he wrote] , being I beleive censured by divers for my scattered lette rs in ye Transactions about such things as no body el s would have let come out wthout a substantial dis cours. I could wish I could retract what has been done , but by that , I have learnt what 's to my convenience , wch is to let what I write ly by till I am out of ye way . Collin s ' s expressed frar that Leibniz's method would prove more general left hi m wholly unmoved . Wi th serene confidence he de­ scribed what his method could acco mplish in a passage quoted ab o ve at the end of C h ap ter 4 . " This m ay see m a b o ld asserti on . . . , " he continued , " b ut it' s plain to me by ye fo untain I d raw i t fro m , tho ugh I will no t undertake to prove it to o thers . " 83 Meanwhile he co mpleted the second response to Leibniz's ques­ tion s , the Epistola poste rio r, one week after Lei bniz left London fo r Hanover. Wickins trans cribed the copy sent to London , p ro bably the la s t episode in his caree r as Newton 's amanuensis . The letter beg an with an auto biog raphi cal passage on New ton 's discovery of the bino mial theo rem an d the various abo rted plan s for p ublication , a precious passage fro m a man not much given to self-revelation . Inexorably , the pa tte rn of the letter drew him into De meth od is and hi s flu xi onal method , whi ch he discus sed ag ain in tantalizing in­ co mpleteness . More even than the Ep istola prio r, the se cond letter was a veritable treatise on infinite series , but twice , as he ap82 83

S ee the discus sion of these notes in Hoffman , Leibniz in Pa ris, Newton to Collins, 8 No v . 1 676 ; Corres 2, 1 79-80 .

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proached the fluxi onal method , he drew back and con cealed criti cal passages in ana grams . Thu s he mentioned that De meth odis con­ tained a method of tangents similar to Slu se' s though more general, and that it dealt with questions such as maxi ma and mini ma . "The foundation of these operations is evident enough , in fa ct; but be­ :cause I cannot proceed with the explanation of it no w , I have p re­ ferred to conceal i t thus: 6accdre 1 3eff7i319n4o4qrr4s8t1 2 u x . On this foundation I have also tried to simplify the theories whi ch concern the squaring of curves , and I ha ve arrived at certain general Theo­ rem s . " 84 He went on to state what he called the first theore m of his method of squaring curves , illu strated by three examples , before he tu rned the letter back to infinite series . Indeed , this was to reveal a good deal and to hin t at much more . What the anagram concealed was a general statement in New ton ' s terminology of the fundamen­ tal theorem of the calculu s: " D ata aequatione quotcunque fluentes quanti tates involvente , fluxi ones invenire; et vice versa" (given an equation involving any number of fluent quantities to find the flu x­ ions , and vice versa) . Toward the end of the letter a similar, longer anagram concealed statements of two methods to solve equations containing flu xions . Leibniz recognized in general terms what the anagrams concealed; after all , he was re ading De ana lysi about the ti me Newton was composing them . Leibniz did not re ceive the Ep istola posterior until the following June . As with the earlier one , Oldenburg had recognized i ts impor­ tance and refu sed to send it u ntil he heard Leibniz was settle d in Hano ver and until he had a reliable carrier. On 1 1 June 1 677 , imme­ diately upon receiving it, Lei bniz penned a response fi lled with prai se . In i t , he co mmunicated the essence of hi s differential cal cu­ lu s , asked so me probing questions such as only an expert could hav e fo r mula ted , and vi rtu ally i mplored fu rther exchange . A month late r, when he had had ti me to digest the letter, he wrote again . Oldenburg warned hi 1n in Augu st that Newton was preoc­ cupied with other affairs . 85 And in September Olden burg died . Both of Leibniz' s letters were forw arde d to Newto n . It i s i mpo ssi­ ble to i magine that he did not recogni ze the significance of their con tents . Perhaps the long delay had arou sed hi s su spicions , though there is no ev idence to warrant a proj ection of later attitudes back onto 1 677 . The fa ct is that Newton had made his decision five years earlier . There is no good reason to think that he would have com­ municated to a German mathematician he had never met what he had been unwilling to let Collins publish five years earlier . With Oldenburg dead , he did not reply , and the correspondence lapsed . 84 85

Con-es 2, 1 34 . Original Latin , p . 1 1 5 . Leibniz t o Oldenburg, 1 1 June and 1 2 July 1 677; Oldenburg t o Leibniz, 9 Aug. 1 977; Co n-es 2, 2 1 2- 1 9, 23 1 -2 , 235 .

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An unpleasant paranoia pervaded the Episto la posterio r. The auto­ biographical p assage insisted on the pressure of C ollins and Ol den­ b u rg to publish , and he concealed two vital passages in an agrams . Two days after he sent i t , he wrote to Oldenbu rg again: " P ray let none of my mathematical papers be p rin ted wthout my sp eci al li­ cence . "86 Leibniz was probably not the obje ct of the parano ia at this time . Rather Newton was o bsessed with his co rresp ondence on colors and allo wed his frus tration with i t to influence his resp onse to Leibni z . In so doing , he sowed the seeds of unlimited turmoil . In 1 676 , Leibniz had not published his calculus . He had no t co mm uni­ cated i t . A free and open co mmunication fro m Newton would have plunged him , undeserv edly , in to a cruel dilemma . Before he had established any clai ms of his own , he would have learned that ano ther mathematician had invented subs tan tially the same method befo re hi m . Since the co rrespondence passed thro ugh Oldenburg , he would hav e learned it publicly . One can only speculate what the ou tco me would have been - and hope it wo uld have been less dis­ cred itable to both than what d id finally happen . As far as Ne wton is con cerned , such a letter would have secured what his fu tile conceal­ men ts gave away , an unassailable claim to prior inven tion of the calculu s . I n 1 676 , Leibniz and Newton stood in differen t personal posi­ tions . In effect , Leibniz had arrived where Newton had been ten years earlier. He had j u st invented the calculu s . He understood its significance . The excitement in his letters exp res sed in his fashion what New ton had acted out a decade earlier b y staying up all night an d igno ring his meals . Meanwhile , Newton had mo ved on to other things . On the one hand, there was the to rment (as he per­ ceived it) of the co rresp ondence on colo rs . He conclude d the Ep is­ to la p rio r with a parag raph about it; he wrote letters abou t it imme­ d iately before and imme diately after the Ep istola posterior. Optics was not where his heart lay in 1 676 , ho wever; it too was another , diversion . Both of the letters fo r Leibniz p rotested his la ck of inter­ est in the mathematical exchange. He was pres sed fo r ti me , he insisted , an d could not explain things fully . " Fo r I w rite rather sho rtly because these theories long ago began to be distas teful to me , to such an exten t th at I have now refrained fro m them for nearly five years . "87 In th e Ep isto la posterio r, he noted (correctly) that he had not co mpleted the treatise, De meth odis, which he began in 1 67 1 , "nor has my mind to this day returned to the task of add ing the res t . "88 In the covering letter that he sent to Oldenburg with the Ep istola posterio r, he gave more blunt vent to his d istaste . 86 88

8 7 Corres 2 , 39 . Original Latin , C o rres 2, 1 63 . orres 2, 1 33-4 . Original Latin , p . 1 1 4 . C

p.

29 .

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" I hope this will so fa r satisfy M . Leibni tz that it will no t be necessary for me to write any more about this subj ect. For hav ing other things in my head , it p ro ves an unwelco me interruption to me to be at thi s ti me put upon considering these things . "89 The manu scrip t remains fro m the 1 670s support Newton ' s as sertions . Oldenburg ' s death had little to do with the b reak in the co rresp on­ dence with Leibniz. Newton participated under p rotest from the beginning . Collins and Wallis continued to press hi m to publish . Wallis , who was alw ays su spicious of fo reign designs to steal En­ glish inventions , inserted pas sages fro m the two Ep istolae in hi s Algebra (1 685) . Newton only wanted to be left alone . S urrendering to hi s exasperation with the need to explain and discuss , he wi th­ drew within hi s shell . Leibniz went on to publish the calculu s . New ton eventually harvested the bitte r fruit hi s own neuro ses had plan ted . He fo rced Lei bniz to share it with him . Meanwhile , optics had refused to leave hi m alone . In the au tumn of 1 674 , Oldenbu rg received a le tter cri ti cizing Newton' s o riginal paper fro m Francis Hall (or Linu s , as he latinized hi s name) , an English Jesuit who was a professor at the English college in Liege . It inaugu rated an extended exchange with Linus and his pupils which lasted into 1 678 and p roved to be for Newton the most trying yet . Ten years earlier, Linus bad conferred i m mortality on Robert Boyle by challenging his concep t of ai r pressure and p ro­ voking the experiments that led to Boyle ' s law . He was seventy­ nine years old in 1 674 . The tone of hi s letter suggests that senility had set in. Citing experi ments he clai med to have done thi rty years before , he confi dently denied that the proj ected spectrum could have appeared on a clear day as Newton described it and opined that clo uds near the sun had mi sled hi m . 9 0 He misunderstood the descrip tion of the spectrum and thought it was parallel to the length of the p rism . New ton took two months to bring hi mself even to ackno wledge the letter, and then he refused to answer it on the grounds that it deserved no answer and that he had " long since determined to concern my self no fu rther about ye pro motion of Philosophy . " As an afterthought, he suggested that Linu s be told " (but not fro m me) " that th e experiment was performed on a clear day and the spectrum was transverse to the axi s of the pris m . 9 1 The matter did no t come to rest there . Early in 1 675 , New ton visited London in connection with the royal dispensation he needed to remain in his fello wship wi thout taking o rders . On 1 8 Febru ary , 89 90 91

Corres 2, 1 1 0 . Linus to Oldenburg, 6 Oct. 1 674; Corres 1 , 3 1 7- 1 9 . S ee Conor Reilly , " Francis Line, Peripatetic (1 595- 1 675) , " Osiris , 1 4 ( 1 962) , 222-53 . New ton to Oldenburg , 5 Dec. 1 674; Corres 1 , 328-9 .

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he attended his fi rst meeting of the Royal Society and became a full-fledg ed member by signing the register. He attended two other meetings during his s tay . I t is evident from later remarks that the visit made a deep impression . Far fro m finding him self the object of critici s m , he was covered with attention . While he was there , a second letter arrived fro m Linus , as vehement as the firs t in denying the possib ility of a p roj ected spectrum as New ton described it. The So ciety ordered th e experimen t to be performed before them to te stify to the matter of fact , tho ugh none other than Robert Hooke assured them that the exp eriment was beyond question . Indeed Newton heard, or tho ught he heard, Hooke accep t his th eory of colors . 9 2 He met Robert Boyle , whose works he had read with care, and conversed with hi m . Though the i m mediate aftermath of the visit to London was six months of further silence , Newton' s realiza tion o f th e respect i n whi ch h e w a s held prepared the way fo r so mething else . Linus p rovided the p roxi mate cau s e . In the autumn he w rote anew demanding that his second letter to Oldenb urg be printed les t p�ople conclude he had been mistaken . When Newton had seen the letter in London, he had deemed it no t to be worth an answer . Upon the receipt of Linus ' s de mand in November, however, he wrote out explicit ins tructi ons on how the experi ment sho uld be performed , cited all the others who had confirmed his description , and asked the Royal Society to try it a t a meeting i f they had not yet done s o . Emboldened by his recollection of the spring , he added something more . I had some thoughts of writing a fu rther discours about co lo urs to be read at one of your Assemblies, but find it yet against ye grain to put pen to paper any mo re on l subject. But however I have one dis­ course by me of yt subj ect written when I sent my first letters to you about colours & of wch I then gave you notice . This you may co m­ mand wn you think it will be convenient if ye cu sto me o f reading weekly discou rses still continue. 93 Althou gh the le tter does not s urviv e , Oldenburg mu s t have in­ formed hi m it was then conveni ent . Th e familiar ro utine had to be performed . Two and a half weeks later, on 30 Novemb er, he had not yet sent the papers because, when he reviewed them , "it came into my mind to write another little scrible to accompany the m . " 94 Perhaps the convenience of Wickins , who was p ressed into service as amanuensi s , contribu ted to the delay . What he finally sent on 7 December contained two items , a " D iscourse of Observations , " which was vi rtu ally identical to Parts I , I I , and I I I of Book I I of the 92 93

Newton to Oldenburg, 7 Dec. 1 675 (the "Hypothesis of Light") ; Co n-es 1, 362-3 . 94 Con-es 1 , 359 . Newton to Oldenburg, 1 3 Nov. 1 675; Corres 1 , 358 .

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Opticks p ublished nearly thirty years later, and " An Hypothesis explaining the P rope rties o f Light disco ursed of in my severall P apers . ' ' The first of the two dated fro m 1 672 , tho ugh Ne wton may have revised the ea rly version into its final form in 1 6 75 . 95 I shall not repeat what I have said before abo u t i ts conten t . In many respec ts , the "H ypothesis o f Light" was also no t new . He had begun to d raft it in 1 672 as part of his reply to Hooke , and things like it had app eared already in his essay " Of Colours " in 1 666 . We need to read Newton ' s co mment abo ut it in a co vering le tter to Oldenburg against this backg round . sr. I had formerly purpo sed never to w rite any Hypothesis o f light & colo urs , fearing it might be a means to ingage me in vain disputes: but I hope a decl ar' d resolution to answer nothing that looks like a controversy (unles possibly at my own ti me upon some other by occasion) may defend me fro m yt fear. And therefore considering that s uch an Hyp othesis would much illustrate ye papers I p romis 'd to send you, & having a little time this last week to spare: I have not scrupled to describe one so far as I could on a sudden reco llect my thoughts about it , not concerning my self whether it shall be thought probable or imp robable so it do but render ye papers I send you , and others sent formerly, mo re intelligible . You may see by the scratch­ ing & interlining 'twas done in hast , & I have not had time to get it transcrib' d . . . 96 For the first ti me , Newton undertook to reveal his thoughts about the ulti mate cons titution of nature; he did no t find it a task that he could do lightly . In the introduction to the "H yp othesi s , " as in the covering le tter,

95

96

I n addition t o t h e draft that definitely dated from 1 67 2 (Add MS 3970 . 3 , ff. 5 1 9-28) th ere is a second auto graph (ff. 501 - 1 7) an d a transcript by Wickins with several corrections and additions in Newton ' s han d , which was the copy sent to London an d later retu rned (ff. 549-68) . I do not know of any way to date the last two with assurance. New ton ' s letter of 30 Nov . 1 6 75 certainly asserted that he was merely sending a paper completed earlier . Since he wa s prone to similar remarks , such as his comment on the " little scrible" that accom panied it, which fended off pos sible criticism by implying a distance between him an d his wo rk, I am not inclined to accept his word without further evidence. Co rres 1, 36 1 . In fact, in addition to the copy sent to London an d later returned (Add MS 3970 . 3 , ff. 538-47) , there is the inevitable transcript in Wickins ' s hand with two sheets missing (ff. 573- 8 1 ) , which could have been made after the other was returned , of course, but probably was not . There is also a d raft (ff. 475 , 476 , 534 , 533, in that order) and drafts for it (ff. 5 35-6) . The draft even in cluded a version , which did not hedge as much as the final statement , of the disclaimer sent to Oldenburg in the covering letter . On various issues asso ciated with the "Hypothesis of Light" see Philip E. B . Jourdain , " New ton ' s Hypothesis of Ether and of Gravitation fro m 1 672 to 1 679 , from 1 679 to 1 693, from 1 693 to 1 726 , " The Monis t, 25 ( 1 9 1 5) , 79- 1 06 , 234-54, 4 1 8-40; A. I . Sabra, " Newton an d the ' Bigness ' of Vibrations, " Isis , 54 ( 1 963) , 267- 8; and Roger H. S tuewer, "Was New ton ' s ' Wave-Particle D uality' Consistent with Newton ' s Observ ations? " Is is , 60 ( 1 969) , 392-4 .

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Newton insisted that he sent it merely to illu strate hi s op tical papers . He did not assu me it; he did not concern himself whether the properties of light he had discovered cou ld be explained by this hypothesis or by Hooke ' s or by another . "This I thought fitt to E xpresse , th at no man may confound this with my other dis­ cou rses, o r mea su re the certainty of one by the other, o r think me oblig ' d to answer obje ctions against thi s scrip t . For I desire to de­ cline being involved in such tro u b lesome & in significant D i s­ putes . " 97 It i s qu ite imp o s si ble to reconcile the actu al "Hypothesis" with Newton 's dep recations o f it, ho wever . For one thing , it pre­ sented far more than an explanation o f optical pheno mena . F o r another , a feeling of intensity pervaded it. In it, Newton presented hi mself in hi s preferred role , not of positive scientist , but of natu ral philosopher confronting the entire sweep of natu re . For ten years he had contemp lated the o rder of things in soli tude . Now he was di sclosing , partially , to a limited audience , where ten years of sp eculation had carried him . N o amount of feigned indi fference and hard words about insignifi cant dispu tes could ob scu re the signifi­ cance the enterpri se held for him . A s far as li ght was concerned , the "Hypothesis" presented with one excep tion an o rthodox mechanical philosophy . The one excep­ tion was a p rincip le of motion that he ascribed to the light corpus­ cles themselves . He assigned reflections and refractions to the cau sa­ tion of a uni versal aether which stand s rarer in the pores of bodies than in free sp ace and causes corpu scles o f light to change directions by its p ressu re. A mechanism of vib rations in the aether exp lained the periodi c phenomena of thin film s . The "Hypothesi s of Light " contained mu ch mo re than an exp lanation of optical pheno mena , however. The fi rst half o f it presented a general system of nature ba sed on the same aeth er . All of the crucial phenomena that ap­ peared in his " Quaestiones" a decade earlier appeared now in the "Hypothesi s" either to be exp lained by aetherial mechanisms or to offer illust rati ve analogies . For examp le , the p ressu re of the aether explained the cohesion of bodies , and su rface tension illu minated an aethereal mech anism . In passing , Newton described an experiment with stati c electri city which played a maj o r role in the early history of that science. He set a disk of gla ss in a brass ring wh ich held it about an eighth or a si xth of an inch fro m the table . Placing the glass over some tiny bits of paper, he ru bbed it b ri skly with cloth until the papers began to move; after I had done rubbing the Glas s , t he papers wo uld continu e a pretty while in various motions , someti mes leaping up to the Glass & resting there a while , then leaping downe & resting there , then leaping u p & 97 Co n-es 1 , 364 .

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perhap s downe & up againe, & this so meti mes in lines see mmg per­ pendicular to the Table, So metimes in ob lique ones , Sometimes also they wo uld leap up in one Arch & downe in another, divers times together, without Sensible resting between; So mti mes Skip in a bow from one part of the Gl asse to another witho ut touching the table, & Sometimes hang by a corner & turn often about very nimbly as if they had been carried ab out in the midst of a whirlwind, & be otherwise variously moved, every paper with a divers motion . 98 Newton saw no way to explain these moti ons except by an aether conden sed in the gla ss whi ch was vap orized and set in motion by the rubbing . Because the aether condensed con tinually in bodies such as the earth , there is , according to the "H ypothesis , " a constant do wn­ ward stream of it which impinges on gross bodies and carries them along . Newton explicitly exten ded this explanation of gravity to the sun and suggested that the resulting movement of aether holds the planets in clo sed orbits . The passage contains the first known hint of the concept of universal gravitation in Newton ' s papers; he did not fail to refer to it when Hooke cried plagiary in 1 686 . The "Hypothesis of Light" refuses to be p resen ted solely as a mechanical syste m of nature , howev e r . If it showed the enduring influence of the mechanical philosophy , it was an ambiguous do cu­ ment which contained vestiges of other influences that had begun to bear on Newton' s conception of nature . O ne of the features that distingui shed it was the prominent role of chemical phenomena , which had played no part in the " Q uaesti ones " ten years earlier. They epitomized the new influences that would , in the years ahead , carry hi n1 on beyond hi s position i n 1 675 . I shall return t o them i n a different context . Whatever his announced in tention to avoid disp utes , the new papers plunged Newton directly into a new round of corresp on­ dence, of explication , and very qui ckly of controversy . The papers were read at the Royal Society immediately , the "H ypothes�s" on 9 and 1 6 Dece mber, and after a Chri stmas recess and two meetings monopolized by discussions stemming fro m the "Hypothesis , " the " Discourse of Observations " fro m 20 January to 1 0 February. Like the paper of 1 672 , they caused a sensati on . The Ro yal Society re­ quested the immediate publication of the " D iscourse , " which New­ ton declined . Questions also arose . The static-electric experimen t in the "Hypothesis" caught the Society' s eye , but their first effort to reprodu ce it failed . They applied to Newton for instructions , of cou rs e , and he had to write two letters about i t . 99 98 Co n-es 1 , 364-5 . 99 New ton to Oldenburg , 2 1 Dec. 1 675 and 1 0 J an . 1 676; Co rres 1 , 404 , 407-8.

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M o re significant was Hooke . Whether deliberately o r through in adv erten ce , Newton had introdu ced hi m into the "Hyp othesis " rathe r pro minen tly , both in the introduction , which j ustified the whole enterp rise by a referen ce back to Hooke's critiq ue of 1 672 , an d in th e discussion of diffraction at the conclus ion . Newton owed his very know ledge of diffra ction to Hooke' s discourse at a mee ting of the Royal So ciety early in 1 675 , where Hooke p resen ted it as his own new discovery . Newton had re marked then tha t diffraction was only a new kind of refraction - which co rresp onde d exactly to the way he later explained it in the " Hyp o thesis . " Seeing the in­ vader fro m Cambridge de molish yet another of his prizes , Hooke had b ridled and rep lied " that th ough it should be but a new kind of refrac tion , yet it was a n e w one . " 1 00 In the "Hypothesi s , " Newton was unkind enough to point out, in connection with a few experi­ ments on diffraction , th at it was not a new one after all . Faber' s book on light had men ti oned i t , and Faber had learned about it fro m G rimaldi . 1 0 1 Small wonder th at Hooke rose when the reading of the "Hypothesis" was co mpleted to assert " that the main of it was contained in his l\,ficrographia , which M r . Newton had only carried fa rther in some particu lars . " 1 02 Without much thought about the provo cation he had offered , Newton erupted in ang er at the charge . Since Oldenburg ' s letter reporting th e incident does not surv ive , we do not know exa ctly what Newton heard , or for that matter exactly what happened. Indeed , none of Oldenburg ' s le tters to New ton from this period surviv e , pos sibly a suspicious circumstance becau se Hoo ke believed that Oldenburg , with who m he was at swords ' points , had deliber­ ately fo men ted trouble . Even minutes of the Royal So ciety do not o ffer an independent account; Oldenburg kept them . It is not hard to bel ieve that an incident occurred , ho wever. Hooke w as a prickly personality in his own right, and he had reason to feel aggrieved with Newton . What the "Hyp othes is" poured on his wounds was mo re embalming fluid th an bal m . For his p ains he now received a fu rthe r dose of gall . Hooke's hypothesis of light , Newton asserted, was merely an einbroidery on Descartes ' s . His own was enti rely differen t , to the exten t that the experi ments , which were new to Hooke , on which Newton based his treatmen t of thin films under­ mined everything Hooke had said about the subj ect . The more he wrote , the ho tter Newton became . True , he had learned about colors in thin films fro m Hooke . Hooke had confes sed that he did 1 00

101

New ton to Oldenbu rg , 7 Dec . 1 675 (the "Hypothesis of Light") ; Corres 1 , 384 . This is, of course, Newton ' s description of the episode. See Roger H . Stuewer, " A Critical Analysis of Newton ' s Work on Diffraction, " Isis , 6 1 (1 970) , 1 88-205 . 1 02 Birch , History, 3, 269 . Con-cs 1 , 384.

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no t know how to measure the thickness of the films , ho wever, "& therefore seing I was left to measure it my self I suppose he will allow me to make use of what I tooke ye pains to find out. " 10 3 Three weeks o f thinking abou t it left Newton even mo re incensed. Initially , he was inclined to grant that he got the idea of vib rations in the aether fro m Hooke . Now he retracted that as well ; it was a common idea . " I des ire Mr Hooke to shew me therefore , I say not only ye summ of ye Hypothesis I wrote , wch is his insinuation , but any part of it taken out of his Micrographia : but then I expect too that he ins tance in what's his own . " 10 4 The handling of Newton's first letter ten ds to confirm Hooke's suspicions that Oldenbu rg egged hi m on . Though Oldenb u rg read a passage from it about the electrical experi ment to the Royal So ci­ ety on 30 December, he neither read the co m men t on Hooke nor told Ho oke about it. Hooke heard it with su rprise at the mee ting on 20 Janua ry. A t th at point, he took matters into his own hands and wrote directly to Newton the same day . He feared Newton had been misinformed about him , a " sini ster practice" which had been u sed against him befo re . He p rotested hi s disapp roval o f conten­ tion , his desire to embrace truth by who mever disco vered , and the value he placed on Newton 's "excellent D isquisitions , " whi ch p ro­ ceeded fa rther than anything he had done . Finally , he proposed a co rrespondence in which the two could discuss philosophical mat­ te rs priva tely . "This way of contending I believe to be the more philo sophicall of the two , fo r thou gh I confess the collision of two hard-to-yield con tenders may produ ce light yet if they be pu t to­ gether by the ears of other's hands and incentives , it will p roduce rather ill conco mitant heat which serves for no other u se but . . . kindle co le [sic]. " 1 0s Newton replied in kind , calling Hooke "a true Philosophi cal spirit . " "There is nothing wch I desire to avoyde in matters of Philosophy more then conten tion , " he ag reed , "nor any kind of 1 03 1 04

1 05

Newton to Ol denbu rg , 21 Dec . 1 675; Corres 1, 406 . New ton to Ol denburg, 1 0 Jan. 1 676; Corres 1 , 408 . By the time he finished the second letter , he h ad cooled down a good bit; he asked to have his service p resented to Hooke, " for I suppose there is nothing bu t misapp rehens ion in wt has lately happend" (Corres 1 , 41 1 ) . Corres 1 , 4 1 2- 1 3 . Hooke made an entry in his diary the same day i n which h e referred to Newton ' s letter "s eeming to qu arrell from Oldenburg fals su ggestions . . . . Wrot letter to Mr. Newton about Oldenburg kindle Cole" ( The Diary of Robert Ho oke M. A . , M. D. , F. R . S. 1 672- 1 680, ed . Henry W. Robinson an d Walter Adams [Lon don , 1 935 ], p . 2 1 3) . The phrase, "kindle cole, " which appeared both i n the letter and in the diary, is eluci­ dated by a use that Cromwell made of it in a sp eech to Parliament on 25 J an . 1 658; " I speak of m en going about that cannot tell what they wou ld have, yet are willing to kindle coals to disturb others " ( The Letters and Speeches of Oliver Cromwell with Elucidations by Th omas Ca rly le, 3 vols . , ed . S . C . Lomas [London , 1 904 ], 3, 1 8 1 ) .

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conten tion more th en o ne in print . . . " Accepting the o ffer of a private correspondence , he went on to p raise Hooke's contribution to optics . " What Des-C artes did was a good step . You have added much several ways , & especially in taking ye colours of thin plates into p hilo sophical consideration . If I have seen further it is by standing on ye sholders of Giants . " 1 06 Senti men ts too lo fty drift away fro m hu man reality . A lack of warmth was evident on both sides . Nei ther man endeavo red to institute the philoso phic corre­ sp ondence bo th p rofessed to want , and thei r basic an ta gonism re­ mained undissolved . Another co rrespondence refused to go away , the one ins tituted by Linus , which provided the occasion for the two papers of Decem­ ber . That same month , a letter fro m Liege wri tten by John Gas­ coines , a pupil of Linus , info rmed Oldenbu rg and New ton that Linus was dead but th at Gascoines in tended to defend his p rofes­ sor's h ono r . As Linus had done , he denied the basic experiment. Newton replied with fu rther instructions on how to p e rform the experi men t and with a heated rej o inder to what he took as an i m plication that he had dealt un derhandedly with Linu s . 1 0 7 A month later, when he saw his own letter in the Ph ilosoph ica l Transa c­ tions with Linus ' s second letter (written a year befo re) , he recalled the whole ep isode and whipped hi mself into a fine fu ry , first at Linus for failing to co mprehend the experimen t , and then at Gas­ coines for impugning his hono r . 1 0 8 Finally on 27 Ap ril 1 676 , the Ro yal Society had the basic experi men t of the p rismatic spectru m performed befo re them to confi rm its outcome - a mere four years and th ree months after th ey had begun to discuss the questions it raised . The success of the experimen t did no t settle the ma tte r with Liege . In June , at the ti me of the Ep isto la p rio r, a new le tter arrived fro m a thi rd co rresponden t, Anthony Lucas , ano ther English Jesuit, who m Gascoines had recruited to take up cud gels too heavy for hi m . Lucas began by conceding the sole point that hi therto had been in contention: a p rism does proj ect a spectru m elon gated per­ pendicularly to the axis of the pri s m . His spectru m did n o t have the same p ropo rtions as Newton ' s , however. Its length was not five ti mes i ts breadth , but only th ree o r three and a half times . Fu rther1 06 Newton to Hooke , 5 Feb . 1 676 ; Con-es 1 , 4 1 6 . I do not accep t the interpretation that the last phrase was a deliberate, oblique reference to Hooke ' s twisted physiqu e. As New ton said once before in regard to Hooke, he avoided oblique thrus ts . When he attacked , he lo wered his head and charged . 1 0 7 Gascoines to Oldenburg, 1 5 Dec. 1 675 ( N . S . ) , Con-es 1 , 393 - 5 . Newton to Oldenbu rg, 1 0 Jan . 1 6 76; Con-es 1, 409- 1 1 . 1 0 8 Newton to Oldenburg , 29 Feb . 1 676; Corres 1 , 42 1 - 5 .

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more , Lucas p roceeded to rela te nine othe r experiments he had perfo r med to test Newton ' s theory . Far from confirming it, their outco me seemed to negate i t . 1 0 9 Through fo ur years of discussion , Newton had challenged opponents to b ring forw ard experiments instead of hypotheses . To be sure , Lucas ' s letters betrayed no p ar­ ti cula r acu men ; the exp eri ments he presented were not well de­ signed . 1 1 0 In regard to them , as Newton insisted on the careful examination of the i mport of his o wn experiments , he made another penet rating pronouncement whi ch modified the seeming empirici s m of other assertions in o rder to cover the other flank of the methodological debate . " For it is not number of E xp1s, but wei ght to be regarded; & where one will do , what need of many? " 1 1 1 Nevertheless , what Lucas sent was a reasoned letter whi ch deserved a reasoned response . Newton greeted it quite other­ wise . A s the corresponden ce continued , he became increas ingly agi­ tated and i rrational . He convinced hi mself that the Lieg ois (papi sts , of course) had formed a conspiracy to engage hi m in perpetual disp utation and to undermine his credit . He refu sed to di scuss Lu­ cas ' s experim ents but insi sted that Lucas discuss hi s , in particular the exp erimentum cru cis . When Lucas obliged and did not find the experim entum cru cis to be conclusive or for that matter to work as Newton described i t , Newton grew angrier yet . Nor was he molli­ fied when Lu cas suggested that the real question was not the length of the spectru m but the theory of colors . The Questi on in hand is this [he sto rmed] . Whether ye Image in ye Expe riment set down in my first letter about colours . . . could be five ti mes longer then b road as I have there exp rest it , or but three or at most three and a half as Mr Lucas has represented it . To this I desi re a direct answer: which I hope will be so free as (wthout putting me to any fu rther wayes of justifying myself) may take off all susp i­ cion of my misrepresenting matter of fact. And again: Tis ye truth of my experiments whi ch is ye bu siness in hand . O n this my Theo ry depends , & which i s of mo re con sequence , ye credit of my being wary , accurate and faithfull in ye rep orts I have made . . 1 1 2 " I see I have made my self a slave t o Philosophy , " h e exclaimed to .

1 09

1 1° 1 11 112

Lucas to Oldenburg , 27 May 1 676 (N . S . ) ; Corres 2, 8- 1 2 . See S . M . G runer , " Defending Father Lucas: A Consideration of the New ton-Lucas Dispute on the Nature of the Sp ectru m , " Centaurus, 1 7 (1 973) , 3 1 5-29 . Cf. Rosenfeld , "La Thforie des couleurs de Newton . " New ton to Oldenburg , 1 8 Aug. 1 676; Co rres 2, 79 . Newton to Oldenburg , 28 Nov . 1 676; Corres 2, 1 83- 5 . New ton replied to Lucas 's letter of 27 May on 18 August (Corres 2, 76- 81 ) . Lu cas 's second letter, to which New ton 's present one replied , was dated 23 O ctober (N . S . ) (Co rres 2, 1 04-8) . His letters had an unfortunate way of arriving just as Newton was engaged in writing to Leibniz.

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Oldenburg in desperation , " b ut if I get free of Mr Linus' s b uisiness I will resolutely bid adew to it eternally, excepting what I do for my p rivat sati sfacti on or leave to co me out after me. For I see a man must either resolve to put out nothing new o r to b eco me a slave to defend i t . " 1 1 3 Recall that at the ti me he wrote , Newton' s " s lav ery" con s i s ted o f five replies t o Liege , to taling fou rteen p rin ted pages , over a period of a year. Recall also that he had co mpleted the Ep istola posterior less than a month before . The length of the spectru m , which became for Newton the sym­ bol of his honor, has been the o bject of con siderable historical co m ment . Lucas might have used a prism made of glass with a different d ispersive power than Ne wton ' s . Had Newton not been so dog matic , he could have pursued the source of the discrepancy and disco vered the possibility of the achromatic lens . The case is by no means p roved , however. Newton had to push hard to get L ucas to measure distances and angles with a rigo r that even app roached his own. He always suspected that Lucas proj ected hi s sp ectrum with a p rismatic angle les s than 60 degrees , which could have ac­ counted for the difference . In the end , Lucas did concede the ques­ tion o f the length of the spectru m . 1 14 Along the way , he also men­ tioned that the faces of his prism were concave (which would have tended to decrease the p roportions of the spectrum) , and he indi­ cated that he usually perfo rmed the experiment between six and seven in the morning . The last admis sion was parti cularly da mning . It called everything Lucas had said into question . Lucas had clai med to project his spectru m eighteen feet . With a horizontal sun , he would have had to proj ect it upward at an angle of a b o ut 45 de­ g rees . While a roo m that would both allow such a traj ectory and enable an observer to receive the spectrum on a screen perpendicu­ lar to its t rajectory in o rder to measure it properly is not i mpossi­ ble , it is also not com mon . Newton ' s dogmatism in the exchange is beyond challenge , but it p robably o bscured Lucas' s slo ppiness in­ stead o f a new discovery . 1 1 5 When a third letter fro m Lucas arrived in February 1 67 7 , Newton decided on a different mode of answer. D uring the previou s year, he had mentioned the po ssibility of a major publication on optics designed to confirm hi s theory and hence to settle all disputes . At lea st once , he had considered publishing all the correspondence his papers had p rovoked . Several pieces of ev idence indicate that New­ ton was planning such a volume during 1 677 . In March , Collins mentioned that D avid Loggan , who was resident in Trinity College 1 1 3 Newton to Ol denbu rg, 18 Nov . 1 676; Co rres 2, 1 82-3 . 1 1 4 Lucas to Oldenbu rg, 2 Feb . 1 677 ( N . S . ) ; Corres 2, 1 9 1 . 1 1 5 In regard to Newton ' s dogmatism in the exchange, see Johannes Lohn e, " N ewton 's ' P roo f of the Sine Law and His Mathem atical P rincip les of Colou rs, " Arch ive for Histo ry of Exact Sciences, 1 ( 1 96 1 ) , 389-405 .

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277

as he p rep ared the engravings fo r hi s Cantabrigia illustrata ( 1 690) , had told him he had drawn Newton ' s picture fo r an engraving to be included in a book on light and colors . 1 1 6 In December, after Lu cas had enquired about the p rinting of his letters , Newton mentioned hi s plan to Robert Hook e, who had su cceeded Oldenburg as secre­ tary of the Royal Society. "Mr Oldenburg being dead I intend God willing to tak e care y1 they be printed acco rding to his mi nd , amongst some other things wch are going into ye P ress . " 1 1 7 A few printed sheets with part of his paper of February 1 672 and notes whi ch expli cate certain passages have recently been di scovered . 1 18 There i s every reason to beli eve they constituted the beginning of the p roposed volume. Then fi re struck his chamber destroying p art of the collection of papers. 1 1 9 Though.. he tri ed briefly to get new copies , he finally abandoned the project . Fourteen years later , Abraham de l a Pryme , a student i n Johns , re corded in hi s diary a story he had heard . Febr: [1 692] What I heard to-day I must relate . There is one Mr . Newton . . . fellow of Trinity College , that is mighty famous for his learning , being a most excellent mathematician , philosopher, divine , etc . . . . but of all the books that he ever writt there was one of colours and light, established upon thousands of experiments , which he had been twenty years of making , and which had cost him many 1 1 6 Collins to Newton, 5 March 1 677; Co rres 2, 200- 1 . Oldenburg appeared to refer to an intended publication in his letters to Newton on 2 Jan . 1 677 (Corres 2, 1 88) and to Leibniz on 2 May 1 677 (Corres 2, 209) . 1 1 7 Ne wton to Hooke, 1 8 Dec . 1 677; Co rres 2, 239 . 1 1 8 Derek Price found the sheets , which ha d been used as stuffing in the bin ding of a book (" Newton in a Church Tower: the Discovery of an Unknown Book by Isaac Newton, " The Yale University Library Gazette, 34 [ 1 960] , 1 24-6) . They are both reproduced and discussed fully in I. Bernard Cohen , " Versi ons of lsaac Newton's Firs t Published Paper, " Arch ives international es d'h isto ire des sciences, 1 1 ( 1 958) , 357-7 5 . The evidence for Newton's plans to publish at this time is fully laid out in A. Rupert Hall, " Newton' s First B ook (I) , " ibid. , 13 ( 1 960) , 39-54 . 1 1 9 The evi dence for a fi re at thi s ti me is stron g . Newton wrote to Lucas on 2 Feb . 1 678 for copies of Lucas's letters . The letter ha s disa ppeared, but in reply Lucas referred to " your losse " (Lucas to Newton , 14 Ma rch 1 678 [N. S. ]; Co rres 2, 2 5 1) . Newton also wrote to Hooke on 5 March and to Aubre y about June , mentioning the loss of pa pers (Corres 2, 253 , 266) . In hi s memorandu m of a conversation with Newton on 3 1 Aug . 1 726, Con­ duitt recorded Newton's recollection of a fi re. " When he was in the mi dst of his dis cov­ eries he left a candle on his table amongs t his pa pers & went down to the B owling g reen & meeting somebody tha t diverted hi m from returning as he intended the ca ndle sett fire to hi s papers & he could never recover them I asked him wether they related to his opticks o r to the method of ftuxions & he said he believed there were some relating to both " (Keynes MS 1 30. 1 0, ff. 4-4v ; Keynes MS 1 30. 4 , pp. 1 4- 1 5) . Humph rey Newton said that he ha d hea rd of " his Opticks being burnt" before he ca me to Cambridge in 1 683. W ri ting in 1 683 for a book pu blished in 1 685, Wallis mentioned the loss of papers by fi re (Op era mathematica , 3 vol s . [Oxford , 1 693-9 ] , 2, 3 90) . Stukeley also mentioned tha t H u mphre y Newton ha d tol d him of several sheets o f optics being set on fire by a ca ndle , and he added a s tory about the burning of a work on chemis try, a pparently at another ti me ( S tukeley to Conduitt, 1 5 July 1 727; Keynes MS 1 36 , p . 1 0) .

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a hundred of pounds . This book which he valued so much , and whi ch was so much talk' d off, had the ill luck to perish and be utterly lost j ust when the lea rned author was almost at putting a concl usion at the same , after this manner. In a winter morning , leaving it amongst his other papers on his studdy table, whi lst he went to chappel, the candle whi ch he had unfortunately left burning there too cachd hold by some means or other of some other papers , and they fired the aforesayd book, and utterly consumed it and sev­ eral other valuable writings, and that which is most wonderful did no fu rther mischief. But when Mr. Newton came from chappel and had seen what was done , every one thought he would have run mad, he was so troubled thereat that he was not himself for a month after. A large account of this his system of light and colours you may find in the transactions of the Royal Society , which he had sent up to them long before this sad mischance happened to them. 1 2 0 De la P ryme' s story is usually attached to Newton' s established breakdown in the autu mn of 1 693 , which a story Huygens heard, also involving a fire, tends to support . However, the da te of de la Pryme ' s entry antedates the breakdown of 1 693 by more than eigh­ teen months , and the past perfect tense in the final sentence does not seem to place the fi re in the recent past . It could refer to the well-authenticated fire th at halted an optical publication in the winter of 1 677- 8 . 1 2 1 There i s a hiatus in Newton' s corre spondence fro m 1 8 December to February, though hi s correspondence in this period was very light in any case . Twice at lea st in earlier corre­ spondence , with Hooke and with Huygens , he had lost p artial con­ trol of hi mself as the heat of hi s own vehemence swept him away , and the tone of hi s letters to Lucas implies a comple te los s of control th at is compatible with a breakdown. As later in 1 693 , Newton was in a s tate of acute intellectual tension throughout the 1 670s , . not just from answering obje ctions to hi s optics , but even more fro m other studies then foremost in his mind, studies which exci ted hi m keenly . Another parallel with 1 693 may also be rele­ vant . The crisis in his relations with Fatio de D uillier at that ti me had its counterp�rt in Wickins ' s decision to leave Trinity . 1 22 When he discarded the intended publication , Newton wrote two further letters to Lucas on the same day , 5 March 1 678 , one answer­ ing Lucas' s first two and the other answering his third (of February 120 The Diary of Abraham de la Pryme, ed . Charles J ackson (Durham , 1 870) , p. 23 . 121 Edleston concluded that P ryme's story referred to a fire between 1 676 and 1 683 (Edleston, p p . lx-lxiii) . 122 The Steward 's Books show that in 1 676 (i . e. , the year that ended on Michaelmas 1 676) Wickins , who had hitherto resided in the college most of each y ear, was in Cambridge only 25 1 12 weeks . In 1 677, he resided in college 13 1 i2 weeks , in 1 678 6 w eeks . He was never there that long again before his final resignation in 1 684.

Publication and crisis

279

1 677) . Even the earlier le tters , fu rious as they were , co uld not have prep ared Lucas for the flood of paranoia that now burst over h i m . Do men use t o press one another into Disputes? O r a m I bound to satisfy you? It seems you thought it not enough to prop ound Obj ec­ tions unless you might in sul t over me for my inability to an swer them all , or durst not trust yo ur own j udgement in cho osing ye best . But how know you yt I did not think them too weak to requ ire an answer & only to grati fy your impo rtunity co mplied to answer one or two of ye best? How know you but l other prudential reasons might make me averse from contending wth you? But I fo rbeare to explain these things fu rther fo r I do not think this a fit Subj ect to dispute about, & therefore have given these hints only in a private Letter. of wch kind you are also to esteem my former answe r to you r second . I ho pe you will consider h o w little I desire t o explain yo ur proceedings in public & make this use of it to deal candidly wth me for ye future . 1 2 3 Arrogant and b rutal , th e two letters made it clear th at n o thing less than the public hu mil iation of his an tagon ists could satisfy N ew­ ton - hateful letters , did not our know ledge of th e ci rcumstances incline us to sy mp athy with the au tho r's anguish . The co rrespondence gave one last spasm before it died . In May , Newton ackno wledged a letter fro m Lu cas , th ough h e probably did not an swer i t . 1 2 4 Later that month , he heard there was ano ther wai ting fo r him in London . Mr

Aubrey I understand you have a letter from Mr Lucas for me . Pray forbear to send me anything more of that nature . 1 2 5

With that he b rought his co rre spondence abo u t colors to an end . Oldenbu rg was dead . Newton had ceased to co rre spond with Col­ lins . As far as he could, he isolated himself. To the b est of ou r kno wledge he wro te only two letters , on e to A rthu r Storer (which has not su rvived) and one to Robert Boyle , between June 1 678 and December 1 67 9 . 1 2 6 To the end of his days Newton re membered this 1 2 3 Co rres 2 , 254-60 , 262-3 . 1 2 4 New ton to Hooke , 1 8 May 1 678; Co rres 2, 264 . 1 2 5 Newton to Aubrey , ca . June 1 678; Co rres 2, 269 . I believe that this letter was not sent. I t exists only a s a n autog raph d raft, and it looks very much like a sentence i n a letter sent to Aub rey about June (Corres 2, 266-8) . Thou gh not so su ccinct - or rather, because it is not so su ccinct - the one sent o ffers a fu rther exhibit in Newton ' s display of paranoia, if another is needed . 1 26 I do not believe in the authen ticity of the supposed letter to Maddock on 7 Feb . 1 679 (Co rres 2, 287) , which su rfaced in a funeral sermon in the middle of the eighteenth century . Its tone of l ight irony resembles Newton ' s style about as much as a gazelle res em bl es a tiger.

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withdrawal as a cons cio u s decision and considered that it had marked an epoch in his life . " Its now about fifty years , " he wrote to Mencke in 1 724 , " since I began fo r the sake of a quiet life to decline co rrespondencies b y Letters about Mathemati cal & Philo­ sophical ma tters fi n d in g them tend to dis p u tes and contro versies . . " 1 2 7 .

1 27 A

draft of a letter; Keyn es MS 1 1 1 . The letter app arently sent (Con-es 7, 255) did not include this passage.

8

Rebellion

EWTON'S

repeated protestation that he was engaged in other studies supplied an ever-p resent theme to his co rres pondence of the 1 670s . Already in July 1 672 , only six months after the Royal Society discovered him as a man sup remely skilled in opti cs , he wrote to Oldenbu rg that he doubted he would make furth er trials with teles copes , " being desirou s to pros ecute s o me other s u bjects . " 1 Th ree and a half years later , he put off the composition of a general treati se on colors because of unspecified obligations and some ' 'buisines of my own wc h at present alm ost take up my time & thou ghts . " 2 Ap parently the other bu siness was not mathematics , because later in 1 676 he hoped the se cond letter fo r Lei bniz would be the last . " Fo r having other things in my head , it p roves an unwelcome interruption to me to be at this time put upon consider­ ing these things . " He was not only p reo ccup ied , he was almost frantic in hi s imp atience. " Sr , " he concluded the letter , " I am in great hast , Yours . . . " 3 In g reat haste be cause of what? Surely not because of ten lectures on algebra that he purp o rtedly delivered in 1 676 . And not becau se of pupils or collegial duties , fo r he had none of either . Only the pursuit of Truth could so drive Newton to di straction that he resented the interruption a letter offered . New­ ton was in a state of ecstasy again . If mathemati cs and optics had lost the capacity to do minate him , it was because other studies had supplanted them . One of the studies was chemistry . Collin s mentioned his ab­ sorption in it twice in letters to Gregory . 4 Years later , when he chatted with Conduitt about his early life in Cambridg e, Newton himself ment ioned that Wi ckins help ed in his " chymi cal experi-

N

1 Newton to Oldenburg , 8 July 1 672; Co rres 1, 2 1 2 . On 21 September he wrote to Olden­ burg that, in re sponse to Oldenburg ' s request , he ha d begun to draw up a list of experi­ ments to an swer the eight queries he ha d posed in July; "before it was fini shed falling upon some other business , o f wh I ha ve my ha nds full, I was obliged to lay it aside, & now know not when I shall take it again into Consideration" (Corres 1, 237-8) . 2 Newton to Oldenburg , 25 Jan . 1 676; Co rres 1 , 4 1 4. Ten da ys later, in accepting Hooke ' s offer of a philosophical correspondence , he added tha t he had grown ti red of optics and doubted that he would ever a gain become interested enough to spend much ti me on it (Co rres 1, 4 1 6) . In fact, he never did. On 1 1 May 1 676, he told Oldenburg that he hoped to use the two papers he ha d sen t in December , if he could " get some ti me " to write the other di scourse, about the colors revealed by the pri s m , to accompany them (Co rres 2, 6) . 3 Newton to Oldenburg , 24 Oct. 1 676; Co rres 2, 1 1 0 . 4 Collins to Gregory, 29 June , 1 9 O c t . 1 675 ; Co n-es 1 , 345 , 356 . 28 1

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ments . "5 His interest in it developed somewhat later th an his in­ terest in natu ral philosophy . When he com po sed the " Q uaestiones qu aedam phil osophicae" in the mid-1 660s , he ente red al most noth­ ing that one would call chemistry , even thou gh Robert Boyle was one of the maj o r sources of hi s new mechanical philosophy . When he extended hi s notes on a number of the headings under " Q uaes­ tiones " in a new noteb ook , howeve r , chemistry did begin to ap­ pear, and the notes indic ate that B oyle supplied hi s intro du ction to the su bject. 6 The fo rmat of the new notebook with its series of headings repeated Newton ' s practice of recording newly acquired knowledge in an ordered fashion . A decade and more later, he was still en tering a few i tems unde r the se headings fro m fu rthe r read­ ing . 7 At about the time of hi s earliest chemical no te s , around 1 666 , he also composed a che mical glossary b ased largely on Boyle , who m alone he cited in it . In the glossary , unde r a number of hea dings , he set down basic informati on needed by a chemi st . He began with " Ab straction , " the ev aporation o r distillation of a solu tion to obtain the salt in i t , and proceeded throu gh quite a nu mbe r of terms such as " Amalgam , " " C rucible , " "E xtraction , " " S ubli mation , " and so o n , giving a brief su mmary o f operative info rm ation that he had culled . 8 Unde r " Testing , " he desc ribed a means to re fme gol d and silver by he ating the m with lead. New­ ton ' s ability to o rganize what he learned so that he could retrieve it w as a signi ficant aspect of his geni u s . Years later, he described exa ctly the same process in a paper he p repared at the Mint and used s o me of the same language he had entered fifty years earlier under " Testing . " 9 A s an example o f the level o f detail Newton dem anded and o f the sophi stication his early kno wledge of chemistry attained , consider his entry under " Fu rnace" (Fi gure 8 . 1 ) . 5 Condu itt ' s memo randum of 3 1 Aug. 1 726; Keynes MS 1 30 . 1 0, f. 3" . O ne of the anecdotes ' that Conduitt collected indicated that N ewton s " fu rnace at Cambridge" was an item of interest shown to visitors (Keynes MS 1 30 . 5 , sheet 1 ; Keynes MS 1 30 . 6 , book 1 ) . 6 Add MS 3975 . The early sections o f this no tebook, o n colors , cold and heat , com pression and elas ticity , and fire and fla me, did not contain an y chemistry . With the exception of the fi rs t , they d rew p rimarily on Boyle, and their frequent citation of his Origine of Formes and Qualities placed them in 1 666 at the earliest . Later sections , " Of Formes & Transmutations wrought in them " (pp . 6 1 -6) an d " Of S al ts , & Sulphureous bodys , & Mercury . & Met­ ' tall s " (pp . 71 -80 , 88- 1 00) , which also cited Boy le s Origine , were definite I y chem ical in conten t . 7 Add M S 3975 , pp . 38-41 , experiments on freezing, January 1 670; p . 46 , a n e xperiment o n degrees of heat and expansion of a i r and water by heat , 1 0 March 1 693 ; p . 5 1 , a formula fo r ' phosphorus , early 1 680s; pp. 65-6, notes fr o m the Earl of S an dwich s trans lation of The Art of Metals (1 674 edition) ; pp. 88- 1 00 , no tes from Boyle, Certain Physi ological Essays (1 669 edition) ; p . 1 62 , under a later heading , "The medicall virtues of S aline & other ' Praeparations , " notes from Boyle s Essay of . . . Effiuviums ( 1 673) . 8 Bodleian Library , Oxford , MS Don. b . 1 5 . 9 Mint 1 9 . 2 , f. 293 .

Figure 8 . 1 . N ewton ' s drawing of various chemi cal fu rnaces . 1 , 2, 3 , a n d 4 . Wind furnaces . 5 . Reverberatory furnace . 6 . Atha­ nor, Piger Henri cus, or Furnace Ace diae . (C ourtesy of the J oseph Halle S chaffner Collecti on, University of C hicago Library . )

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Fu rn ace . As 1 ye Wind fu rnace (for calcination , fusion, cementation & c) wch blows i t selfe by attracting ye ai re through a na rro w pas­ sage 2 ye distillin g fu rnace by naked fi re , fo r t hings y 1 re.q u ire a strong fi re fo r distillation . & i t differs not much from ye Wind furnace only ye glas se re sts on a cro s s e barr o f i ron under wch bar is a ho le to put in the fi re , wch in ye wind fu rnace is put in at ye top . 3 The Reverberatory furnace where ye fla me only ci rculating under an a rched ro of acts up on ye b ody . 4 ye Sand furnace when y e ves sel i s set in S and or sifted a shes heated by a fi re made underneath . 5 B alneu m o r B alneu m M ariae when ye body is set to distill o r digest in hot wat e r . 6 B alneu m R o ris or Vap o rosum ye glasse hanging in the steame o f b oyling wat er I nstead of this may bee u sed ye heat of hors dung (cald venter Equ inus) i : e: brewsters g rains wheat b ran , S aw dust, chopt hay or stra w , a little moi stened cl ose p ressed & cove red . O r it may in an egg shell bee set under a hen . 7 Athano r , Pige r Henricu s , o r F u rnace A cediae fo r long digesti ons [ye] v e s sel being set in sand heated wth a Turret full o f Charcoale wc h is contrived to b u rne only at the [botto] m the upp e r co ales continually sinking downe for a supply. Or the sand may b e heated [by a] Lamp & it i s called the Lamp F u rnace . T hese are made of fi re stone s , o r b ri ck s . 1 0

Putting the skills developed in Grantham to use , Ne wton u su­ ally built his own furnaces . Manifestly , he knew what he was about . Not all of the entries in the glossary confined themselves to straightforward prosaic chemi stry - or " rational chemistry , " as tho se call i t who wish to pretend that Newton did no t leave behind a vast collecti on of alchemical manuscripts . He included qu ite a few entries on mercury , inclu ding mercury sublimate which " opens" copper , tin , and silver, but not gold. "Yet per­ haps , " he added , " there may bee Sublimates made (as by su blim­ ing com1non sublim ate & Sal Armoniack well powdered to­ gether) wch besides notable operations on other metall s , may act upon Gold too . " 11 One entry described Boyle ' s menstru u m p eracu­ tum , which dissol ved gold and even carried some gold with it i n distilla tion . B oyle invested the menstruum p era cutum with alchemi­ cal signific anc e; Newton ' s entry i mplied that he did too . Anti­ mony and its power to purify gold appeared . As with the refin­ ing of gol d by lead, Ne wton later employed hi s kno wledge of refining by antimony in an emotionally charged me morandu m when the standard of his coinage was impu gned at th e trial of 10 11

MS Don. b . 1 5 , f. 3 . The brackets fill in a damaged corner of the manuscript . Cf. Joseph Halle S chaffner Collection , University of Chicago Library , MS 1 075-2 . M S Don. b . 1 5 , f. 4.

Rebellion

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the pyx in 1 7 1 0 . 1 2 H is early glossary also included in stru ctions to make regulus of anti mony, regulus of Mars , and "Regulu s Martis Stellatu s , " the star regulus of Mars , which would soon figure pro minently in an explicitly alchemical settin g . I n similar fashion , the chemical notebook changed its charact er. N otes from Geo rge Sta rkey ' s Pyrotechny Asserted succeeded th ose from B oyle. Starkey was p rob ably the p seudonymou s Ei renaeus Philalethes , who se numerou s treati ses on alchemy exerci sed en o r­ mou s influ ence on N ewton . Under "Medi cal Ob servations, " he ent ered the recipe for primum ens of B aulm , whi ch had the power to restore youth . The instructions called for a dose in wine every mo rnin g "till ye naile s hair & teeth fall of & lastly the skin be dryed & exchan ged fo r a new one . . . " The hesitant are assu red that the ens started a woman of seventy years menst ru ating again . 13 Its al­ chemical connotation rather than its effect i s the point at i ssue here, however, and a pointing hand drawn in the margin , a devi ce famil­ iar to students of Newton's manu scripts of every sort , asserted its i mportance . One of the last sections of the notebook , added per­ haf s a decade after the initi al set, carried the heading , " Of ye wo rk wt common 8 [gold] . " He d rew the content of the entry from Philalethes ' commentary on Ripley . 14 N o soli d evidence all ows us to date Newton's plunge into alchemy with p reci sion . A number of items suggest 1 669 . The completion of hi s opti cal research befo re hi s appointment to the Lucasian chair may have cleared the way fo r a new intellectual passion . His i mp atience with questions about the theo ry of colors in the 1 670s spran g in part from his total abso rpti on in a new investigati on . The o rder of development of Newton ' s chemical noteb ook was significant . He did not st umble into alchemy , discover its ab su r­ dity , and make his way to sober, " rational , " chemi stry . R ather he started with sober chemistry and gave it up rather quickly for what he took to be the greater profundity of alchemy . The latest notes attri buted to B oyle referred to hi s Essay of . . . Effiu viums of 1 673 . An unattributed recipe fo r making pho spho rus (which b eg an with the heroic instru ction , "Take of U rin one B arrel") und oubtedly stemmed from B oyle ' s investigati on of pho spho ru s in the early 1 680s , but an isolated recipe for a new and unu sual su b st ance is a different matter from notes on su stained reading . 1 5 B oyle him self 12 Ibid . , f. 4v . In the Mint me mora ndu m , Newton said the G oldsmiths Company were of the opinion that gold cou ld not be made finer tha n 24 ca rat s . First he told the m how to do it wi th aqua forti s . "Chymists also tell us , " he conti nued, " that gold may be made finer by Anti mony then by A qua fortis & by conse quence then by the Assay [which used aqua fortis]; but the Goldsmiths know not h ow to refine Gold by Antimony" (Mint 19 . 1 , f. 1 3 Add MS 3975, p . 1 89. 250v) . 1 4 Add MS 3975 , p p . 243- 4. 1 5 Add MS 3975 , p . 5 1 .

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was deeply involved in alchemy in any case , and once they became acquainted , the two men corresponded on the subject until B oyle's death in 1 69 1 . Meanwhile , reading th at began with B o yle in the 1 660s turned overwhelmingly to expli citly alchemical autho rs about 1 669 . His accounts show that on his trip- to London th at year he purchased the great collection of alchemi cal writings , Theatrum ch emicum, in six heavy quarto volumes . He also purchased two furnaces , glass equipment , and chemicals . 1 6 Pos sibly s o me practi­ tioner of the Art introduced Newton to it . Evidence exi st s that Cambridg e had its adepts while Newton was th ere . 17 We are not obliged to lo ok for an alchemical father , how ever . Newton had already found his way alone to a nu mber of studies . With collec­ tions such as Thea trum chem icum available, his independent di scov­ ery of alchemy would have been easy enough . Soli d evidence further shows that however i t began , Newton ' s alchemical acti vity included his personal intro duction into the largely clandestine society of English alchemi sts . His reading in alchemy was not confined to the printed word . A mong his manu­ scripts i s a thick sheaf of alchemical treatises , most of the m unpub­ li shed , wri tten in at least fou r different hands . 18 Since Newton copied out five of the treatises plus some recip es, the collection appears to have been loaned to him for study but then , for what­ ever reason , n ot returned . 1 9 In the late 1 660s , he copied Philalethes ' " Exp o sition upon Sir Geo rge Ripley ' s Epistle to King Edward I V " from a version which differed fr o m publi shed ones though it agreed with two m anu scripts now in the B ritish Library . 2 0 He to ok exten­ sive notes from a m anu script of Philalethes ' " Rip ley Reviv' d " about 1 6 Fitzwilliam notebook. 1 7 Betty Jo Teeter Dobbs , The Fo undations of Newton 's Alchemy . The Hunting of the Greene Lyon (Cambridge, 1 975) , pp. 95- 1 2 1 . M rs . D obbs argues that Barrow and Henry More may have introdu ced Newton to alchemy . On Newton and alchemy see also A . R . and Marie Boas Hall , " Newton ' s Chemical Experiments , " Archives internationales d'histo ire des sciences , 1 1 ( 1 958) , 1 1 3-52; Karin Figala, " Newton as Alchemist , " History of Science, 1 5 ( 1 977) , 1 02-37; R . J . Forbes, " Was Newton an Alchemist?" Chymia , 2 ( 1 949) , 27-36; Douglas M cKie, " Newton and Chemistry, " Endea vo ur, 1 ( 1 942) , 1 4 1-4, and " Some Notes on Newton ' s Chemical Philosophy W ritten upon the Occa si on of the Tercentenary o f his Birth , " The Londo n , Edinburgh & Dublin Ph ilosop h ical Maga zine and Jo urnal of Science, ser . 7, 33 ( 1 942) , 847-70; and Mary S . C hurchill, " The Seven Chapters, with Explanatory . Notes, " Chymia , 12 ( 1 967) , 29-57. 1 8 Keynes MS 67. On f. 68v Newton wrote a paragraph , of a nonalchemical natu re , in a hand that appears to belong to the mi d-1 660s . Elsewhere , he corrected some of the manu scripts against Ashmole ' s Theatrum and numbered some reci pes . That is , he studied the collection intensely . 1 9 Keynes MS 62 contains the material he copied from Keynes MS 67. 2 ° Keynes MS 5 2 . Ronald S . Wilkinson, " Some Bibliographical Puzzles concerning George Starke y , " A mbix, 20 ( 1 973) , 235 , has collated Newton ' s copy with Sloan M S S 633 and 3633 ( which contain s excerpts) .

·

Figure 8 . 2 . A set of alchemical symbols compiled by Newton . (Courtesy of the Provost and Fellows of King ' s C ol lege , Cambridge . )

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ten years befo re it was publi shed . 2 1 During the following twenty­ five years , Newton continued to receive a flo w of alchemical manu­ scripts which he himself copied . 22 These m anu scripts offer one of the m o st intrig uing aspects of his career in alchemy . Where did they come from? The Philalethes manuscripts ci rculated initially a mong the g roup of alchemists asso­ ciated with Sam uel H artlib in London . Hartlib had died well befo re Newton to ok up alchemy , but he m ay have been in touch with remnants of the g roup . Since William Cooper, who kept a shop at the sign of the Pelican in Little B ritain , later publi shed " Ripley Revi v ' d " and at least two other treatises that Newton copied , the contact may have been th rough him . Robert Boyle had known the Hartlib circle as well as Phi lalethes-Starkey , though it seem s clear that Newton fi rst met B oyle in 1 675 . One of the copied m anu­ scripts concluded with letters dated 1 673 and 1 674 from A. C . Faber t o D r . John Twi sden with Twi sden ' s notes o n them and on the manuscript . 2 3 Faber ( A . D . rather than A . C . ) was a physici an to Cha rles I I who publi shed a treati se on potable gold . Twi sden , also a physician in London known fo r hi s defense of Galenic medi­ cine , does not seem a likely clande stine alchemi st , but the notes attri buted to him are tho se of a serious practitioner . There was at least the po ssibility of personal contact and direct transmi ssion in connection with thi s paper . On another m anu script , " Manna , " whi ch i s not in hi s hand , Newton entered two pages of notes and vari ant reading s " collected out of a M . S. communicated to Mr F . by W. S . 1 670 , & b y Mr F . t o me 1 675 . " 2 4 M rs . Dobbs h a s argued pla usibly th at " Mr F . " was Ezekiel Foxcroft , a fell ow of King ' s who di ed in that same year of 1 675 . 2 5 Foxcroft , the nephew o f Benj amin Whi chcote, relative by m arriage of John Wo rthington , and friend o f Henry More (all Cam b ridge Platoni sts) , translated the Rosi cru cian tract , "The Chymical Wedding , " whi ch was publi shed fifteen y ears after his death . Newton read it and made, notes on it at that tin1e. Whether or not "Mr F . " was Ezekiel Foxcroft , the essen­ tial mystery of the alchemical m anu scripts remains unclarified . The 2 1 Keynes MS 5 1 . Thi s manu script i s perplexing since the notes have page references , though they do not correspond to the pa ges i n the pu blished version . The hand is u nmistakably that of the late 1 660s . 22 Keynes MSS 22, 24, 3 1 , 33 , 39, 50, 5 1 , 52, 5 5 , 58 (pa rt only) , 62 , 65 , 66, and a manuscript in the Yale Medical Libra ry are such . Keynes MSS 65 and 66, di fferi ng copies of a treati se by the English alchemist o f Newton's da y , William Y-Worth , a trea tise of which he ha d two other versions in other ha nds, may i mply that Newton wa s i n direct contact with 2 4 Keynes MS 33 . 2 3 Keynes MS 50. Y-Worth . 2 5 Dobbs , Foundations, p . 1 1 2 . Fi gala , " Newton as Alchemist, " pp . 1 03-4, questi ons this identification on the basi s o f the Eton College regi ster, which shows that Foxcroft died in 1 674. Since the Eton register used Old Style dates , however, Foxcroft could ha ve lived into 1 675 , the yea r that Venn ' s Alumni cantabrig iensis gives fo r hi s death .

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man who isolated him self fro m his colleagues in Trinity and dis­ cou raged correspondence from philosophical peers in London app ar­ ently re mained in touch with alchemists ·from who m he recei ved manu sc ripts . The mystery refu ses to be ignored. The manuscrip ts su rvive ­ unp ubl ished alchemical treati ses , copied b y Newton, the original s of which are unkno wn . The not very illuminating reference s to Twisden , Faber, "W . S . " and " Mr F . " excepted, the network of acquaintance that brought them to hi m left vi rtu ally no tangible evidence behind. In March 1 683 , one Fran . Meheu x wro te to New­ ton fro m Lon don about the success of a thi rd alchemist , i denti fied only as " hee , " in extracting three ea rths from the fi rst wate r. 2 6 Meheux' s letter mentioned a continuing co rrespondence , but the letters have disappea red . Meheu x and " hee " have all the substance of sh adows . In 1 696 , an unnamed and equ all y shadowy fi gu re , a Londoner acqu ainted with Boyle and Edmund Dickinson (a well­ kno wn alchemist whom Charles II had p atronized) , visited Newton in Camb ridge to discou rse on alchem y . They did not meet by chance; th e man came to find hi m . Newton recorded the conve rsa­ tion in a memorandu m . 2 7 Al chemy formed the ini tial subject of a co rrespondence with Robe rt Boyle which com menced in 1 676 . His friendship s with John Locke and Fatio de Duillier involved al­ chem y , but b oth of them began only in the late 1 680s . Otherwise nothing . One of the maj or passions of his life , as testi fied by a vast body of p apers which stretched over thi rty years , a pursuit which included contact with alchemical ci rcles as attested by hi s copies of unpublished treatises , remained largely hi dden fro m public view and re mains so today. The experience of another collector, Elias Ashmole , helps in as­ sessing Newton' s manu scripts . In the p reface of Theatntm ch em icu m britannicu m , A shmole declined to name t h e source of hi s trea tises because they prefe rred not to see thei r names in print . 2 8 His diary reco rded a visi t , not wholly unlike that which Newton received in 1 696 , when an unkno wn and mysterious man appeared at his door ready to reveal the Art . 2 9 One re members as well the elu sive Eire­ naeus Philalethes , who cloaked hi s identity in a pseudonym so ef­ fectively that only in thi s present gene ration have we le a rned with reasonable assu rance he was George Starkey . We kno w that New­ ton also compo sed an alchemical pseudonym -Jeo va sanctus unu s , a n anagram o f Isaacus Neuutonus - and as kno wledge of hi s al26 27 28 29

Corres 2, 386 . Keynes MS 26; published in Co rres 4, 1 96-8 . Indeed N ewton w rote two versions of the memorandum; the second is University of Chicago MS 1 075- 3 . Elias Ash mole , Theatrum ch em icum britannicum (London, 1 652) , Prolegom ena . C . H . Josten , Elias Ash mole ( 1 61 7- 1 692) , 5 vols . (Oxford , 1 966) , 4, 1 747- 8 .

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chemical acti vity beco mes kno wn , we may lea rn that Newto n fed treati ses into the same network fro m which he received them . Meanwhile , against the b ackground of deliberate secrecy , we can at least speculate th at otherwise unexp lained events in his life were alchemically mo tivated. In the summer of 1 672 , he spent two week s with " friends" at Stoke Park nea r To wcester, No rthampton­ shire . The re i s no o ther kno wn referen ce in hi s life to Stoke Park or these friends . Between 1 668 and 1 677 , Newton went to London at lea st fi ve times . We kno w very little about the visits , where he stayed and who m he saw , except th at in 1 669 he retu rned lo aded with alchemical gear. In Febru a ry and March 1 675 , it is true, he attended the Royal Society three ti mes and he received visits fro m members at his inn . L ater in the yea r , in j u stifying his condu ct vis-a-vi s Linu s , he recalled an occasion when Ol denburg arrived and fo und an unidenti fied "Gentle man" the re , fro m the context clea rly not a member of the Royal Society . 3 0 Perhaps thi s gentle­ man rather than Ezekiel Foxcro ft w as the " Mr F . " who gave New­ ton a manuscript to copy in 1 675 � perhaps he was the " W . S . " who gave it to " Mr F . " Thi s is only speculation , of course . It is not specula tion that Newton had alchemical manu scripts whi ch he mu st have recei ved from so meone since they did no t , I believe , material­ ize ou t of air. Nor i s it speculation that about 1 669 Newton began to read exten­ si vely in alchemical literatu re . His notes on the reading su rvive , the hand not datable with p recision but unmistaka bly from the general peri od of the late 1 660s and perhaps 1 670- 1 . In her recen t stu dy of Newto n ' s early alchem y , M rs . Dobbs asserts that Newton p ro bed " th e whole vast literatu re of the o lder [i . e . , p re- s even teen th century] alchemy as it has never been probed befo re or since . " 3 1 H e also studied seventeenth-centu ry alchemists , especially Sendivo giu s , d ' Espagnet , and Ei renaeus Philalethes , with equal intensity . Much of Newton ' s attention to alchemy came later. I have carried out a rather carefu l qu antitati ve study of the alchemical manu scripts he left behind, in which I divide them into three chronological groups . Of the to tal , which I estim ate to inclu de well over a mill ion words devo ted to alchem y , ab out one-sixth appear to stem fro m the pe­ rio d before 1 675 . 3 2 In his u sual fa shion , Newton purchased a note3 1 Dobbs , Foundations , p . 88 . 3 0 Newton to Oldenburg, 1 3 Nov . 1 675; Corres 1 , 357 . 3 2 I have had to rely mostly o n N ewton' s hand writing in dating his alchemical manu scripts . There i s some internal evidence, su ch a s the dates i n Keynes MSS 3 3 and 50 , and Mint bu siness on Keynes MSS 13 and 56 , to supplement its evidence . Esp ecially important are the pu blication dates of books cited , though one must collate page references with an actual work to be sure Newton was using it instead o f a manu script . The work o f Theodore Mundanu s , D e q u intessentia ph ilosophorum (1 686) , o ffers maj or ass istance .

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book in which he entered twelve gene ral headings and a number of subhead ings under which to org ani ze the fruits of his read ing ­ headings such as " Conj u nctio et liquefactio , " " Regimen per ascen­ sum in Caelu m & descensu m in terram , " and " Multiplic atio . " In thi s case , he did not carry the plan out beyond a s m all nu mber of entries . 33 His later Index chemicus would supply its lack on a heroic scale . M eanwhile , the reading proceeded ap ace . I have already indic ated the copies he made from the bo rrowed sheaf of alchemic al treati ses , the copy of a treatise by P hilalethes , and notes on ano ther. 34 He also studied Philal ethes ' Secrets Reveal 'd ( 1 669) , his heavily anno tated copy of which su rvives . 35 Very early, he read Sendivogiu s ' No vum lumen chymicu m , on which he took at least two sets of notes . The second set also inclu ded no tes on d 'Esp agnet ' s A rca n um hermeticae p h ilosop h iae opus . 3 6 Al ready he was more than merely a reade r. He divided the p ages of the second set of notes with a vertical line . On one side he entered notes fro m Sendivogius; on the other he made a running co mmentary , which fre quently translated Sendivogiu s ' i m agery into the language of laboratory chemicals , and which also pointed out Sendivogius' defi­ ciencies on occasion . At much the same ti me , he read Michael Maier' s Symbola aurea e mensa e duodecim (on the notes of which he drew fo r his lette r to A ston in May 1 669) , the Op era of the med ie­ val English alche mist , Geo rge Ripley , and Basil Valentine ' s Tri umNewton cited it extensively ; in light of Newton' s other activities in the period 1 686-9, I have taken a referen ce to Mundanus to place a paper in the 1 690s . I tried to introduce a rough qu antitative scale . As a unit of measurement, I used a single page of standard size written in a small han d . Newton wrote the great bulk of his alchemical papers on sheets folded twice to make eight pages about seven inches by nine . Four such pages in a s mall , early hand , chosen at rando m , averaged 750 word s . Since I realized how rough the measure was in any case , I did not spen d fu rther time counting. When Newton wrote in a hand of mediu m size , I converted pages to units by redu cing the nu mber by one-third ; when he used a large han d , I redu ced by one-half. When he wrote on folios, I multiplied by two . I think I understand the limitations of the figures I obtained as well as the reader will . Among other things , the measu rement treats every page of writing, whether it be the record of experiment s , the co mparison of alchemical authors, the copy of a manu­ scrip t, or Newton ' s o wn composition , as equal. Manifes tly , they involved greatly differ­ ing amounts of time. In order to get a rough measurement of his alchemical activ ity , I divided the papers into three chronological groups: early (to the middle of the 1 6 70s) , mid dle (from the late 1 670s to 1 687) , and late (the early 1 690s) . By my count, there are 1 , 443 units in the papers I ha ve been able to see . Using the indications of length in the Sotheby catalogue, I estimate that there are about 85 units I have not seen , or a total of 1 , 528 units , or nearly 1 , 200, 000 words . I have not in cluded the closely allied material in drafts of some Queries for the Opticks, which would swell t�e total considerably . Of those I counted , a little under 20 percent belonged to the early period , roughly 30 percent to the middle, and roughly 50 percent to the late. 34 Keynes MSS 62 , 52 , 5 1 . 33 Yah uda MS Var. 260 . 35 University of Wisconsin Library. 36 Yah uda MS . Var . 259, Keynes MS 1 9 . Cf. also Babson MS 925 .

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phal Chariot of Antimony . 37 Notes on van Hel mont , who did not exe rci se much influ ence over Newton, date fro m a few years later, perhaps 1 674 or 1 675 ; additions to these notes about ten yea rs later suggest ag ain how reading notes did not become dead repo sito ries but functioned as l iving parts of continuing investigation s . 38 Notes on individual authors fail to convey the extent of his early reading , ho wev er . At one point , he made a l ist of the most important items in the six-volume Theatntm ch em icum . 39 On a fol ded sh eet with the ti tle , " O f proportions , " th at i s , of the proportions of ing redients in the Work , he cited th e opinions of nineteen separate au thorities such as Morienu s , Hermes , Thomas Aqu inas , B acon , R o sinus , the Tu rba ph ilosophontm, the Scala, and the Rosa ry . The list drew , not only on th e Th ea tntm ch emicu m , but also on the o ther two g reat collecti ons of alchemical literature then available , the three-volume A rs au riferae and the Musa eu m hermeticum . 4 0 Already he had begun what would becom e one of the great private collections o f alchemi­ cal literatu re . When he died , his library contained one hundred and seventy-five alchemical books plus numerous pamphlets not listed sep arately . Al chemical works made up about one-tenth of his total library . 4 1 Rather qu ickl y , as one would expect , he develo ped stan­ dards o f j u dgment. He cro ssed out one short passage among his early notes fro m the Th ea tmm ch emicum and dis missed it summarily: " I believe th at this author is in no w ay adept . " 4 2 Always striving to extract the general fro m the specific , Newton set out at once to redu ce the multitudinou s testi mony of his reading to the one true process . He drew up a set of forty-seven axio ms th at su mmarized the Work with references to th e au thorities on which he based them . 43 He also began to look for common referents behind the manifold extrav agant imagery of his sources . Concerning Magnesia o r the green Lion [he wrote in a list o f Notae, which included s i m ilar expl ications o f other alchemical ter m s] . It is called p ro metheu s & the Chamel eon . Also And rogyn e , an d virgin verdant earth in which the Sun has never cas t its ray s althoug h he is its father an d the moon its mother: Also co mm on me rcu ry , dew of h eaven which makes the earth fertil e, nitre o f the wi s e . In stru ct de A rb . sol . [Instructio de arbore solari] Chap . 3 . It is the Satu rnin e s ton e . 44 37 Notes on Maier: Keynes MS 29 and a manuscript in the library of St. Andrew's Univer­ sity. There are later notes on Maier in Burndy MS 1 4 , and from the 1 690s v ery extensive notes in Keynes MS 32 . Notes on Ripley: Keynes MS 1 7 and in the third , fi fth, sixth, and seventh item s of the manus cript in the Countway Medical Library o f Harvard Universit y . Notes on Basil Valentine: Keynes M S 6 4 ; British Library , Add M S 44888 . 8 3 Eighth item in the Countway MS; the notes are not solely fro m van Helmont . Later notes 4 0 Ibid . , second item, ff. 5-6v. 39 Ibid . , third item, f. 1 0v . on ff. 2 1 , 24\' . 1 4 S ee the list of the library in John Harrison , The L ibrary of Isaac Newton (Cam bridge, 43 Countway M S , fourth item . 4 2 Yah uda MS 259 , no . 9 . 1 978) . 44 Ibid . , third item , f. 7 . Cf. Keynes MS 1 9 .

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Similar but m uch longer passages , in which he listed as many as fi fty different i m ages fo r the same ingredient o r p rod uct o f the Wo rk , were to be common features of Newton 's later alchemi cal m anu scripts . 45 Whatever else alchemy meant to him , Newton was always con­ vinced that the treatises he read referred to changes that m aterial substances undergo . It was hi s goal to penetrate the j ungle of luxu­ riant i mag ery in o rder to fi nd the process common to all the g reat expositi ons of the A rt . To assert as much is not to say that the chemistry he pursued would have been acceptable in the scientifi c academies of hi s day , or that scienti sts of the tw entieth centu ry would be willing even to recognize it as chemi stry . Nevertheless , h e did understand that chemical processes , not mysti cal experience exp ressed in the idiom of chemi cal pro cesses , fo rmed the content o f the A rt . Thu s h i s reading in the li terature o f alchemy p roceeded hand in hand with laboratory experimentation . The progress m ade of late in penetrating the maze of his alchemi cal endeavor has rested on the co rrelation of his su rviving experimental notes with the alchemical m anu scripts . 4 6 Most o f the experimental notes derive from 1 678 and after. There are some undated ones in hi s chemi cal notebook , ho wever , which seem definitely to have belonged to the late 1 660s and early 1 670s . His earliest experiments , based on B oyle and showing p erhaps the i nfluence of Michael Maier as well , attempted to extract the mer­ cury from v ariou s metals . In the intellectu al world of alchemy , mercury - not common quick silver, but the mercury o f the phi­ losophers - was the common . fi rst m atter from whi ch all metals were fo rmed . Liberating it fro m its fi xed fo rm in metals , cleansing 45 "Now till s g reen earth is the Green Ladies o f B . Valentine ye beautifully green Venus & the g reen Venereal Emrauld & green earth of Snyders w1h wc h he fed his lunary � & by vertue o f wc h Diana was to bring forth children & out o f w h saith Ripley the blood o f ye green Lyon is dra wn in ye be gi nning of ye work. Nam viridis et vegetabilis nostri argenti vi vi s ubstantia est Basilisci prulosopru ci pabulum saith Mundanus p 1 80. The spi rit of this earth is f fire in wc h Pontanus digests his feculent matter, the bloo d o f infants in wc h ye 0 & J) bath the mselves , the u nclean g reen Lion wc h , saith Ripley, is ye mean o f joynin g ye tinctu res of 0 & J) , the broth �h Medea poured on / two serpents , the Venus by mediation o f wc h 0 vulgar & the � o f 7 eagles saith Philalethes must be decocted, & in wc h ye same � digested alone will give you the Philosopruc Lune & 0, & the Spirit of Grasseus where he saith y1 in the via humida the 4 Elem 1s become b y steps one � & this � is divi ded into 0 & � w h must be reconjoyned by mediation of ye S pirit to give a third thing" (Keynes MS 46 , f. 1 ) . Thi s passage was composed in the 1 690s ; cf. Keynes MSS 46, f. J V ; 38 , ff. 4- 4v , 9v ; 40, f. 23v ; 56, ff. 7, 12, 1 2v ; and Burndy MS 1 5 . I thi n k I could find hundreds of others . Especially see nu merous entries in the Index chemicu s (Keynes MS 30) . All o f these passa ges dated from the 1 690s ; they were the fruit o f more than t wenty yea rs of alchemical stu dy . The early one, cited in the text , indicates that Newton's stu dy aimed in this direction from the beginning . 46 Dobbs, Fo undations, a n d Figala , " Newton as Alchemist. "

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it of contaminating feces , was equivalent to vivifying it and making it fit for the Work . The two im ages found here , images of puri fica­ tion and of vivification , which included generation by male and female , perv aded the al chemical literature th at Newton read . New­ ton ' s first method of extracting mercury involved the dissolution of quicksilver in aqua fortis (nitric acid) . When he added lead filings to the solution , a running 1nercury , which he took to be the mercury of lead , was released and a white powder precipitated . He did it with tin and copper as well , b ut he did not entirely like what he s aw . " I know not whither y1 � come out of ye liquor or of