John Herivel - The Background To Newton's Principia (1965, Oxford University Press).pdf

T H E B A C K G R O U N D TO Although the scientific and matticmatical manuscripts of the Portsmouth Collection have been available to scholats for almost a centuryj it b only comparatively recently that they have begun to be used for the study of the development of Newton’s thought in Physics and Mathematics. The present i»ok is thus ilie firs: to be devoted exclusively to the develop^ ment ofNewton^s dynamical thought leading up to the composition of the FriTidpia. Part T contains studies of various aspects of this development* Part II the text of the manuicripts from the Portsmouth Collection on which these studies are based. Cer­ tain of these manuscripts, especially the ver^^early ones on ci rc ular mo lion* provide an insight into the nature of Newton’s genius in dynamics W'hich it would he impossible to guess at from the finished form of the Frini:tfija itself.

NEWTON’S P R IN C IPIA

,: r

THE BACKGROUND TO NEWTON’S PK/NC/PM

.

X

»

^ Study of Newton's Dynamical Researches

A

\

nuxl-'tt^

^4 f '*J

^ |^lW i^■ i^lllgi■i

I'Ji^aii. On.lA /■ *«/»»«/ #« i»4(n fftn/^t ».. .* *pu§fm. •'Tf r »■;•» . S M . i nrit r t i t

y -

/

m itm -n r

r i'( E f :«

'{^,

BY ............................ .

^/.

^’ ■ /

f e « * » » - C ‘»

fd o ti

«#»■>/,J».'

4 1 :ft f t / e t d ' f »v,--

^

H f ■#jif ^NMI ^ < ■!•*

><-r »9 :

■\

..fvJinr*

m n*4> m

■ ,/./

ft : f - f d f ih.

in the Years 1664-84

fy jfd tr r ff * /f

»<■«/>,-/••I*

JOHN H E R I V E L

' ■

^

*/*»■»

••■(^^'*^

'-

,

/ f f. t/> I f r'm.

y-

■.

,

/ y // .f^ .

•

■'

^

d **

f

rtf

/>’

. ■ ■ ■ ‘ r^rf S"

f d f t t / ’.

r t

"^.

0 *~

/ '

,.y,„ r. / /<^/.«'ie; fr^ 4 f„^y,.,^ '-jd /wtJhyfi t'X ’f'f y.-,pf:.,/'X

f ')t*ift:^'/f.>tf(

( e>ft^ tt >t ( /» ^)f .i / ¥ e />t f

> «. « | { « I m W !1 * • : « « ; / ■ «

/ ( > / ^ ”’ f

^

y

/ «• r < ^ / ^ r T

A i itf j f

•f fp'Ptfe^'S I

,.

-J'f/^ t / 'Ityfff. (*<»•■ >'I'* d^ef /^ ^ r r f.fM

>.f L i» S * l‘

'*/

/

• ■.

✓

f^4yX

.'^ trv ,.

Z^V A«^^vy

3^^ v4 »

V. OXFORD lui^A^rMEl

^e-4t

f

t

/■ *"»T~^*f

A T T EI E C L A R E N D O N 1 965

I. T h e first page o f the so-called lectures ife M o tii, 1684

PRESS

Oxford University Press, Ely House, London W . t (GLASGOW CAPE TOW N BOMBAY

NKW YORK SALISBURY

TORONTO IBADAN

CALCUTTA MADRAS KUALA LUMPUR

MELBOURNE

WELLINGTON

NAIROBI LUSAKA ADDIS ABABA KARACHI

LAHORE

HONG KONG

Oxford University Press 1965

P R IN T E D

TN

G R F. A T

B R I TA 1 N

PREFACE

DACCA

I n the summer of 1958, while preparing a course of lectures on the his­

tory of dynamics from Copernicus to Newton, I was struck by how little seemed known of the grozvth of Newton’s dynamical thought. It was surprising, for example, to find that almost more seemed to be known of the growth of dynamical thought among Newton’s precursors in Oxford and Paris than in the case of Newton himself. Although this first im­ pression was somewhat modified on reading Rouse Ball’s Essay on Newton's Principia, I was left with the impression that much still remained to be found. In particular, I noticed how seldom Ball referred to aetual documents in the Portsmouth Collection, and this despite the fact that the collection had been catalogued some years previously, and the mathematieal and physical papers deposited in the University Library, Cambridge. For example, while I was struck with Ball’s sugges­ tion that the proof given at the end of the Scholium to Prop. 4, Theor. 4, of the represented Newton’s first discovery o{ the law of centrifugal force, I was surprised he made no mention of having searched for the original document in the University Library. At the first opportunity, then, in April 1959, I visited Cambridge to search for this and other early documents. A study of the dynamical papers among the early mathematical docu­ ments sufficed to confirm the correctness of my original impression, and this was then greatly strengthened by an examination of the Waste Book Feb. i66p. Here was a wealth of documentary evidence far beyond anything hinted at in Rouse Ball’s Essay, and it seemed to me then that if one could interpret and order this evidence it might be possible to pre­ sent some sort of connected account of the growth of Newton’s dyna­ mical thought prior to the composition of the Principia and thus help fill up one of the large gaps in the history of dynamics. The reader may judge to what extent this aim has now been realized. I should like to express my indebtedness to those who in one way and another have made this work possible. To E. Ashby, D. R. Bates, and W. B. Gallie for enabling me to enter the field of the History of Science professionally in the first place. And to W. B. Gallie again and G. Davie for their encouragement during my first year as Lecturer in the History and Philosophy of Science at Queen’s University. T o the late H. W. Turnbull for forbearing to tell me that Newton’s Vellum

PREFACE

MS had been already deciphered, thus enabling me to get to grips with Newton’s earliest researches in dynamics. Also for Turnbull’s criticisms of my own provisional solution to this manuscript, and to those of a referee in the journal Isis which taught me the necessity of inter­ preting Newton in terms of his own approach to dynamics as opposed to the present-day reformulation of that subject. In common with other historians of science I have been indebted to the late Alexandre Koyre for the inspiration of his writings. I was also very fortunate to enjoy his friendship and encouragement, especially the opportunity of lecturing under his auspices on Newton in Paris in 1961. I am particularly indebted to two other colleagues; to I. B, Cohen for first suggesting that I write this book, and for his continued interest in its composition; and to A. C. Crombie for reading an early draft and encouraging me to persist with others. Also to E. S. de Beer for criticism and advice on the reproduction of texts and to E. W. Bower for help with translations from Latin. The research on which this book is based would have been impossible without generous grants from the late Research Committee of the Senate and the present Research Committee of the Council of the Queen’s University, Belfast. Thanks are due too for typing assistance to Margaret Sheridan, Freda Silcock, Rosemary Clarke, and particularly Ann Greer who bore the brunt of the many typings and retypings of earlier drafts. I must finally acknowledge my indebtedness to the staff of the Anderson Room of the Cambridge University Library, especially F. J, Gautrey, for their unfailing courtesy and assistance, and to the Clarendon Press for much sound advice and for their skill in printing a difficult text. Most of all to my wife and children for their continued forbearance over a book which would never be finished. J. H. Department of History and Philosophy of Science The Queen's University Belfast September, ig6j

ACKNOWLEDGEMENTS I AM grateful to the Librarian of the University of Cam­ bridge for permission to reproduce pages from manu­ scripts in the Portsmouth Collection, to Flermann et C ‘°, Paris, for permission to quote in extenso from my article ‘Galileo’s Influence on Newton in Dynamics’ in Melanges Alexandre Koyre (Paris, 1963), and to the Editors of Isis and the Archives Internationales de VHistoire des Sciences for permission to use articles from their respective journals. J. FI.

CONTENTS List of Plates

XI

General Introduction Short Titles and Abbreviations

XV I

PART I newton’s dyn am ical

researches,

1664-1684

1. The main line of development of Newton’s dynamical thought 1.1 Newton’s first steps in dynamics 1.2 Newton before the problem of circular motion 1.3 The solution to the problem of Kepler-motion 1.4 The final synthesis of Newton’s dynamical system

7 14 22

2. The influence of Galileo and Descartes on Newton’s dynamics 2.1 The influence of Galileo 2.2 The influence of Descartes

35 35

3. Newton’s concept of conatiis

54

4. Tests of the Law of Gravitation against the moon’s motion

65

5. The motion of extended bodies 5.1 Kinematical aspects 5.2 Centre of motion in the dynamical sense 5.3 Dynamics of a single rotating body

77 77

5.4 Collision between two rotating bodies 5.5 Discussion in later researches

82

6. Order of composition and dating of manuscripts 6.1 Order of composition of earliest manuscripts 6.2 Dating of earliest manuscripts 6.3 Order of composition of later manuscripts 6.4 Dating of later manuscripts 6.5 Paget’s propositions 6.6 The status of manuscript VIII

42

78 81 84 87 87 91

93 96 102 108

CONTENT'S PART II NEWTON D Y NA MI C A L MANUS CRI PTS ,

L I S T OF P L A T E S

1664-1684

Introduction I. Extracts from early notebook II. Dynamical writings in the Waste Book III. The Vellum Manuscript IV a. On circular motion

ivb. On motion in a cycloid V. The Laws of Motion paper VI. Extracts from MS. Add. 4003 VII. Extracts from correspondence prior to 1684 VIII . The Kepler-motion papers

119

I.

121 128

2.

facing p. 12

3-

ition

183 192 198 208 219 236 246

IX. The tract de Motu xa. Drafts of definitions and laws of motion

257 304

xb. Drafts of definitions XI. The lectures de Motu of 1684

315 321

Note regarding Previous Publication ot Newton Dynamical Manuscripts

327

Bibliography

328

Index

331

13

4-

184

5-

292

GENERAL INTRODUCTION W i t h certain notable exceptions,^ the whole history of dynamics from

We shall, I think, in this as in other subjects, get the best view of the matter if we look at the natural growth of things from the beginning. Aristotle, Politics, Book 1, Chapter 2 , Section 2 Translated b y T . a . S i n c l a i r

Newton to Einstein can be thought of as an exploitation, albeit in­ finitely ingenious and resourceful, of the definitions, principles, and propositions in the Principia. It is hardly surprising, therefore, that Newton’s achievement in dynamics has largely been equated with the Principia itself, an opinion reinforced by Newton’s own claim^ to have composed the whole work within the space of a year and a half apart from a handful of propositions discovered in 1679 and 1684. Admittedly, some indication of much earlier work in dynamics epitomized by the story of the applet has come down to us through the accounts of Pemberton^ and Whiston,5 and Newton’s own account^ of his researches in mathematics and ‘philosophy’ during the Plague Years. But this early work in dynamics, if such there were,^ and whatever its exact nature and extent, was inevitably so dwarfed by the towering achievement of the Principia that it has seemed till recently of quite negligible interest and importance. In the nineteenth century a certain shift in this attitude towards the Principia first became noticeable. Rigaud’s Essay on the Publication of Newton^s Principia,^ though largely concerned with the events im­ mediately preceding the appearance of the great work, touched also on some earlier researches of Newton, especially the important tracts de Motu published for the first time in an Appendix to Rigaud’s Essay. Edleston, in his Correspondence of Newton and C o t e s , added some further important details, notably the presence in the University Library, * For example, the theory of rotating bodies. Curiously, when Newton came to write Book III of the Principia he either ignored, or had forgotten, the early, and particu­ larly brilliant work on rotating bodies found in M S . V. ^ Quoted in Brewster, D . [i], vol. i, p. 471. 3 See for example Stukeley, W . [i], p. 19. Attention seems first to have been drawn to this very important account of the story of the apple by Pelseneer, J. [2]. T h e relevant passage is reproduced below on p. 65, n. 7, Chapter 4. * Pemberton, H. [i]. Preface. T h e relevant passage is reproduced below at Chapter

4. P- 6 5 . ^ Whiston, W . [i], pp. 35-38. T h e relevant passage is reproduced below at Chap­ ter 4, p. 65. ^ Portsmouth D raft Memorandum, Portsmouth Collection, Section i, division X I, number 41, as reproduced below at Chapter 4, p. 66. ’’ e.g. Patterson, L. D ., [i], [2] has questioned the veracity of N ew ton’s account. * Rigaud, S. P. [i]. '' A term first used, apparently, by Rigaud. Edleston, J. [i].

xiv

GENERAL IN T R O D U C T I O N

GENERAL I N T R O D U C T I O N

Cambridge, of the original texts of all Newton’s university lectures be­ tween 1669 and 1687,' especially those apparently read in the Michael­ mas term 1684 under the title de Motu Corporum. Brewster, in his Memoirs of Newton,^ though concerned with the whole sweep of New ­ ton’s life and work, and not only his researches in mathematics and dynamics, drew attention to some important new material relating to dynamics.^ Finally, in Ball’s Essay on Newton’s Principia'^ the whole question of Newton’s researches in dynamics prior to the composition of the Principia came much more to the fore. Admittedly, Ball’s most important contribution was his discovery of the greater parts of the correspondence of 1679 between Hooke and Newton. But apart from invaluable chapters on this correspondence, and on the tract de Motu, Ball also had a chapter devoted to Newton’s earliest dynamical researches. Since Ball’s Essay various studies of Newton have appeared, notably the biographies of More,^ Sullivan,^ and Andrade,^ and Cohen’s study of Newton and Frankl i n. None of these works, however, is exclusively, or even predominantly, concerned with Newton’s researches in dynamics, so that up to date Ball’s Essay represents the only study of this aspect of Newton’s intellectual genius. Ball’s Essay, however, in spite of its unique status and many valuable features, suffered from one serious defect: although composed after the cataloguing of the Portsmouth Collection, it contained surprisingly few references to original documents, especially the earliest ones. This defect may well have been due not to any de­ ficiency in Ball qua historian of science but simply to lack of time.^® On the other hand, there are certain indications that Ball had not entirely freed himself from the prevailing attitude towards the Principia. For example, he finally dismissed one important question relating to the early researches as being ‘mainly a matter of antiquarian interest’ ." Since the publication of Ball’s Essay there has been something approaching a revolution in the attitude of historians of science towards dynamics. Beginning, perhaps, with Duhem’s celebrated researches into

late medieval dynamics, increasing emphasis has been placed on the filiation and growth of dynamical thought as opposed to the final pro­ duct of such thought. So that today the importance of the study of the growth of Newton’s dynamical thought would no longer be questioned. In the absence of original manuscripts, however, this study would in­ evitably be largely a matter of speculation eked out by the few meagre details supplied by the above-mentioned accounts of Pemberton, Whiston, and Newton himself. Fortunately there are a considerable number of ‘early’^ Newton dynamical manuscripts in that part of the original Portsmouth Collection now lodged in the Cambridge Univer­ sity Library. Although many of these manuscripts have already been published in part or in whole, almost half Newton’s early dynamical writings, especially those in the Waste Book, have remained unpublished. These latter writings are now made available in company with those previously published, so that Newton scholars and others interested in the early development of Newton’s dynamical thought have here before them what is to my knowledge the whole extant corpus of Newton dyna­ mical manuscripts prior to the composition of the Principia. Some explanation is called for on the structure of the present work. Part I consists of an extended commentary on the manuscripts repro­ duced in Part II. In the first chapter an attempt is made to give a con­ nected account of the main line of development of Newton’s dynamical thought up to the final draft of Book I of the Principia. The remaining chapters of Part I are devoted to detailed studies of various topics which could not be incorporated directly in Chapter i, including a final chapter on the order of composition and dating of manuscripts. The method followed in the reproduction of texts is explained in the introduction to Part II, especially the division of footnotes into critical and exegetical. Exegetical footnotes relating to matters allowing of a potentially wide variation of opinion, such as those relating to the growth of Newton’s dynamical thought, have been mostly restricted to a direct reference to the appropriate section of Part I, or to such a reference preceded by a brief, unsupported expression of opinion. In this way it is hoped that other scholars anxious to form their own opinions on the many un­ certain questions raised by these manuscripts will be free to do so un­ biased by the arguments advanced in Part I.

' Apart from those for 1686 which appear to he missing. See below, Chapter 6.4, pp. 98-102. ^ Brewster, D . [i]. 3 For example, the document referred to on p. xiii, n. 2. ^ Ball, W . W . R. [t]. 5 T h e remainder of the correspondence has since been discovered by Pelseneer and Koyre. See especially Koyre, A . [2]. More, L . T . [i]. ’ Sullivan, J. W. N . [i]. ® Andrade, E. N . da C . [i]. ^ Cohen, I. B. [i]. See the remarks at the end of first paragraph on p. i of Ball, W. W. R. [i]. ” Ibid., p. 17, end of top paragraph.

* i.e. prior to the composition of the Principia.

xv

PART I

SHORT T I T L E S AND ABBREVIATIONS (a) P U B L I S H E D W O R K S Correspondence

The Correspondence of Isaac Newton (Cambridge Univer­ sity Press, for the Royal Society of London, vols. 1-3, 19 59 ,

i9 6 0 ,

NEWTON’S DYNAMICAL RESEARCHES 1664-1684

19 6 1).

(Euvres de Descartes publiees par Charles Adam et Paul Tannery, 13 vols. (Paris, 1897-1913). Dialogo Sopra i Due Massimi Sistemi Del Mondo Galileo, Dialogue (Florence, 1632). Discorsi e Dimonstrazioni Mathematiche Intorno a Due Galileo, Discorsi Nuove Scienze (Leyden, 1638). Le opere di Galileo Galilei, Edizione Nazionale, 20 vols. Galileo, Ed. Naz. (Florence, 1890-1909). Huygens, Horologium Christiani Hugenii Zulichemii, Const. F. Horologium Oscillatorium (Paris, 1673). (Euvres completes de Christiaan Huygens, 22 vols. (The Huygens, (Euvres Hague, 1888-1950). Descartes, (Euvres

{b) M A N U S C R I P T S C.U.L. Vellum MS. Waste Book

Cambridge University Library. Newton MS. (C.U.L. MS. Add. 3958, folio 45). Newton MS. (C.U.L. MS. Add. 4004).

1 T H E M A I N L I N E OF D E V E L O P M E N T OF NEW TON’S D YN AM ICAL THOUGHT 1.1. N e’WTon ’s F irs t

S teps

in

D ynamics

N e w t o n ’ s interest in dynamics would inevitably have awakened very early, and the boy who constructed a mill turned by the wind or the force of a mouse, or a water clock to measure the time,^ or the youth who attempted by jumping to gauge the force of the great storm of 1658^ would already have begun to ponder the two basic concepts of dynamics — force and movement— ^though it is improbable that he would have had any contact with the subject on the theoretical side before his entry to Cambridge in 1661.^ At some stage in his undergraduate studies Newton would inevitably have encountered Aristotle’s theory of local motion directly or at second hand through a work such as that of Magirus [i] from which extracts are found in one of the early notebooks. In any case the passage ‘On Violent Motion’ in the same notebook^ proves that he had already begun to free himself from any such Aristote­ lian or scholastic influences by the second half of 1664.5 For in spite of the unmistakably medieval flavour of the writing, the upshot of the argument was to reject the possibility of the motion of a projectile as due either to the surrounding air (Aristotle), or to a force impresst (im­ petus school), and to And the cause of the continued motion of the body in its natural gravity, that is, presumably, in its inertia. References to Galileo^ and Descartes'^ close to this passage ‘On Violent Motion’ suggest that it was under their influence that Newton began to develop his own • See Brewster [i], vol. i, p. 9.

^ Ibid., p. 15. M S . I, § 2. 5 See below, Chapter 6.2, for question of dating of earliest manuscripts. * M S . I, § 6. ’’ See, for example, M S. I, § 3. 3 Ibid., p. 17.

8i)S205

B

MAIN LINE OF DEVELOPMENT OF

1.1

views on dynamics.^ This development was evidently very rapid, for an entry in the Waste Book^ dated 20 January 1664 (O.S.) discloses a recognizably modern quantitative approach to the problem of the inelastic collision of two bodies. This entry, like others in the Waste Book and elsewhere among the earliest dynamical manuscripts, is remarkable for the confident way it deals with the kinematical side of the problem. It is probably impossible to disentangle the various influences which went to shape Newton’s approach to kinematics. He could well have been influenced by Aristotle himself, or by certain of the late medieval thinkers of the Oxford school such as Calculator ,3 or by Galileo directly, or indirectly through Torri­ celli or Barrow. Also his interest in kinematics (and later in dynamics) could have arisen out of his researches in fluxions or vice versa .4 In any case, kinematics evidently presented no problem to Newton from the Waste Book onwards, and at various points in the earliest dynamical manuscripts we find clear definitions of movement as the passage of a body from one point of extension to another, or change of place, velocity being the intensity of motion and proportional to the distance covered in a given time. With one exception the measurement of velocity always involved comparisons between the velocities of two different bodies, or of different velocities in the same body, thus avoiding the ‘forbidden’ division between the two dissimilar quantities space and time.^ This was equally true of other quantities such as momentum or force, a typical example at a much later stage being provided by the enunciation of the law of centri­ * See below, Chapter 2, for the influence of Galileo and Descartes on Newton in dynamics. 2 See M S . Ilb . 3 Boyer [i], p. 194, notes that the terms fluxion and fluent employed by Newton had appeared earlier in the work of Calculator. C f., for example, Prop, i and 2 of the paper ‘T o Resolve Problems by M otion ’ {Correspondence, vol. iii, no. 348) with §§ 2, 3, respectively, of M S . V. ^ T h e notion of a proportion between dissimilar quantities was unthinkable to both Greek and medieval philosophers. For example, Bradwardine defines proportion as ‘the mutual relationship of two things of the same kind’, Clagett [i], p. 465. And although the Merton College kinematicists ultimately achieved a perfect definition of instantaneous velocity in the case of non-uniform motion, it is evident that they still thought of this velocity not as a ratio but as representing the distance which would have been covered by the body in a certain time, ibid., p. 167. T h e notion of velocity as a separate quantity is evident in the section on ‘Uniform M otion’ in the Third D ay of Galileo’s Discorsi, where he goes so far as to represent individual speeds by lines of diflferent lengths. But it is still a question always of comparing speeds, and neither in this section nor elsewhere in the Discorsi does one find a definition of an individual speed as a ratio o f distance to time.

1.1

N E W TO N ’ S D Y N A M I C A L T H O U G H T

3

petal force for circular motion in Prop. 4, Theor. 4, of the Principia. The unique exception to this comparison rule is found in the note to Axiom 26 of the Waste Book.^ There Newton considers a single body, and after noting that in the case of uniform motion the distance and time may be represented by two of the sides of a right-angled triangle, he proceeds to state that in the general case of non-uniform motion a crooked line will result from plotting space against time, the (instanta­ neous) velocity at any point being given by the inverse gradient of the space-time curve. This brief note must be regarded as representing, simultaneously, the hidden culmination^ of the train of thought which went back to the definition of instantaneous velocity by the Merton College school in the first half of the fourteenth century,^ and, for New­ ton, the dividing of the ways: if he had followed the line indicated in this note it would inevitably have led him to the construction of a dynamics based on analytical geometry and the method of fluxions as opposed to the actual development based on synthetic geometry and the method of prime and ultimate quantities expounded in Section I of Book I of the Principia. The long and difficult process of ‘transformation’ and reformu­ lation of the results and methods of the Principia in the first half of the eighteenth century might then have been avoided. Almost inevitably the problem of collisions would have been that to which Newton first applied his exact, quantitative definition of velocity. It is probable, too, that his first ideas on this problem were based on the discussion given in Part 2 of Descartes’s Principia Philosophiae including the fundamental definition of the motion of a body as jointly propor­ tional to its size and speed.^^ If so, Newton must have very soon corrected the error in Descartes’ law of conservation of motion. For in the dis­ cussion of inelastic collisions in the Waste Book^ we find that for Newton, unlike Descartes, the unchanging total quantity of motion was the alge­ braic sum of the separate motions taking account of the directions or determinations^ of the movements of the individual bodies. In the case of inelastic collisions all that was required for a complete solution was the size of each body and the magnitudes and directions of their respective velocities before the collision. To find the magnitude and direction of the velocity of the composite body after collision was ‘ M S . Ild . ^ Assuming, of course, that Newton was not forestalled in this definition of velocity as a ratio by some previous writer such as Barrow. ^ See, for example, Clagett [i], Chapter 4, especially §§4, 5. ♦ Op. cit.. Art. 36. 5 See M S . Ilb . * See M S . lie , Def. 4.

M A I N L I N E OF D E V E L O P M E N T O F

1.1

then a matter of elementary algebra. Noteworthy, however, and very characteristic, is the systematic way in which Newton allows for all possible cases. Of greater interest is the discussion in the Waste Book of the direct collision of two elastic spheres.^ There the mutual reflection of the spheres was due to some ‘springing motion’ in the bodies or the matter crowded between them. And as the spring was more dull or vigorous so the bodies would be reflected more or less vigorously. During the actual process of collision the spheres would ‘relent’ from their spherical shapes and be pressed into ‘sphaeroidicall’ figures up to the point where they momentarily came to rest when the pressure between them would be greatest.2 But if the bodies were ‘absolutely solid’ the collision would take place instantaneously, no relenting from the spherical shape being possible.2 In the direct collision of two equal, perfectly clastic spheres moving with equal speeds in opposite directions the relative velocity of separa­ tion would necessarily equal that of approach.^ This would still re­ main true for all sizes and velocities of the bodies.^ No indication of the proof of the general result was given. Evidently it corresponded to Newton’s experimental law of collisions for the special case in which the coefficient of restitution equalled unity. Later it was employed in the remarkable solution to the problem of the collision of two rotating bodies.5 By the middle of the seventeenth century the centre of interest in dynamics in continental Europe had shifted from the problem of motion, to which it must have seemed that Galileo had provided the definitive solution in his Discorsi, to the problem of collisions, a topic to which Galileo had contributed little. Attention had been focused on this latter problem by its treatment in Part 2 of Descartes’s Principia Philosophiae. In England any previous interest in this problem would have been stimulated by Huygens’s successful application of his own theory of collisions before various members of the Royal Society, including Laurence Rooke, Wallis, and Wren, at the time of his visit to London in i66i.^ This theory, as we now know, had been fully worked out as early as 1656.7 Finally, in 1668, the Royal Society sought to bring to a * See M S . lid . Ax. 7-10 . 3 Ibid., at end.

^ M S . I Id, Ax. Q. M S . I Id, Ax. 10.

5 See M S . V , §§ 9, 10. * See Huygens’s CEuvres, vol. xvi, pp. 172-3, 181. 7 Ibid., pp. 10, 137, n. I.

1.1

N E W TO N ’ S D Y N A M I C A L T H O U G H T

head the various inconclusive discussions of previous years by inviting papers on the subject from Wren, Wallis, and Huygens.^ The true importance of the problem of collisions was very different from that which would have been imagined by the Scientific Establish­ ment of the day. It was, in fact, twofold: it provided material for the pro­ longed controversy between the supporters of Leibniz’s measure of motion, the vis viva, on the one hand, and the supporters of the Cartesian measure on the other.^ Even more important, perhaps, it provided an ideal testing ground for the earliest formal development of Newton’s concept of force. It is this latter aspect which concerns us here. There were, as we have seen, three basic elements in Newton’s treat­ ment of the problem of collisions in the Waste Book. First the assump­ tion (following Descartes) that the proper measure of the quantity of motion in a body was proportional jointly to its size and speed. Next Descartes’s law of conservation of total quantity of motion suitably emended as regards direction of motion. Lastly a perfect understanding of the physical process of elastic collisions with its momentary distortion of the colliding bodies. It was this last, physical, aspect of the collision process which figured in Newton’s first definition of force in the Waste Book as ‘the pressure or crowding of one body on another’.^ We can thus assume that the pressure between colliding bodies was one of the first kinds of force seriously considered by Newton. This pressure, however, had the grave disadvantage that it was impossible to assign any numerical quantity to mirror its extremely rapid variation from zero up to a maximum and down to zero again. In fact, in the extreme case of ‘absolutely solid’ bodies the process of collision was instantaneous.^ Newton’s powerful drive towards a quantitative treatment of dynamics would then have led him to search for a new quantitative definition of force. Hence the various Axioms in the Waste Book^ equivalent to the single definition/orre oc change in quantity of motion produced. We cannot be sure how Newton arrived at this fundamental definition of force. But it seems probable that it had some close connexion with the immediately preceding enunciation of the principle of inertia:^ * See Birch [i], vol. ii, pp. 315, 320. Huygens’s paper, transmitted to the Royal Society early in January 1669, was un­ accountably, and much to H uygens’s annoyance, not published in the Transactions like those of Wren and Wallis. Hence the paper ‘Regies du Mouvement dans la Rencontre des Corps’ appearing in thQ Journal des Scavans in March 1669. ^ For various aspects of this controversy see Costabel [i], Chapter 2; Dugas [i], Chapter 14, § 7; Dugas [2], Part III, Chapter 2. ^ M S . lie , Def. 9. M S. Ild , Ax. 9. 5 Ibid., A x. 4, 5, 6, 23. ^ Ibid. Ax. i, 2.

M AI N LIN E OF DEVELOPMENT OF

1.1

(1) If a quantity once move it will never rest unless hindered by some external cause. (2) A quantity will always move on in the same straight line (not changing the determination nor celerity of its motion) unless some external cause divert it. From this enunciation it followed that any change in the state of rest or motion of a body was due to some external cause. If this external cause were identified with force then force would be that which produced change in a body’s state of motion or rest. This is precisely the identifica­ tion found later in the Waste Book where force is said to be the power of the cause which changes a body’s motion.* But the measure of the force corresponding to a given change in velocity had still to he assigned. The simplest assumption would have been to measure it in terms of actual change of velocity. This, however, would have ignored the role of a body’s bulk epitomized by the fact that it is more difficult to induce a given change in the velocity of a large than a small body. Given the role of quantity of motion in collision pro­ cesses, Newton would inevitably have assumed that the measure of force depended in some way on the change in quantity of motion produced. The simplest assumption— force directly proportional to change in quantity of motion— would have been tried first, and would then have been found to possess the signal merit of ensuring conservation of quantity of motion in collision processes on the assumption that the forces between the colliding bodies were equal and opposite.^ In some such way Newton was led to his fundamental, quantitative definition of force. It represented the first important step towards his final achievement in dynamics. But at this point a formidable difficulty seemed to bar the way to any further progress: how to relate forces thus defined with the actual, physical forces encountered in Nature, such as the pressure between bodies, or the tension in a rotating sling. Newton’s ability to surmount this difficulty in the case of centrifugal force marked his first decisive breakthrough in dynamics and one which lifted him to a new plane of achievement in the subject only attained by Huygens^ besides. * M S . He, A x. 104. 2 M S . Ild , A x. 7, 8. ^ In the treatise D e V i Centrifuga, written by 1659. This, however, was first pub­ lished in 1703 among the Opuscula Posthuma. Various results based on the law of centrifugal force were published without proof in the Appendix to the Horologium Oscillatorium of 1673. See section on centrifugal force in Huygens’s (Euvres, vol. xvi, pp. 237-328. A detailed comparison between the respective approaches of Huygens and Newton to the problem of circular motion would be very valuable.

1.2

1.2.

N E W TO N ’ S D Y N A M I C A L TH O U G H T N ew ton

before

the

P roblem

of

C ircular

M otion

It is worth pausing for a moment to consider how fortunate the exis­ tence of uniform circular motion was for Newton, and how important his successful treatment of it for the whole future development of his dynamics. Apart from motion in a circle, the only relatively simple kinds of movement available for study by Newton were rectilinear, parabolic, and elliptical. The first two occurred in motion under gravity at the Earth’s surface, and had already been fully explored, at least in their kinematical aspects, by Galileo. Both bulked large in the growth of Newton’s dynamical thought, especially uniformly accelerated rectilinear motion, the paradigm case for all other more complicated motions. But neither of these motions admitted of any development of the concept of force. On the other hand, the elliptical motion discerned by Kepler in the unruly movements of the planet Mars was far too difficult and complex a case for Newton to treat first. In contrast, the problem of uniform circular motion was at once not impossibly difficult and yet of sufficient complexity to call for a real advance in his concept of force and his method of applying it to motion in a curved path. The first known discussion by Newton of the problem of circular motion is found at Axiom 20 of the Waste BookJ The case considered is that of a ball moving on the interior of a hollow spherical surface. Accord­ ing to the principle of inertia there is a constant tendency for the ball to continue on in the instantaneous direction of its motion at any point, i.e. along the tangent to the circle. And the fact that it does not so continue but moves instead in a circle argues the continuous action on it of a force. This force can only arise from pressure between the ball and the surface. But if the surface presses the ball, the ball itself must press the surface. From which it follows^ that all bodies moved circularly have an endeavour from the centre about which they move. This first discussion of the physical origin of centrifugal force bears evidence of Descartes’ treatment of the same subject in Part 2 of his Principia Philosophiae.^ Having thus attained to a clear physical under­ standing of the problem Newton was now faced with the task of provid­ ing it with an exact, quantitative treatment. His first tentative steps toward such a treatment are found at Axiom 22 of the Waste Book.^ There » M S . Ild . 3 See below. Chapter 2, § 2, section on circular motion.

® Ibid., A x. 21. M S . Ild , A x. 22.

MAIN LINE OF DEVELOPMENT OF

1.2

he notes that the whole force hy which a body co endeavours from the centre m in half a revolution is more than double the force that is able to destroy or create its motion, that is the quantity of its motion. For in moving from any point of the circle to the diametrically opposite point the ‘resistance’ of the surrounding surface (equal to the contrary pressure exerted by the body on the surface) is able to destroy the original quantity of the body’s motion along the tangent at the first point and create an equal quantity of motion in exactly the opposite sense at the second point. Two things are noteworthy here: first a small point of capital im­ portance. If a complete revolution of the circle had been considered no conclusion could have been drawn since the final state of the body would then have been the same as its original one. On the other hand, for half a revolution the position is entirely different. The line of action of the motion remains the same, certainly, but not its sense, and New­ ton, unlike Descartes, was well aware of the importance of sense or de­ termination in defining the motion of a body. Next there is the phrase ‘whole force . . . in half a revolution’. The exact sense of this phrase will become clear later. At present it need only be noted that it is a matter of the effect of a continuous force during an extended interval of time. In fact, without understanding the exact meaning of the argument its general drift is sufficiently clear, namely, that the whole effect of the force on the hall during half a revolution is exactly to reverse its sense of movement and consequently to change its motion by an amount equal to double its original quantity. Equally clear is the attempt to effect a connexion between force in the physical sense on the one hand, and force in the sense of change of quantity of motion on the other. In this ingenious way Newton was able to introduce number into his treatment of circular motion. Admittedly the result obtained was no more than approximate. It was a question not of an equality but of an inequality. The problem now was to improve on this inequality and replace it, if possible, by an exact result. As long as the circle itself was retained further progress was impossible. It was necessary, like Des­ cartes, but in a very different sense, to escape from the tyranny of the circle only to return to it again later. And that Newton was able so to escape was entirely due to the fact that he had replaced the stone in the Cartesian sling by a ball moving on an inner spherical surface. For now it was possible to imagine the body moving along a square inscribed in the circle as opposed to the circle itself, as on folio i of the Waste Book.^ I M S . Ila, § 2.

1.2

N E W TO N ’ S D Y N A M I C A L THOUGFIT

At each corner it would be reflected from one side into the next and so on ad infinitum. In this particular case an exact relation could be deduced involving the magnitude of the shock encountered by the ball at each corner of the square, namely: total sum of shocks at 4 corners force of movement of ball

sum of sides of square radius of circle

The same argument could be applied to regular inscribed polygons of any order invariably leading to the result: total sum of shocks at all corners force of movement of ball

sum of all sides radius of circle

Newton then continued: and so if [the] body were reflected by the sides of an equilateral circumscribed polygon of an infinite number of sides (that is by the circle itself) the force of all the reflections are to the force of the body’s motion as all those sides {id est the perimeter) to the radius.

The last proportion was evidently ztt thus explaining the cancellation of a 4-F elsewhere in the Waste Book^ and its replacement by a 6 + , that is to say ztt. In addition, it is evident now that the expression force of all reflections meant sum of all the forces of reflection in a complete revolu­ tion, thus explaining the earlier phrase of doubtful meaning the whole force . . . hi half a revolution. This replacement of an inequality by an exact result was a brilliant achievement. Nevertheless the result obtained in the form given is not very useful. For example, the sum of all the forces of reflection during a complete revolution is considered but no mention is made of the centri­ fugal force itself. Again, and even more serious, although the sum of the forces of reflection remains finite in passing to the limit of an inscribed polygon with an infinite number of sides all the individual forces tend towards zero! In addition these forces of reflection continually change their direction, so that it is rather doubtful if any valid physical sense can be attributed to their sum. These difficulties may explain the exis­ tence of another paragraph immediately preceding the above proof where one finds in essence the following result: If a ball moves in a circle, then the force by which it tends away from the centre acting on some other body in a straight line (like the force of gravity) will create in this other body, in the time for motion in the circle through ‘ M S . I Id, Ax. 24.

10

M A I N L I N E OF D E V E L O P M E N T OF 1.2 a distance equal to the radius of the circle, motion equal to the quantity of motion of the ball.

The equivalence of this result referring to movement in a straight line to the result obtained for movement in a circle follows without difficulty, and doubtless Newton would have divined it immediately. Thus Newton obtained a quantitative result concerning the motion of a body in a straight line under the action of a constant force equal to the centrifugal force for a given circular motion. Noteworthy is the fact that up to this point he had not obtained a formula giving the dependence of centrifugal force on the speed and the radius of the circle. This com­ pletes the treatment of the problem of circular motion in the Waste Book. The scene now shifts to MS. I ll, devoted, for the most part, to calcula­ tion of the ratios of the force of gravity to the centrifugal forces of the diurnal and annual movements of the Earth. These calculations are all based implicitly, nevertheless certainly,^ on the following formula: A body moves from rest in a straight line under the action of a force equal to that acting on an equal body moving in a circle, radius R, with speed V. Then in the time of movement in the circle through a distance R the other equal body will move from rest in a straight line through a distance \R.

To consider Newton’s probable derivation of this formula we must return to the result immediately above his exact quantitative ‘polygonal’ treatment of circular motion on folio i on the Waste Book. The known behaviour of a body in this case then leads to the ‘\R! formula. For if jR be the radius of the circle, and V the speed of the ball, the result given by Newton implies that the second body acquires motion M V in time RjV, M being the bulk of the ball. If, as in MS. I ll , the second body happens to have the same bulk as the ball, it will clearly acquire speed V in time RjV, and by the Merton^ Rule the distance moved in this time will be (^-R/V)V = \R— precisely the result used by Newton in MS. I ll, and most probably derived by him in the manner just indicated. T o cal­ culate the ratio of the weight of a body to any given centrifugal force it was then only necessary to compare the distances moved by the body under the action of the two forces in the same interval of time. At first the value assumed for the rate of fall due to gravity was based on a result given by Galileo in the Dialogue.^ Later becoming dissatisfied with this result, perhaps as the result of some crude experiment, he redetermined the rate of fall due to gravity by a very ingenious use of certain results * See § 2 o f Commentary and Interpretation to M S . III. * Familiar to Newton if only through Galileo’s Dialogue or Discorsi. ^ See M S . I, § 6; M S . I l l , Appendix A .

1.2

N E W TO N ’ S D Y N A M I C A L T H O U G H T

11

relating to the motion of vertical and horizontal pendulums. He thus arrived at the excellent figure of 196 inches of fall in one second. The assumption that the distances moved in a given time were pro­ portional to the forces, and the extension of Galileo’s V law from the special case of motion under gravity to that under any constant force, would both have appeared self-evident to Newton, requiring no justi­ fication. Later they played a central role in his treatment of motion in an ellipse where he extended them still further to a case where the forces acting were no longer constant. ^ Having learnt how to calculate the ratio of weight to centrifugal force, Newton then discovered that the centrifugal force of a body at the equator due to the diurnal motion could be safely neglected compared with its weight. This was his first great practical discovery in dynamics. The ancient arguments of Ptolemy against the rotation of the Earth were theoretically sound but in practice the effect was negligible and would have none of the catastrophic consequences imagined by Ptolemy. This was a result of great importance and must have powerfully reinforced Newton’s belief in the Copernican system. Yet it in no way implied a knowledge of the dependence of centrifugal force on radius and speed. One calculation in MS. I I P indeed proves that Newton had reached this result sometime after his first calculations on that manu­ script, and it is even possible that he first derived this result by the following proof found at the end of the Scholium to Prop. 4, Theor. 4, of the Principia.^ In any circle suppose a polygon to be described of any number of sides. And if a body, moved with a given velocity along the sides of the polygon, is reflected from the circle at the several angular points, the force, with which at every reflection it strikes the circle, will be as its velocity: and therefore the sum of the forces, in a given time, will be as the product of that velocity and the number of reflections; that is (if the species of the polygon be given), as the length described in that given time, and increased or diminished in the ratio of the same length to the radius of the circle; that is, as the square of that length divided by the radius; and therefore the polygon, by having its sides diminished in infinitum coincides with the circle, as the square of the arc de­ scribed in a given time divided by the radius. This is the centrifugal force, with which the body impels the circle; and to which the contrary force, where­ with the circle continually repels the body towards the centre, is equal.

The approach to the problem of circular motion in this proof is evidently the same as that found in the Waste Book. And the possibility'^ ' See below, § 3 of present chapter. See Ball [i], p. 13.

3 As first suggested by Ball.

^ See M S . I l l , Appendix B, § 2. ^ See Ball [i], p. 13.

PLATE 2 12

M A I N L I N E OF D E V E L O P M E N T OF

1.2

that it represents the actual culmination of the discussion in that manu­ script is strengthened by Newton’s statement in his letter of 14 July 1686 to Halley that he came across it when ‘turning over some old papers’. However, the first undoubtedly early proof of the law of centrifugal force is that given in MS. IVa. Once again, as in the polygonal treatment, there is the basic Cartesian notion of the natural tendency of the body to follow the inertial path along the tangent. But whereas the ‘polygonal’ treatment was based on the idea of replacing the circle by an inscribed polygon, and the con­ tinuous force by a series of sharp impulsive ‘forces of reflection’, the circle is now retained throughout, the outward conatus or endeavour from the centre being measured by the deviation it would produce be­ tween the actual and inertial paths in a given small interval of time. This deviation, D B (MS. IVa, Fig. i), is thus the distance through which the conatus would impel the body if free to act in the absence of any constraining force. By Galileo’s F law it then follows that the same conatus acting on the body in a straight line, like the force of gravity, would move it through a distance 2tt^R in the time of a complete revolu­ tion in the circle. That this result is equivalent to the ‘ l-R’ formula of MS. HI follows immediately by a second application of the F law. So the two treatments, the polygonal and the deviational, agree. But where­ as the derivation of the law of centrifugal force by the polygonal method is, as we have seen, rather difficult, it follows easily from the deviational treatment. For if the given conatus moves the body through a distance 2tt^R in the time T of a complete revolution, it will move the body through a distance 27 t^Rx {i ITY in unit time. From the proportionality arguments for constant forces it follows that the conati of divers circular motions will vary with the result given by Newton at §2 of MS. IVa. In this deviational treatment of circular motion we thus find renewed use of the paradigm case of motion under gravity, including application of the extension of Galileo’s F law; in addition there is the first intro­ duction of the important notion of measuring force or conatus in terms of the deviation DB in a given time. Two things are noteworthy: first, Newton’s apparent belief in conatus as a real species of potential force; z/the body were free then the conatus would impel it through the distance DB in the given small interval of time. This is precisely the same notion of conatus found in Def. 6 of MS. VI, where it is said to be ‘an impeded force or a force in so far as it is resisted’. In the case of circular motion this is an entirely false notion

' si ^I -1

: 5 * ^ ■ ^

4

V

: 2

t

.

. - i

tS I

. V

o o cq

I

•

I

O

I i I■

“I

u ^ " ^ .N ^ L i S V i .

:

,

it- ; fA "

^ '4

..:r

. ,ir

5t

' *‘ J-JkJf w

-

;■

^

1 V- I ' ?c ' ‘ >^ r • V - A-j 11- c1

t I

• _.^ 4 t| •> ^<- i ^

Cl- t f' ^ d < . Ii ^ F ^ ^

nI .

. -

■ v ‘ -I

i t ^ T-1 ' ' -V - f 3 ' ’V ' C - t*

V. ¥ ** i" ‘ ^ ' N '• "K

V

t

plate

3 1.2

I iS hfm A H 4.

0

Qiuniuf ^

^aAvw h.*Juf Sf#>~j u ^ L i Uk,

|0t.J

(f^

*9

«. £
fi

ht-^OVL,

***

^ <Xt itO ih«./ f„^V f

9

C

^

n

i

t

h

u./-

?

*: ' ' T V ’T ufjuJArv-u*^

%

J, <

a.1 34 ^ e . 0A'--»/4 ;^ at.O . /•

Ai0lA

/

T«;

r
/M' jk .;„■ «» ^ .2 ^ ) ' /l3£/( ’

2/tf #<4-^iCii*, (rr%»jlfii«.^^

/V^ \ ' * ‘1 ^f.r■ f fTi'fc'% 3'WlHfi*P^-Hf.

t ^ n^fciur- M .

4

.

.

C-t«

..I w " ^

4»

.

ft...

s.v^Mi-swrf - V , ' ^%n4u^ Ca^Oi-ivtf WtWi^Pt' O. Ci«.i^

Wct^«4 ; ^ a.

4 »Ut 4 w i ^ ; #* 9tx

«*Jrr ^vn.0
^y». ^vr- |XC» plliS^S

1 ^

Am

JUH,‘9!o4»*Ufff A'/^ff'Ujt

-it^

pW b<^<^SJ^

tf’ i«t Ht *mAc ^r\A^ :-t*t

f*^

. ft^u»,'ii! ■ifck **A pTAM-im. 9 c4>t3M»i. p\APg4'

i'lif

\ficCA»*t ^»^*“# **‘ i <- v4 - ci>kUt •\i,\‘ fjrrpYPL »•» ^ , ^#itC '**.

*C 4 rmL ' tL «* «9^^’
/k

!-> "*'” 7

)«#/*f. <9 t • • .i l I iA jk c iSfi

oMiWy t\4 npfi^ ,
i g0 S s ^ 'if^ M

^ u.l~

i|rt. “#

•

.!,< .CM^m ffi ^4 ^x 5

k v r^ k 4 i a tv v r i

J J^C 0 i j 4 ikmk'a^ J*tf^/r^Sk i i ktUkg pnffirrL'0n.^ »^

A page from an early Newton manuscript on circular motion (MS. IVa)

^ K et^ l^ *)

*

N E W TO N ’ S D Y N A M I C A L TH O U G H T

13

dynamically, the only true tendency being that along the tangent. That Newton ultimately came to realize the illusory nature of centrifugal, as opposed to centripetal, force would seem* to follow from his treatment of orbital motion from the tract de Motu onwards. For now it is a devia­ tion BD equal to the original one, but in the opposite sense, in terms of which he compares the centripetal forces for different circular motions. In the second place, it is noteworthy that Newton does not set the conatus proportional to the distance DB. He states only that the conatus is of such a quantity that it would impel the body through a distance D B in the time corresponding to AD . Such an identification would, of course, have been false, since the conatus is finite while the deviation D B is very small. Nor was there any call to consider the actual relation between the deviation and the conatus, since the constancy of the latter made it possible to deduce the distance moved in any finite time given the dis­ tance moved in the ‘time’ AD . This was sufficient for comparing the magnitude of the conatus to that of any other constant force, such as the force of gravity, or the conatus of some other circular motion. But this method of comparing forces by the distances moved in a finite interval of time no longer applied in the problem of Kepler-motion owing to the variation in the magnitude of the force acting on the body. The derivation of the law of centrifugal force represented the most important achievement of Newton’s early researches in dynamics. It was particularly important in two respects; in the first place it made possible his derivation of the inverse square law of gravitation via Kepler’s third law and its test against the moon’s motion. Lacking any direct documentary evidence of this test one cannot be entirely certain that it did take place, but there is a considerable amount of indirect evidence largely consistent with this possibility.^ In the second place, the long process leading from the first tentative approach to the problem of circular motion to the final derivation of the law of centrifugal force must have provided Newton with invaluable experience in the dynamical treatment of motion in a curved path. Lacking this experience of circular motion at the time of Hooke’s intervention in 1679, is per­ haps rather unlikely that he would then have been able to solve the vastly more difficult problem of motion in an ellipse.^ ' But see below, Chapter 3, p. 62. ^ See below, Chapter 4. ^ Assuming that Newton first solved the problem o f Kepler-motion synthetically rather than analytically. See n. 2, p. 17, below for a discussion of the possibility of an original analytic?.! solution.

M AIN LINE OF DEVEL OPM EN T OF

14

1.3.

T he

S olution

to

the

P roblem

of

1.3

K epler- M otion

The precise dating of the various elements of the early researches will probably always remain a matter of speculation. But it seems likely that the first test of the law of gravitation against the moon’s motion took place at Woolsthorpe in the summer or autumn of i666d and that New­ ton’s return to Cambridge in the following spring marked the end of the first period of his creative work in dynamics. There follows a long break in the story from 1667 to 1679 during which there is no evidence of any further development in Newton’s dynamical thought.^ Brewster sup­ posed that this apparent loss of interest in the subject resulted from his disappointment at the failure of the first test,3 but it seems more prob­ able that it was due simply to the renewal of his optical researches and the consequent total absorption even of all Newton’s creative scientific energy. In any case, it is hardly surprising that these early researches in dynamics should have been followed by a long gap of years during which his interest in the subject lay almost entirely dormant. For one can regard it as the inevitable working of a mind so deeply intuitive as Newton’s, an infallible index of genius of the rarest kind, that this in­ tensely difficult and creative work in dynamics should have been fol­ lowed by a long fallow period during which the experience gained in the early researches could be quietly and unhurriedly absorbed and assimi­ lated. So that when the time was ripe, at the height of his powers, with a mind purged by the most intense emotional experience of his life, he could turn, at Hooke’s prompting, to the supreme problem of elliptical motion and find ready to hand just those tools needed to carry the work through to a final conclusion. It is possible, however, that Newton’s interest in dynamics revived temporarily some years before Hooke’s intervention in 1679 following the receipt of his presentation copy of Huygens’s Horologium Oscilla’ See below, Chapter 6.2. 2 M S . V may well have been completed in this period. But it contains no new developments important for New ton’s later researches in dynamics apart from the separation of the law for composition of motions from that for resolution of velocity. Also the treatment of the problem of the collision of two rotating bodies in §§9, 10 must be regarded as the natural completion of the study of collisions in the Waste Book. T h is problem played no part in New ton’s later researches in dynamics. Its importance lies rather in the brilliant example it provides of his genius in dynamics. ^ Brewster [i], vol. i, p. 26, no doubt on the authority of Whiston and Pemberton who both state that this was the reason for Newton laying aside his researches on gravitation at this time.

1.3

N E W TO N ’ S D Y N A M I C A L T H O U G H T

15

torium in 1673. In his letter of thanks to Huygens he refers* to the use­ fulness of the formulae relating to centrifugal force given (without proof) at the end of that work, instancing a notion of his own that the invariable aspect of the moon from the Earth was due to the greater conatus of its motion relative to the Earth compared with that relative to the sun. This revival of interest, however, must have been at most short-lived, and if any more extensive work in dynamics was contem­ plated it remained unfulfilled. Newton, as we know, was at this time heavily engaged in defending his publication in optics against Hooke and others.2 And the animosity, ill will, and misunderstanding to which the controversy on light gave rise left him with a strong, almost patholo­ gical, aversion to ever publishing anything in ‘philosophy’ again and even with some aversion to the subject itself. For example, in 1675 we find Collins complaining in a letter^ to a friend that Barrow and Newton were so busy with chemical experiments that they had no time for mathematics (and, by implication, philosophy). So that Newton’s statement^ that for some years previous to 1679 he had been endeavouring ‘to bend himself from philosophy to other studies’ may have been very close to the truth. If Newton had died before December 1679 he would have held today a very high place in science for his researches in light and mathematics. But the researches into dynamics and gravitation on which his popular fame largely (and perhaps justly) rests would never have been made, and his early researches in dynamics would have been either entirely ignored or regarded today as of antiquarian interest only. On the 27 November 1679 he returned to Cambridge after an absence of almost six months, much the longest since his return in 1667 after the plague. His mother had died in the preceding May or June.s She had caught a fever when nursing her other son Benjamin Smith at Stamford Bridge and Newton had hurried to her side to nurse her him self.After his mother’s death the ‘ See relevant extract of New ton’s letter of 23 June 1673 to Huygens at M S . V ila . ^ See More [i], Chapter 4. 3 See para. 4 of Collins’s letter of 19 October 1675 to James Gregory reproduced in Rigaud [2], vol. ii, p. 280. In his letter of 28 November 1679 to Hooke. 5 According to the records of Colsterworth Parish (as reported by T u m o r [i]) a Mrs. Hannah Smith was buried there on 4 June 1679. In all probability this was New ton’s mother, nee Hannah Ayscough, who married (secondly) Barnabas Smith in 1645. ^ Conduitt (K M S 130(8) Library, K in g ’s College, Cambridge) describes how Newton ‘ attended her with a true filial pity, sate up whole nights with her, gave her all her physick himself, dressed all her blisters with his own hands, and made use of that manual dexterity for which he was so remarkable to lessen the pain which always accompanied the dressing (the torturing remedy usually applied in that distemper) with as much readiness as he ever had employed it in the most delightful experiments’.

16

M AIN LINE OF DEVELOPMENT OF

1.3

whole business of winding up and settling her estate devolved on Newton who had thus no time ‘to entertain philosophical meditations, or so much as to study or mind any thing else but country affairs’.^ Fortunate Newton, that he could thus escape all creative intellectual thought for so long a period and return to Cambridge with a mind entirely refreshed. For soon after his return he was to be faced by the decisive challenge of his career. This came about, as it were by chance, as a result of the cele­ brated letter of the 24 November from Robert Hooke. The ensuing correspondence between Newton and Hooke is well known thanks to the fortunate recovery of all the letters and to the writ­ ings of Rouse Ball,2 Pelseneer,^ and Koyre.^ Here we are concerned only with Newton’s famous ‘imagination’ of an experiment to discover the diurnal motion of the Earth.s In the light of other evidence^ it then seems certain that it was Hooke’s criticism of the spiral path proposed by Newton for the falling body which revived Newton’s interest in dyna­ mics, and this time so thoroughly that he went on to construct the two propositions corresponding to Kepler’s first and second laws, probably before the end of December 1679. When Newton first gave his entire attention to the problem of motion in an ellipse one thing, at least, would have been clear to him; a body, a planet, could only trace out so unnatural, non-inertial a path as an ellipse under the constant action of a force. What was so difficult was to determine the direction and magnitude of this force at every point of the ellipse. He may, of course, have had some preliminary notions of the answers to these questions. He may have believed, or guessed, that the force on a planet was entirely directed to the sun. He may also have expected that the magnitude of the force would vary inversely with the square of the distance.'^' But there w^as a world of difference between guessing the answers to these questions and proving them. T o Newton was given the privilege of proving that on the basis of Kepler’s laws it necessarily followed that the force on any planet was entirely directed * Letter of 28 November 1679 to Hooke. 2 Ball [i], Chapter 3. 3 Pelseneer [i]. ^ Koyre [2]. 3 T h e relevant passage from New ton’s letter of 28 November 1679 to Hooke is reproduced at M S . V H c. ^ Especially New ton’s statement in his letter of 14 July 1686 to Halley: ‘This is true, that his [Hooke’s] letters occasioned m y finding the method of determining figures, which when I had tried in the ellipsis, I threw the calculation b}", being upon other studies.’ 7 A possibility suggested by Bullialdus, and referred to specifically by Newton in the postscript of his letter of 20 June 1686 to Halley. For the passage on the inverse square law see Bullialdus [i], p. 23.

1.3

N E W TO N ’ S D Y N A M I C A L T H O U G H T

17

towards the sun, varying inversely as the square of the distance there­ from. In this respect, if in no other, our sympathies are entirely with Newton in his cri de coeur to Halley: Now is this not very fine ? Mathematicians, that find out, settle, and do all the business, must content themselves with being nothing but dry calculators and drudges. . . .' What is possibly Newton’s first solution to the problem of Keplermotion is found in Prop, i and 3 of MS. VIH.^ Before attempting to estimate the magnitude of the force he would need to have known its direction, so that he probably discovered Prop, i first. Dynamically the proof of this proposition rests on the notion of replacing the actual * Postscript to New ton’s letter of 20 June 1686 to Halley. 2 See below, Chapter 6.6, for a discussion of the status of this manuscript. Account must also be taken of the possibility that New ton’s first solution to the prob­ lem of Kepler-motion was analytic rather than synthetic. This would seem to be implied by the well-known passage in the Portsmouth D raft Memorandum : ‘By this M ethod I in­ vented the Demonstration of K epler’s Proposition in the year 1679, and almost all the rest of the Difficulter Propositions of the Book of Principles in the years 1684, 1685, and part of the year 1686.’ T h e M ethod in question has usually been assumed to be that of the fluxions, for example by Rouse Ball (Ball [i], p. 7). But it must be remembered that the Portsmouth D raft Memorandum was most probably written after the outbreak of the controversy with Leibniz over the Calculus, so that any references in it to the Method of Fluxions have to be taken with some reserve. T h e potentially most reliable indication of the actual method employed b y Newton is contained in his letter of 14 July 1686 to Halley. There he states: ‘T h is is true, that his letters occasioned m y finding the method of determining figures, which when I had tried in the ellipsis, I threw the calcu­ lation by, being upon other studies; and so it rested for about five years, till upon your request I sought for that paper; and not finding it, did it again, and reduced it into the propositions shewed you by M r. Paget.’ Unfortunately the first vital part of this passage is ambiguous. I f the method referred to was the analytical method for determining figures sketched out in principle in Prop. 41 of Book I of the Principia, then he should have stated that he tried it on the inverse square law instead of the ellipse. On the other hand, if he was referring to Prop. 6, Theor. 5, by means of -which given the figure the cen­ tripetal force for any curve (including the ellipse) may be calculated in principle, then he should have used the phrase method of determining forces. Lacking any documentary evidence we can therefore only surmise the actual method employed. I think it is entirely possible that Newton did develop the method of determining figures in 1679. On the other hand, there is no reliable evidence that he either then or later proved in detail that an inverse square law would lead to a conic, with an ellipse as one particular case. A nd in fact if he had done so, then it is rather surprising that he did not give this case as a corollary to Prop. 41, Book I, Principia. It is possibly significant that in the much simpler case of an inverse cube force referred to in Corollary 3 of that proposition no details are given of the actual orbit. And this remains true for the justification pro­ vided for that corollary by Newton to David Gregory at the time of the latter’s visit to Cambridge in 1694. See Correspondence, vol. iii, pp. 348-9. Possibly relevant too is the well-known detail in Conduitt (see Ball [i], p. 26) that New ton’s failure to reproduce his original proof after H alley’s departure was due to his having drawn the axes of an ellipse ‘instead of two conjugate diameters somewhat inclined to one another’. This must surely have referred to a synthetic proof. So that whether or not he achieved an analytic proof in 1679, he would seem at least to have achieved a synthetic one.f f

See additional note, on p. 34.

18

MAIN LINE OF DEVELOPMENT OF

1.3

continuously varying force by a series of equally spaced sharp impulses. The actual path is then replaced by a polygon made up of the rectilinear, inertial, motions of the body between successive impulses. Given the initial velocity of the body the first impulse will turn it sharply into a determinate new direction, and the position at the end of the first inter­ val of time is then given by the beautiful construction based implicitly on the law of composition of independent motions enunciated in Hypo­ thesis 3. The position of the body at successive instants then follows in precisely the same way and the proof is completed by the passage to the limit of vanishingly small intervals of time. The development of the law of composition of motions is considered elsewhere^ and nothing need be said of the use of the principle of inertia— this could confidently have been predicted a priori. But the replacement of the continuously acting force by a series of equally spaced impulses calls for comment. At first sight this recalls the original poly­ gonal treatment of circular motion,^ the impulses in the present treat­ ment corresponding to the discrete ‘forces of reflection’ in the former. In reality, there is a great difference between the two problems. In the first it was the circle which was given, the ‘forces of reflection’ arising from the impacts at the corners of the inscribed polygon. Now it was the force (and therefore the impulses) which was given as a function of the time, the polygonal approximation to the actual path being then determined in terms of the initial position and velocity of the body. In fact, this step-wise development of the approximate path given the force as a function of the time represents the first example of the determinacy of Newtonian dynamics. From Kepler’s second law and the converse of Prop, i it then followed that the force on the planet was entirely directed towards the sun. New­ ton was thus free to take up the problem of the motion of a body in an ellipse under the action of a force directed to one of the foci. His solution to this problem is contained in Prop. 3 of MS. V III. As in Prop, i he replaces the continuous force of attraction to the focus by a series of focally directed sharp impulses separated by a constant interval of time A^, the corresponding positions of the body on the ellipse being P, Y (MS. V III, Fig. 6) etc. If the force of attraction were suddenly to cease when the body is at P, it would continue its inertial path along the tangent reaching a point X in time Af. Instead it is acted on by one of * See below, § 4 of present chapter, section on Corollary i to laws of motion, 2 See above, § 2 of present chapter.

1.3

N E W TO N ’ S D Y N A M I C A L T H O U G H l '

19

the impulses which turns its motion into the chord P F , Y being the point on the ellipse reached in time A^. The distance X F , having been generated by the impulse at P, must therefore be proportional, and parallel, thereto. The same will be true for any other set of points^, x, y. From this it follows that the force at P ; force at p ~ X Y : xy. By application of certain geometrical properties of the ellipse together with the fact that the rate of description of area is constant, so that A P Y F = ApyF, it follows that the ratio of the forces at P and p will be inversely as PF^ to pF-. From the dynamical point of view the new element in Prop. 3 com­ pared with Prop. I rests in the assumption that the deviation gives a direct measure of the varying force of attraction. It will be convenient to examine this assumption later in the light of the slightly different use of the deviation in the tract de Motu, the lectures de Motu, and the first edition of the Principia. One other feature of the proof calls for com­ ment: the peculiar mixture of tangents and chords. In reality this invali­ dates the claim that the motions will be along the chord P F , and then along the next chord, and so on. For the motion along P F resulted from the impulse at P acting on the inertial motion along the tangent at that point. Whereas when the body reaches F it will be moving along the chord. Of course, the difference between the tangent and the chord is very small and ultimately disappears in the limit. But to remove this difference beforehand would abolish the deviation X Y and with it the proof itself. The process envisaged by Newton can therefore only be applied, if at all, to individual points of the ellipse. So that the analogy between the method adopted in this proposition and that in the first is more apparent than real. Newton could have followed another more consistent method based on a thoroughgoing use of a polygonal approxi­ mation. This will be discussed later when considering the whole question of his use of deviation as a measure of force. In the first and later versions of the tract de Motu^ the propositions corresponding to Kepler’s second and first laws of planetary motion reappear as Prop, i and Prob. 3, Prop. 3, respectively. The proof of Prop. I follows on exactly the same lines as Prop, i of MS. V III. As to the inverse square law for motion in an ellipse under a force directed to one focus, Newton now derives it as a special case of the general formula for centripetal force for motion in any curved path given in Prop. 3. Only this proposition need be considered since it contains all the dynamical I M S . IX .

20

MAIN LINE OF DEVEL OPM EN T OF

1.3

arguments, the special mathematical arguments appropriate to an ellipse being concentrated in Prob. 3. Once again the force at any point P of the curve is calculated in terms of the deviation. But now both the curve and the force are regarded as continuous throughout and the deviation found by drawing a line parallel to S P (MS. IX, Fig. 3) through a point O close to P cutting at R the tangent to the curve at P. RQ is then the deviation. For a given small interval of time between P and O, R Q will be proportional to the magnitude of the actual force, Pp, at P ; and for a given force it will be proportional to AP. S o that altogether Fp{Aty, i.e.

oc RQ/(Aty.

The actual result given then follows in the limit assuming the constancy of description of area about the centre of force S. In the lectures de Motu^ and the first edition of the Principia the derivation of the inverse square law exactly follows those in the various versions of the tract de Motu apart from a marginal reference to the second law of motion in justification of the assumption that the deviation is proportional to the force acting. The central role of deviation in Newton’s treatment of force for motion in a curved path is now sufficiently evident. He first used it in his second, deviational, treatment of circular motion, and then modified it to derive the inverse square law for Kepler-motion. Thereafter it reappeared in his rather different derivation of the same result in the tract de Motu, the lectures de Motu, and the first edition of the Principia. In each case Newton’s handling of the deviation was superb and always led to the right result. But the arguments advanced in its support seemed less compelling and call for a critical examination. There were two unsatisfactory elements in Prop. 3 of MS. V III. The first was the simultaneous use of chords and tangents. This invalidated the continuous process envisaged and really necessitated a new argument at each point of the ellipse. Next there was the assumption that the devia­ tion, being generated by the force (impulse) at P, would be proportional, and parallel, to the force. This was justified by an appeal to Hypothesis 2 that ‘the alteration of motion is ever proportional to the force by which it is altered’. But the motion generated by the impulse at P is propor­ tional to the velocity generated rather than the displacement, that is to X Y j A t as opposed to X Y . If then we set the actual accelerative force at > M S . X I.

1.3

N E W TO N ’ S D Y N A M I C A L T H O U G H T

21

P equal to the corresponding impulse will be FpAt so that the correct equation should read X Y j A t oc FpAt, i.e.

X Y oc Fp{At)\

However, since the interval At is everywhere the same, a comparison between any two points of the ellipse gives

XY _Fp xy ~ F p ' Newton’s measure of the impulse in terms of the deviation thus in­ evitably led him to the correct result for the force. By removing the double use of chords and tangents the whole proof can then be placed on an entirely satisfactory basis. Imagine the given ellipse divided up by a large but finite number of points reached at equal small intervals of time. It may then be asked; what set of focally directed impulses at these points will enable the body to describe this polygon ? The process is now similar to that considered in the ‘polygonal’ treatment of circular motion; but whereas in the latter case the body can obviously follow a regular inscribed polygon indefinitely as a result of continued reflections at the corners, now the motion in the polygon will only persist if the impulses at the corners are arranged so as to be of exactly the right magnitude and direction— there is no question of the body being reflected naturally at the surface of the containing ellipse. T o find the magnitude of these impulses— corresponding to the ‘forces of reflection’ in the case of the circle— it is then only necessary to calculate the change in velocity at each corner; and it follows easily that for a given point P this is propor­ tional to X Y j A t , thus leading to the same result as before though now by an entirely unobjectionable method. Viewed in this light Newton’s own proof appears as a peculiar mixture of the polygonal (the chord) and deviational (the tangent) methods used in his two early treatments of circular motion. It likewise contains a peculiar mixture of the discrete (the impulses) and the continuous (the accelerative force). Though it appears to be a deviational proof, it is in fact much closer to a polygonal treatment in terms of which it derives its true justification. The success of this proof in spite of its deficiencies could be regarded as an indication that Newton was confident beforehand what the correct result should be; but it could also be regarded as one of the best examples of his intuitive grasp of dynamical processes.

22

MAIN LINE OF DEVELOPMENT OF

1.3

Newton’s use of deviation in Prop. 3 of the tract de Motu and the cor­ responding propositions of the lectures de Motu and the first edition of the Principia remains to be considered. Here there is no trace of impulses and polygons. Both the path and the force are regarded as continuous throughout. The assumption deviation oc (A^)^ for a given force is en­ tirely justified. The deviation is now the distance moved away from the inertial path under the action of a force effectively constant in the given small interval of time. Likewise the assumption that the deviation will be proportional to the magnitude of the force given the small interval of time This is obviously true, being the same proportionality argu­ ment used in the early researches though now extended to the action of variable forces in very small intervals of time. But the justification of this assumption in the lectures de Motu and the first edition of the Principia by an appeal to the second law is less satisfactory. For the second law of motion concerns the motive force and the change in motion, whereas now it is a matter of the accelerative force and the distance moved. This epitomizes a certain confusion in Newton’s thought between the two kinds of force, motive and accelerative. It was unfortunate that his second law referred to the former while his calculations always concerned the latter. The use of the versed sine in place of the deviation in the second and third editions of the Principia may possibly indicate that he himself felt a certain dissatisfaction with his use of the latter concept, especially in the crucial theorem on the general expression for force for motion in a curved path. Curiously, however, his justification of versed sine as a measure of force appears in Coroll. IV of Prop, i, and therefore depends on a polygonal rather than a continuous treatment, that is on the im­ pulse rather than the actual force. From this impasse there was no escape except through a thoroughgoing application of the differential calculus.

1.4.

T he

F inal

S ynthesis

of

N e w t o n ’s

D ynam ical

S ystem

Newton’s construction of the propositions corresponding to Kepler’s first two laws of planetary motion represented his most important single achievement in dynamics and the very cornerstone of the Principia. How then explain his apparent loss of interest in dynamics again until 1684? Later he said himself that ‘he threw the calculations by being on other studies’ I and certainly there must have been much else to claim his attention after so long an absence from Cambridge. The question of ' New ton’s letter of 14 July 1686 to Halley.

1.4

N E W TO N ’ S D Y N A M I C A L T H O U G H T

23

his first successful test of the law of gravitation against the moon’s motion is possibly relevant here. If this was made in 1679, as stated by Pemberton, I then his loss of interest at this point would almost seem to argue an inability to appreciate the importance and significance of his own discoveries. On the other hand, the fact that Pemberton’s account is probably erroneous in another respect^ gives us leave to doubt the veracity of his account in regard to the date of the test of the inverse square law. And if this test actually took place later, most probably in 1684,3 the lull between 1679 and 1684, though still puzzling, becomes much less incomprehensible. Another possibility to be considered is that Newton was held up in 1679 through an inability to prove that a gravitating sphere attracts external particles as though all its mass were concentrated at its centre. But against this we know that he certainly was not prevented from composing the tract de Motu in 1684 by the lack of the same proposition.^ The remainder of the traditional account is well known.s How Halley visited Newton at Cambridge in August 1684 to inquire what figure a body would describe under an inverse square law of force. How he was overjoyed to learn that Newton knew the answer and had dXxQzdyproved it. And how Newton, being unable to find the original proof, promised to rework the solution and send it to Halley at the Royal Society, which promise was then fulfilled in November of the same year when Paget carried the proposition or propositions in question from Cambridge to London. Finally, the registration at the Royal Society, some time before February 1685, of Newton’s Propositiones de Motu followed by the pub­ lication of the Principia in 1687. So much has generally been agreed to. Opinions, however, have differed on the exact nature of the propositions carried by Paget to London, and on their connexion with the tract de Motu, the lectures de Motu of the Michaelmas term 1684, and the propositions de Motu regis­ tered at the Royal Society. There is also the possibility that the marginal entry ‘Octob 1684/Lect. i ’ in MS. X I was added considerably later and should have read ‘Octob 1685/Lect. i ’,^ and that the actual contents ' Pemberton [i], Preface. ^ See below, Chapter 4, p. 75. ^ A s suggested below, Chapter 4, p. 75. * In his letter of 20 June 1686 to Halley Newton stated: ‘ I never extended the duplicate proportion lower than to the superficies of the earth, and before a certain demonstration I found the last year, have suspected it did not reach accurately down so low'.’ T h e demonstration in question was presumably the proposition referred to here. 5 See, for example. Ball [i]. Chapter 4. See below'. Chapter 6.4, pp. 98-102.

24

M AIN LINE OF DEV ELOPMENT OF

1.4

of these lectures were considerably different from the text given in M S .X I.i However, the available evidence seems to point fairly unambiguously to the following course of events in 1684 and early 1685. After Halley’s first visit to Newton at Cambridge, most probably in May 1684,2 and not in August as generally supposed, Newton reconstructed the two missing propositions and then added a number of new ones. The result was Version I of the tract de Motu probably composed by the end of July 1684.^ Version II of this tract, a fair copy of Version I, was later sent to Halley in fulfilment of Newton’s original promise of May.^ But for some reason or other, possibly connected with his fear of Hooke’s re­ actions, he at first held it up. Instead he pushed on with his researches, the result being Version III of the tract. Following Halley’s second (or third) visit in November he sent Version II of the tract to London via Paget. The Propositiones de Motu entered in the Register Book of the Royal Society some time before 23 February 1685 were copied from this paper. There seems some doubt whether the so-called lectures de Motu of 1684 were actually read in the Michaelmas term 1684 as opposed to 1685; but what is in any case certain is that up to the beginning of Section I of Book I of the Principia these lectures (MS. XI) contain an original version which, as emended in the manuscript itself, agrees almost exactly with the first edition of the Principia of which it must be re­ garded as the final draft. Also that Versions I and III of the tract de Motu and M SS. Xa, b^ enable us to follow in very considerable detail the creation of this part of the original version of the lectures de Motu, and thus of the corresponding part of the Principia. In other words they enable us to follow the creation of the final synthesis of Newton's dynamical system. This process will now be studied in detail following the actual order in the Principia itself. Wherever possible reference will be made to the appear­ ance of particular concepts in the earlier manuscripts, thus stressing the striking element of continuity between the earlier and later researches. D efinitions

1. Quantity of matter In the early manuscripts, especially in the Waste Book, the term quantity or body refers indiscriminately both to a body and its magnitude, * A s suggested in a private communication by D . T. Whiteside. 2 See below, Chapter 6.4, p. 97. ^ Ibid., p. 97. * See below. Chapter 6.5, pp. 105-8. ® For the order of composition of these manuscripts see below, Chapter 6.3.

1.4

N E W TO N ’ S D Y N A M I C A L T H O U G H T

25

bulk, or size.^ At this stage no consideration is given to bodies of equal size but different mass. So that it is tacitly assumed that all the bodies considered are of the same material. Earlier drafts of the enunciation of this definition are found in Def. i of MS. Xb and in Def. i of the original and revised versions of MS. XI. These are all essentially the same as the final enunciation, the differences between the total definitions residing largely in the explanatory passages following the enunciations. Here we find a smooth progress towards the final definition, the first references to the possibility of freely pervading medium and to the experimentally confirmed proportionality between weight and quantity of matter occurring in the revised version of MS. X I. A brief discussion of the use of pendulum experiments to prove this proportionality oecurs at Def. II of MS. Xa, followed by a much fuller discussion at Def. 7 of MS. Xb where the use oipondus as an alternative to quantity of matter or mass, even in the absence of any gravitational effects, is brought out much more elearly than in the Principia. Rather surprisingly a clear indication of weight-mass proportionality is found already at § 5 of MS. I. Newton’s definition of mass or quantity of matter has occasioned con­ siderable controversy.2 Mach ,3 for example, found the definition un­ fortunate in that it involved a circular argument, density itself being defined by the mass per unit of volume. Crew,^ on the other hand, main­ tained that in Newton’s age density was synonymous with specific gravity, so that it was reasonable for Newton to define quantity of matter in terms of it. In this case, however, the proportionality between quantity of matter and weight would necessarily follow, whereas in Def. i of the Principia Newton asserts that this proportionality has been proved by experiment. Mach’s criticism was therefore justified; there was an essen­ tial circularity involved in Newton’s definition of quantity of matter in terms of density, and this is fully borne out by Def. 6 of MS. Xb where density is said to be proportional to the quantity of matter and the volume conjointly. Further confirmation for this view is provided by the passage towards the end of the second paragraph of Def. 13 of MS. VI where Newton sets the weight (gravitas) of a body equal to its size (molis) multiplied by its specific gravity. However, this flaw in the definition of quantity of matter in no way ’ See, for example, M S . lie , Def. 1-3. ^ See, for example, Cajori [2], p. 638, n. i i . ^ M ach [i], p. 188. * Crew [i], p. 124.

26

MAIN LIN E OF DEVEL OPMENT OF

1.4

interfered with the development of Newton’s thought in the Principia, for when he had to compare the quantities of matter of two bodies he did so by making an appeal to his second law of motion. ^It is this law, therefore, rather than Def. i, which must be regarded as giving the real, operational definition of mass in Newton’s dynamics. As to the original development of the concept, we can be confident that the obvious fact of inertia, or Vis Insita, would inevitably have led Newton to attach a measure to it. For example, collision experiments with bodies of equal size, but of different material, revealed that the inertias of such bodies were not entirely determined by their sizes. Two bodies of equal size could have quite different capacities to sustain their states of rest or motion. That he then attempted to give a definition of mass independently of the second law was understandable, though per­ haps a trifle unfortunate.

2 . Quantity of motion An equivalent definition is found first in Defs. 2, 3 of the Waste Book (MS. lie), where the same explanation is given in terms of the motions of the separate parts of the body. Earlier drafts of the definition in the Principia are found at Def. i i of MS. Xa, Def. 2 of MS. Xb, and Def. 4 of the original version of MS. XI.

3 . Innate force or inertia of a body The origin of this concept went back at least to MS. VI, where it is clearly referred to under the heading of force in Def. 5 as ‘motus et quietis causale principium' or internum principium quo motus vel quies corpori insita conservatur, et quodlibet ens in suo statu perseverare conatur et impeditum reluctatur or in Def. 8 of the same work as Inertia est vis interna corporis ne status ejus externa vi illata facile mutetur. The first formal definition of Vis Insita occurs in Version I of the tract de Motu. This and later definitions are listed below: MS. IXa, Def. 2 Et vim corporis seu corpori insitam [appello] qua id conatur perseverare in motii suo secundum lineam rectam. ‘ See, for example, Principia, Book III, Section V I, Prop. 24, Theor. 18.

1.4

N E W TO N ’ S D Y N A M I C A L TH O U G H T

27

Identical definitions are given in the other Versions of the tract, MS. Xa, Def. 12 Corporis vis insita innata et essentialis est potentia qua id perseverat in statu suo quiescendi vel movendi uniformiter in linea recta, estque corporis quantitati proportionalis, exercetur vero proportionaliter mutationem status et quatenus exercetur did potest corporis vis exercita . . . una species est vis centrifuga gyrantium. Ibid., Def. 13, cancelled Vis motus seu corpori ex motu sua adventitia est qua corpus quantitatem totam sui motus conservare conatur. Ea vulgo dicitur impetus estque motuiproportionalis, et pro genere motus vel ahsoluta est vel relativa. MS. Xb, Def. 3 Materiae vis insita est potentia resistendi qua corpus unumquodque quantum in se est perseverat in statu suo vel quiescendi vel movendi uniformiter in directum: estque corpori suo proportionalis neque dijfert quicquam ah inertia massae nisi in modo conceptus nostri. Exercet vero corpus hanc vim solummodo in mutatione status sui facta per vim aliam in se impressam. . . . MS. X I (Original Version), Def, 5 This version differs from that immediately above only in the replace­ ment of conceptus nostri by concipiendi at the end of the first sentence, and the introduction of the following two sentences immediately after­ wards : Per inertiam materiae fit ut corpus omne de statu suo vel quiescendi vel movendi difficultate deturhetur. Unde etiam vis insita nomine significantissimo vis inertiae did possit. The version in the first edition of the Principia differs in no important respect from that in the lectures de Motu. A cursory examination of these definitions reveals nothing of par­ ticular interest beyond the obvious identity of the various terms vis insita, vis corporis, vis inertiae, or inertia. Admittedly, there is the quali­ fication of this force from MS. Xa onwards as a power by which the inertial state of rest or motion is maintained, and from MS. Xb onwards this power becomes a power of resistance. But a closer examination of Newton’s description of the circumstances under which this power is exercised reveals a remarkable transition in his view of vis insita or inertia. Up to and including MS. Xa this is the force or power by virtue of which a body maintains its inertial state of rest or motion; in vulgar parlance, the impetus of the body, as noted in Def. 13 of MS. Xa.

28

M AIN LINE OF DEV ELOPMENT OF

1.4

That this was Newton’s actual belief is confirmed by the corresponding enunciations of the principle of inertia, each of which contains a refer­ ence to vis insita. In MS. Xa occurs the first hint of an impending transition in his thought. There he states that the vis insita is exercised in proportion to the change in state. In MS. Xb the transition is com­ plete. Now the vis insita is exercised only (solummodo) in changes of state. It would seem, therefore, to have been relegated to a species of potential force, having no effect as long as the state of rest or of uniform motion continued, being called into action only in changes of state. That this was indeed Newton’s new view of the matter is proved conclusively by the absence of any reference to vis insita in the enunciation of the principle of inertia from M S. Xb onwards. In these manuscripts we there­ fore have before us the record of how Newton, on reflection, freed himself from what was apparently his previous, essentially medieval belief in the necessity of some interior force or impetus to maintain an inertial state of uniform motion. From now on such a state of motion (or rest) was a true state in the Cartesian sense, entirely self-sufficient, and the principle of inertia a true principle, something which had to be regarded as given, a natural fact, having no explanation and requiring none.

4 . Impressed force Earlier drafts of this definition are found at Def. 15 of MS. Xa, Def. 4 of MS. Xb, and Def. VI of MS. XI. The qualification that the impressed force consists in the action only appears first in MS. Xb, and the refer­ ence to the force of motion in MS. XI.

5-8 Centripetal force Significantly, the term centripetal as opposed to centrifugal is not found in the early dynamical manuscripts. It occurs first at Def. i of Version I of the tract de Motu, and in identical form at Def. i of Version III of the same work. Thereafter it recurs in identical form at Def. 16 of MS. Xa apart from the addition of the examples of this force given in Def. 5 of the Principia. At Def. 5 of MS. Xb the division into three distinct quantities absolute, accelerative and motive first occurs, while in MS. X I (original and final versions) these different quantities are assigned to separate definitions. The definition in the Principia is effectively identical with that in the original version of M S. X I apart from one rather large emendation in Def. 10 of that manuscript (Def. 8 of Principia).

1.4

N E W TO N ’ S D Y N A M I C A L T H O U G H T

29

Scholium to definitions The first reference to space in the later manuscripts occurs at Law 3 of Version III of the tract de Motu. Another reference in the same work is found in the addendum (compared with Version I) to the Scholium to Theorem 4. The first formal definitions of absolute and relative space and time occur in Def. 1-4 of MS. Xa. In Def. 7, 9 of the same manu­ script occur rudimentary drafts of the definitions of place and absolute motion in MS. XI. Definition 9 above also contains the germ of the methods for distinguishing between relative and absolute motions in­ cluding that depending on centrifugal effects. There is no indication of later expansions of these early drafts, the version in the original of MS. X I being effectively identical with that in the Principia. Much has been written of the metaphysieal views on absolute space (and time) expressed in the Scholium to the Definitions of Book I of the Principia^ especially in the light of the Leibniz-Clarke controversy.^ A beginning has also been made by Fiersz^ to the study of the influences which went to shape these views. The discovery of MS. Add. 4003 provides important material which will make possible a new approach to the question of the early influences on Newton’s metaphysical outlook in general, and those on space in particular. This study, however, will only be possible in the light of a detailed commentary of MS. Add. 4003 against the total philosophical background about the period 1660-70.3 Laws of motion First law. This law, inherited by Newton from Galileo and Descartes,-^ formed the central element of his dynamical thought and method. Fortunately it is well documented, appearing in some form or other in most of the major documents throughout the period 1664-84. The various expressions given to the principle by Newton in these manuscripts are listed below. MS. II, Ax. I , 2. If a quantity once moves it will never rest unless hindered by some external cause and a quantity will always move on in the same straight line (not changing the ' For a summary account see Alexander [i]. For a more recent discussion see Toulmin [i]. ^ Fiersz [i]. ^ A n excellent beginning to this has now been made by Hall and Hall [i]. See pp. 7 6 85 of their introduction to text and translation of M S . 4003. * See below, Chapter 2.

30

MAIN LINE OF DEVELOPMENT OF

1.4

celerity or determination of its motion) unless some external cause divert it.

MS. II, Ax. 100. A body once moved will always keep the same celerity, quantity and deter­ mination of its motion.

MS. VI, § 4. et multo magis quod corporis sine impedimentis nioti velocitas non did potest uniformis, neque linea recta in qua motus perficitur. MS. V III, Hyp.

I.

Bodies move uniformly in straight lines unless so far as they are retarded by the resistance of the medium or disturbed by some other force.

MS. IXa, Hyp. 2. Corpus omne sola vi insita uniformiter secundum rectam lineam in infinitum progredi nisi aliqiiid extrinsecus impediat. MS. IXc, Lex I. Sola vi insita corpus uniformiter in linea recta semper pergere si nil impediat. MS. Xa, Lex i. Vis insita corpus omne perseverare in statu suo quiescendi vel movendi uni­ formiter in linea recta nisi quatenus viribus impressis cogitur statum ilium mutare. MS. XI, Lex I. Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum suum mutare. A noteworthy feature of these enunciations is the absence of any reference to a state of rest as opposed to that of motion until MS. Xa immediately preceding the lectures de Motii. This represented a closer approach to Descartes’s formulation of the principle. The only previous reference to rest occurs in Def. 5 of MS. VI where force is said to be ‘the causal principle of motion or rest’. Interesting, too, is the reference to Vis Insita appearing in Versions I and HI of the tract de Motu and in MS. Xa but absent from later enun­ ciations. A possible explanation of this has already been given above in the discussion of the genesis of the definition of inertia. Second law. The germ of this law is found in the early researches. For the definition of force as change in momentum in the Waste Book corre­ sponds to the first part of the second law, and the second part corresponds to the statement in the same manuscript that ‘A body must move that way which it is pressed’.^ ^ M S . lie , Ax. 120.

1.4

N E W TO N ’ S D Y N A M I C A L TH O U G H T

31

Curiously, there is no reference to the second law in MS. V apart from the statement at the end of § i ‘that the force is equivalent to that motion which it is able to beget or destroy’. It first reappears as Hypothesis 2 of MS. V III: The alteration of motion is ever proportional to the force by which it is altered.

The presence of this hypothesis is accounted for by Newton’s explicit use of it in justification of his assumption of proportionality between ‘force’ and deviation. There is no trace of the second law in the first two versions of the tract de Motu. But in the third version we find it enunciated in Law 2 as follows: Mutationem status movendi vel quiescendi proportionalem esse vi impressae et fieri secundum lineam rectam qua vis ilia imprimitur. It is employed there in the proof of Lemma i, and in Theorem 4 where it ensures that equal forces produce equal deviations (in equal times). In MS. Xa (Law 2) the enunciation is identical with that given above apart from the replacement of status movendi vel quiescendi by motus. In the same manuscript a reference is added to the use of the first and second laws by Galileo in his derivation of the vertical and parabolic motions of bodies under gravity. In the lectures de Motu the law retains the same form as in MS. Xa apart from the introduction of motrici between vi and impressae, and there is no further change in the Principia. Third law. The assumption of the equality of action and reaction in the direct collision between two bodies is found in Def. 7, 8 of the Waste Book (MS. Hd), and for oblique collisions in Ax. 119, 121 of the same work (MS. He). There is no trace of this law after its early indication in the Waste Book before M S. Xa where it is enunciated as Law 3 in the form: Corpus omne tantum pati reactione quantum agit in alterum. Both in MS. X I and in the first edition of the Principia it appears in the form: Actioni contrariam semper et aequalem esse reactionem: sive, corporum duorum actiones in se mutuo semper esse aequales et in partes contrarias dirigi. The most important use of this law, and undoubtedly one of the rea­ sons for its explicit formulation in the lectures de Motu and the Principia, was its role in the proof of Corollary 3 to the laws of motion.

32

M AIN LINE OF DEVEL O PME NT OF

1.4

Corollaries to laws of motion Corollary i. It is easy to confuse the dynamical rule for composition of motions with the kinematical rule for combining two relative velocities or for finding the components of a given velocity in two oblique directions. Newton himself seems to have fallen into this confusion in the Waste Book where he gives the correct rule for the composition of two motions as a corollary (without proof) to an apparently incorrect rule for resolving velocity.^ In MS. V this confusion has disappeared. In § 2 of that manuscript he gives a perfectly correct rule for finding the re­ solved part of a given velocity in any oblique direction, followed in § 3 by the rule for composition of motions. His formulation of this rule in MS. V is potentially of great interest. As suggested in Chapter 2.1 it affords a possible indication of Galileo’s influence on Newton. The same rule appears as Hypothesis 3 of MS. VIII. Now the two motions are on an equal footing— the possible indication of Galileo’s influence contained in MS. V having disappeared— and the same is true of the rather obscure formulation in Hypothesis 3 of Version I of the tract de Motu. In Version III of the same work the status of the rule suddenly changes. Previously it had been a hypothesis, now it has be­ come a lemma (Lemma i) furnished with a proof based implicitly on the second law of motion. In MS. Xa it reappears as Lemma i, the enuncia­ tion and proof being effectively identical with those in Version III of the tract de Motu. The same is true of MS. X I and the Principia apart from the change of title from Lemma to Corollary. Corollary 2. There is no trace of this corollary before MS. X L Subject to a substantial emendation of the last three paragraphs this agrees with Corollary 2 of the Principia. Corollary 3. This represents Newton’s emended form of Descartes’s law of conservation of motion. Its probable role in the early develop­ ment of Newton’s dynamical thought has already been noticed in § i of the present chapter. The first formal enunciation occurs in MS. X I and is effectively identical with that in the Principia. The outline in the last paragraph of the method of treatment of the problem of the collision of two non-spherical bodies, with the possibility of rotation arising from the collision, must be read in the light of the detailed solution to this problem provided by Newton in MS. V. Corollary 4. This appears first as Law 4 of Version HI of the tract de Motu, being employed in the remarkable discussion of the motion of I M S. Ilg .

1.4

N E W T O N ’ S D Y N A M I C A L TH O U G H T

33

the solar system in the addendum (compared with Version I) to the Scholium to Theorem 4. The same result appears as Law 5 in MS. Xa, and as Corollary 4 to the laws of motion in MS. X L The proof given in this latter manuscript is essentially the same as that in the Principia. In the first part of this proof, that relating to the purely kinematical case in which there are no collisions between the bodies, an appeal is made to Lemma X X III. This is the result first proved by Newton at Ax.-Prop. 28 of the Waste Book (MS. Ilf) for the special case of motion in a plane. The immediately following Ax.-Prop. 29 provides justification for the claim in the Principia that the result may also be proved in loco solido. Corollary 5. This appears first as Law 3 in Version III of the tract de Motu and then in identical form as Law 4 of MS. Xa apart from the addition of the example of motion relative to a ship. In MS. X I and the Principia the enunciation is the same as before but now the law has become a corollary. Corollary 6. This first appears in MS. X I and then in identical form in the Principia. Scholium to laws of motion The version of this in MS. X I is effectively identical with that in the Principia. No trace of earlier drafts has been found apart from the inter­ esting reference to Galileo and the first two laws of motion at the end of Law 2 in MS. Xa. Method of prime and ultimate ratios This method, given immediately after the scholium to the laws of motion, formed the final element of Newton’s formal dynamical system in the Principia. With one important exception to be noted presently there appear to be no earlier drafts of this section before MS. XI, of which the original version differs substantially from that in the Principia only in the absence of the two corollaries to Lemma X and the final paragraph to the scholium. However, this absence of earlier drafts is of no importance dynamically since the whole section is of purely mathematical interest apart from Newton’s generalization of Galileo’s law of falling bodies at Lemma X. In Newton’s early dynamical in­ vestigations this law played a vital role in M SS. I l l and IV. The first formal enunciation of the generalization of the law to curvilinear motion occurs at Hypothesis 4 of Version II of the tract de Motu. In Version HI 858205

D

34

DEV ELOPMEN T OF D Y N A M I C A L T H O U G H T

1.4

of the same work its status has changed from a hypothesis to a Lemma (Lemma 2), the proof being based on an application of the result that the area under a velocity-time graph equals the space described. For a proof of this result Newton appeals to the authority of Galileo. In MS. Xa the same enunciation is given at Lemma 2, but the proof has not been preserved beyond the first few lines. In Lemma X of MS. X I and the rrincipia the proof is contained largely in the preceding Lemma apart from the result relating to the space described which is now unsupported by a reference to the authority of Galileo.’' ' Curiously, Newton himself supplies no proof of the result. Since Galileo’s proofs in the Dialogue and the Discorsi were both defective, being based on the method of in ­ divisibles, the foundation of the Principia was at this point insecure. It would be in­ teresting to know when the first published correct proof of the result was given. T h at given by Beeckman was almost certainly the first in point of time, but it remained un­ published till the rediscovery and publication of his Journal by de Waard. P'or Beeckman’s proof see Beeckman [i], vol. i, p. 262.

Additional note

f

to p. l y :

Since writing note 2 on p. 17 I have become aware through Lohne [i] of two other relevant passages in Newton manuscripts. T h e first passage occurs in M S . Add. 3968 b, fol. l o i : after referring to the correspondence of 1679/80 with Hooke, Newton con­ tinues: ‘Whereupon I computed what would be the Orb described by the Planets for I had found before by the sesquialterate proportion of the tempora periodica of the Planets with respect to their distances from the sun that the forces which kept them in their Orbs about the Sun were as the squares of their mean distances from the Sun reciprocally, and I found now that whatsoever was the law of the forces w'hich kept the Planets in their Orbs the areas described by a Radius drawn from them to the Sun would be proportional to the times in which they were described. And by the help of these Propositions I found that their Orbs would be such Ellipses as Kepler had described.’ T his would seem to point unequivocally to New ton’s discovery o f the proposition corresponding to Prop, i of the tract de M otii in the winter o f 1679/80. But it could imply equally well a direct (analytic) or converse (synthetic) proof of the proposition corresponding to Kepler’s first law (motion in an ellipse). In the other passage (M S. Add. 3968 b, fol. 405) Newton states unequivocally: ‘By the inverse M ethod of fluxions I found in the year 1677 [ = 1679/80] the demon­ stration of Kepler’s Astronomical Proposition viz. that the Planets move in Ellipses, which is the eleventh proposition of the first book of the Principles.’ Nevertheless, I am still inclined to agree with Lohne [i] and Whiteside {History o f Science, i (1962), p. 20) that Newton never possessed an analytic proof that an inverse square centripetal force implied motion in a conic, including an ellipse as a special case.

T H E I N F L U E N C E OF G A L I L E O A N D D E S C A R T E S ON N E W T O N ’ S D Y N A M I C S 2.1.

T he

Influence

of

G alileo

I t has always been assumed that Newton was indebted in dynamics to Galileo, and this would remain highly probable even in the absence of any corroborative evidence. For by 1661, the year of Newton’s entry to Cambridge, Galileo’s writings, especially in dynamics, were widely known throughout Europe including England, witness the translation of his Dialogue in Tome I of Salusbury’s Mathematical Collections pub­ lished in the same year. To my knowledge, the only explicit indication of such an influence in the Principia occurs in the celebrated passage at the beginning of the Scholium to the laws of motion. There Newton states: Per leges duas primas & Corollaria duo prima adinvenit Galilaeus descensum graviumesseinduplicata ratione temporis, & motumprojectiliumfieri in Parabola, conspirante experientia, nisi quatenus motus illi per aeris resistentiam aliquantulum retardantur. In certain respects this statement is entirely clear and unexceptionable. Galileo did discover both the law of falling bodies and the parabolic path of a projectile, basing his derivations of them on an inertial principle and a method of compounding motions identical with that found in the first corollary to Newton’s laws of motion. To what extent, however, did Galileo’s inertial principle agree with Newton’s principle of inertia, and how far if at all did Galileo either recognize or use the second law of motion? Finally, are we to understand from this state­ ment of Newton’s that he felt himself indebted to Galileo for all these elements of his dynamics ? Before attempting to answer these questions two things are necessary: one must first decide what were Galileo’s original contributions to dynamics; and one must then look for documen­ tary evidence of the influence of these contributions on Newton. There can be no doubt that Galileo was the first to derive the para­ bolic path of a projectile. His derivation' was based on the combination of two independent and non-interfering motions, the one uniform and horizontal, appropriate to a ball moving freely on a horizontal plane, the ‘ Discorsi, Fourth Day, Theor. i. Prop. 1.

36

I NF LU EN C E OF GALILEO AND D E S C A R T E S

2.1

other vertical and uniformly accelerated, appropriate to free fall under gravity. It likewise seems certain that although Galileo may not have been the first to derive the fundamental formula soc for uniformly accelerated motion, having been possibly anticipated by Beeckman,' yet he was the first to publish it and to realize its true physical signifi­ cance.^ It is equally certain that Galileo never enunciated the principle of inertia,3 and indeed could not have done so correctly, since ‘horizontal’ motion was for him always at the surface of the earth, equidistant from its centre, and therefore in reality circular and not rectilinear. This was clearly his belief in the Dialogue, and although there is no indication of a continuing belief in ‘circularity’ for the greater part of the kinematical and dynamical writing in the Discorsi, a passage towards the end'^ makes it evident that he had in no way modified this particular belief in the interval between the composition of the two works. Nevertheless, al­ though Galileo’s principle of inertia was thus restricted to a very special terrestrial case, this restriction did not obtrude itself in his vivid physical discussion of inertial motion on a horizontal plane, especially in his discussion of the motion of a projectile. And Newton would have been powerfully impressed and influenced by this discussion. It is not at all clear how far, if at all, Galileo’s understanding of force had progressed along the road leading to the second law of motion. New­ ton himself, as we have seen, seems to imply that this law was known to Galileo, and various historians of science, including Whewell,s Dugas,^ and Singer,^ have all to differing extents adopted the same point of view. Koyre, however, is rightly very guarded in his estimation of Galileo’s views on force.® In particular, there seems to be only one passage in Galileo which gives any hint of the second law. This occurs in the ^ For Beeckman’s derivation, and the part played in it by Descartes, see Duhem [3], pp. 570-4; Koyre [i]. Part II, pp. 25-40. ^ For Galileo’s treatment of the law of falling bodies, see especially Koyre [1], Part II. 3 In this connexion see particularly Koyr^ [i]. Part III, Appendix A. * See Salviati’s statement found in Ed. N a z ., t. viii, p. 274. 5 Whewell, W ., [i], vol. ii, p. 29: ‘T o see that a transverse force would produce a curve was one step; to determine what the curve is, was another step, which involved the discoverj'^ of the second law of motion. T h is step was made by Galileo.’ * Dugas, R., [2], p. 206, where New ton’s statement at the beginning of the Scholium to the laws of motion is reproduced and interpreted as New ton’s homage to his prede­ cessors. ’ Singer, C ., [i], p. 236: ‘New ton’s second law . . . is involved in Galileo’s theory of projectiles.’ What he means by ‘involved’ is not clear. * See especially Koyr^ [i]. Part III, Chapter 3.

2.1

ON N E W TO N ’ S D Y N A M I C S

37

Discorsi, in the well-known passage elaborated by Viviani at Galileo’s request.^ In this passage, which contains a derivation of the theorem that the velocity acquired from rest on an inclined plane depends only on the vertical height fallen, there is certainly an assumption which could be interpreted^ as implying that the distance moved in given time on an inclined plane will vary with the component of the weight of the body along the line of greatest slope. But whether or not Galileo’s use of this assumption should be regarded as an indication that he possessed an inkling of the second law of motion seems to me a very moot question. In any case it would still have to be decided whether Newton knew of, or was influenced by, this particular passage in the Discorsi at the time of his earliest researches in dynamics. Among the dynamical problems touched on by Galileo in the Dialogue was that of circular motion.^ His treatment of this was rather uneven. For although he believed that a stone on a revolving wheel tended to be thrown off along the tangent, yet his argument for supposing it might be prevented from doing so because of its weight was unconvincing. In any case, his failure to reach a true understanding of inertial movement, to separate it completely from terrestrial phenomena and imagine it taking place in empty space, inevitably precluded him from attaining that clarity of conception of circular motion found in Descartes’s treat­ ment in Part II of the Principia Philosophiae. Against this, however, we have to set Galileo’s suggestive argument, in favour of the centrifugal tendency decreasing with increase of radius for a given rate of rotation.+ Apart from the topics already considered mention must be made of Galileo’s realization that the efficacy of a body’s motion varied directly with its bulk and velocity. This notion is found both in the Dialogue^ and in the Discourse on Bodies in Water.^ But although, as is well known, Galileo had given much thought to the problem of percussion, and even promised a treatment of it in the Discorsi,"^ he never published anything * See Ed. N a z., t. viii, p. 214 et seq. (Crew and Salvio [i], p. 180 et seq.). ^ Could be interpreted because in the key phrase ‘since the forces are in the same ratio as these distances’ (Crew and Salvio [i], p. 184) the term employed by Galileo (or Viviani) for ‘force’ was ‘ momenti' which may or may not have implied the same thing as ‘Vimpeto’— the force impelling a body to fall— which in turn may or may not have had the same significance as force for Newton or for us. ^ See pp. 200 et seq. of Salusbury’s translation of the Dialogue in Santillana [i]. Ibid., pp. 231 et seq. ^ Ibid., pp. 229-30. ^ A t Def. 5, A x. 2, 3. See Salusbury’s translation of this work in Drake [i]. ^ See Ed. N a z., t. viii, p. 293 (Crew and Salvio [i], p. 271).

38

IN F LU E N C E OF GALILEO AND D E S C A R T E S

2.1

on the subject,^ and his projected treatment of it must remain a matter of speculation. Bearing in mind these achievements, and limitations, of Galileo in dynamics, let us now consider what evidence there is for his influence on Newton’s earliest researches in the subject. First there is the reference to Galileo in MS. I. This comes after a critical discussion of Aristotle’s theory of projectiles in the same manuscript and must date close to the beginning of Newton’s study of dynamics. Next there are two references to Galileo in MS. I l l of which the original data must in any case have been taken by Newton from the Dialogue.^ Here is indisputable evidence of Galileo’s influence on Newton in dynamics at a very early stage. And although Newton was probably little indebted to Galileo, if at all, for his first clear realization of the concept of centrifugal force, it seems very probable that the peculiar formula on which the calculations in MS. I l l are based was derived by Newton from the formula given on folio I of MS. II by application of the Merton Rule.3 In that case, assum­ ing that he was not already familiar with this rule through a study of medieval works,^ it is possible that he took it either from the Dialogue or the Discorsi. Some independent support for this view is provided by Version III of the tract de Motu of 1684 (MS. IXc). For although his proof of Lemma 2^ of that work uses the fact that the distance travelled equals the area under the s-t graph, he gives no proof of the result himself, appealing instead to that given by Galileo— ut exposuit Galilaeus. Apart from this possible indication of a debt of Newton to Galileo in respect of the Merton Rule, MS. I l l contains repeated applications of Galileo’s law, the first, but by no means the last, occasion on which Newton made use of this fundamental result.^ * W ith the exception of the short section entitled ‘O f the Force of Percussion’ at the end of the early treatise ‘On Mechanics’ (see pp. 179-82 of the translation of this work in Drake and Drabkin [i]). This, however, in no way advances the problem of per­ cussion or collision from the dynamical point of view. See, however, the unpublished dialogue ‘Della forza della percossa’ intended by Galileo to supply the treatment of the subject promised in the Discorsi. For a French translation of this dialogue together with an illuminating discussion of its content and historical significance see the article Remarques sur le dialogue de Galilee ‘D e la force de la percussion' by S. Moscovici, Revue d'Histoire des Sciences, 16, 7 -1 3 7 (1963), ^ See M S . I l l , Appendix A. •'* See above. Chapter 1.2, p. lO. * A possibility by no means to be ruled out, though one of the certain sources of New ton’s knowledge of Aristotelian physics, Magirus [i], while it contains an interesting commentary on the problem of projectiles, at no point goes so far as a dis­ cussion of uniformly accelerated motion. ^ T o appear later as Lemma X in the Principia. ^ See, for example, New ton’s generalization of Galileo’s law in 'I’heor. 3 of the tract de M otu (M S. TX).

2.1

ON N E W TO N ’ S D Y N A M I C S

39

In MS. II Newton gives a correct statement of the law of composition of independent motions as a corollary to what appears to be an incorrect rule for resolving velocityP Later, in MS. V, this connexion between the kinematical rule for resolving velocity and the dynamical rule for com­ pounding motions has disappeared. Now they are given separately, and correctly, in succeeding sections. Only the dynamical rule need concern us. This Newton formulates as follows: If a body move towards B with the velocity R, and by the way hath some new force done to it which had the body rested would have propelled it to­ wards C with the velocity S. Then making AB : A C : : R : S, and com­ pleting the parallelogram BC the body shall move in the diagonal AD and arrive at the point D with this compound motion in the same time it would have arrived at the point B, with its single motion. At first sight this enunciation of the parallelogram law seems to differ from later ones, such as that found in Corollary i to the laws of motion in the Principia, only in its relative verbosity— which in turn could be accounted for by the superior conciseness of Latin over English. But a closer inspection reveals one small, yet possibly significant, difference between this enunciation and all later ones: namely, in all later enuncia­ tions both motions are treated on precisely the same footing, whereas here one motion is regarded as prior to the other. Initially the body was moving from A to B with velocity R when some ‘new force’ was done to it. Immediately there comes to mind Galileo’s marvellous derivation (in the Fourth Day of the Discorsi) of the parabolic path of a projectile. There, too, there was initially a certain motion of the body, along the table, to which was added at the moment of its reaching the edge a ‘new^ force’, that due to the weight of the body downwards. And to find the subsequent movement of the body thereafter it was necessary to com­ pound the two motions, the one horizontal and inertial, the other vertical and accelerated. Admittedly, there was a fundamental difference be­ tween the ‘forces’ considered by Newton and those implicit in Galileo’s problem. In the latter case there was no force in the horizontal direction, and the vertical force w^as constant, producing acceleration, whereas here the forces considered by Newton were really impulses producing (almost) instantaneous changes in motion. Nevertheless, I think that * See M S . Ilg . * See Fig. 2 of M S . V. 3 Notice here New ton’s use of the word ‘new ’. It affords a good illustration of the peculiar exactness and completeness of his dynamical thought. Another force had acted on the body previously to produce the original motion in the direction A B .

40

I NF LU EN CE OF GALILEO AND D E S C A R T E S

2.1

this difference between the present enunciation and all later ones provides a precious indication of a possible, if not probable, influence of Galileo on Newton in his formulation of the parallelogram law based on the compounding of independent motions: given that Galileo was the first to make real use of this novel and entirely un-Aristotelian notion there would always have been a strong a priori possibility of its having influenced Newton, and this possibility must now be somewhat streng­ thened. I MS. II contains what are undoubtedly the earliest of Newton’s extant researches in dynamics. There we find an enunciation of the principle of inertia which could only have been taken by Newton from Part II of Des­ cartes’s Further strong indications of Descartes’s influence are found in Newton’s concept of centrifugal force in the same manuscript, and in his treatment of momentum and collisions. On the other hand, there is no indication whatsoever in the earliest^ dynamical writings in the Waste Book of any Galilean influence: no treatment, for example, of uniformly accelerated motion, or of motion on an inclined plane. All the indications^ therefore, point towards a predominant prior influence of Descartes on Newton in respect both of the principle of inertia and the concept of centrifugal force. And this, of course, is just what one would expect given the great superiority in respect of both completeness and clarity of Descartes’s treatment of these two topics compared with that of Galileo. On the other hand, Newton’s treatment of centrifugal force in MS. IVa where he first derives the law of centrifugal force by a method entirely different from that which he had followed in MS. II, shows traces of a possible influence of Galileo. Like Galileo’s argument in favour of the centrifugal tendency decreasing with the ratio it is based on the notions of measuring this tendency by the deviation be­ tween the actual circular path and the inertial path along the tangent.^ It remains to consider the second law of motion. Unmistakable traces of the genesis of this law are found in MS. II. There Newton defines force as the change in motion produced. Likewise the change in direction of motion produced in a body is in the direction of the force ‘ Assuming, of course, that Newton had access to the Discorsi at an early date. ^ T h e reasons for this are given in the immediately following section on Descartes’s influence on Newton. ^ A s opposed, that is, to the somewhat later writings such as those on fol. i of the same manuscript where there is a reference to Galileo in connexion with the synchron­ ism of the simple pendulum. See M S . Ila, § 4. On the other hand a rather similar kind of deviation figures in Descartes’s dis­ cussion of centrifugal force in Art. 58, Part 3, o f his Principia Philosophiae.

2.1

ON N E W TO N ’ S D Y N A M I C S

41

acting on it— ‘a body must move that way which it is pressed’. But this definition of force, its connexion with motion, and its effect on direction arose directly out of his discussion of the problem of collisions. The influence, if any, was therefore from Descartes, not from Galileo, and will be discussed in the immediately succeeding section. T o summarize, the one ingredient of his early dynamical researches which Newton undoubtedly owed to Galileo was the latter’s t^ law of falling bodies. This continued to play a vital role in the growth of New­ ton’s dynamics up to and including the composition of the Principia. It is enshrined in Newton’s generalization of the law in Lemma X of that work. In the same Lemma he implicitly assumes the Merton Rule the proof of which he had previously attributed to Galileo. In all probability he took this originally from the Dialogue or the Discorsi sometime prior to the composition of the Vellum MS. Probably, if not certainly, Newton owed to Galileo the notion of com­ pounding non-interfering independent motions as first used by the latter in his proof of the parabolic path of a projectile in the Discorsi. Without this notion Newton could not have formulated his parallelogram law for the composition of independent motions. A possible indication of such an influence is provided by the peculiar nature of an early formu­ lation of the parallelogram law. This law, too, played a vital role in the growth of Newton’s dynamical thought— ^without it, for example, the original proof of the proposition corresponding to Kepler’s second law of planetary motion would have been impossible. Finally, there is no indication in the earliest researches of any influence of Galileo on Newton in respect of the principle of inertia, centrifugal force, or the genesis of the second law of motion. In the last case the possibility of any influence is effectively ruled out by the actual form of the law and its undoubted origin in Newton’s study of momentum and collisions, whereas some influence in the case of centrifugal force is possible at a later stage. As to the total impact of Galileo on Newton in science in general and in dynamics in particular, this is something which goes beyond the present volume and for which there is no evidence in the purely dyna­ mical manuscripts. In all probability it was profound. Certainly the whole cast of Newton’s thought, his humility before Nature, his drive towards exact quantitative results, his delight in experiment, was alto­ gether Galilean, and if he recognized any master in science apart from Archimedes it could only have been Galileo.

42

2.2

IN F LU EN CE OF GALILEO AND DE S C A R T E S T he

Influence

of

2.2

D escartes

Newton’s aversion to certain elements of Descartes’s physics has long been recognized. For example, the Principia was evidently intended to provide not only a new ‘explanation’ of the solar system based on Kepler’s laws and Newton’s own dynamical principles, but also a rebut­ tal of Descartes’s theory of vortices. Moreover, as Koyre has demon­ strated, ^a careful reading of the Principia, especially between the lines, proves how strong was Newton’s aversion to certain elements of Des­ cartes’s I n fact it would not be too strong to say that Newton eventually came to detest Descartes both as a physicist and a philosopher, his aversion to Descartes even extending to mathematics with his wellknown belief that a geometrical proof could only be regarded as well founded if presented in synthetic rather than analytic form. And yet in spite of Newton’s strong, one might almost say passionate, dislike of Descartes and all his works, there has always been a strong possibility that Newton was largely indebted to Descartes for the most important single element of his dynamics, the principle of inertia! In questions of the influence of one thinker on another, however, possibilities are rarely of much help. Other possibilities are almost always equally open. Here, for example, apart from the possible, even probable, influence on Newton of Galileo’s imperfect, ytt physically very powerful, discussion of inertial movement, there is also the possibility of his having taken the true and complete notion of this concept from Gassendi.^ Failing other evidence beyond the fact of Descartes’s memorable enunciation of the principle of inertia in Part II of his Principia Philosophiae further pro­ gress towards establishing Newton’s debt to Descartes in respect of this element of his dynamics is therefore impossible. Additional evidence, however, is fortunately available. First there is MS. VI in which Newton displays a most intimate acquaintance with Parts II and III of Descartes’s Principia. And since he was then already interested in dynamics— witness his statement of the principle of inertia on p. 19 of that manuscript^— he could hardly have failed to notice the specifically dynamical articles in Descartes’s work. So that the fact that he pointedly omits any reference to these articles is then of no sig­ nificance apart from the indication it affords of how absolutely he had ^ See Koyre [6]. * Petri Gassendi, D e motu impresso a motore translato, Paris, 1642. See especially Chapter 15, p. 59, and Chapter 16, p. 62. Translations of the two relevant passages are given by Koyre [i]. Part III, pp. 152, 153. See M S . V I, § 4.

2.2

ON N E W TO N ’ S D Y N A M I C S

43

set his face against Descartes at the time of composing this manu­ script. At first sight this seems a very promising start. Parts II and III of Descartes’s Principia contain effectively all his published^ writings on dynamics, and MS. V I bears every indication of being an early work. Newton must therefore have been familiar at an early stage with Des­ cartes’s dynamics, thus improving the prior odds in favour of his having been influenced by them. But not very much, for we know that Newton’s knowledge of dynamics was growing rapidly around the beginning of the year 1665, ^so that to provide any real help towards determining Descartes ’ s influence MS. V I would need to be dated about the same time. In fact, no date is given, and some internal indications of date point towards the second half of 1666 rather than the beginning of 1665.3 Apart from MS. VI, the only other early references by Newton to Descartes’s Principia occur in MS. I. These were probably entered sometime in the second half of 1664, about the beginning of Newton’s work in dynamics,^ so that once again the probability of an influence by Descartes on Newton in dynamics becomes somewhat strengthened. At this point we can summarize the position as follows: given that Descartes’s enunciation of the principle of inertia in Articles 37 and 39 of his Principia Philosophiae was the first to be published apart from that of Gassendi, there would always be a distinct possibility of Newton having been indebted to Descartes for at least this element of his dyna­ mics; and the likelihood of such an influence is increased by the fact that Newton did make a study of Descartes’s Principia, a study, more­ over, which began very early and was ultimately of a profoundly detailed nature— as MS. Add. 4003 proves. Nevertheless, to proceed further, to render Descartes’s influence on Newton in dynamics seem not only possible but probable, some much more definite evidence is needed. This is fortunately provided in MS. II. On the basis of the new evidence supplied by the dynamical writings in this manuscript, certainly among the earliest of Newton’s which we possess, and in any case very close to the actual beginning of his researches in dynamics,® it first becomes * But by no means all his thought on dynamics, of which much valuable evidence is supplied by the Correspondence. See G abbey [i], for the first comprehensive account of Descartes’s dynamical thought as a whole. ^ One of the earliest dvnamical entries in the Waste Book is dated 20 January 1664 (O.S.). S e e M S . Ilb . 2 See below, Chapter 6.2, p. 93. * Ibid., p. 91. 5 A t various points between p. 92 and p. 121 of M S . Add. 4003 there are entries relating to motion, gravity and levity, violent motion, and Aristotle’s theory of

44

IN F LU E N CE OF GALILEO AND D ES C A R T ES

2.2

possible to attempt a serious assessment of the actual influence of Des­ cartes on Newton in dynamics. A word of explanation is called for on the method of treatment followed. The dynamical topics in Parts II and III of Descartes’s Principia are considered in three separate sub-sections, the first two devoted to the principle of inertia and circular motion, respectively, and the third to topics other than these two. In each section an account is first given of Descartes’s method of treatment followed by Newton’s treatment of the same topic, or topics, in MS. II. No attempt is made in any of these subsections to draw any conclusions relative to Descartes’s influence on Newton, since it seemed best to separate the evidence from the argu­ ment based upon it; in any case, although it is convenient to consider the various dynamical topics separately in the first place, the question of Descartes’s influence on Newton in dynamics must ultimately be con­ sidered as a whole, in the light of all the available evidence, as in the final subsection. Principle of Inertia Descartes. In Art. 37 of Part II of his Principia Descartes introduces the first part of the principle of inertia, not casually, in an ad hoc manner, but as a special case of a general philosophical principle mirroring the immutability of God, namely:

2.2

ON N E W T O N ’S D Y N A M I C S

45

if it is moved, why we should think that it would ever of its own accord, and unimpeded by anything else, interrupt this motion. To this inertial tendency of rest and motion Descartes then adds in Art. 39 the supremely important qualification that III. Any particular part of matter, regarded in itself, is never inclined to prosecute motion in any curved lines, but only in straight lines. Which law, states Descartes, follows like the preceding one from the immutability and simplicity of the operation by which God conserves motion in matter. Namely, not as it was at some previous instant, but as it is at the very instant itself. So that although the movement of a body takes place in time and not instantaneously,' and may in fact be curved and not rectilinear, nevertheless at each instant the determination^ of its motion is never in a curved line but always along a straight line— the tangent to the curved path at the instant in question. And this law is supported by the fact that a stone loosed from a sling moves always along the tangent line to its original circular path. Newton. Newton’s first enunciation of the principle of inertia in MS. II occurs at Axioms i and 2. There he states: IV. If a quantity once move it will never rest unless hindered by some external cause. [Ax. i] and

I. Any particular thing, in so far as it is simple and undivided, remains always to the best of its ability in the same state, nor is ever changed [from this state] unless by external causes.^

V. A quantity will always move on in the same straight line (not changing the determination nor celerity of its motion) unless some external cause divert it. [Ax. 2]

For example, a square-shaped piece of matter always remains square unless something intervenes to change its shape, and

Later, at Ax. 100, we find the statement of the following general philo­ sophical principle:

II. If [a body] is at rest we do not believe it is ever set in motion, unless it is impelled thereto by some [external] cause. Nor that there is any more reason

VI. Every thing doth naturally persevere in that state in which it is unless it be interrupted by some external cause.

projectiles. On p. 9 3 V , the same page on which occurs a reference to Descartes’s Principia, there is a cometary observation dated 17 December 1664. T his would seem to point towards the beginning of a serious interest by Newton in dynamics around the second half of 1664, not long before the dynamical entry on fob 10 of the Waste Book dated 20 January 1664 (O.S.). See below’, Chapter 6.2. ^ A ll translations of passages from the Principia have been made direct from the (first) Latin edition as given in Adam and Tannery, v. viir. We cannot be sure which edition Newton used, though it may well have been that found in his library, the third. But he would presumably have used a Latin version, given his avowed ignorance of French. It is worth noting that there are certain important differences between the first Latin edition and Picot’s French translation. For example, the enunciation of the first part of the principle of inertia in the Latin version contains no explicit reference to the uniformity of the motion, whereas in Art. 37, Part II, of the Picot translation the motion is explicitly stated to be ‘de meme force’.

‘Hence’ he adds ‘Axiom ist & 2nd’, and VII. A body once moved will always keep the same celerity, quantity and determination of its motion. Circular motion Descartes. In Art. 38 of Part II of his Principia Descartes cites the case of the circular motion of a stone in a sling as an example of the ' In this connexion see the discussion in Koyre [i]. Part 3, Appendix B. * In Art. 39, Part II, Principia Descartes states; ‘ omne id quod movetur . . . determina-

tum esse ad motum suum continuandimi versus aliquam partem, secundum lineam rectani’ . Newton, in the Waste Book, uses the term determination in the same sense.

46

IN F LU EN CE OF GALILEO AND D E S CA R T E S

2.2

instantaneous tendency of a body to move rectilinearly along the tangent to its actual curved path. From which it follows that: VIII. Every body which is moved circularly tends perpetually to recede from the centre of the circle which it describes.^ This is something, moreover, which we actually sense with our hand when we whirl a stone in a sling. If this were the only discussion of circular motion given by Descartes, one would be inclined to assume that he did not believe in a centrifugal tendency in circular motion distinct from the inertial tendency along the tangent. After all he specifically states that the second tendency follows from the first. And this deduction is entirely just: motion along the tangent is necessarily motion away from the centre, something which Galileo, too, was well aware of. However, it is clear from the more de­ tailed discussion of circular motion in Art. 57, 58, 59 of Part III of the Principia that Descartes really believed in two separate tendencies, the one centrifugal, the other along the tangent. He opens this discussion in Art. 56 by noting that the attempt of second-class matter to escape from the centre of its vortex due to its circular motion does not imply any conscious striving, but only a tendency so to move which may be impeded by other causes. In fact, as he notes in Art. 57, the simultaneous action of various causes is the rule rather than the exception, so that one can talk of a body simulta­ neously having divers tendencies or conatus in different directions. It is at this point that he specifically instances the case of the motion of a stone in a sling. Under the combined effect of all the causes acting on the stone it moves in a circle. But if one thinks only of the motive force in the stone itself— Descartes is clearly identifying this as one of the causes acting on the stone— the corresponding tendency to motion is always along the tangent to the circle. And although the sling prevents the stone from following this tendency it cannot prevent the tendency itself— IX. tum.

ac quamvis funda hunc effectum impediat, non tamen impedit cona-

Again, if we do not consider the whole motive force of the stone along the circle, something wFich he obviously distinguishes from the motive force in the stone itself, but consider only that part prevented by the sling, ^ Loc. cit., towards end of article.

2.2

ON N E W TO N ’ S D Y N A M I C S

47

then we say that the stone at every instant has a tendency to recede along the radius from the centre of the circle. Descartes therefore definitely gives the impression that he believes in two separate conati: one along the tangent due to the motive force in the stone itself, and one along the radius away from the centre. Both tendencies remain potential only, their realization being prevented by the sling. But this, as he wfisely notes, does not prevent the existence of the tendencies themselves. This interpretation of Descartes’s views is then confirmed by his discussion of the motion of an ant on a rotating ruler in Art. 58, and of a ball in a rotating tube in Art. 59. In each case he clearly believes both in a tangential and a centrifugal tendency or endeavour. One can surmise that any doubts he may have had of the reality of the centrifugal conatus would have been removed by the second, characteristically ingenious, example of the ball in the rotating tube. For how can the ball move away from the centre of rotation in the absence of some centrifugal tendency or conatus} His discussion of the ensuing accelerated centrifugal motion from the centre also gives the impression that he looked on this conatus very much in the nature of a force. This is confirmed when he returns to the example of a stone in the sling at the end of Art. 59: X. atque ista tensio, a sola vi qua lapis recedere conatur a centro sui motus exorta, exhibet nobis istius vis quantitatem. As to the ‘quantity’ of this ‘force’ Descartes makes no attempt to calcu­ late it beyond noting that it increases with the speed of rotation. Newton. In Axioms and Propositions 20, 21 of MS. II we have what must represent something very close to Newton’s first thoughts on circular motion. There he considers a sphere moving within a spherical or cylindrical surface: XI. Axiom 20. If a sphere Oc move within the concave spherical or cylindrical surface of the body circularly about the centre m, it shall press upon the body def, for when it is in c (supposing the circle hhc to be described by its centre of motion, and the line eg a tangent to that at 0) it moves towards g or the determination of its motion is towards g, therefore if at that moment the body ed/should cease to check it it would continually move in the line eg (ax. I, 2) obliquely from the centre m, but if the body de/oppose itself to this endeavour keeping it equidistant from m, that is done by a continued checking or reflection of it from the tangent line in every point of the circle chh, but the body def cannot check and curb the determination of the body co unless they continually press upon one another. The same may be understood if the body adb be restrained into circular motion by the thread. . . .

48

IN F LU EN CE OF GALILEO AND D E S C A R T E S

2.2

XII. Axiom 21. Hence it appears that all bodies moved circularly have an endeavour from the centre about which they move, otherwise the body Oc would not continually press upon edf. Notice that for Newton the sphere is at any instant moving in the direction of the tangent line; this is its determination or endeavour, which endeavour is only prevented by the continual check exerted by the surrounding surface. And this check can only be exerted by the surface on the sphere if the two press upon each other. Hence there must always be an opposite endeavour of the sphere away from the centre. Thus for Newton at the time of composing these two axioms a body moving circularly had two endeavours, one along the tangent, one away from the centre along the radius. In the somewhat later MS. IVa we still find the same belief in an endeavour or conatus away from the centre in circular motion. Other topics The remaining dynamical topics considered in Descartes’s Principia come under the general heading of force, resistance, momentum, and conservation of momentum. All these enter into his treatment of colli­ sions between two bodies. From our childhood, states Descartes, we labour under the delusion that more action is required for motion than rest. This, he suggests, springs from the fact that various forces, including gravity, oppose the motions of our limbs causing fatigue, whereas we are not aware of the action of gravity responsible for rest in our own and other bodies. In fact, no more action^ is required to produce motion in a body than is required to bring it to rest again by destroying this motion. For example, the action required to set a boat in still water in motion will be the same as that required to bring it to rest ;2 apart, that is, from a small diflFerence due to that portion of the original action used to set the surrounding water in motion. It is not clear in what sense Descartes believed this last result would help to remove the original prejudice in favour of motion rather than rest. Nor does it seem to play any further part in his Principia. But it is per­ haps important for the connexion it implies between action and move­ ment, between force and momentum. At Axiom 3 of MS. II we find the following statement: There is exactly required so much and no more force to reduce a body to rest as there was to put it upon motion: et e contra. ' T h e term used by Descartes i s ‘flcn’o’.

2.2

ON N E W TO N ’ S D Y N A M I C S

49

followed immediately by its generalization: So much force as is required to destroy any quantity of motion in a body so much is required to generate it; and so much as is required to generate it so much is also required to destroy it. Although Descartes believed that the motion conferred on a body by the action of some external agent or force resided entirely in the body not as a substance, but as a mode or state of the body, yet this state of motion had a certain attribute consisting in a tendency to maintain it­ self, as followed, of course, from the principle of inertia. This tendency of a body to persevere in its motion— adpergendum secundum lineam rectarrC^— ^was to be compared with the analogous tendency to remain at rest which revealed itself in the form of a resistance to motion. The process of collision between two bodies, one in motion and one at rest, could then be thought of as a contest between the force of motion of the moving body and the force of resistance of the other. In MS. II Newton develops his discussion of colliding bodies in two ways. One is based on the motion of the elastic forces brought into play by the deformation of the colliding bodies, while the other makes use of the term perseverance of bodies in their state*^ (of motion or rest) and regards the collision process as a mutual intervention or hindrance between these powers. In the case of a collision between a moving and a stationary body Descartes supposed that the ‘resistance’ of the latter could exceed the force of motion of the former.3 In this case, if the two bodies were soft the moving body would be brought to rest, its motion being absorbed in some unspecified manner; whereas if the bodies were hard it would be reflected without loss of motion. On the other hand, if the force of motion of the moving body exceeded the resistance of the stationary one the two would move on together the total quantity of motion being con­ served, the quantity of motion in a body being proportional to the mag­ nitude of the body and its velocity. As for the conservation, this was necessary to ensure that the total quantity of motion in the universe was unaffected by such encounters between bodies in accordance with the general law of conservation derived from the immutability of God’s working in the universe.^ In MS. II the quantity of motion, or simply the motion, of a body was likewise proportional to its bulk or magnitude and its velocity. Unlike 2 M S . lie . A x. 104.

* Ibid., Art. 40. 3 See Art. 40, Part II, Principia.

^ See Art. 26, Part II, Principia.

8G8205

* Ibid., Art. 36, 42.

E

50

IN F LU EN CE OF GALILEO AND D E S CA R T E S

2.2

Descartes, however, Newton realized that the total momentum in a collision must be calculated algebraically, paying due regard to determina­ tion as well as to magnitude. So that it was now impossible that the moving body should be reflected from a stationary one which itself con­ tinued at rest. A comparison between Descartes’s and Newton’s treatment of these topics at once reveals a most remarkable similarity in the case of the principle of inertia. There were two peculiar, and quite characteristic, features of Descartes’s enunciation of this principle: in the first place, he enunciated it not as a single principle, but in two separate parts, the first affirming the inertial property of motion or rest in the absence of external disturbances, the second the tendency of motion always to take place in straight lines. Secondly, Descartes deduced both parts of the principle from the nature of God and His working in the universe; the first as a special case of a more general philosophical principle mirroring the immutability of God, the second from the immutability and simplicity of the operation by which God conserved motion. Both these peculiar features figure in Newton’s presentation of the principle: he too divides the principle into two parts, the first affirming the inertial property of motion in the absence of external disturbances, the second its rectilinearity. Given that Descartes was the first, and apparently the only, person to enunciate the principle of inertia in two parts, then the fact that Newton does likewise, and that his parts closely follow those of Descartes, must make it at least probable that he modelled his enunciation on that of Descartes. Admittedly, there are certain differences between Newton’s enunciation and Descartes’s: Newton omits any references to rest as a state of a body^ in Axiom i, whereas he refers both to determination and celerity^ in Axiom 2. How­ ever, any residual doubt as to Descartes’s influence is removed by Axiom 100. For the general philosophical principle in that axiom is nothing less than a faithful English paraphrase of the principle given in Latin at Article 37 of Part II of Descartes’s Principia. For good measure, to make assurance doubly sure, there is New ton’s laconic note following his enunciation of the philosophical principle ‘Hence Axiom ist and 2nd’, * Whereas in Def. 5 of M S . V I we find: 'vis . . . est internum principium quo motus vel quies corpori insita conservatur'. ^ T h e fact that celerity is inserted in the text is just possibly significant. .\s already noted in n. i, p. 44, there is no reference to uniformity of motion in Art. 37 of Part II of the first Latin edition of the Principia.

2.2

ON N E W TO N ’ S D Y N A M I C S

51

precisely the argument used by Descartes to derive the first part of his principle of inertia. I conclude, therefore, that Newton took his first known enunciation of the principle of inertia in Axioms i and 2 of MS. II directly from Articles 37 and 39 of Part II of Descartes’s Principia. The evidence for this as it stands, without any further support, seems to me quite con­ clusive, the probability of Newton having reproduced by chance the peculiar features of Descartes’s enunciation vanishingly small. On the basis of this evidence, for example, I deduce that Descartes’s Principia must have been available to Newton around the beginning of the year 1665. Nevertheless, it is comforting to have the reference to the Principia Philosophiae in M S. I, and to find a copy of Descartes’s work in Newton’s library. Knowing that Newton’s first enunciation of the principle of inertia in MS. II was modelled on that given in Part II of Descartes’s Principia, we can now confidently look for further indications of a like influence in the other topics. First there is Newton’s approach to the problem of circular motion in Axiom 20 of the same manuscript. As with Des­ cartes, the treatment is entirely qualitative, only later does he begin to feel his way towards an exact quantitative result. Here he is concerned with the basic physical approach to the problem, and this approach exactly follows that of Descartes resting on a double tendency, or endeavour, of the body, the first along the tangent to the circle at any point, the second away from the centre. This second endeavour away from the centre is evidenced by the pressure of the ball on the spherical containing surface, corresponding to the tension in the string in the case of motion in a sling. ^ Given that Descartes’s treatment of circular motion differed from all previous treatments by his very clear and explicit insistence on tw^o tendencies, the one tangential, the other centrifugal, and knowing that Newton must have been familiar with this treatment when composing Ax.-Prop. 20, then the fact that Newton bases his appreciation of circular motion on the same two tendencies points strongly to his having modelled his approach on that of Descartes. This conclusion, already probable, is then strengthened by the fact that Newton does not content himself with merely stating the tendency for motion along the tangent, but justifies it, like Descartes, by an appeal to the principle of inertia: it was pre­ cisely this appeal which constituted Descartes’s supreme contribution to the problem of circular motion, which enabled him to make a clean * T h e last sentence of M S . Ild , .\x.-Prop. 20, obviously refers to this case.

52

IN F LU EN CE OF GALILEO AND D E S C A R T E S

2.2

break with the tyranny of the Platonic circle, ^ and which provided the sole basis for its further treatment by Newton. Illuminating also is the comparison between Newton’s use of the concept endeavour^ with Descartes’s use of the term conatus. For Des­ cartes the conatus away from the centre remained potential only, being prevented from taking effect owing to the action of the sling. Compare this notion with Newton’s definition of conatus in Def. 6 of MS. V I: Conatus est vis impedita sive vis quatenus resistitur. That the conatus here defined is the same as the endeavour used in Ax. 20 of MS. II is evident from the treatment of centrifugal force in MS. IV based on the use of the ‘deviation’ between the actual, circular path and the natural, inertial one along the tangent. There Newton states that if the conatus a centra were not impeded (as it actually is) the body would move freely along the tangent. The laws of collision given by Descartes in Part II of his Principia were notoriously full of error. 3 Nevertheless they represented the first widely known published attack^ on the problem based on the notion of momentum and its conservation, and as such would inevitably have influenced Newton’s approach, if only indirectly. But Newton’s treat­ ment of collisions in MS. II certainly antedated the celebrated papers of Wren, Wallis, and Huygens,^ so we are entitled to assume that any influence of Descartes on Newton in that manuscript was direct. In the light of what we already know of Descartes’s influence in MS. II, effectively certain in the case of the principle of inertia, highly probable in that of circular motion, we are justified in looking for an influence also in the case of collisions. An independent indication of such an influence is provided by the close similarity between Descartes’s explanation of the collision process as a contest between the persistence of motion of one body and the resistance to motion of the other, and one of the explana­ tions put forward by Newton in MS. II.^ Whether or not Newton took * A s opposed, for example, to Galileo, and even more strikingly to Beeckman, with his belief that both rectilinear and circular motions could be inertial. ^ For a discussion of New ton’s concept of conatus see below. Chapter 3. 3 See, G abbey [i]. Chapters 3, 4, for an exhaustive and sympathetic account of these rules. * A s opposed to that of J. Marcus Marci whose treatment of collisions in his De/)ro/)ortione motus seu regula sphygmica (Prague, 1639) seems largely to have escaped notice. * T h e papers of Wren and Wallis appeared in 1669, that of Huygens (in French) in 1670; whereas some o f New ton’s work on collisions in M S I lb is dated 20 January 1664 (O.S.). See the section Other topics above.

2.2

ON N E W TO N ’ S D Y N A M I C S

53

the concept of momentum itself directly from Descartes must remain uncertain— for this was a commonly accepted notion which went back at least to Buridan and which is found, for example, in Galileo’s Dia­ logue. In any case it seems probable that Newton took the idea of using momentum (and its conservation) in collision processes directly from Descartes. The arguments here advanced, if sound, point to a very important influence of Descartes on Newton in dynamics, direct in the case of the principle of inertia, circular motion, and collisions, indirect in the case of Newton’s concept of force. Newton’s ‘homage’ to Galileo at the beginning of the Scholium to the laws of motion would then need to be taken with considerable reserve. We would feel that at this point he should also have made an honest avowal of his debt to Descartes. That he did not do so would then provide one more example of the depth and intensity of his aversion to Descartes.

N EW TO N’ S CONCEPT OF

N E W T O N ’ S C O N C E P T OF C O N A T U S I t seems likely that Newton’s first views on conatus or endeavour^ resulted from his study of Parts 2 and 3 of Descartes’s Principia Philosophiae.^ In Art. 39, Part 2, of the Principia Descartes discusses the motion of a stone in a sling in the light of the principle of inertia ad­ vanced in Art, 37, explaining the constant endeavour of the stone to escape from the centre, as evidenced by the tension in the string, in terms of its continuous inertial tendency to escape along the tangent at every point of its path. Descartes thus seems to imply that the centrifugal endeavour derives entirely from the inertial tendency along the tangent, whereas in his later, and more detailed, discussion of circular motion in Arts. 57-59 of Part 3 of the Principia he appears to believe in two separate endeavours, the one along the tangent as before, and another, indepen­ dent, endeavour away from the centre along the outward radius. This centrifugal endeavour, like the tangential one, is normally potential only. Nevertheless, Descartes seems to have supposed it capable of having the effects of a force, for example, in his ingenious example of a particle free to slide in a rotating tube.^ There the particle acquires an increasing outward motion along the tube, and how can this possibly occur in the absence of some cause or force ? So that when he returns at the end of Art. 59 to the original case of a stone in a sling it is not sur­ prising to find him employing the term vis in reference to the centrifugal endeavour: ^atque ista tensio, a sola vi qua lapis recedere conatur a centra sui motus exorta, exhihet nobis istius vis quafititatem’ . There is a striking, and I believe significant, similarity between Descartes’s ultimate views on endeavour and circular motion, and those first advanced by Newton in MS. II.C onsidering the motion of a ball within a spherical surface, he postulates, like Descartes, both a constant endeavour to escape along the tangent, and a centrifugal endeavour along the outward radius. This latter, centrifugal tendency is evidenced by the pressure of the surface on the ball, equal and opposite to that of the ball on the surface, and corresponds to the tension of the string for motion in a ‘ Endeavour will consistently be used in place of conatus apart from the phrase conatus a centra. ^ T h e best evidence for this is supplied by M S . Add. 4003 from which the extracts in M S . V I are taken.

Principia Pliilosophiae, Part 3, Art. 59.

Especially in AIS. Ild, Ax. 20, 21.

CONATUS

55

sling.^ That Newton, like Descartes, regarded the centrifugal tendency as a species of impeded force, normally potential, yet capable on occasion of being imagined as active, is likewise clear. In Def. 6, MS. V I, he actually defines endeavour as an ‘impeded force, or a force in so far as it is resisted’ in MSS. I l l and IV he employs the terms vis a centra and conatus a centra^ respectively, when referring to the centrifugal endeavour or tendency resulting from circular motion; and on folio i of MS. II he derives an exact relation involving the centrifugal endeavour for uniform circular motion by calculating the momentum acquired in a certain interval of time by a body constantly impelled in a straight line by a force equal to the given centrifugal endeavour.^ A similar procedure is imagined in MS. IVa,'^ and both there and on folio i of MS. II he com­ pares the resulting rectilinear motion to that under gravity. There can be little doubt, therefore, that Newton, like Descartes, originally regarded conatus a centra as a species of force very much on a par with the force of gravity. The situation is quite different in Newton’s treatment of orbital motion in Book I of the Principia., and in his discussion of the motion of the moon in Book III. Now the notion of centrifugal endeavour so prominent in the early treatment of circular motion has been replaced by that of centripetal force. A striking example of this change in outlook, of this shift from centrifugal to centripetal force, is provided by the treatment of uniform circular motion in Prop. IV, Theor. IV, of Book I. Once again, as in MS. IV, the central notion is that of deviation, but whereas this was formerly measured outwards between the circle and the tangent, now it is measured inwards between the tangent and the circle. In the absence of any reference to centrifugal force in the treatment of orbital motion in the Principia one might be tempted to assume that Newton had simply seen the error of his ways, had given up the old, false notion of a centrifugal endeavour separated from, and independent of, the inertial, tangential endeavour, and replaced it by the true notion of an impressed centripetal force. After all, one might argue, there is no such thing as a true centrifugal force within the framework of dynamics * A case referred to by Newton at the end of A x. 20, M S . Ild ; ‘the same may be understood if the body adb be restrained into circular motion by the thread . . .’ . ^ ‘ Conatus est vis inipedita sive vis quatenus resistitur.’

^ 'the force by which it endeavours from the centre n would bepet so much motion in a body' (italics mine).

* 'hie conatus corpora, si modo in directujn ad modum gravitatis continuo urgeret, impelleret per spatia . . . .’ M S . IVa, second para., beginning.

56

N E W T O N ’ S CONCEPT OF

CONATUS

presented in the Principia where the only unambiguous measures of both motions and forces are relative to absolute space, motion in a straight line with uniform speed relative to this space being likewise the one true type of inertial motion. O f course, relative to a rotating frame of reference centrifugal forces will make their appearance and bodies will appear to tend to move outwards away from the centre of rotation. But such forces and motions are not real, are merely engendered by measur­ ing them relative to the rotating frame, and are in reality no more than disguised manifestations of inertial motion relative to absolute space. Unfortunately for this simple attitude there is a great deal of evidence pointing towards a continued belief by Newton in centrifugal force right up to and possibly beyond the Principia. The first indication of such a continued belief in centrifugal force or conatus a centra is provided by the well-known letter of 23 June 1673 from Newton to Oldenburg (for Huygens). In the first paragraph of this letter^ Newton draws attention to the formulae for centrifugal force given (without proof) at the end of Huygens’s Horologium Oscillatorium, instancing a notion of his own to explain the invariable aspect of the moon from the earth on the grounds of her greater endeavour from the earth compared with her endeavour from the sun. There can be no doubt^ that Newton is here referring to MS. IVa, and he gives no indication that he has in any way altered the view of conatus a centra found in that manuscript. There are no references to centrifugal, as opposed to centripetal, force in any of the versions of the tract de Motu. Hardly surprising, since this tract evidently represents a first draft of Newton’s treatment of orbital motion in the lectures de Motu and the Principia. However, several references to centrifugal force occur in MSS. X probably com­ posed just prior to the lectures de Motu. In Def. 14 of MS. Xb centri­ fugal force is instanced as one species of the ‘force of a body’ to perse­ vere in its state of movement or rest. It is not stated whether the motion in question is rectilinear or not. But that this is intended is evident from a rather more detailed definition of the ‘force of a body’ in Def. 12 of MS. Xa, where centrifugal force is now instanced as a species of the ‘force of a body’ to persevere in its state of rest or of motion in a straight line with uniform speed. Why Newton should thus have instanced cen­ trifugal force as an example of the ‘force of a body’, i.e. its inertia, as opposed to a force derived from the tendency to persevere in a straight ' See M S . V ila .

^ As first pointed out by Hall [2].

3

N E W T O N ’S

C O N C E P T

OF

CONATUS

57

line (along the tangent) is not at all clear. The absence of any reference to centrifugal force in the preliminary definitions of the lectures de Motu and the Principia is perhaps an indication that Newton himself was not altogether happy about the above definitions in terms of the ‘force of a body’. The other reference to centrifugal force, at Def. 9 of M S. Xa, is evidently a preliminary draft of the well-known passage relating to rotary motion in the Scholium to the Definitions preceding Book I of the Principia. T o distinguish, says Newton, between relative and abso­ lute rotation, it is necessary to examine the effects, if any, of the rotation. In the case of relative rotation, such as that between the water and the bucket, there are no effects. In the case of absolute motion there is always some real effect, such as the curved surface of the water, or the tension in the string connecting the two rotating globes. And in all cases these real effects bear witness to an endeavour to recede from the axis of rotation. As Newton expresses it: This ascent of the water shows its endeavour to recede from the axis of its motion; and the true and absolute circular motion of the water, which is here directly contrary to the relative, becomes known, and may be measured by this endeavour. Soon after the Principia was presented to the Royal Society in June 1686 a bitter controversy broke out over Hooke’s claim that Newton owed him the notion of the inverse square dependence of gravitation as applied to the motion of the planets.^ In the ensuing correspondence between Newton and Halley^ there are a certain number of references to centrifugal force. With one exception, these all originate in Newton’s reference, in his letter of 20 June 1686 to Halley, to the first paragraph of his letter of 23 June 1673 to Oldenburg, and to the original paper (MS. IVa) from which was taken the example of the usefulness of centri­ fugal force quoted in the first paragraph of the 1673 letter. Nowhere in his description of this paragraph from the letter to Oldenburg, nor in his resume of the principal results of the original paper, does Newton give any indication of having changed his view of centrifugal force from that expressed in this letter and paper. A second reference to the 1673 letter occurs in Newton’s letter of 27 July 1686 to Halley. After reporting the discovery of a copy of the * T h e best general account of this controversy is that given in Koyr^ [2]. ^ See Correspondence, vol. ii, Letters 285-91.

58

N E W T O N ’ S CONCEPT OF

CONATUS

letter in question, backed up by a quotation of the first paragraph in extensOy he continues: Thus far this letter concerned the Vis Centrifuga. . . . Now from these words it’s evident, that I was at that time versed in the theory of the force arising from circular motion, and had an eye upon the forces of the planets, knowing how to compare them by the proportions of their periodical revolu­ tions and distances from the centre they move about: an instance of which you have here in the comparison of the forces of the moon arising from her men­ strual motion about the earth, and annual about the sun. So then in this theory I am plainly before Mr. Hooke. The remaining reference to centrifugal force occurs in Newton’s letter of 14 July 1686 to Halley. Writing of Halley’s experiment of a pendulum clock at St. Helena to demonstrate the diminution of gravity at the equator due to the diurnal motion he says: The experiment was new to me, but not the notion; for in that very paper which I told you was written some time above 15 years ago and to the best of my memory was written 18 or 19 years ago, I calculated the force of ascent at the Equator arising from the earth’s diurnal motion in order to know what would be the diminution of gravity thereby. In the light of the evidence so far presented what answer is to be given to the question of whether Newton continued to believe in the physical reality of centrifugal force up to and beyond the composition of the Principia ? On the one hand we have the striking absence of any reference to centrifugal force in the treatment of orbital motion in the Principia'y on the other hand we have the repeated references to this concept in the discussion of rotation in the Scholium to the Definitions, and in the correspondence of 1686 with Halley. Perhaps the best approach to some sort of understanding, and explanation, of this apparent para­ dox is to attempt to trace through the development of Newton’s views on conatus a centra from the beginning, paying special attention to a pos­ sible transition from the concept of centrifugal to that of centripetal force. There can be no doubt, as already indicated, that Newton’s treatment of circular motion in the earliest researches was firmly based on the notion of conatus a centro, or centrifugal force. The question then naturally arises; if he carried out a test of the inverse square law of gravitation during the Plague Years, ^more probably in 1666, what part, if any, did the notion of centrifugal force then play? One plausible ’ For a discussion of this question see below, Chapter 4.

NE W TO N ’ S CONCEPT OF C O N A T U S

59

assumption would be that centrifugal force played no role at all; in which case this test would have marked the transition from a belief in centrifugal to one in centripetal force, at least in the case of orbital motion.^ But how then explain the references to centrifugal force in the letter of 23 June 1673 to Oldenburg, especially that to the endeavour of the Moon from the Earth? It is natural to regard these references as evidence of a continued belief by Newton in centrifugal force for circular orbital motion; in which case one seems forced to assume that there was either no test of the law of gravitation during the Plague Years, or that it involved the notion of centrifugal force. This latter assumption is entirely plausible: Newton could well have regarded the stability of the Moon in her orbit about the Earth, and of the planets in their orbits about the Sun, as the direct result of a balance between a centripetal pull of the Earth, or the Sun, inwards, and a centrifugal force of the Moon, or the planets, outwards. This was the manner in which Borelli^ attempted to explain the stability of circular orbits, and Newton could possibly have been influenced by Borelli’s discussion prior to his first test— cer­ tainly he had read it by 1686.2 In any case, apart from any possible influence of Borelli,'^ it would have been fairly natural for Newton to approach the stability of a circular orbit in this way, especially if we assume, as seems plausible, that his derivation of the inverse square law of gravitation immediately prior to the first test was based on considera­ tion of the force of a globe rotating within a spherical surface. For in this case the centrifugal force of the globe, as evidenced by its pressure on the spherical surface, was exactly balanced by an equal and opposite pressure of the sphere on the globe, and this could have suggested a similar balance between the conatus a centra terrae of the Moon and the ' T his was originally my own view. ^ Borelli [i]. For a discussion of Borelli’s views see Koyre [3]. 3 And probably much earlier. There was a copy of Borelli’s work in New ton’s library. He refers to it in the unfortunate postscript to his letter of 20 June 1686 to Halley; ‘he has published Borell’s hypothesis in his own name; and the asserting of this to himself, and completing it as his own, seems to me the ground of all the stir he makes. Borel did something in it; and wrote modestly.’ ^ Or of Descartes, who in Art. 140 of Part 3 of his Principia explains the stability of a planet’s (circular) orbit by a balance between two opposing conati, one outward due to the circular motion of the planet about the sun, and the other inward due to the action of the surrounding vortex material. Replace the latter conatus by a natural inward tendency to the sun and there results Borelli’s hypothesis. Descartes’s view of the matter is referred to by Newton at the foot of p. 3 of M S . Add. 4003: ‘ S ed postea

tamen in Terra et Planetis ponit conatum recedendi a sole tanquam a centro circa quod moventur, quo per consimilem conatum Vorticis gyrantis in suis a sole distantijs librantur Art. 140 part 3.’ See M S . V I, § 2.

60

N E W T O N ’ S CONCEPT OF

CONATUS

3

inward pull of the Earth. A plausible explanation could then also be given of Newton’s peculiar suggestion (in his letter of 23 June 1673 to Oldenburg for Huygens) that the invariable aspect of the Moon from the Earth was due to her much greater endeavour from the Earth compared with that from the Sun. At first glance this theory seems absurd, for if the endeavour of the Moon from the Earth is greater than from the Sun, this would seem a good reason for the Moon to depart from the Earth rather than the Sun. But if we assume that Newton measured the inward pull of the Earth on the Moon by the outward endeavour of the Moon from the Earth, then the greater value of this endeavour compared with that from the Sun would now be a measure of the much stronger hold of the Earth on the Moon compared with that of the Sun. And this in turn could conceivably explain the invariable aspect of the Moon from the Earth. A similar interpretation applied to the original argument in MS. IVa would then tend to show that this manuscript was composed after the first test, the notion of universal gravitation being present in Newton’s mind when he composed it though absent from the manuscript itself— perhaps for reasons of secrecy, or because the agreement between theory and experiment in the first test, and in the test implicitly con­ tained in this manuscript, still left him in considerable doubt of the adequacy of the inverse square law of universal gravitation as an explana­ tion of the stability of the lunar and planetary orbits.* Given Newton’s familiarity with the notion of centrifugal force, his possible use of it to explain the stability of circular orbits, and his definite reference to it in the letter of 23 June 1673, it would have been natural for him to attempt to use it when he first seriously turned his attention to the problem of Kepler-motion in December 1679. There is even some evidence that he had actually contemplated applying the idea of centrifugal force to the problem of motion in an ellipse at a very early stage. For beneath his discussion of circular motion on folio i of the Waste Book occurs the following entry:

N E W TO N ’ S CONCEPT OF

CONATUS

61

that his attention would soon have become directed to centripetal rather than centrifugal forces. In the case of motion in a uniform circular orbit, as in the Moon about the Earth, or the approximate orbits of the planets about the Sun, it is natural to attach equal importance to the in­ ward pull towards the centre and the centrifugal force outwards. But the situation is vastly different in the case of the elliptical motion of a planet about the Sun at one focus. Now the problem of paramount in­ terest, one to which Kepler had tried in vain to give an answer, is the nature of the force exerted by the Sun on the planet. Having first proved the astounding result that this force must be directed solely towards* the focus, it remained to find the variation in its magnitude at different points of the ellipse. In some such way we must suppose Newton’s treat­ ment of motion in an ellipse, and then general orbital motion, became dominated by the notion of centripetal force directed to a point. It would, of course, have been possible for him to have introduced . fictitious cen­ trifugal force equal and opposite to the centripetal force. But this would have been entirely gratuitous. For it was now clear that the whole process of orbital motion under central forces could be looked on as a continual drawing away of the body from its natural, inertial path along the tangent into its actual curved path. The problem therefore was: given the orbit to determine the force, and vice versa. As Newton him­ self expressed it: 3

for the whole burden of philosophy seems to consist in this— from the pheno­ mena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena.^

There is no evidence, however, that he succeeded in solving the prob­ lem of Kepler-motion by this method in 1679. Instead it seems certain

Given the dominant role of centripetal force in the treatment of orbital motion in Book I of the Principia, and the expected absence of any use of, or reference to, the notion of centrifugal force, I think we need not regard the references to centrifugal force and planetary motion in the 1686 correspondence with Halley as evidence for Newton’s real views on this subject at that time. These are surely to be found in the Principia. And there is really no paradox here since Newton in his correspondence with Halley was not concerned to state his actual views on the subject, but only to describe his views of 1673 and earlier, and to attempt to con­ vince Halley that these proved his understanding of the problem of planetary motion, including a knowledge of the inverse square law of

* Remember W histon’s statement that N ew ton’s disappointment over the first test made him suspect ‘that this Power [that restrained the moon in her orbit] was partly that of Gravity, and partly that of Cartesius’ vortices’.

* T h e force could equally well have heen directed away from the focus, but in that case the orbits would have been hyperbolic rather than elliptic. ^ Preface to first edition of Principia.

If a body be moved in an Ellipse that its force in each point (if its motion in that point be given) would [ ?] be found by a tangent circle of equal crooked­ ness with that point of the Ellipse.

62

NE W T O N ’ S CONCEPT OF

CONATUS

gravitation, prior to that of Hooke. Opinions will differ on whether or not Newton was justified in citing these views as evidence of such an under­ standing, and whether there was not a certain element of special plead­ ing in his argument.^ It is also a moot point how' far so acute a mind as Halley, and one so well versed in the Principia, would have been persuaded by Newton’s case. Certainly it would have been most interesting to know how Newton would have replied to a request from Halley for an explana­ tion of his references to centrifugal force in the light of the treatment of orbital motion in the Principia. Nevertheless, although we can thus easily dispose of the apparent paradox posed by these references to centrifugal force in connexion with planetary and lunar motion in the 1686 correspondence with Halley, there still remains the other reference (in the letter of 14 July 1686) to the centrifugal force arising from the earth’s diurnal motion, and the various uses of the term in the discussion of rotary motion in the Scholium to the Definitions of the Principia. In the Scholium to the Definitions preceding Book I of the Principia Newton discusses the causes and ejfects which distinguish true from relative motions. The causes are the forces impressed to produce true motion ; whereas true motion in a body can only be generated by the action of such forces, motion relative to a given body may be produced by forces impressed on other bodies, or may even be absent in spite of the action of impressed forces, as when equal bodies are acted on simul­ taneously by equal forces. As to the effects which distinguish real from relative motion, these are ‘the forces of receding from the axis of circular motion’. Evidently, then, he has in mind the conatus or ‘endeavour of receding from a centre or axis of rotation’, a phrase actually used in the same passage. These centrifugal forces are in turn demonstrated by certain observable effects, such as the concave surface of the rotating mass of water in the bucket, or the tension in the string between the two rotating globes. Or, he might have added, by the diminution in the force of gravity at the equator due to the Earth’s supposed diurnal motion, as referred to in the letter to Elalley of 14 July 1686. I do not propose to enter here into the question of the proof supplied by these and other observable effects for the existence of absolute rota­ tion. The point at issue is the existence of centrifugal forces arising from the rotation of extended bodies. Once again, as in the case of orbital motion, especially circular motion, there can be no doubt of the answer: relative to Newton's own system of dynamics there is no need for centrifugal ' For a discussion of this see below, Chapter 4, pp. 72-73.

3

N EW TO N’ S CONCEPT OF

CONATUS

63

force, something which can be seen most clearly by following the apparent development of such forces from a state of rest. For example, if a stone is attached to a slightly elastic spring whose other end is fixed, and is suddenly set in motion by a blow at right angles to the length of the spring, then the ensuing motion of the stone, and the resulting tension in the spring, can both be explained in terms of the laws of motion and the elastic properties of the spring. No introduction of centrifugal force is necessary here any more than in the discussion of the oscillatory motion resulting from the projection of the stone along, instead of at right angles to, the spring; nor is the ensuing tension in the spring any more (or less) mysterious in the first case than in the second. It is only when we imagine a rotary motion as pre-existing from eternity^ rather than having been created at some instant of time, that there is any mystery about the resulting forces in the rotating body. It can be argued, of course, that centrifugal forces make their appear­ ance relative to rotating frames of reference. For example, a body attached to a rotating plate experiences a ‘force’ outwards; if released it will move along the outward radius relative to the plate. But this is only to say that when released the body will move uniformly in a straight line along the instantaneous tangent to its original path relative to absolute space; from which its observed motion relative to the plate can immediately be deduced. So that it is true to say that the body has a centrifugal endeavour relative to the rotating plate. But the ‘results’ of the rotation, for example, the pressure of the body on its support, will be the same whether we look at the plate from the standpoint of the rotating frame or from that of absolute space, it is only their explanation which is different: relative to absolute space their origin and presence is explained in terms of the laws of motion in precisely the same way as the tension produced in a slightly elastic spring by a body moving (approximately) in a circle. Relative to the rotating frame their presence (but not their origin) can be explained in terms of supposed centrifugal forces corresponding to the undoubted centrifugal endeavours. If it is accepted that centrifugal forces have no place within the New­ tonian framework of absolute space and time, being no more necessary for an explanation of the effects of rotation than in the case of orbital motion, it has still to be asked why Newton retained these forces for rotating bodies while dispensing with them for orbital motion. Initially, of course, the notion of introducing centrifugal force in rotating bodies would have been a perfectly natural one. For we know that Newton

64

N E W T O N ’ S CONCEPT OF

CONATUS

originally believed in the existence of centrifugal forces for single bodies or particles describing uniform circular motions, and in the case of the uniform rotation of an extended body all its component parts describe uniform circular motions about centres lying on the axis of rotation. But whereas the study of orbital motion after the solution of the problem of Kepler-motion revealed no properties which could not be accounted for purely in terms of centripetal forces, this was apparently not the case for the effects produced in the rotation of extended bodies. It was therefore natural, though as we have seen inconsistent, for Newton to continue to employ the concept of centrifugal force in the latter case while dispensing with it in the first. A word finally about the case of centrifugal force instanced by Newton in the 1686 correspondence with Halley, that referring to the diminution of gravity to be expected at the equator due to ‘the force of ascent at the equator arising from the Earth’s diurnal motion’. Newton’s first calcula­ tion of this force of ascent is to be found in MS. III. It was therefore one of the earliest problems in circular motion to be treated by him. It was likewise the only one in which the notion of conatus a centro was entirely justified, since he would undoubtedly have thought of the problem from the point of view of an observer on the Earth’s surface. For such an observer the conatus a centro could become a real phenomenon provided the rotation of the Earth were sufficiently large, or the gravity of the Earth sufficiently small. Things would then fly off the surface as envis­ aged by Ptolemy, but upwards, towards the sky! It would be interesting to know if Newton had this example of conatus a centro in mind when developing his exact theory of circular motion in MS. IVa.

T E S T S OF T H E L A W OF G R A V I T A T I O N A G A IN S T THE M O O N ’S MOTIO N R e f e r e n c e s to a test of the law of gravitation during the Plague Years occur in the following accounts of Whiston,^ Pemberton,^ and Newton^ himself:

Whiston: What the Occasion of Sir Isaac Newton’s leaving the Cartesian Philosophy, and of discovering his amazing Theory of Gravity was, I have heard him long ago, soon after my first Acquaintance with him, which was 1694, thus relate, and of which Dr Pemberton gives the like Account, and somewhat more fully, in the Preface to his Explication of his Philosophy: It was this. An Inclination came into Sir Isaac’s Mind to try, whether the same Power did not keep the Moon in her Orbit, notwithstanding her pro­ jectile Velocity, which he knew always tended to go along a strait Line the Tangent of that Orbit, which makes Stones and all heavy Bodies with us fall downward, and which we call Gravity} Taking this Postulatum, which had been thought of before, that such Power might decrease, in a duplicate Proportion of the Distances from the Earth’s Center,^ Upon Sir Isaac’s First Trial, when he took a Degree of a great Circle on the Earth’s Surface, whence a Degree at the Distance of the Moon was to be determined also, to be 60 measured Miles only, according to the gross Measures then in Use. He was, in some Degree, disappointed, and the Power that restrained the Moon in her Orbit, measured by the versed Sines of that Orbit,® appeared not to be quite the same that was to be expected, had it been the Power of Gravity alone, by which the Moon was there influenc’d. Upon this Dis­ appointment, which made Sir Isaac suspect that this Power was partly that of Gravity, and partly that of Cartesius’s Vortices,^ he threw aside the Paper of his Calculation and went to other Studies. Pemberton: The first thoughts, which gave rise to his Principia, he had, when he retired from Cambridge in 1666 on account of the plague. As he sat alone in a garden,7 he fell into a speculation on the power of gravity; that as * Whiston [i], vol. i, pp. 35-36. ^ Pemberton [i]. Preface. ^ Catalogue of Portsmouth Collection (Cambridge, 1888), Section i, Division xi, number 41. Ball [i], p. 6, was of the opinion that this was written some years after the 1686 correspondence with Halley, ‘perhaps about 1714 ’. See, for example, Bullialdus [i], p. 23. * Implying, rightly or not, that Newton measured the force on the moon in terms of her ‘fall’ towards the Earth in a given interval of time. * Various references to Descartes’s vortex theory in M S . I prove New ton’s familiarity with the theory at an early date. 7 W e are reminded here of the incident of the falling apple of which the most cir­ cumstantial account is that given by Stukeley [i], p. 19;

66

T E S T S OF THE LAW OF G R A VI T AT IO N

this power is not found sensibly diminished at the remotest distance from the center of the earth, to which we can rise, neither at the tops of the loftiest buildings, nor even on the summits of the highest mountains; it appeared to him reasonable to conclude, that this power must extend much farther than was usually thought; why not as high as the moon, said he to himself? and if so, her motion must be influenced by it; perhaps she is retained in her orbit thereby. However, though the power of gravity is not sensibly weakened in the little change of distance, at which we can place ourselves from the center of the earth, yet it is very possible that, so high as the moon this power may differ much in strength from what it is here. To make an estimate, what might be the degree of this diminution, he con­ sidered with himself, that if the moon be retained in her orbit by the force of gravity, no doubt the primary planets are carried round the sun by the like power. And by comparing the periods of the several planets with their distances from the sun,’ he found, that if any power like gravity held them in their courses, its strength must decrease in the duplicate proportion of the increase of distance. This he concluded by supposing them to move in perfect circles concentrical to the sun, from which the orbits of the greatest part of them do not much differ. Supposing therefore the power of gravity, when extended to the moon, to decrease in the same manner, he computed whether that force would be sufficient to keep the moon in her orbit. In this computation, being absent from books, he took the common estimate in use among geographers and our seamen, before Norwood had measured the earth, that 6o English miles were contained in one degree of latitude on the surface of the earth. But as this is a very faulty supposition, each degree containing about 69J of our miles,^ his computation did not answer expectation; whence he concluded, that some other caused must at least join with the action of the power of gravity on the moon. On this account he laid aside for that time any farther thoughts upon this matter. Newton: I found the method [of fluxions] by degrees in the years 1665 and 1666. In the beginning of the year 1665 I found the method of approximat­ ing Series and the Rule for reducing any dignity of any Binomial into such a series. The same year in May I found the method of tangents of Gregory and Slusius, and in November had the direct method of fluxions, and the ‘A fter dinner on 15th April 1726 the weather being warm we went into the garden and drank thea under the shade of some apple trees, only he and myself. Am idst other discourse, he told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. It was occasioned by the fall of an apple, as he sat in a contemplative mood.’ * B y means of Kepler’s T hird Law of Planetary Motion. ^ Cajori [i], p. 150, points out that this implies i mile == 5280 ft whereas it seems more probable that Newton would have taken i mile = 5000 ft. See n. 3, p. 68, below. 3 For example, that of ‘Cartesius’s Vortices’ as remarked by Whiston. A t this point Whiston is therefore more detailed than Pemberton, thus strengthening the possibility of his account having been independent o f the latter’s.

A G A I N S T THE MOON’ S MOTION

67

next year in January had the Theory of colours, and in May following I had entrance into the inverse method of fluxions. And the same year’ I began to think of gravity extending to the orb of the Moon, and having found out how to estimate the force with which [a] globe revolving within a sphere presses the surface of the sphere, from Kepler’s Rule of the periodical times of the Planets being in a sesquialterate proportion of their distances from the centers of their Orbs I deduced that the forces which keep the Planets in their Orbs must [be] reciprocally as the squares of their dis­ tances from the centers about which they revolve: and thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, and found them answer pretty nearly. All this was in the two plague years of 1665 and 1666, for in those days I was in the prime of my age for invention, and minded Mathematicks and Philosophy more than at any time since. Whiston stated that he had his account from Newton soon after making his acquaintance in 1694, and it is possible, if not probable, that he made a note of it at the time. But since Whiston refers to the ‘like account’ of Pemberton it would be dangerous to regard the two accounts as entirely independent, and since Pemberton’s account was derived from Newton towards the end of the latter’s life^ it is natural to regard Newton’s own earlier^ account as the most reliable of the three. Nevertheless, the accounts of both Whiston and Pemberton contain certain circumstantial details having an authentic ring difficult to resist.'^ For example, Whiston’s account gives a strong impression that Newton’s recollection of his first test of the law of universal gravitation arose from a chance reference by Whiston to the ‘Cartesian philosophy’ he was then studying at Cambridge.s Newton possibly replied that he too had formerly studied the same philsophy,^ and then proceeded to describe the occasion of his leaving that philosophy and discovering his own ’ T h is could refer either to 1665 or 1666. From the context 1666 would seem to be the more probable year. It is also the one given by Pemberton. ^ ‘For it was in the very last years of Sir Isaac’s life, that I had the honour of his acquaintance’ ; Pemberton [i], Preface, p. 2. 3 See n. 3, p. 65 above. Against the view that Pemberton’s account is worthless, having been taken from Newton in extreme old age, it could be argued that memories of childhood and youth are sometimes much more vivid in old age than in middle life. O n this view it is just conceivable, though perhaps improbable, that Newton remembered his first test of the law of gravitation in his old age having previously forgotten it in 1686 at the time of the controversy with Hooke. 5 ‘We at Cambridge, poor Wretches, were ignominously studying the fictitious H ypo­ theses of the Cartesian, which Sir Isaac Newton had also himself done formerly, as I have heard him say’ : Whiston [i], immediately before quotation on p. 65 above. ^ As proved conclusively by M S . Add. 4003.

68

T E S T S OF THE L AW OF G R A V I T A T I O N

4

theory of gravitation. Of particular interest here is the reference to Newton’s disappointment at the failure of his test having lead him to suspect that This power [by which the moon was influenced] was partly that of Gravity, and partly that of Cartesius’s Vortices. Evidently he had not entirely freed himself from the persuasive argu­ ments of Part 3 of Descartes’s Principia Philosophiae\ Equally telling is Pemberton’s statement that Newton took the common measure of 6o miles to a degree of latitude at the surface of the earth ‘being absent from books’. One discrepancy between Newton’s account and those of Whiston and Pemberton has been the subject of much discussion, namely Newton’s statement that he found the force necessary to keep the moon in her orbit to answer ‘pretty nearly’ to the force of gravity, as opposed to Whiston’s statement that Newton was ‘in some degree disappointed’, and Pember­ ton’s statement that ‘his computation did not answer expectation’. The extent of this discrepancy obviously hinges on the meaning to be attached to Newton’s phrase ‘pretty nearly’, and this in turn depends on the actual value derived by Newton for the ratio of the force of gravity at the distance of the moon’s orbit to the same force at the surface of the Earth, or of the corresponding figure for the ‘fall’ of the moon in her orbit in one minute of time. Assuming that he took the period of the moon in her orbit as approximately 27 days 8 hours, ^the radius of the moon’s orbit as sixty times that of the radius of the Earth,^ and the figure of 60 miles to a degree of latitude at the surface of the Earth mentioned by both Whiston and Pemberton, the fall of the moon in her orbit in one minute would have come out at approximately 13*9 or 13-2 feet, according as he set one mile equal to 5280 or 5000 feet. After a thorough investigation of contemporary evidence Cajori concluded that Newton was more likely to have set one mile equal to 5000 rather than to 5280 feet.^ Since this finding is confirmed by Newton’s use of 5000 feet to a mile in M SS. I l l and IVa, whereas he nowhere in the early manuscripts uses the figure of 5280, it seems most probable that the actual figure arrived at by Newton for the ‘fair of the moon in her orbit was approximately 13*2 feet in one * In M S . IV a it is taken as 27 days, 7 hours, 43 minutes. ^ T h e figure adopted in M S . IVa. ^ See Cajori [i], pp. 150-6. Cajori could only find one author prior to 1666 (Oiightred) using 1° = 60 miles of 5280 ft.

A G A I N S T THE MOON’ S MOTION

69

minute.^ This has to be compared with the true figure of approximately 16 feet in one minute, corresponding to a fall of the same distance in one second at the surface of the Earth, a figure derived by Newton in MS. III." It seems probable, therefore, that in this particular respect the accounts of Pemberton and Whiston are rather more reliable than that of Newton himself. Nevertheless, in spite of this discrepancy, and in the absence of any further evidence, there would seem no very good reason to doubt that Newton carried out a test of the inverse square law of gravitation during the Plague Years, more probably in 1666, the year mentioned by Pemberton, and likewise the more probable year from Newton’s own account.3 Some further items of evidence must now be considered, some of which support, while some oppose, the supposition of a test during the Plague Years. One item of evidence supporting the bare possibility of some test or other of the law of gravitation at an early period is supplied by Newton’s treatment of the problem of circular motion in the early dynamical manuscripts.'^ This, as we have seen, commences in the Waste Book and culminates in MS. IVa devoted to circular motion and containing an explicit statement of the law of centrifugal force. Without this law Newton could not have derived the inverse square law of gravitation by way of Kepler’s third law of planetary motion. Whereas the certain knowledge that he was in possession of this law at an early period,s possibly as early as 1665 or 1666, must necessarily increase our confi­ dence in the account in the Portsmouth Draft Memorandum. One striking detail of this account increases our confidence still further, namely, New­ ton’s reference to his having found out how to estimate the force with which [a] globe revolving within a sphere presses the surface of the sphere. This was exactly the situation considered by Newton in his first re­ connaissance of the problem of circular motion in the Waste Book.^ * Assuming, of course, that he was thinking in terms of such a ‘fall’ produced by the pull of the Earth’s gravitation as opposed to a balance between this latter force and an outward centrifugal conatus. In the latter case 13-2 would represent the distance out­ ward through w hich the moon would have moved in one minute under the action of this conatus. See Chapter 3 above for New ton’s concept of conatus. ^ T h e original figure employed in that manuscript, about half the figure finally de­ duced, would have led to an intolerably large discrepancy between theory and observa­ tion for the motion of the moon. ^ See n. i, p. 67 above. * See above, Chapter 1.2. 5 See below. Chapter 6.2. * See M S . Ild , Ax.-Prop. 20.

70

T E S T S OF THE LA W OF GR A VI T AT IO N

Nevertheless, although he went on to derive an exact result involving the centrifugal force or conatus at folio i of that manuscript, there is appa­ rently no trace of his having completed this line of approach by deriving the actual dependence of the centrifugal force on the radius and the speed as implied by the account in the Portsmouth Draft Memorandum. At this point a suggestion of Rouse Ball’s^ comes to the rescue, namely, that the proof of the law of centrifugal force at the end of the Scholium to Prop, IV, Theor. IV, of Book I of the Principia^ was possibly the actual one employed by Newton to calculate the force exerted by the globe on the sphere. That this proof corresponds to the situation referred to by New­ ton in the Portsmouth Draft Memorandum is at once evident: equally evident is the similarity between the method here employed— replace­ ment of the actual circular path by an inscribed polygon, and the actual continuous force by a series of sharp forces of reflection at the corners of the polygon— and the method followed by Newton in his treatment of circular motion on folio i of the Waste Book. One further item of evidence is supplied by Newton at the end of the letter of 14 July 1686 to Halley in which he enclosed the proof in question with instructions to insert it after the Scholium to Prop. IV, Theor. IV, of Book I of the Principia\^ there he mentions that he met with this proof ‘in turning over some old papers’. In the light of this mutually interlocking evidence it seems reasonable to suppose that a test of the law of gravitation did take place during the Plague Years, being based on a derivation of the law of centrifugal force by the ‘polygonal’ method, and in all probability also on the figure of 60 miles of 5000 feet to a degree of latitude at the equator. Otherwise one seems forced to conclude that Newton not only misinformed both Whiston and Pemberton, but also fabricated the very circumstantial account in the Portsmouth Draft Memorandum, presumably after re­ freshing his memory of his early researches in dynamics by referring to the Waste Book. Let us turn now to the evidence provided by the Newton-Halley correspondence of 1686+ arising out of Hooke’s claim that Newton was indebted to him for the notion of the inverse square law of gravitation as applied to the motions of the planets. Hooke’s claim stung Newton to the quick, more especially, one feels, because he was well aware that ' See Ball, [1], p. 13. * Reproduced above, at Chapter 1.2, p. i i . ^ A nd thus conveniently ‘compass’, as Neii'ton thought, the dispute between him­ self and Hooke. See especially Koyre [2].

A G A I N S T THE MOON’ S MOTION

71

he was indebted to Hooke for at least the initial impetus which led to his thorough investigation of the problem of Kepler-motion in the winter of 1679-80. Evidently he would dearly have loved to have been able to present Halley (and the Royal Society) with incontrovertible dated evidence of his knowledge of the inverse square law long prior to any public or private statement of it by Hooke. But he was unable to do so, having to content himself with two items of evidence, neither of them very impressive. The first, and much less important, item is contained in the postscript of his letter of 20 June 1686 to Halley. There Newton quotes the following passage from his letter of 7 December 1675 to Oldenburg: And as the earth, so perhaps may the sun imbibe this spirit copiously to con­ serve his shining, and keep the planets from receding further from him; and they that will may also suppose that this spirit affords or carries with it thither the solary fuel and material principle of light.. . . Newton himself made no large claim on the basis of this passage, con­ tenting himself with noting that it proved his concern with the problem of planetary motions at this time; as for the inverse square dependence of the force of gravity supposedly implicit in the passage, it was ‘But an hypothesis, and so to be looked on only as one of my guesses’. The other, and much more important, item is found in the following passage in the body of the same letter to Halley: That in one of my papers writ (I cannot say in what year, but I am sure some time before I had any correspondence with Mr Oldenburg, and that’s above fifteen years ago)^ the proportion of the forces of the planets from the sun, reci­ procally duplicate to their distances from him, is expressed, and the propor­ tion of our gravity to the moon’s conatus recedendi a centra terrae is calculated, though not accurately enough. That when Hugenius put out his Horol. Oscil., a copy being presented to me, in my letter of thanks to him, I gave those rules in the end thereof a particular commendation for their usefulness in Philo­ sophy, and added out of my aforesaid paper an instance of their usefulness, in comparing the forces of the moon from the earth, and earth from the sun; in determining a problem about the moon’s phase, and putting a limit to the sun’s parallax, which shows that I had then my eye upon comparing the forces of the planets arising from their circular motion, and understood it. Newton was later to substantiate his claim to have written this letter to Oldenburg by discovering a copy in the hand of his friend John Wickins, quoting the first paragraph, that relating to centrifugal force, in * In his letter of 14 July 1686 to Halley Newton refers again to this paper, stating ‘to the best of my memory [it] was w’rit 18 or 19 years ago’.

72

T E S T S OF THE L A W OF G R A VI T AT IO N

his letter of 27 July 1686 to Halley. The existence of the paper referred to by Newton is likewise certain, since it may be identified with MS. IVa.^ Given Newton’s very detailed and accurate synopsis of this paper in the above passage it seems likely that he had the actual paper before him when composing his letter of 20 June 1686 to Halley. In any case it is evident that nowhere in this letter does he go beyond the actual con­ tents of the paper. In particular, he is very careful to refer always to the forces of the planets from the sun, or of the moon from the Earth. Like­ wise he nowhere explicitly claims that this paper proved his familiarity with the inverse square law of gravitation as opposed to the inverse square dependence of the forces of the planets from the sun. Such a claim would, in fact, have been entirely unjustified, since an examination of the paper reveals that although he compares the ratio of the force of gravity at the Earth to the moon’s endeavour to recede from the sun, and proves that the endeavours of the planets vary inversely as the squares of their distances from the sun, yet he makes no mention of the forces on the planets towards the sun, or of the force on the moon towards the Earth, so that there is apparently no indication whatsoever of the notion of universal gravitation.2 And this is particularly evident in the omission of any comment on the discrepancy between the calculated ratio of gravity at the surface of the Earth to the moon’s endeavour to recede from the Earth, and the ratio to be expected on the basis of the inverse square law of gravitation. Nevertheless, whether intentionally or not, Newton’s references to the ‘duplicate proportion’ of the forces of the planets from the sun, and his note that the ratio of the force of gravity to the moon’s endeavour to recede from the Earth was not calculated ‘accurately enough’, would inevitably have given Halley the impression that Newton thought this paper proved his early familiarity with the ‘duplicate proportion’ of universal gravitation. For this was the invariable* * As first pointed out by Hall [2]. ^ Though David Gregory evidently formed a different impression on his visit to Newton at Cambridge in M ay 1694: ‘I saw a manuscript [written] before the year 1669 (the year when its author M r Newton was made Lucasian Professor of Mathematics) where all the foundations of his philosophy are laid: namely the gravity of the M oon to the Earth, and of the planets to the Sun. And in fact all these even then are subjected to calculation. I also saw in that manuscript the principle of equal times of a pendulum suspended between cycloids, before the publication of H uygens’ Horologium Oscillatorium.' (University Library, Edinburgh, Gregory C43, quoted in translation, Correspondence, vol. iii, p. 332.) Since there » a paper on cycloidal pendulums in the Portsmouth Collection (M S. IVb) written in the same hand, and on the same size of paper as M S . IVa, it seems very prob­ able that the latter was the manuscript referred to by Gregory in the above passage.

4

A G A I N S T THE MOON ’ S MOTION

73

meaning of the phrase elsewhere in the same and other letters of the correspondence. ^ Admittedly, the fact that this paper contains no trace of the concept of universal gravitation does not prove that Newton had not entertained such a notion prior to its composition. As Hall rightly remarks: ‘With Newton, however, the argument from silence is never s tr o n g ,O n the other hand, the continual reference to endeavour from the centre in circular motion, and the absence of any reference to centripetal force is in striking, and I think significant, contrast to the position in the Principia. For there the terms centrifugal force, conatus, or endeavour from the centre, nowhere appear in Newton’s treatment of orbital motion in Books I and III .3 Instead the emphasis throughout is on centripetal force in its capacity of drawing a body away from its natural, inertial path along the tangent, and forcing it to follow its actual curved orbit about the centre of force. This approach to the problem of orbital motion is particularly evident in the discussion of gravitation in terms of the moon’s motion about the Earth at the beginning of Book III. One seems forced to conclude that if Newton really intended to give Halley the impression that this paper implied not only an interest in the prob­ lem of planetary motions, as it certainly did, but also an understanding of that problem including a knowledge of the ‘duplicate proportion’ of universal gravitation and its application to the motion of the moon, then he was apparently following a line of argument quite different from that found in the Principia presented to the Royal Society a short time previously.'^ Given that this paper contained no explicit evidence for a knowledge of the inverse square law of gravitation, and that Newton seems scarcely to have been justified in giving the impression that it implied such a knowledge, why then did he make no mention of the early test of the law of gravitation referred to in the accounts of Whiston, Pemberton, and Newton himself? One possible explanation, of course, would be that there was no such test apart from the somewhat inaccurate comparison of ' For example, in the letter of 27 M ay (twice), of 20 June (three times), 14 July (once), 27 July (once). 2 Hall [2]. 2 This, however, unfortunately does not prove that Newton had entirely given up the concept of conatus. There are the references to this concept in New ton’s discussion of absolute motion in the Scholium to the laws of motion. See above. Chapter 3, p. 62. * But it must be remembered that the first test of the law of gravitation was possibly based on the notion of a balance between the inward pull of the Earth and an equal and opposite outward conatus of the moon. See above. Chapter 3, p. 59.

74

T E S T S OF THE LA W OF G R A VI T AT IO N

the force of gravity at the surface of the Earth to the moon’s endeavour to recede from the Earth contained in MS. IVa. In that case one has to assume that both the reported use of 6o miles to a degree at the surface of the Earth, and the method of determining the law of centrifugal force referred to in the Portsmouth Draft Memorandum were figments of Newton’s imagination. For the measure of the Earth’s radius employed in MS. IVa was the same as that employed in MS. I ll, in turn derived by Newton from Galileo’s Dialogue^ and though it happens to differ little from 6o miles of 5000 feet to a degree at the surface of the Earth,* this seems no reason for supposing that Newton would have confused the two measures. Likewise the method of deriving the law of centrifugal force in MS. IVa, based on the use of the ‘deviation’ as a measure of the endeavour to recede from the centre together with an application of Newton’s extension of Galileo’s P law of falling bodies, could never be confused with the polygonal approach to the same problem begun in the Waste Book and completed in the proof at the end of the Scholium to Prop. IV, Theor. IV, of the Principia. On the other hand, if we reject this possibility, and assume that there was a true test of the inverse square law during the Plague Years, it is then hardly surprising that Newton made no mention of it in his cor­ respondence with Halley. For it is evident he was looking for dated evidence— ^witness his emphasis on the letter of 23 June 1673 to Olden­ burg— and the original calculations of the first test, even if still extant in 1686, would probably have been undated— like almost all early Newton dynamical manuscripts including MS. IVa— and therefore of no help in establishing a watertight case against Hooke. In the absence of further evidence, especially of any documentary evidence of the calculations of the first test, or of the original of the proof given at the end of the Scholium to Prop. IV, Theor. IV, of the Principia, it is impossible to make a definite choice between these two possibilities. On the one hand, the assumption that there was no true test of the law of gravitation during the Plague Years argues for a degree of duplicity on Newton’s part, both in casual conversations with Whiston and Pemberton (in the latter case long after Hooke’s death), and in his unpublished account in the Portsmouth Draft Memorandum, difficult to credit even allowing for a continued desire to establish his indepen­ dence of Hooke in the matter of universal gravitation. On the other * A radius for the Earth of 3500 Italian miles of 5000 feet gives one degree at the equator equal to 6 1 -i miles.

4

A G A I N S T THE MOON’ S MOTION

75

hand, it seems possible, even probable, that Newton was guilty of some doubtful pleading in his letter of 20 June 1686 to Halley, reminding us forcibly of Hooke’s bitter entry in his diary on 15 February 1688/9: At Halys met Newton— ^vainly pretended claim yet acknowledged my information. Interest has no conscience. . . . On the basis of Pemberton’s account it has generally been assumed that Newton made his first successful test of the law of gravitation based on Picard’s figure for the dimensions of the earth in 1679, following Hooke’s intervention. In that case, given his successful solution to the problems posed by Kepler’s first two laws of planetary motion, it is certainly extraordinary that he should have lost interest in dynamics again till the time of Halley’s first visit to Cambridge in May 1684.* There is, however, one aspect of Pemberton’s account in flat contra­ diction with other, more trustworthy evidence, namely his statement that having discovered the proposition relating to motion in an ellipse Newton went on to compose nearly a dozen propositions relating to the motion of the primary planets about the sun, the Principia being com­ posed from scarce any other material than those few propositions in the space of one year and a half. For elsewhere Newton himself stated that the Principia was entirely composed between December 1684 and May 1686 apart from two propositions discovered in December 1679, and twelve discovered in June and July 1684. The twelve propositions in question fit remarkably well with the contents of the tract de Motu^ which was thus most probably composed by Newton around June and July 1684 following Halley’s first visit in May of that year. The likeli­ hood of an error in Pemberton’s account at this point thus makes it permissible to doubt his assertion that Newton made his successful test of the law of gravitation in 1679. In that case the most probable alterna­ tive date would be in 1684 or 1685, at some time following Halley’s visit in May 1684. In fact, from Conduit’s account^ we know that Newton was unable to find the proposition of 1679 on the occasion of Halley’s visit. After Halley had left he would very probably have made a further careful search through all his old papers in dynamics for the missing proposition, and it may have been in the course of this search that he came across the paper containing the calculations of his first test of the law of gravitation during the Plague Years. If he had reworked these using Picard’s figure the resulting excellent agreement between ' A s opposed to August. See below, Chapter 6.4, p. 97. ^ See below, Chapter 6.4, p. 96. ^ Given in Brewster [i] vol. i, p. 297.

76

T E S T S OF THE LAW OF GR A V I T A T I O N

theory and observation would have given him just the stimulus he needed to rework the 1679 proposition, and then carry his dynamics through to their final completion— as we know he did. In any case, it seems reasonably certain that the successful test was carried out before the summer of 1685^ at the latest, for in Version III of the tract de Motu, in the penultimate paragraph of the Scholium to Prob. 5 we find the following: Nam virium centripetarum species una est gravitas: et computanti mihi prodijt vis centripeta qua luna nostra detinetur in motu suo menstrua circa terram, ad vim gravitatis his in superficie terrae, reciproce ut quadrata distantiarum a centra terrae quamproxime. * See below, Chapter 6.4, pp. 97-102, for a discussion of the date of composition of Version II I of the tract de M otu (M S. IXc).

T H E M O T I O N OF E X T E N D E D B O D I E S 5.1.

K inem atical

A spects

T he first dynamical entries in the Waste Book deal only with the move­ ment, in modern parlance, of a mass-point or particle, or an extended body devoid of rotation. But knowing Newton’s powerful drive to­ wards greater generality it is not surprising to find that he soon turns his attention to the general motion of extended bodies. His first recon­ naissance into this new territory is based on the concept of centre of motion (or rotation) as a point ‘which rests when a body is moved with any circular but not progressive motion’h and on a peculiar criterion for determining whether one body can be said to move towards another, namely, that a body is said to move towards another body either when all its parts move towards it, or else when some of its parts have more movement towards it than others have from it.^ This criterion is but an ansatz, and soon exhausts its usefulness, though its free creation by Newton reminds us that his genius comprised both that of the pure and the applied mathematician, whereas the concept of centre of motion continues to play a central role until it finally coalesces with the other, dynamical centre of motion in his treatment of the collision of two rotating bodies. ^ From this criterion it follows that a line rotating about its mid-point has no progressive motion, for the movement of any point of it towards any fixed line is exactly balanced by the opposite motion of the mirror points away from the same.'^ It follows likewise that when the centre of motion (or rotation) is not at the mid-point ‘the whole line moves the same way which the longest part doth’.s Also that if a line rotates but has no progressive motion (in the sense of Def. 12) its centre of motion must coincide with its middle point.^ And ‘by the same reason the middle point of a parallelogram, parallelipiped, prism, cylinder, etc. (are at the) centre of their motion’. Having made a beginning of the study of the pure rotation of an extended body he next considers the translational motion of such a body. The simplest case is that in which the body is moved parallel to itself J M S . lie , Def. 10. 4 M S . Ild , A x. ir. ’ Ibid., Ax. 14a.

^ Ibid., Def. 12. 5 Ibid., Ax. 12.

3 M S . V . §§ 9, 10. ^ Ibid., A x. 13.

78

THE MOTION OF EXTENDED BODIES

5.1

(pure translation) when all its points describe parallel lines each with the same determination and velocity as that of the body as a whole. * But if the body moves both ‘straightly forward and circularly’ its centre of motion has the same determination and velocity as the body.’ For example, any motion of a line (in a plane) may be compounded of a certain pure translation of the line parallel to itself followed by a rotation about its middle point. In the first movement the determination of the motion coincides with that of the mid-point, i.e. the centre of motion. In the second there is no progressive motion (by Ax. 11). A reference to Ax. 17 and the corresponding diagram will show that Newton’s proof of the result is identical with the modern proof. In the general case where the progressive part of the motion is no longer restricted to be rectilinear but is in any ‘crooked line’ the centre of motion has the same determination and velocity as the body.^ For this has been proved true in the special case (Ax. 17), and a ‘crooked line’ may be thought of as made up of an infinite number of straight lines. Alternatively, at any point of the ‘crooked line’ the motion may be con­ ceived to be along the tangent line. Unlike the case of translational velocity, for which a quantitative measure has been given, his treatment of circular movement has so far been entirely qualitative. A quantitative measure of the concept of circular movement remains to be given; this is found in MS. V where he states that ‘the angular quantity of a body’s circular motion and velocity is more or less accordingly as the body makes one revolution in more [or] less time . . .’.5 The rule for resolving angular velocity is also given in the same paper.^ Another kinematical result relating to rotational motion is found in Prop. 38 of the Waste Book. 5 .2 .

C entre

of

M otion

in

the

D ynam ical

S ense

We have already noticed Newton’s definition of centre of motion in the kinematical sense as a point ‘which rests when a body is moved with any circular but not progressive motion’. Elsewhere in the Waste Book he uses the same term in another quite distinct, dynamical sense. This ’ M S . Ild , A x. 14b.

^ Ibid., A x. 17.

^ Ibid., A x. 18.

* Evidently the (Archimedean) notion of approximating to a curve by a set of broken lines was second nature to Newton. It was also one of the most powerful instruments of his dynamical method as used, for example, in his earliest treatment of centrifugal force or in his proof of the proposition corresponding to Kepler’s second law. Charac­ teristic too is his method of proceeding from the simple (motion in a straight line) to the complex (motion in a curved line). 5 Loc. cit., § 5. * Ibid., § 6.

5.2

THE MOTION OF EXTENDED BODIES

79

centre of motion is first defined as a point such that if the body be rotating about an axis through it there will be the same quantity of motion on both sides of any plane through this axis.^ It is probable that this definition dated from a period when Newton still measured circular movement in terms of quantity of motion, not having developed the concept of endeavour away from the centre on which the following, final definition of centre of motion in the dynamical sense is based: ‘in every body there is a certain point called its centre of motion about which if the body be anyway circulated the endeavours of its parts every way from the centre are exactly counter poised by opposite endeavours’.2 He has therefore introduced a notion of dynamical balance similar to that found in practical applications of the theory of rotating bodies. In fact, if one slightly amends Newton’s definition to read ‘the endeavours of its parts every way from the axis of rotation’ then it easily follows that the point in question must coincide with the centre of mass or gravity of the body in question,^ a result well known to be necessary if the bear­ ings are to be free of periodic forces due to the rotation of the body. It is interesting to note that on occasion Newton first uses the term centre of gravity replacing it later by centre of motion.^ It is as if he had divined that the two points are identical. And as usual his physical intuition is correct. On the other hand, it is possible that both this interchangeable use of the terms centre of motion and centre of gravity and the final definition of centre of motion originally arose from his consideration of the centre of motion of a pair of bodies. This he defines as a point so placed between the bodies that if it be thought to rest and the bodies circulate around it any way they shall have equal quantities of motion.^ This is clearly based on the definition of centre of motion given in the immediately preceding paragraph. But he later ‘proves’ that the centre of motion for the pair of bodies as thus defined is at a point dividing the line joining their respective centres of motion in the inverse proportion of their quantities (i.e. their masses).^ For if the bodies circulate around this point always keeping opposite one another, they will have equal motions, and consequently equal endeavours from a\ so that if they be joined to a ‘the one hinders the other from forcing the centre any way so that it shall stand in equilibrio between them and (by Def. 10) is * M S . lie , Def. 10.

2 M S . V , § 4.

* For example, in Def. 10 of M S . lie . ^ M S . Ild , A x. 25.

2 See M S . V , n. 8. 5 Ibid., Def. i i . ’ See Fig. lo to M S . Ild .

80

THE MOTION OF EXTENDED BODIES

5.2

therefore their centre of motion’. The reference to Def. lo implies that he is here thinking of the centre of motion more in the kinematical sense as a point ‘which rests when a body is moved with any circular but not progressive motion’. Thus in this proof of the equivalence of the centre of gravity of two bodies and their common centre of motion we have at once a peculiar mixture of centres of motion in the kinematical and dynamical senses, and also what seems to be a bridge between the original dynamical defi­ nition in terms of quantity of motion and the final definition in terms of endeavour. But although his thinking on the question is obviously still provisional and heuristic yet his thought is always extraordinarily clear. Like good philosophy it is meaningful and persuasive even where one cannot follow every detail of the argument. And always his dynamical intuition is leading him towards the right goal, that final identification of the kine­ matical and dynamical centres of motion found in MS. V. Having thus defined the centre of motion of a pair of bodies Newton proceeds to prove a number of theorems relating to the motion of this point. These theorems provide a remarkable example of his drive to­ wards ever-increasing generalization. First he proves that the centre of motion of a pair of non-colliding bodies moving uniformly in one plane itself describes a straight line, ^and with uniform velocity.^ Then he proves that the same result holds when the lines of motion of the two bodies are no longer coplanar.^ Finally he shows that the result is unaffected by collision.4 The proof of the last theorem assumes that the quantities of motion of two bodies relative to their centre of gravity are equal and opposite, a generalization^ of the special case where the centre of motion is stationary,6 After this final result he characteristically notes^ that he is now able to find the position of the centre of motion of the two bodies at any time, and since their distance apart is also known the two spheres on which the bodies must lie at any instant are known, so that ‘there wants there­ fore only their determination to be known that their places in the sphere be found’.^ A good example of Newton’s tendency to give an exact for­ mulation of the solution in principle to some general problem. Newton’s discussion in the Waste Book of the centre of motion of a M S . Ilf, Prop. 28. Ibid., Prop. 31 , 2. M S . Ilf, Prop. 27,

^ Ibid., Prop. 30. 5 Given without proof. ’ Immediately after Prop. 32.

THE MOTION OF EXTENDED BODIES

5.3

pair of bodies ends with the case in which the two bodies are attached to a weightless rod lying perpendicular to the direction of motion of a third, impinging body.^ In this case although the attached bodies them­ selves will no longer describe straight lines, but will check one another and thus describe ‘crooked lines (perhaps Troichoides)’, yet their centre of motion will continue serenely to describe a straight line as when the bodies simply rested on the rod, and with the same speed. It is clear, then, that at this early stage Newton had already divined the dynamical importance of the centre of motion ( = centre of gravity) of a pair of bodies moving under some sort of mutual interaction. So that we see shadowed forth, as it were, the role which the centre of mass of the Solar System will play later in the Principia.

5.3 .

D ynam ics

of

a

S ingle

R otating

B ody

We have already noticed Newton’s semi-quantitative definition of the angular velocity of a rotating body.^ Now we are concerned with the dynamical aspect of such motion. One must, says Newton, distinguish between the angular quantity of a body’s circular motion and velocity, and the real quantity of such motion which is ‘more or less accordingly as the body has more or less power and force to persevere in that motion’.^ Which real quantity of circular motion divided by the body’s bulk (i.e. mass) is the real quantity of its circular velocity. But how can a quantita­ tive measure be assigned to this real quantity of circular motion? By the following method whose beauty is matched only by its inevitability. Suppose, says Newton, that the body is rotating about an axis E F (MS. V, Fig. 3) and let it strike a second, spherical body of equal bulk so placed that it captures all the motion of the rotating body. ‘Then hath the globe gotten the same quantity of progressive motion and velocity which the other had of circular.’ Newton attaches special importance to the distance between the axis of rotation E F and the line of motion of the centre of the sphere, terming it the ‘radius of circulation’. A simple analysis'^ based on application of conservation of momentum, angular momentum about the point of contact, and energy reveals that Newton’s ‘radius of circulation’ equals the radius of gyration, k, of the rotating body about EF, and that the real quantity of circular velocity, equal by defini­ tion to the velocity of the sphere, is kw where co was the original angular velocity of the rotating body.

Ibid., Prop. 29.

8 Ibid.

81

' M S Ilf, Prop. 37. 3 M S . V , § 5.

8.'^820r>

See above, end of § i of present chapter. See M S . V , n. 16.

82

THE MOTION OF EXTENDED BODIES

5.3

These three quantities, real quantities of circular motion and velocity, and radius of circulation later play a vital role in Newton’s solution to the problem of the collision between two rotating bodies. ^

Imagine a body rotating with angular velocity^ R about an axis A C acted on by a new force which if applied alone would cause the body to rotate about another axis CB with angular velocity S. Newton then gives a rule for finding the new axis of rotation of the body and its angular velocity about it. Although the rule only holds if the moments of inertia of the body about the three axes in question are equal, it is interesting for the indication it affords of Newton’s realization of the vector nature of angular momentum. Free rotation of an extended body In § 8 of the same MS. V Newton gives a wonderfully just physical appreciation of the free rotation of an extended body. In the first place every such body keeps the same real quantity of circular motion so long as it remains undisturbed. This is the principle of inertia for rota­ ting bodies. Moreover, it continues to rotate about the same axis which always remains parallel to itself provided the endeavours of its four quarters away from the axis of rotation balance. This condition is obviously necessary though apparently not sufficient. But if this balance does not obtain then although there will be a tendency for the body to draw ever nearer to such a balance it will never actually attain to it. And as the actual axis of rotation moves continually in the body, so it will also move continually space some kind of ‘spiral motion’, always drawing nearer and nearer to a ‘centre of parallelism with itself’ but never attaining it. ‘Nay ’tis so far from ever keeping parallel to itself that it shall never be twice in the same position.’ It is to be regretted that Newton did not proceed to give a quantitative treatment of this problem. It is clear that his unerring intuition had led him to an almost perfect physical appreciation of the problem. C ollision

b e t w e e n

T wo

R otating

B odies

We consider next Newton’s treatment of the problem of the collision of two rotating bodies, beginning with the special case where one body is immobile. It is likely that this special case, found towards the end of ' In §§ 9, l o of M S . V .

THE MOTION OF EXTENDED BODIES

83

the dynamical writings in the Waste Book, represented Newton’s first orientation towards the much more difficult general problem. One body immobile

Combination of circular motions

5 .4 .

5.4

^ See Fig. 5 to M S . V.

A spherical body colliding at right angles with an immobile plane will be reflected back along its original path, the pressures on each side of the vertical through the centre being equal and so producing no tendency to motion on one side or the other of the vertical.^ If a body collides at one corner with an immovable plane its motion parallel to the plane will be unaffected by the collision.^ From the dis­ cussion it is clear Newton had in mind a smooth plane such that a body ‘might slide upon it without losing any motion’. In the case just considered all the points in the body in the line Op perpendicular to the plane through the point of contact O shall move away from the plane immediately after impact with the same velocity they had before impact.^ This result follows from the facH that in rotary motion all the points of any line (and in particular Op) have the same velocity in the direction of this line, together with the assumption that the velocity of the point of contact from the plane immediately after impact is the same as that towards the plane immediately before. Both bodies rotating Owing to the unfamiliar manner of his approach Newton’s treatment of this problem in §§ 9 and 10 of MS. V is at first sight incomprehensible. However, by solving the problem by modern methods and then expressing everything in terms of the parameters used by Newton in § 5 of the same manuscript his solution may be shown to be correct.^ It remains, therefore, to consider the method by which he arrived at it. Here there are two distinct factors, his assumption that the relative velocity of the two points of contact merely changes its sign as a result of the collision, and his method of distributing the resulting change in relative velocity among the translational and real angular velocity components of the two points of contact. There is no indication of the origin of the first assumption. But know­ ing Newton’s invariable habit of passing from the simple to the complex we can be fairly confident that he derived it by direct generalization ' M S . Ilf, Prop. 33.

* Proved in Prop. 38.

* Ibid., Prop. 39. 5 See M S . V , n. 22.

^ Ibid., Prop. 40.

84

THE MOTION OF EXTENDED BODIES

5.4

from the case of the head-on collision of two elastic spheres considered in the Waste Book.^ No indication is given either of the origin of his method of distri­ bution based on factors of easiness or difficulty of change of the four distinct velocities in question. As to the factors themselves, for a given impulse P acting on the body of quantity (— mass) A the change in translational velocity will be p

5.5

THE MOTION OF EXTENDED BODIES

85

Among the early researches we have noticed the remarkable set of theorems relating to the motion of the centre of gravity of a pair of moving

bodies.^ We have also noticed Newton’s development of the concept of centre of motion in the dynamical sense culminating in his use of this concept in his solution to the problem of the collision of two rotating bodies. Moreover, his interehangeable use of the two terms, and his proof of their equivalence in the case of two bodies, would seem to indicate that he had divined that they were identical in the case of a single body also. After the early researches there is no trace of either concept before Version 3 of the tract de Motu (MS. IXc). In Law 4 of that work he states that the communal centre of gravity of a set of mutually interacting bodies maintains its state of rest or motion with uniform velocity in a straight line. In the lectures de Motu this law becomes Corollary 4 to the laws of motion, the proof supplied being based on a generalization of the result for two bodies previously derived in the Waste Book. But whereas there it was presumably derived by Newton with no particular aim in view, but simply as an application of his new-found facility in dynamics, in Version 3 of the tract de Motu we have a clear indication of the physical motivation behind its introduction. For in the Scholium to Theorem 4 of that paper Newton notes that the centre of gravity of the solar system will either be at rest or move uniformly in a straight line, thus providing a proof a priori of the Copernican system. The concept of centre of motion in the dynamical sense found in Newton’s treatment of rotating bodies in MS. V plays no part in the Principia. But very remarkably, and significantly, it appears in the original Definition 3 of MS. X L There Newton defines the centrum materiae as the intersection of two axes of the body, an axis having previously been defined (Def. 2) as a line about which the body can revolve in free space maintaining always the same position relative to the parts of the body. This axis is clearly the same as that referred to in § 8 of MS. V, and the centre the same as the centre of motion defined in § 4 of that work. Moreover, Newton notes that the centre of matter as thus defined will coincide with the centre of gravity provided the latter is calculated according to the masses rather than the magnitude of the various parts of the body. Thus Newton explicitly confirms in MS. X I that identification of the two points, centre of motion and centre of gravity, previously hinted at in MS. V and in the Waste Book. The presence of these two Definitions 2 and 3 in the original version of MS. X I thus provides a remarkable

^ See M S . lid , A x. lo.

* See § 2 above.

8F = _^,

SO that for given P, SF is proportional to ijA. Again, the same impulse P acting at the point B will produce a change in angular velocity SQ given by P F = AG^ SQ, so that the real quantity of angular velocity D — GO. will change by an amount 8D = AG' For given P the change in D is therefore proportional to FjAG. These are the factors given by Newton. He could well have arrived at the first factor in much the same way as given here, but failing a knowledge of the principle of angular momentum he must have derived the second factor by some general argument: clearly the greater F the greater the change produced in the angular velocity, and so on. . . . Given the various resistance factors it remains to decide on the proper distribution formula. The one given by Newton has the double advantage of being not only the simplest formula imaginable, but also the correct one. Having provided this impressive proof of his dynamical genius, Newton proceeds to draw on all the preceding sections of the paper (apart from the first) to present the solution in principle to the problem of the collision of two rotating bodies. The next section entitled ‘Some Observations about Motion’ is in the hand of Newton’s friend John Wickins. Newton, however, was almost certainly the author. It shows how clearly he had understood the con­ ditions actually governing collisions between rotating bodies in practice as opposed to the ideal conditions assumed in §§ 9 and 10. 5 .5 .

D iscussion

in

L ater

R esearches

86

THE MOTIO N OF EXTENDED BODIES

5.5

example of the continuity and persistence of Newton’s thought in dyna­ mics between the early and later researches. Apart from the reference to axes and centre of matter in Definitions 2 and 3 there is no further reference to rotating bodies in the lectures de Motu. Nor is there any indication of any further development of Newton’s thought on this topic in the Principia itself. On the contrary, his erroneous treatment of the precession of the equinoxes^ would seem to point to a definite retrogression in his thought on this subject com­ pared with the original treatment of it in the problem of the collision of two rotating bodies. There is, however, a very interesting reference to rotation at the end of his enunciation of the principle of inertia in MS. Xa: there he continues Motus autem uniformis hie est duplex^ progresswus secundum lineam rectam quam corpus centro suo aequabilite lato descrihit et circularis circa axem suum quemvis qui vel quiescit vel motu uniformi latus semper manet positionibus suis prioribus parallelus. It would seem, therefore, that originally Newton had in mind a prin­ ciple of inertial rotatory motion besides that of translatory motion. ' See Book 3, Prop. 39, Prob. 20, Principia (3rd ed.).

O R D E R OF C O M P O S I T I O N A N D D A T I N G OF M A N U S C R I P T S relative degree of certainty attainable in the following discussion is inevitably subject to great variation; at some points the true order of composition of the manuscripts is unmistakable, at others no more than probable, while the almost total lack of dated entries renders the question of dating much more uncertain than the order of composition. Thus it is not claimed that the order of composition and dating here suggested is other than uncertain, though I think it is the most probable one in the light of the evidence considered. If further evidence becomes available it may, of course, at some points invalidate, or perhaps strengthen, the order here suggested. Newton’s dynamical manuscripts prior to the composition of the Principia fall into two distinct sets; the first set comprises the earliest manuscripts, all most probably composed before Newton’s letter of 23 lune 1673 to Huygens, and the majority almost certainly much earlier; the second set comprises the remaining manuscripts, one possibly dating from the winter of 1679-80, and the remainder all probably written between May 1684 and the summer of 1685. Given the gap be­ tween these two sets, nothing is lost by regarding them as independent, and they will therefore be treated separately in sections 6.1, 6.2 and 6.3, 6.4 respectively. The topics treated in the last two sections also refer to the later manuscripts but could not conveniently be included in sections 6.3, 6.4. T he

6.1.

O rder

of

C om position

of

E arliest

M anuscripts

Given its unmistakable medieval flavour, and the serious qualitative discussion of the old problem 'a quo moveantur projecta\ there can be no doubt that the passage ‘On Violent Motion’ in MS. I preceded all the dynamical writings in MS. II with their recognizably ‘modern’ approach to dynamics firmly based on the principle of inertia and quanti­ tative measures for motion and force. The same is then presumably true of the small number of other dynamical entries in MS. I. Internal evidence^ points unmistakably towards the following order • See para. 3 of general introduction to M S . II. It is reproduced here for complete­ ness’ sake.

88

ORDER OF CO M P O S IT I O N AND

6.1

of composition in MS. II; Def. 1-14, Ax.-Prop. 1-26, Ax. 100-122, Ax.-Prop. 27-40. Given that the order of composition for these entries corresponds to their order of entering in the Waste Book, it is then natural to assume this is true also for the remaining entries on folios 1,10, and 38 respectively. No internal contradiction results from this assump­ tion for the last pair of entries; in fact in both cases there is some sup­ porting evidence. P'or the calculations on folio 10 are restricted to the specially simple case of totally inelastic collisions, whereas the physical discussion in Ax.-Prop. 7-10 deals with the much more difficult case of perfectly or partly elastic collisions. Again, the entries on resolution of velocity and composition of motions on folio 38 are the first to treat of these topics in the Waste Book, and follow the preceding dynamical entries on folio 15 after a considerable gap. Lacking further evidence it is therefore reasonable to assume that the most probable order of com­ position of the entries on folios 10 and 38 is given by their position of entry in the Waste Book. A similar assumption, however, cannot be made for the treatment of circular motion on folio i which evidently came after the first tentative discussion of the problem in Ax.-Prop. 20. Striking confirma­ tion of this is provided by the alteration of the figures 4-]- in Ax.-Prop. 24 to 6 - f , corresponding to the exact result 2 tt obtained in folio i. As noted previously,^ the peculiar formula of MS. I l l may be derived from a result obtained on folio i by an immediate application of the Merton Rule. Even if it cannot be absolutely certain that this w^as the actual method of derivation employed by Newton, nevertheless the fact of the derivation itself can scarcely be in doubt. In particular, it seems inconceivable that Newton would have accidentally directed attention in both cases to so ‘odd’ a time as that for circular motion through a distance equal to the radius. MS. I l l must therefore be placed after folio i of MS. II. The position of MS. IVa after MS. I l l is likewise certain. This rests on the values employed in these manuscripts for the rate of fall under gravity. Originally, on the left-hand side of MS. I ll, Newton employed a wildly erroneous figure about half the true value. Later, on the righthand side of the same manuscript, he recalculated the rate of fall arriv­ ing at a figure of 196 inches of fall from rest in one second. This agrees almost exactly with the figure of 16 ft of fall in one second employed in MS. IVa, so that we can be certain that MS. IV followed the completion of MS. III. ‘ See above, Chapter 1.2, p. 10.

6.1

DA TI N G OF M A N U S C R I P T S

89

MS. IVa is particularly memorable for its explicit statement of the dependence of centrifugal force on the radius and period of revolution of the motion— that is, for the law of centrifugal force. Although MS. IVa as a whole must have been composed after MS. I ll, there is good evidence that the law of centrifugal force was discovered by Newton sometime between the calculations of centrifugal force on the left-hand side of MS. I ll, and the calculations of the rate of fall due to gravity on the right-hand side. In the first place, it seems very unlikely that if Newton had already been in possession of the law of centrifugal force he would have based his calculations of centrifugal force on the lefthand side of the manuscript exclusively on the formula. At least one application of the '^R', or equivalent, formula is certainly necessary to compare the force of gravity with a given centrifugal force; there­ after, however, the law of centrifugal force provides a far more con­ venient method of comparing other centrifugal forces with the original one and thus with the force of gravity. This is precisely the method em­ ployed in MS. IVa. It seems probable, therefore, that the discovery of the law of centrifugal force followed the calculations on the left-hand side of MS. III. That it preceded the calculations on the right-hand side of the same manuscript is made probable by the use of this law (at 2.4) to compare the centrifugal forces due to the annual and diurnal motions of the Earth. This single use of the law of centrifugal force in MS. I l l may well point to its first employment by Newton. His subsequent use of the result obtained for the above ratio is equally interesting. He com­ bines it with the figure 144 for the ratio of the force of gravity to the centrifugal force due to the diurnal motion. This figure, however, is based on the original, erroneous value for the rate of fall under gravity. It would thus seem probable that this particular application of the law of centrifugal force was made not only after the original calculations on the left-hand side but before the calculations on the right-hand side. The actual position of the calculation itself, in the ‘middle’ of the manu­ script, ‘between’ the two sets of calculations, supports this interpretation. The position of Newton’s first test of the law of gravitation relative to M SS. HI and IVa must now be considered. This test must have been based on a figure of around 16 ft of fall per second under gravity as opposed to the figure about half that value employed originally in MS. I ll ; otherwise the disagreement between theory and observation would have been intolerably large. So that we can be certain that if there were a test it must have taken place after the commencement of MS. III.

90

ORDER OF CO M P O S IT I O N AND

6.1

While its position relative to MS. IVa is less certain it is perhaps rather more likely to have taken place after the composition of that manuscript. For although MS. IVa contains a comparison between the force of gravity and the conatm recedendi of the moon from the Earth, and a deduction via Kepler’s Third Law of the inverse square dependence of the conati of the planets from the sun, any reference to a force of gravity on the moon towards the Earth, or on the planets towards the sun, or of universal gravitation, is conspicuously absent. It is as if the one thing absent from a stage entirely set for its reception w^ere the notion of universal gravita­ tion itself. On the other hand, as Hall has rightly remarked in discussing this curious absence of any reference to universal gravitation in MS. IV a : ‘With Newton the argument from silence is never strong’, and it could well be that MS. IVa was composed after the test, any reference to universal gravitation having been intentionally suppressed in a paper quite possibly originally intended for publication. On the other hand, the chances of MS. IVa having been composed after the first test are somewhat improved if we allow for the possibility that this test may have been based on the notion of a balance between a centrifugal force out­ wards and a gravitational pull inwards. ^ If MS. IVa were composed before the test, then the original discovery of the law of centrifugal force in the ‘middle’ of MS. I l l would most probably have resulted from the treatment of circular motion in MS. IVa, as opposed to its presumably distinct derivation by the ‘polygonal’^ method just prior to the test itself. Just as M SS. I l l and IVa must be regarded as the continuation of the original discussion of circular motion in MS. II, so MS. V must be regarded as both a resume and extension of certain other subjects treated in the same manuscript. For example, the true method of resolu­ tion of velocity in § 2 of MS. V must be compared with the erroneous method given at folio 38 of MS. I I ; the law of composition of independent motions in § 3 corresponds to the first enunciation of the same law on folio 38; the definition of centre of motion in § 4 is a final form of the earlier drafts in Ax.-Prop. 11-18. Finally, the solution to the problem of the collision of two extended, rotating bodies in §§ 9, 10, represents the culmination of the discussion of collisions at various points of MS. II, especially that of the collision between one moving and one stationary, extended body in Ax.-Prop. 39; this proposition gives every indication of representing Newton’s first orientation towards his treatment of ' See above, Chapter 3, p. 59.

^ See above, Chapter 1.2, pp. 8 - 1 1.

6.1

DATING OF M A N U S C R I P T S

91

the completely general collision problem. MS. V must therefore be placed after Ax.-Prop. 39 of MS. II. MS. V I remains to be considered. Given its largely philosophical, non-technical nature, it is not surprising that there is little or no indica­ tion of its order of composition relative to the other manuscripts. There are, perhaps, some indications that it was composed, in part at least, before MS. V ; for example the definition of place in § i of that manu­ script— There is an uniform extension, space, or expansion continued every way without bounds: in which all bodies are, each in several parts of it: which parts of space possessed & adequately filled by them are their places [italics mine] is reminiscent of Def. i of MS. V I : Place is that part of space which a thing entirely fills. Nevertheless, the only really firm indication of the early nature of this manuscript is provided by the handwriting which places it definitely among M SS. II-V , and certainly before 1673.

6.2 .

D ating

of

E arliest

M anuscripts

The following indications of date for the earliest manuscripts are available: 1. The various cometary observations between 4 December 1664 and I April 1665 found among the ‘original’ * entries in MS. I including the dynamical entries, especially the passage ‘On Violent Motion’. This would seem to point towards a date of composition for this passage to­ wards the second half of 1664, and in any case before 20 January 1665 given the much more advanced nature of the discussion of collisions in MS. Ilb. 2. The marginal entry 20 January 1664 [= 1665 N.S.] on folio 10 of MS. II. 3. //the dependence of the period of a simple pendulum on the square root of its length, expressed at § 5 of MS. Ila, was taken by Newton from Galileo’s Discorsi,^ the entries on folio i of MS. II would have been composed after 1665, the year of publication of Tome II of Salusbury’s Mathematical Collections, of which Part I contains an English translation of the Discorsi.^ * A s opposed to abstracts of books. ^ It is found at p. 139 of the Discorsi (ed. Naz.) but not in the Dialogue. ^ T o my knowledge there was no Latin version of the Discorsi prior to Salusbury’s English translation.

92

ORDER OF C O M P O S IT I O N AND

6.2

4. Certain indications of the place and date of the first test of the law of gravitation. First there is the statement in Pemberton’s account* that The first thoughts, which gave rise to his [Newton’s] Principia, he had when he retired from Cambridge in 1666 on account of the plague. As he sat alone in a garden, he fell into a speculation on the power of gravity. Also that Newton took the figure of 60 miles to a degree at the surface of the earth ‘being absent from books’. The statement that Newton was sitting in a garden when the notion of universal gravitation first entered his head agrees with the various accounts of the story of the falling apple of whieh the most circumstantial and compelling is that of Stukeley.^ Aceording to this account Newton was sitting in the shade of the apple tree, so that the falling of the apple w'ould seem to point towards late summer or autumn. Again, Newton’s account in the Portsmouth Draft Memorandum^ asserts that the first test took place during the Plague Years. The actual year intended, 1665 1666, is not absolutely clear, but in the context of the whole account 1666 seems the more probable candidate. This is also the year given in the above passage by Pemberton. On the basis of this evidence it would seem that if a test of the law of gravitation took place during the Plague Years it was certainly away from Cambridge, and probably in the summer or late autumn of 1666 rather than 1665. 5. Indications of the date of composition of M SS. IVa, b. According to Gregory these manuscripts were composed sometime before Newton’s election to the Lucasian Chair,^ that is sometime before 1669; while to the best of Newton’s memory MS. IVa was composed eighteen or nineteen years prior to 1686, that is in 1668 or 1667.5 6. There are no explicit indications of the date of MS. V. But in view of what has been said of its order of composition in the preceding section it must have been composed after Ax.-Prop. 39 of MS. II, and it seems not improbable that it was composed like M SS. IV before 1669. 7. Indications of the dates of composition of MS. VI. {a) Various references to Descartes’s Principia in MS. D make it probable that the * Given above at Chapter 4, p. 65. ^ Given above at Chapter 4, p. 65, n. 7. ^ Given above at Chapter 4, p. 66. * See Correspondence, vol. iii, p. 332. T h e passage in question is given above in n. 2, p. 72 of Chapter 4. See his letter of 14 July 1686 to Halley. See, for example, that at M S . I, § 3.

6.2

D A TI N G OF M A N U S C R I P T S

93

profound study of that work on which MS. VI is based was under way towards the second half of 1664. {b) Newton’s proud, but rather juvenile­ sounding claim at p. 16: Et sic plurimorum solidorum turn longitudine turn latitudine infinitorum quanti­ tates solidas positive et exacte determinare possum. would seem to refer to his inverse method of fluxions possibly dis­ covered in May 1666.* In which case this part of the manuscript would have been composed after that date. The results obtained for the probable order of composition and dating of the earliest manuscripts are displayed in the following chart. Second half 1664 c. January 1665

MS. I MS. Ilb MS. lie.

MS. Ild-g 1665-6?

MS. Ila

I

1665-6?: < 1669

MS. Ill

I

MS. V

MS. IVa . . . b

<1669?

MS. VI c. 1665-9?

6.3.

O rder

of

C om position

of

L ater

M anuscripts

Given the uncertainty regarding its early status it will be best to consider MS. V III separately.^ The remaining manuscripts^ are then all apparently ‘early’ (i.e. prior to the Principia) and their true order of composition can be established with certainty apart from a small element of doubt in the case of M SS. Xa and Xb. It will be convenient to express certain relations of order between the manuscripts in equation form by means of the symbols < and = signi­ fying ‘composed earlier than’ and ‘identical with’ respectively. We then have: M S. IXa < M S. IXc. (0 * See Portsmouth D raft Memorandum reproduced above at Chapter 4, p. 66. ^ In § 6 of present chapter. 3 Apart from M S . IX b which might have been copied from IXa after the Principia. However, there are good reasons for identifying this manuscript with the original of the propositions carried by Paget from Cambridge to London in 1684. See § 5 of present chapter.

94

ORDER OF C OM P OS IT IO N AND

6.3

This is almost inevitable given the more advanced state of dynamical thought in the second manuscript. It is proved conclusively by a compari­ son of the texts of Prob. 3 in the two manuscripts. These are found to be identical apart from a certain number of emendations in M S. IX c; in almost all cases the original text is legible, and is then identical with that in MS. IXa: M SS. Xa, b < MS. X I Orig. (2 ) There can be no doubt of this; for example, Def. 1-4 and Def. 9 of MS. Xa constitute what is evidently a rough preliminary draft of the finished, and much more detailed, discussion of absolute and relative space and time in the Scholium to the Definitions at the beginning of MS. X I Orig., and Def. 5 of MS. Xb is evidently a preliminary draft of Def. 7-10 of MS. XI Orig. Prob. 3 MS. IXc = Prob. 3 MS. XI Orig. Lemma i MS. IXc = Lemma i MS. Xa.

(3) (4 )

Consider next the hypothesis MS. X I Orig. < MS. IXc.

(Hypothesis i)

In that case given equation (3), and the relationship between Prob. 3 in M SS. IXa and IXc used to establish equation (i), it has to be assumed that Prob. 3 of MS. IXc was first copied from MS. IXa and then emended so as to agree with the text of MS. X I Orig. This emendation, though of course not the original copying from MS. IXa, must necessarily have been carried out between the date of composition of MS. XI Orig. and its emendation— that is, sometime before the presentation of the Principia to the Royal Society in April 1686. So that the possibility of a date of composition and emendation of MS. IXc as late as 1694 can definitely be ruled out. This is important, given that M S. IXc may possibly be in the hand of David Gregory who is known to have visited Newton at Cambridge in 1694. If Prob. 3 of MS. IXa was emended in the light of MS. X I Orig., then one would have expected all the addenda in MS. IXc compared with MS. IXa to have been taken from MS. X I Orig. But this is not the case either for the definitions, laws, or lemmas. On the contrary, both the definitions and the laws give every sign of being rather distant drafts of those in MS. X I Orig. For example, there is a reference to vis insita in Law 2 which is missing from the version in MS. XI Orig. It can be argued that this omission corresponds to Newton’s ultimate realization

6.3

DATIN G OF M A N U S C R I P T S

95

that no explanation was required for the principle of inertia.' Also, there is no law in MS. IXc corresponding to the third law in MS. X I Orig. Finally, and perhaps most conclusive evidence of all against Hyp. I, there is the identity between Lemma i of MS. IXc and that in MS. Xa. Since MS. Xa was undoubtedly composed before MS. XI Orig., we are forced to assume that although Prob. 3 of MS. IXc was emended in the light of the corresponding proposition in M S. X I Orig., nevertheless Lemma i was taken not from the very similar version in the same manuscript, but from a fragmentary document preceding MS. X I ! There can therefore be no doubt that Hypothesis i is false, so that MS. IXc < MS. IX Orig.

(5)

This latter equation seems to fit all the evidence.^ The order of MSS. Xa, b relative to each other and to MS. IXc remains to be considered. The relative order of M SS. Xa, b cannot be established with certainty. But what indications there are point towards MS. Xa < MS. Xb. For example, whereas in (a) there are definitions of locus (7), quies (8), motus (9), and velocitas (lo), in (b) only the headings are given, as if Newton were referring to the definitions already given in the other manuscript. Again Def. 17-19 of (a) reappear neither in (b) nor in M S. X L Finally, the actual order of the definitions in (b) follows that in MS. XI Orig. much more closely than does the order in (a). The equation MS. Xa < MS. Xb

(6)

therefore seems probable. Next, given that the definitions in MS. IXc are less detailed than the corresponding ones in both MS. Xa, b, and less numerous, and that MS. Xa contains one law^ not found in MS. IXc, the equation MS. IXc < MS. X (7) seems again probable. Finally a comparison of MS. X I (Emended) with the First Edition of the Principia proves conclusively that MS. X I Orig. < First Edition Principia.

(8)

* See above, Chapter 1.4, p. 28. * T h e fact that M S . IXc, unlike M S . X I, contains applications of the results obtained for elliptical motion to the problem of planetary motion provides a good indication that M S . X I was always intended as the beginning of a larger work in which the motion of the planets would find its proper place, as in Book III of the Principia. ^ T h at numbered 3, corresponding to the third law of motion in the Principia.

96

ORDER OF C O M P O S IT I O N AND

6.3

This latter equation, in company with equations (i), (2), and (5), establishes the ‘early’ status of all the manuscripts so far, the probable order of composition being: IXa < IXc < Xa < Xb < XI Orig, < First Edition Principia. 6 .4 .

D ating

of

L ater

M anuscripts

Newton’s letter of 23 February 1685 to Aston contains the following well-known passage: I thank you for entering in your Register my notions about motion. I de­ signed them for you before now, but the examining several things has taken a greater part of my time than I expected, and a great deal of it to no purpose. And now I am to go into Lincolnshire for a month or six weeks. Afterwards I intend to finish it as soon as I can conveniently. The ‘notions about motion’ referred to in this passage must be those contained in the paper Isaaci Newtoni Propositiones de Motu, apart from the Principia the only work of Newton’s on dynamics in the Archives of the Royal Society. Since this paper represents an emended version of MS. IXa the latter must have been composed before February 1685, and possibly considerably earlier, judging by Newton’s statement ‘I designed them for you before now’. A possible indication of the actual date of composition is provided by a document^ originally forming part of the Portsmouth Collection in which Newton states that the propositions in the Principia were all composed after the end of December 1684 apart from the ist and n th of Book I, in December 1679, and the 6th, 7th, 9th, loth, 12th, 13th, and 17th of Book I, and the ist, 2nd, 3rd, and 4th of Book II, in June and July 1684. Disregarding the order, this list tallies with the problems and propositions in MS. IXa, apart from Theor. 4 of MS. IXa and Props. 8, 9, 12, and 13 of Book I. However, Props. 8 and 9 are no more than special applications of Prop. 6, on the law of force to a given point under which a body describes a given orbit (— Prop. 3, MS. IXa); and Props. 12, 13 extend to the case of the hyperbola and parabola what has already been proved for the case of an ellipse in Prop, i i ( = Prob. 3, MS. IXa). The fact that apart from Theor. 4, and the two propositions discovered by Newton in December 1679, the remaining theorems and problems of MS. IXa are included among the propositions Newton admitted to * Reproduced in Brewster [i], vol. i, p. 471. Present whereabouts unknown.

6.4

DATING OF M A N U S C R I P T S

97

having discovered in June and July 1684 makes it not improbable that MS. IXa was composed about that time. But in that case what of the belief that Newton took up his researches in dynamics following Halley’s visit to him at Cambridge in August 1684?^ Not that there is any reason to doubt that Halley was responsible for renewing Newton’s interest in dynamics. But whether or not he paid a visit to Newton in August 1684,2 there is some evidence that his first, decisive visit took place in May of that year. This is the month referred to in Conduitt’s account,3 while Newton himself refers to Halley visiting him in ‘Spring 1684’.4 Again, Halley relates^ that his interest in the subject was aroused by a discussion with Wren and Hooke in January 1684, and that Wren promised either of them a book if they found a solution to the problem of motion in an ellipse within two months. Why then did Halley, having failed to win the book by March, wait till the following August before paying his visit to Newton ? It seems more likely that he would have tried to visit him earlier, for example in May. T o summarize, in the light of the available evidence it seems most probable that MS. IXa was composed around June and July 1684 following Halley’s first visit to Newton at Cambridge in May of the same year. Given that MS. IXc was composed before MS. X I Orig.,^ and that the emended version of the latter manuscript is effectively identical,^ as far as it goes, with the corresponding part of Book I of the Principia, there can be no doubt that M S. IXc was composed some considerable time before the presentation of the Principia to the Royal Society in April 1686, and even possibly before October 1684, the date given on folio i of MS. XI. Without further investigation, however, it would be dan­ gerous to assume October 1684 preceded the actual date of composition of MS. X I, as we shall now see. * A s implied, apparently, by H alley’s letter of 29 June 1686: ‘T h e August following when I did myself the honour to visit you, I then learnt the good news that you had brought this demonstration to perfection.’ ^ It is just possible that Halley paid three visits to Newton in Cambridge. T h e first in M ay, the second in August, the third in November. It seems unlikely that Newton would have given the impression that he had ‘brought the demonstration to perfection’ at the time of the first visit, especially considering he could not then find the original paper. But he could have given this impression in August after the composition of Version I of the tract de M otu. ^ Reproduced in Brewster [i], p. 297. * M S . Add. 3968b, fol. l oi . ® In his letter of 29 June 1686 to Newton. ^ See preceding discussion of order of composition of later manuscripts. ’ Apart, that is, from a number of emendations and insertions, none of great moment. 858205

II

98

ORDER OF CO M P O S IT I O N AND

6.4

MS. X I of the present work forms part of MS. Dd-9-46 C.U .L. The latter manuscript consists of 102 folios written recto only apart from a small number of addenda on the verso of certain folios. The folios them­ selves are in a state of great disorder, but after rearrangement they may be grouped in four sections a, y, each consisting of a number of propositions, &c., following in unbroken sequence without gaps. MS. XI consists of a only. and ^2 ^^e at first identical but then diverge. Either or ^2 could have been the original continuation of a, but the writing, which is the same in a and and different in points to having followed a. This is confirmed by the division of a and jSj into a number of ‘lectures’ by marginal entries not found in ^2- Oo the other hand, there is no doubt that y followed though like and unlike ^2, it is divided up into a number of ‘lectures’ by marginal entries. Folios I of a and y are dated Octob. 1684 and 1685 respectively. Finally, the emended version of a^2 is effectively identical with the correspond­ ing part of Book I of the First Edition of the Principia of which it must have provided a final draft. A regulation relating to the Lucasian Chair of Mathematics required Newton to give a course of public lectures each year.* Those from January 1669 to October 1683 are extant, the number of lectures in different years varying between three and ten, with ten the most fre­ quent number. So that given the division of and y by marginal entries into a number of lectures, nine in the first case, ten in the second,^ there seems no reason to doubt that each represents one course of Newton’s public lectures. Also considering that the lectures up to October 1683 are complete, and that there is a set dated September 1687,3 the dates actually given on and y, namely October 1684 and October 1685, must be regarded as plausible. Nevertheless, two awk­ ward facts bar the way to an easy acceptanee of the correctness of these dates. First, as already noted, the presumed set of lectures y follows ^2 rather than /Sj, and this in spite of the fact that the second set was evidently numbered 10, i i , 12, . . . originally in place of i, 2, . . ., thus giving the impression that Newton regarded the two sets as forming a single whole, as was the case for the four courses of Optics Lectures from January 1669 to October 1672 which are numbered as if they ' See Edleston [i], pp. xci-xcviii, for a useful synopsis of these lectures. - In each case the last lecture is incomplete. ^ See Edleston [i], pp. xcviii and 209. Also the footnote to p. 27 of Ball [i].

6.4

DA TIN G OF M A N U S C R I P T S

99

consisted of only two sets of fourteen and sixteen lectures respec­ tively.* Then there is the reference in a. Coroll. 4, to the laws of motion to Lemma 21. In the Principia the reference at this point is to Lemma 23, and there is no doubt this was the lemma Newton had in mind. But when we examine y we find that although the lemma in question is actually numbered 23 there is some indication that it was originally numbered 21. This is supported by traces of corresponding changes in the numbers of the three following lemmas, and is then confirmed absolutely by a reference to Lemma 21 (instead of 23) in Lemma 25. Evidently Lemma 23 in y was originally numbered 21, probably due to a simple error over the number of the immediately preceding lemma which happens to occur in jSj. So it seems that Newton was referring in a. Coroll. 4 of the laws, to a lemma appearing in section y. This is difficult to understand, for a comparison of the common beginning of and ^2 reveals that jSg must have been composed after and y, which follows ^2y would have been composed still later. Since a would certainly have been composed before jSj, we have to assume a reference in a to a lemma in y composed after oc. It will be argued later that y was composed some considerable time after a, thus making the forward reference to Lemma 23 (alias 21) even more difficult to understand. I shall return to this problem presently. A possible explanation of the significance of the two continuations jSj, ^2 fo a, and the faet that y follows ^2 ^rid not is provided by the following passage from Newton’s letter of 20 June 1686 to Halley. The Proof you sent me I like very well. I designed the whole to consist of three books, the second was finished last summer being short and only wants transcribing and drawing the cuts fairly. Some new Propositions I have since thought on which I can as well let alone. The third wants the Theory of Comets. In Autumn last I spent two months in calculations to no purpose for want of a good method, which made me afterwards return to the first Book and enlarge it with divers Propositions some relating to Comets, others to other things found out last Winter. The third I now design to suppress. Judging by this passage the original version of Book I of the Principia was finished some time before the summer of 1685 but was later altered to incorporate certain new material, some of it ‘relating to comets’ and some found out the ‘last winter’— in the context necessarily the winter of 1685-6. The actual incorporation of this new material may have pro­ ceeded in stages, but it evidently did not begin till after the two fruitless * See Edleston [i], loc. cit.

- jSa is at first identical with

as amended.

100

ORDER OF C OM P OS IT IO N AND

6.4

months devoted to the problem of comets in the autumn of 1685, and it possibly took place early in the year 1686. In the light of this passage, of the fact that oc^2y represents a final draft of part of Book I of the First Edition of the Principia^ and of the existence of the two alternative continuations jSj, ^2 to the following hypothesis naturally suggests itself: Hypothesis. corresponds to the ‘original’ draft, and ajSgy to the ‘revised’ draft of part of Book I of the Principia. A comparison of with appears to support this hypothesis. For contains only three propositions and one lemma from Art. IV of Book I of the Principia, and nothing from Art. V. Instead, after Prop. 19 of jS^, corresponding to Prop. 21 of there follows Prop. 20 corresponding to Prop. 30, Prob. 22, of jSg, that is the first proposition of Art. V I of Book I of the Principia. In other words, if we suppose represented the begin­ ning of the original version of Book I of the Principia, then it was lacking most of Art. IV and all of Art. V of the final version. But Art. V is devoted to the problem of determining conics from various sets of given data excluding a knowledge of foci or axes, precisely the problem posed by cometary orbits. In fact, although there appear to be no direct references to Art. V in the treatment of cometary orbits in Part III of the First Edition of the Principia, Lemma 5 there corresponds to Prop. 22, Prob. 14, of Art. V, Book I. Returning now to ajSj, y regarded as two sets of lectures, this same hypothesis provides an explanation of why y follows ^2 instead of jSj. For we need only suppose that the additions to found in ^2 were made after the lectures a^i had been read, in which case it is not par­ ticularly surprising that Newton should not have continued with the original continuation of (of which the manuscript appears not to have been preserved) but with part of the final continuation ^27 * 5 that he started this second set of lectures at y and not earlier, in jSg, for example at the beginning of Art. V, is perhaps a little surprising but raises no insuperable difficulties. Finally, if we hold to this hypothesis the section y must have been composed several months after the composition of oc. So that the full reference to Lemma 21 could not possibly have been made at the time of composing a. We are therefore forced to assume some insertion at this point of a. On examination of the text it seems almost certain that the whole of the phrase ab Lemmate X X P was present originally with the * A t 1. 4, proof of Coroll. 4 to laws of motion, Principia (First Edition).

6.4

DATIN G OF M A N U S C R I P T S

101

possible exception of the numeral X X L This gives the appearance of being written slightly more heavily than the surrounding text. Not only so, there is a noticeably large gap between the right-hand side of X X I and the extreme left of the immediately following word. In fact, by measuring the intervals between successive words it appears that the gap immediately to the right of X X I is the largest in the body of the text of a. The supposition of an insertion as required originally by the hypothesis thus gains powerful backing from the text itself. We are now in a position to re-examine the problem of the dates of the lectures represented by and y, especially that of the first set. The only plausible dates for the first lecture of are October 1684 or October 1685, for October 1686 is ruled out as being after the transition from ajSj to ajSay. If the lectures were given in October 1684 they would need to have been composed between that date and the preceding May.^ Assuming MS. IXa was composed around June and July this seems just possible, since all the most difficult theorems of are already found in the earlier manuscript. Relevant also is the following passage from the minutes of a meeting of the Royal Society on 10 December 1684: Mr. Halley gave an account that he had lately seen Mr. Newton at Cam­ bridge who had shewed him a curious treatise de Motu\ which upon Mr. Halley’s desire, was, he said, promised to be sent to the Society to be entered upon their register. Mr. Halley was desired to put Mr. Newton in mind of his promise for the securing of his invention to himself till such time as he could at leisure publish it. Since the propositions carried by Paget to Halley in November 1684 were almost certainly those contained in Version I of the tract de Motu, it is a little difficult to believe that this was the work referred to here. In which case he was presumably referring to part of the first draft of Book I of the Principia, i.e. to part or all of But this does not prove that what he had seen was already being given by Newton for his public lectures of the Michaelmas term 1684. In fact, there are various internal indications that was originally conceived as part of the Principia, being used later as lecture material only as an afterthought. For example, there is the title itself De Motu Corporum Liber Primus, the phrase unde caveat Lector . . . at the beginning of the last sentence following the Definitions, and the sentence at the end of the succeeding Scholium: Hum enim in finem Tractatum sequentem composui. None of ‘ August would have left an impossibly short time even for Newton.

102

ORDER OF C O M P O S IT IO N AND

6.4

these sound right in the text of a work originally intended as a set of public lectures as opposed to a book. On the whole, therefore, it seems rather more probable that the lec­ tures ajSj were given in the Michaelmas term 1685, beginning in October. This would leave open the actual date of commencement of ajSj though it would necessarily have been some considerable time before the sum­ mer of 1685, and perhaps even as early as November 1684. As for the lectures y, it seems most likely that they were begun in October 1686; for October 1685 would have given very little time for the rather large insertions found in /S.^y compared with ^8^after the two fruitless months spent on the problem of cometary motion in the autumn of that year. In any case, apart from the question of the actual texts of the lectures delivered by Ne\Mon in the years 1684-6, there is good reason to be­ lieve that a represents the first part of the original draft of Book I of the Principia, in which case MS. X I would have been composed some time before the summer of 1685. 6 .5 .

P a g e t ’s P r o p o s i t i o n s

Opinions have differed on the nature of the propositions carried by Paget from Newton to Plalley in November 1684.^ Brewster^ was of the opinion that there was only one proposition, that corresponding to Kepler’s first law of planetary motion, while Edleston^ thought it more probable that the propositions in question corresponded to those entitled Isaaci Newtoni Propositiones de Motu in the Register Book of the Royal Society. Ball’s^ views on the nature of ‘Paget’s Propositions’ are not very clear, though he presumably would not have equated them with the tract de Motu, since he believed that all three versions of this work were composed in December or January 1685. Opinions have likewise dif­ fered on the identity of the ‘curious treatise de Motu' referred to by Halleys at the meeting of the Royal Society of 10 December 1684. Rigaud^ identified this work with the tract de Motu, whereas Edleston thought Halley’s reference was to Newton’s Public Lectures of the Michaelmas term 1684. More recently Hall and HalL have given it as their opinion that it ‘is now impossible to decide which (if any) of the surviving drafts was intended and how much was communicated by Newton to his friends * As described in Extract C below. ^ Brewster [i], vol. i, pp. 298-9, esp. n. i, p. 299. ^ Edleston [i], p. Iv, n. 75. ■* Ball [i], pp. 30-32. ^ See Extract A below. Rigaud [i], pp. 14-16. ’ Hall and Hall [i], p. 237.

6.5

DA TING OF M A N U S C R I P T S

103

in London at any given date between August 1684 and March 1685’. In any case, they judged the matter to be ‘trivial’ since ‘comparison of the various pxQ-Principia drafts makes it perfectly clear from internal evi­ dence that all of them precede the Michaelmas term lectures, which themselves are headed in Newton’s own hand ‘Octob 1684’.^ Nevertheless, for completeness’ sake, and because of the unique historical interest attaching to Newton’s first public announcement of his decisive discoveries in dynamics, it seems worth while to re-examine the above and related questions in the light of all the available evidence, a thing so far apparently not attempted.The first known public reference to Newton’s researches in dynamics was made at a meeting of the Royal Society on 10 December 1684. At this meeting^ (A) Mr. Halley gave an account that he had lately seen Mr. Newton at Cambridge who had shewed him a curious treatise, de Motir, which upon Mr. Halley’s desire, was, he said, promised to be sent to the Society to be entered upon their register. Mr. Halley was desired to put Mr. Newton in mind of his promise for the securing of his invention to himself till such time as he could at leisure publish it. Various references to certain ‘papers’, evidently on motion, are found soon afterwards in letters between Newton and Flamsteed. On 27 December 1684 the latter wrote: I am obliged by your kind concession of the perusal of your papers, tho I believe I shall not get a sight of them till our common friend Mr. Hooke and the rest of the town have been first satisfied. In his letter of 5 January 1685 to Newton, Flamsteed wrote again: If you will give me leave to guess at your design I believe you are endeavour­ ing to define the curve that the comet described in the aether from your Theory of Motion and later in the same letter: Sir I have not had the happiness of Mr. Paget’s company this Christmas tho’ he promised it me; the hard weather perhaps prevented him as it did me from going to London so I have not yet had the happiness of the perusal of your papers. I am very well pleased how^ever to hear that you intend to oblige us with the publication of them next term when I hope to have the use of them [not] being obliged to any but your self for it. * This, however, may be in error for Octob. 1685. See above, § 4. ^ Further evidence has come to hand since the account given in Herivel [2], [3]. ^ Birch [i], vol. iv, p. 347.

104

ORDER OF CO M P O S IT IO N AND

6.5

In his letter of 12 January 1685 Newton excused the non-arrival of the papers by reason of Paget’s sickness but promised to instruct the latter to transmit them to Flamsteed ‘as soon as he has a convenient opportunity’. Finally, from Flamsteed’s letter of 27 January 1685 we learn of his safe reception of the papers in question from Paget. The next relevant reference occurs in Newton’s letter of 23 February 1685 to Aston, one of the then Secretaries of the Royal Society; (B) I thank you for entering in your Register my notions about motion. I designed them for you before now, but the examining several things has taken a greater part of my time than I expected, and a great deal of it to no purpose. And now I am to go into Lincolnshire for a month or six weeks. Afterwards I intend to finish it as soon as I can conveniently. In connexion with the above passage it is worth noting that the ver­ sion of the tract de Motu in the Register Book of the Royal Society has the date-line 10 December. Next there is the following well-known passage in Halley’s letter of 29 June 1686 to Newton at the time of the controversy with Hooke on the inverse square law of gravitation: (C) The August following when I did myself the honour to visit you, I then learnt the good news that you had brought this demonstration to perfection, and you were pleased, to promise me a copy thereof, which the November following I received with a great deal of satisfaction from Mr. Paget; and thereupon took another jour[ne]y down to Cambridge, on purpose to confer with you about it, since which time it has been entered upon the Register Books of the Society. And in Newton’s letter of 14 July 1686 to Halley: (D) . . . and so it rested for about 5 years, till upon your request I sought for that paper; and not finding it, did it again, and reduced it into the pro­ positions shewed you by Mr. Paget. Relevant too is the following passage in Halley’s letter of 11 December 1686 to Wallis: (E) You were pleased to mention some thoughts you had of communicating your conclusions concerning the opposition of the Medium to projects moving through it; the Society hopes you continue still inclined so to do, not doubting but that your extraordinary talent in matters of this nature, will be able to clear up this subject which hitherto seems to have been only mentioned among Mathematicians, never yet fully diseussed. Mr. Isaac Newton about 2 years since gave me the inclosed propositions, touching the opposition of the Medium to a direct impressed Motion, and to falling bodies, upon supposition

6.5

DA TI N G OF M A N U S C R I P T S

105

that the opposition is as the Velocity; which tis possible is not true: however I thought any thing of his might not be unacceptable to you, and I beg your opinion thereupon, if it might not be (especially the 7th problem) somewhat better illustrated. Finally there is the following passage in Newton’s letter of 13 February 1687 to Halley: (F) Dr. Wallis has sent up some things about projectiles pretty like those of mine in the papers Mr. Paget first shewed you. . .. Since the version of the tract de Motu entitled Isaaci Newtoni Propositiones de Motu is the only dynamical work by Newton in the Register Book of the Royal Society, there can be no doubt that these propositions corresponded to the ‘notions on motion’ referred to in extract (B). This version of the tract de Motu must then have been sent to London some time before 23 February 1685, and judging by Newton’s remark in the same letter ‘I designed them for you before now’ they were possibly ‘composed’ considerably earlier. As for the statement ‘I intend to finish it as soon as I can conveniently’, the ‘it’ in question presumably referred to the Principia of which the second book was apparently com­ pleted in the summer of 1685.^ It appears that Newton sent certain propositions relating to motion in a resisting medium to London ‘about 2 years’ before i i December 1686, that is around the winter of 1684-5. According to Newton (F) they were shown to Halley by Paget; whereas Halley stated (E) that they were given to him by Paget. Since the tract de Motu contains two pro­ positions relating to motion in a resisting medium, and a version of this tract was certainly sent by Newton to London before 23 February 1685, it is possible that the propositions referred to in extracts (E) and (F) were identical with those in the tract de Motu. This possibility is then greatly strengthened by the reference in (E) to the ‘7th problem’. For the last ‘proposition’ in the tract de Motu, the second of two on motion in a resisting medium, appears in that work as ‘Prob. 7’. It seems very probable, therefore, that both Newton and Halley were referring to the two propositions on motion in a resisting medium at the end of the Royal Society copy of the tract de Motu, and that consequently the original of that copy was brought to London by Paget around the winter of 1684-5, either given to Halley or ‘first shewed’ him by Paget. ^ See passage at beginning of last paragraph of New ton’s letter of 20 June 1686 to Hallev.

106

ORDER OF C OM P OS IT IO N AND

6.5

Conclusive evidence in support of this hypothesis is supplied by Version II of the tract de Motu in the Portsmouth Collection at Cambridge. At first sight this version of the tract de Motu appears to be little more than a fair copy of Version I. But a closer examination reveals a number of peculiar features: 1. Compared with Versions I and III Version II was composed with extreme care as regards writing, punctuation, and Latin quantities. It also contains a list of contents not found in the other versions. 2. Version II consists of a number of single sheets and one double sheet. If, as seems probable, the double sheet was originally used to contain the others, on the cover sheet were part of the list of contents, and the back sheet was blank apart from the following note: ‘Letter, Book No. 7, P. 234, Mr. Locke’s letter to Mr. Oldenburg le.’ The letter referred to, on poisonous fishes, is in the Archives of the Royal Society. 3. Where the Royal Society version of the tract de Motu differs sub­ stantially from Version I it agrees with Version II, namely: {a) Both the Royal Society copy and Version II lack the Scholium to Prop. 2 found in Versions I and III. This Scholium notes that the case considered in Coroll. 5, where the square of the periodic time is assumed proportional to the cube of the radius (leading to an inverse square dependence of the force on the radius), holds both for the revolutions of the major planets about the sun, and for the satellites of Jupiter and Saturn. ^ {h) At the end of the table of contents in Version II appear two lemmas numbered 3 and 4 and not found in Version I. The same lemmas now numbered 2 and i, respectively, appear in the Royal Society copy. It appears, therefore, that Version II was copied from Version I with some particular end in view (i); that it passed at some time through the hands of an official of the Royal Society (2); and that the Royal Society copy of the tract de Motu w^as either taken from Version II, or from some other manuscript differing similarly from Version I (3). Since there is already a very strong presumption that the proposition referred to in (E) and (F) formed part of the original of the Royal Society copy, the '■ A modified form of the same Scholium was eventually introduced by Newton after Prop. 4, Theor. 4, of the Principia to ‘compose’ the dispute between himself and Hooke (see N ew ton’s letter of 14 July 1686 to Halley). T h e omission of this Scholium in the version of the tract de M otu sent to London in 1684 is therefore just possibly an indication that Newton anticipated some reaction by Hooke in the matter of the inverse square law.

6.5

DA TIN G OF M A N U S C R I P T S

107

possibility now inevitably suggests itself that Version II of the tract de Motu represents the actual papers carried by Paget from Cambridge to London. And one further peculiarity of that version not so far alluded to— the absence in it of the two last propositions— provides with (E) final confirmation of the correctness of this supposition. If extract (C) is omitted, the remaining evidence agrees internally and with the conclusion arrived at above. There is now every reason to identify the papers (on motion) referred to in the Flamsteed-Newton correspondence of the winter 1684-5 with Version II of the tract de Motu, in which case it probably reached London prior to 27 December 1684. Judging by extract (A) it would also seem to have reached London after 10 December, for there now seems every reason to identify the ‘curious treatise de Motu' referred to in Halley’s account with Version II of the tract de Motu: this latter work was entered in the Register Book of the Royal Society, and no other reason for this suggests itself than that given in (A), namely, to ‘secure’ Newton’s invention(s in dynamics) to himself‘till such time as he could at leisure publish it’ ( = his invention) — as in fact he later did in the Principia. This pleasing picture, however, is somewhat marred by the evidence in extract (C). According to this some proposition or propositions were brought by Paget to London in November 1684, that is before the meeting of 10 December. And the reference in extract (A) to Halley having ‘lately seen Mr. Newton in Cambridge’ seems to tie up with Halley’s statement in (C) that he took another journey down to Cambridge after receiving the proposition(s) from Paget in November. One possible solution to this apparent conflict of evidence is that there were two sets of propositions carried by Paget to London, the first before 10 December 1684, probably in November, and the second set, after 10 December, consisting of the complete Version II of the tract de Motu. But arguments can be advanced against this solution. For example, Newton’s reference to his reduction of the demonstration of the inverse square law to the proposition showed Halley by Paget sounds more like the tract de Motu than a separate demonstration. Again, Halley states in (C) that the copy of the demonstration of the inverse square law which he had received ‘with a great deal of satisfaction from Mr. Paget’ in November was later ‘entered upon the Register Books of the Society’ and this was true only of the demonstration contained in the Royal Society copy of the tract de Motu. On the other hand, if there was only one set of Paget’s propositions

108

ORDER OF C O M P O S IT I O N AND

6.5

(C) implies that it was sent to London before lo December 1684. And in that case the account in (A) would seem to be somewhat misleading — provided of course we continue to identify the ‘curious treatise de Motu' with Version II of the tract de Motu. Lacking further evidence it is impossible to decide definitely between the existence of one or two sets of Paget’s propositions. It seems certain, however, that if there was only one set then it must have con­ sisted of the propositions in the complete Version II of the tract de Motu, and this would likewise have been true of the second set in the other case. It also seems certain that in both cases these propositions were brought by Paget to London before 23 February 1685, and probably before 27 December 1684, and shown to various people including Halley and Flam­ steed, and possibly Hooke; that they were later copied into the Register Book of the Royal Society and then retained in London, probably by Halley, until the last two propositions were sent by him to Wallis with his letter of i i December 1686; and that finally, at some later date, by some unknown route, the remaining propositions found their way back into the Portsmouth Collection. 6.6.

T he

S tatus

of

M anuscript

VIII

As already noted, ^the existence of the Locke copy, with its endorse­ ment ‘Mr. Newton Mar. 1689-90’ throws doubt on the early status of the Newton copy. It is then natural to assume^ that the original (or originals) of both copies was composed around March 1690 to provide Locke with a more straightforward exposition of the two vital proposi­ tions from which followed the inverse square law of attraction on the basis of the known elliptical orbits of the planets about the sun. But there are certain difficulties in the way of accepting this assumption of the late status of the original MS. V III. These will now be considered. An examination of the copy in the Locke papers, hereafter referred to as the ‘Locke copy’, makes it certain that this must have been taken from some earlier manuscript.^ For the handwriting is certainly not Newton’s,4 and the complicated and detailed nature of the argument, especially the mathematical parts, makes it inconceivable that even ' M S . V I I I , Introduction. ^ T his was the view taken by Brewster [i], vol. i, p. 339, and Ball [i], p. 116. ^ It will appear later that this earlier manuscript could not have been the Newton copy. * According to Cranston [i], p. 337, the handwriting is that of Locke’s valet and amanuensis Brownover.

6.6

D A TI N G OF M A N U S C R I P T S

109

Newton could have dictated it extempore. This is confirmed by the presence of only two corrections in the whole copy, something again almost inconceivable in the extempore dictation of so long and difficult a paper. In fact, one of these corrections, the substitution of ‘times’ for ‘lines’,! provides good support for the copy hypothesis; whereas this seems a rather unlikely error in dictation, an examination of the other copy, in Newton’s handwriting, referred to hereafter as the ‘Newton copy’, reveals how easy it would have been for an unwary copyist to misread Newton’s ‘times’ as ‘lines’, an error even committed at one point by Newton himself!^ The establishment of an original manuscript from which the Locke copy was taken has been laboured at some length because of its impor­ tance for the remainder of the argument. Hereafter this original manu­ script, or manuscripts, since there may well have been several, will be referred to as the ‘Original M S .’ Consider now the hypothesis that the Original MS. was composed specially for Locke’s benefit, probably around March 1690, but in any case certainly after the composition and publication of the Principia. The assumption then is that Locke had asked Newton for a more straightforward derivation of the inverse square law than that given in the Principia. Newton did supply such a proof. One can even agree with Brewster^ that it is somewhat more ‘popular’ than the one in the Principia. Though of course, given the essential depth of the proposition in ques­ tion, the great ‘distance’ between it and the axioms of dynamics, not even Newton could make the proof simple. It is also an entirely dijferent kind of proof; whereas the proof in the Principia follows by application of Prop. 6, Theor. 5, Book I, applicable to any curve, to the special case of motion in an ellipse under a force to one focus, the present proof begins and ends with an ellipse, and, in particular, makes no use of Newton’s beautiful generalization of Galileo’s law of falling bodies, as in Prop. 6, Theor. 5. What is surprising, however, is that Newton felt any need after the publication of the Principia to supply Hypotheses 1-3 and Prop. i. Why did he not simply refer at the appropriate places in Prop. 2 to Laws i, 2, Coroll, i, of the Laws, and Prop, i, Theor. I, Book I, of the Principia, to which Hypotheses 1-3 and Prop, i evidently correspond? He had, after all, a recent precedent for so ’ A t Prop. 2, para. 4, word 15. A t the end of the enunciation of Prop, i the same error of ‘lines’ for ‘times’ occurs, this time uncorrected. ^ Six words from the end of the penultimate sentence of Prop, i . ^ Brewster [i], vol. i, p. 340.

no

ORDER OF C OM P OS IT IO N AND

6.6

6.6

DATIN G OF M A N U S C R I P T S

111

doing in his correspondence with Huygens of the previous autumn.^ Then he referred at need to the Principia, instead of working out every­ thing ab initio. Even if it is granted, however, that for some unknown reason Newton now felt unwilling to quote references from the Principia,^ it would still have been surprising if he had not consulted it when composing, at Locke’s request, a new and more comprehensible version of two of the propositions in that work. Yet such would appear to have been the case. First there is Prop, i : apart from the detailed justification of the con­ struction for the position of the body at the end of the second moment of time, the proof of this proposition in the Locke copy is essentially the same as that in Prop, i, Theor. i, of Book I of the Principia. Yet it would be impossible to regard it as in any sense a translation of that proposi­ tion from Latin into English. This confers a unique status on Prop. I in the Locke (and Newton) copies. For example, a comparison of Theor. i of the tract de Motu with Prop, i, Theor. i, of Book I of the Principia reveals that apart from the transition to the limit at the end the proof in the Principia is an almost exact copy of that in the tract de Motu. This in itself is not surprising. The proof in the tract de Motu is well nigh perfect, unmatched in all Newton’s dynamical writings for its simplicity, elegance, and display of profound physical intuition. Little wonder that he felt no need to modify it in any essential detail in the Principia. Yet we now have to assume that in March 1690 he had somehow become dissatisfied with this proof. Or, alternatively, that he did not have a copy of the Principia available when composing the Original MS., a rather improbable hypothesis, especially if the Original MS. was composed after Newton’s return to Cambridge in February 1690^ following the dissolution of the Convention Parliament, as seems most probable. To explain the difference between the form of Prop, i in the Original MS. and that found in other versions of the same proposition, it has been necessary to assume that Newton had either become tem­ porarily dissatisfied with the previous form of that proposition, or else did not have a copy of the Principia available when composing the

Original MS. But another explanation suggests itself: that Newton not only had not a copy of the Principia available, but could not have had, that work not yet having been composed. It is the three hypotheses, the same in both the Locke and Newton copies, and therefore presumably in the Original MS. also, which seem to point to this conclusion. First, the very term ‘hypothesis’ itself is a little surprising. We know that as Newton grew older he increasingly tended to avoid use of this term.^ An interesting example of this tendency is provided by the tract de Motu. In Version I of that work there are four hypotheses, of which the first is the principle of inertia. In Version III there were also origi­ nally a number of hypotheses, of which the first was again the principle of inertia. But later the term ‘hypothesis’ was cancelled and replaced by ‘Lex’, the term exclusively used thereafter. The use of the term ‘hypothe­ sis’ as applied to the principle of inertia in the Original MS. (Hypothe­ sis i) therefore represents a small reversion towards the more primitive viewpoint of the summer of 1684.2 Newton, however, might have used the term ‘hypothesis’ in a paper intended to show how a certain result (the inverse square law) followed logically from certain initial assumptions or hypotheses. After all, it was precisely the strict deducibility of Newton’s system of the world which so impressed the philosopher Locke .3 Nevertheless, the actual form of the first two hypotheses, corresponding respectively to the first and second laws of motion, is rather surprising in the author of the Principia little more than two years after the publication of that work. Why, for example, does he omit any reference to rest as opposed to motion in Hypothesis i ? Once again this represents a small reversal to the level of the tract de Motu in no version of which is there a reference to the state of rest, although it could be countered that Newton makes no use of the assumption of rest in the Original MS. But the omission in Hypothesis 2 of any reference to the change of motion being in the direction of the force cannot be disposed of so easily. For this part of the second law is not only of great intrinsic importance but is actually used at one point in Prop. 2.‘^ It is difficult to assess the improbability of Newton making so serious an omission in the enunciation of the second

^ See the various references to the Principia in the two letters from Newton to Huygens reproduced as items 341, 2, of vol. iii of the Correspondence. ^ One possible explanation, that Locke himself never possessed a copy of the Prin ­ cipia, can be eliminated, since a copy of that work, with manuscript corrections in New ton’s own hand, was found among Locke’s possessions. It is of course not certain that Locke was in possession of this copy in March 1690. See More [i], p. 356, foot.

’ See especially Koyre [4]. ^ See § 4 above for a discussion of the date of composition of Version I of the tract de

M otu. ^ See, for example, the quotation from Locke’s essay ‘On Education’ given in foot­ note 6 of p. 76 of vol. iii of the Correspondence. * See reference to Hypothesis 2 towards the end of penultimate paragraph of this proposition in the Newton copy.

112

ORDER OF C O M P O S IT I O N AND

6.6

law of motion after his final synthesis of dynamics in the Principia. One simply feels that it is not the sort of omission he would have made. Finally, there is the evidence supplied by Hypothesis 3 against the supposition that the Original MS. was composed around 1689-90. This hypothesis is clearly a form of the parallelogram law for the com­ position of independent motions. In Version I of the tract de Motu it occurs as Hypothesis 4. But in Version HI of that work it has suffered a most important and significant change of status. From being an hypo­ thesis, something ipso facto incapable of proof, it has become a lemma, something derived from more primitive assumptions, in this case the second law of motion. Thereafter it naturally maintained its new status, appearing as Coroll, i to the laws of motion in the lectures de Motu and the Principia, and as such was referred to in Newton’s correspon­ dence with Huygens in the autumn of 1689. Once again, therefore, we find a reversion in the level of Newton’s thought to that of Version I of the tract de Motu (or earlier). This time, however, the reversion be­ comes far more difficult to account for: that the author of the Principia, a philosopher in his own right, always most punctilious about the exact meaning of his terms, suspicious of hypotheses in general, and with Ockham eschewing unnecessary multiplication of causes— Causas rerum naturalium non plures admitti debere, quam quae et vera sint et earum Phaenomenis explicandis sufficiunt^— that Newton should have given as an hypothesis in 1689-90 what two years previously he had given as a corollary in the Principia is surely difficult to accept. All the more so remembering that for Newton qua mathematician the reduction in the number of independent axioms in his system of dynamics effected by the abrupt change in the status of this law would necessarily have been a most significant and memorable event. To summarize the argument so far: the presence and form of the three hypotheses and Prop, i in both the Locke and Newton copies raises certain difficulties for the supposition that the Original MS. was composed by Newton specially for Locke’s benefit, probably around March 1690, and in any case definitely after the publication of the Principia. Of these difficulties much the most serious are those raised by Hypothesis 3: it is at this point one feels the onus most definitely rests on the supporters of the 1689-90 hypothesis to defend their case. Never­ theless this hypothesis encounters further difficulties arising from certain differences between the Locke and Newton copies. ' Hypothesis i, Book III, Principia.

6.6

D A TI N G OF M A N U S C R I P T S

113

In the first place, it is certain that the Locke copy could not have been taken from the Newton copy. If one examines the very small number of cancellations in the Newton copy then in two cases one finds that the original versions, later cancelled, exactly agree with the Locke copy,i whereas in no case is the converse true. So that it seems certain that the Newton copy was taken either from the Locke copy or the Original MS., and not vice versa. As to the more substantial differences between the two copies, these are as follows; 1. The detailed justification in the Locke copy for the ‘construction’ in Prop. I is missing from the Newton copy. 2. The proof of Lemma i in the Newton copy represents an emended and improved version of that in the Locke copy: in particular, as noted below (n. i), a small, yet nevertheless real, non-sequitur in the Locke copy has been corrected in the Newton copy. 3. The Newton copy contains an additional proposition, numbered 2, not found in the Locke copy. This is a special case of the last propo­ sition. 4. Certain parts of Prop. 3 in the Newton copy represent emended, and on the whole improved and clarified, versions compared with the corresponding parts of Prop. 2 in the Locke copy. In the light of these differences between the two copies, and bearing in mind that the Locke copy could not have been taken from the Newton copy, consider again the hypothesis that the Original MS. was com­ posed specially for Locke’s benefit at some time after the publication of the Principia. There are then two mutually exclusive possibilities: either Newton sent the Original M S., or one of the Original M SS., to Locke in the form of a letter to be retained by Locke, or not. If he did so, then the existence of the Newton copy is at once explained as the copy ' T h e first, and more striking, o f these two cases is found in the last paragraph of Lemma i. Originally Newton had begun this paragraph as follows: ‘For joyn i y and d raw /F parallel to C D and because F f and F E are bisected in C and £)’ exactly as in the Locke copy (and presumably also in the original manuscript). Later he emended this to read: ‘ For joyn P f and d ra w /E parallel to C D and because F f is bisected in C , F E shall be bisected in D , and therefore z P D shall be equal to the sum of P F and P E that is to the summ o f P F and P f , that is to A B , and therefore P D shall be equal to A C . ' Clearly the Locke copy contained a small non-sequitur removed in the Newton copy. For the fact that P D is half the sum of P E and P F does not follow directly from the fact that F E and F f are both bisected, as stated in the Locke copy, but from the bi­ section of F E alone, itself derived of course from the bisection of F f, as stated in the Newton copy. T h e other case is found at the beginning of the last sentence of Lemma 3 where Newton had originally written: ‘But PX'^ was to G//® as . . .’ as in the Locke copy, replacing it by: ‘But Y X I was to P X ^ as . . . ’. T his emendation, like the previous one, results in a certain logical improvement in the presentation of the argument.

114

ORDER OF CO M P O S IT I O N AND

6.6

retained by Newton. But in this case why was it not an exact copy, as one would have expected ? Why the omission of the justification of the construction in Prop, i ? This justification, found nowhere else outside the Locke copy, represents an important clarification of the proposition in question and was presumably introduced specially for Locke’s benefit. Why then omit it from the copy} Again, the improvements in the Newton copy compared with the Locke copy are difficult to understand. Why not incorporate them in the manuscript sent to Locke One is almost forced to assume there were two Original M SS., one of which was sent to Locke, the other retained by Newton from which at some time after the transmission of the first to Locke Newton made the new and improved version represented by the Newton copy. Other small factors against the supposition that Newton sent one of the Original MSS. to Locke are the absence of such a manuscript, or of any covering letter, or of a letter of acknowledgement from Locke. Also that if Locke was in possession of an original manuscript in Newton’s hand why bother to have a copy made of it— remember the Locke copy is certainly not in Newton’s hand. Now consider the other possibility. In that case either the Locke copy was taken from the original ‘in Newton’s presence’, or else it was taken from the Original MS. which was later returned to Newton. In either case the Original MS. must have remained with Newton. In which case one has to assume that Newton, after the publication of the Principia, went to the trouble of copying out the manuscript again very carefully, emending and improving it, and omitting the justification of the construction in Prop. i. It seems, therefore, that either of these two mutually exclusive possibilities regarding the actual origin of the Locke copy are fraught with certain difficulties arising from the existence of the Newton copy and the peculiar differences between it and the Locke copy. This completes the case against the assumption of late status for the Original MS. V III. It remains to consider alternative explanations of the composition of the Original MS. and the Newton copy. Two such explanations suggest themselves: the Original MS. may have been composed in the winter of 1679-80 following Hooke’s intervention, thus containing Newton’s first proofs of the propositions in question: or it may have been composed in 1684 after Version I of the tract de Motu representing the original of 2i first set of propositions^ carried by Paget ' Assuming that there were two sets. See § 5 above, p. 107.

6.6

DATIN G OF M A N U S C R I P T S

115

from Cambridge to London in November 1684 in fulfilment of Newton’s promise to Halley. To assess the relative probabilities of these two alter­ native explanations certain comparisons will now be made between the Newton copy and Version I of the tract de Motu. The sections of the Newton copy involved in these comparisons are the same in the Locke copy, and thus presumably occurred in unchanged form in the Original MS. 1. The form of the principle of inertia given in Hypothesis i com­ pared with that given in Hypothesis 2 of Version I of the tract— Corpus omne sola vi insita uniformiter secundum lineam rectam in infini­ tum progredi nisi aliquid extrinsecus impediat— throws no light on the question of the relative order of composition of the two manuscripts. The reference here to the resistance of the medium is without signifi­ cance considering the reference to resisting mediums in Def. 3 of Ver­ sion I of the tract, and the fact that Prob. 6 of that work deals with the actual motion of a body in a resisting medium. 2. Hypothesis 2 of the present manuscript is clearly an early form of the second law of motion. An equivalent and closely similar form of this law first appears in Version III of the tract. The presence of this hypothesis could therefore be interpreted as evidence in favour of the present manuscript post-dating Version I of the tract. 3. The presence of Hypothesis 3 (the parallelogram law of compo­ sition of velocities) in the present manuscript also provides some slight evidence in favour of it post-dating Version I of the tract, since an equally clear formulation of the law first appears in Version III, the form given in Hypothesis 3 of Version I— Corpus in dato tempore virihus conjunctis eo ferri quo virihus divisis in temporibus aequalibus successive— being much more obscure. On the other hand one must remember the very clear formulation of the same law in the much earlier MS. V. 4. The proof here given in Prop, i of the result equivalent to Kepler’s second law of planetary motion differs in no important respect from those given in the three versions of the tract de Motu. But the fact that the reference to the planar nature of the motion in the present manu­ script is absent from all three versions of the tract could be regarded as evidence that the present manuscript post-dated these works. 5. The partial approximation to the ellipse by a broken line in Prop. 2 and 3 shows unmistakable evidence of the influence of the procedure adopted in Prop. i. No trace of such an influence is found in Version I of the tract. It seems difficult to account for this unless

116

ORDER OF C O M P O S IT I O N AND

6.6

one regards it as an indication of the way Newton first arrived at his solu­ tion to the problem of elliptical motion, and it is even more difficult to understand why he should have reproduced these details in a paper written after Version I of the tract, unless, of course, one assumes that Newton was trying to make the proofs easier for Halley, as opposed to Locke, a rather improbable assumption. 6. In Prob. 3 of Version I of the tract Newton derives the inverse square law for elliptical motion under a force to a focus by applying the general result of Theor. 3 to the special case of motion in an ellipse, whereas here he concerns himself solely with the problem of motion in an ellipse. Again it is difficult to understand why Newton should have composed the present paper after Version I of the tract. The evidence supplied by the above comparison fails to decide de­ finitely between the two explanations, though there is perhaps a slight balance of evidence in favour of the 1684 hypothesis. Against this we have the argument based on Prop, i already employed in consideration of the 1689-90 hypothesis. For if this proposition existed already in a perfectly acceptable form in Latin in Version I of the tract de Motu why did Newton not take it directly from that work? The same argument could even be applied to the last proposition, though perhaps with less force, since the proof in the tract de Motu was given in the form of an application (Prob. 3) to elliptical motion of the general theorem given in Theor. 3, whereas if there were two sets of Paget proposi­ tions the assumption is that the first set was devoted to motion in an ellipse. The above comparison between M SS. V III and Version I of the tract de Motu was based on certain elements the two copies have in common. It remains to consider the substantial differences between them noted already in connexion with the 1690 hypothesis, namely; 1. Certain improvements in the Newton copy compared with the Locke copy. 2. The omission in the Newton copy of the justification of the con­ struction in Prop. I of the Locke copy. 3. The existence in the Newton copy of an additional proposition not found in the Locke copy. These differences raised certain difficulties for the supposition that the Locke and Newton copies were composed after the Principia. But these difficulties no longer apply if it is assumed that the Original MS. was composed before the Principia. For in each case it may be assumed that

6.6

DA TIN G OF M A N U S C R I P T S

117

the Newton copy was also composed before the Principia shortly after the Original MS. of which it represented an emended and somewhat improved version. The omission in the Newton copy of the con­ struction justification in Prop, i could then be explained in one of two ways: either it was not in the Original MS., being introduced first in the Locke copy as an aid to Locke’s understanding; or else it was in the Original MS. and was simply omitted as unnecessary in the Newton copy. Finally, the existence of the additional proposition would have been quite natural if the Newton copy had been composed in 1679, and might or might not have been in the Original MS. It would perhaps have been a little less natural in 1684. Whereas its presence in a paper composed in 1690 is difficult to credit. T o summarize, three explanations of the composition of the Original MS. V III have been considered. It may have been composed: 1. In the winter of 1679-80 following Hooke’s intervention, thus containing the original proofs of the proposition in question. 2. In 1684 following Version I of the tract de Motu representing the original of a first set of Paget propositions. 3. After the Principia, around March 1690, specially for Locke’s benefit. The evidence for explanation 3 rested on Locke’s known interest in Newton’s Principia, the existence of the Locke copy, and the admittedly more ‘popular’ nature of the propositions compared with the correspond­ ing ones in the Principia. But there were several items of internal evi­ dence against this explanation all pointing towards the early status of the Original MS. One of the strongest of these was the use of the term ‘hypothesis’ applied to the parallelogram law of composition of motion. As regards explanations i and 2, the relevant evidence failed to de­ cide definitely in favour of one or the other, though there was perhaps a small balance in favour of the second explanation. The possibility of other explanations must naturally be allowed for.^ ^ T h e above interpretation of M S . V I I I follows closely that given in Archives Internationale d ’Histoire des Sciences, 16 , 13-22 (1963). T h is interpretation has been criticized by A . R. and M . B. Hall (ibid., pp. 23-28) who hold to the traditional inter­ pretation (previously advanced by Brewster and Ball) of this M S . as having originated in the request from Locke for clarification of the corresponding proposition in the Principia. D . T . Whiteside has also drawn m y attention to a Latin translation o f the same M S . in Whiston [2], Lecture X IV /X V . Whiston notes that he gives the deduction ‘ Qualem nempe earn e charta M S . ipsius Newtoni olim acceperim'. T he fact that Newton apparently said nothing to Whiston about the origin of the M S . does not seem to tell much for or against the above interpretation.

PART

II

NEWTON DYNAMICAL MANUSCRIPTS 1664-1684

INTRO D UCTIO N T h e text aimed at throughout is a faithful ‘compromise’ reproduction

of the original manuscripts. Thus the use of capital letters, archaic forms of spelling, and the punctuation follow the original manuscripts exactly, whereas contractions such as ‘ye’ for ‘the’, ^ for par, &c., are every­ where extended as being a matter of handwriting and not of language. All extensive, legible cancelled passages are given in full in critical foot­ notes, but certain cancellations of words and short phrases judged unimportant have been omitted silently. Marginal entries in the original manuscript other than footnotes are indicated in critical footnotes apart from certain titles of sections entered in the text. Newton’s own footnotes are given at the foot of the relevant page separated where necessary by a rule from other footnotes. Alternatives in the manuscript are indicated by a bar in the text. Everything within square brackets represent additions to the text apart from [ ] representing a word or words missing from the manuscript, or [?] representing a word or words illegible in the manuscript. A word or words inserted within square brackets and suc­ ceeded by a ? indicate an editorial conjecture. All manuscripts, or sections thereof, are supplied with an introductory note giving details of location, probable order and date of composition where possible, and a brief description of contents. Any indications of the growth of the original texts are given at the end of these notes. Other notes are of two kinds: critical notes, labelled alphabetically, and given as footnotes; exegetical notes, labelled numerically, and given together at the end of each manuscript or section of manuscript. These latter notes fall under the following headings: (i) Cross-references to the occurrence of the same or cognate topics elsewhere. For the most part these have been restricted to the more important/omard' references, a complete list of contents of manuscripts according to topic being given in the Index. (2) References to the works listed in the Bibliography.

120

IN T R O D U C T IO N

(3) Comments not involving expressions of opinion, for example, those referring to the modern equivalent of certain terms used by Newton. (4) Non-controversial interpretations of certain technical dynamical or mathematical questions, for example the majority of the notes to the commentary to MS. III. The remaining notes, for example those referring to the growth of Newton’s dynamical thought, all relate to matters allowing of a poten­ tially wide variation of personal opinion. For the reasons already given in the General Introduction these latter notes are confined wherever possible either to a direct reference to the appropriate section of Part I, or to such a reference followed by a brief unsupported expression of opinion. With two possible exceptions, all the manuscripts here reproduced certainly belong to the period 1664-84. The first of these exceptions, that represented by MS. XI, is comparatively unimportant. For even if that manuscript does not contain the text of public lectures given by Newton in the Michaelmas term 1684, it must still have been composed by the autumn of 1685, and in any case it certainly represents the direct continuation of the earlier drafts given by M SS. IX, Xa, and Xb. The other possible exception, that represented by MS. V III, is a much more serious one. For the existence of the Locke copy, with its endorsement ‘Mr. Newton Mar 1689-90’, points to the possibility that it may have been composed after the Principia. Nevertheless, for the reasons given in Part I, Chapter 6.6, it seems to me on balance more probable that the original of both the Locke and Newton copies was composed either in December 1679 or in 1684. In any case, the treatment here afforded to MS. V III should help other interested scholars to decide for themselves on the status of this manuscript and its original.

I E X T R A C T S FRO M E A R L Y N O T E B O O K MS. Add. 3996 consists of a small notebook of 140 leaves bound in worn leather bearing within the inscription ‘Isaac Newton Trin. Coll. Cant. 1661’. A careful general description of contents, together with a repro­ duction of certain passages, especially on optical subjects, has been given by Hall [i]. Apart from extensive quotations from certain philosophical works, among which those from Magirus [i] seem of particular interest as one of the most likely sources of the scholastic background to Newton’s physical thought, the book contains original entries by Newton covering a wide variety of topics listed at folios 87, 87V. With few exceptions, largely of a psychological or physiological nature, such as ‘fantasy’, ‘soule’, or ‘sleepe’, these topics all treat of physical matters, the most substantial entries being on motion, optics, and properties of matter. It seems likely that the majority of these entries were made in 1664 or early 1665 at the latest (among them are various cometary observations dated between 9 December 1664 and i April 1665) that the present manu­ script probably provides the best available evidence of Newton’s preparation for his decisive discoveries in optics and dynamics during the plague years. Another notebook (U .C.L. MS. Add. 4000) provides parallel evidence for his preparation for the mathematical discoveries during the same period. The text is clean throughout apart from a small number of unimpor­ tant deletions. Unlike the later manuscripts, the punctuation in the long passage on violent motion is very deficient and has been supplied and emended where necessary to facilitate understanding of what is in any case a rather difficult piece of writing. Text (1) * Note that the mean distances of the primary Planets from the Sunne are in sesquialiter proportion to the periods of their revolutions in time. 2 (2)3

of violent motion

[i] Violent motion is continued either by the aire or by a force impresstor by the natural gravity in the body moved. Not by the aire, since the aire crowds more uppon the thing projected before, than behind, and must therefore rather hinder it. For you may observe in water that a thing moved in it doth carry the same water behind it along with it as in a cone, or at least the water is moved from behind it with but a small forced as you may

122

E X T R A C T S FROM EARLY NOTEBOOK

I

observe by the motes in the water. Suppose {a) [Fig. i] to be the body moved, (b.d.i.f.) to be the water moved behind (a) to give it place, (r) the water behind {a) following it and going along with it.s Then if the water at (/) ran so violently against the backsid of [a) it would beate away the water at (r) with violence, but the water is moved very slowly from behind (a) if it be moved away [at all]: as you may perceive by the motes in the water. The like must hapen in aire. If you say no I answer must

I

E X T R A C T S FROM EARLY NOTEBOOK

123

aire well be more compressed at (/) than at (b) and consequently when let loose againe it will dilate it self and so begin a new motion. I answer how comes the aire to be more crowded behind the ball than before it since (a) will communicate as much force on (b) as it receives from (r), and the fore parte of the aire will croud no more on the latter parte than the ball will croude on it. Againe whence is it that a piece of leade will move farther and with more force than a piece of wood of the same bignesse since the aire will have the same influence on both.*® (2)**

of violent motion

This motion is not continued by a force impresst because the force must be communicated from the mover into the moved either by some corporeall efflux or incorporeall one or nothing. If by corporeall attomes we are still at a losse how those attomes must continue theire one motion. If by an incorporeall efflux it must be by either spirit or some quality. If by a spirit how comes the spirit to be so easly united to the body and not to slip through it and when united to it how comes the spirits to cease so soone and the spirit to leave it and hence every little attome must have soules in store to cast away upon every body they meete with. If a quality then qualitas transfigurat de subjecto in subjectum}^ and this quality cannot be the motion of the mover since it and the mover are separated at once from the thing moved. In a word how can that give a power of moveing which it selfe hath not. then move (a) forwards in water.^ So if hot leade drop into water that parte which is behind will be pointed the fore parte round which would be otherwise if the aire pressed as much on it behind as before. Thirdly, how can the aire continue the motion of a globe on it axis.7 Fourthly in the former figure the aire is supposed to have the same propensity* to motion which the ball {a) is supposed to have, that is will move no longer than it is propelled on. Then I say the water at (r) cannot move the ball unless the ball do not at the same time move {b), that {b) may [move] (g) and {h), and (g) may move (d), and (d) move (/), and (f) move (/), and (/) move (r), and force it to rush upon the ball. And consequently at the same instant (r) must [move] the ball, and the ball move (r), which cannot be. But suppose the aire and the ball were detained from motion by some outward agent, yet kept the same respect to one another in situation as they did in theire flight then as soone as they were both let loose againe the aire would have as much power to move the ball as it had when they were in theire former flight. If it be answered that the

of violent motion Therefore it must be moved after its separation from the mover by it one gravity. *3 Which will be cleare by seeing whether there can be motion in a vacuum and what that motion is and so comparing it with motion in plena. That there may be motion in vacuo let us suppose (a, b) [Fig. 2] to be a body as a piece of Aire, {c, d, e) to be three globes, (/, gy h, i) and all the space about the globes and that aire to be inane.*^ Now in the chapter de vacuo we have shewed that these three globes would be really separate and not touch one another. You will grant that halfe the globes are in places, and consequently may move. Suppose then the halfe of (i:) in the aire move towards {d) we aske whether that part in vacuo would move along with it or stay behind and be separated frome it. If the first we have our desire, if the last wee ask what should separate it from it. Not the vacuum since that is accounted nothing. But you may say that it is not truly motion for the upper parte of (c) to be carried to {d). We answer that

124

E X T R A C T S FROM EARLY NOTEBOOK

I

where there is action (for such is the passing of (c) to (d)) and where there are new respects acquired to the same bodys there must be motion, but the upper part of (c) hath neither the same respect to the aire nor to (d) which it had before it began to pass towards d. If this going oi cto d be

I

E X T R A C T S FROM EARLY NOTEBOOK

125

the aire is uniforme. And we judge a thing to be moved when we see it come nigher or goe farther from some thing which our senses can per­ ceive and so we judge not a thing to be moved in respect of the aire but of the earth or something. (3) ‘^ Cartes defines motion 2nd parte Pr to be the Transplantation of one part of matter or one body from the vicinity of those bodys which immediately touch it and seem to rest, to the vicinity of others. (4) 21 How much longer will a pendulum move in the Receiver22 than in the free aire. Hence may bee conjectured what bodys there bee in the receiver to hinder the motion of the pendulum. (5) 23 The gravity of bodys is as their solidity, because all bodys descend equall spaces in equall times consideration being had to the Resistance of the aire etc.

not motion I aske what is it, But this is onely to [strive ?] about termes and if it please you not to call it motion call of motion it what you will but it is that which we aimed to prove and there is but this difference twixt it and motion inpleno^ that the one is environed with such mater as is impenitrable and consequently the mater must be crouded out of the moving bodys way before or rather at the same time that the body moves, it must needs impede the motion and be continually thrusting against and resisted by the body before it: but in vacuo it meets with nothing impenitrable to stay it. Tis true God is as far as vacuum extends but he being a spirit and penetrating all matter^^ can be noe obstacle to the motion of matter noe more than if nothing were in its way. Let me aske why one should be motion more than another since in pleno [it] is so stopped by one body rubbing uppon another and in vacuum it hath its liberty. Can the same thing (viz. a being environed with bodys) at the same time give a being to motion and yet destroy it, *7 wherefore to be in pleno cannot be essentiall to motion. And if it were, things would be more properly sade to move where there is most body or they find most resistance to theire motion and so more properly in water than in aire etc. But if it is objected by A risto tleth a t a vacuum is uniforme and everywhere alike and a body hath the same respects to a vacuum in all places alike but there is no motion with [out ] some mutation of circumstances. And so in vacuo no motion, I answer as to our senses

(6y^ ‘According to Galilaeus, a iron ball of 100 Florentine (that is 78 at London Adverdupois Weight) descends an 100 braces Florentine or cubits (or 49,01 ells, perhaps 66 yds) in 5" of an hower.’ 1. At fol. 2 9 . It is possible that Newton first encountered Kepler’s laws of planetary motion through a reading of Streete [i]. For example, some astro­ nomical entries on fol. 2 7 V are headed ‘Out of Streete’. 2. See Newton’s reference in the Portsmouth Draft Memorandum (repro­ duced above at Part I, Chapter 4, p. 66) to ‘Kepler’s Rule of the Periodical times of the Planets being in a sesquialterate proportion of their distances from the centers of their Orbs’. 3. Beginning at fol. 98 and continuing (from the beginning of section 2) at fol. 113. This considerable discussion of violent motion is memorable as being the earliest extant extended piece of writing by Newton on a dynamical subject. It is also of great interest for the evidence it provides for the medieval background to Newton’s thought. The tone of the whole passage is immistakably scholastic, being cast almost in the form of a medieval disputation. Equally unmistakable, however, is the extreme clarity of expression in striking contrast to that of many of Newton’s contemporaries. Noteworthy, too, is the characteristically detailed nature of his observations of the vortices produced in fluid motion, and especially of the behaviour of small motes in the water at the ‘stagnation point’ immediately behind the moving body. At the top of fol. 88, before the first of the topics listed on fols. 87, 87a, Newton has written: 'Amicus Plato amicus Aristoteles magis arnica veritas.' In this passage on violent motion he has already parted company with the Aristotelian belief that projectiles were moved by the circumambient air, though he is largely con­ cerned (in § [i]) with refuting the theory of antiperistasis which he may possibly have confused with Aristotle’s own rather different theory of the motion of a projectile. Just as Aristotle’s cosmological views became largely irrelevant after the publication of Galileo’s Siderius Nuncius, so Aristotle’s theory of motion ceased to have any true scientific value after the publication of Newton’s Principia

126

E X T R A C T S FROM EARLY NOTEBOOK

I

in 1687. Nevertheless it is gratifying to find Newton taking his place at the end of the long and distinguished line of critics of Aristotle’s theory of motion. At the beginning of the passage three distinct causes of the continuation of violent motion are mentioned: the air, a ‘force impresst’, or the ‘natural gravity’ of the body. The first section is devoted to a refutation of the first cause, the second cause is rejected at the beginning of the second section, and the remainder of the passage is devoted to a defence of the third and, in Newton’s view, correct cause. 4. This ‘small force’ (of the body’s motion) is clearly intended to be measured relative to the moving body. 5. That is the region of the backward stagnation point relative to the moving body. 6. The meaning of this sentence as it stands is obscure. Possibly Newton in­ tended to supply ‘the water’ between ‘answer’ and ‘must’. 7. An argument found, for example, in Buridan. See Duhem [3], p. 37. For the best-documented and most detailed account of medieval dynamics see Clagett [i], and for the best brief account Crombie [i], vol. ii. Part I, § 3. 8. This like propensity (towards cessation of motion) followed from the Aristotelian belief that all motion required a cause failing which it would im­ mediately cease. 9. The sense of this and the preceding passage is obscure. Most probably Newton meant to imply that the air would have no more power to move the ball when they were released than when they were restrained. 10. This argument, based on the assumption that the same force (— influence) will have the same effect on two different bodies, one heavy, one light, provides a good indication of the primitive state of Newton’s dynamical thought at the time of composing this passage. It should be compared with Buridan’s much juster argument against the Aristotelian theory on the grounds that a feather would then go further than a heavy body. See Duhem [3], p. 39. 11. The arguments in this section against the continuation of motion by a ‘force impresst’ do not seem so conclusive as those in the previous section. 12. One would have expected non migrat in place of transfigurat. 13. That is, the body’s natural gravity. An echo, perhaps, of the medieval distinction between natural gravity = weight, and unnatural gravity = im­ petus or force. See, for example, Duhem [i], p. 114 et seq. 14. It appears later that Newton somehow intended the spheres to be partly in vacuo and partly in air. Originally he had written earth throughout in place of Aire which would have made the supposed arrangement more plausible. 15. As opposed, for example, to Descartes’s belief {Principia, Pt. 2, Art. 18) that a vacuum in a vessel would be unthinkable since the walls of the vessel would then necessarily touch. In the chapter de vacuo referred to (at fol. 8qv) Newton bases his belief in the possibility of empty space between bodies on the existence of atoms not infinitely divisible, so that if the bodies are nearer than the least atom there would necessarily be empty space between them. 16. The view, for example, of Henry More: for a discussion of More’s views on space, spirit, and matter see Koyre [5], especially Chapter 6. The influence of More on Newton remains to be established. A beginning has been made by Koyre and by Fiersz [i]. 17. Reminiscent of Ockham’s argument against Aristotle based on the collision of two stones, when one and the same portion of air would have simultaneously to move the two stones in opposite directions

I

E X T R A C T S FROM EARLY NOTEBOOK

127

18. Physics, Book IV, Chapter 8. 19. At fol. 117. Whereas Newton here gives Descartes’s definition of motion in the true and philosophical sense without comment, later, in MS. VI, he criticizes the same definition at length and to great effect, ultimately replacing it by his own definition of absolute motion relative to absolute space. There are a number of other references to Descartes in the present notebook, including one to ‘page 54 Princip. Philos. 3rd’ at fol. 9 3 V in connexion with Descartes’s Vortex Theory. On the same folio occurs a cometary observation dated Sat. December, 4th 1664. 20. That is, Descartes’s Principia Philosophiae first published in Amsterdam, 1644. There was a copy of the third (Latin) edition of this work in Newton’s library. See de Villamil [i], p. 74. 21. At fol. 117. 22. References to ‘Mr. Boyle’s Receiver’ occur at fols. 9 6 V , 9 9 . 2 3 . At fol. 1 2 1 V . This passage would seem to point towards a realization of the distinction between the concepts of mass and weight. Solidity would have to be equivalent to density in the sense of the quantity of matter per unit of volume. For a discussion of Newton’s concept of mass and density see above. Part I, Chapter 1 . 4 , p. 2 5 . 2 4 . At fol. 1 2 1 V . The figure of 1 0 0 braces in 5" reappears in the Vellum Manuscript (MS. Ill) and was taken by Newton from the Dialogue of Galileo. See MS. Ill, Appendix A, para. (i).

II

II DYNAMICAL WRITINGS IN THE WASTE B O O K the religious commonplace book of Newton’s stepfather, the Rev. Barnabas Smith, whose signature, dated 12 May 1612, appears on the fly-leaf, the Waste Book (MS. Add. 4004) probably passed to Newton on the death of his stepfather {c. 1656) along with the latter’s library. Some of the pages have theological entries followed by notes in Smith’s handwriting, and the original index is also extant. When first examined by the Syndics of the University of Cambridge appointed to catalogue the Portsmouth Collection the book was in a state of great confusion with many folios missing. It has now been restored and appears to be complete. It contains many entries by Newton apart from the dynamical ones, those on mathematics being of particular importance. The dynamical entries occur as follows: O r ig in a l l y

At folio I (Ila) At folio 10 (Ilb) Definitions, at folio lov. (lie) Ax.-Prop. 1-26 at folios iov-12 (Ud) Ax. 100-22 at folios 12-13 (lie) Ax.-Prop. 27-40 at folios 13V-15, 39 (llf) At folio 38 (Ilg) The occurrence of references to the Definitions in both Ax.-Prop. 1-26 and Ax. 100-22 proves that lib was composed before lie and Ild. Likewise references to certain of Ax.-Prop. 1-26 in Ax. 100-22 prove that lie was composed before lid . Finally in He there are references to all the preceding sections so that the order of composition of Ilc-e follows the alphabetical order, that is, the order of entering in the Waste Book. It is then natural to assume the same to be true for the remaining entries on folios 1,10, and 38. No internal contradiction results from this assumption for the last pair of entries; in fact, in both cases there is some supporting evidence. For the calculations on folio 10 are restricted to the specially simple case of totally inelastic collisions, whereas the physical discussion in Ax.-Prop. 7-10 deals with the much more difficult case of perfectly or partly elastic collisions. Again, the entries on resolution of velocity and composition of motions on folio 38 are the first to treat of these topics in the Waste Book, and follow the preceding dynamical entries on

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

129

folio 15 after a considerable gap. Lacking further evidence it is therefore reasonable to assume that the most probable order of composition of the entries on folios 10 and 38 is given by their position of entry in the Waste Book. A similar assumption, however, cannot be made for the treatment of circular motion on folio i, for this must have come after the first tenta­ tive discussion of the problem in Ax.-Prop. 20. Striking confirmation of this is supplied by the alteration of the figures 4 + in Ax.-Prop. 24 to 6 + , corresponding to the exaet result 2tt obtained in folio i. Given the date ‘Jan. 20th 1664’ [O.S.] oeeurring in the margin of folio 10 it is probable that all the dynamieal entries in the Waste Book were made on or after that date. On the other hand it is possible that some of these entries represent final versions of earlier drafts composed earlier in January 1665 or towards the end of 1664.! I la At folio I. The heading Dens on this folio in Barnabas Smith’s hand, against the normal alphabetical order, is confirmed by the index, and is explained by the original nature of the book. There can be no doubt that the first two dynamical entries on this folio must have been made later than the discussion of circular motion beginning at Ax.-Prop. 20. It is just possible that Newton originally left the first page blank either out of a feeling of respect for his stepfather’s memory, or on religious grounds, whereas the mathematical entry on the verso, bearing the date Septem­ ber 1664, the earliest in the Waste Book, could well have been the first entry in the book. Apart from a few unimportant erasures the text is clean and shows no signs of being worked over. Given the rather complicated nature of the argument on circular motion it is perhaps unlikely that it was written in extempore. Text [i.]^ If the ball b [Fig. i] revolves about the center n the force by which it endeavours from the center n would beget soe much motion in a body as there is in h in the time that the body h moves the length of the semidiamiter bn. (as if b is moved with one degree of motion through b7i in one seacond of an hower then its force from the center n being continually (like the force of gravity) impressed upon a body during one second it will generate one degree of motion in that body.) Or the force from n in one revolution is to the force of the bodys motion as : : periph : rad. t T h e dating of the earliest dynamical manuscripts is considered above in Part I, Chapter 6.2.

130

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

II

Demonstration. [2.]“ If e/ fg = g h ~ he — zfa — zfb = zgc = zed. And the globe move from a to b then zfa : ab : : ab : fa : : force or pression of b upon fg at its reflecting : force of 6’s motion, therefore ^ab = ab be cd da : fa : : force of the reflection in one round (viz: in b, c, d, and a) : force of

II

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

131

[4.] If a body undulate in the circle bd all its undulations of any altitude are performed in the same time with the same radius. Galileus.^ [5.] As radius ab to radius they undulate.5

: : so are the squares of there times in which

[6.] If c circulate in the circle egef [Fig. 2], to whose diameter ce, ad = ab being perpendicular then will the body b undulate in the same time that c circulates.6 a

[7.] And those bodies circulate in the same time whose lines drawne from the center a to the center d are equall.7 [8.] And ad : dc : : force of gravity to the force of c from its center d.^

Figure i.

Vs motion, by the same proceeding if the Globe b were reflected by each side of a circumscribed polygon of 6, 8,12, 100, 1000 sides etc. the force of all the reflections is to the force of the bodys motion as the sume of those sides to the radius of the circle about which they are circumscribed. And so if [the] body were reflected by the sides of an equilaterall circum­ scribed polygon of an infinite number of sides (i.e. by the circle it selfe) the force of all the reflections are to the force of the bodys motion as all those sides {id est the perimiter) to the radius. [3.] If the body b moved in an Ellipsis^ that its force in each point (if its motion in that point bee given) [will ?] bee found by a tangent circle of Equall crookednesse with that point of the Ellipsis.

[9.] Coroll : hence may the force of gravity of the motion of things falling were they not hindered by the aire may very exactly [be] founds (viz. [}] cd : ad : : force from d : force from a. 1. For an interpretation of this and the following subsection see above, Part I, Chapter 1.2, pp. 7-1 r. An equivalent result is derived by an entirely different method in MS. IVa. It seems probable that Newton used this result to derive the peculiar formula employed in the calculations of MS. III. See § 2 of the ‘Commentary and Interpretation’ to that manuscript. 2. This demonstration must have followed Newton’s first estimate of the fo r c e o f the body's endeavour fr o m the centre in h a lf a revolution given in Ax.-Prop. 22. Particularly interesting in this connexion is the cancellation of the figure 4 + in Ax.-Prop. 24 and its replacement by 6 + corresponding to the 2^ of the present section. For a similar ‘polygonal’ treatment of circular motion see the demon­ stration of the law of centrifugal force at the end of the S ch o liu m to Prop. 4, Theor. 4, Book, I, P rin cip ia . Ball ([i], p. 13) suggested that this latter demon­ stration was the one employed by Newton to calculate ‘the force with which a ball revolving within a sphere presses the surface of the sphere’ prior to his

132

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

II

derivation and first test of the inverse square law of gravitation against the moon’s motion during the Plague Years. See above, Part I, Chapter 1.2, especi­ ally pp. 8-11. 3. Although the results derived in the preceding subsections are exact, they give no hint as yet of the dependence of centrifugal force on speed and radius: this dependence is first given (implicitly) in MS. IVa. Nevertheless Newton is already pondering the more difficult problem of motion in an ellipse. Given the reference to Kepler’s third law of planetary motion noted in § i of MS. I, it is not improbable that he had in mind the elliptical paths of the planets about the sun. 4. See, for example, the Dialogue (edit. Santillana [i], p. 246). 5. This result does not appear in Galileo’s Dialogue. It is found, however, in his Discorsi (edit. Crew and Salvio [i], p. 96). See above. Part I, Chapter 6.2, p. 91, for the possible indication of date thus provided. 6. An early, if not the earliest, correct statement of this result appeared with­ out proof in Huygens’s Horologium Oscillatorium. See Theorem IX of the appendix on centrifugal force at the end of Part 5 of that work. The problem of the circular pendulum does not seem to have been considered by Galileo. There is a passing reference to it by Descartes in the Cogitationes Privatae (Qiuvres, vol. X, p. 224) probably inspired by an entry in Beeckman’s Journal (Beeckman [i], vol. i, pp. 256-7). The use of a conical pendulum as a regulator for clocks was demonstrated by Hooke at meetings of the Royal Society at the beginning of the year 1667 (see Birch [i], vol. ii, pp. 150 et seq.). Later Huygens contested Hooke’s claim to have been the first to employ such a method. It seems prob­ able that Huygens had employed the circular pendulum in clock regulation as early as 1658 (see Huygens, CEuvres, vol. vii, pp. 390-1). An important reference to the circular pendulum is also found in Hooke’s paper ‘Concerning the in­ flection of a direct motion into a curve by a supervening attractive principle’ read before the Royal Society at their meeting of 23 May 1666. See Birch [i], vol. ii, pp. 90-92. 7. Horologium Oscillatorium, Theorem VII. 8. Ibid., Theorem X, for a special case of the result given by Newton. This special case is the one employed in MS. HI. See § 3 of the ‘Commentary and Interpretation’ of that manuscript. 9. Characteristic of Newton’s invariable search for exact quantitative results.

I lb At folio 10. The date, ‘Jan. 20th 1664’ (O.S.), in the margin is the only one occurring among the dynamical entries in the Waste Book. This whole section is devoted to the problem of collisions between two perfectly inelastic bodies. It is assumed throughout that the bodies move together after collision the total directed motion (= momentum) being conserved. Subsection 2 gives the impression of being a revised and more far-reaching version of the cancelled subsection i. It is possible, there­ fore, that in spite of the small number of emendations the subsections 2-8 were written in extempore. There are no errors in the calculations apart from the small slip indicated at note (f).

II

D Y N A M I C A L W R I T I N G S IN THE W A S T E B O O K

133

Text Of Reflections. [i.]'* Suppose that the bodys a, h, [Fig. i] have noe zis elastica to reflect the one from the other but at theire occursion conjoine and keepe together as if they were one body. Then first if Theire bulke and motion be equall then at theire meeting they shall rest. (2). If {b) have more motion than (fl) all the motion of {a) shall be lost Figure i. and soe much of {b)s as (a) had and they shall both move towards c shareing the difference of the motion proportionally twixt them. Demon: Suppose the motion of ehf =■ motion of a (2). If a rest and b hit it they shall both move towards c shareing the motion of b twixt them. O f Reflection [2.]*^ Suppose the Bodys a, b [Fig. 2] doe not reflect one another but conjoyne*= at their meeting and soe move or rest together, a = the body a^\b = the body fee; c = body ced\ G d = body fedc. m — motion of a, n — motion of b, p = motion of c, q = motion of d — n-\~p, before re­ Figure 2. flection. e = motion of a, f = motion of b, g ^ motion of c, h = motion of d after reflection, r = swiftnesse of a, ^ = swifness of b, c or d, before reflection, t — swiftness of a, u =^swiftnes of b, c or d after occursion. 0 the point of theire occursion Axiome first [3.] Two bodys {b, c) [Fig. 2] being alike swift the motion of b : motion of c : : b : c. for equall parts have equall motion. Therefore b : c : : a\\ the parts of b : all such parts of c : : motion of b : the motion of c. [4.] Prop: first. If before the occursion of a [Fig. 2] and d a rest then shall e^ h = q.~ and since t = v, tis alsoe e : q : : a : a-\-d.^ Or e ~ h == f/—

-also a-\-d

= -A a-\-d a\ d [5.] Prop; 2*^1 If a meete d, and have lesse motion than it, then.

134

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

q— m = e-\-hA for suppose m — n. then should a and occursion did not p = q— ni force them towards k.

II

rest after

[6.] Prop, 3: Suppose the center of gravity^ in d [Fig. 2], y i n a . z and / the [points] in which the bodys a and d touch in theire meeting 0 the point of theire meeting, {a) the magnitude of the body («). (d) the magni­ tude of (d) m — motion of a before meeting, n = motion d before meet­ ing. the time in which a or d<^ moves to 0 = time in which they both move to y [Fig. 3]. p = motion oi a, q — motion of d after occursion

II

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

135

cdm-\-bam dn-\-an j , , 7j , 7 1 cdmA-bam ---- ------ ~ — ----- or cdm^bam = bdn-\-ban and — — -— — n. ba a bd\--ba bd-yba : cd^ba : : m n. tm vn : b : \ tbd-\-tba : vcd-\-vba. cdvA-abv , aetA-det — e and v = td-\-ta obA-cd [8.i°] If a and d [Fig. 6] meete one another that c must be negative that is ^

negative the pointy must be taken on the same

side of [0 as z]

Figure

3.

a. This sub-section deleted. b. At this point there is a marginal entry, ‘Jan 20th/i664’. c. Succeeded by either moving or resting together deleted. d. See Diagram 2. e. Immediately after this occurs the equation: zo x a ■ fo x d : :m : n equivalent to the assumption motion proportional to magnitude of body X distance moved in given time. f. In error for m : p : : zo : oy.

m-^n = p+q- a : d :: p : q, or a-\-d :d : : m^n (= p + q ): q am-\-an Q,-\- d

am X zo-{-an X zo p. m [ p : zo : zy^. am-\-dm

dm^dn CL-[- d

oy.

[7.]^ a = magnitude of the body a. [Figs. 4 and 5] = mag: of body d. 0 = the point of concourse : z , f = the points of contact, at 0. zo — h, fo — c, op — e. t — time in which the bodys move from z and / to o. V — time in which they move from 0 to p. m = motion of a before occdnfi cursion. n = motion of it afterwards, ba : cd : : m = motion of ba dn d before occursion. a : d motion of d after occursion.

1. Meaning a is the magnitude or bulk of the body a. From ‘Axiome first’ it is clear that both bodies are assumed to be of the same material so that the question of mass and density does not arise. 2. For the definitions of these and other terms in subsections 3-5 see sub­ section 2. 3. This assumes the motions of two equivelox bodies to be proportional to their bulks. See MS. lie, Def. 3. 4. The first indication that Newton had corrected Descartes’s law of conserva­ tion of motion in respect of the direction of movement. 5. The reason for the division of the body d in Fig. 2 into two parts b and c now becomes evident. 6. Although the centre of gravity plays no part in the discussion its appear­ ance here indicates that Newton was thinking of extended bodies rather than idealized particles. 7. The calculations in this subsection, with their introduction of times and distances for measuring speed, may have been intended to serve as a basis for an experimental test of the law of conservation of motion. 8. In conformity with MS. He, Def. 3, this assumes that motion is propor­ tional conjointly to bulk (magnitude or quantity) and distance moved in given time.

136

DYNAAIICAL W R I T I N G S IN THE

W ASTE BOOK

II

9. T h is is eq u ivalent to the assum ption speed proportional directly to distance and inversely to tim e. A lth o u g h N ew to n n orm ally com pares speeds b y the dis­ tances m o ved in the same tim e, as in M S . l i e , D e f. 2, he does giv e the m ore general expression in vo lvin g the ‘ fo rb id d en ’ division o f one kind o f ejuantity

II

DY N A M I C A L W R I T I N G S IN THE

WASTE BOOK

137

Alsoe if the cube Inniqyz [Fig. 3] 8 move the length of op 5; and the piramis [Fig. 4] tvwx — 7 move the length of rs —- 3 in the same time; then, as opxlmqyz : rsXtvwx : : 40 : 21 : : the motion of Imqyz to the

(distance) b y another (time) in the note to M S . I l d , A x .-P r o p . 26. H ere he is careful on ly to com pare quantities o f the sam e kin d on each side o f the equation. 10. T h is su bsection again dem onstrates N e w to n ’s clear realization o f the im portance o f direction. H is easy m astery o f the significance o f e n egative is also striking.

lie Commencing at folio lov. Four diagrams occur at the top margin opposite Defs. i, 2 having no connexion with these definitions but similar to the diagrams found at the foot of folio 10. Apart from the substitution of ‘quantity’ for ‘body’, and the cancellations of the original version of Def, 3 and the end of Def. 10, there are few alterations in the text which could thus have been a write-out of an earlier draft. Some support for this possibility is supplied by the absence of definitions numbered 6 and 8, and by the fact that in Def. 3, of which the preceding passage would seem to represent an earlier version, the term ‘quantity’ is used throughout in contrast to the original use of ‘body’ in the immediately preceding and succeeding definitions.

Figure i.

Text Definitions. When a Quantity^ is translated/passeth from one parte of Extension^ to another it is saide to move^

2.3 One Quantity® is soe much swifter than another, as the distance through which it passeth is greater than the distance through which the other passeth in the same time. One'^ Quantity® hath soe much more motion^ than another, as the summe of the spaces through which each of its parts moveth is to the summe of the spaces through which each of the parts of the other quantity® moveth'^ in the same time, supposeing each of the parts in both Quantitys'^ to be equall and alike to one another'' and moved in the same p osition .O r 3. One Quantity hath so much more motion than another, as the distance through which it moveth drawne into its quantity, is to the distance through which the other moveth in the same time, drawne into its quantity. As if the line ah [Figs, i and 2] move the length of be and ef the length of eh in the same time, the motion of ah is to the motion of cd, as abxbc — abed, to e fx eh — iehk.

Figure 3.

motion of tvwx. Or the motion of one quantity to another is as their powers to persever in that state.s 4. Those Quantitys'i are said to have the same determination^ of their motion which move the same way, and those have divers which move divers ways. 5. ^'^A quantity® is reflected when meeting with another quantity it looseth the determination of its motion by rebounding from it. As if the bodys

138

D Y N A M I C A L W R I T I N G S IN THE

fiOOK

II

a [Fig. 5], h meete ones another in the point c they are parted either by some springing motion in themselves or in the matter crouded betwixt them, and as the spring is more dull or vigorous/quick soe the bodys will bee reflected with with [sic] more or lesse force: as if it endeavour to get liberty to inlarge it selfe with as greate strength and vigor as the bodys a, b, pressed it together, then the motion of the body a

II

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

139

on both sides any plaine with which [mn) is coincident**; also the line mn drawne through it is called an axis of motion.* 11 .*3 The center of motion in 2 divers bodys is a point soe placed twixt these bodys that (if it bee conceived to rest and) if the bodys bee moved about it with circular motion they shall both have an equall quantity of motion, the line about which they move is the axis of motion.

d

Figure 5.

Figure 7.

from b will bee as greate after as before that reflection, but if the spring have but halfe that vigor, then the distance twixt a and b, at the minute after the reflection shall bee halfe as much as it was^’ at the minute beefore the reflection. 7.9 Refraction is when the body ^passing obliquelythrough the surface ed [Fig. 6] at the point b meets with more opposition on one side of the surface than on the other and soe looseth its determination; as if it turne towards a. 9.1° Force is the pressure or crouding of one body upon another. 10.^^ The center of motion* in the same body is such a point with in a quantity! which rests when a body is moved with any circular but not progressive motion!; **that'^ if it be conceived to rest as at a [Fig. 7] and some line as (nrn) be drawne through it, about which (as about an axis) the quantity (dklp) revolving there shall bee the same quantity of motion

12. ’^ A Body is said to move toward another body either when all its parts move towars it or else when some of its parts have more motion towards it than others have from it. Otherwise not. 13. * Bodys are more or lesse distant as the distances of their "^centers of motion*" are more or lesse. or as their distances might bee acquired with more or less motion. a. Replacing body deleted. Quantity, as appears from Def. 3, either means body, or its quantity, or bulk, a term sometimes used by Nezvton, as in M S. V, § i. b. The succeeding passage dozen to position deleted. It is clearly an earlier version of Def. 3. c. Succeeded by supp deleted. d. Replacing bodys deleted. e-e. Substituted for Or struck out and zvritten at the end of the line. f. Marginal entry noe motion is lost in reflection. For then circular motion being made by continuall reflection would decay.* g. Succeeded on the left-hand side, as though at the beginning of a nezv definition, by A quantity is said to bee refracted deleted. h. Succeeded by before deleted. i. Replacing gravity deleted. j-j. Written in above the passage within a quantity. k -k . Deleted. 1. This definition entered in the margin. m-m. Replacing parts deleted.

140

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

II

1. See reference to extension, space, or expansion in MS. V, § i. Newton’s metaphysical views on the nature of extension are given in MS. Add. 4003. Certain extracts from this manuscript are given in MS. VI. 2. Similar definitions of motion are given in MS. V, § i, and MS. VI, Def. 4. 3. Cf. the definitions of velocity in MS. V, § i ; MS. VI, Defs. 13, 14. With one remarkable exception (see note to MS. I Id, Ax.-Prop. 26), Newton always compares either the velocity of one body with another, or two different velocities in the same body. 4. A more sophisticated definition of ‘motion’ is given in MS. V, § i, sent. 4. Normally motion for Newton means quantity of motion momentum) but on occasion he uses it in the sense of movement, as in MS. V, § i, sent. 3. 5. This statement refers back to that at the beginning of the Definition and implies that for Newton the pow er o f a body to persevere in its state is proportional to its motion. See MS. lie, Ax. 100. 6. Cf. Descartes, P r in cip ia Ph ilosop h iae, Art. 39, Part 2: omne id quod m ovetur . . . determ inatum esse a d motum suum continuanduni versus aliquam p artem , secundum lineam rectam .

7. Notice how Newton follows his definition of reflection by a vivid illustra­ tion of the process as in certain of the definitions in P rin cip ia , Book I. 8. Cf. same notion in MS. I Id, Ax.-Prop. 20. 9. Recalling, possibly, his early experiments in optics. 10. The force here defined is force in the instantaneous, physical sense. Else­ where, as in MS. I Id, Ax.-Prop. 3-6, he uses the term in the sense of an impulse measuring the total effect of a variable, physical force. It is this latter type of m otive force which features in the second law of motion. For a discussion of the development of Newton’s concept of force see above, Part I, Chapter i.i. 11. For a further discussion of the movement of extended bodies beyond that in Defs. 10-13, see MS. Ild, Ax.-Prop, i i et seq. 12. If the deletion of the remainder of the passage is ignored there results a definition of centre of motion in the dynam ical sense. Whereas on omitting the cancelled section there results a definition of centre of motion in the kinem atical sense. For a further discussion of this centre of motion see MS. Ild, Ax.-Prop. 11-18, and for the centre of motion in the dynamical sense MS. V, § 4. Newton’s kinematical centre of motion corresponds to, though it does not always coincide with, the modern instantaneous centre of motion, whereas his dynamical centre of motion as defined in MS. V, § 4, may be shown to coincide wdth the centre of mass or gravity of the body (see MS. V, n. 8), so that his alternative use of the term ‘gravity’ for ‘motion’ in the present definition was entirely just as applied to the dynamical centre of motion. 13. This gives the extension to two bodies of the immediately preceding definition of the dynamical centre of motion for a single body. See MS. Ild, Ax.-Prop. 25. 14. References to this definition occur in MS. Ild, Ax.-Prop, i i, 12.

I ld The Axiom-Propositions commence at folio lov immediately after Def. 12 and continue recto and verso to Ax. 100 at folio 12. They recom­ mence with number 27 after Ax. 122 on folio 13V and continue recto and verso to Ax.-Prop. 37 at foot of folio 15. 'I'here is then a break till folio 39

II

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

141

where the set is completed at Ax.-Prop. 40. The stop after ‘Axiornes’ in the heading would seem to indicate that the original heading was ‘Axiomes’ only. This is confirmed by various references to certain of Ax. 1-26 in Ax. 100-22, for example, those to Ax. i, 2 in Ax. 100. The term Propo­ sition first occurs in a reference to Prop. 28 in Ax.-Prop. 29. Again, there is a reference in the revised version of Ax.-Prop. 28, 30 to Prop. 25 which had previously been referred to as Ax. 25 in Ax.-Prop. 27. It can be assumed, therefore, that the original title of numbers 1-26 inclusive was ‘Axioms’, the term ‘Proposition’ having been added after the compo­ sition of those following Ax. 100-22. The whole set of Axioms and Propositions has therefore been divided into two sections Ild and Ilf, separated by Ax. 100-22 at lie . The members of the present set treat of the following topics: Principle of inertia (i, 2) Force and motion (3-6, 23) Collision between two bodies (7-10) Kinematics of an extended body (11-18) Circular motion (19-22) Centre of motion of a pair of bodies (24-25) Kinematics of a single particle (26) Several inversions of number order point towards the original exis­ tence of an earlier version. With the exception of a number of considerable deletions (all noted) the text is almost entirely clean.

2'ext Axiomes. and Propositions 1.1 If a quantity once move it will never rest unlesse hindered by some externall caus. 2. A quantity will always move on in the same streight line (not changing the determination nor celerity of its motion) unlesse some externall cause divert it. 3.2 There is exactly required so much and noe more force to reduce a body to rest as there was to put it upon motion: et e contra. 4.3 Soe much force as is required to destroy any quantity of motion in a body, soe much is required to generate it; and soe much as is required to generate it soe much is alsoe required to destroy it. 6. If the s[ame] force move 2 unequall bodys {a and h) the swiftnesse of

142

D Y N A M I C A L W R I T I N G S IN THE

WASTE BOOK

one body {a) is to the swiftnesse of [the ot]her as h is to fore the motion of both bodys shall bee equall.

II

and there­

5. If two [equal ?] bodys [b, c) bee moved by uneqiiall forces, as the force moveing (b) is to the force [movjing c, so is motion of b, to the motion of c, so is the swiftnes of b, to that of c. 7. If two bodys [a and b moving?] against one another the same way towards O [Fig. i], {a) overtaking {b) none of theire motion shall be lost, for {a), presses [(6) as much] as {b) presses {df^^ and therefore the motion of (i!)) shall increase [as much] as that of (<2) decreaseth. a

d\ Figure i.

h

ct Figure 2.

8. ^ If two quantities [a and b) [Fig. 2] move towards one another and meete in O, Then the difference of theire motion shall not bee lost nor loose its determination. For at their occursion they presse equally uppon one another and^P^ therefore one must loose noe more motion than the other doth; soe that the difference of their motions cannot bee destroyed. If one body {a) overtake another body {b) they both moveing towards O then they shall always move together. If the body (c) move against an immoveable quantity {d) it shall not bee rebounded for c having urged d with 9. *=If two equall and equally swift bodys {d and c) meete one another they shall bee reflected, soe as to move as swiftly frome one another after the reflection as they did to one another before it. For first suppose the sphaericall bodys e, f [Fig. 3] to d f 1 ^ have a springing or elastic force soe that meeting one another they will relent and be pressed into a sphaeroidicall figure, and in that moment in which Figure 3. there is a period put to theire motion towards one another theire figure will be most sphaeroidical and theire pression one upon the other is at the greatest, and if the endeavour to restore theire sphaericall figure bee as much vigorous and forcible as theire pressure upon one another was to destroy it they will gaine as much motion from one another after their parting as they had towards one another before theire reflection. Secondly suppose they be sphaericall Def. 3^^.

Axiome 4th.

Axiom 4th.

II

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

143

and absolutely sollid'^ then at the period of theire motion towards one another (that is at the moment of theire meeting) theire pression is at the greatest, or rather ’tis done with the whole force by which theire motion is stopt (for theire whole motion was stoped by the force of theire pressure upon one another in this one moment, and there cannot bee/ succeede divers degrees of pressure twixt two bodys in one moment). Now so long as neither of these 2 bodys yeild to one another they will retaine the same forcible pressure towards one another : that is soe much force as deprived the bodys of their motion towards one another soe much doth now urge them from one another and therefore^'^ they shall move from one another as much as they did towards one another before theire reflection. 10.5 There is the same reason when unequall and unequally moved bodys reflect, that they should separate from one another with as much motion as they came together. 11 If a line df [Fig. 4] bee moved not with d a Progressive but onely a Circular motion its middle point (n) shall rest. For if it move let it move towards r soe that, when the point (d) is in p and / in {q), then (w) shall be moved to {s). 11. If aline {ce) [Fig. 5] be bisected in (a) about which the line (ce) doth circulate and that Figure 4. point bee fixed. Then the whole line hath noe progressive motion. For making ab = ad, bf, ag, and dh bee parallel, and perpendic to fh, then is vb = dp and vf-\~ph = bf-\-dh = zag. Wherefore the point c moveing towards n the point d shall move soe much towards the line/A as the point b doth from it, and all the points in {a, c) or the line ac move as much to the line fh as all the points in {ae) or the line (ae) moves from it soe that the whole line ce stays in equilibrio neither moveing to nor from//?, by the 12th Defin. 12. Hence when the center of a line (a) is not in the midst of a line (/« e) [Fig. 5] the whole line moves the same way which the longest parte doth, for supposeing ca = ae then the line ce is in equilibria (par ax. ii) but if {m c) moves towards (fh) and be added to (c e) then (m e) moves towards ce (by def. 12). 13. When (ce) [Fig. 5] moves circularly but maketh noe progression its axiome 3''.

144

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

II

midle point shall rest and is therefore the center of its motion, for if the middle point move let it bee to r from a soe that the line ec bee moved into the place [w t) then let {w t) move about the fixed center r into the place xi’, then xs and wt are equally distant fromfh (by def. 13 and ax. 11) and alsoe (In) and [ce) are equally distant from the [line ?] f h : but xs and In are not equally distant from/A: therefore neither are wt and ce equally

II

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

145

determination from ac which d hath from h (for if ac be understood to move parallell to its selfe into the place ef, all its points describe parallel lines and therefore have the same determination one with another and each with the whole body (by axiom 14) therefore gh hath the same determination from ac which the point d hath from the point b. 14(b). A Body being moved parallell to its selfe all its points describe parallel lines and each of them have the same determination and velocity with the body, for (by axiom 2)^ they must all bee streight ones which if they intersect the body will not be moved parallel to its selfe. etc. 16.*’^ If a body move forward and circularly its center of motion shall allways bee in the same streight line. For the body hath allways the same determination (ax. 2) and the center of its motion hath the same deter­ mination with the body (ax. 15) therefore it hath always the same determination, and soe will move continually in the same streight line.

distant from fh. and therefore the line ce had progressive motion when it passed into wt. i4(a).^ By the same reason the middle point of a parallelogram parallelipipedon, prisme, cilinder, circle, sphaere, elipsis, sphaeroides [are at the ?] center of theire motion. 15.'^ A Body hath the same determination of its motion which the center « of its motion hath. As if the line ac Q f . ^ [Fig. 6] move into the place the center of its motion b moveing into the place d, then let gh move about the center d untill it be parallell to ac, as into ef, soe that the point a fall into the point e. Now since gh by turning about the center d hath noe progressive motion (by Def. 10) tis plaine that gh and ef have determination from ac but ef hath the same

If a body move streightly forward and circularly its center of motion shall have the same determination and velocity that '4he body hath'\ For suppos ac [Fig. 6] to be moved into the place and its center of motion b into the place d, then let it turne about the center d into the place ef parallel to ac soe that the point whieh was in {a) bee now in {e). Now since [gh) by moveing into the place (/
146

DY N A M I CA L W R I T I N G S IN THE

W ASTE BOOK

11

II

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

147

call surface of the body edf circularly about the center m, it shall press upon the body def for when it is in c (supposeing the circle bhc to be described by its center of motion and the line eg a tangent to that circle at 0) [it] moves towards g or the determination of its motion is towards therefore if at that moment the body efd should cease to check it it would continually move in the line eg (ax. i, 2) obliqly from the center m.

7/ a

\

/

/ /

\

nv

\

\

\

/

but if the body def oppose it selfc to this endeavour keeping it equi­ distant from m, that is done by a continued checking or reflection of it from the tangent line in every point of the circle ebh, but the body edf cannot check and curbe the determination of the body eo unless they continually presse upon one another. The same may be understood if the body adb [Fig. 9] bee restrained into circular motion by the thread {om)W

/ Figure 8.

one to another soe are theire velocitys one to another (ax. lo, def. 2) and their motions are to one another as theire bulkes drawne into the Radij of those circle (which theire centers of motion describe) are to another (Def. 3). A s ; ec : CO : : velocity of eb : velocity of ao and eh xec \aoXco : : motion eb : motion of ao. 20.^ If a sphaere or [Fig. 8] move within the concave shaeicall of cilindri-

2 1. Hence it appeares that all bodys moved circularly have an endeavour from the center about which they move, otherwise the body oe would not continually presse upon edf. 22. '^ The whole force by which a body eo indevours from the center m in halfe a r e v o lu t io n is more than double to the force which is able to generate or destroy its motion, that is to the force with which it is moved: for supposeing it have moved from (c) by {h) to {h) [Fig. 8]. Then the resistance of the body def (which is equall to its pressure upon def) is able to destroy its force of moveing from e to g and to generate in it as much force of moveing from [b) to {h) the quite contrary way.

148

D Y N A M I C A L W R I T I N G S IN THE

WASTE BOOK

II

25.™ Having [the] cent[er] of motion of the 2 bodys ob and de [Fig. 10] to find the common center of both [their motions] draw a line oe from the centers of their motions o and e, and div[ide it at a ?] so that the body ob is to the body de as the line ae to the line oa: that is so that obxoa = ae X de.^^ For then if they move about the center a being always opposite to one another they have equall motion (ax. 19) and consequently have"

II

D Y N A M IC A L

W R IT IN G S

IN

TH E

W A ST E B O O K

149

velocity of a is to the velocity oi e 2is, a b x d f to ed X be.^^ For supposeing 7 .1 • eb X ed * 1 / , 1 * r • that gp — eb then is ep = — ^ — And (by def. 2) the velocity of a is to , r 7 . . eb Aed 7 7- 7 7 7 * 1 1 the velocity oi e : : ab : ep : : ab : ■— ^ - : : a b xfd : ebx ed. Alsoe the fd motion of a is to that motion of e (by def 3^) : : a xa b : eXcp : : aXab: , eb Xed i n 7 7 • 1 . eXep : : aXab : e x : : a x a b x f d : eX eb xed : : a X its velocity :

e x its velocity.**^ a Figure 10. Figure 12.

an equall endeavour from the center a (ax. 24) soe that if they bee joyned to center (a) by the lines ae and ao, the one hindereth the other from forcing the centre a any way soe that it shall stand in equilibrio betwixt them and (by def. 10)1^ is therefore theire center of motion. 24." If two bodies {cb and de) [Fig. ii] move about a center (a) then making be ~ a, de — b, ac — c,ae — d, the time in which be makes halfe a revolution call e, that in which de doth the same call /, the pressure of cb from the center a in halfe a revolution call g, and that of de call h : the motion of eb in halfe a revolution call k, and that of de call /, then k : I : : 24. If two bodys {eb and de) [Fig. 11] move about a center a thenP The whole force by which the body eb tends from the center a in one revolu­ tion (being equall to 6-f^7 times the force by which the body is moved (ax. 22)^^ is to the motion of that body eb as the whole force by which the body de tends from the center a in one revolution (which is equall to 6 + ^7 times the force by which the [body] de is moved, or which is able to stop its motion (ax. 22)) is to the motion of the body de. Vide Axioma 23""’. 26. If the body a [Fig. 12] move through the space ab =r- b m the time d = be, and the body e through the space cd — e in the time /. then the

Note^o that when the motion is uniforme that is when a body moves over the same space in the same time (which will ever bee when the motion of that body is neither helped nor hindred) then in a right angled triangle ab may designe the space through which a body moveth in the time eb. Otherwise, when ’tis not uniforme the proportion of the time in which a body moves to the distance through which it moves may be designed by lines drawne to a crooked line, as the time by [Fig. 13] or ih, the distance by gh or fi, the velocity by the proportion of nh to hi, ni being tangent to the crooked line at i, etc.

150

D Y N A M I C A L W R I T I N G S IN THE

WASTE BOOK

II

23.’' ®If the body bacd [Fig. 15] acquire the motion q by the force cT, and the body/the motion / by the force then d : q : ; ^ for suppose the body rscb = / to acquire the motion w by the force d, then (ax. s) ^ S ■ '■ w : p. but q = zv (by ax. 4) therefore d : g : : q : p.

II

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

151

result fo r ce oc change in m otion produced given later in A x .-P r o p . 23. A n alm ost identical expression o f the present n um ber occurs at M S . H e, A x . X14. 3. See M S . H e, A x . 115. 4. N o tice for N e w to n absolutely solid m eant p erfectly elastic, whereas for Descartes in his P rin cip ia P h ilosop h iae it im plied p erfectly inelastic. See, h o w ­ ever, G a b b e y [i]. C h a p ter 4. 5. T h e result given in this n um ber is true for a p erfectly elastic collision, as follows, for exam ple, either from N e w to n ’s ow n law for relative velocities w ith e =- I , or b y sim ultaneous application o f conservation o f m om en tu m and co n ­ servation o f energy. B u t it is not clear how N e w to n supposed that the general result follow ed from the very special case considered in the p revious num ber. T h e general result is em p loyed in M S . V , §§ 9, 10. N o tic e that the term ‘m o tio n ’ is here em p loyed in the sense o f m o vem en t rather than in the usual sense o f cjuantity o f m otion ( — m om entum ). 6. In troduced to distin guish it from the later section o f the sam e num ber. T h is latter section is clearly the ‘A x io m 14 ’ referred to in n u m ber 15, and therefore presum ably bore the same n um ber in som e previous draft o f the present version.

F'igurc 15. a. F o l. I I begins. b. D ele te d in fa v o u r o f the succeeding version. c. E n title d in the margin O f the separation o f bo dys after reflection d. D eleted in fa v o u r o f the succeeding version. e. F o l. I I V begins. T h is zvhole section deleted. f. W hole section deleted.

g. E n title d in the margin T h e center o f m otions deter and velocity. h-h. R ep lacin g It w o u ld have had did the b o d y m o ve parallel to it selfe deleted. i. Su cceed ed by i f ac w o u ld be m oved parallel to it selfe into the place o f e f it w ou ld keep the same determ ination and q u an tity o f m otion deleted. j. S ucceed ed by in an y poin t o f the crooked line its determ ination is in the tan gen t deleted. k. E n title d in the margin O f endeavor from the center. l . I n error f o r g. m. E n title d in the margin T o find the center o f m otion in two bodys. n. F o l. 12 begins. o. D eleted in fa v o u r o f the succeeding version. p. T h e r e follow s a n u m ber o f dashes in the m anuscript as if N e w to n were pon derin g the correct continuation. q. S u cceeded b y a p artly illegible deleted portion containing a p ro of o f the same result som ew hat less elegant than the final version. r. M a r g in a l entry let this follow the s ' ax: s. B en ea th I f 2 b o dys be m oved w ith equall or uneq deleted.

7. T h e reference to A x . 2 is curious in this apparently kinem atical proposition. N e w to n m ay have had in m ind a b o d y m o vin g w ith o u t rotation un der the action o f no forces. 8. N u m b e r 17 evid e n tly represents an im proved, com posite version o f the present n um ber and the p revio u sly cancelled n um ber 15. In an y case the state­ m ent in the first sentence is u n tr u e : the centre o f m otion need n ot m o ve in a straight line b u t m ay describe instead som e ‘ crooked lin e’ , as in n um ber 18. 9. N o tice the identical nature o f the dem onstration em p loyed to prove this proposition and that giv en in the ‘ m o dern ’ p ro o f that any m o vem en t o f a b o d y in a plane m ay be replaced b y a translation follow ed b y a rotation. 10. F o r exam ple, a circle, as in N e w to n ’s polygo n al treatm ent o f centrifugal force at fol. i (M S . Ila ). 1 1. A s in the case o f circular m otion. See n um ber 20. 12. T h e case considered b y D escartes in A rt. 38, Part 2, P rin cip ia Philosop hiae. 13. T h is first ten tative attem p t to introduce number into the treatm ent o f circular m otion is carried to a precise conclusion in fol. i (M S . H a). 14. T h e exact m ean in g o f the p recedin g part first becom es evid en t in fol. i (M S . H a). 15. S o that a is at the centre o f mass o f the tw o bodies. 16. T h e reference is obscure. D e f. 10 refers to one b o d y only, whereas the dynam ical definition o f centre o f m otion o f a pair o f bodies in D e f. 11 contains no reference to equilibrium ( -- rest). N e w to n p ro b ab ly had in m ind the re­ quirem ent that the centre o f m otion should be at rest, som eth ing w h ich fo l­ low ed in this case from the equal and opposite endeavours o f the tw o bodies from the poin t under consideration. H ow ever, the im portan t th in g w as his

divination o f the special dyn am ical significance o f the poin t (centre o f mass) thus

1. N o tic e h o w in this and the su cceed ing n um ber N ew to n enunciates the

defined. M a n y propositions relating to the m otion o f this p oin t are giv en in

p rinciple o f inertia in tw o separate parts. See A x . 100. See also above Part I,

A x .-P r o p . 27 et seq. T h e s e propositions, in turn, m u st be regarded as the fore­

C h a p ter 2.2, for a discussion o f the evidence p rovid ed here and elsewhere in

runners o f C o ro ll. 4 to the laws o f m otion in M S . X I and in the P rin cip ia .

the W aste B o o k for the influence of D escartes on N ew to n in dynam ics. A co m ­

17. R ep la cin g a cancelled 4 k

w h ich w o u ld correspond to the tentative

parison o f this and other enunciations o f the p rinciple o f inertia up to and in ­

approxim ation found in n um ber 22. W hereas 6 + corresponds to the exact result

clu d in g that in the P rin cip ia is given above in Part I, C h a p ter 1.4, p p. 29.

2TT derived at fol. i, good evidence that the present en try was originally m ade

2. T h is and the three follo w in g num bers are each special cases o f the general

before those on fob i.

152

D Y N A M I C A L W R I T I N G S IN THE

WASTE BOOK

II

18. N o tic e in A x . 22 N e w to n equates the fo r c e by vAiich a body is m oved to the force able to generate or destroy its m otion. B y A x .-P r o p . 23 (found out o f order at end o f the present set) this latter force w ill be proportional to the m otion created or destroyed. T h is explains the final reference to A xio m a 2 3 ”” '. 19. T h e first occasion on w h ich qu a n tity o f m otion (as originally defined in D e f. 3) is set jo in tly proportional to bu lk and velocity. 20. W h a t follow s represents an extraordinary excep tion to N e w to n ’s cu sto ­ m ary m anner o f dealing w ith velocities b y com parison o f one w ith another rather than directly, as here, as the gradien t o f the space-tim e graph. See above. Part I, C h a p ter i . i , p. 3.

lie At folio 12 immediately after Ax.-Prop. 23, and thereafter without a break to folio 1 3 V. Several inversions of number order, especially in the last four axioms, point towards the existence of a previous version. With the exception of the last four, the present set of axioms constitute a closely interconnected whole whose significance and purpose is at first sight difficult to grasp. The clue to their interpretation is provided by Ax. 1 18. This axiom is evidently equivalent to the general result relating force and motion generated or destroyed given in Ax.-Prop. 23, itself the generalization of the special cases given in Ax.-Prop. 3-6. Now it can be proved that Ax. 1 18 is connected, directly or indirectly, to almost all the results derived in the preceding axioms, and these results can in turn ultimately be derived from a small number of basic assumptions regarded as self-evident by Newton. Ax. 100-18 therefore amount to an attempted proof of the necessity of the result given as an axiom in Ax.-Prop. 23. In addition, and quite independently, though in the same spirit, the principle of inertia enunciated previously at Ax.-Prop. i, 2, is derived from a certain more general philosophical principle at the beginning of Ax. 100. As an aid towards comprehension an outline of the argument cul­ minating in the proof of Ax. 118 will now be given. At Ax. lo i, itself a revised draft of that part of Ax. 100 following the discussion of the principle of inertia, Newton ‘proves’ that the ‘hindrance impediment, resistance or opposition’ required to destroy the velocity of a given body is double that required for an ‘equivelox’ (= equally fast) body of half the size. Or generally, that the resistance or hindrance required to destroy a body’s velocity is proportional to its size (Ax. 102). Likewise for the generation of velocity, the ‘power or efficacy vigor strength or virtue of the cause’ required to generate new velocity is proportional to the body acted on (Ax. 103). A particular type of hindrance to the motion of one body is provided by the motion of another, and two equal, equivelox bodies offer equal and opposite hindrances to each other (Ax. 105). In

II

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

153

this case the hindrance to the motion of one results from the pow er of the other to ‘persevere’ in its velocity or state. This power is usually called ‘the force of the body’. The endeavour of the force of one body to hinder the progress of another is performed by pressure (Ax. 106). For two equivelox bodies, their forces (equal to the forces with lehich they are moved or their respective powers to persevere on their motion) are propor­ tional to the bodies (i.e. their sizes) (Ax. 107 via Ax. 102). At this point a sharp distinction must be drawn between the force of a body (as defined previously) and the force (acting) on a body. The velocities acquired by equal bodies are proportional to the force acting on them (Ax. 108). The same is true for the destruction of velocity. For example, if the velocity of a is triple that of an equal body h, the force required to deprive a of its velocity will be triple that required to deprive b of its (Ax. 109). Hence the force of a ’s motion is triple that of A’s (Ax. 1 10 using the definition of force of a body’s motion in Ax. 106). The same holds generally for equal bodies (Ax. i i i ) . The concept of a body’s motion (= momentum) is now introduced: ‘A body is said to have more or less motion as it is moved wdth more or less force, that is, as there is more or lessforce required to generate or destroy its whole motion' (Ax. 1 12). The motion of a body or the force of its motion is proportional jointly to its bulk and the distance it moves in a given time (Ax. 1 13, via Ax. 107, 111). But if the same force acts on two bodies a, b, with a = 36, then the velocity generated in b is triple that generated in a (Ax. 1 16, via Ax. 108). And generally for equal forces as « is to 6 as the velocity generated in b to the velocity generated in a (Ax. 117). Hence the motion generated in any two bodies is proportional to the forces acting on them. Likewise if motion is destroyed in both or generated in one and destroyed in the other (Ax. 118, via Ax. 108, 113, 117). The total structure of the argument leading up to the demonstration of Ax. 1 18 is exhibited in the following chart. Only general results have been included. Text Ax: 100. Every thing doth naturally persevere in that state in which it is unlesse it bee interrupted by some externall cause, hence axiome ist and 2^ and [2?] A body once moved will alw^ays keepe the same celerity, quantity and determination of its motion. IT two equall bodies {bepq and r) [Fig. i] meete one another wdth equall celerity (unlesse they could pass through one the other by penetration of dimensions) they must mutually hinder their perseverance in their states and (since the one hath noe advantage over the other they

154

DYNAAIICAL W R I T I N G S IN THE

WASTE BOOK

II

must equally hinder the one the others perseverance in its State having both of them an equall power to persever in their state, likewise if the body aocb be = and equivelox with r they meeting would equally

II

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

155

which can onely deprive cbqp of its whole velocity and motion by hinder­ ing its perseverance in its state, can also only deprive aocb of its whole velocity and motion, for that cause hath the same advantage over both the bodys. Now if I add the opposition {a) which can onely deprive cbpq of its motion to the opposition {b) which can onely deprive aobc of its motion the whole opposition {a^b = 2a = 2b) can onely deprive both the bodys aobc~\-cbqp of their motion when they are joyned into one {aopq) for a : cbpq : \ b : aobc : : a-\-b : cbqp-\-aobc : : 2a : aopq. Also neither the opposition a nor b alone can deprive aopq of its motion for then the parte*^ {a or b) would be equall to the whole {a-\ b — 2a — 2b).

/// Figure i.

By the same reason aopq and cbqp loosing equall velocity the impediment of aopq must be double to the opposition of cbqp.

hinder or oppose the one the others progression or perseverance in their states and therefore the power of the body aopq (when tis equivelox with r) is double to the power that r hath to persever in its state, that is the efficacy force or power of the cause which can reduce aopq to rest must bee double to the power and efficacy of the cause which can reduce r to rest, or the power which can move the one must be double to the power which can move the other soe that they bee made equivelox.' Hence in equivelox bodies the powers of persevering in their states^ are proportionall to their quantitys. Hence may bee perceived what is meant lo i. Supposing the bodies aobc and cbqp [Fig. i] to be equall and equi­ velox; Then that cause hindrance impediment resistance or opposition

1 0 2 . Since^ because aopq [Fig. i] is double to cbqp and both of them equivelox therefore the opposition which can deprive aopq of its motion must be double to that which can deprive cbpq of its motion; by the same reason^ it will follow that in equivelox bodys as one body {a) is to another [b) soe must the resistance which can deprive that body (a) of its velocity/motion bee to the resistance which can deprive {b) of its whole velocity/motion so is the resistance which can deprive («) of some parte of its velocity, to the resistance which can deprive {b) of the same quantity of velocity, soe that a and b bee still equivelox. °Now it may be perceived how and why amongst bodys moved some require a greater some a lesse opposition to deprive them of theire whole velocity or of some parte of it.*' 103. By the same reason alsoe If two bodys rest or bee equivelox: then as the body {a) is to the body {h) so must the power or efficacy vigor strength or virtue of the cause which begets new velocity in (a) bee to the power virtue or efficacy of the cause which begets the same quantit\of velocity in {b), soe that a and b bee still equivelox.

156

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BO O K

II

104. Hence it appeares how and why amongst bodys move some require a more potent or efficacious cause others a lesse to hinder or helpe their velocity. And the power of this cause is usually called force. And as this cause useth or applieth its power or force to hinder or change the perse­ verance of bodys in theire state, it is said to Indeavour to change their perseverance. 105. ^ If the equall and equivelox bodys a [Fig. 2] and b meete (unlesse they could passe the one through the other by penetrating its dimentions)

II

D Y N A M I C A L W R I T I N G S IN THE

io 8.j’7 Tis knowne by the light of nature that equall forces shall effect an equall change in equall bodies. Therefore if the forces^, h, k, m be equall, and the bodys a, b equall and rest, then let {a) bee moved by the force

k Figure 3.

ct

b

157

For let there be 2 other bodys a and d equivelox to them soe that a meeting b, and d meeting c they would equally hinder one anothers progression then is, a = b, and c — d (ax. 105) and a { — b) \ d { = c) : : the force which can stop a (= to the force of b) to the force which can stop d (= to the force of c). [vide ax. == 102)^

a

d

WASTE BOOK

b

/

■9 Figure 4.

c

Figure 2.

they must necessarily hinder the one the others progression, and since these bodys have noe advantage the one over the other the hindrance on both parts will be equall, likewise if the bodies d~\-a and b \-c bee equall and equivelox they must equally hinder one another’s progression. But the body {b) (being lesse than the body (6-fc)) and equivelox with [d-[-a) cannot hinder the progression of the body d-\-a soe much as the body (6+c) can; for then the power of [b) being part of the power of the body b-\-c would bee equall to the whole power of therefore that b-\-c and d-\-a being equivelox doe equally hinder the one the others pro­ gression tis required that they be equall. 106. g Now if the bodys a and b meete one another the cause which hindereth the progression of a is the power which b hath to persever in its velocity or stated and is usually'^ called the force of the body b and as the body b useth or applyeth this force to stop the progression of a it is said to Indeavour to hinder the progression of a which endeavour in body is performed by pressure and by the same reason the body b may bee said to endeavor to helpe the motion of a if it should apply its force to move it forward; soe that it is evident what the Force and indeavor in bodys are. 107. ' If the bodys b and c be equivelox then as 6 : c : : the force with which b is mov ed (or the power of b to persever in its velocity or [helpe ?] or to hinder another body from persevering in its velocity) to the force of c.

g ; and b h y h ,a and b shall be equivelox: Also (since tis noe greater change for (a) to acquire another part of motion now it hath one than for it to acquire that one when it had none)^ if {a) bee againe moved forward by the force k, its velocity shall be double to the velocity of b, and if it bee againe moved forward by the force m its velocity shall be triple to that of b, etc. Whence as the force moving (a) is to the force moving (b) soe is the velocity acquired in (a) to the velocity acquired in (b) by that force. 109. By the same reason if a = b and the velocity of a be triple to the velocity of b, that force can deprive a of its velocity which is triple to the force which can deprive b of its velocity. Or in generall so is the lost velocity of a to the lost velocity of b as the force which deprives a of some or all of its velocity, to the force which deprives b of some or all of its velocity.^ 112.6m body is saide to have more or lesse motion as it is moved with more or lesse force, that is as there is more or lesse force required to generate or destroy its whole motion. 1 13." If a body {a) [Fig. 3] move through the space ab = r in the time c, and the body / [Fig. 4] through .§/ = v in the time h then, time c : time h X cib h : : line ab — r : - — = ak, and the body a would move through the c space ak in the same time {K) in which the body / moves through the space jg. Therefore the velocity of a is to the velocity of / as the line b V nb ak — : line fg : : h xa b : c x fg (def. 2). Then I ad the body r t o f soe that/'+/=-:= a. Since/and rare equivelox, (ax. 107) as / : f ^ r ^ a :

158

D Y N A M I C A L W R I T I N G S IN THE

WASTE BO O K

II

II

D Y N A M IC A L

W R IT IN G S

IN

TH E

W A STE B O O K

159

am m = force or motion of /, to -y = force o f/ + r. againe since a = /-|-r

by that force. Tis axiome the 4th.^ And (by ax; 113) the bodys will have equall motion.

(ax. i n ) as the velocity of a; to the velocity o f / + r : : h xa b : c x fg : :

1 18." If the body / be moved by the force q, and r by the force 5, (to find (t?) the celerity of / and {w) that of q). I add (f) to / , soe th a t/+ / — r, and that (/) and (/+ /) are moved with equall force, then p-\-t ( = r)

n = the force or motion of a, to

~ = to the force of hxab f f-[-r. Soe that n x f x c x f g = m x a x h x a b . Soe that haveing any 7 of these the 8th may bee found, but suppose the bodys moved in equall times that is if c = h, then the rest of the termes may bee found by, m x a x a b = n X fx fg etc. that is as f x fg is to a X ab soe is the motion (m) of the body / to the motion (n) of the body a etc. If the body (a and b) bee equall and the celerity of a triple to that of b, then if the force d can deprive b of its motion the force can deprive a of its motion. But if there bee less force {},d--p) it cannot de­ prive a of its motion for soe the parte (3<^—/) would be = to the whole 2,d; if there be more force (3<^+/>) it will doe more than deprive the body a of its motion (i.e. move it the contrary way) otherwise the parte (3^/) would be equall to the whole (3^^+^).'^ Therefore the force which can deprive a of its motion must bee triple to the force which can deprive b of its motion and consequently (def.’' 106) the force wherewith a is moved is triple to the force wherewith b is moved H I . By the same reason as the celerity of the body a ( = b) is to the celerity of b soe is the force wherewith a moveth to the force wherewith b moveth.

1 14. '! There is required soe much and noe more force to reduce a body to rest than there is to move it: et a contra. And 1 15. Soe much force a[s] is required to generate any quantity of motion in a body, so much is required to destroy it, and e contra. For in loosing or to [«c] getting the same quantity of motion a body suffers the same quantity of mutation in its state, and in the same body equall forces will effect a equall change. I i6.f If the bodys a — 36, and a and b are m o v e d w i t h the same force d then the celerity of b is triple to the celerity of ap for 36 moved by 2>d is equivelox to b moved by dp^ but since 36 = a, therefore a moved by ^d is equivelox to b moved by d. And (ax. 108) as the celerity of a moved by d to the celerity of a moved by 3 / soe is i to 3 soe is the celerity of a moved by d to the celerity of b moved by d. I I 7. By the same reason. Any bodys / and g being moved by the same force, as (/) is to {g) soe is the celerity of {g) to the celerity of (/) acquired

: p : :v ; ^ t h e celerity of p-\-t (ax: 117) alsoe (ax: 108) ^ : q : :w : ^

=

or qrw = pvs. that is the celerity of / is to the celerity of r as qr VO

is to ps. And by ax. 113 the motion of / is to the motion of r as the force of / to the force of ^s r. And by the same reason if the motion of / and r bee hindered by the force q and the motion lost in / is to the motion lost in r, as q is to s. or if the motion of / be increased by the force q, but the motion of r hindered by the force s] then as to ^ : : so is the increase of motion in /, to the decrease of it in r (ax: i i i ) 121.^ If 2 bodys / and r meet the one the other, the resistance in both is the same for soe much as / presseth upon r so much r presseth on / . ’<’ And therefore they must both suffer an equall mutation in their motion. 119. If r [Fig. 5] presse / towards zv then / presseth r towards i ’.’®Tis evident without explication. 120. A body must move that way which it is pressed.*^ 122.2'’ Therefore if the body / come from c [Fig. 5] and the body r from d soe much as / ’s motion is changed towards w, so much the motion of r will be changed towards v. vide prop. [?] a. T h e succeeding passages dozen to the beginning o f A x . 1 0 1 deleted. b. F o l. I 2 v begins. c. M a r g in a l entry referring to 1 0 2 , 1 0 j . W h a t force is required to beget or destroy equall velo city in unequall bodys.

160

D Y N A M I C A L W R I T I N G S IN THE

WASTE BO O K

II

II

DYNAAdICAL W R I T I N G S IN THE

W ASTE BOOK

161

d. P reced ed B y th ey same reason that deleted.

forces equal to the latter actin g sim idtaneously, the extension to a force n tim es

e -e . D eleted .

another b ein g im m ediate. N o w he defines a force n tim es another as the latter

f. M a r g in a l entry. W h a t resistance in bodys.

force actin g n tim es in succession. T h e result given finally is eq uivalent to that in

g. M a r g in a l entry. W h a t force In deavor and Pression is. h. Com m encing above usu ally an d betzveen the lines occurs soe that a b o d y is

A x .-P r o p . 5. 10. Im p licit here is the identification o f m otion (in the sense o f q u an tity o f

said to be m o ved w ith m ore or lesse force® w h ich m eetin g w ith another b o d y can

m otion) and the fo r c e o f the body's m otion. T h is identification is also used at the

cause a greater or lesse m u tation in its state, or w h ich requireth m ore or less

en d o f the su cceed in g axiom .

force® to destroy its m otion.

1 1 . N o tic e the inversion o f order at this point.

i. M a r g in a l entry. W h a t force or M o tio n is in E q u iv e lo x bodys. j. M a r g in a l entry referring to i o 8 ,

l o g . W h a t v elo city acquired or lost in

equall b o dys b y un equ all forces.

12. T h e p recedin g sentence is really a parenthesis, the argum en t fo llo w in g on from the end o f the first sentence.

k. S u cceed ed a fter tzvo d eleted an d largely illegible lines by the follozcing d eleted

13. T h a t is a cted on by the sam e force. 14. T h e assertion 36 m o ved b y (the force) 3d eq u ivelo x to b m o ve d b y d im ­

passage. T h e force w h ich the b o d y (a) hath to preserve it selfe in its state shall

plies that the force on b, that is d, is reproduced sim ultan eously on each o f

bee equall to the force w h ich p u t it in that s ta te ; n ot greater for the effect cannot

b, b, b. T h is is his im plied definition o f an y force 3d.

exceede the [cause ?] for there can be n o th in g in the effect w h ich w as n ot in the

15. ‘ O f ’ in error for ‘ o n ’ . T h e reference to A x . 113 is e v id e n tly to the last

cause nor lesse for since the cause o n ly looseth its force aw ay b y co m m u n ica tin g

equation m X a X a b = n x f x f g w h ich im plies that the force o f a b o d y ’s m otion,

it to its effect there is no reason w h y its [«c] should [not] be in the effect [w hat ?]

that is its q u an tity o f m otion, or sim p ly its m otion, is proportional co n jo in tly to

is lost in the cause. H en ce appeares the tru th o f the 3"^^ and 4 “^axiom e.

its b u lk and velo city. 16. C o rrespo n din g to the third law o f m otion.

l.

F o l. 1 3 begins.

m . M a r g in a l entry. W h a t m otion in bo dys. n. M a r g in a l entry. A

generall theorem o f the p roportion o f v e lo city and

m otion o f g iv en b o d y m o vin g th rou gh g iv e n spaces in giv en tim es. o. M a r g in a l entry referring to n o , i n . W h a t force required to bege t or des­ tro y u n equ all celerity in equall bodys.

17. A ssu m in g force is m easured b y change o f m otion p roduced , as in A x . 1 18. 18. A c tio n and R eaction equal and opposite! 19. C o rrespo n d in g to that part o f the second law o f m otion w h ich states that the change o f m otion is in the direction o f the force acting. 20. T h is and A x . 1 19, 1 2 1 sho w that N e w to n had b egu n to thin k o f the m ore

p. I n error f o r A x .

general problem o f oblique collisions to w h ich the com plete solution is given in

q. M a r g in a l entry referring to 1 1 4 , i i 5. O f hin derin g and help in g m otion.

M S. V.

r. M a r g in a l entry referring to 1 1 6 , 1 1 7 . W h a t celerity acquired or lost b y equall forces in u nequall bodys. s. S u cceed ed by a p a r tly illegible deleted passage, evid en tly the beginning o f a fir s t attem pt a t p roving the result in question. t. In error f o r u. M a rg in a l entry. W h a t velo city and m otion gotten or lost b y un equall forces in u nequall b odys. A G en erall T h e o rem . V.

M a r g in a l entry. O f the m utuall force in reflected bodys.

1. N e w to n ’s conven ien t term for ‘eq u ally s w ift’ . 2. In A x . 106 this pow er o f p ersevering in its state is term ed the ‘force o f the body’. 3. T h e extension o f the previous definition for a - - zb to the case a ^ nb w ith n integral is im m ediate, after w h ich it can be farther extended to the case p a --qb w here p and q are integral. 4. N o tic e here, as at the b eg in n in g o f A x . 100, how N e w to n follow s D escartes in id en tifyin g m o vem en t as a state rather than a process. 5. N o tic e the different m eanings o f these tv'o kinds o f fo r ce. 6. W h ere he states that for eq u ivelo x bodies a, d, a - d = force w h ich can stop

a : force w h ich can stop d. 7. W hereas A x . 10 0 -7 dealt w ith unequal, e q u iv e lo x bodies A x . 1 0 8 -1 1 w ill be concerned w ith equal bodies havin g different speeds. 8. A v e r y plausible assum ption. B u t it w o u ld n ot be true in the Special T h e o r y o f R elativity. 9. P reviously, in .Ax. l o i , he had defined a force double another as ecjual to tw o

I lf At folio 13V immediately after Ax. 122. The fact that there are no inversions of order in this set of Axioms and Propositions, whereas such inversions occur in Ax.-Prop. 1-26 and Ax. 100-22, is perhaps an indi­ cation that the present set were original having no previous draft. Some confirmation for this is afforded by the comparatively large number of cancellations, and the manner in which Ax.-Prop. 27 breaks off abruptly in the middle of a calculation. Against this is the total absence of error in any of the algebraic calculations, some of which are extremely heavy, though this could simply be regarded as the algebraic counterpart of the impressive standard of arithmetical accuracy displayed in the calculations of MS. I l l which are certainly extempore. Ax-Prop. 27-30 deal with the motion of the centre of motion of two non-colliding bodies, establishing that in all cases this motion is rectilinear and uniform, and Ax.-Prop. 31,2 extend this result to the ease of two colliding bodies. The remaining propositions are concerned with various collision problems. The last four seem to represent the orientation of Newton’s thought towards the general problem of the collision of two Hr.K^ori M

162

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BO O K

II

II

D Y N A M I C A L W R I T I N G S IN THE 11/^.b’ TE il O O A

163

rotating bodies to which the complete solution is given in MS. V. The fact that these four propositions were entered separately at folio 39, in a hand rather more mature than the last of the preceding propositions, is probably an indication that they were not only entered but also composed at a somewhat later date. Text 27.® If two bodys b and c [Fig. i] move from (0) their center of gravity they shall have equall motion.^ For suppose h moved into the place d\ h

d

o

e

c

Figure i

then putting, c : b :: do

bxdo c

oe (ax. 25) the body c must be then

moved into the place e. Alsoe c :b : :bo

= oc (ax. 25) therefore

b x b o — b xd o b xb d , that is cXec bxbd. But (ax. or) c c 26 cXec : bdx b : : the motion of c, to the motion of d, and therefore c and d have equall motion towards o. If b-2 the body b [Fig. 2] move through the places d, f, r and the body c move through the places e, g, r and their center of [motion] 0, p, q, r, the line opqr shall be a streight line. For nameing the lines be — g, cr = h, ec

—

BX

br

k,bd ^ X . d e : x:ce. Then ce — ^ . Supposing be parallel to dv d g k -g x parallel to ps then k : g : : k —x ; dr. b-{-c : c : : de : pe : : dv : k hx , ex hx ps : : ve : es also k : h : - - = cv, and ^ ve. Therefore k d k ps

cgk~cgx

and es

cekx— cdhx ex cr— ce = er — h — bdk^edk cekx— cdhx bdk-\-cdk

Or

rs

dh— ex d

bdhk A- edhk— bkex— cdhx bdk~\~dck

CP bA-c : c : : be : CO : : a : h : -A - : : bh A-ch :cp : :rs : CO. c r : CO ^ b-i-c b+ c ^ ^ sp : : bdhk-\-cdhk~bkex— dhcx : cdgk— cdgx. Whence

cdgkbh — edgxbh+ edgkeh— cdgxch = bdhkcg

cdhkcg — kexbcg— dhexeg.

Figure 3.

2 8 . If two bodys b and c [Fig. 3] move in the lines br and cr. The body c moveing through the space eg in the time vs, and th[rJough^/e in the time nv, and through kr in the time nr. and the velocity of the body b is to the velocity of c as if : e, and as the line eg to the line be, or as ck to br,'^ then when the body c is in the place g, b will bee in e, and when c is in k, b will be in r. to find the line which the center of their motion describes, viz. dfo. Then nameing the quantitys br = a, cr f, be g. e : d : :

164

D Y N A M I C A L W R I T I N G S IN THE

WASTE BO O K

II

da ef— da a : - — ck. kr = — ^— . If o be the center of motion‘dof the bodys at k and r, then b+ c :b^ : :

e

eb^-ec

^

passe through o. againe making gk = v. then d \ e \ \v

■' ev

er. and

II

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOO K

165

q moves over the space qa in the time vw, and ap in the time wt. Also suppose another body b (= q) and equivelox to {q) that is to move over , .^.^1 qa\ . . . vw\ when (c) is in [g) or [k), the space bc\>= i/tt,^m thp time \ er] ap] wt] v/ (b) will bee in {e) or (r) and {q) in (a) or (p). Then drawing four^ streight

7

if be parallel to f i parallel to em, then a: g : -7^ := er : — em. and d ad r

ev

gni — gr— mr =

fev -a d

mr, gr — gk-\-kr =

—^ 4

ev-\-ef— ad fev adev4 -adef— aadd~feev^ —— ------ ■'— = ------ '---- A______ , e ad ade

S in ce/is the center of motion in the bodys at ^and e ’tis b~fc : c : ; ge ; jg. b \ c : c :: eg : fg : : em = gl

gev egev = f i : : gm = etc : ad abd-\~acd

cadev+ cadef— caadd— cfeev ev-\-ef— ad gr = gk+kr = bade-\-cade

ev-fef~ ad , abd—bef bev-\-cev4 -cef— cad go = gr— or — -----^--------1---------- £ — ---- ------ — L____. e be-\-ec eb-^ec go—gi — 10

adbev^ cfeev bade-\-cade

abdv^cefz . CO ~ cr— or abd-\-acd

fec-f-abd eb-\-ec

C9

b-\-c \ c : \ g : — = cd. Now if the lines oi : if : : oc : cd. Then the o-)-c line o[f\d must bea streight line: but of ; // : : adb^efe : ege : : oc : cd : : ; eg : :fec-\-abd : ecg. therefore the line d[f]o is a streight line, which which [^fc] may bee found by the two points d and o. The demonstration is the same if the body b moved from a^ to b. 29.* If two bodys q and c [Fig. 4] be moved in divers plines, then find the shortest line {pr) which can bee drawne from one line {cr) to the other line {qp) in which those bodys are moved, and that Xmtpr shall bee perpendicular to both the lines cr and qp, viz. Aqpr — Arps = pre — recto, then draw qb equall and parallell to pr and draw br = qp. Then shall the plaine qbrp be perpendicular to the plaine bcr. Suppose alsoe eg 1 vw i the body c moves over the space gk | in the time wt 1 and that the body kr ] tr ]

lines qc, ag, pk if : c^ :: be : cd :: eg : fg :: rk : ok\ the points d,f, and 0, shall be the centers of motion'^ of the bodys b and c, when they are in the places b and c, e and g, r and k, and (prop. 28) therefore the line dfo is a streight line. Likewise if it bee q-\-c : c : : qc : Ic : : ag : mg \ :pk : nk, then the points I, m, n are centers of motion to the bodys {q and c) being in the places (q) and (c), a and g, p and k. Then drawing the lines Id, mf, no (twixt the neighbouring centers of motion) since b-j-c : c^ : : q^ c : c :: be : cd :: qc : Ic therefore Z qbc = ZJdc and by the same reason Zgfm = Agea and Z krp = Akon. Wherefore all the lines qb, ae, pr, Id, mf, no are parallell to one another. And b-\-c \ c^ \ \ be : dc : : qb : Id : : eg :fg : : ea {= qb) : mf : : kr : ko : : pr — {bq) : no, soe that Id = mf = no and since these line line [«r] Id, mf, no are parallell equall in the same plaine Idon, and stand upon the same streight line do,^ the line [Imn) in which their other ends I, m, n are are [sic] terminated (i.e. in which are all the centers of motion'' of the bodys (c and q)) must bee a streight line.

166

D Y N A M IC A L

W R IT IN G S

IN

TH E

W ASTE B O O K

II

'Fhe demonstration is the same if {q) moved from {p) to {q).

II

D Y N A M IC A L

W R IT IN G S

IN

TH E

W A STE B O O K

167

b

30.j’'" Suppose the bodys b and c moved towards r; [Fig. 5] so that when (h (c h is in { e then f is in ( and theire centers of motion describe the line dq. [k Then the motion of theire centers of motion shall be uniforme. For if pw parallel to nt parallel to ey parallel to fs parallel to he pr : er : :pw : ey : :

Figure 5.

nt : fs^ \ : nq : fq. that is pr : ep (= er—pr) : : nq : fn — (fq— nq) and therefor since the motion of the body in epr is uniforme, the motion of theire centers of motion in the line fnq must be uniforme, that is have allway alike velocity. The demonstration is the same in all other cases. 28 and 3 0 . Supposing the things suppose in the 28th prop by schem 30th it may be thus done er : br :: ey : be :: fs \ dc. [Fig. 6] Also ep : eb :: eg : gk : \ ey : yw and eg : ek : : ey : aw : : gy (= cy— eg) : kw (= ew— ek) : : gs : kt : : sy : tw : : be : bp cs et : : de--fs : de— nt etc. Makeing fs parallel to dc parallel to fit. and mf parallel to qn parallel to ct, then is mf — cs and qn ~ ct. and/y = me, and nt — qc ~ is. Then be : bp :: ey : cw : '.eg : ck (so is the velocity of h to the velocity of c) : : gy (= ey— eg) : kw (= cw— ck) : : sy \ tw (for c^ h : :: eg : ef : : gy: sy : :kp : np : : kw : tw) : : mf (= c y —sy) : qn {— ew— tw). Againe br : er : :hc : ey : : dc :fs ^ me (for b-\-c : : : be : dc : : eg :fg : : ey : fs) whence be (-= br—er) : br : : dm (= dc — mc) : de. Also br : pr : : be : pw : : dc : nt (-- qc). Therefore be : bp (-- br - pr) : : dm : dq (— dc qc). That is

be : bp : : dm : dq : : mf : qn. and consequently the points dfn are in one streight line, and since the motion of (b) is uniforme and be : bp \ : dm : dq : : df : fn, the motion of the center d is uniforme 28 and 30"’“ Or Thus the bodys {b and c) being in b and c [Fig. 7], e and g, r and k, in the same times, and dn being described by their centers of motion. Also making de parallel to fs parallel to ey, and m/parallel to en. Then be :br : : ey : cr : : eg : ck (for the motions of b and c are uniform)

168 ;

DY N A M I CA L W R I T I N G S IN THE

:g y { ^ cy~cg)

g s : : k r : kn)

[: :]

: kr (= m f (=

c r - c k ) : :gs : kn cs — cg-\^gs)

(for b + c : c : :g e : g f ck-\-kn). Againe hr

II

: :gy : : er : :

(prop. 25))- Therefore be : br : : dm : d e : : m f \ en, and consequently the points dfn are in one streight line, also since be : br : : d f : dn the center of motions motion must bee uniforme. be : e y : : dc : fs = me

(for

: cn

W ASTE BO O K

b + e : e : : g e : g f : : ey : f s : : be : de

II

D Y N A M I C A L W R I T I N G S IN THE

169

WASTE BOO K

motion of the bodys {b) and (r) when they are in the places g and it is in the line kp . The demonstration is same in all cases.

e,

and

32.^ If the bodys {b and e) reflect at q [Fig. 9], to e and^, and the centers of their motion describe the line kdop the velocity of that center (0) after reflection shall bee equall to the velocity of that center {d) before reflec­ tion. For from the center {d) draw the line a/perpendicularly to kp and

Figure 9. Figure 8.

A

31.° If two bodys (h and e) meete and reflect one another at q [Fig. 8] their center of motion shall bee in the same line {kp) after reflection in which it was before it. For the motion of b towards d the center of their motion is equall to the motion of e towards d, by prop. 2 5 . then drawing bk J_ kp, and em _j_kp, then e d : b d : : em : bk, therefore the bodys b and e have equall motion towards the points k and m, that is towards the line kp. And at their reflection so much as (r) presseth fb) from the line k p ; so much [b) presseth (r) from it (ax. 121). Wherefore they must have equall motion from the line kp after reflection, that is drawing g p _]_ kp and 7 te _L kp, (e) and (g) have equall motions towards (n), (p ), then drawing the line eog tis ne : n o * \ : g p \go. Therefore e z n d g have equall motion from the point 0^^ which by prop. 25'*^ must? therefore be the center of * In error for f’o.

suppose the line a f to have the same celerity which (the point d ) the center of motion hath before reflection, soe that when the bodys (after reflection) are in e and g, the line a f may bee in hr. Also drawing ab parallel to kp parallel to fe parallel to eh parallel to rg parallel to kp. Then since d is the center of motion in [b) and (c), the bodys {b) and (c) have equall motion towards d, but b d : ba : ; de : f e . Therefore the bodys {b) and (c) have equall motion towards a and / that is towards the line adf. Now when the bodys reflect, so much as the body b presseth the body e from the line a f (or st) or towards p soe much the body e presseth the body b from the same line, or towards k (by ax. 1 19) therefore the bodys b and e have equall motion from the line af, after reflection (by ax. 121) that is when [they] are at e and g they doe equally move from the points h and then drawing eg, tis eh :e o : \rg :go. Therefore*^ the bodys doe equally move from the point (o) which (by ax. 25) must bee their center

170

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

II

of motion, and since the motion of the line [ a f or hr) is uniforme (by supposition) and the point o is in the line hr, and also in k p (by prop. 31) its motion must be uniforme. Note that by this and the 31st prop. I can find the center of motion of two bodys at any given time; and by prop. 9, or def. I can find their distance, 17 and by prop. 25 their distance from their center of motion that is the 2 spheres in whose perimeters they must be found; There wants therefore onely their determination to bee knowne that their places in the sphaere bee found'^

II

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

171

from the Globe {g).~^^ Let the same be supposed of h km . Then suppose (for distinctions sake) (^) have 8 parts of force, with which it presseth f h . Then must/A presse with 8 parts of force; that is (since^ = g h or c d = dn) with 4 parts on the point h, and with 4 on the point/upon the body efk, so that e f k must presse with 4 parts of force viz. (since it presseth equally on the the [«V] points e and k ) it presseth on k with 2 parts of force and with the other 2 at e on the body aem, so that aem presseth with 2 parts of force, viz. with one on the point m with the other on the point

; Soe that the body {hkm ) hath 7 parts of pression upon the point n, 4 at 2 at k , and one at m, and since the pressure of all the points h, k , and m is directly towards n, n will be pressed by it all, but the Globe causeth but one part of pression upon a. Now if these bodys f h , ek, aem, be understood continually to diminish and come nearer to the line adn untill it bee coincident with it, the pressure of the body g upon a and n will still bee the same that is as i is to 7, and so is the line dn to da. By the same reason it may be generally pronounced a d : dn : : pressure of the body g upon n : to the pressure of it upon a.

a

h,

Figure 10.

33. '^Suppose the body dcgk [Fig. 10] to be immoveable, the surface deg being plaine. Also let the shaerical body am n bee moved in the perpen­ dicular ch, so as to be reflected in c. Then since the side am hath as much force to weigh or press towards {d) as the side an to press towards e by reason that the center of its motion is in the line ch, the body must be in equilibria neither pressed towards d nor e but reflected back in the line ch. The same may be said of any bodys whose motive center is in the perpendicular to the reflecting point. 34. ®Take an = 2bn = e^cn — Sdn [Fig. ii] soe that a d : dn : : y : i Then draw the perpendiculars eb, f c , g d , hm . And set a body (aem) upon the points a and m and let {efk) stand on the points e and k which are in the perpendiculars eb and h km , and f h on the points / and h, and lay a Globe {g) on the midst off h . Suppose also the bodys aem, e f k , f h , to have noe pressure on one another or on the line [ajti) except what they receive

35. Or if the bodys a and n bee [Fig. 12] supposed united by the line and another body {g) moving towards them hit perpendicularly upon the line an 2X d\ as dn to a d so is the pressure of g upon a, to its pressure upon n, so is the motion in (<2) to the motion in {n) which is generated in them by those pressures of g , that is which they received from g , at the moment of reflection, and which they might continually enjoy as in fig. 6 [Fig. did not there union by the line adn [Fig. 12] hinder. By the same reason if g reflect not twixt the bodys [Fig. 15a, b] then a d : d n : : pression of { n ) towards ^ : the pression of a towards r : : motion of w to ^ : motion of a to r.

{and)

172

D Y N A M I C A L W R I T I N G S IN THE

WASTE BO O K

II

Note that here a and n are taken for the centers of motion in a and n and adn for a line which passeth through them. [36a]*^ AH" things put as before that a and n are loose, then [Fig. i6] qd : p d : : motion of n : motion of a received from^ (pr. 34) Therefore (by

O-

o

d

II

D Y N A M I C A L W R I T I N G S IN THE ps

=

and

Therefore

dt

=

^

_

nXpdXqp

^

—

dt

n xpd

173

W ASTE BOO K

x

A qdA -qp

qr

qp

_[_qr. Now if in the same time

^*2 ^ ^ qp

that the body g moved from {h) to {d) in the same time suppose that a and n move to r and s, and the point d to the point t, and the body g to

Figure 12.

CL

@

n

________________d __ 1

.s

Figure 15 h.

the point e, then must et = dh (by ax: Let the whole motion of ^ (at h towards d) be called m. the whole motion of a and and [«c] w to r and s be called then the motion of g after reflection = m — y and tn : m—y

: ;

hd

tis

dt

: de. Or

=

=

de

^

zhdxm — hdxy

axqrxqdxdq

tn

nxpdx

And since the whole motion of a X qr-{-n x p s

Figure 14.

prop, i i ^ y ^ n x p d : a x q d : :velocity of (a) : velocity of {n). Orsupposeing that the bodys a and n move through the distances qr and p s in the same time then, n x p d : a x q d : : qr : p s. and producing h d to [t in) the line rs, since qr parallel to d t parallel t o p s , qp = ri : is p s — qr : : qd ~ rv : p s X q d — qr qp

X dq

vt.

Wis d t

and since

m

a

and

n

, axqdxqr a X q r A ------ ^ yZ 4 or, ^ pd

hd,

qrxqd

pq

toward

=

et

qp r

and

s

_ pdxy -v---------------

qr.

is equall to qr.

axpq

^ ^ d 'A m — h d x y

q d x q d x .y

dqxpdxy

^p d x y

m

nxpqxpq

axpqxpq

axpq

etc .25

Or thus 36. {g)

If the bodys/globes a and n [Fig. 17] doe rest, but soe that the body moving perpendicularly to qp to them and reflecting on the line qp

174 D Y N A M I C A L W R I T I N G S IN T H E W A S T E B O O K k

II

II

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BO O K

175

(which is supported by but not fastened to the bodys a and n and ought now to be conceived a line onely) doth move them by communicating its whole motion to them which it selfe looseth, tis required what motion a and n shall receive from g . Suppose that in so much time as g moveth to [d ) before reflection in so much time it moveth from d to e after reflection and in so much time the bodys a and n move the one to r the other to s, and that then the point of reflection {d) is moved to (^)a Then naming the given quantitys g d = b, qd = c, qp ~ e. dp = e — c = f . qr — z the whole motion of {a) and {n) to (r) and {s) call x x , that is a X qr-\r n X p s = xxd ^ or if : w : ; p w : zvq, (then w is the center of the bodys motion) and if w l parallel to d t parallel to qr parallel to p s, then (Iw) is a line described by their center of motion. Then is ( )xx = X /w.27 Now, as the motion of a to the motion of n (prop. 34) so is [dp) to {dq)\ Therefore (ax. 113) as the velocity of {a) to the velocity of « : : n x pd : a X qd

caz

qr ps that is fn \ ca \ ;

fn

ps. but

a z ^ n X p s = xx;

that is

az -f

caz

or

/ and IS

ps fx x ae

cxx fie

af-]-ac caz fn

ae

qr.

cxx en

cfnxx— ccaxx anee

efnxx— cfnxx-^ccaxx anee

td ~ qr

Also (by ax:

fxx

fn xx— caxx ri : is : : rv : tv. therefore, ane tv =

and

fxx

ffnxx~\-ccaxx anee

gd = et - b Therefore ed

ffnxx^ccaxx— banee anee

Now since g hath soe much motion before reflection as all three bodys {g-fa-\~n) have after-ward, therefore (ax. T-if) gh = xx-{-gde; Or, gb— xx g

st Figure 17.

That is Or,

ffnxx^ccaxx— banee anee

2abeegn = aeenxx-\-gffnxx-\-ccgaxx. XX

zabeegn aeen -^gffn -{-gcca

176 D Y N A M I C A L W R I T I N G S I N T H E W A S T E B O O K Or, calling

« + « — d\ then

II

zabeegn = wl. adeen -\-gdffn -\-gccad

Soe that by this equation the point I twixt the bodys at r and ^ being then their center of motion may bee always found. Note that the line qr

/ ^ I

/ / Figure i8.

and ps must be described by the centers of motion of 2 bodys on divers sides of the point d that is ar by center of motion of the body [a) or {ad), and ps by that center of {n) or [dn). 37. Now when a and n or ad and dn are united together (as in the 2^ fig) [Fig. 18] they cannot separate the one from the other, and therefore since

II

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

177

(when they are not equivelox) rs is longer than pq, the one cannot be at ^ when the other is at r, but they will check the one the others motion soe as their centers of motion shall not describe streight lines (as {qr) or {ps)) but crooked ones (perhaps Trochoides as qmk, phi). Yet the com­ mon center of their motion w or I shall retaine both the same determina­ tion and velocity that it would did the bodys move parallell to themselves or were they not united (by ax. Soe that if the conjoyned bodys (fig. z'^) [Fig. 18] move to m and h in the same time that they would have moved to r and s were they [free ?] their center of motion (/) when they

are at m and h is the same that it would be were they at r and 5 and there­ fore may be found by the former rule. viz. aeedn-\-gdffn-\-gccad : zaeegn \ :b : w l: ; the velocity of the point I to the velocity of the body g before reflection. Vide pag 39. Vide'''^ pag: 15. But here observe that unlesse the reflecting line adn bee drawn through the point (w) the center of motion in the whole body aiwn [Fig. 19] the determination of the motion of adn will not be the same with the determination of the motion of g before reflection (as in the first figure) [Fig. 19] but verge from it (as in the 2^ fig) [Fig. 20] that is wl and gdi will not bee parallell. For since the chiefe resistance of the body adn is from its center of motion (prop. 32)^^ that is from w towards d, and not from i towards d, the body g will find more opposition on that side towards the center w, than on the other side towards a and therefore at its reflection it must incline toward v (ax. 120) and not returne in the line dg. But if the body awn presse g towards v then g presseth the body awn towards the contrary parte as from w towards I (ax. 119) and not from w towards m, if wm parallel to gd. But if the line adn pass through the point w (as in fig i®‘) then-^^

178

D Y N A M I C A L W R I T I N G S IN THE

WASTE BO O K

II

II

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

179

38. If the superficies ahr (fig. 3^^) [Fig. 21] circulate [about r\ all its points in the line cd move with equall velocity from c towards d, ffor make ,^sfr = Tier = recto and Asrf = / ert and draw ta _L 6c then is the motion of the point e from c to the motion of the point/from c as ae to

\

\

but ae =

(for Arsf similis Aret therefore er X ae

CY /K s f

— et also Aaet

erxsf

and ae = sf) fr fr therefore the motion of e from c is equall to the motion of/ from c

similis A er/therefore er \fr : :et :aeor,

et

39. If the body g reflect on the immoveable surface {dv) at its corner o (fig 4th) [Fig, 22] its parallell motion (viz. from d to v) shall not bee hindered by the surface dv, (viz. if the center of g'& motion w^ere distant from the perpendicular dn an inch at one minute before reflection it shall bee so farr distant from it at one minute after reflection), ffor dv is noe ways opposed to motion parallel to it, and a body might slide/move upon it without looseing any motion, and if at the first moment of contact the body g should loose its perpendicular and only keep its parallel motion it would (perhaps) continue to slide upon it and not reflect.^^

Figure 21.

40. The body g reflecting on the plaine vd [Fig. 22] at its corner o all its points in the line op vd shall move from the plaine vd with the same velocity which before reflection they moved to it. ffor the point 0 (prop. 9) moves with that velocity backwards which it before did forwards (viz. to vd) and all the other points (prop. 38) move with the same velocity from it.

180

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

II

a. F o l. i j v begins. b. F rom here to beginning o f A x .-P r o p . 28 deleted. c. M a r g in a l title T w o bodys b ein g u n iform ely m oved in the same plaine their center o f m otion w ill describe a streight line. d. S u b stitu ted fo r grav ity deleted. e. Throughout x x stands fo r x^. f. M a r g in a l entry T h e y doe the same in divers plaines. g. F o l. 1 4 begins. h. S u b stitu ted f o r grav ity deleted. i. S u cceed ed by their other ends (the centers o f m otion o f c and a) bun m ust bee in the same straight line Imn, w h ich line deleted. j. M a r g in a l title O f the v elo city o f the center o f m otion. k. T h is section deleted. l.

M a r g in a l entry T h e 2 8 and 30*'’ prop done otherwise.

m . T h is section deleted. n. M a r g in a l entry O r thus opposite last line o f preceding section. o. M a rg in a l title T h e center o f m otion is in the same straight line before and after reflection. p. F o l. 14 V begins. q. M a r g in a l title T h e center o f m otion in finite bo dys hath the same v elo city before and after reflection. r. M a r g in a l entry T h is ou gh t to be p roved b y the 3 4 “’ and 35'^, and the 36*’’ b y this concern in g the im presse o f {g) on (qdp). s. M a r g in a l entry O f the advantages o f force in divers positions to som e center. t. T h is section is unnum bered in the m anuscript an d is deleted. u. F o l. 1 5 begins. V.

S ucceed ed by T h e n (b y ax. 9th) et

is lost or gotten g X g d -

=

g d and since b y reflection noe m otion

a X a r + n X w s + g X f/e. A n d (b y prop. 113 ) as the

v elo city and b y prop. 34 the m otion o f « is to a deleted.

w. F o l. Jp begins. T h e zchole passage dozen to the beginning o f A x .-P r o p . 3 8 deleted. 1. T h e p ro o f assum es that the centre o f g ra v ity is at rest. It is true gen erally i f for m otion is su b stitu ted ‘ m otion relative to centre o f g r a v ity ’ . T h is general result is used in A x .-P r o p . 31, 2 tho u gh N e w to n there appeals to A x .-P r o p . 25, instead o f n u m b er 27. 2. In this first approach to the p roblem o f the m otion o f the centre o f mass o f a pair o f bodies, to w h ich he u ltim ately in A x .-P r o p . 32 gives the m o st general solution, N e w to n characteristically considers the sim plest case in w h ich the bodies o rigin ally coincide at r. 3. T h e calculation breaks o ff at this poin t. T h e last equation reduces to dh

he. Sin ce d je = x jee, this gives x h — kce, w h ich is equivalent to bdjbr

- cejer.

T h is last result w ill be satisfied if b oth bodies m ove un iform ly and reach r sim u l­ taneously. In that case the result erjeo = rsjsp, from w h ich the final equation resulted, w ill be true and the p oin t p w ill lie on or, so that the centres o f m otion w ill be collinear. In the n ext section N e w to n e x p licitly introduces the assum p ­ tion o f u n iform ity o f m otion for bo th bodies. F a ilin g this the proposition does n ot hold. 4. S o that b reaches r as <: reaches k. 5. T h is ratio should have read h -^ c : c and the same error occurs elsewhere in the calculations. T h e final result, how'ever, is unaffected.

II

D Y N A M I C A L W R I T I N G S IN THE

W A S T E BOOK

181

6. Probably in error for r. 7. Error for 3. 8. As in the previous proposition Newton writes b f c ; c in place o i b A c : a. Once again the consistent use of the error ultimately makes no difference. 9. Since eg and bk are divided at /and n in the same ratio. 10. As before these should read b y e : c and b -V c : b, respectively. 11. Newton’s search for brevity and elegance finds its goal in this version. 12. Rather by the generalization of Prop. 27 to the case where the centre of motion is not at rest. 13. This does not strictly follow' from the previous assumption. One cannot argue from a com ponent of the relative motion to the tota l relative motion. 14. Once again a more appropriate reference would seem to be to the con­ verse of Prop. 27. 15. The meaning seems to be this; originally, when the bodies were at b and c their relative motions towards the line a f W'ere equal. This will be true at the moment of impact of the position of the line a f at that time (assuming its motion uniform). It will likewise be true immediately after impact and will thus continue to be true thereafter. It must be remembered that the motions towards a f are relative motions. 16. Once again this does not seem to follow from eh : eo \ : rg : go and motion towards h == motion towards r. He should first have added motion tow^ards n = motion towards p . 17. Assuming the bodies to be perfectly elastic. 18. Notice the reference to ‘Prop.’ 25 whereas in the previous number it is to ‘Ax.’ 25. 19. A similarly brilliant summing up is given in § 11 of MS. V . 20. That is, the bodies are assumed light. 21. Between this and diagram 12 occurs diagram 13 not specifically referring to any of the results obtained. Its relevance, however, to the general result at the end of Prop. 34 is obvious. 22. In error for ‘axiom’. 23. The point dividing the line joining r, s in the ratio qd : dp has a constant velocity equal to that of the point d of the line qp immediately after impact, et is therefore the measure of the velocity of g relative to qp immediately after impact. 24. Omitting the constant time factor. 25. The calculation breaks off at this point. 26. Omitting the constant time factor. 27. Meaning { a - f n ) x l z c . 28. Since no motion is generated or destroyed by their interaction. 29. The significance of this reference is obscure. 30. The above argument seems of doubtful validity. If the direction of motion of g before impact is perpendicular to the face of the other body then the impulse between them on impact will likewise be perpendicular to this face, and the subsequent motions of both g and zc should be parallel to the original line of motion of g. 31. Newton probably divined this result (which does not seem obvious) by approximating to the actual circular arcs described b y/ and e by the lines/i and et. The component of velocity of e from c w'ould then be proportional to ea. 32. For example in the case of optical reflection at glancing incidence.

182

D Y N A M I C A L W R I T I N G S IN THE

W ASTE BOOK

II

Ilga

Ill

If the body b [Fig. i] move in the line bd and from the point d two lines da^ dc be drawne the motion of the body b from ad is to its motion from dc as ab parallel to dc is to cb parallel to ad^

THE V E L L U M M A N U S C R I P T

Coroll 1.2 the body b receiving two divers forces from a and c and the force from ba is to the force from be as ba to be then draw ad parallel to be and cd parallel to ab and the body b shall be moved in the line bd.

a

Fi gure i.

a. At fol. 38. 1. Assuming that the motion from ad (or dc) means the component of motion perpendicular to ad (or be), the result given would only be true in the special case of ad perpendicular to dc. For a correct statement relating to resolu­ tion of motion (or velocity) see MS. V, § 2. 2. The statement in this so-called corollary is correct, though it cannot be derived from the previous, incorrect result. It is given as an independent result in MS. V, §3.

MS. Add. 3958, folio 45, consists of a torn legal parchment on which a lease is engrossed, the calculations appearing on the back. Attention was first drawn to this manuscript by the late H. W. Turnbull in a letter of 3 October 1953 to the Manchester Guardian and some extracts were published by A. R. Hall [2]. The manuscript itself has been repro­ duced photographically in Isis 52, part 4 (1961), with a commentary by Herivel [4], and at pp. 46-54 of vol. iii of the Correspondence. The present interpretation follows closely that given in the Isis article which appears to be substantially the same as that in vol. iii of the Correspondence. As first noticed by Turnbull, the original data on which the calcula­ tions are based appears to have been taken by Newton from Galileo’s Dialogue, probably from Salusbury’s translation. ^ From the hand­ writing and other indications it seems probable that the manuscript was composed very early, most probably in 1665 or 1666.2 In this manuscript Newton calculates the ratios of the force of gravity to the centrifugal forces due to the diurnal and annual motions of the earth. Beginning with two primitive calculations possibly preceding any clear understanding of centrifugal force, he then calculates the two ratios making implicit use of the following result; a body moves in a straight line under the action of a force equal to the centri­ fugal force for a given motion in a circle, radius R \ then in the time for motion in the circle through distance R, the body will move in the straight line through distance \R.

At first, the value assumed for the rate of fall due to gravity is that given on p. 200 of Salusbury’s translation of the Dialogue, namely 100 cubits in 5 seconds. Later, becoming dissatisfied with this result, perhaps as the result of some crude experiment, he sets out to redetermine the value himself experimentally, making ingenious use of the following: (i) in a conical pendulum of semi-angle 45° the force of gravity equals the centrifugal force; (2) the time of complete revolution of a conical pendulum equals that for a simple pendulum of length equal to the ver­ tical depth of the conical pendulum below the point of support. In this way he finally arrives at a value of corresponding to a fall of 196 inches in one second, about double the figure assumed originally.

184

THE V E L L U M M A N U S C R I P T

III

hence explaining the final doubling of the value originally obtained for the diurnal and annual ratios, namely, i : 144 and i ; 3749, respectively. C

o m m e n t a r y

a n d

I

n t e r p r e t a t io n

Newton’s arguments in this manuscript can be divided into three groups, primitive calculations, calculations based on the correct formula for centrifugal force, and calculations of the acceleration, due to gravity. They will be considered separately in that order. A rough grid is used for reference purposes. For example, the calculation 133225 : 500c is found at (2, 4), 2 divisions down and 4 divisions across from the top left-hand corner. d e s c r ip t io n

of

c a l c u l a t i o n s

I. Primitive calculations At (2, 3) occurs the statement: ^Terra sub Equatore movet 165000004 cubit [.yfc] in 6 horis\ Basing himself on Galileo’s law of falling bodies^ expressed for example at (6, i), where y represents time in seconds, and x distance in braces or cubits, and assuming with Galileo that a body falls 100 cubits in 5 seconds (i, i)h e then calculates (3, 1-2) it will fall through 165 X lo^ cubits in 2031-1 seconds. This leads to the ratio 90 X 60 X 60/2031=159-5

( L 3-4)

and the statement (i, 5-9) ‘the force from gravity is 159-5 tiiues greater than the force from the Earth’s motion at the equator’. The use of 90 is puzzling. I suggest that he confused degrees with hours when working with the quarter circumference (see diagram at (2, 4), and so substituted 90 in place of the correct figure of 6 hours. He would then seem to compare the magnitude of two ‘forces’ by the times taken to move the body through a distance equal to a quarter of the earth’s circumference. A somewhat similar comparison is made in the second primitive calculation ^ x i 360/23-935 = 15, (3, 3) where 360 is now 6 hours expressed in minutes, and 23-935 (5> time in minutes for free fall through half the distance of the quarter circumference of the earth. II. Calculations based on correct formida for centrifugal force Half-way down the left-hand side of the manuscript occur a series of statements concerned with the centrifugal forces of the diurnal and

^ ' f 't

The Vellum manuscript

\

i t'

Ill

THE V E L L U M M A N U S C R I P T

185

annual motions of the earth. All these are based, implicitly, on the fol­ lowing expression^ of the centrifugal force law: A body moves in a straight line under the action of a constant force equal to the centrifugal force for a given uniform motion in a circle, radius R\ then in the time required for motion in the circle through distance R, the body will move in the straight line through distance In terms of this expression, referred to hereafter as (N), and of Galileo’s law, referred to hereafter as (G), these statements on centri­ fugal force become clear. First he calculates (bottom right-hand corner) that ‘The Earth in about 83677 minutes moves the length of the solar distance’ (given immediately above as 525 X 10^ braces) (7, 1-5); then 'v is g ra v ita tis in 83677 minutes m ovebit corpus p e r distantiam 100826500737600 braces’ (G) (4, 3)-(5, 4). 'v is terrae a sole m ovebit corpus p e r distan tiam 262 5 X i o"7( = 525 X 10^/2) braces in 83677 minutes’ (N). 'v is g ra v ita tis in 60 seconds m ovebit corpus p e r dista n tia m 14400 braces’

(G)'v is terrae a sole in 60 seconds m ovebit corpus p e r d ista n tia m 2625 X 10^/ 7001840329 = 3749014354 (braces)’ (G) (4, 3)-(5, 3) and (5, 5). ‘so that the force of a body from the sun is to the force of its gravity as one to 3749 or thereabouts’ (6, 3).

Immediately below follow the statements for diurnal motion. The figure 229-09 equals the time in minutes for a point on the equator to turn through a distance equal to the Earth’s radius _ 24 _ ^ _ 2 tt

818 hours (4, 5).

II

'v is terrae a centro m ovebit corpus in 229-09 minutes p e r distantiam 5250000 braces’ ( == 3500 miles of 3000 braces (7, i)) (N). 'v is g r a v ita tis in 229-09 . . . minutes 755747081 braces’

(G) (3, 4)‘So that the force of the Earth from its centre is to the force of gravity as one to 144 or thereabouts. (755 . . . /525000 = i43'9) (4, 4)~(4> S)Immediately beneath this result we find ‘Or rather as i : 300 : : and beneath this

vis a centro terrae: vim g r a v ita t is '

186

t h e

v e l l u m

III

m a n u s c r i p t

. . 1:7500 : : vis terrae a sole : vim gravitatis.^ In order to understand this doubling of results we must turn to the final group of calculations on the acceleration due to gravity. III. Calculations of the acceleration due to gravity At (3, 8) we find ‘Grave cadit 2| yds in i second or 10 yds in 2 seconds’ and immediately beneath secundam Gallil. caderet 4 cubits id est 3 yds in i second’. This disagreement between the two values of ‘g\ one quoted by Galileo, the other possibly obtained from a rough experiment by Newton himself, may have led him to undertake the calculations now to be de­ scribed. Besides the previous assumptions (G) and (N), implicit use is made of two further assumptions, hereafter referred to as I and II respectively: I. The time of revolution of a conical pendulum equals the period of oscillation of a simple pendulum of length equal to the depth of the ‘centre’ of the conical motion below the point of support. As Newton expresses it:^ ‘Pendulum gyrans et undulans si sint aeque profunda in eodem tempore redeunt.' II. In a conical pendulum, the ratio of the gravity of the body to the centrifugal force due to its circular motion equals the tangent of the angle of inclination of the string to the horizontal. In particular, the only case here considered by Newton, if the string is inclined at 45° to the horizontal, so that the ‘depth’ of the conical pendulum equals the radius of the circle described, the force of gravity equals the centrifugal force. Based on I and II we find two independent calculations of g. First calculation. At (4, 8) there is a faint drawing of a conical pendu­ lum of length 81 inches. The string is inclined at 45° to the horizontal as follows from the calculation 812/2 = 57'28 (4, 9). This pendulum makes 1512 ‘ticks'^ in hora' (3, 8). The figure 3024 immediately beneath could refer to the number of to and fro vibrations of the equivalent simple pendulum. Beneath the diagram we find ‘gravitas movebit corpus per 28-64 inches in

hours. . . .’

Ill

THE V E L L U M M A N U S C R I P T

187

The division by two was an afterthought which will be explained presently. Since the pendulum string is inclined at 45° to the horizontal the force of gravity on the body will equal the centrifugal force (II), and the above statement follows from an application of (N) to the given circular motion; 28-64 inches is and 0,0002. • . /2 hours should be the time for motion along the circumference through a distance R, namely, 1/277 X 1/1512 hours. At (3, 5) we find 44:7 : : 1/756 exactly twice the result just quoted. This explains the final division of the time by two. The calcu­ lation of the time in hours is carried out at (3, 6), and then converted to minutes, seconds, and second seconds, finishing with the result 45-38961102, and, on division by two, 22-6958 [«c] second seconds (5, 8). Newton next applies {g) to calculate the distance moved under gravity in one second. First he uses 0-756 seconds, approximates this by I and finds the result 50-915 inches (5, 7) quoted as: ‘A heavy thing in falling moves 50 inches in one second’ (5, 8-9). Later, realizing his initial mistake, he corrects the time to second, and deduces beneath that the body falls 200 inches in one second. Immedi­ ately beneath this he states: ‘or rather 196 inches = 5 yds.’ There is no trace of a recalculation of the figure 200 inches in one second, and I suspect the 196 derives from the second calculation. Newton may well have devised a mechanism for maintaining steady motion in a conical pendulum with the string at a given inclination. Otherwise he would have been familiar with the difficulty of initiating and maintaining a given motion accurately. In any case he felt the need for a second independent experiment; the corresponding calculation will now be described. Second calculation. At (6, 8) we find: ‘A pendulum of 6o| inches vibrates 3024 in an hour -j- 1/30.’ The figure 3024 refers to the number of to and fro vibrations. He begins to use this result at (6, 4) with 7:22: : 1033:3024, continuing dowm to (7, 6) but then breaks the calculation off. Next he states (7, 8) ‘a pendulum of 56-I- vibrating (.^) 1512 times in 1,005 (0 - This time the calculation is carried right through. At (3, 5-6) we find 44:7 : : 1003/1512,'° the time for motion in the corresponding conical pendulum, revolving 1512 times in 1-005 hours, through a distance equal

188

THE V E L L U M M A N U S C R I P T

III

to the radius (I). This expression equals 0-000105745 hours (3, 7), and is converted to 0-380682 seconds at (4, 7). Using II and (N), he arrives at the result: gravity moves a body through |(56|) —- 28] inches in 0-380682 seconds. The distance x moved through in one second is therefore given b y : 1:0-14491802376 : : X1281 (G) (4, 7-8). This is first calculated roughly at (6, 5) approximating to 0-579672 by 0-58, and then more fully at (5, 6)” leading to the result x = 195 (immediately under conical pendulum diagram). D iscussion

The value for ‘g' obtained in the second calculation, corresponding to 196 inches = i6|^ ft. of fall in one second, is roughly twice the value (of 4 cubits = 3 yards) assumed in the calculations of centrifugal force. This presumably explains the doubling of the original results. The calculations of ‘g^ would then be later than those of centrifugal force; independent evidence for this is furnished by the manuscript itself at (7, 5) where part of the abortive calculation based on the pendulum of length 6o| is seen to jump over the last three zeros of the solar distance. It is natural to compare the value 300 for the ratio (force of gravity/ centrifugal force of diurnal motion) to the value 350 given by Newton in MS. IVa. Allowing for the differing units and values for 'g' in the two manuscripts, these two ratios may be proved equivalent. The significance of the ‘primitive’ calculations remains to be consi­ dered. Rejecting the possibility of Newton having forgotten the true formula for centrifugal force, once found, these calculations must be dated prior to the second group. Moreover, his manner of finding the ‘force from the Earth’s motion’ purely in terms of the uniform motion along the circumference (allowing for the error of 90 in place of 6), and the omis­ sion of any reference to the centre compared with his later use of ‘the force of the Earth from its centre’, makes it probable that the first primitive calculation was made before Newton had any true concept of centrifugal force. The second primitive calculation, in which the (correct) time for the diurnal motion through the quarter circumference is compared with the time for free fall through half that distance, may date from a period of uncertainty preceding the true concept of centri­ fugal force, and the great discovery of ‘the force with which a globe re­ volving within a sphere presses the surface of the sphere’.

III A ppendix

THE V E L L U M M A N U S C R I P T

189

A

Units and distances I am indebted to the late II. W. I ’urnbull for pointing out that almost all the original data for Newton’s calculations in this manuscript must have come from Galileo’s Dialogue, most probably from p. 200 of Salusbury’s translation. On the corresponding page (239) of Santillana’s edition of Salusbury (Santillana[i]) we find: (1) value of 'g' given as 100 cubits or brace (translated yards by Salus­ bury) in 5 seconds (i, 1-2); (2) the distance of the lunar concave from the centre of the earth as 196,000 (Italian) miles = 56 semi-diameters of the earth, yielding a value for radius = 196,000/56 = 3500 miles ((i, i) almost indeci­ pherable) ; (3) 17,280 miles = 51,840,000 cubits yielding value 3000 cubits == i (Italian) mile (i, 4-6). In the second, later, manuscript IVa he takes I mile = i.e. I passus = then I passus = i.e. 3 braces =

1000 passus 3 brace 5 pedes (feet) 5 pedes.

His value of in this manuscript is 16 pedes = i 6 x f brace in one second. Comparing this with the value 4 braces per second originally used in the present manuscript, the ratio of the force of gravity to the centrifugal force of diurnal motion should come out at 144 X

48 . = 345^6 20

in exact agreement with the value i6/y og (— 350) obtained in the second manuscript, IVa. A ppendix

B

Miscellaneous calculations A number of calculations remain unaccounted for; these will now be considered. I. At (i, 8-9) occur the figures 21-6 7-54 56-

190

THE V E L L U M M A N U S C R I P T

III

Beneath we find a calculation equivalent to the result 1512 = i / x ^\^X56. The pairs of figures quoted seem therefore to refer to his calculation of the number of revolutions per hour (1512) of the conical pendulum of length 81 inches used in the first determination of ‘g\ 2. At (2, 4) we find the statement: T 33225:5000 :: vis terrae a centro ad vim ejus a sole' and immediately beneath ‘a centro 26,645 that a sole' The figure 133225 is arrived at (immediately above) as 365 X 365, while 5000 is the solar distance taking the earth’s radius as unit. This calcula­ tion must therefore have been made at a time when Newton had realized his formula for centrifugal force was equivalent to RjT^ (a form he uses explicitly in MS. IVa). Having thus found the ratio of the centrifugal forces due to the diurnal and annual motion, he proceeds at (4, 5-6) to use it as a check of his original calculation multiplying by 144 to obtain 3836,88 in sufficient agreement with the figure 3749 (or more accurately 3841 (6, 3) already found. 3. There remain calculations at (4, 3-4) ^/656250oooo == 81009,2, 3600)81009,2 = 22 (4.5) 44:7: (6.5) 23,9357:229,09 (i, 5-6) 1000000:4345 None of these play any part in the main calculations. Each of them can be given a reasonable interpretation apart from the last. 1. See Appendix A above. 2. See above Part I, Chapter 6.2. 3. Newton himself does not use the term acceleration. It is used here for convenience, sometimes abbreviated as ‘g'. 4. Cf. Appendix A above for the origin of the units and dimensions used by Newton. 5. That the distance moved in a straight line under the action of gravity varies as the square of the time. Newton extends the law to the case of a con­ stant force equal to the centrifugal force. 6. An equivalent result, corresponding to the time for a complete revolution, is given explicitly in MS. IVa. In the present manuscript Newton is put to considerable trouble to calculate the time for motion through a distance equal to the radius. This would seem to indicate that the present manuscript is the earlier one. See above Part I, Chapter 6.1. 7. For uniform speed V, the time for motion in the circle through distance R = RjV. In this time, under acceleration V~/R, the body will move in a straight 1 j/2 R line through distance “ (—j = .

HI

THE V E L L U M M A N U S C R I P T

191

8. See para. 3 of MS. IVa. 9. Newton may have limed his conical pendulum by observing its passage across the ends of a diameter. The time between successive passages would then have been equal to the time for a single ‘tick’ of the corresponding simple pendulum. 1005 I 10. Newton’s method of writing - X— .

1512 2TT

11. But there appears to be a small error in this calculation. 12. See Appendix A above. 13. See Portsmouth Draft Memorandum as reproduced above at Part I, Chapter 4, p. 66.

IVa

IV T h e s e two papers are juxtaposed in section 5 of MS. Add. 3958. The close similarity of the handwriting in both papers, and the fact that they are written on paper of the same size, possibly indicate that this juxtaposition corresponds to a close connexion between the original manuscripts as regards either times of composition or drafting. This possibility is reinforced by David Gregory’s statement relating to his visit to Newton at Cambridge in 1694.

ON C I R C U L A R MOTION

193

throughout and gives the impression of being a final draft possibly intended for publication. Text I. Corporis A [Fig. i] in circulo A D versus D gyrantis, conatus a centro tantus est quantus in tempore A D (quod pono minutissimum esse) deferret a circumferentia ad distantiam DB: siquidem earn distantiam in eo tempore acquireret si modo conatu non impedito libere moveret in tangente AB.

I saw a manuscript [written] before the year 1669 (the year when its author Mr Newton was made Lucasian Professor of Mathematics) where all the foundations of his philosophy are laid: namely the gravity of the Moon to the Earth, and of the planets to the Sun. And in fact all these even then are sub­ jected to calculation. I also saw in that manuscript the principle of equal times of a pendulum suspended between cycloids, before the publication of Huy­ gens’s Horologium Oscillatorium.

It seems probable (as suggested at fn. i of p. 301 of vol. i of the Corre­ spondence) that Gregory was here referring to the present manuscripts.

IV a ON C I R C U L A R M O T I O N MS. Add. 3958 (5), folios 87, 89 (left half), among the papers listed as ‘Early Papers by Newton’ in the Catalogue of the Portsmouth Collection (Cambridge, 1888). The text of the manuscript with translation and commentary was published first by Hall [2], and then at pp. 297-303 of vol. i of the Correspondence. As pointed out in Hall’s paper and in the Correspondence, this manuscript is undoubtedly the one referred to by Newton in his letter of 20 June 1686 to Halley. For example, the ratio 10000 to 56 referring to the ratio of the maximum distances of the sun and moon from the Earth was originally written 100,000 to 559^ as in the present manuscript. As noted above, it was also most probably the manuscript referred to by David Gregory in a memorandum referring to a visit to Newton at Cambridge in May 1694 (U.L. Edinburgh, Greg. C 43) reproduced at p. 331 of vol. iii of the Correspondence. According to this memorandum the manuscript was composed before Newton’s appointment to the Lucasian Chair of Mathematics in 1669. Apart from a small number of unimportant deletions the text is clean

Jam cum hie conatus corpora, si modo in directum ad modum gravitatis continue urgeret, impelleret per spatia quae forent ut quadrata temporum:^ ut noscatur per quantum spatium in tempore unius revolutionis A D E A impellerent,^ quaere lineam quae sit ad B D ut est quadratum periferiae A D E A ad AIH. Scilicet est B E . B A : : B A . BD (per 3 elem). Vel cum inter B E ac DE ut et inter B A ac D A differentia supponitur infinite parva, substituo pro se invicem et emergit DE .D A \ : D A .DB.-\ Faciendo denique DA'i (sive D E x D B ). ADEA^ ADEA^ : : D B .— >obtineo lineam quaesitam (nempe tertiam proDE portionalem in ratione periferiae ad diametrum) per quam conatus recedendi a centro in directum constanter applicatus propelleret corpus in tempore unius revolutionis.^ f Notice that in this and succeeding manuscript Newton writes D E jD A , etc., as

D E .D A in place of D E : D A as in the Waste Book. 858205

O

194

ON C I R C U L A R MOTIOxN

IVa

Verbi gratia cum ista tertia proportionalis aequat 19,7392 semidiametros si conatus accedendi ad centrum^ virtute gravitatis tantus esset quantus est conatus in aequatore recedendi a centro propter motum terrae diurnum: in die periodico propelleret grave per 19I semidiametros terrestres sive per 69087^ milliaria. Et in bora per 120'’”^*. Et in minuto primo per 1/30™^^ sive per 100/3 passus, id est 500/3 pedes. Et in minuto se[cun]do per 5/108®^*^, sive per 5/9 digit. At revera tanta est vis gravitatis ut gravia deorsum pellat 16 pedes circiter in i" hoc est 3505 vicibus longius in eodem tempore quam conatus a centro circiter, adeoque vis gravitatis est toties major, ut ne terra convertendo faciat corpora recedere et in aere prosilire. 2. Coroll. Hinc in diversis circulis conatus a centris sunt ut diametri applicatae ad quadrata temporum revolutionis,^ sive ut diametri ductae in numerum revolutionum factarum in eodem quovis tempore. Sic cum Luna revolvit in 27 diebus 7 horis et 43'7 sive in 27,3216 diebus (cujus quadratum est 746I) ac distat 59 vel 60 semidiametris terrestribus a terra. Duco distantiam lunae 60 in quadratum revolutionis Imnaris i ; ac distantiam superficiei terrestris a centro i, in quadratum revolutionum 746I, et sic habeo proportionem 60 ad 746I, quae est inter conatum Lunae et superficiei terrestris recedendi a centro terrae. Itaque conatus superficiei terrestris sub aequatore est I2-| vicibus circiter major quam conatus Lunae recedendi a centro terrae. Adeoque vis gravitatis est 4000 vicibus major conatu lunae recedendi a centro terrae, et amplius.^ Et si conatus ejus a terra efficit ut cum eadem facie terram semper respiciat;^ Hujus Lunaris et terrestris systematis conatus recedendi a sole debet esse minor quam conatus Lunae recedendi a Terra, aliter luna respiceret solem, potius quam terram. Sed ut de hac re justiorem aestimationem faciam sit 100000 distantia systematis Lunaris a sole, & y distantia lunae a terra. Et cum luna conficit i3revoi^s]g J28'r 52' in anno stellari, sive 13,369 revolutiones (cujus quadratum est 178,73): duco distantiam solis 100000 in quadratum ejus revolutionis i, et distantiam Lunae y in quadratum ejus revolutionum 178,73 et fit 100000 ad 178,733;, ita conatus terrae a sole ad conatum Lunae a terra. Unde constat quod distantia lunae a Terra debet esse major quam 100000/178,73 sive 559^ respectu distantiae solis 100000. Et inde Solis maxima parallaxis in orbita lunari non erit minor 19''*' et solis horizontalis parallaxis in terra non minor 19'^^ puta cum O et }) distant 90“^’ab Apogaeis. Pone vero parallaxim esse 24" et erit distantia

IVa

ON CIRCULxVR MOTION

195

lunae a terra 7o6|,^2 gt conatus ejus recedendi a terra ad conatum terrae recedendi a sole ut 5 ad 4 circiter. Et sic vis gravitatis erit 5000'“^vicibus major conatu terrae recedendi a Sole. Sit Magni orbis |diam 100000, terrae Jdiam. xEritque 3651x365!^; (sive 132408 ita conatus hominis a terra, ad conatum ejus a sole.^s Denique in Planetis primarijs cum cubi distantiarum a sole reciproce sunt ut quadrat! numeri periodorum in dato tempore conatus a sole recedendi reciproce erunt ut quadrata distantiarum a sole. Verbi gratia est in $, ©, ut i, 27’ , qof, sive ut i, 3I, 6f, 15I, 183^, 614^-, reciproce. Vel directe ut 614; 173; 91; 39; 3^; i. 3. Pendulum gyrans et undulans si sint aeque profunda in eodem tempore redeunt.^7 4. Si pendulum gyrans et undulans sint aeque profunda, arcus vibrantis a perpendiculo descriptus est ut’^ chorda arcus quern gyrans descripsit in eodem tempore. Translation The endeavour from the centre of a body A revolving in a circle A D towards D is of such a magnitude that in the time [corresponding to movement through] A D (which I set very small) it would carry it away from the circumference to a distance DB\ since it would cover that distance in that time if only it were to move freely along the tangent without hindrance to its endeavour. Now since this endeavour, provided it were to act in a straight line in the manner of gravity, would impel bodies through distances which are as the square of the times to know through what space they would [be] impell[ed] in the time of a single revolution AD EA, I ask for a line which may be to BD as the square of the circumference AD E A to the square of AD. Now BE jBA ~ B A jB D (by [Book] 3 [of Euclid’s] Elements). But since the difference between B E and DE, and also be­ tween B A and D A is supposed infinitesimally small, I substitute one for the other in each case and it follows DEjDA = DAjDB. And then by making DA^ (or D E x D B ) to ADEA^ as D B to ADEA^jDE I obtain the required line (namely the third proportional of the circumference to the diameter) through which its endeavour of receding from the centre w'ould impel the body in the time of a complete revolution when applied con­ stantly in a straight line.-

196

ON C I R C U L A R MOTION

IVa

P'or example, since that third proportional equals 19.7392 semi­ diameters, if the endeavour of approaching to the centre^ [of the Earth] in virtue of gravity were exactly equal to the endeavour of receding from the centre at the equator due to the diurnal motion of the earth: then in a periodic day it would impel a heavy body through 19I terrestrial semi­ diameters, that is through 69087"^ miles: and in an hour through 120 miles. And in a first minute through i/30 mile or through 100/3 paces, that is 500/3 feet; and in a second minute through 5/108 feet or 5/9 inches. But actually the force of gravity is of such a magnitude that it moves heavy bodies down about 16 feet in one second, that is about 350^ times further in the same time than the endeavour from the centre [would move them], and thus the force of gravity is many times greater that what would prevent the rotation of the earth from causing bodies to recede from it and rise into the air. 2. Corollary: Hence the endeavours from the centres of divers circles are as the diameters divided by the squares of the periodic times,^ or as the diameters multiplied by the [squares of the] numbers of revolution made in any given time. So that since the Moon revolves in 27 days 7 hours and 43 minutes or in 27-3216 days^ (whose square is 746I) and is distant 59 or 60 terrestrial semidiameters, from the Earth. I multiply the distance of the moon 60 by the square of the Lunar revolution, i, and the distance of the surface of the Earth from the centre, i , by the square of the revolutions, 746^, and so I have the ratio 60 to 746^, which is that between the endeavour of the Moon and the surface of the Earth of receding from the centre of the Earth. And so the endeavour of the surface of the Earth at the equator is about 12-|- times greater than the endeavour of the Moon to recede from the centre of the Earth. And so the force of gravity [as at the surface of the Earth] is 4000 and more times greater than the en­ deavour of the Moon to recede from the centre of the Earth.^ And if the Moon’s endeavour from the Earth is the cause of her always presenting the same face to the Earth,^ the endeavour of the lunar and terrestrial system to recede from the Sun ought to be less than the endeavour of the Moon to recede from the Earth, otherwise the Moon would look to the Sun rather than to the Earth. But that I may make a more exact estimate in this matter, let 100,000 be the distance of the lunar system from the Sun, and y the distance of the Moon from the Earth. And since the Moon completes 13 revolutions 4 sig. 12 gr. 52' in a stellar year, or 13-369 revolutions (whose square

IVa

ON C I R C U L A R MOTION

197

is 178-73) I multiply the distance of the Sun, 100,000, by the square of its revolution, i, and the distance of the Moon [from the Earth], y, in the square of her revolutions 178-73 and make 100000 to 178-733^ as the endeavour of the Earth [and Moon] from the sun to the endeavour of the Moon from the Earth. From which it follows that the distance of the Moon from the Earth ought to be greater than 100000/178-73 or 5 5 9 i compared with the distance of the Sun, 100000. And hence the greater solar parallax in the lunar orbit will not be less than 19 minutes, and the Sun’s horizontal parallax not less than 19 seconds,” reckoned when the Sun and Moon are distant 90° from their apogees. Assume the parallax actually to be 24 seconds and the distance of the Moon from the Earth y o b f a n d the endeavour of the Moon to recede from the Earth to the endeavour of the Earth to recede from the Sun will be about 5:4.^^ And so the force of gravity will be 5000” times greater than the endeavour of the Earth to recede from the Sun. If the great orbit’s semidiameter be 100000, X the Earth’s semidiameter, then the endeavour of a man from the Earth to his endeavour from the Sun will be 365^X 365|^c or 132408 to unity ^5 Finally since in the primary planets the cubes of their distances from the Sun are reciprocally as the squares of the numbers of revolutions in a given time:’^the endeavours of receding from the Sun will be re­ ciprocally as the squares of the distances from the Sun. For example in Mercury, Venus, Earth, Mars, Jupiter, Saturn as ^7, i, 2 ^ , 27I, qof, or as I , 3f, 6f, 15!, 183^, 614I, reciprocally. Or directly as 614; 173;

91; 39; 3 i ; I3. If a rotating pendulum and a pendulum swinging to-and-fro are of the same depth they return in the same time.” 4. If a rotating pendulum and a pendulum swinging to-and-fro are of the same depth, the arc of swing described from the perpendicular is proportional to the chord of [twice] the arc which the rotating pendulum describes in the same time.^^ a. The sense seems to require im pellerentur . b. Preceded by lit distantia gyrantis a loco initiali sive deleted. 1. As follows from the extension of Galileo’s law for bodies falling under gravity to the case of a body moving under the action of any constant force or conatus. The same extension has already been used in MS. III. 2. Since the ratio in question equals 2tt^, approximated to by 19,7392 in the succeeding paragraph, it follows that the distance moved by the body in a straight line under the action of the conatus in the time of a complete revolution equals zn^R. Ry a further application of the t^ law it follows that the distance

198

ON C I R C U L A R MOTION

IVa

m o ved in the tim e for m otion along the circle th rough distance R equals ztt^R x

(R I zttR-Y — R / z , the result em p loyed in M S . I I I . 3. In contrast to the endeavour o f receding from the centre due to circular m otion. N e w to n has ev id e n tly in m in d the p ossibility o f a balance betw een the tw o o pp osin g tendencies. 4. N o tic e

19 -7 3 9 2 X 3 5 0 0 = 69087-2 so that the figure assum ed for the

radius o f the E arth is 3500 m iles. T h e value ultim ately derived for the ratio o f the endeavour due to gravity to that due to the diurnal m otion proves that these m iles are Ita lia n , so that N e w to n is taking over the figure for the radius o f the E arth used in M S . I l l , itself taken from G a lile o ’s D ialogue (see note 5 below and

IVb

ON MOTION IN A CYCLOID

199

Oscillatorhim. This would certainly have been very plausible if it were not for the statement by Gregory quoted above. The importance at­ tached by Newton to the possible derivation of a value for from the time of vibration of a cycloidal pendulum in the last paragraph of the paper is interesting. It should be read in the light of the other method of determination in MS. III. The text is relatively clean and shows no sign of any major develop­ ment.

M S . I l l , A p p e n d ix A , Section 2). 5. A s show n in A p p e n d ix A to the interpretation o f M S . I l l , this result agrees w ith that derived there on the basis o f the uncorrected figure for ‘g ’ , cor­ responding to 20/3 ft o f fall from rest in one second, the radius o f the E arth b ein g set equal to 3500 Italian miles. See above. Part I, C h a p ter 4, p . 68, for the bearing o f this result on the problem o f N e w to n ’s first test o f the inverse square law o f gravitation. 6. I f the distance m o ved in the tim e, T , o f a com plete revolution is p ro po r­ tional to R , then b y the

law the distance m o ved in unit tim e (the natural basis

o f com parison for different m otions) w ill be proportional to R jT ^ . 7. In M S . A d d . 3996 C . U . L . (from w h ich are taken the extracts given in M S . I) the len gth o f the lunar m onth is given at p. 20 as alm ost 27 days 8 hours. 8. T h e large discrepancy betw een this figure and the correct figure o f about 3600 results from the erroneous figure ad opted for the radius o f the earth. T h is , in turn, differs som ew hat from the figure o f 60 m iles to a degree o f latitude m en ­ tioned in the accounts o f Pem berton and W h iston. See above. Part I, C h a p ter 4, p. 68. 9. A s su ggested b y N e w to n in his letter o f 23 June 1673 to H u ygen s (see extract at M S . V i l a below). 10. S in 19 ' — 559J/100000. 11. A ssu m in g radius o f m o o n ’s orbit == 60 x radius o f Earth. 12. 5 5 9 ^ X 2 4 /19 = 706-74. 13. R atio o f endeavours = 24/19 — 5/4. 14. A ssu m in g previous result that vis g ra vitatisjcona tu s L u n a e > 4000. 15. P resu m ab ly in error for (365J)^x 1000000 • In M S . I l l the solar distance is taken as 50000 X radius o f Earth. 16. K e p le r ’s T h ir d L a w o f Planetary M o tio n . 17. A result already qu oted in M S . I la , § 6, and used in M S . I I I . 18. A s su ggested in Correspondence, vol. i, p. 303, note 14, this result is true i f for ‘the chord o f the arc’ is read ‘ the chord o f tw ice the a rc’ .

IV b ON M O T I O N

IN A C Y C L O I D

MS. Add. 3958(5), folios 90-91V, with diagrams at fol. 89 (right half). The text of this manuscript together with a translation have already been published in Hall and Hall [i], pp. 170-80. They consider that it may have been composed after Newton’s receipt of Huygens’s Horologium

In Fig; prima Sit D CE [Fig. i] Trochoides ad circulum B C Y pertinens quae planum horizontal tangat in C insistens ei normaliter. Inque curva D C grave descendat a D ad C dilapsum per puncta 8, P, et tt . Et agantur SYS, PVR, ttxQ parallelae ad DE, etc. I Dico quod gravitatis efiicacia sive descendentis acceleratio in singulis descensus locis, D, S, P, etc est ut spatium describendum DC, SC, P C etc. Scilicet obliquitas descensus minuit efficaciam gravitatis ita ut si gravia duo descensura sint ad C alterum B recta per diametrum BC, alterum oblique per chordam YC: Minor erit acceleratio gravis Y propter obliquitatem descensus idque in ratione Y C ad B C ita ut ambo gravia simul perveniant ad C.^ Est autem B C parallela curvae in D,^ ac Y C parallela ipsi in S,^ ideoque acceleratio gravis in D est eadem cum acceleratione gravis descendentis in B C ut et acceleratio gravis in S eadem cum acceleratione gravis descendentis per YC.^ Quare descen­ dentis acceleratio in D est ad accelerationem ejus in S ut B C ad YC, sive ut eorum dupla D C ad SC. Q.E.O.

200

ON MOTION IN A CYCLOID

IVb

2 Gravia^ in Trochoide descendentia, alterum a Z), alterum a quolibet alio puncto P, simul pervenient ad C. Nam ut sunt longitudines DC, PC, ita accelerationes sub initio motus in Z) et P : quare spatia primo descripta puta Dd et P/>‘=erunt in eadem ratione.^ Unde dividendo est DC. PC :: dC.pC. Quare accelerationes in delp permanent in eadem ratione, et '^etiamnum generabunU velocitates descendentium in eadem ratione, efficientque ut gravia pergant describere spatia dh et p-n in

IVb

ON MOTION IN A CYCL OID

201

2 Quod OP et OR in directum jacent propter parallelismum rectarum DS, BT. Adeoque omnis recta perpendicularis ad I'roch DC tanget Troch AD et contra. 3 Quod est PO — TB — DS = ^ curvae DR, adeoque zPQ sive PR = DR. Et recta P P + curva RA = toti curvae DRA = Rectae CA. 4 Quare si ARP sit filum datae longitudinis,^ cui pondus P appenditur ita undulans intra trochoides AD et AE ut filum ab ipsis paululum prohibeatur ne in rectum protendatur, quemadmodum videre est in parte AR, ubi se applicat ad Trochoidem: Tunc pondus P undulabit in Trochoide DCE, adeoque quamlibet utcumque longam vel brevem undulationem in eodem tempore perficiet. 5 Patet etiam quod undulationes in circulo FCI centre A descripto modo sint perbreves (puta los^ hinc inde vel minus) sunt ejusdem temporis proximo ac in Trochoide DCE.^ Nam undulatio in utroque casu fit circa centrum A nisi quod filum paululum incurvatur in uno casu; quae curvatura quam parva sit ex eo percipies quod R tantum supra rectam DE esse debet imaginari quantum P cadit infra.

F'igure 2.

eadem ratione. Adeoque spatia SC'" et ttC erunt in ilia ratione idque continue donee utrumque simul in nihilum evanescat. Quare gravia simul attingent punctum C. Potuit etiam hoc inde ostendi quod posito DC. SC ; : PC. vC sit D 8 . P tt : : ^/BS.^'RO : : velocitas post descensum ad profunditatem B S ad velocitatem post descensum ad profunditatem RO.^

3 Itaque si grave undulet in Trochoide undulationes quaelibet erunt ejusdem temporis. In ffig: seda Super diametrum Trochoidis DE [Fig. 2] erige perpendiculum BA— BC et a puncto A hinc inde describe duas semi-Trochoides AD, AE tangentes rectam DE in D et E, adeoque ejusdem magnitudinis cum Trochoide DCE. Jam puncto Q in BD ad arbitrium sumpto, fac arcus BT ac DS aequales longitudini DO et comple parallelogramma BOPT ac DORS. Et constat.7 Quod OP normaliter insistit Trochoidi DC in P et quod OR tangit Trochoidem DRA in R.

In Figura tertia 1 Stantibus jam ante positis cum gravis a D [Fig. 3] per P ad C descendentis velocitas in loco quolibet P est uts radix altitudinis BV^ hoc est ut linea B T pro designanda ilia velocitate exponatur^^ eadem BT. 2 Dein sit Pp particula spatii DC in ejusmodi particulas infinite multas et aequales divisi, et agatur par parallela ad PTV secans semicirculum in a et rectam TC in r. Jam propter parvitatem, curvarum portiunculae Pp ac Ter pro rectis haberi possunt, adeoque Pp ac Tr erunt aequales propter parallelismum et inde t T erit datae licet infinite parvae longitudinis; ductaque semidiametro TK, triangula Tar, TKB erunt similia, siquidem latera unius sunt perpendiculariter posita ad alterius latera correspondentia viz To ad TK, T t ad TB et ar ad BK. Quare est BT. TK : : rT. ra sive BTxTa = TKxTr. Adeoque cum TKacrT pro datis habenda sunt, erunt BT ac Ta reciproce proportionalia. Cum itaque tempus et velocitas quibus datum spatium ut Pp describitur sunt reciproce propor­ tionalia, et P T pro velocitate exponitur exponi potest etiam Ta pro tempore. Atque ita si spatii Dp pars quaelibet Pp describitur in parte tempo[re] Ta, describetur totum spatium Dp in toto tempore A^a.

I

3 Hinc posito quod semicircumferentia BTC designat tempus in quo

202

ON MOTION IN A CYCLOID

IVb

spatium D C percurriturp' ut noscas in quo tempore pars D P describetur age P T parallelus D B et arcus B T designet tempus. 4 Potest etiam tempus per angulum^CT vel per inclinationem descensus Pp aut per longitudinem DO^^ designari. 5 Caeterum ut tempora perpendicularis descensus cum temporibus descensus in hac curva conferantur, pone quod grave B per V ad C

IVb

ON MOT ION IN A CYCL OID

203

sed etiam quod descendet a B ad F in tempore B T vel a i? ad C in tempore B C ; et contra, Tempora etiam descensus ab alijs curvae punctis ut P exhinc noscuntur siquidem partes proportionales in aequalibus temporibus peraguntur.i2

Sed praecipuum est quod ex dato tempore in quo pendulum datae longitudinis vibrat, datur tempus in quo grave ad datam profunditatem descendet.^ Nam si grave decidat a B ad C in t e m p o r e f a c BC^ .B T C ‘ : : B C .Ay, ac decidet ah A ad y in tempore BTC^^ quod est unius semivibrationis. Posito a u t e r n = i,ooo, calculus dabityly 1,2337, per cujus quadruplum 4,9348 descendet in tempore unius vibrationis, hoc est per ^AC fere, et per 19,7392 sive ig §A C fere in vibratione replicata. Nota [i] quod motus gravis a D ad C descendentis persimilis est motui puncti in rota uniformiter mota quod describit Trochoidem, respectu velocitatis. 2 Quod pendulum ex argento vivo confectum diutius perseverat in motu. Translation In the first figure Let D C E [Fig. i] be a trochoid belonging to the half circle B C Y and touching the horizontal plane, to which it is normal, at C. Suppose a heavy body descends in the curve D C from D to C falling through the points 8, P, and tt. And draw 8 Y S, PV R , iryO parallel to D E etc.

Figure 3.

descendit, et cum linearum B T quadrata sunt ut lineae BV: exponatur B T pro designando tempore descensus ad V. Adeo ut si grave descendat ad C in tempore BC, descendet ad V in tempore BT. 6 Jam cum descensus aeque a D ac i? sub initio sunt ad Horizontem perpendiculales, manifestum est quod utrumque grave D ac B incipit aequaliter’*^descendere, etsi D confestim in obliquum fertur. Adeoque lineae per quas tempora descensuum designantur ita debent inter se constitui ut initialiter exhibeant aequalia tempora descensuum aeque altorum et postea recte exhibebunt tempora descensuum aeque altorum ut fiunt sensim inaequalia. Et hinc patet arcum et ejus chordam B T (cum sint initialiter aequalia) non modo recte designare haec tempora seorsim, sed et inter se conferre. Ita ut posito quod grave defertur a D ad C in tempore BTC , non modo sequetur quod deferetur ad P in tempore BT,

I. I affirm that the efficacy of gravity, or the acceleration of descent, in the singular points D, S, P, etc of the descent are proportional to the spaces to be described DC, 8C, P C etc. Clearly the obliquity of the descent diminishes the efficacy of gravity so that if two weights are about to descend to C, the one B directly by the diameter B C , the other C obliquely by the chord YC : the acceleration of the weight Y will be less on account of the obliquity of the descent in the ratio Y C to B C so that both weights will arrive simultaneously at C.^ But B C is parallel to the curve at D,^ and Y C is parallel to it at 8,^ so that the acceleration of the weight in D is equal to the acceleration of the weight descending in BC, and the acceleration of the weight in 8 the same as the acceleration of the weight descending in YC.^ Therefore the acceleration of descent at D is to the acceleration at S as B C to YC, or as the ratio of their doubles [i.e.] D C to SC. Q.E.O.

204

ON MOTION IN A CYCLOID

IVb

2. Weights descending in the cycloid, one from Z), the other from any other [given] point P, reach C simultaneously. For as are the distances DC, P C so are the accelerations at the beginning of the motions at D and P : therefore the spaces described to begin with, for example Dd and Pp, will be in the same ratios [as before]. Whence by divisionPC/PC = dCIpc. Therefore the accelerations at d and p remain in the same ratio, and so now generate velocities of descent in the same ratio, which ensure that the weights continue to describe spaces dS and prr having the same ratio. And so the distances SC and ttC will be continually in the same ratio until both simultaneously dwindle to nothing. Therefore the weights reach the point C simultaneously. It can also thereby be proved that given DC/SC = PC/ ttC then DS/P tt = V(PN)/V(PO) = velocity after descent through depth PiS is to velocity after descent through distance RQ.^ 3. And so if a heavy body oscillates in a cycloid all oscillations whatsoever will occupy the same time. In the second figure On the diameter of the cycloid DE [Fig. 2] erect a perpendicular B C and on both sides of the point A describe the half-cycloids AD , A E touching the line D E at D and E, and of the same size as the cycloid DCE. Having chosen the point Q anywhere in BD, make the arcs B T and D S equal in length to D O and complete the parallelograms B O P T and DORS. We have then^ 1. That Q P stands normally to the cycloid D C in P and that OR touches the cycloid D R A in R. 2. That O P and OR are collinear on account of the parallelism of the lines D S and B T . And so every line perpendicular [i.e. normal] to the cycloid D C touches the cycloid A D and conversely. 3. T h a tP P = 7’P = D S = i curve P P , and that 2PO or P P ~ P P . And the line P P + cu rve RA = whole curve DRA = line CA. 4. Therefore if A R P is a string of given length,^ to which a weight P is attached oscillating between the cycloids A D and AE, so that the string is just prevented from stretching in a straight line as can be seen in the part A R where it is applied to the cycloid: then the weight P will oscillate in the cycloid DCE, so that howsoever long or short the oscillation may be it will be executed in the same time. 5. It also appears that the oscillations in the circle P C I described about

IVb

ON M O TI O N IN A CYCLOID

205

centre A, provided only they are very small (for example lo-'’ or less on each side) are approximately of the same duration as in the cycloid DCE. For the oscillations in both cases are about the centre A except that the string in one case curves slightly inwards; and how small this curvature is can be seen from the fact that P is to be thought of as far above the line DE as P falls below it. In the third figure 1. On the same assumption as before when a heavy body [falls] from P [Fig. 3] to C via P the velocity of descent in any place whatsoever, P, is as the square root of the length PF,^ that is as the line BT^^\ and so BT may be used for denoting that velocity. 2. Then let Pp be a small element of the space DC which is divided into infinitely many equal elements of the same length, and let par be drawn parallel to PTV cutting the semicircle in a and the straight line 7'C in t . Then on account of their smallness the elements Pp and Ta oi the curve can be taken as straight, and so Pp and Tr will be equal by parallels and then t T will be [equal] to the given arbitrarily small length; then the radius T K being drawn, the triangles Tar, T K B will be similar, so that the sides of one are set perpendicularly to the corresponding sides of the other, namely, Ta to TK, Tr to TB and ar to B K . Therefore is B T / T K = rT/ra or P P x Ta = T K x Tr. Therefore since T K and rT are to be taken as given, B T and Ta will be reciprocally proportional to each other. Since therefore the time and velocity in which a given space such as Pp are described are reciprocally proportional, and B T [from i] represents the velocity, Ta can then represent the time. And thus if any part whatsoever Pp of the space Dp is described in the part of time Ta, the total space Dp will be described in the total time A^a. 3. This being so since the semicircumference BTC represents the time in which the space DC is traversed, to know in what time the part DP is described draw PT parallel to DB and the arc BT will represent the [required] time. 4. The time can also be represented by the angle BCT or by the inclina­ tion of the descent Pp, or by the length DO.^^ 5. Moreover that the time of the perpendicular descent may be compared with the time of descent on the curve, assume that a body descends from B to C via V ; then since the square of the line B T is proportional to the line BV, BT may be taken to represent the time of descent to V. So that

206

ON MOTION IN A CYCLOID

IVh

if the body descends to C in the time BC, it descends to V in the time BT. 6. Now since the descents from D and B are initially both perpendicular to the horizontal, it is clear that both D and B begin to descend equally, although D is immediately forced in an oblique direction. And so the lengths representing the times of descent should be so constituted relative to one another that initially they exhibit equal times of descent by lengths as equal one to the other, and immediately afterwards they represent equal times of descent as becoming unequal. And this proves that the arc B T and the corresponding chord B T (since they are initially equal) do not only rightly represent these times separately, but also in the correct proportions one to the other. And so assuming that the body is translated from D to C in time B T C , it follows not only that it is translated to P in the time BT, but also that it descends from 5 to F in time B T , or from B to C in time BC, and conversely. Moreover the times of descent from other points of the curve such as P are likewise known provided only proportional parts are covered in equal times. But especially important is the fact that given the time in which the pendulum vibrates a given distance, there follows the time in which a heavy body falls through a given depth. For if the heavy body falls from jB to C in time B C make BC^jBTC^ = BCjA y, and it falls from A to y in the time BTC,^^ which is one half oscillation. Putting A C — looo calcu­ lation gives Ay 1-2337, through whose quadruple 4*9348 it descends in the time of one oscillation, that is through ^AC approximately, and through 19-7392 or through ig^AC approximately in a double oscilla­ tion. Note [i] that as regards velocity the motion of a heavy body falling from Z) to C is very similar to the motion of the point of the uniformly moved wheel which describes the cycloid. 2. That a pendulum made from quicksilver continues longer in motion. a. In error for Y. b. Followed by a quolibet puncto P descenclet ad C in eodem tempore ac si descendisset a D cancelled. c. Followed by [} quantum vis parva cogit] deleted. d-d. Inserted in text in place of cancelled generant/o//o?a’wg descendentium. e. Folloiced by DS, ttP et eorum residua cancelled. f. Followed by Nam punctum R tunc baud recedit a puncto A et filum vix omnino incurvatur cancelled.

IVb

ON MOTION IN A CYCLOID

207

g. Follozved by quadra cancelled. h. Follozved by illius B V radix [?] est B T si modo diameter cancelled. i. In error for B. j. Follozved by arcus B T designet tempus in quo spatium DP cancelled. k. Follozved by in eodem tempore cancelled. l. Followed by Nam si grave descendat a D ad C in tempore DTC — 15708, decidet ab 5 ad C in tempore BC = 10000. Et facto BC^.BTC^ : : BC . Ay = 24674 [decidet?] ad y in tempore BTC, hoc est dum semissis undulationis peragitur, adeoque in tempore unius undulationis decidet per 4Ay sive per spatium 98696 cancelled. 1. Newton may have been familiar with the result through Galileo. It is given in Theor. 6, Prop. 6, of the Discorsi. 2. That is, the tangent to the curve at D is parallel to BC. 3. A standard property of the cycloid (-- trochoid). 4. This is true only of the tangential component of the acceleration. Newton throughout ignores the normal component. 5. Understood to be described in the same short interval of time. 6. Since the ratio of the accelerations of the two weights remains the same throughout the motion, the same will likewise be true of the speeds so that Speed at S/Speed at tt = DCjPC. Now D CIPC^ hC/zrC, from which it follows by use of he = zC Y ,C Y ^ = CS.CB and analogous relations that DC/PC = f(BS/RQ) ~ Speed atS/Speed atvr. That the velocities of vertical descent through distances BS, RQ are proportional to fB S , fR Q follows from Galileo’s law of falling bodies. 7. Results I to 3 follow easily from standard properties of the cycloid. 8. According to the preceding argument the length of the string would need to be equal to that of the arc AD of the cycloid ARD. 9. Compare the result quoted at the end of paragraph 2 of the 2nd section on Fig. I . 10. Since BT^ = BV.BC. 11. Compare Fig. 2. 12. The significance of this statement is obscure. 13. From Galileo’s law of falling bodies.

V

THE

L A W S OF M O TIO N

PAPER

209

V

Whence its velocity towards them is in such proportion as its distance from them that is, as AB , A C , AD , A E etc.

T H E LAIVS O F M O T I O N PAPER

3.^' If a body A [Fig. 2] move towards B with the velocity jR,^ and by the way hath some new force done to it which had the body rested would have propelled it towards C with the velocity S. Then making A B :A C : : R : S, and completing the Parallelogram B C the body shall move in the Diagonall A D and arive at the point D with this compound motion in the same time it would have arrived at the point B with its single motion.

MS. Add. 3958 (5), folios 81-3, listed in the catalogue of the Portsmouth Collection (Cambridge, 1888) among ‘Early Papers by Newton’. There exists also a rough autograph draft of the same manuscript. There can be little doubt that this is an early manuscript written well before the Principia. For a discussion of the question of its order of composition among the early manuscripts see above Part I, Chapter 6.1. The hand­ writing is Newton’s apart from the first lines of section i, the marginal title of § II, and the passage headed ‘Some observations on Motion’, which are all in the hand of Newton’s friend and one-time chamber fellow John Wickins. For some information regarding Wickins see Newton Correspondence, vol. ii, p. 447, note 2. The text of the manu­ script has previously been published in full at pp. 60-64 of vol. iii of the Correspondence and by Hall and Hall [i], and the text of §§ 5, 9, 10 together with an interpretation by Herivel [5]. The original is clean throughout apart from a very small number of unimportant deletions and was possibly intended for publication. Text The Laws of Motion How solitary bodys are moved Sect, i.^ There is an uniform extension space or expansion continued every way with out bounds: in which all bodys are each in severall parts of it : which parts of space possessed and adequately filled by them are their places. And their passing out of one place or part of space into another, through all the intermediate space is their motion.^ Which motion is done with more or lesse velocity accordingly as tis done through more or i^lesse space in equal times or through equall spaces in more or lesse time. 2But the motion it selfe and the force to persevere in that motion is more or lesse accordingly as the factus of the bodys bulk into its velocity is more or lesse.^ And that force is equivalent to that motion which it is able to beget or destroy.*^ 2.<^’5 The motion of a body tends one way directly and severall other ways obliqly. As if the body A [Fig. i]^ move directly towards the point B it also moves obliquely towards all the lines B C , BD, B E etc. which passe through that point B : and shall arrive to them all at the same time.

B

D

4. ®In every body there is a certaine point, called its center of motion about which if the body bee any way circulated the endeavours of its parts every way from the center are exactly counterpoised by opposite endeavours.^ And the progressive motion of the body is the same with the motion of this its center^ which always moves in a streight line and uniformly when the body is free from occursions with other bodys. And so doth the common center of two b o d y s w h i c h is found by divid­ ing the distance twixt their propper centers in reciprocal! proportion to their bulk.’^^And so the common center of 3 or more bodys etc.*^ And all the lines passing through these centers of motion are axes of motion. 5. ^ The angular quantity of a bodys circular motion and velocity is more or lesse accordingly as the body makes one revolution in more [or] lesse timers but the reall quantity of its circular motion is more or lesse accord­ ingly as the body hath more or lesse power and force to persevere in that motion; which motion divided by the bodys bulke is the reall quantity of its circular velocity. Now to know the reall quantity of a bodys circular motion and velocity about any given axis E F [Fig. 3]; suppose it hung upon the two end E and F of that axis as upon two poles: And that another globular body of the same bignesse, whose center is 858205

P

210

THE

L A W S OF M O T IO N

PAPER

A, is so placed that the circulating body shall hit it in the point B and strike it away in the line B A G (which lyeth in the same plane with one of the circles described about the axis EF) and thereby just loose all its owne motion. Then hath the Globe gotten the same quantity of progressive motion and velocity which the other had of circular, its velocity being the same with that of the point C which describes a circle touching the line BG. The radius D C of which circle I may therefore call the radius of

circular motion or velocity about the axis EF. And the circle described with the said Radius of Circulation in that plane which cuts the axis E F perpendicularly in the center of motion I call the Equator of circulation about that axis, and those circles which passe through the poles, meri­ dians etc.i^ 6. " A body circulates about one axis (as P C [Fig. 4]) directly and about several other axes (as AC, BC, etc.) obliquely. And the angular quantity of its circulations about those axes {PC, A C, B C etc.) are as the sines {PC, AD, BE, etc.) of the angles which those axes make with the Equator {EG) of the principall and direct axis (PC). *7 7. ^ If a body circulates about the axis A C [Fig. 5] with the angular quantity of velocity R : and some new force is done to it, which, if the body had rested, would have made it circulate about another axis BC, with the angular quantity of velocity S. Then in the plane of the two axes, and in one of those two opposite angles (made by the axes) in which the two circulations are contrary one to another (as in the angle ACB),

THE

LAWS

OF M O T IO N

PAPER

211

1 find such a point P from which the perpendiculars {PK, PH) let fall to those axes bee reciprocally proportional to the angular velocitys about those axes (that is P K .P H : :R:S). And drawing the line PC, it shall bee the new axis about which the compound motion is performed. And the summe of C H j C P x R and C K j C P x S when the perpendiculars P H and P K fall on divers sides of the axis PC, otherwise their difference, is the angular quantity of circulation about that axis; which in the angle A C P jB C P tends contrary to the circulation about the axis ACjBC.^^

8. * Every body keepes the same reall quantity of circular motion and velocity so long as tis not opposed by other bodys.^^ And it keeps the same axis too if the endeavour from the axis which the two opposite quarters twixt the Equator and every meridian of motion have, bee exactly counterpoised by the opposite endeavours of the 2 side quarters, and then also its axis doth always keepe parallel to it selfe.^® But if the said endeavours from the axis bee not exactly counterpoised by such oppo­ site endeavours: then for want of such counterpoise the prevalent parts shall by little and little get further from the axis and draw nearer and nearer to such a counterpoise, but shall never bee exactly counterpoised. And as the axis is continually moved in the body, so it continually moves in the space too with some kind or other of spirall motion, always draw­ ing nearer and nearer to a center or parallelisme with it selfe, but never attaining to it. Nay tis so far from ever keeping parallel to it selfe, that it shall never be twice in the same position.-^ How Bodys are Reflected-9. J Suppose the bodys A and ot [Fig. 6] did move in the lines DA and ey. till they met in the point .6: that B C is the plane which toucheth them in the point of contact B : that the velocity of the Body A towards the

212

THE

L A W S OF M O TIO N

PAPER

V

said plane of contact is B and the motion A B ; and that the change which is made by reflection in the velocity and motion is x and Ax. Suppose also that from the body A its center of motion two lines are drawne the one A B to the point of contact the other A C to the plane of contact; that the intercepted line B C is F: that the axis of motion which is per­ pendicular to the plane A B C and its equator are called the axis and equator of reflected circulation: that the radius of that equator is G : that the reall quantity of velocity about that axis is D, and the motion

V

THE

L A W S OF M O TIO N

PAPER

213

Which mutations x, and v tend all of them from the plane of Contact. And these four rules I gather thus: The whole velocity of the two points of contact towards one another perpendicularly to the plane of contact is \Q (arising partly from the bodys progressive velocity 5 and ^ and partly from their circular D and 3): And the same points are reflected one from another with the same quantity of such velocity. So that the whole change of all that their velocity which is perpendicular to the plane of contact is 0 . Which change must bee distributed amongst the foure opposed velo­ citys B, /3, D and 8 proportionably to the easinesse (or smallnesse of resis­ tance) with which those velocitys are changed, that is, proportionably to I

j

A’

oc’ AG’

F

(f)

ay

Soe that

1+1+JL+. 0

A that is

a AG

Q

=

AP ~

ay

8_

aP

O

F

I

A

AG ^

— y and

±

ay

(f>Q r 1 — P [ - v]

ayP ^ ii.i Now if any two reflecting bodys A and a, with the quantity of

A D : that the change which reflection makes in that velocity and motion is y and A y ; And that the correspondent lines and motions of the other body a are a^,
n

And

Q

2DF

28(f)

G

y

0.

Observing that at the time of reflection if in either body the center of motion doth move from the plane of contact, or those parts of it nearest the point of contact doe circulate from the plane of contact: then the said motion is to bee esteemed negative and the signe of its velocity B, D or 8 must bee made negative in the valor of Q. 10A The velocitys B, D and 8 and they only are directly opposed and changed in Reflection; and that according to these rules Q — X

^

^

_V

AGP-y

and

— v

y^p-"-

AGP

^

their progressive and angular motions, and their position at their meeting and consequently their point and plane of contact etc be given to know how those bodys shall bee reflected, first find B and fl by Sec. 2. Then the lines F and cf) and the axis of reflected circulation by Sec. 9 and their Radij G and y by Sec. 5. Then their angular quantity of velo­ city about the axes of reflected circulation by Sec. 6, and the reall quantity D and 8 by Sec. 5. Then P and Q by Sec. 9. Then x, i, y and v by Sec. 10. Then the bodys new progressive determinations and velocitys by Sec. 3. Then the angular quantity of that circulation (y and 1^), which is generated by reflection by Sec. 5. And lastly the new axes and angular quantity of velocity about them by Sec. 7. Some Observations about Motion’" ^^ Only those bodys which are absolutely hard are exactly reflected accord­ ing to these rules. Now the bodys here amongst us (being an aggregate of smaller other bodyes) have a relenting softnesse and springynesse, which makes their contact be for some time and in more points than one. And the touching surfaces during the time of contact doe slide one upon another more or lesse or not at all according to their roughnesse. And few or none of these bodyes have a springynesse soe strong as to force them one from another with the same vigor that they came together.

214

THE

L A W S OF M O TIO N

PAPER

V

Besides, that their motions are continually impeded and slackened by the mediums in which they move. Now he that would prescribe rules for the reflections of these compound bodies, must consider in how many points the two bodies touch at their meeting, the position and pression of every point, with their planes of contact etc.: and how all these are varyed every moment during the time of contact by the more or lesse relenting softnesse or springynesse of these bodies and their various

V

THE

L A W S OF M O TIO N

PAPER

215

For completing the square Bj3 , the body a shall move in the Diagonall ocC, and arrive at C but at the same time it would have arrived at j8 without reflection, see the third section. dou bled . 25

5. Motion may be gained by reflection. For if the body a return with the same motion back again from C to a. The two bodyes A and a after reflection shall regain the same equall motions in the lines A D and aS (though backwards) which they had at first. a. b. c. d. e.

Marginal entry o f Place m otion, v e lo city and force. At this point Newton took over from Wickins. Marginal entry W ith w h at v e lo city a b o d y m oves severall w ays at once. Marginal entry H o w tw o progressive m otions are jo yn e d into one. Marginal entry o f centers and axes o f m otion and the m otion o f those

centers. f. g.

Marginal entry o f circular m otion and v e lo city about Marginal entry W ith w h at v elo city a b o d y circulates

those axes. about severall axes at

once. h. i.

Marginal entry H o w tw o circular m otions are jo y n e d into one. Marginal entry In w h at cases a circulatin g b o d y perseveres in

the same

state and in w h at it doth not. j. k. l.

Marginal entry Som e nam es and letters defined. Marginal entry T h e R u le for Reflection. Marginal entry T h e conclusion.

m . T h is section w ritten in W ick in s ’s hand. 1. T h is definition o f m otion, based on the p recedin g definition o f space and place, corresponds to the definitions o f absolute space, tim e, place and m otion

Figure 7.

slidings. And also what effect the air on other mediums compressed betwixt the bodies may have. 2. There are some cases of Reflections of bodies absolutely hard to which these rules extend not: As when two bodies meet with their angular point, or in more points than one at once; Or with their superficies. But these cases are rare. 3. In all reflections of any bodies what ever this rule is true; that the common center of two or more bodies changeth not its state of motion or rest by the reflection of those bodies one amongst another. 24 4. Motion may be lost by reflection. As if two equall Globes A [Fig. 7] and a with equall motions from D and S done in the perpendicular lines D A and Sa, hit one another when the center of the body a is in the line DA. Then the body A shall loose all its motion and yet the motion of a is not

given in D e f. 2 -4 o f the

Scholium

to the D efinitions o f Book I o f the

Principia.

See M S . V I for som e indications o f the genesis o f N e w to n ’s view s on this subject. 2. A s noted p reviously ( M S . l i d , n. 20), N e w to n alm ost always w orked w ith the ratio o f one v elo city to another, rather than w ith a single velo city. B u t the definition o f m otion in the im m ed iately fo llo w in g sentence is v e ry close to a direct one. 3. See M S . H e, D e f. 3. 4. T h is m ethod o f m easuring force b y the corresponding m otion created or destroyed had already been em p loyed b y N e w to n in M S . II, especially at A x .Prop. 3 -6 , 23 o f M S . I ld . 5. T h is , and the su cceed ing section, should be com pared w ith the entries on fob 38 o f the

Waste Book

(see M S . H g). A

proof oi the

rule for co m po u n din g

in dependent m otions appears first in L e m m a i o f Version H I o f the tract

Motu

de

(see M S . IX c).

6. T h e s e and subsequen t diagram references are taken from the original m anuscript. 7. In all know n later form ulations o f the result b o th m otions are assum ed to have been generated sim ultaneously. F o r the possible significance o f this fact see Part I, C h a p ter 2 .1, p p . 39-40. 8. T h e dynam ical definition o f centre o f m otion corresponds to that already

THE

216

LAWS' OF M O TIO N

PAPER

centre o f m otion w ill coincide w ith the centre o f mass o f the b o d y. F o r i f the b o d y is assum ed to be m ade up o f particles rn-i, from a given axis th rou gh the centre o f m otion are

w hose directed distances

r^, rz,...,

TH E

V

f?iven in D e f. lo o f M S . l i e . F ro m the present definition it follows easily that the

then the fa ct that

the various endeavours from the axis balance requires that ^ t-gnig a>l — o, cog S

L A W S

OF

M O T IO N

217

PAPER

17. A s follow s im m ediately b y resolution, regarding angular v e lo city as a (polar) vector. 18. T a k e directions and sense o f rotations o f R and S as in diagram 5, so that in the angle A C B the resulting m otions are opposite. A ssu m in g that the reaction (if any) to the applied couple acts through C , the new total angular m om en tum

b ein g the angular v e lo city o f the p article m,, about the axis. Since cOg w ill be the

w ill be equal to the vector sum o f the original angular m om en tu m p lus that due

same for all parts o f the b o d y it follow s that '^ n ig T g = o. I f Xg, y>g are the re-

to the applied couple. T h e resulting rotation w ill therefore b e about the axis C P ,

S

Ji R sin P C H == / , S sin P C K ,

solved parts o f r , w ith respect to an y tw o m u tu ally perpendicular axes through the centre O in a plane p erpendicular to the axis, the above equation requires V nigXg ~ 2 ^^hy’ s -

and its m agn itude

w ill be given b y

O) so that the centre o f mass o f the b o d y m u st lie on the

LgV.

ly R cos P C H ! /a 6" cos P C K .

axis throu gh O . I f this is true for every axis th rough O , as required in the defini­ tion, then O m u st be at the centre o f mass o f the body. 10. A s follow s from the principle o f inertia. 1 1 . A s p roved b y N e w to n in A x .-P r o p . 2 7 -3 0 o f M S . I lf. 12. T h e definition given in A x .-P r o p . 25 o f M S . I ld . 13. A n extension n o t given in M S . I I . T h e p ro o f is g iv en in the P r in cip ia at 14. A x is o f m otion b ein g defined b y im plication at the begin n in g o f the

16. T o elucidate the m eanin g o f the various term s in the p recedin g section, let us solve the collision prob lem b y ‘m o d ern ’ m ethods.

Then P

19. T h is statem ent follow s, o f course, fio m the p rinciple o f angular m o m en ­ tum . 20. T h is corresponds, although it does n ot appear to be actually equivalent, to the m otion

of principal axes of rotation.

tion o f this statem ent. H is treatm ent o f the p ro blem o f the collision betw een that such a justification w o u ld have been w e ll w ith in his powers.

15. See A x .-P r o p . 19 o f M S . I ld .

P

22. T o elucidate N e w to n ’s treatm ent o f this p roblem in the tw o follow in g sections it is necessary to reform ulate it in m odern terms.

be mass o f either body,

In ad dition to the term s defined b y N e w to n in § 9 l e t ;

be angular v elo city o f the rotating b o d y before im pact,

H, Cl' b e the (clockwise) angular v e lo city o f A before and after im pact,

m o m en t o f inertia o f rotating b o d y about axis E F ,

oj, <x)' be

be speed in du ced in A ,

B ',

= CD,

the (anticlockwise) angular v elo city o f a before and after im pact,

be the velocities o f A and a respectively aw ay from the plane o f contact

after im pact,

= m agn itu d e o f im pulse experienced b y A . = M V.

I , i be the m om ents o f inertia o f A , a respectively about the axes o f rotation th rough A , cx.

P x r ~ I c j, since the im pulse P actin g on the rotating b o d y m u st be equal and o pposite to that on A , and the m o m en t o f this im pulse about E F equals the

T h e fo llo w in g equations then ho ld;

Conservation of linear momentum perpendicular to plane of contact:

en su ing dim in u tion o f the angular m om en tu m o f the rotating b o d y about

EF. M V ^ Iz

-- h .

tw o rotating bodies in the present m an uscript m akes it seem en tirely possible

section.

u> I V r

- h

2 1. I t is to be regretted that N e w to n d id n ot provide a quantitative ju stifica­

C oroll. 4 to L a w s o f M o tio n .

Let M

T h e results given b y N e w to n then fo llo w on the assum ption ly O th erw ise th ey w o u ld seem to be untrue.

9. See A x .-P r o p . 17, M S . I ld .

I v P jz , assum ing (i) the rotating b o d y is brou gh t to rest, (2) that

en ergy is conserved. T h is later assum ption im plies that the collision is elastic.

A B ~ ol^

and

V = rcu.

ol^

'-A B '.

(

i

)

Conservatioyi o f angular momentum about B (through w h ich the force o f im pact m ust act) for A :

F ro m these equations w e deduce

I = Mr^,

=

sim ilarly for or.

m

\BAF -

I Q '- B 'A F ;

(2)

ioj—^af ~ ico'

(.3)

C onservation o f energy f o r elastic collision (from section ‘ Som e observations about S o that N ezvton's 'radius o f circular motion or velocity about the axis E F ' equals our radius o f gyration k : w h ile the quantities o f circular m otion and v e lo city o f the rotating b o d y, b y definition equal to V and M V respectively, are eq ual to roj ~ koj, and M r o j = M k co, respectively. In other w ords, N e w to n ’s real qu a n tity o f

m o tio n ’ it is clear the collision is assum ed to be ela stic):

I C P j z f - A B ^ l z + ioj'^jz-roc^^lz = I C l'^ lz - fA B '^ jz ^ ia o '- lz - y a ^ '^ lz .

(4)

F ro m these four equations it follows that

circular v e lo city is that o f the b o d y on its equator o f circulation, and the real qu antity o f circular m otion, w h ich he takes, b y definition, as the m easure o f the b o d y ’s p ow er to persevere in this rotation, is equal to the momentum o f a single particle o f mass M

B —C IF ^

to ~ B

'

C

l

'

F

(5)

i.e. the relative v elo city o f separation after collision equals the relative v e lo city

rotating w ith the b o d y around the equator o f circulation.

o f approach before, the assum ption m ade b y N ew to n . M o reover, referring to

T h is measure o f the p ow er to persevere in rotation should be contrasted w ith

note 16, the value o f this relative v e lo city — l ( z B z ^ — z D F j G + z S f l y ) — Q j z

the m odern m easure o f angular m o m en tu m ; the tw o differ b y a factor k.

(allow ing for an un im portant difference in sign).

THE

218

L A W S OF M O TIO N

PAPER

O n solvin g i, 2, 3, and 5 w e ultim ately find:

VI

O

{B' + B)

( 6)

AP’

E X T R A C T S FROM MS. A D D . 4003

.0

(7)

cxP’ QF

(0^-0)

(8)

IP '

(co~o)') —

(9 )

ip ’

where Q , P have the values given b y N ew to n . Sin ce I = A G ^ , i ^ ay^ the last tw o equations m ay be written ( 8)'

{yoj —ycxj') =1^ Q i

(9)'

ayP

N o tin g that B ' F B ,

give the changes x, ^ in the corresponding linear

velocities, and that G O '— G i l , yco~ya>' give the changes y , v in the real quantities

o f circular m otion (cf. note 16) w'e find

O AP

=

.r,

Q ocP

C

FQ AGP

y,

fQ ayP

as given b y N ew to n . 23. T h e first part o f this section proves how clearly N ew to n had understood the conditions actu ally go vern in g collision betw een bodies in practice as opposed to the ideal conditions assum ed in §§ 9, to. 24. A s p roved for the case o f tw o bodies in Prop. 32, M S . I lf. 25. T h is rather surprising result follow s from the assum ptions o f conservation o f en ergy and m o m en tu m and that the collision is such that A

receives no

m otion p erpen dicular to D B . It also seems to in dicate that N e w to n had n ot realized the -vectorial nature o f the law o f conservation o f m om entum . 26. A good exam ple o f the reversibility o f classical N ew ton ian dynam ics.

U .L.C. MS. Add. 4003 consists of a small book of forty pages in Newton’s hand in Latin. The text of the manuscript together with an introduction and translation has been published by Hall and Hall [i]. After a short introduction there are four definitions of certain basic physical concepts followed by a long philosophical discussion. Then come fifteen more definitions, followed by a section entitled Propositiones defluido non elastico containing two propositions, five corollaries, and a concluding Scholium. Although the presence of this section on hydrosta­ tics may be an indication that the preceding part was originally intended by Newton as an introduction to a projected treatise on hydrostatics, the true importance of the book lies elsewhere. In the first place, it contains what is to my knowledge the only known extended writing by Newton on a purely philosophical subject ; as such it throws most inter­ esting light on Newton’s standing as a philosopher. In the second place, it provides conclusive evidence both for Descartes’s influence on Newton, and for the nature of Newton’s reaction to Descartes’s philosophy. And finally it throws important light on Newton’s view^s on space, time and body. The philosophical discussion between Def. 4 and 5 may be divided up as follows, (i) pp. 3-10: An attack by Newton on the definition of motion in the true and absolute sense given in Part 2 of Descartes’s Principia Philosophiae. (2) pp. 11-19: An exposition of Newton’s own views on space, time and motion. (3) pp. 20-31 : A criticism of Descartes’s identification of body and extension, and of his assertion of an absolute dichotomy between mind and body, together with Newton’s own views on the nature of body. Extracts will largely be restricted to those of dynamical interest, and those bearing on Newton’s theory of space. For a discussion of the question of dating see above, Part I, Chapter 6.2. Apart from a small number of unimportant deletions the text is clean throughout. Text I. De^ Gravitatione et aequipondio fluidorum et solidorum in fluidis

scientiam duplici methodo tradere convenit. Quatenus ad scientias Mathematicas pertinet, aequum esl ut a contemplatione Physica quam

220

E X T R A C T S FRO M MS. ADD. 4003

VI

maxime abstraham. Et hac itaque ratione singulas cjiis propositiones e principijs abstractis et attendenti satis notis, more Geometrarum, stricte demonstrare statui. Deinde cum haec doctrina ad Philosophiam naturalem quodammodo affinis esse censeatur, quatenus ad plurima ejus Phaenomena enucleanda accommodatur, adeoque cum usus ejus exinde praesertim elucescat et principiorum certitudo fortasse confirmetur, non gravabor propositiones ex abundanti experimentis etiam illustrare S ita tamen ut hoc laxius disceptandi genus in Scholia dispositum, cum priori per Lemmata, propositiones et corollaria tradito non confundatur. ffundamenta ex quibus haec scientia demonstranda est sunt vel definitiones vocum quarundam; vel axiomata et postulata a nemine non concedenda. Et haec e vestigio tradam.

Definitiones. Nomina quantitatis, durationis et spatij notiora sunt quam ut per alias voces definiri possint. Def: 1.3 Locus est spatij pars quam res adaequate implet.'^ D e f: 2. Corpus est id quod locum implet.*^ Def: 3. Quies est in eodem loco permansio. D ef: 4. Motus est loci mutatio.'^ Nota. Dixi corpus implere l o c u m, h o c est ita saturate ut res alias ejusdem generis sive alia corpora penitus excludat, tanquam ens impenetrabile."^ Potuit autem locus did pars spatij cui res adaequate inest, sed cum hie corpora tantum et non res penetrabiles spectantur, malui definire esse spatij partem quam res implet. Praeterea cum corpus hie speculandum proponitur non quatenus est substantia Physica sensibilibus qualitatibus praedita sed tantum quatenus est quid extensum mobile et impenetrabile; itaque non definivi pro more philosophico, sed abstrahendo sensibiles qualitates (quas etiam Philosophi ni fallor abstrahere debent, et menti tanquam varios modos cogitandi a motibus corporum excitatos tribuere) posui tantum proprietates quae ad motum localem requiruntur. Adeo ut vice corporis Physici possis figuras abstractas intellegere quemadmodum Geometrae contemplantur cum motum ipsis tribuunt, ut fit in prop 4 & 8, lib i Elcm Euclid .5 Et in demonstratione definitionis decimae, lib i i debet fieri; siquidem ea inter definitiones vitiose recensetur et potius inter proposi­ tiones demonstrari debuit, nisi forte pro axiomate habeatur.

VI

E X T R A C T S FROM MS. ADD. 4003

221

Definivi praeterea motum esse loci mutationem,'^ propterea quod motus, transitio, translatio, migratio etc videntur esse voces synonymae. Sin malueris esto motus transitio vel translatio corporis de loco in locum. Caeterum in his definitionibus cum supposuerim spatium a corpore distinctum dari et motum respectu partium spatij istius, non autem respectu positionis corporum contiguorum determinaverim ne id gratis contra Cartesianos assumatur, ffigmenta ejus tollere conabor. 2. ^ Jam vero quam confusa et ration! absona est haec doctrina’ non modo absurdae consequentiae convincunt, sed et Cartesius ipse sibi contradicendo videtur agnoscere. Dicit enim Terram caeterosque Planetas proprie et juxta sensum Philosophicum loquendo non mover!,^ eumque sine ratione et cum vulgo tantum loqui qui dicit ipsam mover! propter translationem respectu fixarum (Art. 26, 27, 28, 29 part 3). Sed postea tamen in Terra et Planetis ponit conatum^ recedendi a Sole tan­ quam a centro circa quod moventur, quo per consimilem conatum Vorticis gyrantis in suis a Sole distantijs librantur Art 140 part 3.3° Quid itaque? an hie conatus a quiete Planetarum juxta Cartesium vera et Philosophica, vel potius a motu vulgi et non Philosophico derivandus estP^' 3. *^ Ex utraque harum consequentiarum^^ patet insuper quod e motibus nullus prae alijs dici potest verus absolutus et proprius, sed quod omnes, sive respectu contiguorum corporum sive remotorum, sunt similiter philosophici, quo nihil absurdius imaginari possumus. Nisi enim concedatur unicum cujusque corporis motum physicum dari, caeterasque respectuum et positionum inter alia corpora mutationes, esse tantum externas denominationes: sequetur Terram verb! gratia conari recedere a centro Solis propter motum respectu fixarum, et minus conari rece­ dere propter minorem motum respectu Saturni et aetherei orbis in quo vehitur, atque adhuc minus respectu Jovis et aetheris circumducti ex quo orbis ejus conflatur, et iterum minus respectu Martis ejusque orbis aetherei, multoque minus respectu aliorum orbium aethereae materiae qui nullum Planetam deferentes sunt propiores orbi annuo Terrae; respectu vero proprij orbis non omnino conari quoniam in eo non movetur. Qui omnes conatus et non conatus cum non possunt absolute competere dicendum est potius quod unicus tantum motus naturalis et absolutus Terrae competit, cujus gratia conatur recedere a Sole, et quod translationes ejus respectu corporum externorum sunt externae tantum denominationes. 4.35 Denique ut hujus positionis absurditas quam maxima pateat, dico

222

E X T R A C T S FROM MS. ADD. 4003

VI

quod exinde sequitur nullam esse mobilis alicujus determinatam velocitatem nullamque definitam lineam in qua movetur. Et multo magis quod corporis sine impedimentis moti velocitas non dici potest uniformis, neque linea recta in qua motus perficitur. Imo quod nullus potest esse motus siquidem nullus potest esse sine aliqua velocitate ac determinatione. 5. ^^ Quin imo sequitur motum Cartesianum non esse motum, utpote cujus nulla est velocitas, nulla determinatio et quo nullum spatium, distantia nulla trajicitur.'^ Necesse est itaque ut locorum determinatio adeoque motus localis ad ens aliquod immobile referatur quale est sola extensio vel spatium quatenus ut quid a corporibus revera distinctum spectator, Et hoc lubentius agnoscet Cartesianus Philosophus si modo advertat quod Cartesius ipse extensionis hujus quatenus a corporibus distinctae ideam habuit, quam voluit ab extensione corporea discriminare vocando genericam. Art. 10, 12 et 18 part 2 Princip. Et quod vorticum gyrationes, quibus vim aetheris recedendi a centris, adeoque totam ejus mechanicam Philosophiam deduxit, ad extensionem hance genericam tacite referuntur.'^ 6. *'^ De extensione jam forte expectatio est ut definiam esse vel substantiam vel accidens aut omnino nihil. At neutiquam sane, nam habet quendam sibi proprium existendi modum qui neque substantijs neque accidentibus competit. Non est substantia turn, quia non absolute per se, sed tanquam Dei effectus emanativus, et omnis entis affectio quaedam subsistit; turn quia non substat ejusmodi proprijs affectionibus quae substantiam denominant, hoc est actionibus, quales sunt cogitationes in mente et motus in corpore. Nam etsi Philosophi non definiunt substantiam esse ens quod potest aliquid agere, tamen omnes hoc tacite de substantijs intelligunt, quemadmodum ex eo pateat quod facile concederent extensionem esse substantiam ad instar corporis si modo moveri posset et corporis actionibus frui. Et contra baud concede­ rent corpus esse substantiam si nec moveri posset nec sensationem aut perceptionem aliquam in mente qualibet excitare. Praeterea cum ex­ tensionem tanquam sine aliquo subjecto existentem possumus dare concipere, ut cum imaginamur extramundana spatia aut loca quaelibet corporibus vacua; et credimus existere ubicunque imaginamur nulla esse corpora, nec possumus credere periturum esse cum corpore si modo Deus aliquod annihilaret, sequitur earn non per modum accidentis inhaerendo alicui subjecto existere. Et proinde non est accidens. Et multo minus dicetur nihil, quippe quae magis est aliquid quam accidens

VI

E X T R A C T S FROM MS. ADD. 4003

223

et ad naturam substantiae magis accedit. Nihili nulla datur Idea neque ullae sunt proprietates sed extensionis Ideam habemus omnium clarissimam abstrahendo scilicet affectiones et proprietates corporis ut sola maneat spatij in longum latum et profundum uniformis et non limitata distensio.2” Et praeterea sunt ejus plures proprietates concomitantes hanc Ideam, quas jam enumerabo non tantum ut aliquid esse sed simul ut quod sit ostendam. 7.2’' Denique spatium est aeternae durationis et immutabilis naturae, idque quod sit aeterni et immutabilis entis effectus emanativus. Siquando non fuerit spatium, Deus tunc nullibi adfuerit, et proinde spatium creabat postea ubi ipse non aderat, vel quod non minus rationi absonum est, creabat suam ubiquitatem. Porro quamvis fortasse possumus imaginari nihil esse in spatio tamen non possumus cogitate non esse spatium; quemadmodum non possumus cogitate durationem non esse, etsi possibile esset fingere nihil omnino durare. Et hoc per extramundana spatia manifestum est, quae (cum imaginamur mundum esse finitum) non possumus cogitate non esse, quamvis nec a Deo nobis revelata sunt, nec per sensus innotescunt nec a spatij s intramundanis quoad existentiam dependent. Sed de spatijs istis credi solet quod sunt nihil. Imo veto sunt spatia. Spatium etsi sit corpore vacuum tamen non est seipso vacuum. Et est aliquid quod sunt spatia quamvis praeterea nihil. Quin imo fatendum est quod spatia non sunt magis spatia ubi mundus existit quam ubi nullus est, nisi forte dices quod Deus cum mundum in hoc spatio creabat, spatium simul creabat in seipso vel quod Deus si mundum in his spatijs posthac annihilaret, etiam spatia annihilaret in seipsis. Quicquid itaque est pluris realitatis in uno spatio quam in altero, illud corporis est et non spatij, quemadmodum clarius patebit si modo puerile illud et ab infantia derivatum praejudicium deponatur quod extensio inhaeret corpori tanquam accidens in subjecto sine quo revera nequit existere. 8.22 Cum autem aqua minus obstat motibus trajectorum solidorum quam argentum vivum et aer longe minus quam aqua, et spatia aetherea adhuc minus quam aerea rejiciamus praeterea vim omnem impediendi motus trajectorum et sane naturam corpoream penitus rejiciemus. Quem­ admodum si materia subtilis vi omni privaretur impediendi motus globulorum, non amplius crederem esse materiam subtilem sed vacuum disseminatum. Atque ita si spatium aereum vel aethereum ejusmodi esset ut Cometarum vel corporum quorumlibet projectilium motibus

224

E X T R A C T S FROM MS. ADD. 4003

VI

sine aliqua resistentia cederent crederem esse penitus inane. Nam impossibile est iit fluidum corporum non obstet motibus trajectorum, puta si non disponitur ad motum juxta cum eorum motu velocem (Part 2 Epist 96 ad Mersennum),23 quemadmodiim suppono. Hanc autem vim omnem a spatio posse tolli manifestum est si modo spatium et corpus ab invicem differunt; et proinde tolli posse non est denegandum antequam probantur non differre, ne paralogismus, petendo principium admittatur. Sed nequa supersit dubitatio, ex praedictis observandum venit quod inania spatia in rerum natura dantur. Nam si aether esset fluidum sine poris aliquibus vacuis penitus corporeum, illud, utcunque per divisionem partium subtilitatum, foret aeque densum atque aliud quodvis fluidum, et non minori inertia motibus trajectorum cederet, imo longe majori, si modo projectile foret porosum; propterea quod intimos ejus poros ingrederetur, et non modo totius externae superficiei sed et omnium internarum partium superficiebus occurreret et impedimento esset. Sed cum aetheris e contra tarn parva est resistentia ut ad resistentiam argenti vivi collata videatur esse plusquam decies vel centies mille vicibus minor: sane spatij aetherei pars longe maxima pro vacuo inter aetherea corpuscula disseminate haberi debet. Quod idem praeterea ex diversa gravitate horum fluidorum conjicere liceat, quam esse ut eorum densitates sive ut quantitates materiae in aequalibus spatijs contentae monstrant turn gravium descensus turn undulationes pendulorum.^^ Sed his enucleandis jam non est locus, 9. Def. 5. Vis est motus et quietisms causale principium. Estque vel externum26 quod in aliquod corpus impressum motum ejus vel generat vel destruit, vel aliquo saltern modo mutat; vel est internum principium^^ quo motus vel quies corpori insita conservatur, et quodlibet ens in suo statu perseverare conatur et impeditum reluctatur. Def. 6. Conatus est vis impedita sive vis quatenus resistitur.28 Def. 7. Impetus est vis quatenus in aliud imprimitur. Def. 8. Inertia29 est vis interna corporis ne status ejus externa vi illata facile mutetur. Def. 9. Pressio est partium contiguarum conatus ad ipsarum dimensiones mutuo penetrandum. Nam si possent penetrare cessaret pressio.3° Estque partium contiguarum tantum, quae rursus premunt alias sibi contiguas donee pressio in remotissimas cujuslibet corporis duri mollis vel fluid! partes transferatur. Et in hac actione communicatio motus mediante puncto vel superficie contactus fundatur.

VI

E X T R A C T S FROM MS. ADD. 4003

225

Def. 10. Gravitas est vis corpori indita ad descendendum incitans. Hie autem per descensum non tantum intellige motum versus centrum terrae sed et versus aliud quodvis punctum plagamve, aut etiam a puncto aliquo peractum. Quemadmodum si aetheris circa Solem gyrantis conatus recedendi a centro ejus pro gravitate habeatur, descendere dicetur aether qui a Sole recedit.^i Et sic analogiam observando, planum dicetur horizontale quod gravitatis sive conatus determination! directe opponitur. Caeterum harum potestatum, nempe motus, vis, conatus, impetus, inertiae, pressionis, et gravitatis quantitas duplici ratione aestimatur; utpote vel secundum intensionem earum vel extensionem. Def. II. Intensio potestatis alicujus praedictae est ejus qualitatis gradus. Def. 12. Extensio ejus est spatij vel temporis quantitas in quo exercetur. Def. 13. Ejusque quantitas absoluta est quae ab ejus intensione et extensione componitur. Quemadmodum si quantitas intensionis sit 2, et quantitas extensionis 3, due in seinvicem et habebitur quantitas absoluta 6. Caeterum hasce definitiones in singulis potestatibus illustrare juvabit. Sic itaque motus intensior est vel remissior^^ quo spatium majus vel minus in eodem tempore transigitur, qua quidem ratione corpus dici solet velocius vel tardius moveri. Motus vero magis vel minus extensus est quocum corpus majus vel minus movetur, sive qui per majus vel minus corpus diffunditur. Et motus absoluta quantitas est quae componitur ex utrisque velocitate et magnitudine corporis moti. Sic vis, conatus, im­ petus, et inertia intensior est quae est in eodem vel aequali corpore major; extensior est quae est in majori corpore; et ejus quantitas abso­ luta quae ab utrisque oritur. Sic pressionis intensio est ut eadem super­ ficiei quantitas magis prematur, extensio ut major superficies prematur, et absoluta quantitas quae resultat ab intensione pressionis et quantitate superficiei pressae. Sic denique gravitatis intensio est ut corpus habeat majorem gravitatem specificam, extensio est ut corpus grave sit majus, et absolute loquendo gravitatis quantitas est quae resultat ex gravitate specifica et mole corporis gravitantis. Et haec quisquis non dare distinguit, ut in plurimos errores circa scientias mechanicas incidat necesse est. Potest insuper quantitas harum potestatum secundum durationis intervallum nonnunquam aestimari: qua quidem ratione quantitas 858205

Q

226

E X T R A C T S FROM MS. ADD. 4003

VT

absoluta erit quae ex omnibus intensione extensione ac duratione componitur. Quemadmodum si corpus 2 velocitate 3 per tempus 4 movetur: totus motus erit 2X3X4, sive I2[s^c].-^^ Def. 14, Velocitas est motus intensio, ac tarditas remissio ejus. Translation is proper to expound a knowledge of the heaviness and equili­ brium of fluids and of solids in fluids by a double method. In so far as this knowledge relates to mathematical topics, it is proper that I should depart as much as possible from physical considerations. And so on this account I have decided on a rigorous demonstration of particular propositions of the subject from self-evident abstract principles in the manner of mathematicians. But then since this subject is considered to be somewhat akin to Natural Philosophy, in so far as it is employed in elucidating many of the phenomena thereof, and also that its usefulness may be particularly evident and the certainty of its principles perhaps confirmed, I shall not hesitate also to illustrate propositions by abundance of experiments yet in such a way that this freer type of argument assigned to scholia may not be confused with the form expounded in lemmas, propositions and corollaries. The foundations from which this science is to be deduced are either definitions of certain terms or else axioms or postulates admitted by all. And these I set forth herewith. I.

Definitions. The names quantity, duration and space are too familiar to be definable in other terms. Def. 1.3 Place is a part of space that a thing fills evenly.**^ Def: 2. Body is that which fills a place. D e f: 3. Rest is continuance in the same place. D e f: 4. Motion is change of place.^ Notes. I have said that a body fills a pl ace, that is so completely that it utterly excludes other things of the same kind or other bodies, like an impenetrable thing.^ On the other hand, place could have been defined as a part of space in which a thing exists uniformly, but since bodies only and not penetrable things are here considered, I have preferred to define it as a part of space which a thing fills. Moreover, since the body discussed here is put forw'ard not as a physi­ cal substance endowed w'ith sensible qualities but only as something

VI

E X T R A C T S FROM MS. ADD. 4003

227

extended, mobile and impenetrable, I have defined it not in a philosophical manner, but in abstracting sensible qualities (which unless I am mis­ taken philosophers ought also to abstract, and assign to the mind as vari­ ous modes of thinking excited by the motion of bodies) I have posited only properties requisite for local motion. So that instead of physical bodies you may understand abstract figures in the same way as they are regarded by Geometers when they assign motion to them, as is done in Prop. 4 and 8 Book I of Euclid’s Elements.s And as ought to be done in the demonstration of Definition 10 of Book II, since it is taken as a defini­ tion in error and ought rather to be demonstrated among the proposi­ tions, unless perhaps it should be regarded as axiomatic. Moreover I have defined motion to be change of place, because motion, transition, translation, migration etc. seem to be synonymous words. If you prefer it let motion be the transition or translation of a body from place to place. In addition, since in these definitions I shall suppose space to be given apart from bodies, and motion in respect of the parts of this space, and I shall not determine it in respect of the position of neighbouring bodies, lest this is taken as gratuitously contrary to the Cartesians I shall [first] endeavour to dispose of his [Descartes’] imaginings. 2. ^ Now truly not only the absurd consequences of this doctrine"^ prove how confused and foreign to reason it is, but Descartes himself by con­ tradicting himself seems to acknowledge this to be the case. For he says the Earth and other planets are at rest speaking truly and in the philo­ sophical sense,s and that he wEo says it is moved on account of its trans­ lation relative to the fixed stars lacks reason and speaks only in the vulgar sense (Art. 26, 27, 28, 29, Part 3). But later he nevertheless posits in the Earth and the planets an endeavour*^ to recede from the Sun as if from a centre about which they are moved, by means of which and the like endeavour of the revolving vortices they are poised at their distances from the Sun (Art. 140 Part 3).!° What then? Is this conatus to be derived from the Planets’ rest, according to Descartes true and philo­ sophical, or from their vulgar and non-philosophical motions?” 3 . From both of these consequences'’ it appears moreover that of motions no one can be said to be true, absolute, and more proper beyond the others, but that all in respect of both neighbouring and remote bodies are equally philosophical, than which w^e are able to imagine nothing more absurd. For unless a unique physical motion is allowed

228

E X T R A C T S

FROM

MS.

ADD.

4 0 0 3

VI

to any particular body, additional changes of its situation and position among other bodies being regarded as external denominations only, it will follow, for example, that the earth will have an endeavour to recede from the centre of the sun on account of its motion relative to the fixed stars, and a less endeavour on account of its lesser motion relative to Saturn and the aetherial orbit in which it is carried, and so much the less again relative to Jupiter and the surrounding aether composing its orbit, and less again relative to Mars and its aetherial orbit, and much less relative to other aetherial orbits which carry no planets and are closer to the annual orbit of the Earth; and indeed relative to its own orbit no endeavour to recede at all since it is not moved in that orbit. And since all these endeavours and non-endeavours are incapable of absolute agree­ ment, it must rather be affirmed that there is only one natural and absolute motion appropriate to the Earth thanks to which it endeavours to recede from the Sun, and that its translations relative to external bodies are external denominations only.J*^ 4.^5 Finally that the absurdity of this position may appear most evident, I say that it then follows that any moving thing has no determinate velo­ city and no definite line in which it is moved. And much more that the velocity of a body without impediment to its motion cannot be said to be uniform, nor the line straight in which its motion takes place. Nay rather that there can be no motion since there is none without a certain velocity and determination. 5.I*’ Indeed it follows that Descartes’ motion is not motion, seeing that it has no velocity, no determination and that by it no space and no distance is covered. Jt fg necessary therefore that the assignment of places (and thus likewise local motion) should be referred to a certain immobile entity, such as extension or space alone in so far as it is regarded as something truly distinct from bodies. And this the Cartesian Philo­ sopher would more easily acknowledge if he would but notice that Descartes himself had the notion of such extension as distinct from bodies, which he intentionally distinguished from bodily extension by calling it generic, Art. 10, 12, and 18 Part 2. Principia. Also that the gyration of vortices (from which he deduced the force of aether to recede from the centres and thus his whole mechanical philosophy) are referred implicitly to this generic extension. 6.^9 Now concerning extension it may perhaps be expected that I would define it as either a substance or accident or nothing at all. But by no

VI

E X T R A C T S FROM MS. ADD. 4003

229

means, for it has a certain mode of existence proper to itself which befits neither substances nor accidents. It is not substance, then, because on the one hand it does not subsist absolutely by itself, but like an effect emanating from God and as a certain affection of each thing; on the other hand because it does not sustain the peculiar effects which characterise substance, that is actions, such as thoughts in mind and motion in body. For although Philosophers do not define substance as a thing capable of some action, nevertheless they all tacitly understand this of substances, on which account, for example, it is evident that they would readily admit extension to be substance in the manner of body if only it were able to be moved and enjoy the actions of body. And conversely, they would scarcely admit body to be substance if it were incapable either of being moved, or of exciting sensation or perception in some mind or other. Moreover, since we can clearly conceive extension existing alone without any subject, as when we imagine extra-mundane spaces or cer­ tain places devoid of bodies; and since we believe it to exist wherever we imagine absence of bodies and cannot believe it to perish with a body (if God were to annihilate that), it follows that it does not exist in the manner of an accident inhering in a certain subject. And hence it is not an accident. And much less can it be said to be nothing, in as much as it is more thinglike than an accident, and approaches more closely to the nature of a substance. There is no idea attached to nothing, but of extension we have the clearest idea of all by abstracting corporeal effects and properties, so that there remains only a stretching out of place uniform and unlimited in length, breadth, and d e p t h . A n d besides there are many properties of space concomitant with this idea, which I shall now enumerate that I may show not only that space is some­ thing but what it is. 7.2* Then finally space is of eternal duration and immutable nature, and is thus because it is an effect emanating from an eternal and immutable being. If ever space had not been, God would then have been nowhere, and then he either created space later where he was not present himself, or else, and no less absurd, he created his own ubiquity. Moreover although we may perhaps imagine that there is nothing in space yet we cannot think space non-existent; just as we cannot think that there is no duration [of time], even though it might be possible to imagine that nothing at all endures. And this is made manifest by extra-mundane spaces, which (since we imagine the world to be finite) we cannot think to be non-existent, although they are neither revealed to us by God, nor

230

E X T R A C T S FROM MS. ADD. 4003

VI

become known through the senses, nor depend for their existence on intra-mundane spaces. But such spaces are usually believed to be noth­ ing. Yet truly they are spaces. Space even if it were empty of bodies yet would not be empty of itself. And [space] is something because there are spaces although there may be nothing else besides. Again it is necessary that spaces are not more space where the world is than where there is no world, unless perhaps you would say that when God created the world in this space he simultaneously created space itself, or that if God later annihilated the world in these spaces he were also to annihilate these spaces themselves. And so whatever is more real in one space than in another belongs to body and not to space, as will appear more clearly provided only that puerile prejudice derived from childhood is given up according to which extension inheres in a body as an accident in a subject without which it cannot actually exist. 8.22 However, since water opposes the motion of solid projectiles less than quicksilver, and air much less than water, and aetherial spaces that mueh less again than air-filled ones, if we reject all force of impeding the motion of projectiles we must surely also entirely reject any corpo­ real nature [in empty space]. For example, if subtle matter were deprived of all force of impeding the motion of globes, I would no longer believe it to be subtle matter but rather disseminated vacuum. And so if aerial or aetherial space were of such a kind that it yielded without any resistance to the motions of comets or bodies of any kind, then I would believe it to be entirely empty. For it is impossible that a corporeal fluid should not oppose the motions of projectiles, assuming (as I suppose) that it is not set in motion with the same speed (Part 2 Letter 96 to Mersenne).2 > But it is evident that all this force [of opposing the motion of projectiles] can be removed only if space and body differ from each other; and then the possibility of this removal must not be denied before they are proved not to differ, lest we commit a paralogism by positing the principle in question. But lest any doubt remain, from what has been said it follows that empty spaces exist in the natural world. For if aether were a fluid with­ out any empty pores, entirely corporeal, it would, however subtilised by division of parts, be equally dense as any other such fluid, and would yield not with less but with much more inertia to the motion of projectiles if these were porous; because then it would enter the intimate pores, and run against and impede not only the total external surface but also all the surfaces of the internal parts. But since the resistance of aether

VI

E X l ’R A C TS FROM MS. ADD. 4003

231

is on the contrary so small that compared to the resistance of quick­ silver it seems to be more than ten or one hundred times smaller, clearly by far the greatest part of aetherial space must be regarded as emptiness spread between aetherial corpusculcs. And the same may be conjectured besides from the divers heaviness of these fluids, which is as their den­ sities or the quantities of matter contained in equal spaces, as is shown both by the descent of heavy bodies and the vibration of pendulums.2+ But this is not the place to go into these matters. 9. Def. 5. Force is the causal principle of motion and rest,25 and is either something externaF^ which, impressed in a certain body, either generates or destroys its motion, or at least to some extent changes it; or it is the internal principle27 by which the motion or rest imprinted on the body is conserved, and by which every entity endeavours to persevere in its actual state, and opposes itself to any impediment. Def. 6. Conatus is an impeded force, or a force in so far as it is resisted.28 Def. 7. Impetus is force in so far as it is impressed on something else. Def. 8. Inertia2^ is the internal force of a body [ensuring] that its state is not easily changed by any external force. Def. 9. Pressure is the conatus of adjacent parts [endeavouring] mutually to penetrate their [respective] boundaries. For if they were able to penetrate [each other] the pressure would c e a s e . And it belongs to adjacent parts only, which [nevertheless] press all other adjacent parts, until the pressure is transmitted into the remotest parts of the given body whether it be hard, soft, or fluid. And communication of motion springs from this action of contact across a point or surface. Def. 10. Gravity is the intrinsic force imparted to a body inclining it to descend. By descent, however, is here to be understood not only the motion towards the centre of the earth, but also towards any other point or region, or also [motion] away from any point. For example, if the conatus of the ether revolving about the sun to recede from the centre is taken as its gravity, ether receding from the sun is said to descend.31 And, analogously, it is to be observed that a plane may be said to be horizontal because it is directly opposed to the determination of the gravity or conatus. Moreover the quantities of these powers such as motion, force, conatus, impetus, inertia, pressure and gravity may be estimated in two ways; according, namely, to their intensity or their extension. Def. II. The intensity of any of the aforementioned powers is the measure of its quantity.

232

E X T R A C T S FROM MS. ADD. 4003

VI

Def. 12, Its extension is the quantity of space or time in which it acts. Def. 13. And its absolute quantity is that compounded from the intensity and extension. For example, if the intensity of the quantity be 2, and the quantity of its extension 3, multiply one by the other and the absolute quantity is given by 6. Further it is proper to illustrate these definitions in the case of particu­ lar powers. Thus motion is more intense or remiss^^ according as a greater or lesser space is traversed in the same time, for which reason a body is usually said to be moved faster or slower. Motion is greater or less ex­ tended in so far as the body moved is greater or smaller, that is in so far as it is spread through a greater or a smaller body. And the absolute quantity of motion is that which is compounded from both the velocity and magni­ tude of the body moved. Force, endeavour, impetus, and inertia is more intense which is greater in the same or an equal body; is more extensive which is in a greater body; and its absolute quantity is that which derives from both. So the intensity of pressure is more when it consists of a greater pressure on the same quantity of surface, and is [more] extensive as it presses a greater surface, and the absolute quantity is that which results from the intensity of pressure and the quantity of the sur­ face pressed. And so again the intensity of gravity is greater in the body having a greater specific gravity, and its extent [is greater] in the larger body, and speaking absolutely the quantity of gravity is that which results from the specific gravity and the bulk of the gravitating body. And anyone who does not clearly distinguish these things must neces­ sarily fall into many errors concerning the science of mechanics. It is possible in addition that the quantities of these powers may be sometimes calculated in respect of the duration of intervals: in which case the absolute quantity will be that compounded from all of intensity, extension and duration. For instance if a body 2, with velocity 3 is moved for time 4, the total motion will be 2X3 X4, or 12^3 [^/c!]. Def, 14. Velocity is the intensity of motion, and retardation its remis­ sion. 1 . C o m m en c in g at the b egin n in g o f the m an uscript w h ich lacks a title. 3. C o m pare this and the three im m ed iately su cceed in g definitions w ith D e f. 7, 5, 8, 9, respectively, o f M S . X a.

Principia Philosophiae, d id

not

regard im pen etrability as an essential p ro perty o f bo dy. L eib n iz, likewise, d is­ agreed w ith D e sca rte s’s view o f im pen etrability as a secondary cjuality.

233

5. T h e significance o f these references is obscure. 6. A t p. 3. 7. T h e double theory o f m otion ad vanced b y D escartes in his

sophiae,

Principia Philo­

especially in A rt. 25, Part 2, and A rts. 28, 29, Part 3, o f w h ich N ew to n

has ju s t giv en a v e ry detailed and careful account. 8. N e w to n was p ro b ab ly unaware that D escartes had argued in

Le Monde

for the m otion o f the earth and the planets b u t had w ithdraw n the w o rk from publication on learning o f G a lile o ’s condem nation b y the In quisition. T h e fact that

Le Monde

was based on a single theory o f m otion w h ich resulted in the

m o vem en t o f the planets, whereas the

Principia

contained a double theory o f

m otion w h ich resulted in the planets b ein g really at rest, th o u gh th ey m o ved in the v u lga r sense, has usually b een taken as an indication that D escartes d e ­ velop ed his peculiar definition o f m otion in the true and philosophical sense in order to escape the fate o f G alileo , W h ile this no d o u b t supp lied the

original in ­

cen tive for D escartes to develop a theory o f proper m otion, insufficient attention seems to have been directed to other possible m otives. I f one com pares w ith the

Principia it is clear that the

Le Monde

form er can o n ly be regarded as a prelim inary

draft o f the latter, w ith its far m ore detailed developm en t o f D e sca rte s’s view s on m atter and m otion. In particular, D escartes’s identification o f m atter w ith extension in evita b ly led to an en tirely undifferentiated, featureless universe unless som e further factor w as in troduced to accoun t for the separateness and in d iv id u ality o f bodies. T h is elem ent could o n ly be supp lied b y m otion, and abso lute m otion at that, since m otion in the vulgar sense was ju st as subjective and relative as any o f the secondary qualities. T h a t this is w h at D escartes had in

Principia itself. In A rt. 23, P art I I (C ousin '’que toutela diversite des formes qiii s'y rencontrent depend du

m in d is m ade quite clear in the edition), he states

mouvement local'; in

A rt. 24 he shows that m otion in the vulgar sense is subjective,

‘qu'en mime temps elle se meat et ne se meut point'. F in ally, in A rt. 25, he puts forw ard a definition o f m otion w h ich enables us ‘lui attribuer une nature qui soil determinee'. 9. T h e centrifugal conatus accom p an yin g circular m otion p layed as central a role in D escartes’s vortex theory (see especially A rt. 140 o f P art 3 o f Principia Philosophiae) as centripetal force was ultim ately to p lay in N e w to n ’s ow n theory

that depen din g on our p oin t o f view w e can say o f a th in g

o f gravitation. 10. I t is striking how close this explanation o f the stability o f a circular orbit is to that advanced b y Borelli (see Borelli [i]). It is o n ly necessary to replace D e s ­ cartes’s endeavour to the centre due to the surrounding v ortex m aterial b y B o relli’s natural ten den cy tow ards the central b o d y. It is also n otew orth y that in D e f. 10 o f this present m an uscript N e w to n actually instances as a case o f gravity the

conatus

potential

o f ether m o vin g circularly about the sun aw ay from

the sun. 1 1 . T h e drift o f this and the su cceed in g argum ent against D escartes w ould

absence o f m otion cannot produce any real effect. conatus from the centre. T h is cannot, therefore, be

seem to b e :

B u t there is a real

effect— the

due to the absence

o f m otion in the philosophical sense. It could, how ever, be due to

2. N o such experim ental illustrations, how ever, are actually given.

4. A s opposed to D escartes, w ho, at least in his

E X T R A C T S FROM MS. ADD. 4003

VI

the presence o f

circular m otion, though D escartes o n ly takes this to b e present in the vu lgar sense. H en ce the latter (vulgar) m otion is to be preferred to the former, w h ich leads to a contradiction — the im proper m otion p ro ducin g the real effect. 12. A t p. 6, follow in g a series o f argum ents against D e sca rtes’s double theory of m otion. 13. T h e tw o consequences referred to were ( i) the im possibility o f true

234

E X T R A C T S FROM MS. ADD. 4003

VI

m otion in the interior o f a b o d y given that any interior part was always in co n ­ tact w ith the same im m ediate n eighb o u rin g parts, (2) the existence o f in n u m er­ able true m otions in a b o d y, as follow ed from the assertion (A rt. 25, P a rt 2,

Principia Philosophiae)

that the true m otion o f a b o d y w as shared b y all its

interior parts, even if these were in relative m otion am ong them selves. In the present extracts N e w to n seems o n ly to be referring to the second o f these tw o consequences. 14. C om pare this passage w ith N e w to n ’s use o f centrifugal force in D e f. 9 o f M S . X a for draw ing a distinction betw een relative and absolute m otion. See above, Part I, C h a p ter 3, for a discussion o f N e w to n ’s views on centrifugal

conatus. 15. A t p. 8. T h is extract has been given for its

implicit

reference to the

p rinciple o f inertia; the absence o f any explicit references to any o f the p urely dynam ical articles o f D esca rtes’s

Principia

is a curious feature o f a m an uscript

otherwise so w e ll furnished w ith detailed chapter and verse. It is p o ssib ly an indication o f h o w absolu tely N e w to n had set his face against D escartes at the tim e o f com posin g the present m anuscript. 16. A t p. 10. C o m in g at the end o f a w hole set o f argum ents against D e s ­ cartes’s double theory o f m otion, this passage is m em orable as contain in g N e w to n ’s first assertion o f the necessity o f som e absolute frame o f reference against w h ich to m easure the position and m otion o f bodies. 17. T h is result follow ed for N e w to n from the im possibility o f determ ining accurately the true position o f a b o d y in the Cartesian sense at an y previous instant o f tim e, itse lf due to the im possib ility o f determ ining the corresponding positions of all the surrounding particles. 18. T h e last sentence allows o f the possibility o f a continual belief b y N e w to n in the m echanism o f D esca rtes’s vortex theory; see, for exam ple, the reference to ‘ C artesiu s’s V o rtic e s’ in W h isto n ’s account (reproduced above, in Part I, C h a p ter 4, p. 65). 19. A t p. 1 1 . T h is passage is a good exam ple o f N e w to n ’s purely philosophical w ritin g. It is also interesting to com pare the v ie w here expressed o f space as

'Dei effectus emanativiis’

w ith N e w to n ’s later v ie w o f space as the sensorium o f

G o d , and w ith the view s p u t forw ard b y C larke in his controversy w ith L eib n iz. F o r a discussion o f N e w to n ’s view s on space in the ligh t o f this co ntroversy see A lexan d er [i], p p . x x x ii-x l. 20. A process o f abstraction strongly rem iniscent o f that adopted b y D e s ­ cartes in his

Principia Philosophiae.

2 1. A t p. 19. A good sum m ary o f the properties o f absolute space advanced in p recedin g sections. 22. A t p. 30. In tb e previous paragraph N e w to n had refused to allow the capacity o f bodies to excite perceptions in m ental beings as secondary, able to be stripped aw ay in the Cartesian process o f reduction to the essence. T o rem ove this facu lty, he said, w ou ld be as serious as to rem ove that other fa cu lty o f m utual interaction o f bodies, and this w o u ld reduce bodies to em p ty space. T h e present passage discusses the consequences o f such a reduction. I t should be com pared w ith a rather sim ilar passage in the I I I o f the tract

de Motu

Scholium

to Prob. 5 o f V ersion

(M S . IX c).

23. N e w to n m u st here be referring to vol. ii o f C lerselier’s edition o f D e s ­ cartes’s

Correspondence. A t

para. 5 o f letter 96 o f this volu m e D escartes explains

h o w he conceives o f a b o d y m o vin g w ith o u t resistance in a m edium w h en the velo city o f the latter is ex actly equal to that o f the bo dy.

E X T R A C T S

VI

FROM

MS.

ADD.

4 0 0 3

235

24. A t first sight this clear realization o f the distin ction b etw een mass and w eigh t w o u ld seem to go against a su pp osedly early date o f com position of the m anuscript. B u t the extract in § 5 o f M S . I points to N e w to n h a vin g m ade this distinction v e ry early, in all p ro b ab ility before the end o f 1664. 25. W hereas in M S . I I force w as the causal prin ciple o f m otion only, no m en ­ tion b ein g m ade o f rest. 26. R em in iscen t o f the ‘ external causes’ responsible for ch an gin g the inertial state o f a b o d y in the general p hilosophical prin ciple (A rt. 37, Part 2,

Philosophiae)

Principia

from w h ich D escartes derived the p rinciple o f inertia.

27. C o m pare this wdth the

vis corporis said

to be responsible for a b o d y ’s en ­

deavour to continue in its state o f m otion in D e f. 2 o f V ersion I o f the tract

Motu

de

(M S . IX a).

28. T h u s m aking it clear that for N e w to n at the tim e o f com posin g this work

conatus, in clu d in g the conatus due

to circular m otion, was a real species o f p oten ­

tial force. See above, Part I, C h a p ter 3, for a discussion o f N e w to n ’s concept o f

conatus. 29. T h e use o f inertia as an alternative for the internal force, or principle, o f a b o d y responsible for m aintaining its state o f m otion or rest is interesting. I t is n o t used in M S . I X and first appears in M S . X l b , D e f. 3. 30. T h e

conatus w o u ld

then beco m e actual.

3 1 . T h is exam ple reinforces the im pression that N e w to n believed at this tim e in the real existence o f cen trifugal

conatus.

See above. Part I, C h a p ter 3.

32. T h e use o f these term s, and the previous definitions o f in ten sity or e x ­ tension, p ro vid e an indication o f the possible influence on N e w to n o f the late scholastic doctrine o f the intension and remission o f form s, especially in its application to kinetics b y the M e rto n C o llege S ch ool. See M a ie r [i] for an illu m in atin g discussion o f this question. F o r a discussion o f the contribution o f m edieval p h iloso ph y to the birth o f dynam ics see C la g e tt [i]. Chapters i i and 12. 33. A similar notion is that of the at A x .-P r o p . 22.

total force in a given time found

in M S . I l d

VII E X T R A C T S FROM C O R R E S P O N D E N C E PRIOR TO 1684 T h e passages following appear to be the only important discussions of dynamical topics in extant Newton letters prior to 1685. This is rather surprising, given what we know of Newton’s earliest researches in dynamics, and is in marked contrast to the fairly large number of mathe­ matical discussions in the same period. But on examining these latter we find that most if not all of them arose as the result of some definite query, which queries in turn were occasioned by the widespread knowledge of, and interest in, mathematics, as in the case of Collins. Dynamics, on the other hand, was a much more esoteric subject a knowledge of which was restricted almost entirely to a very small number of original thinkers, so that it is not surprising that Newton had seldom occasion to discuss it in letters with his correspondents. In any case, given his extreme (and understandable) secrecy in regard to a subject in which he may possibly have sensed he was to make his sup­ reme contribution to knowledge, it is rather doubtful if he would have gone any further in revealing his actual discoveries in dynamics than in the first paragraph of the letter of 23 June 1673 Oldenburg. The omission of this paragraph from the version later forwarded by Olden­ burg to Huygens, if not actually indicative of this secrecy, at least provides a memorable symbol of it.

537

CORRESPONDENCE PRIOR TO 1684

VII

that the greatest distance of the sun from the earth is to the greatest distance of the Moon from the earth, not greater than 10000 to 56 and therefore the parallax of the Sun not less than Parallax of the M oon; Because were the sun’s distance less in proportion to that of the Moon, she would have a greater conatus from the sun than from the earth. I thought also sometime that the moons libration might depend upon her conatus from the Sun and Earth compared together, till I apprehended a better cause. 1. C f. Correspondence, vol. i, p . 290. T h is letter p layed an im portant role in the N e w to n -H a lle y correspondence o f 1686 arising out o f H o o k e’s claim s regarding the inverse square law. 2. N e w to n ’s presentation co p y o f H u y g e n s ’s H orologium O scilla to r him . 3. T h e rem ainder o f the extract was un acco u n tably om itted from the co py o f the letter forw arded b y O ld e n b u r g to H u ygen s. T h e w h ole passage was reproduced b y N e w to n in his letter o f 27 Ju ly 1686 to H alley. In a previous letter (14 July 1686) he had claim ed that the argum ents p u t forw ard in this paragraph had been taken from an earlier paper ‘w rit 18 or ig years a go ’ . T h e re can be no do u bt that the paper in question is that reproduced in M S . IV a . F o r the possible bearing o f the present passage on N e w to n ’s view s on centrifugal force, and on the p roblem o f the test o f the inverse square law o f gravitation during the Plague Y ears, see above, Part I, C h a p ter 3, p . 56, and C h a p ter 4, p. 7 1 , respectively.

V llb E

x t r a c t

C

o l l in s

'

from

N

e w t o n ’s

L

etter

of

20 J

une

1674

to

Sr V ila E

x t r a c t

O

l d e n b u r g

’'

f r o m

N

e w t o n

’

s

L

e tt e r

of

23 J

u n e

1673

to

Sr I received your letters with M. Hugens kind present, ^ which I have viewed with great satisfaction, finding it full of very subtile and usefull speculations very worthy of the Author. 13 am glad that we are to expect another discours of the vis centrifuga, which speculation may prove of good use in naturall Philosophy and Astronomy as well as mechanicks. Thus for instance if the reason why the same side of the Moon is ever towards the earth be the greater conatus of the other side to recede from it; it will follow (upon supposition of the Earths motion about the Sun)

I thank you for your kind present.^ Mr Andersons book is very in­ genious, and may prove as usefull if his principles be true. But I suspect one of them, namely that the bullet moves in a Parabola. 3 This would be so indeed were the horizontal celerity of the bullet uniform, but I should think its motion decays considerably in the flight. Suppose for instance a bullet shot horizontally from A moves in the line AE, and A I being perpendicular to the horizon in it take A F, AG , AH , A I etc in proportion as the square numbers i, 4, 9, 16 etc: and its certain that if in one moment of time the bullet descend as low as F, in the next moment it shall descend as low as G, in the third as low as H etc. And therefore drawing the horizontal! lines FB, GC, HD, 1 E\ the bullet at the end of the first moment will be somewhere in the line F B suppose at B, and at the end of the second moment it will be somewhere in the line

238

CORRESPONDENCE PRIOR TO 1684

VII

G C suppose at C etc. But that FB, GC, HD and IE are in Arith­ metical! progression (which is the condition of the Parabola) seems not probable; for if it were so, the celerity of the bullet would increas becaus the spaces A B , BC, CD, D E described in equall times are the latter bigger than the former; whereas I should rather think that the celerity decreases very considerably. And perhaps this rule for its de­ creasing may pretty nearly approach the truth, v iz: Letting fall the per­ pendiculars B K , CL, DM, etc to make IK , K L , L M etc, a decreasing Geometricall progression. If you should have occasion to speak of this to the Author, I desire you would not mention me becaus I have no mind to concern my self further about it.

VII

CORRESPONDENCE PRIOR TO 1684

239

once a day about its center C from west to east according to the order of the letters BDG\ and let A be a heavy body suspended in the Air and moving round with the earth so as perpetually to hang over the same point thereof B. Then imagin this body A let fall and it’s gravity will give it a new motion towards the center of the Earth without diminishing the old one from west to east. Whence the motion of this body from west to east, by reason that before it fell it was more distant from the center of

1. C f. Correspondence, vol. i, p. 309. 2. C o p ies o f the same book were sent b y C o llin s to James G r e g o r y and W a llis. T h e form er gave his opinion o f the w ork in a letter to C o llin s o f 8 O cto b e r 1674 (see R ig au d [i], vol. ii, letter 215). G r e g o r y fo u n d A nderson [and StreeteJ’s book ‘ p itifu l s tu ff’ . L ik e N e w to n he criticized the assum ption that the horizontal m otion w o u ld be uniform . W allis, in his letter o f 24 A u g u s t 1674 to C o llin s (see R ig au d [i], vol. ii, letter 339), su ggested that A nd erso n m igh t have taken his principles from Prop. 8, C ap . 10, o f W a llis’s o w n M ech a n ica . H e also criticized the assum ption o f the u n iform ity o f the horizontal m otion in practice as opposed to theory, referring to the S ch o liu m to the above proposition in w h ich the effect o f the resistance o f the air was noted. 3. It is curious that now here in an y o f the extant dyn am ical m anuscripts does N e w to n discuss the ideal, parabolic p ath o f a projectile. T h e first reference to G a lile o ’s deduction o f the result occurs in M S . X a , L a w 2, and is repeated at the b egin n in g o f the S ch o liu m to the laws o f m otion in the P rin cip ia . I t w o u ld be interesting to kn ow h o w N e w to n first learnt o f the parabolic path, w h eth er d irectly from the translation o f the D iscorsi in S a lu sb u ry’s translation, or in ­ directly, perhaps throu gh T o rricelli [i]. Likew ise, w h ether he had th o u gh t o f the effect o f air resistance him self, or was p ro m pted to do so b y the discussion o f this and other p ertu rb in g factors in G alileo [D iscorsi, ed. N a z., p. 274].

VIIc E

x t r a c t

TO H

^

f r o m

N

e w t o n ’s

L

e tt e r

of 2 8 N

ov e m be r

1679

ooke

I am glad to heare that so considerable a discovery as you made of the earth’s annual^ parallax is seconded by Mr Flamstead’s Observations. In requital of this advertisement I shall communicate to you a fansy of my own about discovering the earth’s diurnal motion. In order thereto I will consider the Earth’s diurnal motion alone without the annual, that having little influence on the experimt I shall here propound. Sup­ pose then BD G [Fig. i] represents the Globe of the Earth carried round

the earth then the parts of the earth at which it arrives in its fall, will be greater then the motion from west to east of the parts of the earth at which the body arrives in it’s fall: and therefore it will not descend in the perpendicular A C , but outrunning the parts of the earth will shoot forward to the east side of the perpendicular describing in it’s fall a spiral line AD EC, quite contrary to the opinion of the vulgar who think that if the earth moved, heavy bodies in falling would be outrun by its parts and fall on the west side of the perpendicular. The advance of the body from the perpendicular eastward will in a descent of but 20 or 30 yards be very small and yet I am apt to think it may be enough to determin the matter of fact. Suppose then in a very calm day a Pistol Bullet were let down by a silk line from the top of a high Building or Well, the line going through a small hole made in a plate of Brass or 'Einn fastened to

240

CORRESPONDENCE PRIOR TO 1684

VII

the top of the Building or Well and that the bullet when let down al­ most to the bottom were setled in water so as to cease from swinging and then let down further on an edge of steel lying north and south to try if the bullet in setling thereon will almost stand in aequilihrio but yet with some small propensity (the smaller the better) decline to the west side of the steel as often as it is so let down thereon. The steel being so placed underneath, suppose the bullet be then drawn up to the top and let fall by cutting clipping or burning the line of silk, and if it fall con­ stantly on the east side of the steel it will argue the diurnall motion of the earth. But what the event will be I know not having never attempted to try it. If any body may think this worth their triall the best way in my opinion would be to try it in a high church or wide steeple the windows being first well stopt. For in a narrow well the bullet possibly may be apt to receive a ply from the straitned Air neare the sides of the Well, if in its fall it come nearer to one side then to another. It would be convenient also that the water into which the bullet falls be a yard or two deep or more partly that the bullet may fall more gently on the steel, partly that the motion which it has from west to east at its entring into the water by meanes of the longer time of descent through the water, carry it on further eastward and make the experiment more manifest.

VII

CORRESPONDENCE PRIOR TO 1684

241

least if the falling Body were supposed in the plaine of the equinoxciale supposing then the earth were cast into two half globes in the plaine of the equinox and those sides separated at a yard Distance or the like to make Vacuity for the Desending Body and that the gravitation to the former Center remained as before and that the globe of the earth were supposed to move with a Diurnall motion on its axis and that the falling body had the motion of the superficial! parts of the earth from whence it

\E

1. C f. Correspondence, vol. ii, p p . 3 0 1 -2 . F o r an illum in atin g account o f the w hole N e w to n -H o o k e correspondence o f the w in ter 16 79 -8 0 see K o y r e [2]. 2. T h e r e can be su rely no d o u b t that it was H o o k e ’s reference to his ow n su p ­ posed discovery o f the parallax due to the earth’s an m ial m otion w h ich p ro m pted N e w to n ’s su ggested experim ent for the d isco very o f the diurnal m o tio n . N ew to n , o f course, had lon g before (M S . IV a) learnt h o w to calculate the centrifugal force due to the E a r th ’s rotation, or the 'force o f ascent at the equator, arising from the earth’s diurnal m o tio n ’ as he expressed it in his letter o f 14 J u ly 1686 to H alley. T h a t it required an external stim ulus to persuade him to consider the m atter again is perhaps an indication o f a certain lack o f interest in dyn am ics in the p eriod 1667 to 1679.

V lld E

x t r a c t

TO N

^

f r o m

H

o o k e ’s

L

etter

of

9 D

e cem be r

1679

e w t o n

But as to the curve Line which you seem to suppose it to Descend by (though that was not then at all Discoursed of) Vizt a kind of spirall which after sume few revolutions Leave it in the Center of the Earth my theory of circular motion^ makes me suppose it would be very differ­ ing and nothing att all akin to a spirall but rather a kind Elleptueid. At

F igu re 2.

was Let fall Impressed on it, I conceive the line in which this body would move would resembles An Elleipse:for Instance Let A B D E [Fig. 2] represent the plaine of the equinox limited by the superficies of the earth; C the Center therof to which the lines of Gravitation doe all tend. Let A represent the heavy Body let fall at A and attracted towards C but Moved also by the Diurnall Revolution of the earth from A towards BD E etc. I conceive the curve that will be described by this desending body A will be A F G H and that the body A would never approach neerer the Center C then G were it not for the Impediment of the medium as Air or the like but would continually proceed to move round in the Line A F G H A F G etc. But w[h]ere the Medium through which it moves has a power of impeding and destroying its motion the curve in which it would move would be some what like the Line A IK L M N O P etc and after many resolutions would terminate in the Center C. But if the Body litt [5/c] fall be not in the aquinochill plain as here at London Ja 51°. 32' the elleipsed will be made in a plain inclined to the plaine of the Equinox’

242

CORRESPONDENCE PRIOR TO 1684

VII

51.32 soe that the fall of the Ball will not be exactly east of the per­ pendicular'^ but South East and indeed more to the south then the east;s as lett N L O S [Fig. 3] represent the Meridian of London and O the equinox L London and P L the parrallel in which it moves about the Axis N S : the body let fall at L would desend in the plaine L C supposed at Right angles with the plaine of that Meridion N LQ SR and not in the superficies of the cone P L C whose apex is C the Center of the earth

V II

C O R R E SPO N D E N C E

PRIOR

TO

1 6 8 4

243

3. T h is w ould seem to rule out the p ossib ility o f the p ath actu ally bein g an ellipse. 4. T h e p erpendicular in question w'ould have been assum ed b y H ooke to rotate w ith the earth. 5. H o o k e ’s reason for this assum ption is n o t clear. N ew to n , how ever, agreed w ith it subject to the qualification that the b o d y was let fall from som e consider­ able h e igh t above the earth.

V ile E xtract' from

N e w t o n ’s L e t t e r

of

13

D ecember

1 67 9

TO PIOOKE

Sr I agree with you that the body in our latitude will fall more to the south then east if the height it falls from be any thing great. And also

Q

s Figure 3.

and whose base is the plaine of the parrallel circle PL. I could adde many other conciderations which are consonant to my Theory of Circular motions compounded by a Direct motion and an attractive one to a Center. But I feare I have already trespassed to much upon your more Usefull thoughts with these my impertinants yet I would desire you not to look upon them as any provacations to alter your mind [to] more mature and serious Resolutions. Goe on and Prosper and if you succed and by any Freind Let me understand what you think fit to impart, any thing from you will be Extremly Valued by Sr Your very Humble Sarvant R O ; HOOKE 1. C f. Correspondence, vol. ii, pp. 305-6. 2. U n fo rtu n ately for H ooke he was n ever able to p ut the theory on a firm, quantitative basis, as was m ade evident b y his failure to w in the book prom ised b y W ren to H a lle y and h im self for a p ro o f o f the inverse square law' for m otion in an ellipse. See H a lle y ’s letter o f 29 June 1686 to N ew to n .

that if its gravity be supposed uniform it will not descend in a spiral to the very center but circulate with an alternate ascent and descent made by it’s vis centrifiiga and gravity alternately overballancing one another. Yet I imagin the body will not describe an Ellipsoeid but rather such a figure as is represented by A F O G H IK L etc. [Fig. 4] Suppose A the body, C the center of the earth, A B D E quartered with perpendicular diameters AD , BE, which cut the said curve in F and G ; A M the tangent in which the body moved before it began to fall and GA^ a line drawn parallel to that tangent. When the body descending through the earth

244

CORRESPONDENCE PRIOR TO 1684

VII

(supposed pervious) arrives at G, the determination of its motion shall not be towards N but towards the coast between A^and D. For the motion of the body at G is compounded of the motion it had at A towards M and of all the innumerable converging motions successively generated by the impresses of gravity in every moment of its passage from A to G: The motion from A t o M being in a parallel to G N inclines not the body to verge from the line GN. The innumerable and infinitly little motions (for I here consider motion according to the method of indivisibles) continually generated by gravity in its passage from A to F incline it to verge from G N towards D, and the like motions generated in its passage from F to G incline it to verge from G N towards C. But these motions are proportional to the time they are generated in, and the time of passing from A to F (by reason of the longer journey and slower motion) is greater then the time of passing from F to G. And therefore the motions gene­ rated in A F shall exceed those generated in F G and so make the body verge from G N to some coast between N and D. The nearest approach therefore of the body to the center is not at G but somewhere between G and F as at O. And indeed the point O, according to the various propor­ tions of gravity to the impetus of the body at A towards M, may fall any where in the angle B C D in a certain curve which touches the line BG at C and passes thence to D. Thus I conceive it would be if gravity were the same at all distances from the center. But if it be supposed greater nearer the center the point O may fall in the line CD or in the angle D C F or in other angles that follow, or even no where. For the increase of gravity in the descent may be supposed such that the body shall by an infinite number of spiral revolutions descend continually till it cross the center by motion transcendently swift. Your acute Letter having put me upon considering thus far the species of this curve, I might add something about its description by points qiiam proxime. But the thing being of no great moment I rather beg your pardon for having troubled you thus far with this second scribble wherin if you meet with any thing inept or erroneous I hope you will pardon the former and the latter I submit and leave to your correction remaining Sr Your very humble Servant IS. NEWTON. I.

C f. Correspondence, vol. ii, p p . 3 0 7-8 . It is chiefly m em orable for the

eviden ce it provides o f the u n developed state o f N e w to n ’s thought on the p ro b ­ lem at w h at was p resu m ably o n ly a short tim e before he arrived at a definitive

VII

CORRESPONDENCE PRIOR TO 1684

245

solution. T h e r e is, for exam ple, no firm indication th at he had actually attem pted a quantitative solution. In this connexion see Pelseneer [i]. H o w ever, N e w to n ’s assertion that he m igh t have added som eth ing to the p roblem ‘ b y poin ts quam

p roxim e' could be interpreted as an in dication that his th o u gh t on the problem had advanced further than he w as prepared to adm it to H ooke.

VllI

VIII T H E K E P L E R - M O T I O N PAPERS T here are two closely similar versions of the present manuscript. One in Newton’s hand,i of which the text is reproduced here, and another among the Locke papers- bearing the title: ‘A Demonstration!that the Planets by their gravity towardsj the sun may move in Ellipses' The two versions will be termed the Newton and Locke copies, respectively. The Locke copy is not in Newton’s hand but according to Cranston^ in that of Locke’s valet and amenuensis Brownover. Given that the Locke copy is endorsed in Locke’s hand ‘Mr. Newi:on Mar 1689/90’ it is natural to assume that the original of both copies was composed after the Principia especially for Locke’s benefit, most pro­ bably shortly before March 1689/90. Nevertheless for the reasons given above in Part I, Chapter 6.6, it is possible that the original was composed before the Principia^ and in that case most probably in 1679 or 1684. The normal procedure has been followed of dividing footnotes into two sets, numerical and alphabetical, the non-trivial variations between the two copies being included in the latter set. Trivial variations are largely due to the erratic use of ‘contractions’ and ‘expansions’ in the two copies. T he

N ew ton

THE KE PLE R- M OT IO N PAPERS

247

Let A [Fig. 2] be the center towards which the body is attracted, and suppose the attraction acts not continually but by discontinued impres­ sions® made at equal intervalls of time which intervalls we will consider as physical moments. Let B C be the right line in which it begins to move ffrom and which it describes with uniform motion in the first physical moment before the attraction make its first impression upon it B

D

C opy

Hypoth. 1.4 Bodies move uniformly in straight lines unless so far as they are retarded by the resistance of the Medium or disturbed by some other force. Hyp. 2.5 The alteration of motion is ever"* proportional to the force by which it is altered. Figure 2.

Hyp. 3.^ Motions imprest^ in two different lines, if those lines be taken in proportion to the motions and completed into a parallelogram, com­ pose a motion whereby the diagonal of the Parallelogram shall be de­ scribed in the same time in which the sides thereof would have been described by those compounding motions apart. The motions^/? [hig. i] and A C compound the motion AD Prop, id If a body move in vacuo and be continually attracted toward[s] an immoveable center, it shall constantly move in one and the same plane,^ and in that plane‘s describe equal areas in equall times.

At C let it be attracted towards the center A by one impuls Ror impres­ sion of force, and let CD be the line in which it shall move after that impuls. Produce B C to I so that C l be equall to B C and draw ID parallel to C A and the point D in which it cuts CD shall be the place of the body at the end of the second moments. And because the bases B C C l of the triangles A B C , A C I are equal those two triangles shall be equal. Also because the triangles A C I, A C D stand upon the same base A C and be­ tween two parallels^' they shall be equall. And therefore the triangle A C D described in the second moment shall be equal to the triangle A B C described in the first moment. And by the same reason if the body at

248

THE K EP LE R -M OT IO N PAPERS

VI] I

the end of the 2^, 3^, 4^^, 5*^ and following moments be attracted 'by single impulses' in Z), E, F, Gi etc. describing the line D E in the 3^^ moment, E F in the 4^*', F G in the 5*^ etc: the triangle AED shall be equall to the triangle A D C and all the following triangles ^AFE, A G F etc. to the preceding ones and^^ to one another. And by consequence the areas compounded of these equall triangles (as ABE, AEG, A B G etc.) are to one another as thetimes^ in which they are described. Suppose now that the moments of time be diminished in length and encreased in

VIII

THE KE P L E R -M O T I O N PAPERS

249

are tangents to the Ellipsis at its two ends A and C and that E M and D N are perpendiculars let fall from the points E and D upon those tangents: and because the Ellipsis is alike crooked at both ends those perpendiculars E M and D N will be to one another as the squares of the arches A E and CD, and therefore E M is to D N as FC^ to F A ‘^. Now in the times that the body by means of the attraction moves in the arches A E and CD from A to E and from C to D it would without attraction move in the tangents from A to M and from C to N. Tis by the force of

number in infinitum, so that the impulses or impressions of the attrac­ tion may become continuall and that the line BCD E FG by the infinite number and infinite littleness of its sides BC, CD, D E etc. may become a curve one: and the body by the continual attraction shall describe areas of this Curve ABE, AEG, A B G etc. proportionall to the times in which they are described. '"W. W. to be Dem.”' Prop. 2." If a body be attracted towards either focus of an Ellipsis and the quantity of the attraction be such as suffices to make the body revolve in the circumference of the Ellipsis; the attraction at the two ends of the Ellipsis shall be reciprocally as the squares of the body in those ends from that focus. Let A E C D [Fig. 3] be the Ellipsis, A, C its two ends or vertices, E' that focus towards which the body is attracted, and AF'E, CFD areas which the body with a ray drawn from that focus to its center, describes at both ends in equal times: and those areas by the foregoing Proposi­ tion must be equal because proportionall to the times: that is the rect­ angle ^ A F x A E and ^ F C x D C must be equal supposing the arches A E and CD to be so very short that they may be taken for right lines and therefore A E is to CD as F C to E'A. Suppose now that A M and CiV

the attractions that the bodies are drawn out of the tangents from M to E and from N to D and therefore the attractions are as those distances M E and ND, that is the attraction at the end of the Ellipsis A is to the attraction at the other end of the Ellipsis C as M E to N D and by conse­ quence as FC*? to FA^. W. W. to be dem. Lemma, i. If a right line touch an Ellipsis in any point thereof and parallel to that tangent be drawn another right line from the center of the Ellipsis which shall intersect a third right line drawn from the touch point through either focus of the Ellipsis: the segment of the last named right line lying between the point of intersection and the point of contact shall be equal to half the long axis of the Ellipsis. Let A P B O [Fig. 4] be the Ellipsis; A B its long axis; C its center; F, / its ffoci; P the point of contact; PR the tangent; CD the line parallel to

250

TH E

K E P L E R -M O T IO N

P A P E R S

VIII

the tangent, and PD the segment of the line FP. I say that this segment shall be equal to AC."^ For joyn i y i ’ and draw/F parallel to CD and because 'is‘ bisected in C,® F E shall be bisected in D and therefore 2^PD shall be equal to^‘ the summ of P F and P E that is to" the summ of P F and Pf, that is to A B and therefore PD shall be equal to AC.^ W.W. to be Dem. Lemma, 2, Every line drawn through either Focus of any Ellipsis and terminated at both ends by the Ellipsis is to that diameter of the Ellipsis which is parallel to this line as the same Diameter is to the long Axis of the Ellipsis. Let A P B O [Fig. 4] be the Ellipsis, A B its long Axis, F ,f its foci, C its center, P O the line drawn through its focus F, and V C S its diameter parallel to PO and P Q will be to V S as V S to AB. For draw fp parallel to Q F P and cutting the Ellipsis in p. Joyn Pp cutting V S in T and draw PR which shall touch the Ellipsis in P and cut the diameter V S produced in R and C T will be to C S as C S to CR, ^as has been shewed by all those who treat of the conic sections'^. But C T is the semisumm of F P and fp that is of F P and F O and there­ fore 7.CT is equal to PO. Also z C S is equal to V S and (by the fore­ going Lemma) zCR is equal to A B . Wherefore PO is to V S as V S to A B . W.W. to be Dem. Corol. A B x P Q ^ VS^^ = ^CS‘J

VIII

THE KE PLE R- M OT IO N PAPERS

251

gent, X Y the subtense'^' produced to the other side of the Ellipsis and Y Z the distance between this subtense and the first line. I say that the rectangle Y X I is to the rectangle A B x P O as YZ'^ to KL^. For let FA be the diameter of the Ellipsis parallel to the first line P F and G H another diameter parallel to the tangent PX , and the rectangle Y X I ^will be to the square of the tangent PX^^ as the rectangle A C F to the rectangle G C H that is as AF'^ to GH^. This a property of the Ellipsis demonstrated by all that write of the conic sections. And they have also demonstrated that all the Parallelograms circumscribedy about an Ellipsis

Lem. 3. If from either focus of any Ellipsis unto any point in the perimeter of the Ellipsis be drawn a right line and another right line doth touch the Ellipsis in that point and the angle of contact be subtended by any third right line drawn parallel to the first line; the rectangle which that subtense conteins with the same subtense produced to the other side of the Ellipsis is to the rectangle which the long Axis of the Ellipsis conteins with the first line produced to the other side of the Ellipsis as the square of the distance between the subtense and the first line is to the square of the short Axis of the Ellipsis. Let A K B L [Fig. 5] be the Ellipsis, A B its long Axis, K L its short Axis, C its center, F ,f its foci, P the point of the perimeter, P F the first line P O that line produced to the other side of the Ellipsis P X * the tan* T h is letter and V , Z written in capitals in the text but in lower case on Fig. 5.

are equall. Whence the rectangle z P E x G H is equal to the rectangle A B X K L and consequently GH is to K L as A B that is (by Lem. i) 2PD to 2PE and ^in the same proportion is P X to YZ^. Whence P X is to G H as Y Z to K L and PX^ to GH^ as YZ'? to K L ‘h^^ But^‘ Y X I was to as SV^ that is (by Cor. Lem. 2) A B x P O to GH^, whence invertedly Y X I is to A B x P O as PX'> to GH‘>and by consequence as Y Z ‘' to K D l W. W. to be Dem."' Prop. I l l If a body be attracted towards either focus of any Ellipsis and by that attraction be made to revolve in the Perimeter of the Ellipsis: the attraction shall be reciprocally as the square of the distance of the body from that focus of the Ellipsis.

252

THE K E P L E R -M O T I O N PA PERS

VIII

Let P [Fig. 6] be the place of the body in the Ellipsis at any moment of time and P X the tangent in which the body would move uniformly were it not attracted and X the place in that tangent at which it would arrive in any given part of time and Y the place in the perimeter of the Ellipsis at which the body doth arrive in the same time by means of the attraction.9 Let us suppose the time to be divided into equal parts and that those parts are very little ones so that they may be considered as physical moments and that the attraction acts not continually but by

VIII

THE K E P LE R -M O T I O N PAPERS

253

Axis of the Ellipsis. And by the third Lemma Y X I will be to A B X PQ as FZ*^ to KL^ and by consequence Y X will be equall to A B x P O x YZ^i XIxKL
AB xpqx xi X KL^

Lb that is as^^^ YZ^ to ^ ysfswadXI XI And because the lines P Y py are by the revolving body described in equal times, the areas of the triangles P Y F pyF must be equal by the first Proposition; and therefore the rectangles P F x Y Z an6.pFxyz are PQ equal, and by consequence Y Z is to as ^P to PP. Whence Y Z ‘i JCl is to ^ yziitiad. as XI

intervalls once in the beginning of every physical moment and let the first action be upon the body in P, the next upon it in Y and so on per­ petually, so that the body may move from P to F in the chord of the arch P Y and from Y to its next place in the Ellipsis in the chord of the next arch and so on for ever.^^ And because the attraction in P is made towards F and diverts the body from the tangent P X into the chord P F ” so that in the end of the first physical moment it be not found in the place X where it would have been without the attraction but in F being by the force of the attraction in P translated from X to F : the line X Y gene­ rated by the force of the attraction in P must be proportional to that force and parallel to its direction that is parallel to PF.^'’ Produce X Y and P P till they cut the Ellipsis in I and O. Joyn F Y and upon F P let fall the perpendicular Y Z and let A B be the long Axis and K L the short

as

p p q iia d .

ppquad. A .1

M ppquad.^

therefore Y X is to yx

XI

p p q u a d .^

XI ^ xi And as we told you that X Y was the line generated in a physical moment of time by the force of the attraction in P, so for the same reason is xy the line generated in the same quantity of time by the force of the attraction in p. And therefore the attraction in P is to the attraction in p as the line X Y to the line xy, that is a s^ ^ ppiuad.^Q^ ppquad.^ XI xt Suppose now that the equal times in which the revolving body describes the lines P F and py become infinitely little, so that the attrac­ tion may become continual and the body by this attraction revolve in the perimeter of the Ellipsis and the lines PO, X I as also pq, xi bePO coming coincident and by consequence equal, the quantities - y ppiund. JCJ

254

THE K E P LE R -M O T I O N PA PERS

VIII

VIII

THE KE P LE R- M OT IO N PAPERS

and ~ ppmiadxt

become

and

And therefore the attrac-

tion in P will be to the attraction in p as pF'^ to PF^, that is reciprocally as the squares of the distances of the revolving bodies from the focus of the Ellipsis. W. W. to be Dem. a. Absent from Locke copy. b. Succeeded by by forces in Locke copy. c. Succeeded by which a right line drawn continually from its own center to the immoveable center of attraction in the Locke copy. d. The Locke copy has lines. e. Succeeded by or impulses in Locke copy. f-f. Absent from Locke copy. g-g. Replaced in the Locke copy by Produce BC to I so that C l be equal to BC. In CA take CR in such proportion to C l as the motion which the impulse alone would have begotten hath to the motion of the body before the impulse was imprest. And because these two motions apart would in the second moment of time have carried the body the one to I by reason of the Equality of C l and B C and the other to R by reason of the aforesaid: proportion: complete the parallelogram ICRD and they shall both togeather in the same time of that second moment carry it in the Diagonal of that Parallelogram to D by Hypoth. 3. h. The Locke copy has parallel lines A C and DI. i-i. The Locke copy has again by single impulses successively, j . Succeeded in the Locke copy by H referring to an additional point in the dia­ gram of the Locke copy compared tvith the Nezoton copy. k-k. Absent from Locke copy. 1. Replacing lines cancelled in Newton copy. m-m. Written out in full in Locke copy. n. This proposition is not found in the Locke copy. o. The Locke copy has CB. p. Replacing P F deleted. q-q. In the Locke copy this passage reads and FE are bisected in C and D, PD shall be Equall (to half the Summ of P f and PE, that is to half the summ of P F and Pf, that is to half A B that is) to CB. r-r. Replacing and Fe are deleted. s. Succeeded by and D deleted. t. Probably inserted. u. u. Succeeded by half deleted. v-v. This passage absent from Locke copy. w. Succeeded by X I the same subtense in the Locke copy. x-x. Written Shall be to PX^ the square of the tangent in Locke copy. y. Replacing subscribed cancelled in Locke copy. z-z. The Locke copy has by consequence as P X to YZ. ai-aj. The Locke copy has But P X ‘ was to GH'‘ as Y X I to S V ‘ and SF'^ (by Cor. Lem. 2) is equal to x P 0 , and therefore Y X I to A B x PQ as YZ'' to K L ''. W.W. to be Dem. bj. Succeeded by PX'' was to GH'' as deleted, as in the Locke copy. Cx. Succeeded by as is manifest by the third Hypothesis in Locke copy.

The Locke copy continues

255

P Y , py are b y the rev o lv ­ ing b o d y described in equal tim esf the areas o f the triangles P YF, pyF m ust be equall b y the first Proposition and therefore the rectangels P F x YZ and P F x yz are E q u a l; and pF is to P F as YZ to yz and pF‘^'""^ to as YZ'"^""^ to di.

N o w because the lines

y^quad.^ and (if yo u m u ltip ly the antecedents alike and the consequents alike) ppm ad ^ ^ H p p q u a d as

yz'"''"^ that

YZ'inad

XI XI XI ABxpqXyz'"'-'"^ . , ■------------------------ that IS as YX xt X KL'^ attraction i n p b y H y p o th : 2 and

to

yx

is « as

^

to

X IX KL'^ • • D

therefore as the attraction in P

1

to the

3.

Sup pose n o w that the E q u al tim es in w h ich the revolvin g b o d y describes the lines

PY

py

and

beco m e infinitely litle so that the attraction m a y becom e

continual and the b o d y b y this attraction revolve in the perim eter o f the E llipsis and the lines

PQ, X I

as also

pq, xi

beco m in g coincident, and b y consequence

pF" and^ PF '' w ill becom e pF'^ and PF'', and therefore A j XI w ill be to the attraction in p as pF" to PF'' that is reciprocally

equal the quantities the attraction in

P

as the squares o f the distances o f the revolvin g b o d y from that focus o f the E lip sis tow ards w h ich the attraction is directed. W h ic h was to be dem onstrated. “ by multiplying the terms of the ratio b y A B and dividing them by K L ''. b by the conclusions of the two last paragraphs. 1. M S . A d d . 3965(1). 2. M S . L o ck e C 3 1, f. 1 0 1 -4 , B odleian L ib ra ry, O xfo rd . 3. See Cran ston [i], p. 337. 4. C o m pare this definition o f the prin ciple o f inertia w ith that in D e f. i o f V . I o f the tract

de Motu.

5. C o m pare this w ith the enunciation o f the same law in V . H I o f the tract

Motu.

de

N o tice the absence o f any reference to a state o f rest.

6. A m ore prim itive en unciation o f this ‘parallelogram la w ’ is given in H y p . 3

de Motu. O n the other hand, a com plete given in § 3 o f M S . V . C o rrespo n din g to Prop, i o f the tract de Motu.

o f V . I o f the tract

en unciation o f the

same law has already been 7.

8. T h e r e is no m ention o f this fact in any o f the enunciations o f the corre­ spo n d in g propositions in the various versions o f the tract

de Motu.

g. T h e sam e assum ption is m ade in all o f N e w to n ’s kn ow n proofs o f Prop, i corresponding to K e p le r ’s second law o f planetary m otion. 10. N o tice the striking sim ilarity betw een the assum ptions in this paragraph and those m ade in Prop. i. 1 1 . N o tice the peculiar m ixture o f m otion

both

along the p o lygo n

and

the

tangent. T h e same sort o f assum ption is m ade, tho u gh less explicitly, in the p ro of o f Prop. 2. 12. T h a t A F is parallel to

PF,

parallel

to the direction o f the force (or im pulse) at

P,

i.e.

is assum ed b y N e w to n w ith o u t proof. It can be show n to follow

from the assum ption (2nd paragraph) that

X

and F are the points arrived at b y

the b o d y in one and the same tim e on the tangent and the ellipse respectively. B u t it w ould be necessary to replace the ellipse b y an inscribed p o lygo n

out and

through­

follow the type o f argum ent (based on H ypo th esis 3) found in the p ro of to Prop. I.

I

R e p la c in g

lines

d e le t e d .

256

THE K E P L E R -M O T I O N PAPERS

VIII

That -iAF is proportional to the magnitude of the force (or impulse) at P is a basic assumption which Newton justifies neither here nor in the tract de Motu. In Prop. 6, Theor. 5, of Book I of the Principia, however, he justifies the same assumption by an appeal to the second law of motion. In Prob. 3 of the tract de Motu Newton derives the inverse square law for motion in an ellipse under a force to a focus as a special case of the general formula for motion in any curve given in Prop. 3. The dynamical argument is therefore concentrated in Prop. 3 , and the mathematical argument, corresponding to the three lemmas in the present paper, in Prob. 3 . In Prop. 3 of the tract de Motu he omits all reference to the two points on the ellipse and the tangent being reached in the same time. Instead he takes a point Q on the curve close to the initial point P, and draws a line through Q parallel to SP intersecting at R the tangent to the curve at P. He then simply states, without any justification, that for motion in a given time from P to Q the centripetal force at P will be proportional to QR. It is clear, then, that the dynamical argument in the present document, though still incomplete, is more detailed than that given in the tract de Motu. See above. Chapter 1.3, p. 19. 1 3 . No such comparison between two distinct points on the curve is made in Prop. 3 of the tract de Motu. Instead he makes an ingenious application of Galileo’s P law. 1 4 . The limiting process here envisaged is effectively a double one. It is necessary, in the first place, because the ellipse has been replaced (partially, at least) by an inscribed polygon; and in the second place, because the results already obtained are merely approximate. Only the second type of limiting pro­ cess is necessary in Prop. 3 of the tract de Motu where the curve is not approxi­ mated to by an inscribed polygon.

IX T H E T R A C T DE M O T U T h e r e are five known versions of the tract de Motu, three in that part of the Portsmouth Collection in the University Library, Cambridge,* one in the archives of the Royal Society of London, and one in the Macclesfield Collection. All are substantially the same, so that the possibility of independent origins can be excluded. For convenience’ sake the versions in the Cambridge University Library will be numbered I, II, III, corresponding respectively to M SS. IXa, IXb, IXc. As shown later, I is undoubtedly the earliest version, and II is essentially a copy of I, as is also the case for the Royal Society and Macclesfield versions. 2 Version III, however, has substantial additions compared with I, and represents a more advanced state of dynamical thought. The full text of I is given in § i together with exegetical and textual footnotes. Certain of these latter footnotes refer to differences between I, II, and III. These form the basis of the general comparisons between I and II, and I and III in §§ 2, 3 respectively. In § 3 the additions in III compared with I are also given. The question of the probable dates of composition of Versions I and III is discussed above in Part I, Chapter 6.4. A tentative identification of Version II with all but the last of the actual propositions carried by Paget from Newton to Halley in late 1684 is given in Part I, Chapter 6.5.

1. Manuscript Add. 3 9 6 5 ( 7 ) , fols. 5 5 - 6 2 V , 6 3 - 7 0 , 4 0 - 5 4 , respectively. This is certainly true in the case of the Royal Society copy, first published in Rigaud [i]. Appendix I, and then in Ball [i], pp. 35-51. I understand on good authority that it is also true in the case of the elusive Macclesfield Version. 2.

1.

V

e r s io n

I De Motu Corporutn in Gyrum

Def Vim centripetam appello qua corpus impellitur vel attrahitur versus aliquod punctum quod ut centrum spectatur. Def 2^ Et vim corporis seu corpori insitam qua id conatur perseverare in motu suo secundum lineam rectam. Def

3^ ‘^ ‘I

Et resistentiam quae est medij regulariter impedientis.

Hypth Resistentiam*^' in proximis novem propositionibus nullam esse, in sequentibus esse ut corporis celeritas et medij densitas conjunctim. s.'isuo.')

S

258

THE T R A C T

DE M O T U

IX

Hypoth Corpus omne sola vi insita uniformiter secundum rectam Jineam in infinitum progredi nisi aliquid cxtrinsecus impediat. Hypoth Corpus in dato tempore viribus conjunctis eo ferri quo viribus divisis in temporibus aequalibus successive. Hypoth [Spatium quod corpus urgente quacunque vi centripeta ipso motus initio describit esse in duplicata ratione temporis.]

IX

THE T R A C T

DE M O T U

259

SDE, ipsi SC D et SEE ipsi SD E aequale erit. Aequalibus igitur tem­ poribus aequales areae describuntur. Sunto jam haec triangula numero infinita et infinite parva, sic, ut singulis temporis momentis singula respondeant triangula, agente vi centripeta sine intermissione, et constabit propositio. Theorem 2^ Corporibus in circumferentijs circulorurn uniformiter gyrantibus vires centripetas esse utv arcuum simul descriptorum quadrata applicata ad radios circulorurn.

Theorema Gyrantia omnia radijs ad centrum ductis areas temporibus propor­ tionals describere. Dividatur tempus in partes aequales, et prima temporis parte describat corpus vi insita rectam A B [Fig. i]. Idem secunda temporis parte si nil impediret" recta pergeret ad c describens lineam Be aequalem ipsi A B adeo ut radijs A S , B S, cS ad centrum actis confectae forent aequales areae A S B , BSc. Verum ubi corpus venit ad B agat vis centripeta impulsu unico sed magno, faciatque corpus a recta Be deflectere et pergere in recta B C . Ipsi B S parallela agatur eC occurrens B C in C et completa secunda temporis parte^ corpus reperietur in G.® Junge S C et triangulum S B C ob parallelas SB, Ce aequale erit triangulo SBc atque adeo etiam triangulo SA B . Simili argumento si vis centripeta successive agat in C, D, E, etc., faciens corpus singulis temporis momentis singulas describere rectas CD, DE, EE, etc., triangulum .SC/) triangulo S B C et Myp. I."

Ivcm. I.*'

Corpora B, b [Fig. 2] in circumferentijs circulorurn BD, bd gyrantia simul describant arcus BD, bd. Sola vi insita describerent tangentes BC, be his arcubus aequales. Vires centripetae sunt quae perpetuo retrahunt corpora de tangentibus ad circumferentias, atque adeo hae sunt ad invicem ut spatia ipsis superata CD, ed, id est productis CD, ct/adFet/ut T in q u a d

—r^E

Uf-quad

J )J)q u a d

Cn j? '

hAquad

-j—

• Hoquor despatijsB D ,b d minu-

tissimis inque infinitum diminuendis sic ut pro \CF, \ef scribere liceat circulorurn radios SB, sb. Quo facto constat Propositio. Cor Hinc vires centripetae sunt ut celeritatum quadrata applicata ad radios circulorurn. Cor 2 Et reciproce ut quadrata temporum periodicorum applicata ad radios. Cor 3 Unde si quadrata temporum periodicorum sunt ut radij circulorum vires centripetae sunt aequales. Et vice versa. Cor 4 Si quadrata temporum periodicorum sunt ut quadrata radiorum vires centripetae sunt reciproce ut radij. Et vice versa.

260

THE T R A C T

IX

DE M O T U

Cor 5 Si quadrata temporum periodicorum sunt ut cubi radiorum vires centripetae sunt reciproce ut quadrata radiorum. Et vice versa. Schol.’i Casus Corollarij quinti obtinet in corporibus coelestibus. Quadrata temporum periodicorum sunt ut cubi distantarium a communi centre circum quod volvuntur. Id obtinere in Planetis majoribus circa Solem gyrantibus inque minoribus circa Jovem ""‘et Saturnum™* jam statuunt Astronomi.

IX

TH E

T R A C T

DE

M O TU

261

Corol. Hinc si datur figura quaevis et in ea punctum ad quod vis centripeta dirigitur, inveniri potest lex vis centripetae quae corpus in figurae illius perimetro gyrare faciet. Nimirumcomputandum est solidum ---- — huic vi reciproce proportionale. Ejus rei dabimus exempla in problematis sequentibus, Prob.

Theor. 3I” Si corpus P [Fig. 3] circa centrum S gyrando, describat lineam quamvis curvam APQ , et si tangat recta PR curvam illam in puncto quovis P et

Gyrat corpus in circumferentia circuli requiritur lex °ivis centripetae*^! tendentis ad punctum aliquod in circumferentia.

R

ad tangentem ab alio quovis curvae puncto Q agatur QR distantiae S P parallela” ac demittatur O T perpendicularisad distantiam-SPidico quod ^pquad

0

'J 'q u a d

vis centripeta sit reciproce ut solidum ------- ----------- , si modo solidi illius ea semper sumatur quantitas quae ultimo fit ubi coeunt puncta P et O. Namque in figura indefinite parva Q R P T lineola OR dato tempore est ut vis centripetal 2 gt data vi ut*^ quadratum temporis atque adeo neutro dato ut vis centripeta et quadratum temporis conjunctim, id est ut vis centripeta semel et area S Q P tempori proportionalis (vel duplum ejus S P x Q T ) bis. Applicetur hujus proportionalitatis pars utraque ad SP'^xOP^ lineolam OR et fiet unitas ut vis centripeta et — — conjunctim, hoc est vis centripeta reciproce ut

SP^xOT^

QR " Lem. 2."'

Q.E.D.

Esto circuli circumferentia S O P A, [Fig. 4] centrum °'vis centripetae®* S, corpus in circumferentia latum P, locus proximus in quern movebitur Q. Ad SA diametrum et S P demitte perpendicula P K , Q T et per Q ipsi S P parallelam age LR occurrentem circulo in L et tangenti PR in R, p*et coeant TQ, PR in Z. Ob similitudinem triangulorum ZOR, ZT P , SPA^^ erit PP® (hoc est QRL) ad ut SA^ ad SP^. Ergo SPq OPLxSP*^ QT^. Ducantur haec aequalia in —~ et punctis P et O QR S P qc Q T^ xSP'i coeuntibus scribatur S P pro RL. Sic fiet . Ergo °‘vis SAi OR Spao centripeta®' reciproce est ut - - j - , id est (ob datum SA"') ut quadratecubus distantiae SP. Quod erat inveniendum. Schol."'^’*'* Caeterum in hoc casu et similibus concipiendum est quod postquam corpus pervenit ad centrum S, id non amplius redibit in orbem

262

THE T R A C T

DE M O T U

IX

sed abibit in tangente. In spiral! quae secat radios omnes in dato angulo vis centripeta tendens ad Spiralis principium est in ratione triplicata distantiae reciproce, sed in principio illo recta nulla positione deterininata spiralem tangit.

IX

Gyrat corpus in Ellipsi veterum: requiritur lex Vis'"' centripetae tendentis ad centrum Ellipseos. Sunto CA, C B [Fig. 5]^* semi-axes Ellipseos, GP, D K diametri conjugatae, PF, Qt perpendicula ad diametros, Q V ordinatim applicata ad diametrum G P et O V PR parallelogrammum. His constructis erit ("‘ex conicis"') P V G ad OV^ ut PC^^ ad et QF® ad OP ut PO^ ad pji'Qij et conjunctis rationibus P V G ad ut PC'^ ad et ad PP ad — • Scribe OR pro PF« et B C x C A pro C D x P F , nec non (piinctis P et O coeuntibus) 2PC pro VG et Ot^xPC'i zBC'^xCA^^ ductis extremis et medijs in se mutuo, liet ~ PC ■ E st'‘ ergo vis centripeta reciproce ut ---- ^ — - id est (ob datum 2BO^x CA^>) ut

PC

hoc est directe, ut distantia PC. Q.E.T. I’er Lem 4.'*-

DE M O T U

263

Prob. 3^^ Gyrat corpus in ellipsi; requiritur lex ’‘‘vis centripetae*' tendentis ad umbilicum Ellipseos. Esto Ellipseos superioris umbilicus S [Fig. 5]. Agatur S P secans Ellipseos diametrum D K in P .y Patet E P aequalem esse semiaxi major! A C eo, quod acta ab altero Ellipseos urnbilico H linea H I ipsi E C parallela, ob aequales CS^ CH aequentur ES, EE adeo ut PPsemisumma sit ipsarum PS, P I id est =''(ob parallelas HI, PR et angulos aequales IPR, HPZY'^ ipsarum PS, P H quae conjunctim axem totum 2 A C adaequant. Ad S P demittatur perpendicularis QT. Et Ellipseos latere recto princi­ pal! |seu

Prob. 2'5

THE T R A C T

j dicto L, erit L

X

OR ad L X P V ut OR ad P V id est ut

PE, (seu AC) ad P C e t L x P V ad G V P ut L ad G V et G V P ad OF^ ut CP^ ad CD^I Et OV'^ ad ut M ad A" et ad ut EP^ ad PF^ id est ut CA'^ ad PF^ sive^ ut CD'^ ad CP'^ et con­ junctis hie omnibus rationibus, LxQR^^ ad ut A C ad PCXL^^ ad G F + C P ^ ad ad ad CB^, id est ut A C x L (seu 2PGO ad P C X G F + C P ^ ad ad sive ut 2PC ad GFe^+ M ad N.^- Sed punctis 0 et P coeuntibus rationes ^^2PC ad ^^GVi^ et M ad N hunt aequali tati sergo et ’^"ex his composita ratio'^'' L x OR Spa SP'^xOT'^ ad Ducatur pars utraque in et fiet L X SP^ = ---- - -— . o/? ^ Ergo"'* °”-vis centripeta®'* reciproce est ut L x S P ^ id est?'* in ratione duplicata distantiae SP. Q.E.I. Schol. Gyrant ergo Planetae majores in ellipsibus habentibus umbili­ cum in centro solis, et radijs ad solem ductis describunt areas temporibus proportionales, omnino ut supposuit Keplerus. Et harum Ellipseon latera recta sunta'*

QR

punctis P e tQ ‘^"spatio quam minimo et quasi

infinite parvo distantibus'"*. Theorem 4'^ Posito quod vis centripeta sit reciproce proportionalis quadrato distantiae a centro,quadratatemporum periodicorum in Ellipsibus sunt ut cubi transversorum axium. Sunto Ellipseos axis transversus A B [Fig. 6] axis alter PD, latus rectum L, umbilicus alteriter S Centro S intervallo S P describatur circulus PM D. Et eodem tempore describant corpora duo gyrantia " Per. Lem. 4."'*

264

THE T R A C T

DE M O T U

IX

arcum Ellipticum PQ et circularem PM , vi centripeta ad umbilicum S tendente. Ellipsin et circulum tangant PR, P N in puncto P. Ipsi P S agantur parallelae QR, M N tangentibus occurrentes in R et N. Sint autem figurae PQR, P M N indefinite parvae sic ut (per schol. Prob. 3) fiat L x Q R = et zSP^^xM N — MV'L Ob communem a centre S distantiam S P et inde aequales vires centripetas sunt M N et OR aequales.22 Ergo QT'i ad MV'^ est ut L ad zSP, et Q T ad M V ut medium proportionale inter L et 2 ^^SP seu PD ad z SP'^k^^ H oc est area S P Q

IX

THE T R A C T

265

DE M O T U

umbilicus unus A, B, C, D loca Planetae observatione inventa et O axis transversus Ellipseos. Centro A radio 0 —A S describatur circulus FG et erit ellipseos umbilicus alter in hujus circumferentia. Centris B, C, D, etc., intervallis Q — B S, Q ~ C S , Q — D S etc., describantur itidem alij quotcunque circuli et erit umbilicus ille alter in omnium circumferentijs atque adeo in omnium intersectione communi F. Si intersectiones omnes non coincidunt, sumendum erit punctum medium pro umbilico. Praxis ___________ Q

-c

ad aream S P M ut area tota Ellipseos ad aream totam circuliA^ Sed partes arearum singulis momentis genitae sunt ut areae S P Q et S P M atque adeo ut areae totae et proinde per numerum momentorum multiplicatae simul evadent totis aequales. Revolutiones igitur eodem tempore in ellipsibus perficiuntur ac in circulis quorum diametri sunt axibus transversis Ellipseon aequales. 25 Sed (per Cor 5. Theorem 2) quadrata temporum periodicorum in circulis sunt ut cubi diametrorum. Ergo et in Ellipsibus. Q.E.D. Schol. Hinc in systemate coelesti ex temporibus periodicis Planetarum innotescunt proportiones transversorum axium Orbitarum. Axem unum licebit assumere. Inde dabuntur caeteri. Datis autem axibus determinabuntur orbitae in hunc modum. Sit S [Fig. 7] locus Solis seu Ellipseos

hujus commoditas est quod ad unam conclusionem eliciendam adhiberi possint et inter se expedite comparari observationes quamplurimae. Planetae autem loca singula A, B, C, D etc., ex binis observationibus, cognito Telluris orbe magno invenire docuit Halleus.26 Si orbis ille magnus nondum satis exacte determinatus habetur,^? ex eo prope cognito, determinabitur orbita Planetae alicujus puta Martis propius: Deinde ex orbita Planetae per eandem methodum determinabitur orbita Telluris adhuc propius: turn ex orbita Telluris determinabitur orbita Planetae multo exactius quam prius: et sic per vices donee circulorum inter­ sectiones in umbilico orbitae utriusque exacte satis conveniant.28 Hac method© determinare licet orbitas Telluris, Martis, Jovis, et Saturn!, Orbitas autem Veneris et Mercurij sic. Observationibus in maxima Planetarum a Sole digressione factis, habentur orbitarum

266

THE T R A C T

IX

DE M O T U

tangentes. Ad ejusmodi tangentem K L [Fig. 8] demittatur a Sole perpendiculum SL centroque L et intervallo dimidij axis Ellipseos describatur circulus K M . Erit centrum Ellipseos in hujus circumferential^ adeoque descriptis hujusmodi pluribus circulis ^'^reperietur in omnium''^ intersectione. "'^Cognitis tandem"^ orbitarum dimensionibus, longitudines horum Planetarum postmodum exactius ex transitu suo per discum Solis determinabuntur.’^^^o

IX

THE T R A C T

DE M O T U

267

rectum Ellipseos. Sit istud L. Datur praeterea Ellipseos umbilicus S Anguli R P S complementum ad duos rectos fiat angulus R PH et dabitur positione linea P H in qua umbilicus alter H locatur. Demisso ad P H perpendiculo S K et erecto semiaxe minore B C est SP^--2KPH^--{-PH^ = SH^-^ = = {SP+PHy^<^d-Lx{SP-\-PH) = 5 P«+ 2SPH-\-PH^-~L (*S'P+PH). Addantur utrobique 2KPH^^-\-Lx{SP-\P H )~ S P ^ -~ P m et fiet L x{SP -\-P H ) = 2SP H ^ 2K P W ^ seu S P +

L T

K

r‘

\

M / I

/

F igu re 8.

Prob. 4'^^ Posito quod vis centripeta sit reciproce proportionalis quadrato distantiae a centro, et cognita vis illius quantitate: requiritur Ellipsis quam corpus describet de loco dato cum data celeritate secundum datam rectam emissum. Vis centripeta tendens ad punctum S [Fig. 9] ea sit quae corpus 7r| in circulo TTx centro S intervallo quovis Sn descripto gyrare faciat.^^ p)e loco P secundum lineam PR emittatur corpus Py^ et mox inde cogente vi centripeta deflectat in Ellipsin PO. Hanc igitur recta PR ianget in P. Tangat itidem recta np circulum in tt sitque PR ad irp ut prima celeritas corporis emissi P ad uniformem celeritatem corporis 7 7 . Ipsis S P et Sn parallelae agantur RO et px haec circulo in x Ellipsi in O occurrens, et a O etx ad SPtX Sn demittantur perpendicula Q T et nT. Est RO ad px ut vis centripeta in P ad vim centripetam in n^^ id est ut Sn^“^^^ad SP^*’^^*^, adeoque datur ilia ratio. Datur etiam ratio O T ad RP^^ et ratio P P a d pn seu x l. et inde composita ratio O P ad x1- De hac ratione duplicata auferOP® nM • atur ratio data OR ad vp et manebit data ratio ^ — ad — id est (per Schol OP XP Prob. 3) ratio lateris recti Ellipseos ad diametrum circuli. Datur igitur latus t T h e archaic form vr for tt sometim es used by N ew ton in this passage has been replaced throughout by tt. vj is also found in Fig. 9.

PH ad PH ut 2SP-\-2KP'°^ ad L. Unde datur umbilicus alter H. Datis autem umbilicis una cum axe transverse SP-\~PH datur Ellipsis. Q.E.I. Haec ita se habent ubi figura Ellipsis est. ffieri enim potest ut corpus moveat‘^3 in Parabola vel H yp er bol a. Ni mi rum si tanta est corporis celeritas ut sit latus rectum L aequale 2SP^2KP,'°^ ffigura erit Parabola umbilicum habens in puncto -S' et diametros omnes parallelas lineae PH. Sin corpus majori adhuc celeritate emittitur movebitur id in Hyperbola habente umbilicum unum in puncto S alterum in puncto H sumpto ad contrarias partes puncti P et axem transversum aequalem differentiae linearum P S et PH. Schol. Jam vero beneficio ^shujus Problematis solutF®, cometarum'“3 orbitas definire concessum est, et inde revolutionum tempora, et ex orbitarum magnitudine, excentricitate, Aphelijs, inclinationibus ad planum Eclipticae et nodis inter se collatis cognoscere an idem cometa ad nos saepius redeat. Nimirum ex quatuor observationibus locorum

268

THE T R A C T

DE M O T U

IX

cometae, juxta Hypothesin quod Cometa movetur uniformiter in linea recta,37determinanda est ejus via rectilinea.3^Sit taAPBD , [Fig. lo] sintque A, P, B, D loca cometae in via ilia temporibus observationum, et S locus solis. Ea celeritate qua cometa uniformiter percurrit rectam A D finge ipsum emitti de locorum suorum aliquo P et vi centripeta mox correptum deflectere a recto tramite et abire in Ellipsi Pbda. Haec Ellipsis determinanda est ut in superiore Problemate.39 In ea sunto a, P,

IX

THE T R A C T

DE M O T U

269

ut ^ffillipseos axis minor ad axem majorem et erit punctum P in Ellipsi atque acta recta S P "’^abscindetur area'"^ Ellipseos E P S tempori proportionalis.'^s Namque area H S N M triangulo S N K aucta et huic aequali segmento H K M diminuta fit triangulo H S K id est triangulo H SC

aequale. Haec aequalia adde°» areae ESH, pffient areaeP* aequales E H N S et EHC. Cum igitur Sector EH C tempori proportionalis sit et area E P S areae E H N SN erh etiam area E P S tempori proportionalis. b, d loca cometae temporibus observationum. Cognoscantur horum locorum e terra longitudines et latitudines. Quanto majores vel minores sunt his longitudines et latitudines observatae tanto majores vel minores observatis sumantur longitudines et latitudines novae. Ex his novis inveniatur denuo via rectilinea cometae et inde via Elliptica ut prius. Et loca quatuor nova in via Elliptica prioribus erroribus aucta vel diminuta jam congruent cum observationibus ssexacte s a t i s . A u t si fortes^ errores etiam num sensibiles manserint potest opus totum repeti. Et ne computa Astronomos moleste habeant suffecerit haec omnia per *^3descriptionem linearum’^3 determinare. Sed’3areas aSP, PSb, bSd temporibus proportionales assignare difficile est. Super Ellipseos axe majore EG [Fig. ii], describatur semicirculus EHG. Sumatur angulus ECH tempori proportionalis. Agatur SH eique parallela C K circulo occurrens in K . Jungatur H K et circuli segmento H K M (per tabulam segmentorum vel secus) aequale fiat triangulum S K N . Ad EG demitteJ^ perpendiculum NQ, et in eo capei^^ PQ ad NO

Prob. 5^2 Posito quod ^nis centripetal^ sit reciproce proportionalis quadrato distantiae a centro*^* spatia definite quae ^“corpus recta®* cadendo datis temporibus describit. Si corpus ^^*non cadit perpendiculariter describet id Ellipsin puta A P B [Fig. 12] cujus umbilicus inferior puta S congruetcum centro.^* Id ex jam demonstratis constat. Super Ellipseos axe majore A B describatur semicirculus A D B et per corpus*^* decidens transeat recta D PC perpendicularis ad axem, actisque DS, PS, erit area A S D areae A S P atque adeo etiam tempori proportionalis. Manente axe A B minuatur perpetuo latitude Figure 12. Ellipseos, et semper manebit area A S D tempori proportionalis. Minuatur

270

THE T R A C T

IX

DE M O T U

latitude ilia in infinitum et orbita A P B jam coincidente cum axe A B et umbilico S cum axis termino B descendet corpus'^^ in recta A C et area A B D evadet tempori proportionalis. Definietur itaque spatium A C quod corpus‘“ de loco A perpendiculariter cadendo tempore dato describit si modo tempori proportionalis capiatur area A B D et a puncto D ad rectam A B demittatur perpendicularis DC. Q.E.F. Schol."3 Priore Problemate definiuntur motus''^ projectilium in acre nostro hacce motus gravium perpendiculariter cadentium ex Hypothesi

IX

THE T R A C T

DE M O T U

271

mento lineae D C proportionale est incrementum lineae AD. Ergo incrementum spatij per incrementum lineae AD, atque adeo spatium ipsum per lineam illam recte exponitur Q.E.D. Prob. Posita uniformi vi centripeta, motum corporis in medio similar! recta ascendentis ac descendentis definire.

H

C

quod gravitas reciproce proportionalis sit quadrate distantiae a centre terrae quodque medium aeris nihil resistat. "''‘Nam gravitas est species una vis centripetae.'''^’ Prob. 6^3ya

Figure 14,

Corporis sola vi insita per medium similare resistens delati motum definire. Asymptotis rectangulis AD C, CH [Fig. 13] describatur Hyperbola secans perpendicula AB, D G in B, G. Exponatur turn corporis celeritas turn resistentia medij ipso motus initio per lineam A C elapso^3 tempore aliquo per lineam"** D C et tempos exponi potest per aream A B G D atque spatium eo tempore descriptum per lineam AD. Nam celeritati proportionalis est resistentia medij et resistentiae proportionale est decrementum celeritatis/^ i^4hoc esC*, si tempus in partes aequales dividatur, celeritates ipsarum initijs sunt‘=*differentijs suis proportionales. Decrescit ergo celeritas in'* proportione Geometrica dum tempus crescit in Arithmetica. Sed <^*tale est decrementum'^* lineae D C et incrementum*'* areae AB G D , ut notum est. Ergo tempus per aream et celeritas per lineam illam recte exponitur. Q.E.D. Porro celeritati atque adeo decremento celeritatis proportionale est incrementum spatij descripti sed et decre“ Lem. **

Corpore ascendente exponatur vis centripeta per datum quodvis rectangulum B C [Fig. 14] et resistentia medij initio ascensus per rectangulum BD sumptum ad contrarias partes. Asymptotis rectangulis AC , CH per punctum B describatur Hyperbola secans perpendicula DE, de in G, g et corpus ascendendo tempore DGgd describet spatium EGge, tempore D G BA spatium ascensus totius EGB, tempore AB^G^D spatium descensus BE^G atque tempore W^G-g^d spatium descensus ^GEe^g\ et celeritas corporis resistentiae medij proportionalis, erit in horum temporum periodis ABED, ABed, nulla, ABE^D, ABe^d; atque maxima celeritas quam corpus descendendo potest acquirere erit BC. Resolvatur enim rectangulum A H [Fig. 15] in rectangula innumera Ak, Kl, Lm, Mn, etc. quae sint ut incrementa celeritatum aequalibus totidem temporibus facta et erunt'* Ak, Al, Am, An etc., ut celeritates totae atque adeo" ut resistentiae medij in fine singulorum temporum " H yp. >**

272

THE T R A C T

DE M O T U

IX

aequalium. j^ffiat A C ad A K , vel A B H C ad A B k K ut vis centripeta ad resistentiam in fine temporis primi et erunt AB H C, KkH C, LIHC, NnHC etc. ut vires absolutae quibus corpus urgetur atque adeo ut incrementa celeritatum, id est ut rectangula Ak, K l, Lm, Mn etc et proinde^ in progressione geometrica. Quare si rectae Kk, LI, Mm, Mn productae occurrant Hyperbola in 17, A, /x, v etc. erunt areae ABrjK, Kr]XL, LXfxM, MfjLvN etc aequales, adeoque turn temporibus aequalibus turn viribus centripetis semper aequalibus analogae. Subducantur rectangula Ak, Kl, Lm, Mn, etc viribus absolutis analoga et relinquentur

areae Bkr], kr]Xl, lAixm, mixvn, etc resistentijs medij in fine singulorum temporum, hoc est celeritatibus atque adeo descriptis spatijs analogae. Sumantur analogarum summae et erunt areae Bkrj, BIX, Bm(i, Bnv etc. spatijs totis descriptis analogae, nec non areae ABrjK, ABXL, AB[xM, ABvN, etc. temporibus.Corpus igitur inter descendendum tempore quovis ABXL describit spatium BIX et tempore LXfxn spatium Xlnv. Q.E.D. Et similis est demonstratio motus expositi in ascensu. Q.E.D. Schol. Beneficio duorum novissimorum problematum innotescunt motus projectilium in aere nostro, ex hypothesi quod aer iste similaris sit quodque gravitas uniformiter et secundum lineas parallelas agat. Nam si motus omnis obliquis corporis projeti distinguatur in duos, unum ascensus vel descensus alterum progressus horizontalis: motus posterior determinabitur per Problema sextum, prior per septimum ut fit in hoc diagrammate.

Lt‘m.

'’ 1

IX

THE T R A C T

DE M O T U

273

Ex loco quovis D [Fig. 16] ejaculetur corpus secundum lineam quamvis rectam DP, et per longitudinem D P exponatur ejusdem celeritas sub initio motus. A punctoPad lineam horizontalem D C demittatur perpendiculum PC, ut et ad D P perpendiculum C l ad quod sit D A ut est resistentia

medij ipso motus initio ad vim gravitatis. Erigatur perpendiculum A B cujusvis longitudinis et completis parallelogrammis, D A BE, C A B H per punctum B asymptotis DC, C P describatur Hyperbola secans DE in G. Capiatur linea n ad EG ut est D C ad CP, et ad rectae D C punctum quodvis R erecto perpendiculo R tT quod occurrat Hyperbolae in T et D R tE -D R T B G et projectile tempore rectae EH in t, in eo cape Rr =

274

THE T R A C T

DE M O T U

IX

D R T B G perveniet ad punctum r, describens curvam Jineam DarFK quam punctum r semper tangit, perveniens autem ad maximum aititudinem a in perpendiculo A B , I'deinde incidens in lineam horizontalem D C ad F ubi areae DFsE, D F S B G aequantur^i et postea semper appropinquans Asymptoton PH CL. Estque celeritas ejus in puncto quovis r ut curvae tangens rL. Si proportio resistentiae aeris ad vim gravitatis nondum innotescit; cognoscantur (ex observatione aliqua) anguli AD P, AFr in quibus curva DarFk secat lineam horizontalem DC. Super D F constituatur rectangulum DFsE altitudinis cujusvis, ac describatur Hyperbola rectangula ea lege ut ejus iina Asymptotes sit DF, ut areae DFsE, D F S B G aequentur et ut sS sit ad EG sicut tangens anguli AFr ad tangentem anguli AD P. Ab hujus Hyperbolae centre C ad rectam D P demitte perpendiculum C l ut et a puncto B ubi ea secat rectam Es, ad rectam D C perpendiculum B A , et habebitur proportio quaesita D A ad C l, quae est resistentia medij ipso motus initio ad gravitatem projectilis. Quae omnia ex praedemonstratis facile eruuntur. Sunt et alij modi inveniendi resistentiam aeris quos lubens praetereo. Postquam autem inventa est haec resistentia in uno casu capienda est ea in alijs quibusvis ut corporis celeritas et superficies sphaerica conjunctim (Nam projectile sphaericum esse passim suppono) vis autem gravitatis innotescit ex pondere. Sic habebitur semper proportio resistentiae ad gravitatem seu lineae D A ad lineam C L Hac proportione et angulo A D P determinatur specie figura DarFKLP: et capiendo longitudinem D P proportionalem celeritati projectilis in loco D determinatur eadem magnitudine sic ut altitudo Aa maximae altitudini projectilis et longitudo D F longitudini horizontali inter ascensum et casum projectilis semper sit proportionalis, atque adeo ex longitudine D F in agro semel mensurata semper determinet turn longitudinem illam DFt\im alias omnes dimensiones figurae DarFK quam projectile describet in agro. Sed in colligendis hisce dimensionibus usurpandi sunt logarithmi pro area Hyperbolica DRTBG. Eadem ratione determinantur etiam motus corporum gravitate vel levitate et vi quacunque simul et semel impressa moventium in aqua. ai- C a n celled in V . I l l a n d replaced by L e x i. b]. E v ery th in g dozen to beginning o f Theorem i deleted. C|. E v id en tly inserted as an afterthought betzveen D e f. 2 a n d H y p . i . di. R ep lacin g the follozcing d eleted version o f the p rin cip le o f inertia C orpora nec m edio im pediri nec alijs causis externis [qua?] m inus viribus insitae ct centripetae exquisite eadem.

IX

4TIE TRA C 4'

DE M O T U

275

e^. Su cceed ed by esse ut corporis celeritas et m edij densitas conjun ctim deleted. fi. Im m ediately beneath this hypothesis is zcritten H y p o th [3 ?]. R esistentiam in proxim is n ovem p ropositionibus n ullam esse in sequentibus esse u t celeritas ct m edij densitas co njun ctim deleted. gL. W ritten in margin. h]. T h e enunciation o f this hypothesis is taken fr o m V . I I , the title only being given in V . I , in the m argin. I t is succeeded in V. I I [but n ot in V . I l l ) by a list o f the enunciations o f a ll the subsequent theorems an d problem s follozved fin a lly by tzvo lemmas ziumbered 3 an d 4 agreeing zcith the corresponding lemmas in V. I II . ii. R eplacin g H y p . 3 deleted. ji. S ucceed ed by insertion celeritatum [qua ?] deleted. ki- In V . II these corollaries are o m itted havin g already figured in the original list o f en unciations o f theorem s and problem s, h . T h is S ch oliu m is om itted from V . II. m i-m i. D ele te d in V . I I I . ni- R eplacin g H y p . 4 deleted. Oi-Oj. R eplacin g gravitas deleted. P i-p i- Inserted into V . I , present in V . I l l , missing fr o m V . I I . cii- R eferrin g to Lem m a 4 o f V . I I I . ri. Sch oliu m om itted in V . I I . Si. Preceded by gravitas deleted. tj. T an d t are interchanged in the D iagram o f V . I I com pared u it h those o f V . I , I I I . A llow a n ce is made f o r this interchange in the subsequent p ro o f in V . I I . Ui. Inserted into V . I , present in V . I l l , missing fr o m V . I I . Vi. M a rg in a l fo o tn o te (b) to C o r. T h e o r . 3 in V . I I I . \\q. R eferring again to Lem m a 4 o f V . I I I . Xi~x.. W ritten in above gravitas deleted. yi- E t lineam O F in x et com pleatur parallelogram m um Q x P R inserted in V . I l l fo r clarification since the same diagram is used as in P roh. 2 zvhile nozv Q R is supposed to be drawn p a ra llel to S P as opposed to C P . W hereas it is actu ally p a ra llel to C P . I t is curious to fin d the same error in the diagram to the corre­ sponding proposiHon {P rop. X I , P rob . V I ) in the F ir st E d itio n o f the P rin cip ia . I n later editions it is corrected. Inserted in I , present in I I I , missing fr o m I I . aa-ag. D eleted in V . I l l an d replaced by p unctis Q et P coeuntibus fit ratio aequalitalis et Qx'^ seu Qr>'' est. ba- A lte r e d fr o m QT'^ to Q t ‘ in V . I I to allozv f o r the interchange o f t fo r T in D iagram 5 previously n oted in fo o tn o te t^. c ,. S ucceed ed by insertion fit in V . I I I . do. T h is an d succeeding signs + a ll im ply m ultiplication. ej-Ca. D eleted in V . I I I . fa-fo. D eleted in V . I I I . D eleted in V . I I I . ha. D eleted an d replaced by aequantur in V. I I I . i j. R ep laced by et in V . I I I . ja-jo. D eleted in V . I I I . ka-ko. D eleted in V . I I I . la. D eleted in V . I I I . mo. S ucceed ed by insertion A eq u a n tu r in V. I I I .

276

THE T R A C T

DE M O T U

IX

H2. Marginal reference (b) to Cor. Th. 3 in V. III. Written in above gravitas deleted in both VI and VIII. P2. Succeeded by insertion reciproce in V. III. Q2. Succeeded by insertion quantitas in V. III. O2-O2.

T2. Given in error in V. II as ~ - in place of QR ^ •’ QR S2. Succeeded by quae ultimo fit ubi coeunt in V. III. t2~t2. Deleted in V. III. U2-U2. Missing from V. II. V2-V2. Replaced by reperientur in M omnium intersectiones in V. II represent­ ing an attempt to remove an imagined obscurity in the text. But the subject of reperietur was intended to be centrum and not intersectiones. V. I ll follows V. I. W2-W2. Replaced in V. II by Turn cognitis. X2. A long passage intervenes at this point in V. I l l before Prob. 4. See B of§j. Y2. Succeeded by ea celeritate qua sit ad celeritatem uniformem corporis m ut recta quaevis PR ad rectam quamvis -mp deleted. Z2. V. I I has ± 2KPH corresponding to the possibility of K lying in H P pro­ duced as is actually the case in the diagram in that version. 83. Succeeded by = 4CH'Gnserted in V. III. b3. ± 2 K P in V .I I . C3. Altered to moveatur in V. III. ds-dg. Altered to soluti hujus Problematis in V. III. eg. Substitutedfor Planetarum in all versions. f g . Succeeded by insertion id adeo ut correctiones respondeant erroribus in V. III. S3~g3- Deleted and replaced by quam proxime. At Si in V. III. hg-hg. R e p l a c i n g Geometricam deleted. V. Illfollozvs V. I, zvhereas V. II has praxin Geometricam. ig . Replaced by Verum in V. III. j g . Demittatur in V. III. k g . Capiatur in V. III. Ig. Succeeded by insertion est in V. III. mg-mg. Altered to Abscindet aream in V. III. ng. Altered to proportionalem in V. III. 03. Altered to addita in V. III. pg-pg. Altered to facient areas in V. III. qg-qg. Replacing gravitas deleted. rg. Succeeded by terrae deleted. S3-S3. Replacing gravia deleted. tg. Replacing grave deleted. Ug. The Scholium in V. I l l is different. It is given belozc in § 3 C. V g. Succeeded by gravium deleted. Wg-Wg. Replacing sequentibus resistentia medij similaris primo absque gravi­ tate dum cum gravitate uniforme consideratur. Nam vis centripetae species una est gravitas deleted apart from vis centripeta allozced to stand. X g. V. II ends here. Yg. Preceded in V. I l l by sub-heading De motu corporum in medijs resistentibus. Z g. Replaced by insertion datae longitudinis, elapso autem in V. III. a^. Succeeded in V. I l l by insertion indefinitam. b4~b4. Replaced by proinde in V. III.

IX

THE T R A C T

DE M O T U

277

C4. Replaced by erunt in F. III. d4~d4. Replaced in V. I l l by proportione priore decrescit. C4. Replaced in V. I l l by posteriore crescit. f4. Number 3 supplied in V. III. g4. Altered to Lex 5 in V. III. h4. Number 3 supplied in V. III. in. Succeeded by insertion nihil in V. III. j4~j4- After considerable emendation, partly illegible, this passage in V. HI reads; Fiat A C ad A K vel ABH C ad A BkK ut vis centripeta ad resistentiam in principio temporis deque vi centripeta subducantur resistentiae et manebunt ABHC, KkHC, LIHC, NnHC etc ut vires absolutae quibus corporibus in principio singulorum temporum urgetur atque adeo ut incrementa celeritatum, id est ut rectangula Ak, Kl, Lm, Mn etc. et proinde in progressione geometrica. Quare si rectae Kk, LI, Mm, Nn productae occurrant Hyperbola in rj. A, p, v, etc erunt areae ABrjK, Kt]\L, LXpM, MpvN etc. aequales adeoque turn temporibus aequalibus num est autem area ABrjK ad aream Bkr) ut K t) ad hkK seu A C ad \A K viribus centripetis semper aequalibus analogae. Hoc est ut vis centripeta ad resistantiam in medio temporis primi. Et simili argumento area rjKLX, XLMp, pMNv, etc. Sunt ad areas 17/e/A, Xlnp, pmnv ut vires centripetas ad resistentias in medio temporis secundi tertij quarti etc. Proinde cum areae aequales AKrj, rjKLX, XLMp, pMNv etc sunt viribus centripetis analogae, erunt areae Bkrj, rjklX, Xlmp, pmnv etc resistentijs in medio singulorum temporum, hoc est celeritatibus atque adeo descriptis spatijs analogae. k4. Immediately beneath in V. I l l occurs Et hae areae [ ?] rectangula numero infinita et infinite parva evadunt coincidunt cum Hyperbolicis deleted. I4-I4. Deleted in V. HI.

Translation The Motion of Revolving Bodies Definition i.^ I call centripetal that force by which a body is impelled or drawn towards any point which is regarded as a centre [of force]. Definition 2.^ And I call that the force of a body or the force innate in a body by reason of which it endeavours to persist in its motion along a straight line. Definition 3.3 And the resisting force that arising from the steadily impeding medium. Hypothesis i.“^The resisting force in the next nine propositions is zero, and in the succeeding ones jointly as the speed of the body and the density of the medium. Hypothesis 2.5 Every body under the sole action of its innate force moves uniformly in a straight line to infinity unless anything extraneous hinders it.

278

THE T R A C T

DE M O T U

IX

Hypothesis 3.^ A body is carried in a given time under the combined action of [two] forces so far as it would be carried by the forces acting separately in succession for equal times. Hypothesis 4. [The space described by a body at the beginning of its motion under the action of any centripetal force is proportional to the square of the time.] Theorem All bodies circulating about a centre [of force] sweep out areas pro­ portional to the times [of description]. Let the time be divided into equal intervals, and in the first interval of time suppose the body by reason of its innate force describes the line A B [Fig. i]. Likewise in the second interval of time if nothing were to impede it'* suppose it would continue straight on to c covering a length Be equal to the line ABy so that the radii A S , B S, cS being drawn to the centre [5 ] the areas A S B , BSc would be made equal. But actually when the body comes to B let the centripetal force act [on it] with a single great impulse, forcing the body to deviate from the line Be and continue on in the line BC. Let eC be drawn parallel to B S meeting B C in C and on the completion of the second interval of time^^ the body will be found at C.^ Join S C and because of the parallels SB, Ce the triangle S B C will be equal to the triangle SBc and so also to the triangle SA B . By a like argument if the centripetal force acts successively in C, D, E etc., causing the body to describe separate lines CD, DE, EE etc. in separate intervals of time, the triangle S C D will be equal to SB C, and S C D itself equal to SDE, and SD E itself equal to SEE, Therefore equal areas are described in equal times. Suppose now these triangles are infinite in number and infinitely small so that the centripetal force acts without a break, to the individual intervals of time corresponding individual triangles, and the proposition will be established. Theorem 2'' The centripetal forces of bodies revolving uniformly in the circum­ ferences of circles are as the squares of the arcs described in the same time divided by the radii of the circles. Suppose the bodies B, b [Fig. 2] revolving in the circumferences of the circles BD, hd describe arcs BD, hd in the same time. By the action of " 1 lyp oth fsis J.

Ivemm.i r.

IX

THE T R A C T

DE M O T U

279

their innate forces alone they would describe tangents BC, he equal to these arcs. The centripetal forces continuously pull back the bodies from the tangents to the circumferences, and consequently are respectively as the distances CD, cd by which the tangents exceed the circumferences, BD^ bd^ BC~ hc^ or as T^^to .I that is, producing CD, cd to F and f, as — to \C F CF cf w speak of spaces BD, bd very small and infinitely decreased so that for \CF, \ef it is permissible to write the radii SB, sb of the circles. Which being done the proposition is established. Corollary i. Hence centripetal forces are as the squares of the speeds divided by the radii of the circles. Corollary 2. And reciprocally as the squares of the periodic times divided by the radii. Corollary 3. Hence if the squares of the periodic times are as the radii of the circles the centripetal forces are equal. And vice versa. Corollary 4. If the squares of the periodic times are as the squares of the radii the centripetal forces are inversely as the radii. And vice versa. Corollary 5. If the squares of the periodic times are as the cubes of the radii the centripetal forces are inversely as the squares of the radii. And conversely. Scholium. The case of the fifth corollary holds for the celestial bodies. The squares of the periodic times are as the cubes of the distances from the common centre around which they revolve. Astronomers are agreed that this holds for the major planets circulating about the sun and for the minor about Jupiter and Saturn. Theorem 3**^ If the body P [Fig. 3] circulating about the centre 5 describes some curved line APO , and if the straight line PR touches that curve in a certain point P, and if from any other point O of the curve OR is drawn to the tangent parallel” to S P and the perpendicular O T is dropped on to the line SP: I say that the centripetal force will be inversely as the SP^xOT^ ratio---- — , provided only the quantity of that ratio is always taken as that which it becomes in the limit when the points P and O coincide. For in the indefinitely small figure O R PT the little line OR varies for a given time with the centripetal force,” and with the square of the time

280

THE T R A C T

DE M O T U

IX

when the force is given,^ and therefore when neither is given varies conjointly as the centripetal force and the square of the time, that is as the centripetal force directly and the area S O P proportional to the time (or the double of this area S P x OT) squared. Let each side of this pro­ portionality be applied to the little line QR and the centripetal force and SP^xQT^ together make unity, that is the centripetal force is inversely

QR

as

SP^xOT^ - . Q.E.D.

QR

Corollary. Hence if any particular figure is given and a point in it to which the centripetal force is directed, it can be found what law of centri­ petal force makes the body revolve in the perimeter of that figure. SP^xO T^ Clearly the ratio — proportional to this force is to be calculated. We shall give examples of this result in the following problems.

IX

THE T R A C T

DE M O T U

281

along the tangent. In a spiral which cuts all its radii at a given angle a centripetal force directed to the eye of the spiral is inversely as the third power of the distance, but at the actual eye there is no line of determinate position which touches the spiral. Problem Given a body revolving in the ellipse of the ancients, there is required the law of centripetal force directed to the centre of the ellipse. Let CA, CB [Fig. 5] be the semi-axes of the ellipse, GP, D K conjugate diameters, PF, Ot perpendiculars to the diameters, Q V the ordinate applied to the diameter G P and Q VPR a parallelogram. By these con­ structions it will follow (by conic sections) that PV. VG 'i^toQV^ as PC^ to and QV^ to Q u as PC^ to PF^,^^ and taking these ratios together PV. VG to OU as PC^ to CD^ and PC^ to PF^, that is VG to

as PC^

CD^xPF^ . Writing OR for P F “ and B C X CA for CD X PF, and also PC~^ (since the points P and 0 coincide) 2PC for VG and multiplying the extreme and middle terms together, there results

to

Problem If a body revolves in the circumference of a circle there is required the law of centripetal force tending to a certain point in the circumference. Let S Q P A [Fig. 4] be the circumference of the circle, S the centre of the centripetal force, P the body carried in the circumference, and Q the next place into which it will be moved. Drop perpendiculars PK^ Q T to the diameter S A and to SP, and through O draw LR parallel to S P cutting the circle in L and the tangent P R in R, and let TQ, P R meet in Z. On account of the similarity of the triangles ZQR, ZT P , SPA, RP^ (that is OR.RL) will be to QT^ as SA^ to SP^, Therefore O R .R L x S P ^ SA^ Multiplying these equal quantities by

SP^

QR

OT^ and writing S P for RL as the

points P and Q coincide, one has . Therefore the ^ ^ SA^ OR SPs centripetal force is inversely as that is (since SA^ is given) as the fifth power of the distance SP. Q.E.I. Scholium.'^ Moreover in this and like cases it is to be supposed that after the body reaches S it will not return again in its orbit but go off

® Lemma 2 .

OVXPC^

2BC^-x CA^

OR

PC

Therefore the centripetal force is inversely as 2BC^ x CA^ is given) as —

2BC^ x CA^ , that is (since PC

that is directly as the distance PC. Q.E.I.

Prob. 3^^ Given a body revolving in an ellipse there is required the law of centripetal force directed to a focus of the ellipse. Let S [Fig. 5] be the superior focus of the ellipse. Draw S P cutting the diameter D K of the ellipse in E. E P evidently equals the semi-major axis AC, for drawing H I parallel to E C from the other focus, H, of the ellipse, ES, E l are equal because of the equal lines CS, CH, so that EP is half the sum of PS, P I that is (on account of the parallels HI, PR and the equal angles IPR, HPZ) of PS, PH which together make up the whole axis 2^C. Drop the perpendicular Q T to SP. And calling the latus rectum of the ellipse |or

j L, L x QR to L x P V will be as QR to P V

“ By Lemma 4 .

282

^rilE 'i 'R A C T

IX

DE MOTLJ

that is as P E (or A C ) to P C and L x P V to G V .V P as L to G V ar.d G V .V P to Q V- as CP~ to CD~. And OV^ to as M to A^, say, and Ox- to OT^ as EP^ to PF^, that is as CA^ to PF-, or“ as CD^ to CB-, and taking all these ratios together L X Oi? to QT^ as A C to P C x L to G V X CP^ to C D ^ x M to N x CD^ to CB^ that is as A C x L (or zBC^) to P C x G V x CP^ to C B - x M to N, or z P C to G V x A l to N. But when the points P and O coincide the ratios z P C to G V and M t o N both equal unity: therefore the combined ratio of all these is L X OR to 0 T-. Multi­ plying both parts by there results L X S P ^ t j ^ r j OR

=

QJl

. There­

fore the centripetal force is inversely as Z>X S P - that is as the square of the distance SP. Q.E.I. Scholium. Therefore the major planets revolve in ellipses having a focus in the centre of the sun and radii drawn to the sun describe areas pro­ portional to the times, all as supposed by Kepler. And the latera recta OTof these planets equal the points Pand O being the least possible and OR ^ as it were infinitely small distance apart. Theorem 4'^ Given that the centripetal force is inversely proportional to the square of the distance from the centre^® the squares of the periodic times in ellipses vary as the cubes of the transverse axes. Let A B [Fig. 6] be the transverse axis of the ellipse, P B the other axis, L the latus rectum, S one of the foci, and suppose the circle PM D with centre S and radius S P be drawn. And in the same time suppose the two revolving bodies describe the elliptical arc PQ and the circular PM , the centripetal force being directed to the focus S. PR, P N touch the ellipse and the circle at P. Draw OR, Af A parallel to P S meeting the tangents in R and iV. But the figures POR, P M N are indefinitely small so that (by the Scholium to Problem 3) we have L X OR — OT^ and zSP^^ X MA^ = MV^. On account of their common distance S P from the centre S and the resulting equality of the centripetal forces MA' and OR are equal.22 Therefore O T- to MV^ is as L to zSP, and O T to M V as the mean pro­ portional between L and zS P or PD to zSPX^ That is the area S PO is to the area S P M as the whole area of the ellipse to the whole area of the circle.24 But the portions of area generated in individual moments are as the areas SPO and SPM , and hence as the total areas, and hence when " By Lemma 4.

IX

THE T R A C T

DE M O T H

283

multiplied by a [certain] number of moments become equal to the whole areas. Therefore complete revolutions in ellipses are performed in the same time as in circles whose diameters are equal to the transverse axes of the ellipses.25 But (by Cor. 5 Theorem 2) the squares of the periodic times in circles are as the cubes of the diameters. And so also in ellipses. Q.E.D. Scholium. Hence in the celestial system from the periodic times of the Planets the proportions of the transverse axes of the orbits become known. It will be permissible to assume [the magnitude] of one axis. Then the others will be given. But if the axes are given the orbits will be determined in this manner. Let S [Fig. 7] be the position of the sun or one focus of the ellipse. A, B, C, D the positions of the planet found from observation and O the transverse axis of the ellipse. With centre A radius Q — A S let the circle EG be described and the other focus of the ellipse will be in its circumference. With centres B, C, D, etc., and distances O — B S , 0 ~ C S , 0 — D S etc., let so many other circles be similarly described and that other focus will be in all their circumferences, and so in their common intersection F. If all the intersections do not coincide, the mean point will have to be taken for the focus. The con­ venience of this procedure is that very many observations can easily be brought together and used to produce a single conclusion. Moreover the individual position A, B, C, D etc., of the planet can be found from pairs of observations, given the great orbit of the Earth, as shown by Halley.26 If that great orbit is not yet determined with sufficient accuracy,27 then knowing it approximately the orbit of any other planet, for example Mars, will be determined more accurately: Then from the orbit of the planet the orbit of the Earth will be determined still more nearly: then from the orbit of the Earth the orbit of the planet will be determined more accurately than before: and so by turns until the intersections of the circles in the focus of both orbits agree sufficiently well.28 By this method the orbits of the Earth, Mars, Jupiter, and Saturn can be determined, but the orbits of Venus and Mercury thus: from observations made at the maximum separation of the planets from the Sun the tangents of the orbits are given. To such a tangent K L [Fig. 8] a perpendicular SL is dropped from the Sun and with centre L, and radius the semi-axis of the ellipse, a circle K M is described. The centre of the ellipse will be in the circumference of this circle,2‘>and so when several circles have been described in this way the centre will be given by the intersection of them all. Finally, knowing the dimensions of the orbits.

284

THE T R A C T

DE M O T U

IX

the longitudes of these planets will be determined later more exactly by their transit across the Sun’s disc.3° Problem 431 Given that the centripetal force is inversely proportional to the square of the distance, and knowing the magnitude of that force, required to find the ellipse which a body describes when projected from a given point with given velocity in a given straight line. Suppose the centripetal force directed to the point S [Fig. 9] be such that a body tt may be made to revolve in a circle ttx centre S with any radius S tt?^ Suppose the body P be projected from the point P along the line PR and is thereupon immediately deflected by the action of the centripetal force into the ellipse PQ. P R therefore touches this ellipse at P. Let the straight line np likewise touch the circle at tt, and let PR be to TTp as the initial speed of the projected body P to the uniform speed of the body tt.33 Draw RQ and px parallel to iSPand Sn, respectively, the one cutting the circle in the other the ellipse in O, and from O and x drop perpendiculars Q T and yT to S P and S t t . Then RQ is to px as the centripetal force in P to the centripetal force at 77,34 that is as S tt^to SP^, and so that ratio is given. The ratios Q T to RP^^ and R P to prr or x'T_ are also given, and hence the composite ratio Q T to x"J- To this ratio squared apply the given ratio QR to xp and there will remain the known ratio

Q T^

QR X'~j2 to --A- that is (by Scholium Problem 3) the ratio of the latus rectum of

XP

the ellipse to the diameter of the circle. The latus rectum of the ellipse is therefore known. Let that be L. The focus S of the ellipse is also known. Let the angle R PH be the complement of the angle R P S to two right angles, and the line P H in which the other focus lies is given as regards position. Then the perpendicular S K to P H having been dropped and the semi-minor axis B C being erected it follows that SP^— 2KP.PH-\^ PH^ = SH^ = = {S P + P H Y -L x{S P + P H ) = 2S P .P H + P H ^ ~ L {S P + P H ). Adding to both sides 2K P .P H + L (S P + P H )-S P ^ -P H ^ there results L x { S P + P H ) = 2 P S .S H + zK P .P H or S P + P H to P H as 2SP-^2K P to L. And so the other focus H is known. But given one focus together with the transverse axis SP-\-PH the ellipse is known. Q.E.I. This is the way when the figure is an ellipse. But it can happen that the body moves in a parabola ora hyperbola.3^ Thus if the speed of the

IX

TH E T R A C T

DE M O T U

285

body is such that the latus rectum L equals 2SP-\-2KP, the figure will be a parabola having its focus in the point S and all its diameters parallel to PH. If the body is projected with even greater speed, it will be moved in a hyperbola having one focus in the point S and the other in a point H chosen on the other side of the point P, and having its transverse axis equal to the difference between P S and PH. Scholium. Now truly with the help of the solution of this problem it is possible to define the orbit of comets, and thence their periodic times, and by comparison of the orbital magnitudes, eccentricities, aphelions, inclinations to the plane of the eclipse, and nodes to recognise if the same comet returns to us frequently. Certainly from four observations of the comet’s position on the hypothesis that the comet moves uniformly in a straight line37its rectilinear path may be determined. 38 Let that [path] be A P B D [Fig. 10], and let A, P, B, D be the positions of the comet at the times of observation, and S the position of the sun. Now imagine the comet projected from the point P with that speed with which it uniformly traverses the line AD , and constrained immediately by the centripetal force to deviate from the straight path and go off into the ellipse Pbda. This ellipse is to be determined as in the above problem .39 In it let a, P, b, d be the places of the comet at the times of observation. The longitude and latitude of these points relative to the Earth are known. By how much greater or less than these are the longitudes and latitudes of observation, so much greater or less than observation are the new longitudes and latitudes to be chosen. From these new [positions] the rectilinear path of the comet is to be found anew, and hence the elliptical path as before. And the four new positions in the elliptical path diminished or increased by the former errors now agree well enough with observation.4^ But if it happens that sensible errors still remain the whole process can be repeated. And that the calculation may not be burdensome to astronomers it will suffice to determine everything by geometrical drawing. But it is difficult to assign the areas aSP, PSb, bSd, proportional to the times. On the major axis EG [Fig. 11] of the ellipse let the semicircle EHG be described. Let the angle E CH be set proportional to the time. Draw SH and C K parallel to it cutting the circle in K . Join H K and (by a table of segments or secants) make the triangle S K N equal to the cir­ cular segment H K M . Drop N Q perpendicular to EG, and in it choose PQ to N Q as the minor to the major axis of the ellipse, and the point P will lie on the ellipse, and the line 5 P having been drawn an area E P S of the ellipse is cut off proportional to the time. For the area H SN M

286

THE T R A C T

DE M O T U

IX

increased by the triangle S N K and diminished by the equal segment H K M makes up the triangle H SK , that is the equal triangle H SC . To these equal parts add the area E S H making the areas EH N S and EH C equal. Since therefore the sector EH C is proportional to the time, and the area E P S proportional to EHNS,^^ the area E P S will also be pro­ portional to the time. Problem 5^2 Given that the centripetal force is inversely proportional to the square of the distance from the centre to define the spaces described in given times by a body falling straight to the centre. If the body does not fall down perpendicularly let it describe an ellipse, A P B say [Fig. 12] whose inferior focus, say S, coincides with the centre [of force]. This follows from what has already been proved. On the major axis A B of the ellipse let the semicircle A D B be described, and through the falling body passes the line D P C perpendicular to the axis, and D S, P S being drawn, the area A S D will be proportional to the area A S P and so also to the time. Retaining the a.xis A B let the width of the ellipse be decreased steadily, and the area A S D will always remain proportional to the time. Let the width be decreased indefinitely and the orbit A P B now coinciding with the axis AB , and the focus S with the end B of the axis, the body descends in the straight line A C and the area A B D comes out proportional to the time. And so the space A C which the body describes in falling perpendicularly from the position A in a. given time may be defined, provided only the area A B D is taken propor­ tional to the time, and from the point D the perpendicular D C is let fall to the line AB . Q.E.F. Scholium. By the preceding problem the motions of projectiles in our air are defined, and the motions of heavy bodies falling perpendicularly on the hypothesis that gravity is inversely proportional to the square of the distance from the centre of the earth and that the medium of the air in no wise resists. For gravity is one species of centripetal force. Problem To define the motion of a body carried along by its innate force alone in a uniformly resisting medium. Let a hyperbola be described with rectangular asymptotes A D C , CH [Fig. 13] cutting the perpendiculars AB , D C in B, G. I.et both the speed of the body and the resistance of the medium at the beginning of the motion be represented by the line A C , and after some lapse of time by

IX

THE T R A C T

DE M O T U

287

the line DC, then the time may be represented by the area A B G D and the space described in that time by the line AD. For the resistance of the medium is proportional to the speed, and the decrease of the speed is proportional to the resistance,44 that is, if the time is divided into equal parts, the speeds at the beginnings of these parts are proportional to their differences. Therefore the speed decreases in geometrical propor­ tion^* as the time increases in arithmetical proportion. But such is the decrease of the lines D C and the increase of the area AB G D , as is well known. Therefore the time can properly be represented by the area and the speed by that line. Q.E.D. Moreover the increment of the space described is proportional to the speed and so to the decrement of the speed; but the increment of the line A D is proportional to the decrement of the line DC. Therefore the increment of the space is represented by the increment of the line AD, and thus the space itself is rightly repre­ sented by that line. Q.E.D. Problem 7^5 Given a uniform centripetal force, to define the motion of a body ascending and descending rectilinearly in a uniformly resisting medium. I'or the ascending body let the centripetal force be represented by a certain given rectangle B C [Fig. 14] and the resistance of the medium at the beginning of the ascent by the rectangle BD taken on the other side. With rectangular asymptotes A C , CH let an hyperbola be drawn through the point B cutting the perpendiculars DE, de in G, g, and the body in ascending in time DGgd covers a distance EGge, and in time D G BA the space of the whole ascent EGB, and in time AB^G^D a distance of de­ scent BE^G, and in time ^D^G^g^d the space of descent ^GEe^g: and the speed of the body, proportional to the resistance of the medium, will be in the ends of these times ABED , ABed, zero, ABE^D, ABe^d ; and so the greatest speed that the body is able to acquire in descending will be BC. For if the rectangle A H [Fig. 15] is split up into innumerable rectangles Ak, Kl, Lm, Mn etc., which are proportional to the increments of the speed made in the like number of equal times, then Ak, Al, Am, An etc. will be as the total speeds and therefore'^ as the resistances of the medium at the end of the individual equal times. So that A C is to A K , or A B H C to A B k K as the centripetal force to the resistance at the end of the first time, and ABH C, KkH C, LIHC, NnHC etc., will be as the absolute forces by which the body is impelled, and so as the increments of the L e m m a [ 3 ].

Hypothesis.

288

THE T R A C T

IX

DE M OTU

speed, that is as the rectangles Ak, Kl, Lm, Mn etc., and hence" in geometrical progression. Therefore if the lines Kk, LI, Mm, Nn produced meet the hyperbola in rj. A, v etc., the areas AB-qK, KqXL, LXjxM, MixvN etc., will be equal, and so analogous both to the equal times and to the constantly equal centripetal forces. Subtracting the areas Ak, Kl, Lm, Mn analogous to the total forces there remain the areas Bkq, kqXl, IXfxm, mfxvn etc., analogous to the resistances of the medium at the end of the individual times, that is to the speeds and so to the spaces described. Taking the sum of analogous parts the areas Bkq, BIX, Bmfx, Bnv etc., will be analogous to the total spaces described, and besides the areas A B qK , ABXL, ABjxM, ABvN etc., will be analogous to the times. The body therefore in descending for a certain time ABXL covers a dis­ tance BIX, and for a time LX(in a distance Xlnv [sic]. Q.E.D. And the demonstration of the motion in ascending is similar to that just given. Scholium. By the help of the last two problems the movements of pro­ jectiles in our air are known on the hypothesis that the air is homogeneous and that gravity acts uniformly and in parallel straight lines. For if the total motion of the body is separated into two, the one ascent or descent, the other a progressive horizontal one; the latter motion will be deter­ mined by Problem 6, and the former by Problem 7, as in this diagram. Let the body be projected from any place D [Fig. 16] along any line DP, and let its speed at the beginning of the motion be represented by the length DP. From the point P let the perpendicular P C be dropped to the horizontal line DC, so that the ratio of the perpendicular C l to D A is as the resistance of the medium at the beginning of the motion to the force of gravity. Erect the perpendicular A B of any length and having completed the parallelograms D ABE, C A B H describe through the point B an hyperbola with asymptotes DC, C P cutting in G. Take the ratio of line n to EG as D C to CP, and at any point R of the line D C erect a perpendicular R tT cutting the hyperbola in T and the line EH

.

j

.

.

,

DvtE—DRTBG

r r ii

1

•

-i

•

m t, and m it take Rr = ------------------ . 1 hen the projectile in time n D R TB G will arrive at the point r, describing a curved line DarFK which always touches the point r, reaching also to its maximum altitude a in the line AB , and from thence encountering the horizontal line D C at F (where the areas DFsE, D F S B G are equal) and afterwards always approaching the asymptote PH CL. And its speed at any point r is pro­ portional to the tangent to the curve rL.

" Lemma 3 .

IX

THE T R A C T

DE M O T U

289

If the ratio of the resistance of the air to the force of gravity is not yet known, the angles AD P, AFr in which the curve DarFk cuts the horizontal line D C may be discovered (by some experiment). On DE let a rectangle DFsE of any altitude be drawn and a rectangular hyperbola of such a form that one of its asymptotes is DF, that the areas DFsE, D F S B G are equal, and that 56” be to EG as the tangent of the angle AFr to the tangent of the angle AD P. From the centre C of this hyperbola drop a perpendicular C l to the line DP, and from the point B where the curve cuts the line Es drop a perpendicular B A to the line DC, and the ratio sought for, namely the initial resistance of the medium to the gravity of the projectile, will be given by D A to C l. All which things may easily be sought out from what has already been proved. And there are other methods of determining the resistance of the air which I gladly pass over. However, after this resistance has been determined in one case it is to be taken in all others proportional jointly to the velocity and the surface of the sphere (for I assume throughout the projectiles to be spherical) whereas the force of gravity is known through the weight. And so the ratio of the resistance to the gravity on the line D A to the line C l is always known. By this ratio and the angle A D P the form of the figure D arFKLP is determined and taking the length D P proportional to the velocity of the projectile on the point D its size; so that the altitude Aa of the maximum elevation of the projectile and the length D F of the horizontal length between the ascent and face of the projectile are always in proportion, and so the length D F measured once in the open determines both the length D F itself and all the dimensions of the figure DarFK which the projectile describes in the open. But in calculating these dimensions logarithms are to be used in place of the hyperbolic area DRTBG. By the same argument may be determined the motion of bodies moving in water under the simultaneous action of gravity or levity and any other force. 1 . See V . H I , D e f. i . It is fittin g that N e w to n should b egin w ith the definition o f vim centripetam giv en that the central proposition correspon din g to K e p le r ’s first tw o laws o f p lanetary m otion is concerned w ith such forces. It is interest­ in g also that now here in an y versions o f the tract de M o tu does N ew to n em p loy the term centrifugal force. 2. See V . H I , D e f. 2. 3. See V . H I , D e f. 3. 4. T h e corresponding version o f this hypothesis in V . H I is at L e x 5. T h e definition o f resistance proportional to the v elocity corresponds to that used in Prob. 6, 7.

290

THE T R A C T

5. See V . I l l ,

Lex

IX

DE M O T U

THE T R A C T

22.

i.

6. A rather obscure form ulation o f the p arallelogram law for the com position o f indep en den t m otions given as

IX

Lemma

Q TJ _

i in V . I I I .

planar nature

o f the m otion

8. N o tic e the absence o f an y ju stification o f this beautiful construction w h ich depends on an application o f the parallelogram law referred to in the correspon d­ in g Prop. I, T h e o r . i , in the lectures

de Motii and

the

Principia.

T h e o n ly place

w here the ju stification is giv en in detail is in the L o ck e c o p y o f the original o f M S. V III. 9. C o rresp o n d in g to Prop. 4, T h e o r . 4, o f the lectures

cipia and

Sin ce th ey are the deviations from the inertial paths p roduced b y ecjual

forces in equal tim es.

de Motu

and the

Prin­

to the sam e p roposition in V . I I I . T h e germ o f the p resent p ro o f has

already been n oted in M S . IV a . B u t n o w all m en tion o f centrifugal

conatus has

L

QT

b u t V (T . z S P ) = 24.

A re a S P Q

^ {L IzS P ) -

M V

M V^ ■ ■ zS P

7. C o rrespo n din g to K e p le r ’s second law o f planetary m otion. N o tic e the absence o f any reference in the enunciation to the

^j{{L K z S F ) ! { z S P - ) ]

\!{{2b~la)';<2a)} = 2b = P D QT

A re a S P M M V 25. In other w ords, if

.'.

77 P D S P

PD z S P ""

QT

PD

M V

zS P '

A rea

of ellipse

2 tt S P ^

A rea o f circle is the com m on tim e to describe areas S P Q and ,

,1-

.

•

.

A

A re a o f ellipse

S P M , the tim es o f revolution in the ellipse and circle are M x — 7------ > A re a S P Q M X ^---------respectively, and these are equal ow in g to the result A re a o f S P M

CD, cd, directed to the centre, and n ot aw ay from it as centripetal forces. I t is also neither here n or in the lectures de Motu nor the Principia is

disappeared, the deviation

A rea o f S P Q

A rea o f ellipse

in M S . IV a g iv in g the measures o f the correspon din g

A re a o f S P M

A re a o f circle

n otew orth y that ( i)

291

DE M O T U

an y ju stification offered for the proportionality betw een the deviation and the

N o te that the specially favourable choice o f P at the end o f the m in or axis does

centripetal force, (2) the fact that the directions o f the ‘ deviation ’ are n ot quite

n ot invalidate the p ro of since the central nature o f the force ensures that the

parallel to those o f the forces is ignored. F o r a discussion o f N e w to n ’s use o f the

rates o f description o f area in b o th the circle and the ellipse are constant. 26. I have been unable to trace the m etho d referred to here b y N ew to n .

‘d eviatio n ’ as a m easure o f force see above Part I, C h a p ter 1.3, p p . 1 9 -2 2 . 10. C o rrespo n d in g to the sam e proposition in V . I l l , and to Prop. 5, T h e o r . 5, o f the lectures

de Motu

27. Im p ly in g that the transverse axis is o n ly kn ow n approxim ately, and likew ise the positions A , B , C , D , so that the various circles w ill not quite

(M S . X I ) .

1 1. N o tic e the difference betw een this m etho d o f constructin g the ‘d e via tio n ’ and that em p loyed in Prop. 2.

intersect in a single point. 28. T h e success o f the m ethod sketched w o u ld depend on the convergence

12. O n ce again, as in Prop. 2, no ju stification is given for this assum ption.

o f the series o f approxim ations w h ich w o u ld in turn be in dicated b y an im pro ve­

13. C o rrespo n din g to the same p roblem in V . I l l , and to Prop. 7, Prob. i, o f

m en t in the closeness o f intersection o f the circles for the E arth and the Planet.

the lectures de Motu. N o n ew points o f dyn am ical interest arise in this and the tw o su bsequ en t problem s.

29. A s follow s at once from the bisector p ro perty o f the tan gen t and the result S P + P H = za .

14. N o tic e the m asterly w a y in w h ich N e w to n deals w ith the m otion o f the

30. T h e im portance o f the transit o f an inferior p lanet across the sun’ s disc

Scholium is m issin g from the correspon din g problem in the lectures de Motu and the Principia. 15. C o rrespo n din g to the same p roblem in \'^ III, and to the Scholium to Prop. 7, Prob. 2, o f the lectures de Motu.

for determ in ing its true apparent m agn itude was recognized b y H orrox as early

16. N o tic e how fundam ental this result was for N e w to n ’s treatm en t o f the ellipse.

( is t edn.).

b o d y after reach ing the centre o f force. C u rio u sly this

17. S in ce the angles

QtV, PFC

are rig h t angles, and

LQVt

=

Z

PCF

by

31. Co rrespo n din g to the same proposition in V . I l l , to Prop. 16, Prob. 8, o f the continuation

to M S . X I , and to Prop. 17, Prob. g. B ook I, P rin cip ia

32. B u t to k n o w this circular m otion its uniform speed m u st have been observed, or the exact form ulae for centripetal force, in clu d in g the mass d e p en ­ dence, m u st be know n. T h is represents a weakness o f the present m ethod.

parallels. 18. T h e fact that N e w to n nam es this central proposition, corresponding to K e p le r ’s first law o f p lanetary m otion, a

problem is

an indication o f the extrem e

logicality o f his th o u g h t and n ot o f any failure to realize its im portance, as b e ­ comes evident from the

as 1639. See the extract in S h ap le y and H o w arth [i], p p . 58 -62 .

Scholium. See the same proposition in V ersion I I I and de Motu. I t w ill be n oticed that apart from the

Prop. 8, Prob. 5, o f the lectures

33. T h u s ensuring that the arcs P Q , ttx are described in equal times. 34. Sin ce these deviations are generated in equal tim es, as noted in note 33 above. 35. T h is ratio b ein g the tan gen t o f the given inclination o f the initial p ath to

SP.

tw o results for ellipses already em p loyed in Prob. 2, viz. that referred to in note

36. It w o u ld be interesting to kn ow how N e w to n arrived at this result.

16, and the co nstan cy o f the area o f circu m scribed parallelogram s, the o n ly n ew

P o ssibly it was on the basis o f the im m ed iately su cceed ing consideration o f the

properties em p loyed are those o f the tan gen t bisectin g the exterior angle

ratio o f z S P h 2 K P to L .

form ed b y the focal lines through its p oin t o f contact, and o f the sum o f the two 19. C o rrespo n din g to the same p roposition in V . I l l , different p ro o f in Prop. 15, T h e o r . 7 , o f the lectures

37. R igau d [i], p. 29, took this to m ean that N e w to n still b elieved in recti­ linear paths for com ets at the tim e o f com posin g the tract de M o tu in spite o f the

focal distances b ein g equal to the len gth o f the m ajor axis. and to the rather

de Motu.

deduction o f the elliptical paths o f the planets in that work. B u t an exam ination o f the su cceed in g argum en t show s that the rectilinear hypoth esis is only em ­

20. M e a n in g the centre o f force as opp osed to the centre o f the ellipse.

p lo yed as a first approxim ation to the actual elliptical orbit. T h e p roblem o f

21. 7 / for a circle b ein g equal to the diameter.

com ets caused N e w to n m uch trouble— witness the fo llo w in g passage in his

PLATE

292

THE T R A C T

DE M O T V

IX

\ U T>';,

letter o f 20 June 1686 to H a lley : T h e third zvatits y e T heory o f C om ets, h i A u tu m n last [r6tS\5] I spent tzvo months in calculations to no purpose fo r zvant o f a good m ethod. . . . H is treatm ent o f the problem in the P rin cip ia ((is t edn.), B ook H I , p p. 4 7 4 -e n d ) is pure!}’ m athem atical and is free from the weakness o f the present m ethod w ith its assum ption o f an exact kn ow ledge o f the central force. It is based on the su bstitu tion o f an approxim ate parabolic path for the actual elliptical orbit o f the com et. T h is was p resu m ably the good m ethod N e w to n had searched for in vain in the autum n o f 1685. 38. A s shown, for exam ple, b y N e w to n in Prob. 56 o f his A rith m etica U n iv er­

salis, a reference g iv e n b y R igau d [i], p. 29. 39. A s poin ted ou t above (note 32), this assumes a kn ow ledge o f the actual value o f the cen trifugal force w h ich cou ld scarcely have been available to

I*

N ew to n .

'“H

J- ijf

40. It is n ot clear w h y the n ew elliptical p osition should be increased or d e­ creased b y the form er errors. N o r is it at all certain that the com plicated double

t .* v ^ vi.

process o f approxim ation here envisaged w o u ld necessarily converge tow ards the p ath o f the com et. 4 1. F ro m the m etho d o f construction o f the ellipse it follows that A rea E H N Q A rea E P Q

a

^

^ b‘

A rea S N Q A rea S P Q Area ^ N S

_

A r ea E P E

~

C-rl-

so that b y subtraction

* «».v£ci '

■—

t

y^u

- Jv*

f

fc cL

*

cXmd*^.*4tx. ree/* (Sc

42. C o rresp o n d in g to the same p roblem in V . H I , to Prop. 21, Prob. 13, o f the

^ *<«.w**< ^

^

continuation jS^to M S . X I , and to Prop. 32, Prob. 24, Book I, Prin cip ia ( is t edn.). 43. C o rrespo n din g to the same problem in V . H I , and to Prop. 2, T h e o r . 2, B ook I I , P rin cip ia ( i st edn.). T h e interest o f this and the succeed ing p ro blem is largely m athem atical. For a detailed interpretation in terms o f the calculus see

fic , r j m . a

^

'T }’0«

Correspondence, vol. ii, pp. 4 5 9 -6 2.

■ y\e/04

44. F ro m the second law o f m otion.

VC A«-

45. C o rrespo n din g to the same p roblem in V . H I , and to Prop. 3, Prob. i. Prop. 4, Prob. 2, o f B ook I I , P rin cip ia ( i st edn.).

A«cc A.v^r^

2.

C

o m p a r is o n

of

V

ersions

I

an d

In the first place, it is clear that V. II must have been composed after V. I. This is conclusively proved by the fact that with one exception wherever there are deletions in V. I, V. II follows the final rather than the original draft. Apart from this the texts of the two versions are so nearly identical wherever they can be compared that V. II must have been copied directly from V. I, or from another manuscript elTectively identical with the emended draft of that version. And the one exception referred to above (note hg-hg) in which V. II follows the original rather than the final draft of V. I, makes it probable that V. II was actually copied from V. I. V. II is also definitely not in Newton’s hand. There are, however, a certain number of substantial differences between the two versions:

-----------

5^'^

^

I u

II

, .j

,

X

Cthpmym.

ii^grW

Ct-rluf-wCCHi

** $ c jc

'

h

^

<^>v« p>.fjp9 Ur ‘^

,

'■

'‘

^^ ^«.0 ^ -h ;"

-

y ..—

r-/’-. ^ c i. ,

-

L

^


^

11^ _ ^ 4-c-r

04

r Jut

sH CH,

id .

rH * T
ir c r , i ./ e n s U P Y r»jp0h'C tf ,

T h e first folio o f the tract de M o tu

5

IX

THE T R A C T

DE M O T U

293

(«) Following the definitions and hypotheses in V. II— identical with those in V. I— there is a complete list of the enunciations of the theorems and problems of V. I. The corollaries to Theorem 2 are given in this list and not after the theorem itself, as in V. I. Also, the enunciations to the two lemmas numbered 3 and 4 are found at the end of the list. These agree with those in V. III. (b) The Scholium to Prop. 2 of V. I is absent from V. II. {c) V. II lacks the last two problems of V. I though the enunciations of these are given in the preliminary list of contents. {d) Unlike the free-hand diagrams in V. I those in V. II were drawn with instruments and differ in certain respects from those in V. I. For example, T and t are interchanged in diagram 5 (note t^), and with one exception (note rg) care has been taken to allow for this in the subsequent proof (note b2). Also in Diag. 9 of V. II K lies on H P produced as op­ posed to between H and P as in V. I. Once again careful allowance has been made for this possibility by the appropriate introduction of + and — signs (notes Zo, bg). (e) Certain insertions in V. I are missing from V. II but given in V. I ll (notes pi-pi, Ui, Zi^Zi). Apart from the differences noted above there are a certain number of unimportant differences between the two versions such as small variations in spelling, punctuation and use of capital letters, the use of ii in place of ij, the absence of footnotes in V. II, and the omission in V. II of a small number of words. T o summarize: with the exception of that part of the list of contents after the definitions and hypotheses, and the absence of the last two problems, V. II was copied from V. I or another effectively identical manuscript. This copying took place before the composition of V. I ll ((e) above) although the enunciations of Lemmas 3 and 4 in V. II were presumably added after the composition of V. III. The copyist was evidently something more than a mere amanuensis, for he had a real acquaintance with mathematics, as evidenced by the changes noted in notes Zo, bg, and also with Latin, as witness his attempts to improve the clarity of the text— see note Vg. For some reason or other he omitted the whole of the Scholium to Theor. 2, possibly an oversight. Judging by the presence of their enunciations in the preliminary list. Problems 6 and 7 formed part of V. II originally, but have since become separated from it. It must be emphasized finally that V. II nowhere goes beyond V. I as

THE T R A C T

294

DE M O T U

IX

regards dynamical content. Its interest, if any, is therefore purely his­ torical. In fact, a case can be made out for identifying it with the originals of the set of propositions carried by Paget from Newton to Halley towards the end of 1684. A discussion of this is given above in Part I, Chapter 6.6.

3.

V ersion

III De Motu Sphaericorum Corporum in fluidis

Except for the small number of variants already noted in the alpha­ betical footnotes in § i, the greater part of V. I l l is a faithful copy of the final emended draft of V, I. There are, however, some substantial additions in V. I l l compared with V. I. Some of these represent impor­ tant conceptual advances in Newton’s dynamical thought, such as the change in status of the parallelogram law from an hypothesis in V. I to a derived lemma in V. III. These additions are reproduced below with relevant footnotes in three sections. Section A consists of the preparatory definitions, laws, and lemmas of V. III. These are largely new although certain of the definitions and axioms of V. I reappear in the same or slightly different guise. The remaining sections B and C represent two large insertions in V. I ll compared with V. I. A Def. 1.^ Vim centripetam appello qua corpus attrahitur vel impellitur versus punctum aliquod quod ut centrum spectator, Def. 2,^ Et vim corporis seu corpori insitam qua id conatur perseverare in motu suo secundum lineam rectam. Def. 3.^ Et resistentiam quae est medii regulariter impedientis. Def. 4.^ Exponentes quantitatum sunt aliae quaevis quantitates proportionales expositis. Lex I.'"' 5 Sola vi insita corpus uniformiter in linea recta semper pergere si nil impediat. Lex 2.^ Mutationem^ status movendi vel quicscendi proportionalem esse vi impressae et fieri secundum lineam rectam qua vis ilia imprimitur. Lex 3.^ Corporum dato spatio inclusorum eosdem esse motus inter se sive spatium illud quiescat sive moveat id perpetuo et uniformiter in directum absque motu circulari. Lex Mutuis corporum actionibus commune centrum gravitatis non mutare statum suum motus vel quietis. Constat ex Lege 2.

IX

4'HE T R A C T

DE M O T U

295

Lex 5.'^ Resistentiam medii esse ut medii illius densitas et corporis moti sphaerica superficies et velocitas conjunctim. Lemma Corpus viribus conjunctis diagonalem parallelogrammi codem tempore describere quo latera separatis. Si corpus dato tempore vi sola m [Fig. i] ferretur ab A ad 5 , et vi sola n ab A ad C, compleatur parallelogrammum A B D C , et vi utraque feretur id eodem tempore ab A ad D. Nam quoniam vis n agit secundum lineam A C ipsi BD parallelam, haec vis, per Legem 2, nihil mutabit C

Figure i.

celeritatem accedendi ad lineam illam B D vi altera impressam. Accedet igitur corpus eodem tempore ad lineam BD sive vis A C imprimatur sive non, atque adeo in fine illius temporis reperietur alicubi in linea ilia BD. Eodem argumento in fine temporis ejusdem reperietur alicubi in linea CD, et proinde in utriusque lineae concursu D reperiri necesse est. Lemma 2.^^ Spatium quod corpus urgente quacunque vi centripeta ipso motus initio describit esse in duplicata ratione temporis. Exponantur tempora per lineas AB , A D [Fig. 2]. Datis Ah, Ad proportionales,^- et urgente vi centripeta aequabili exponentur spatia descripta per areas rectilineas ABT", A D H perpendiculis BF, DH, et recta quavis A F H terminatas, ut exposuit Galilaeus.'^ Urgente autem vi centripeta inaequabili exponantur spatia descripta per areas ABC, ADE, curva quavis A C E quam recta A F H tangit in A, comprehensas. Age rectam AE, parallelis BF', bf, dh occurrentem in G, g, e, et ipsis bf, dh occurrat A F H producta in / et h. Quoniam area A B C major est area A B F, minor area AB G , et area curvilinea A D E C major area AD H minor area ADFJG, erit area A B C ad aream AD E G major quam area ABF' ad aream ADEG, minor quam area A B G ad aream ADH, hoc est

296

THE T R A C T

DE M O T U

IX

major quam area A bf ad aream Ade, minor quam area Abg ad aream Adh. Diminuantur jam lineae A B , A D in ratione sua data usque dum puncta Al, B, D coeunt, et linea Ae conveniet cum tangente Ah, adeoque ultimae rationes A bf ad Ade et Abg ad Adh evadent eaedem cum ratione A bf ad Adh. Sed haec ratio est dupla rationis Ab ad Ad, seu A B ad AD. Ergo ratio A B C ad A D E C ultimis illis intermedia jam fit dupla rationis

A B ad AD, id est ratio ultima evanescentium spatiorum seu prima nascentium dupla est rationis temporum." Lemma 3. Quantitates difFerentiis suis proportionales sunt continue proportionales. Ponatur A ad A —B ut B ad B — C, tt C ad C —^D etc. et dividendo fiet A ad B ut B ad C, et C ad D, etc. Lemma 4. Parallelogramma omnia circa datam ellipsin descripta esse inter se aequalia. Constat ex Conicis. Next come eleven propositions. The first nine of them are headed De motu corporum in medijs non resistentibus. The last two are headed De motu corporum in medijs resistentibus. Except for the following additions, the text is substantially the same as that in Version I.

IX

THE T R A C T

DE M O T U

297

B At the end of the Scholium to Theorem 4 is the following paragraph; Caeterum totum coeli Planetarij Spatium vel quiescit (ut vulgo creditor) vel uniformiter movetur in directum et perinde Planetarum commune centrum gravitatis (per Legem'^ 4) vel quiescit vel una move­ tur. Utroque in casu motus Planetarum inter se (per Legem'' 3) eodem modo se habent, et eorum commune centrum gravitatis respectu spatij totius quiescit, atque adeo pro centro immobili Systematis totius Planetarij haberi debet. Inde vero systema Copernicaeum probatur a priori. Nam si in quovis Planetarum situ computetur commune centrum gravitatis hoc vel incidet in corpus Solis vel ei semper proximum erit. Eo Solis a centro gravitatis errore fit ut vis centripeta non semper tendat ad centrum illud immobile, et inde ut planetae nec moveantur in Ellipsibus exacte neque bis revolvant in eadem orbita. Tot sunt orbitae Planetae cujusque quot revolutiones, ut fit in motu Lunae et pendet orbita unaquaeque ab omnium Planetarum motibus conjunctis, ut taceam eorum omnium actiones in se invicem. Tot autem motuum causas simul considerare et legibus exactis calculum commodum admittentibus motus ipsos definire superat ni fallor vim omnium humani i ng enii .Omi tt e minutias illas et orbita simplex et inter omnes errores mediocris erit Ellipsis de qua jam egi. Siquis hanc Ellipsin ex tribus observationibus per computum trigonometricum (ut solet) determinate tentaverit, hie minus caute rem aggressus fuerit. Participabunt observationes iliac de minutijs motuum irregularium hie negligendis adeoque Ellipsin de justa sua magnitudine et positione (quae inter omnes errores mediocris esse debet) aliquantulum deflectere facient, atque tot dabunt Ellipses ab invicem discrepantes quot adhibentur observationes trinae. Conjungendae sunt igitur et una operatione inter se conferendae obser­ vationes quam plurimae, quae se mutuo contemperent et Ellipsin positione et magnitudine mediocrem exhibeant. C At the end of problem 5 and before problem 6 is the following Scholium: Schol. Hactenus motum corporum in medijs non resistentibus exposui; id adeo ut motus corporum coelestium in aethere determinarem. Aetheris enim puri resistentia quantum sentio vel nulla est vel perquam exigua. Valide resistit argentum vivum, longe minus aqua, aer vero longe

298

THE T R A C T

DE M O T U

IX

adhuc minus.'5 Pro densitate sua quae ponderi fere proportionalis est atque adeo (paene dixerim) pro quantitate materiae suae crassae resistunt haec media. Minuatur igitur aeris materia crassa et in eadem circiter proportione minuetur medij resistentia usque dum ad aetheris tenuitatem perventum sit. Celeri cursu equitantes vehementer aeris resistentiam sentiunt, at ^^navigantes exclusis e mari interiore ventis'*^ nihil omnino ex aethere praeterfluente patiuntur. Si aer libere interflueret particulas corporum et sic ageret, non modo in externam totius superficiem, sed etiam in superficies singularum partium, longe major foret ejus resistentia. Interfluit aether liberrime nec tamen resistit sensibiliter. Cometas infra orbitam Saturni descendere jam sentiunt Astronomi saniores quotquot distantias eorum ex orbis magni parallaxi praeterpropter colligere norunt; hi igitur celeritate immensa in omnes coeli nostri partes indifferenter feruntur, nec tamen vel crinem seu vaporem capiti circumdatum resistentia aetheris impeditum et abreptum amittunt. Planetae vero jam per annos millenos in motu suo perseverarunt, tantum abest ut impedimentum sentiant. Demonstratis igitur legibus reguntur motus in coelis. Sed et in aere nostro, si resistentia ejus non consideratur, innotescunt motus projectilium per Prob. 4 et motus gravium perpendiculariter cadentium per Prob. 5 posito nimirum quod gravitas sit reciproce proportionalis quadrato distantiae a centro terrae. Nam virium centripetarum species una est gravitas; et computanti mihi prodijt vis centripeta qua luna nostra detinetur in motu suo menstruo circa terram ad vim gravitatis his in superficie terrae, reciproce ut quadrata distantiarum a centro terrae quamproxime.^^ Ex horologij oscillatorij motu tardiore in cacumine montis praealti quam in valle liquet etiam gravitatem ex aucta nostra a terrae centro distantia diminui, sed qua proportione nondum observatum est. Caeterum projectilium motus in aere nostro referendi sunt ad immensum et revera immobile coelorum spatium, non ad spatium mobile quod una cum terra et aere nostro convolvitur et a rusticis ut immobile spectator. Invenienda est Ellipsis quam projectile describit in spatio illo vere immobili et inde motus ejus in spatio mobili determinandus. Hoc pacto colligitur grave, quod de aedificij sublimis vertice demittitur, inter cadendum deflectere aliquantulum a perpendiculo, ut et quanta sit ilia deflexio et quam in partem. Et vicissim ex deflexione experimentis comprobata colligitur motus terrae. Cum ipse olim hanc deflexionem Clarissimo Hookio significarem,^^ is experimento ter facto rem ita se

IX

THE T R A C T DE

MOTU

299

habere conlirmavit, deflectente semper gravi a perpendiculo versus orientem et austrum ut in latitudine nostra boreali oportuit. a. In this an d the other lazvs L e x has been substituted f o r H y p . deleted. b. P reced ed by insertion M o tu u m gen itu m deleted.

c. R eplacin g H y p . deleted.

Translation The Motion of Spherical Bodies in Fluids A Definition i . H call centripetal that force by which a body is attracted or impelled towards any point which is regarded as a centre [of force]. Definition 2.^ And I call that the force of a body or the force innate in a body by reason of which it endeavours to persist in its motion along a straight line. Definition 3.^ And the resisting force that arising from the steadily impeding medium. Definition The representatives of quantities are any other quantities proportional to those under consideration. Law 1.5 By its innate force alone a body will always proceed uniformly in a straight line provided nothing hinders it. Law 2.^ The change in the state of movement or rest [of a body] is pro­ portional to the impressed force and takes place along the straight line in which that force is impressed. Law 3.^ The relative motions of two bodies contained in a given space are the same whether the space in question rests or moves perpetually and uniformly in a straight line without circular motion. Law 4.^ By the mutual actions between bodies the common centre of gravity does not change its state of motion or rest. It follows from Law 2. Law 5.'^ The resistance of a medium is proportional conjointly to the density of the medium, the speed, and the spherical surface of the body moved. Lemma A body acted on simultaneously by [two] forces describes the diagonal of a parallelogram in the same time as it would the sides if the forces acted separately. If the body were carried in a given time from A [Fig. i] to B by the force m alone, and from A to C by the force n alone, then completing the

300

THE T R A C T

DE M O T U

IX

parallelogram AB D C, it will be carried by both forces in the same time from A to D. For since the force n acts along the line AC, parallel to BD, this force, by Law 2, will nowise change the speed of attaining that line BD induced by the other force. Therefore the body will attain to the line BD in the same time whether a force is impressed in A C or not, and therefore at the end of that time it will be found somewhere in that line BD. By the same argument at the end of the same time it will be found somewhere in the line CD, and hence it must be found at the intersection of the two lines. Lemma 2.” The distance a body describes from the beginning of its motion under the action of any force whatsoever is in the duplicate ratio of the time. Represent the times by the lines AB , A D [Fig. 2]. Let Ab, A d be proportionah^ to these lines, then when [the body is] acted on by a uniform centripetal force the distances covered will be represented by the rect­ angular areas A B F , A D H terminated by the perpendiculars BF, D H and a certain line AFH , as shown by Galileo. But when acted on by a non-uniform centripetal force the distances covered will be represented by areas A B C , ADE, terminated by a certain curve A C E which A F H touches at A. Draw a line A E cutting the parallels BF, bf, dh at G, g, e and let A F H produced meet these same bf, dh at / and h. Since the area A B C is greater than the area A B F and less than the area A B C , and the curvilinear area A D E C is greater than the area A D H and less than the area ADEG, the ratio of the area A B C to the area AD EG will be greater than the ratio of area A B F to area ADEG, and less than the ratio of area A B G to ADH, that is greater than area Ahfto the area Ade, and less than area Abg to area Adh. If now the lines AB, A D are diminished while maintaining their ratio until the points A, B, D coincide and the line Ae coincides with the tangent Ah, the limiting ratios Abf to Ade and Abg to Adh will coincide with the ratio Abf to Adh. But this ratio is the square of the ratio Ab to Ad, or A B to AD. Therefore the ratio A B C to A D E C intermediate between these limits now becomes the square of the ratio A B to AD, that is the limiting ratio of the evanescent spaces or the first ratio of the vanishing spaces is the square of the ratio of the times. Lemma 3. Quantities proportional to their differences are continually proportional. Assuming A to A ~ B as B to B ~ C , and C to C — D etc., then by division we have A to ^ as /? to C, and C to D etc.

IX

^niE TRAC/r

DE M O T U

301

Lemma 4. All parallelograms drawn about a given ellipse are equal. It follows from conic sections. B Moreover the whole space of the planetary heavens either rests (as is commonly believed) or moves uniformly in a straight line, and hence the communal centre of gravity of the planets (by Law 4) either rests or moves along with it. In both cases (by Law 3) the relative motions of the planets are the same, and their common centre of gravity rests in relation to the whole of space, and so can certainly be taken for the still centre of the whole planetary system. Hence truly the Copernican system is proved a priori. For if the common centre of gravity is calculated for any position of the planets it either falls in the body of the Sun or will always be very close to it. By reason of this deviation of the Sun from the centre of gravity the centripetal force does not always tend to that immobile centre, and hence the planets neither move exactly in ellipse nor revolve twice in the same orbit. So that there are as many orbits to a planet as it has revolutions, as in the motion of the Moon, and the orbit of any one planet depends on the combined motion of all the planets, not to mention the action of all these on each other. But to consider simul­ taneously all these causes of motion and to define these motions by exact laws allowing of convenient calculation exceeds, unless I am mistaken, the force of the entire human intellect.’ ^ Ignoring those minutiae, the simple orbit and the mean among all errors will be the ellipse of which I have already treated. If anyone tries to determine this ellipse by trigo­ nometrical computation from three observations (as is customary), he will have proceeded without caution. For these observations will share in the very small irregular motions here neglected and so cause the ellipse to deviate somewhat from its actual magnitude and position (which ought to be the mean among all errors), and so there will be as many ellipses differing from each other as there are trios of observations em­ ployed. Very many observations must therefore be joined together and assigned to a single operation which mutually moderate each other and display the mean ellipse both as regards position and magnitude. Scholium. Thus far I have considered the motion of bodies in non­ resisting media: so that I may determine the motion of celestial bodies in ether. But as far as I can judge the resistance of pure ether is either nothing or excessively small. Quicksilver resists strongly, water much

302

THE T R A C T

DE M O T U

iX

less, air certainly far less again, These media resist according to their density which is almost proportional to their weight and so they resist (or rather almost resist) according to the quantity of their solid matter. Therefore as the solid matter of air is diminished so is the resistance of the medium and in about the same proportion up to the point that it attains to the tenuity of ether. In swift flight horsemen feel the violent resistance of the air, but '^sailors inside the ship when the winds are excluded!^ feel nothing at all of the ever flowing ether. If air were to penetrate the parts of bodies freely and so were to act not only in the external surface of the whole but also in the surfaces of the individual parts, its resistance would be far greater. Now ether penetrates freely but does not offer sensible resistance. That comets descend below the orbit of Saturn is the opinion of all those sounder astronomers who know how to calculate their approximate distance from the parallax of the great orbit: these comets are therefore carried with immense speed indifferently in all parts of our heavens yet do not lose their tail nor the vapour surrounding their heads [by having them] impeded or torn away by the resistance of the ether. And the planets actually have now per­ sisted in their motion for thousands of years, so far are they from ex­ periencing any resistance. Motions in the heavens are ruled therefore by the laws demonstrated. But the motions of projectiles in our air, ignoring its resistance, are known by Problem 4, and also the motions of heavy bodies falling per­ pendicularly by Problem 5, given of course that gravity is inversely proportional to the square of the distance from the centre of the earth. For certainly gravity is one kind of centripetal force: and my calculations reveal that the centripetal force by which our Moon is held in her monthly motion about the Earth is to the force of gravity at the surface of the earth very nearly as the reciprocal of the square of the distance [of the Moon] from the centre of the Earth. From the slower motion of oscillating clocks in the tops of very high mountains than in valleys it is clear also that gravity is diminished by an increase in our distance from the centre of the earth, but in what proportion has not yet been observed. Moreover the motions of projectiles in our air are to be referred to the immense and truly immobile space of the heavens, not to the movable space which turns round with the earth and our air and is commonly regarded as at rest. The ellipse which the projectile describes in that truly immobile space is first to be found, and then its motion in the mobile space must be determined. Knowing this it is inferred that a heavy body

IX

THE T R A C T DE

303

MOTU

released from the top of a high building will deviate a little from the vertical in falling, and also how much that deflection will be and to what side. And conversely from experimental evidence of this deflection the [diurnal] movement of the earth is inferred. When I myself formerly pointed out this deflection to the most excellent Hook,’^he confirmed the matter to be so by an experiment performed three times, the deflection in the heavy body being always from the vertical towards the east and south as in our northern latitude it should. 1. S e e M S . X a , D e f. 16. 2. See M S . X a , D e f. 12. 3. See M S . X a , D e f. 17. 4. See M S . X a , D e f. 18. 5. See M S . X a , L e x i. 6. See M S . X a , L e x 2. 7. See M S . X a , L e x 4. k

See M S . X a , L e x 5.

9. See M S . X a , L e x 6. 10. See M S . X a , L e m m a i . A p a rt from a reference to L a w 2 n ot fo und in the p resent p ro o f the tw o versions are identical. N o tice that the forces referred to in this lem m a are n ot continuous b u t im pulsive. 1 1 . See M S . X a , L em m a 2. 12. T h e phrase is som ew hat obscure b u t it is probable that b, d are intended to b e any tw o points on the line su ch that A b / A d = A B j A D . T h is is confirm ed e x p licitly in the pen ultim ate sentence o f the proof. T h e s e points are introduced w ith the ultim ate lim itin g process in v i e w ; b, d are to b e kep t fixed w h ile By D ten d to A so that A B j A D remains fixed and therefore equal to A b j A d . 13. See above, Part I, C h a p ter 2 .1. N e w to n , how ever, w as m istaken in su p ­ p osing that G alileo ever gave a satisfactory

proof o f

the M e r to n R u le . It seems

likely that Beeckm an w as the o n ly person to do so in the seven teen th century. H is p ro of rem ained un kn ow n till the rediscovery o f his Journal b y de W aard. 14. C u rio u sly rem iniscent o f the line from L u cretiu s on R o u b ilia c ’s statue o f N e w to n in the chapel o f T r in it y C o llege , C am b rid ge—

Qui genus humanumm

ingenio superavit. 15. C f. M S . V I , § 8, first sentence. 1 6 -1 6 . T h e phrase is obscure. P o ssibly N e w to n th o u gh t o f the a p p lyin g to the

navigantes.

A ltern a tive ly

e mari could

exclusis e mari in nave.

be an error for

17. C o m pare the p reced in g sentence w ith M S . V I , § 8, third paragraph. 18. T h e first explicit reference b y N e w to n to this

experimentiim crucis

o f his

theory o f gravitation. 19. A reference to the correspondence w ith H ooke in the w in ter o f 1679.

Xa

X Xa D R A F T S OF D E F I N I T I O N S A N D LAWS OF M O T I O N Xb D R A F T S OF D E F I N I T I O N S T h e s e two manuscripts, referred to hereafter as MSS. Xa and Xb, clearly preceded the lectures de Motu,-\ constituting preliminary drafts of parts of the Definitions and Axioms of that work. It is also probable that they succeeded Version III of the tract de Motu. The actual order of composition of the two manuscripts seems to be of no great impor­ tance, but it will be convenient to give them in their probable order of composition, namely Xa followed by Xb. Although there is a certain amount of overlap between the two manuscripts, each contains material not found in the other. In particular, Xa alone contains drafts of the discussion of absolute space, time and motion in the Scholium to the Definitions of Book I of the Lectures de Motu (and the Principia), and of the Laws of Motion, whereas Xb consists largely of drafts of the Definitions. Both manuscripts contain a considerable number of insertions and alterations.

M S . Xa^

D e M otu Corportim in m edijs regu la riter cedentibus Definitiones

Def 12 Tempus absolutum est quod sua natura absque relatione ad aliud quodvis aequabiliter fluit. Tale est, cujus aequationem investigant Astronomi, alio nomine dictum Duratio. Def 2^ Tempus relative spectatum est quod respectu fluxionis seu transitus rei alicujus sensibilis consideratur ut aequabile. Tale est tempus dierum mensium et aliarum periodorum caelestium apud vulgus. Def 3 Spatium absolutum est quod sua natura absque relatione ad aliud quodvis semper manet immobile, Ut partium temporis ordo immutabilis est sic etiam partium spatij, Moveantur hae de locis suis et mo\ ebuntur de seipsis. Nam tempora et spatia sunt suiipsorum et rerum t The reasons for this and succeeding assertions a.bout order of composition are fii\ en aho\ e in Part I, Cliapter 6 .3 .

D E F I N I T I O N S AND L AW S OF MOTION

305

omnium loca. In tempore quoad ordinem successionis, in spatio quoad ordinem situs locantur universa. De illorum essentia est ut sint loca et loca primaria mover! absurdum est. Porro vi illata moveatur una pars spatij et vi tanta ad omnes in infinitum partes applicata movebitur totum, quod rursus absurdum est. Def 4 Spatium relativum est quod respectu rei alicujus sensibilis con­ sideratur ut immobile: uti spatium aeris nostri respectu terrae. Distinguuntur autem haec spatia ab invicem ipso facto per descensum gravium quae in spatio absoluto recta petunt centrum in relative absolute gyrante deflectunt ad latus. Def 5'* Corpora in sensus omnium incurrunt ut res mobiles quae se mutuo penetrare nequeunt. Def 6^'^ Centrum corporis cujusque est quod vulgo dicitur centrum gravitatis et axis corporis est linea quaevis recta per centrum transiens. Def 75 Locus corporis est pars spatij in quo corpus existit, estque pro genere spatij vel absolutus vel relativus. Def 8^ Quies corporis est perseverantia ejus in eodem loco, estque vel absoluta vel relativa pro genere loci. Def 9’ Motus corporis est translatio ejus de loco in locum, estque itidem vel absolutus vel relativus pro genere loci. Distinguitur autem ipso facto motus absolutus a relatiVO in gyrantibus, per conatum recedendi a centre, quippe qui ex gyratione nude relativa nullus est, in relative quiescentibus permagnus esse potest, ut in corporibus coelestibus quae ex mente Cartesianorum^ quiescant, tamen a sole recedere conantur. Conatus ille certus semper et determinatus arguit certam aliquam et determinatam esse motus realis quantitatem in singulis corporibus, a relationibus quae innumerae sunt totidemque motus relatives constituunt minime pendentem. Porro motum et quietem absolute dictos non pendere a situ et relatione corporum ad invicem manifestum est ex eo quod hae nunquam mutantur nisi vi in ipsum corpus motum vel quiescens impressa, tali ante vi semper mutantur; at relativae mutari possunt vi solummodo impressa in altera corpora ad quae sit relatio et non mutari vi impressa in utraque sic ut situs relativus conservetur. Def Velocitas^ est quantitas translationis quoad longitudinem itineris certo tempore confecti. Iter vero est quod corporis puncto medio describitur a Geometris dicto centro gravitatis. Loquor de motu progressive.

306

D E F I N I T I O N S AND L A W S OF MOTION

Xa

Def i i ‘° Quantitas motus est quae oritur ex velocitate et quantitate corporis translati conjunctim. Aestimatur autem quantitas corporis^ ^ ex copia materiae corporeae quae gravitati suae proportionalis esse solet. Pendulis aequalibus numerentur oscillationes corporum duorum ejusdem ponderis, et copia materiae in utroque erit reciproce ut numerus oscillationum eodem tempore factarum.^^ Def 12*3 Corporis'^ vis insita innata et essentialis est potentia qua id perseverat in statu suo quiescendi vel movendi uniformiter in linea recta, estque corporis quantitati proportionalis, exercetur vero proportionaliter mutationem status et quatenus exercetur dici potest corporis vis exercita . . . cuius [?] una species est vis centrifuga*^ gyrantium. Def i3*s.e Vis motus seu corpori ex motu sua adventitia est qua corpus quantitatem totam sui motus conservare conatur. Ea vulgo dicitur impetus estque motui proportionalis, et pro genere motus vel absoluta est vel relativa. ^Ad absolutam referenda est vis centrifuga gyrantium^. Def 14*6’^ Vis corpori illata et impressa est qua corpus urgetur mutare statum suum movendi vel quiescendi estque diversarum specierum ut pulsus seu pressio percutientis, pressio continua, vis centripeta, resistentia medij etc. Def 161*7 Vim centripetam appello qua corpus impellitur vel attrahitur versus punctum aliquod quod ut centrum spectatur. Hujus generis est gravitas tendens ad centrum terrae, vis magnetica tendens ad centrum magnetis et vis coelestis cohibens Planetas ne abeant in tangentibus orbitarum.

Xa

D E F I N I T I O N S AND L A W S OF MOTION

307

et relativas ab invicem sedulo distinguere necesse fuit eo, quod phaenomena omnia pendeant ab absolutis. Vulgus autem qui cogitationes a sensibus abstrahere nesciunt semper loquuntur de relativis, usque adeo ut absurdum foret vel sapientibus vel etiam Prophetis apud bos aliter loqui. Unde et sacrae literae et scripta Theologorum de relativis semper intelligenda sunt, et crasso laboraret praejudicio qui inde de rerum naturalium motibus absolutis PhilosophicisJ disputationes moveret.*^ Leges Motus Lex 1^1 Vi insita corpus omne^ perseverare in statu suo quiescendi vel movendi uniformiter in linea recta nisi quatenus viribus impressis’" cogitur statum ilium mutare. Motus autem uniformis hie est duplex, progressivus secundum lineam rectam quam corpus centro suo aequabiliter lato describet et circularis circa axem suum quemvis qui vel quiescit vel motu uniform! latus semper manet positionibus suis prioribus parallelus. Lex 2^2 Mutationem motus proportionalem esse vi impressae et fieri secundum lineam rectam qua vis ilia imprimitur. Hisce duabus Legibus jam receptissimis Galilaeus invenit projectilia gravitate uniformiter et secundum lineas parallelas agente, in medio non resistente lineas Parabolicas describere. Et suffragatur experientia nisi quatenus motus projectilium resistentia aeris aliquantulum retardatur. Ab.

Def 162*'^’ * Momenta quantitatum sunt ipsarum principia generantia vel alterantia fluxu continue: ut tempus praesens praeteriti et futuri, motus praesens praeteriti et futuri, vis centripeta aut alia quaevis momentanea impetus, punctum lineae, lineae superficiei, superficies solidi et angulus contactus anguli rectilinei.

Lex 3^3 Corpus omne tantum pati reactione quantum agit in alterum. Quicquid premit vel trahit alterum, ab eo tantum premitur vel trahitur. Si vesica acre plena premit vel ferit alteram sibi consimilem cedet utraque aequaliter introrsum.'^ Si corpus impingens in alterum vi sua mutat motum alterius et ipsius motus (ob aequalitatem pressionis mutuae) vi alterius tantum mutabitur. Si magnes trahit ferrum ipse vicissim tantum trahitur et sic in alijs. Constat vero haec Lex per Def. 12 et 1424 in quantum vis corporis ad status sui conservationem exercita sit eadem cum vi in corpus alterum ad illius statum mutandum impressa, et vi priori proportionalis sit mutatio status prioris posteriori ea posterioris.

Def 18 Exponentestemporumspatiorum, motuum celeritatum et virium sunt quantitates quaevis proportionales exponendis.^** Haec omnia fusius explicate visum est ut Lector praejudeijs [r/c] quibusdam vulgaribus liberatus et distinctis principiorum Mechanicorum conceptibusimbutusaccederet ad sequentia. Quantitates autem absolutas

Lex 4-3 Corporum dato spatio inclusorum eosdem esse motus inter se sive spatium illud absolute quiescat sive moveat id perpetuo et uni­ formiter in directum absque motu circulari. E.g. motus rerum in navi perinde se habent sive navis quiescat sive moveat ea uniformiter in directum.

Def 17*8 !' Per medij Resistentiam in sequcntibus intelligo vim medij regulariter impedientis. Sunt et aliae vires ex corporum elasticitate, mollitie, tenacitate etc., pendente quas hie non considero.

308

D E F I N I T I O N S AND L A W S OF MOTION

Xa

Lex 526 Mutuis corporum actionibus commune centrum gravitatis non mutare statum suum motus vel quietis. Haec lex et duae superiores se mutuo probant. Lex 627.0 Resistentiam medij esse ut medij illius densitas et sphaerici corporis moti superficies et velocitas conjunctim. Hanc legem exactam esse non affirmo. Sufficit quod sit vero proximo. Corpora vero Sphaerica esse suppono in sequentibus, ne opus sit circumstantias diversarum figurarum considerare. Lemmata Lem. Corpus viribus conjunctis diagonalem parallelogrammi eodem tempore describere quo latera separatis.

D E F I N I T I O N S AND L A W S OF MOTION

Xa

309

a. T here are a considerable number o f unim portant an d p a rtly illegible deletions in the original text o f this definition. b. Introdu ced a t some tim e a fter com position o f present D e f. i S as proved by the renumbering o f a ll D e f. 7 - 1 8 . c. R ep lacin g celeritas m otus deleted. d. Preceded by V is corporis seu deleted. e. T h is section deleted. f - f . T h is section deleted. g. O rig in ally num bered 1 3 an d changed to 1 5 , presum ably in error f o r 1 4 . h . T h is section deleted. i. D eleted , presum ably after introduction o f previous D e f. 1 6 , as the number 1 6 has not been altered. j. W ritten above absolutus. k. Succeed ed by P erinde est ac si quis lunam [(in G em in i) ?] m agn itudin e non apparente sed absoluta inter duo m axim a lum ina [G em in i ?] num erari co n tenderet cancelled. l.

S u b stitu ted f o r sem per deleted.

m . Succeed ed by im pedim entis deleted.

D

n. S u cceed ed by Si m agnes trahit ferrum ipsi vicissim tan tum trahitur deleted an d inserted later belozc. o. T h is section deleted. p. E n ds abruptly here.

Translation

On the M otion o f B odies in uniform ly yield in g m edia B Figu re i.

Si corpus dato tempore vi sola M ferretur ab ^ ad ^ [Fig. i] et vi sola N ab A ad C, compleatur parallelogrammum A B D C et vi utraque feretur id eodem tempore ab A ad D. Nam quoniam vis M agit secun­ dum lineam A C ipsi BD parallelam, haec vis nihil mutabit celeritatem accidendi ad lineam illam BD vi altera impressam. Accedet igitur corpus eodem tempore ad lineam BD sive vis A C imprimatur sive non, atque adeo in fine illius temporis reperietur alicubi in linea ilia BD. Eodem argumento in fine temporis ejusdem reperietur alicubi in linea CD, et proinde in utriusque lineae concursu D reperiri necesse est. Lem. 229 Spatium quod corpus urgente quacunque vi centripeta ipso motus initio describit, esse in duplicata ratione temporis. Exponantur tempora per lineas AB, A D datis Ab Ad proportionales, et urgente vi centripeta aequabili exponentur spatia descripta per areas rectilineas A B F A D H perpendiculis.p

Definitions Definition 12 Absolute time is that which according to its own nature, unrelated to anything else, flow's evenly. It is that whose equation astronomers investigate, and by another name is called duration. Definition 2 Time regarded as relative is that which is uniform in respect of the flux or variation of any sensible thing. Such is the time of days, months, and other periodic celestial phenomena as commonly received. Definition 3 Absolute space is that which by its own nature and un­ related to any other thing whatsoever always remains at rest. As the order of temporal parts is immutable so also is that of parts of space. If these were to move from their places they would move out of themselves. For times and spaces are of themselves and all things the places. All things are located in time as regards order of succession, and in space as regards order of situation. The essence of these is that they are positions and it is absurd that basic positions be moved. For example if one part of space mav be moved by a certain force then if an equal force be applied to all

310

D E F I N I T I O N S AND L A W S OF MOTION

Xa

parts of space to infinity the whole of space will be moved, which is again absurd. Definition 4 Relative space is that which is regarded as immobile in relation to any sensible thing; such as the space of our air in relation to the earth. However these spaces are in fact distinguished from each other through the descent of heavy bodies which in absolute space seek the centre directly but in relative space rotating absolutely are deflected to one side. Definition 5^ By common consent bodies are moveable things unable to penetrate each other. Definition 6“^ The centre of any body is what is commonly called its centre of gravity, and the axis of a body is any straight line through the centre. Definition 75 The place of a body is the part of space in which the body exists, and according to the kind of space is absolute or relative. Definition 8^ A body’s rest is its continuation in the same place, and according to the kind of place is absolute or relative. Definition 9^ The motion of a body is its translation from one place to another, and is consequently either absolute or relative according to the kind of place. But absolute motion is in fact distinguished from relative in circular motions by the endeavour to recede from the centre, which in an entirely relative circular motion is zero, but in a circular motion relative to bodies at rest may be very large, as in the celestial bodies which the Cartesians® believe to be at rest, although they endeavour to recede from the sun. The fact that this endeavour [from the centre of circular motion] is certain and determinate argues some certain and determinate quantity of real motion in individual bodies in no wise dependent on the relations [between the bodies] which are innumerable and make up as many relative motions. For example, that motion and rest absolutely speaking do not depend on the situation and relation of bodies between themselves is evident from the fact that these are never changed except by force impressed on the body moved or at rest, and are always changed after [the action of] such a force; but the relative [motion and rest of a body] can be changed by forces impressed only on other bodies to which the relation belongs, and is not changed by a force impressed on both so that their relative situation is preserved.

Xa

D E F I N I T I O N S AND L AW S OF MOTION

311

Definition lo^ Velocity is the quantity of translation as regards the length of path traversed in a certain time. Where of course the path is that described by the middle point of the body called the centre of gravity by mathematicians, I speak of progressive motion. Definition ii**’ The quantity of motion is that which arises from the velocity and quantity of a body conjointly. Moreover the quantity of a body” is to be estimated from the bulk of the corporeal matter which is usually proportional to its gravity. The oscillations of two equal pendu­ lums with bodies of equal weight are counted, and the bulk of matter in both will be inversely as the number of oscillations made in the same time.” Definition 12” The internal and innate force of a body is the power by which it preserves in its state of rest or of moving uniformly in a straight line. It is proportional to the quantity of the body, and is actually exer­ cised proportionally to the change of state, and in so far as it is exercised it can be said to be the exercised force of the body, of which [ ?] one kind is the centrifugal force” of rotating bodies. Definition 13” The force of a body arising from its motion is that by which the body endeavours to preserve the total quantity of its motion. It is commonly called impetus and is proportional to the motion, and according to its kind is absolute or relative. The centrifugal force of rotating bodies is to be referred to the absolute kind. Definition 14^^ Force impressed on a body is that by which a body is urged to change its state of moving or resting and is of divers kinds such as impulse or pressure of percussion, continuous pressure, centripetal force, resistance of medium etc. Definition ifij” I call centripetal force that by which a body is impelled or drawn towards a certain point regarded as centre. Of this kind is gravity tending to the centre of the earth, magnetic force tending to the centre of the magnet, and the celestial force preventing the planets from flying off in the tangents of their orbit. Definition 17^® By the resistance of the medium in the following I under­ stand the force of a uniformly resisting medium. And there are other forces arising from the elasticity, softness, tenacity etc., of bodies which I do not consider here. Definition 162” The moments of quantities are their principles of gene­ ration or alteration, as time present of the past and future, present

312

DEFIxNri’IONS AND L A W S OF MOTION

Xa

motion of past and future motion, centripetal or any other momentary force of impetus, as point of a line, a line of a surface, a surface of a solid, and a contacting angle of a straight angle. Definition i8 The representatives of times, spaces, motions, speeds and forces are any quantities whatsoever proportional to the things rep re­ sented The aim of explaining all these things at length is that the reader may be freed from certain vulgar prejudices and imbued with the distinct principles of mechanics may agree in what follows to distinguish carefully from each other quantities which are both absolute and relative, a thing very necessary since all phenomena depend on absolute quantities. But ordinary people who fail to abstract thought from sensible appearances always speak of relative quantities, so much so that it would be absurd for wise men or even Prophets to speak to them otherwise [than of relative quantities]. Hence both the sacred writings and theological writings are always to be understood in terms of relative quantities, and he who would on this account bandy words with philosophers concerning the absolute motions of natural things would be labouring under a gross misappre­ hension. Lam of MotionLav^ 1 B y reason of its innate force every body preserves in its state of rest or of moving uniformly in a straight line unless in so far as it is obliged to change its state by forces impressed on it. Uniform motion, however, is of tw'o kinds, progressive along a straight line which the body describes uniformly with its centre, and circular about a certain axis which either rests or with a motion of constant size always remains parallel to its previous position. Law 2^^ The change of motion is proportional to the force impressed and takes place along the straight line in which the force is impressed. By means of these two laws now widely acknowledged Galileo discovered that projectiles under a uniform gravity acting along parallel lines described parabolas in a non-resisting medium. And experiment sup­ ports this unless in so far as the motion of the projectiles is a little retarded by the resistance of the air. Law 32-’ As much as any body acts on another so much does it experience in reaction. Whatever presses or pulls another thing by this equally is pressed or pulled. If a bladder full of air presses or carries another equal

X;i

D E F I N I T I O N S AND L A W S OF MOTION

313

to itself both yield equally inwards. If a body impinging on another changes by its force the motion of the other then its own motion (by reason of the equality of the mutual pressure) will be changed by the same amount by the force of the other. If a magnet attracts iron it is itself equally attracted, and likewise in other cases. In fact this law follows from Definitions 12 and 14^4 in so far as the force exerted by a body to conserve its state is the same as the impressed force in the other body to change the state of the first, and the change of state of the first is proportional to the first force and of the second to the second force. Law 425 The relative motion of bodies enclosed in a given space is the same whether that space rests absolutely or moves perpetually and uniformly in a straight line without circular motion. For example, the motions of objects in a ship are the same whether the ship is at rest or moves uniformly in a straight line. Law 52^ The common centre of gravity of [a number of] bodies does not change its state of rest or motion by reason of the mutual actions of the bodies. This law and the two above mutually confirm each other. Law 6^7 The resistance of a medium is jointly proportional to the density of that medium, the area of the moved spherical body and the velocity. I do not assert this law to be exact. It suffices that it should be approxi­ mately true. Hereafter I actually suppose the bodies spherical lest it be a question of considering the states of different figures. Lemmata Lemma A body acted on simultaneously by [two] forces describes the diagonal of a parallelogram in the same time as it would the separate sides. If the body is carried in a given time by the force M alone from A to B [Fig. i] and by the force N alone from A to C, then the parallelo­ gram A B D C being completed it will be borne by both forces in the same time from A to D. For since the force M acts along the line A C parallel to BD, this force will nowise change the speed of attaining to that line BD impressed on it by the other force. The body will therefore attain to the line BD in the same time whether the force A C is impressed or not, and so at the end of that time it will be found somewhere in that line BD. By the same reasoning at the end of that time it will be found somewhere in the line CD, and thus it must be found in the inter­ section D of both lines.

314

D E F I N I T I O N S AND L A W S OF MOTION

Xa

Lemma 2^^ The space described by a body from the beginning of its motion under the action of any force is proportional to the square of the time. Let the times be represented by lines AB , A D to which Ab, Ad are proportional, then under the action of a uniform centripetal force the spaces described are represented by the rectilinear areas A B F , A D H by the perpendiculars. 1. M S . A d d . 3965 (5a), fols. 25, 26 (right half) 23, 24. 2. C o m pare this and the tw o fo llo w in g definitions w ith the corresponding ones in the

Scholium to

the D efinitions o f the lectures

de Motu.

3. C o m pare this w ith note (a) to D e f. 2 o f M S . V I . T h e r e is no correspon din g definition in the lectures

de Motu

or the

Principia.

4. C o m pare w ith the original D e f. 3, later cancelled, o f the lectures

D E F I N I T I O N S AND L A W S OF MOTION

315

22. C o rrespo n din g to the second law o f m otion, together w ith the germ o f the first part o f the S ch o liu m to the laws o f m otion in the lectures de M o tu and the P rin cip ia . 23. A prim itive draft o f the third law o f m otion. 24. T h e reference here w o u ld seem to be to D e f. 12, 14 o f M S . X b . 25. C o rresp o n d in g to C o ro ll. 5 to the laws o f m otion in the lectures de

M o tu and the P rin cip ia . 26. C o rrespo n din g to C o ro ll. 4 to the laws o f m otion in the lectures de M o tu and the P rin cip ia . 27. C o rrespo n din g to Section I o f B ook I I o f the P rin cip ia . 28. C o rrespo n din g to C o ro ll, i to the laws o f m otion in the lectures de M o tu and the P rin cip ia . 29. C o rrespo n din g to L e m m a X o f Sectio n I Book I o f the lectures de M o tu and the P rin cip ia .

de Motu

M S . Xb^

(M S. X I).

Scholium to

5. C om pare this w ith D e f. 3 o f the

de Motu

Xa

and the

the D efinitions o f the lectures

Principia.

De M otu C orporum

6. C o m pare this w ith D e f. 3 o f M S . V I . T h e r e is no corresponding definition in the lectures

de Motu

or the

Principia.

Definitiones

7. T h e germ o f the celebrated attem p t to distin guish absolute from relative m otion in the

Scholium

to the D efinitions o f (the lectures

de Motu

and) the

Principia. 8. T h is specific reference to the Cartesian p h ilo so p h y is significantly om itted both in the lectures

de Motu

and the

Principia.

9. N o definition o f v elo city is giv en either in the lectures

de Motu

or the

Principia. 10. C o rrespo n din g to D e f. 2, M S . X b , and to D e f. 4 o f the lectures

de Motu.

1 1 . C o m p a re the alternative definition o f qu a n tity o f m atter in D e f. i , o f M S . Xb. 12. F o r a m ore detailed expression o f the same ideas see D e f. 7 o f M S . X b . 13. A m ore detailed treatm ent o f this co n cep t is given in D e f. 3 o f M S . X b . 14. T h e r e is a reference to centrifugal force as an exam ple o f the

exercita in

corporis

the cancelled definition 14 o f M S . X b , b u t no such reference occurs

in D e f. 3 o f that m an uscript nor in the correspon din g definition o f the lectures

de Motu

or the

Principia.

F o r a discussion o f N e w to n ’s concept o f cen trifugal

force see above, Part I, C h a p ter 3. 15. C om pare this w ith the cancelled D e f. 14 o f M S . X b . 16. C o rrespo n din g to D e f. 4 o f M S . X b . 17. C o rresp o n d in g to D e f. 5 o f M S . X b . 18. A n echo o f the corresponding definition in all versions o f the tract

Motu. N o such definition Motu or the Principia.

is given in M S . X b , nor in B ook I o f the lectures

de de

19. N o trace o f this definition, later cancelled, is fo und in M S . X b nor in the lectures

de Motu or the Principia.

20. C o m pare this w ith D e f. 4 o f V ersion I I I o f the tract

de Motu.

N o trace

o f it is fo u nd in M S . X b , or thereafter. 21. C o rrespo n din g to the first law o f m otion.

B u t the reference to

tzvo

types o f uniform m otion, one rectilinear, the other circular, is not found else­ where.

See above. Part I, C h a p ter 5, § 3.

1.2 Quantitas materiae est quae oritur ex ipsius densitate et magnitudine conjunctim. Corpus duplo densius in duplo spatio quadruplum est. Hanc quantitatem per nomen corporis vel massae designo. 2.3 Quantitas motus est quae oritur ex velocitate et quantitate materiae conjunctim. Motus totius est summa motuum in partibus singulis, adeoque in corpore duplo majore aequali cum velocitate duplus est et dupla cum velocitate quadruplus. 3. '^ Materiae vis insita est®potentia resistendi qua corpus unumquodque quantum in se est perseverat in statu suo vel quiescendi vel movendi uniformiter in directum: estque corpori suo proportionaliss neque differt quicquam ab inertia massae nisi in modo conceptus nostri. Exercet vero corpus hanc vim '^solummodo^ in mutatione status sui facta per vim aliam in se impressam estque Exercitum ejus Resistentia et Impetus respectu solo ab invicem distinct!: Resistentia quatenus corpus reluctatur vi impressae. Impetus'^ quatenus corpus difficulter cedendo conatur mutare statum corporis alterius. Vulgus apud resistentiam quiescentibus et impetum moventibus tribuit: sed motus et quies ut vulgo concipiuntur respectu solo distinguuntur ab invicem; neque vere quiescunt quae vulgo tanquam quiescentia spectantur^. 4. ® Vis impressa est "^actio in corpus exercita ad mutandum statum ejus vel quiescendi vel movendi. Consistit haec vis in actione sola neque post

316

DEFINITIONS

Xb

actionem permanet in corpore. Est autem diversarum originum, ut ex impetu, ex pressione, ex vi centripeta.^' 5.9 Vis centripeta est vel actio vel potentia quaelibet qua corpus versus punctum aliquod tanquam ad centrum trahitur impellitur vel utcunque tendit. Hujus generis est gravitas qua corpus tendet ad centrum terrae, vis magnetica qua ferrum petit centrum magnetis, et vis ilia, quaecunque sit, qua Planetae retinentur in orbibus suis et perpetuo cohibentur ne abeant in eomm tangentibus. Est autem vis centripetae quantitas triplex; absoluta, acceleratrix et inotrix. Quantitas absoluta (quae et vis absoluta dici potest) major est ad unum centrum minor ad aliud, nullo habito respectu ad distantias et magnitudines attractorum corporum; uti virtus magnetica major in uno magnete minor in alio. Quantitas seu vis acce­ leratrix est velocitati proportionalis quam dato tempore generat; uti virtus magnetis ejusdem major in minori distantia minor in majori, vel vis gravitatis major prope terram minor in regionibus superioribus. Quan­ titas seu vis matrix est motus proportionalis quern dato tempore produ­ c t ; uti pondus majus in majori corpore minus in minore. Ita se habet igitur vis matrix ad vim acceleratricem ut matus ad celeritatem. Namque oritur quantitas matus ex celeritate dueta in corpus mobile et quantitas vis matricis ex vi acceleratrice ducta in idem corpus, ETnde juxta superliciem terrae ubi gravitas acceleratrix in corporibus universis eadem est, gravi­ tas motrix seu pondus est ut corpus: at si longius recedatur a terra, inque regiones ascendatur ubi gravitas acceleratrix fit minor, pondus pariter minuetur eritque semper ut corpus in gravitatem acceleratricem ductum. Porro attractianes et impulsus eodem sensu acceleratrices matrices nomino. Voces autem attractianis impulsus vel prapensianis cujuscunque in cen­ trum indifferenter et pro se mutuo usurpo, has vires non physice sed mathematice tantum considerando. Unde caveat Lector ne per hujusmodi voces cogitet me speciem vel modum actionis causamve aut rationem physicam alicubi definire, 6. ’'° Densitas corporis est quantitas seu copia materiae collata cum quantitate occupati spatij. 7. ” Per pondus intelligo quantitatem '^’seu copiam materiae movendae'^ abstracta gravitationis consideratione quoties de gravitantibus non agitur. Quippe'' pondus gravitantium proportionalc est quantitati materiae et analoga per se invicem exponare et designare licet. Analogia vero sic colligabitur pendulis aequalibus numerentur oscillationes cor­ porum duorum ejusdem ponderis et copia materiae in utroque erit

Xb

D E F IN IT IO N S

317

reciproce ut numerus oscillationum eodem tempore factarum. Experimentis autem in auro, argento, plumbo, vitro, arena, sale communi, aqua, ligno, tritieo, diligenter factis^ incidi semper in eundem oscilla­ tionum numerum (ob hanc analogiam et defecta vocis commodioris expono et designo quantitatem materiae per pondus etiam in corporibuss quorum gravitatio non consideratur). Locus. 9. Quies. 10. Motus. 11. Velocitas. Quantitas motus est quae oritur ex velocitate et pondere corporis translati conjunctim. Motus additione corporis alterius tanto cum motu fit duplus et duplicata velocitate quadruplus. 14.^-^ Corporis vis exercita est qua id conatur conservare* status sui movendi vel quiescendi partem^ illam quam singulis momentis amittit estque status illius mutationi seu parti singulis momentis amissae pro­ portionalis nec improprie reluctatio vel resistentia corporis dicitur. Hujus^' una species est vis centrifuga gyrantium, a. S ucceed ed by inertia sive deleted. b - b . R eplacing an earlier and p a rtly illegible version. c - c . R eplacin g an earlier and p a rtly illegible version. d -d . R eplacin g m ateriae abstracta consideratione deleted. e. Succeed ed by copiae m ateria deleted. f. S ucceed ed by institutis reperi deleted. g. Preceded by non gravitantibus deleted. h. T h e headings only are given o f this an d the three subsequent definitions. i. W ritten above is singulis m om en tis deleted. j. Su cceed ed by amissam deleted. k. Succeed ed by referenda deleted.

Translatian

On the M otion o f B odies Defmitians 1.2 The quantity af matter is that arising conjointly from its density and magnitude. A body twdce as dense in double space is four fold. This quantity I designate by the name body or mass. 2.^ The quantity afmatian is that arising conjointly from the velocity and the quantity of matter. The total motion is the sum of the motion in

318

DEFINITIOxNS

Xb

individual parts, and so in a body twice as large with the same velocity [the motion] is double and with double velocity quadruple. 3.4 The internal force of matter is the power of resistance by means of which any one body continues so far as it can in its state of rest or moving uniformly in a straight line: and it is proportional to its bodys nor differs at all from the inertia of matter except in our mode of conceiving it. In fact a body only*^ invokes this force in changes of state produced in it by another force impressed on it, and its exercise is Resistance and Impetus which are distinct only in relation to each other; being resistance in so far as the body opposes itself to an impressed force, and impetus^ in so far as the body by yielding with difficulty attempts to change the state of the other body. It is customary to attribute resistance to bodies at rest and impetus to those in motion; but motion and rest as commonly con­ ceived are distinct only in relation to each other: nor do those things truly rest which are regarded as if they rested by ordinary people. 4.8 Impressed force is an action exercised on a body to change its state of rest or motion. This force consists truly in the action only, nor does it remain in the body after the action. However it is of diverse origins, as of impetus, or pressure or centripetal force. 5.9 Centripetal force is a certain action or power by which a body is impelled or drawn or in any way tends towards a certain point as if to a centre: of this ilk is the gravity by which a body tends to the centre of the earth, the magnetic force by which iron seeks the centre of a magnet, and that force, whatsoever it may be, by which the Planets are held in their orbits and perpetually restrained from flying off at a tangent. Moreover, there are three quantities of centripetal force: absolute^ accelerative and motive. The absolute quantity of a centripetal force (which can also be called absolute force) is greater to one centre than another, no attention being paid to the distances and the magnitudes of the other bodies attracted; as the magnetic virtue is greater in one magnet and less in another. The accelerative quantity ox force is propor­ tional to the velocity generated in a given time; as the power of the same magnet it is greater at a lesser distance less at a greater distance, or of the force of gravity is greater near the Earth, less in the higher regions. The motive quantity or force is proportional to the motion it produces in a given time; as of weight more in a larger body less in a smaller one. Therefore motive force is to accelerative force as motion to speed, f'or quantity of motion is derived from speed multiplied by the body moved

Xb

DEFINITIONS

319

and the quantity of accelerative force from the accelerative force multiplied into the same body. Hence near the surface of the earth where the accelerative quantity is the same in all bodies, the motive gravity or weight is as the body; but if one recedes further from the Earth, and ascends into regions where the accelerative quantity is less, the weight is diminished equally, and will always be as the body multiplied by the accelerative gravity. Moreover in the same sense I call attractions and impulses, accelerative and motive. And use the words attraction, impulse or propensity of any sort towards a centre indifferently and inter­ changeably one for the other, considering these forces not in the physical but only in the mathematical sense. Hence let the reader beware lest he think that by words of this kind I define a type or mode of action or cause or physical reason of any kind, 6. ’®The density of a body is the quantity or bulk of matter compared with the quantity of space occupied. 7 . ” By the heaviness of a body I understand the quantity or bulk of matter moved apart from considerations of gravity as often as it is not a matter of gravitating bodies. T o be sure the heaviness of gravitating bodies is proportional to their quantity of matter by which it can by analogy be represented or designated. And the analogy can actually be inferred as follows. The oscillations of two equal pendulums of the same weight are counted and the bulk of matter in each case will be inversely as the number of oscillations made in the same time. But careful experi­ ments made on gold, silver, lead, glass, sand, common salt, water, lignite and twill led always to the same number of oscillations. On account of this analogy and lacking a more convenient word I represent and desig­ nate quantity of matter by heaviness, even in bodies in which there is no question of gravity. 8. Position, 9. Rest. 10. Motion. 11. Velocity. 12. ^^ The quantity of motion is that which arises jointly from the velocity and quantity of matter of the moving body. The motion by addition of another body of the same motion is double and with doubled velocity quadruple.

320

DEFINITIONS

Xb

14.^3 The exercised force of a body is that by which it attempts to pre­ serve that part of its state of rest or motion which it gives up at particular moments and it is proportional to the change or part of its state given up at particular moments, and not improperly is said to be the reluctance or resistance of the body, of which one species is the centrifugal force of rotating bodies. 1. M S . A d d . 3965 (5), fol. 21, fol. 26 (left half). 2. C o rrespo n d in g to D e f. i o f lectures de Motu. 3. C o rresp o n d in g to D e f. 4 o f lectures de Motu. 4. C o rresp o n d in g to D e f. 5 o f lectures d e M otu . T h e latter part represents a considerably em ended version o f an earlier and p artly illegib le version. 5. T h e phrase estque co r po r i suo proportionalis agrees w ith that in the lectures de M otu, rather than the Principia.

6.

T h e possible significance o f solummodo here is discussed above in Part I,

C h a p ter 1.4, p. 28. 7. N e w to n ’s co ncept o f impetus is touch ed on above in Part I, C h a p te r 1.4, p. 27. 8. C o rrespo n d in g to D e f. 6 o f lectures d e Motu. 9. A draft o f D e f. 7 - 1 0 o f the lectures de Motu. 10. N o such definition occurs in the lectures de M o tu or the Principia, althou gh the co ncept is used in D e f. i o f bo th those works. For N e w to n ’s co n ­

XI T H E L E C T U R E S £)£ M O T V OF 1684 manuscript, entitled De Motu Corporum Liber primus, and dated in Newton's hand, in the right hand margin of fol. i, ‘Octob. 1684’, forms part of MS. Dd-9-46 C .U .L. The original text has been subjected to considerable emendation at certain points, including a number of omis­ sions and insertions. As emended, the manuscript agrees so closely with the corresponding part of Book I of the Principia^ that it must represent something very close to, if not identical wdth, the final draft of that work, from which was taken the copy transmitted to the Royal Society in 1686. Consideration of the differences between the Principia and the revised text of the present manuscript falls outside the scope of the present work, and we shall be concerned only with significant differences between the final and original versions. A discussion of the probable date of composition of the present manu­ script, and of its connexion with the other parts of MS. Dd-9-46 C.U .L. is given above in Part I, Chapter 6.4.

T h is

cept o f d ensity see above. Part I, C h a p ter 1.4, p. 25. 1 1 . See the reference to pon dus and to p en d u lu m experim ents at the end o f D e f. 3 o f the lectures de Motu. 12. A return to D e f. 2. 13. T h e r e is no D e f. 13. T h is definition later cancelled. It corresponds to D e f. 3 and is in terestin g for the reference at the end to vis c en tr ifu ga g y r a n t iu m m issin g from the latter definition, at least in its final version.

S ig n if ic a n t

D if f e r e n c e s

betw een

O r ig in a l

and

F in a l

V e r s io n s

Definitions Def. 1-3 in the original version read as follows: I P Ouantitas materiae est copia seu mensura ejusdem orta ex illius densitate et magnitudine conjunctim. Vas idem plus continet aeris vel pulveris cujust is qui compressione magis condensantur. Corpus aiitem diiplo densius in duplo spatio quadruplum est. Hanc quantitatem per nomen corporis vel massae designo. 2. ^ Axis materiae est Tinea quaevis recta circum quam materia servato partium situ inter se, in spatio libero absque impedimentis et incitamentis uniformiter revoTci possit. 3. Centrum inatcriae est axium diiorum conciirsus,^ estque in corpore similari punctum illud quod rulgo centrum gravitatis dicitur. Sed et in materia dissimilari idem est cum centra gravitatis, si modo centrum illud non ex magnitudine sed quantitate materiae determinatur. Verbi gratia centrum gravitatis magnitudiniim globi aiirei et globi lignei punctum illud est quo dividitur distantia inter centra globorum in ratione reciproca magnitudinum.

322

THE L E C T U R E S

DE M O T U

OF 1684

XI

at venini gravitatis centrum centrumque materiae est punctim illud quo dividitur haec distantia in ratione reciproca sen ponderum sen quantitatum materiae in globis. Nam materiam in corpore unoquoque esse ponderi proportionalem reperiper experimentum pendulorump uti posthac explicabitur. Allowing for the dilferent numbering consequent on the original exis­ tence of Defs. 2, 3 not found in the Principia, the next six definitions of the original version were subjected to trivial emendations only. This is true likewise for the next definition (Def. V III of the Principia) up to vim autem absolutam ad centrum (1. 12, p. 4, Principia) after which there is a substantial difference between the original and final version down to the end of the paragraph. The continuation in the original version read: . . . vel si mavis ad corpus aliquod in centro consistens, tanquam efficaciam eius ad propagandas vires acceleratrices de se per regiones omnes in circuitu. Mathematicus est hie conceptus. Nam virium causas physicas jam negligo. In Physica referendae sunt vires absolutae ad earum causam veram sive causa ilia sit corpus aliquod in centro {uti magnes in centro vis jnagneticae vel Terra in centro vis gravitatis) sive alia aliqua quae non apparet. Nam centrorum quae sunt puncta Mathematica, vires j'evera niillae sunt. In Mathesi autem has vires vel abstracte considerare licet, et disputationibus de causa vera omissis, ad centrum ceu principium Mathematicum simpliciter referre, vel corpori alicui in centro, ceu causae sine qua non sunt, atque adeo a cujus efficacia et quantitate pendent concrete tribuere. Thereafter there are no significant emendations in the original text down to the end of the Scholium to the Definitions. Latos of Motion Coroll. II beginning at S ifilio p r ... (1. 7, p. 1^, Principia) and continuing down to the end of the corollary, read in the original version: Pondus autem v ip N trahit filum directe et vi IIN urget planum pG hide vi directum oppositum. Unde tensio fili hujus obliqui erit ad tensionem fili alterius perpendicularis PA3 utpN ad pH. Ideoque si pondus p sit ad pondus A in ratione qua componitnr ex ratione reciproca minimarum distantiarum filorum suorum AM , p N a centro rotae et ratione directa pH ad p N ; pondera idem valebunt ad rotam movendam, atque adeo se mutuo sustinebunt ut quilibet experiri potest. Per similem virium divisionem innotescit vis qua pondus simul urget plana dua oblique quae et vis cunei est. Nam erecto ad lineam p N piano perpendicularo pQ, si corpus p planis pO, pG utrinque incumhat, hoc inter plana ilia consistens rationem habebit cunei inter corporis fissi facies internas: vis

XI

TH E

L E C T U R E S

DE

M O T U

OF

1 6 8 4

323

autem qua urget planum vel faciem pO, eadem erit qua, sublato hoc piano, distenderet filum pN, atque adeo est ad vim qua vel pondere suo vel ictu mallei, impellitur secundum lineam Hp in plana utp N ad pH, et ad vim qua urget planum alterumpG utpN ad pH. Sed et vis cochlea per similem virium divisiofiem colligitur; quippe quae cunetis est a vecte impulsus. Usus igitur Corollarij hujus latissime patet, et late patendo veritatem ejusdem evincit, cum pendeat ex jam dictis Mechanica tota ah Authoribus diversimode demonstrata. Ex hisce enim innotescunt vires Machinarum quae ex rotis, tympanis, trochleis, vectibus, radijs volubilihus, nervis tensis et ponderibus directe vel oblique ascendentibus ac descendentibus, caeterisque potentijs Mechanicis componi solent; ut et vires nervorum ad animalium ossa movenda, quatenus a contractione musculorum tendentur. Apart from the insertion in the Scholium of the phrase Et primus quidem D. Wallisius, dein D. Wrennus et D. Hugenius inventum prodidit there are no further substantial alterations of the original text. Article I : Method of prime and ultimate ratios With the exception of the insertion of the two corollaries to Lemma X found in the Principia and the addition of the final paragraph of the Scholium, there are no substantial alterations of the original text. Article I I : Propositions Prop. I, Theor. i. The original enunciation of this proposition agreed with that of MS. IXc, Prop. i. An interesting addition to the original text is the phrase in eodem piano cum triangulo A S B at the end of the fourth sentence from the beginning. The last sentence of the original version is cancelled and replaced by the passage found in the Principia. Corollaries i, 2 found in the Principia are missing from the original version. Prop. 3, Theor. 3. The first three corollaries to this proposition originally read as follows: Corol. I. Him si corpus unum radio ad alterum ducto describit areas temporibus proportionales, urgetur hoc corpus 7iulla alia vi praeter compositam illam ex vi centripeta ad corpus alterum tendente, et ex vi omni quae agit in corpus alterum et in utrumque aequaliter {pro mole corporum) et secundum lineasparallelas agere intelligitur. Namque additio et subductio virium in hoc Theoremate sit secundum situm linearum, ut in Legum Coroll, i exponitur. Y 2

324

THE L E C T U R E S

DE M O T U

OF 1684

XI

XI

THE L E C T U R E S

DE M O T U

OF 1684

325

CoroL 2. Et ijsdem positis si areae sint temporibus quam proxime proportio­ nates vis ilia communis aut aequaliter agit in corpus utrumque quamproxime, aut agit secundum lineas quamproxime parallelas, aut perquam exigua est si cum vi centripeta ad corpus alterum tendente conferatur.

gulum Y P G ad rectangulum ZA O , id est, ob aequales, PG, AM , AO, ut P Y a d A Z Q.E.D. Prop. 13, Prob. 8. Originally there was the following Scholium to this Proposition:

Corol. 3. Et vice versa si haec tria contingunt, corpus radio ad alterum corpus ducto describet areas quamproxime proportionates temporibus.

S i vis centripeta ageret in omnibus distantijs aequaliter, corpus autem hac vi urgente describeret curvam A B C G E et in A longissime distaret a centro S, perveniret idem corpus ad minimam a centra distantiam in C ubi angulus A S C est n o graduum circiter, deinde ad Augem seu maximam a centro distantiam in D ubi angulus C S D est aequalis angulo A S C , postea ad minimam a centro distantiam in E ubi angulus D SE est aequalis angulo C SD et sic infinitum. Quod si vis centripeta reciproce proportionalis esset distantiae a centro, corpus de loco maximae sui a centro distantiae A descen­ deret ad locum minimae a centro distantiae, puta ad G, ubi angulus A S G est quasi 136 vel 140 graduum, dein hoc angulo repetito ascenderet rursus ad maximam a centro distantiam et sic per vices in infinitum. Et universaliter, si vis centripeta decresceret in majore quam duplicata et minore quam triplicata ratione corpus prius minori quam duplicata ratione distantiae a centro corpus ad maximam a centro distantiam prius rediret quam compleret circulum, sui vis ilia decresceret in majore quam duplicata et minore quam triplicata ratione corpus prius compleret circulum quam rediret ad maximam a centro distantiam. A t si vis eadem decresceret in triplicata vel plusquam triplicata ratione distantiae a centro, et corpus inciperet moveri in curva quae in principio motus fecaret radium A S perpendiculariter, hoc si semeI inciperet descendere, pergeret semper descendere usque ad centrum, si semel inciperet ascendere abiret in infinitum. Prob. 16, Theor. 8. The present manuscript ends at Coroll. 5 of this proposition.

Prop. 4, Theor. 4. The original enunciation agreed with that in MS. IXc, Prop. 2. Originally there were only five corollaries, as in MS. IXc, corol­ laries 3, 7 in the Principia having been inserted later. The Scholium to this proposition was also emended considerably, the original version agreeing with that in MS. IXc. Prop. 5, Prob. i. This was not in the original version. Its insertion necessitated certain alterations in the numbering of the subsequent propositions. The revised numbering will be adhered to thereafter for convenience of reference to the Principia. Prop. 6, Theor. 5. The end of the second paragraph originally agreed with the corresponding portion of the text of Prop. 3 of MS. IXc. Prop. 10, Prob. 5. The corollary to this in the Principia was inserted. Article III: De motu corporum in conicis sectionibus excentricis This article number, and title, was inserted after Prop, i i , Prob. 5, having originally been immediately before Prop. 14, Theor. 6. Lemma 13 of the Principia was inserted. The original lemma 13, corresponding to lemma 14 of the Principia, read Quadratum perpendiculi quod ab umbilico Parabolae ad tangentem ejus demittitur est ad quadratum intervalli inter umbilicum et punctum contactus lit lotus rectum principale ad latus rectum quod pertinet ad diametrum transeuntem per punctum contactus. Sit A O P Parabola, S umbilicus ejus, A vertex principalis, P punctum contactus. P M tangens diametro principali occurrens in M, et S N linea perpendicularis ab umbilico in tangentem. Produce S A ad Z lit sit A Z latus rectum principale. Huic A Z erige perpendicularum Z Y , cui occurrat P Y ipsi A Z parallela, et erit haec P Y latus rectum pertinens ad diametrum transeuntem per punctum contactus P. Id ex conicis patet. Dico igitur quod sit PS^ ad SN'^ ut P Y ad A Z. Nam completis parallelogrammis AG PM , Z Y P O ob aequalia triangula PSN , M S N et similia M SN , M PO, est P S i ad SN^‘ ut PM'^ sell AG'' ad PO'', hoc est {ex natiira Parabolae) ut rectan-

1. H ere and elsewhere the reference is to the First E dition . 2. T h is version was subjected to considerable em endations and cancellations and then finally replaced (at fol. 5v) b y the follo w in g version corresponding to, th o u gh not identical w ith, D e f. i, P rincipia: Quantitas m ateriae est mensura ejiisdem orta ex illius densitate et magnitudine conj un ctim . Aer duplo d ensior in duplo spatio quadruplus est. Idem intellige de n i v e et pulveribus p e r compressionem v e l liquefactionem condensatis. Et p a r est ratio c o r ­ po ru m omnium quae p e r operationes naturae diversim ode condensantur n eg le ct o scilicet a d medium respectu, s iq uo dfuerit, in[ter]stitiapartium libere pervaden s. Innotescit autem quantitas m ateriae p e r corporis cujusque pondus. Nam p on der i p r o portionalem esse r ep e r i p e r experimenta pendulorum accuratissim e instituta, uti p o st ha c docebitur. Eandem v e r o sub nomine corporis v e l massae in sequentibus passim intelligo.

326

THE L E C T U R E S

DE M O T U

OF 1684

XI

3. B o th this and the su cceed in g definition were o m itted from the Principia. T h e y are o f interest fo r the evidence th ey su p p ly o f the co n tin uity o f N e w to n ’s dyn am ical tho u ght. See above, C h a p te r 5, p . 86. 4. C an ce lled and replaced b y intersectio.

NOTE REGARDING PREVIOUS P U B L I C A T I O N OF N E W T O N D Y N A M I C A L MANUSCRIPTS

5. N o tic e h o w this im portan t fact was incorporated in the final version o f D e f. I given in n. 2 above.

I. MS. Add. 3996 has not yet been published in full. Certain extracts other than those given here have been published by Hall [i]. II. Unpublished previously apart from certain extracts relating to centri­ fugal force in Herivel [i]. III. The Vellum MS. has been published previously by Herivel [4], and in vol. iii of C or re sp on de nc e. IVa. Published previously with translation by Hall [2], and in vol. i of C o r re sp on de nc e.

IVb. Published previously with translation by Hall and Hall [i] apart from the Diagrams which they reconstructed from the text. V. Published previously in part by Herivel [5], and completely in vol. iii of C o r re sp on de nc e. VI. MS. Add. 4003 published previously with translation by Hall and Hall [!]• VII. Correspondence. VIII. The Newton Copy MS. first published by Whiston [2] in a Latin ver­ sion and in English incompletely by Ball [i], and completely by Hall and Hall [i]. The Locke Copy has been published previously by King [i], and in vol. iii of C o r re sp on de nc e. IX. The Royal Society Copy (Isaaci N e w t on i P r op os it io ne s d e M otu ) was the first version to be published, by Rigaud [i]. An emended version of the same manuscript was then published by Ball [i] who also gave the substantial differences and additions of Version HI. Hall and Hall [i] have published Version HI with translation giving certain of the differ­ ences between that version and the first two versions in footnotes. Xa. Unpublished previously. Xb. Definitions 1-5 published previously with translation by Hall and Hall [i]. XL Unpublished previously.

BI BLI OG RAPH Y H ai .l , a . R.,

BIBLIOGRAPHY

and

Hali., M .

B.

329

[ i ]. U npublished S cien tific Papers o f Isaac Nezvton

(C am bridge, 1962).

Herivel, A lexander, H . G . [ i ], T h e L e ib n iz -C la r k e Controversy (M anchester, 1956). A ndrade, E . N . da C . [ i ]. S ir Isaac N ew ton (L o n d o n , 1954). B all , W . W . R . [ i ]. A n Essay on N ew to n 's P r in cip ia (L o n d o n , 1893). B eeckman, I. [i\ , J o u rn a l, w ith in troduction and notes b y C . de W aard, 4 vols. (L a H aye, 19 3 9 -53 ).

B irch,

T.

[ i ], A H isto ry o f the R o y a l S o ciety o f Lond on ,

1 7 5 6 -7 )B orelli, J. a .

4 vols. (L o n d on ,

[ i ], T heorica medicearum planetarum a causis physicis deducta

(Florence, 1666).

B oyer,

C . B.

[ i ]. T h e H istory o f the C a lcu lu s an d its C on cep tual D evelop m en t

(N e w Y o rk , 1959).

Brewster, D. [ i ]. M em oirs o f the L ife , W ritings, an d D iscoveries o f S ir Isaac N ew ton , 2 vols. (E d in b u rgh , 1855). Bullialdus, I. [i], A stronom ia P h ilo la ica (Paris, 1645). C ajori, F . [ i ], ‘ N e w to n ’s T w e n t y Y ears D e la y in A n n o u n cin g the L a w of G ra v ita tio n ’ , p p. 1 2 7 -8 8 o f S ir Isa a c N ezcton, a B icenten ary E v a lu a tio n o f his W ork (Baltim ore, 1928). ------- [2] (Revisor), S ir Isaac N ezvton’s M a th em a tica l P rincip les . . . T ran slated into English by Andrezv M o tte (Berkeley, 1934). C lagett, M . [ i ]. S cien ce o f M ech a n ics in the M id d le A g es (M adison and O xfo rd , 1959).

N ew Sciences (D o v er edition o f 19 14 N e w Y o r k edition, no date).

C rombie, a . C . [ i ], A u g u stin e to G a lileo , 2 vols. (Lo n don , 1961). DE V illamil , R . [ i ], Nezvton: T h e M a n (L o n d on , no date). D rake, S. [ i ] (Editor), G a lileo G a lilei, D iscourse on B odies in W ater. T ran sla te d b y T homas S alusbury (U rbana, i960). ------- and D rabkin , I. E. [i] (Translators), G a lileo G a lile i O n M o tio n an d O n M ech a n ics (M adiso n , i960). D ugas, R . [ i ]. L a M eca n iqu e an X V I F siecle (N eu ch atel, 1954). ------- [2], A H isto ry o f M ech a n ics translated b y J. R . M a d d o x (L o n d on , 1957). D lihem, P. [ i ], [2], [3], E tu d es stir L eon a rd de V in ci, Serie i , 2, 3 (Paris, 1906, 1909, 1913). j

[ i ]. Correspondence o f S ir Isa a c Nezvton an d Professor C otes

.

(L o n d on , 1850).

F iersz,

M.

[ i ],

‘U b e r den U rsp ru n g u n d B ed eu tu n g der L eh re Isaac N ew to n s

M a n u scrip t’ , Isis,

52,

4 10 -16

(1961). ------- [5], ‘ N e w to n on R o tatin g B o d ie s’ , ibid. 53, 2 1 2 -1 8 (1962).

K in g , L ord [ i ]. L if e o f L o c k e (2nd ed., London, 1830). K oyre, A . [i]. E tu des G alileen nes, Parts I - H I (Paris, 1939). ------- [2], ‘A n U n p u b lish ed L e tte r o f R ob ert H oo ke to Isaac N e w to n ’ , Isis,

43,

312-38 (1952). ------- [3], ‘L a m ecan ique celeste de J. A . B o relli’ , R ev u e d 'H isto ir e des Sciences, 5,

101-38 (1952). ------- [4], ‘ L ’h yp oth ese et I’experience chez N e w to n ’ , B u ll. S o c. F r a n f. P h il.

48,

59-97 (1956). ------- [5], F ro m the C lo sed W o rld to the In fin ite U niverse (Baltimore, 1957). ------- [6], N ezvtonian S tu d ies (in

press). L ohne, j . [ i ], ‘ H ooke versus N e w to n ’ , C entaurus, 7 , 6 -5 2 (i960). M ach, E . [ i ]. D ie M ech a n ik in ihre Entzoicklung (L e ip zig , 1921). M agirus, j . [ i ], Physiologiae P erip a tetica e L ib r i S e x C u m Com m entariis . .

.

G abbey, W .

a

.

[ i ],

‘ D e sca rte s’ D y n a m ica l T h o u g h t ’ (un published thesis.

R . [i], ‘ Sir Isaac N e w to n ’s N o te Book, 1 6 6 1 -6 5 ’, Cam bridge H isto rica l

9,

2 3 9 -5 0 (1948).

------- [2], ‘N e w to n on the C alcu latio n o f C en tral F o rces’ , A n n a ls o f Scien ce, 6 2 -7 1 (1957)-

------- [2], ‘ L a p om m e de N e w to n ’ , d e l et Terre, 53, 190 -3 (1937).

Pemberton, H . [ i ], A Viezv o f S ir Isaac N ezvton's P h ilosop h y (D u b lin , 1728). R igaud, S. P. [ i ]. H isto rica l E ssay on the F irst P u b lica tio n o f S ir Isaac N ezvton's P rin cip ia (O xford , 1838). Correspondence o f S cien tific M e n o f Seven teen th C en tu ry ,

------- [2] (Editor),

2 vols. (O xfo rd , 1841).

S antillana,

G . de [ i ] (Editor), G a lileo G a lilei, D ialogue on the G reat W orld System s (C hicago, 1953). S hapley, H. and H owarth, H . E. [ i ], A S ource B o o k in A stronom y (N e w Y o rk ,

1929)-

S inger, C . [ i ], A S h o r t H istory o f S cien tific Ideas to i g o o (O xfo rd 1959). S treet, T. [ i ], A stronom ia C arolin a: A Nezv Theory o f C oelestia l M o tio n (L o n ­ S tukeley, W . [ i ]. M em oirs o f S ir Isaac Nezvton s L ife , edited

by

W hite, A . W .

(L o n d o n , 1936).

Q u e e n ’s U n iversity, Belfast, 1964).

J o u rn a l,

M aier, A . [i], ZzveiP roblem e der scholastischen N a tu rp h ilosop h ie, (2nd ed., R om e, 1951)M ore, L . T . [ i ], Isaac N ezvton, a B iography (N e w Y o rk , 1934). P atterson, L. D. [ i ], ‘ H o o k e ’s G ravita tio n theory and its influence on N e w to n ’, Isis, 40 (1949). — — [2], ibid. 41 (1950). Pelseneer, j . [ i ], ‘ U n e lettre inedite de N e w to n ’ , Isis, 12, 2 3 7 -5 4 (1929).

don, 1661).

vom absoluten R a u m ’, Gesnerus, i i , 6 2 -1 2 0 (1954).

Hall , A .

------- [3], ibid., p p . 7 1 - 7 8 . ------- [4], ‘ Interpretation o f an early N e w to n

(C a m brid ge edition, 1642).

C ohen, I. B. [i], F ra n k lin an d N ew ton (Philadelphia, 1956). C ostabel, P. [ i ], L e ib n iz et la dynam ique (Paris, i960). C ranston, M . W . [ i ], J o h n L o ck e (L o n d on , 1957). C rew, H . [ i ]. T h e R ise o f M od ern P h ysics (Baltim ore, 1928). ------- and S alvio , A . de [ i ] (Translators), G a lile o 's D ialogues Concerning Tzco

E dleston,

j . W . [ i ], ‘ N e w to n ’s D is co v e ry o f the L a w o f C en trifu ga l F o r ce ’ , Isis, 5 1 , 5 4 6 -5 3 (i960). ------- [2], ‘ O n the D a te o f C o m p o sitio n o f the F irst Version o f N e w to n ’s T r a c t dc M o t u ', A rch iv es Internationales d 'H isto ire des Sciences, 13, 6 7 -7 0 (i960).

13,

S ullivan , J. W . N . [i], Isaac Nezvton i 6 4 2 - i j 2 y (L o n d o n , 1938). T orricelli, E . [ i ], D e M o tu Projectorum : O p . G eom etrica (Florence, T oulmin, S. [ i ], P h il. R ev . 68, 1 -2 9 (1959).

1644).

I'u R N O R , E. [i]. C ollections fo r the H istory o f the Tozvn and S o k e o f G rantham (L o n don , 1806).

330

inBL10C;RAPHY

W hew ell , W. [ i ], H istory o f the I n d u c ti ve Sciences, 2 vols., 3rd ed. (London,

1857)W h is to n , W. [ i ], M em oirs o f the Life o f Mr. William Whiston by himself, 2 vols.

(London, 1749). ------[2], P r a ele ct io n es P h y s i c o -M a t h e m a t i c a l (Cambridge, 1710).

INDEX Where a manuscript is in Latin references are given only to the translation. Entries involving Newton and another individual are generally given under the latter only. Acceleration, 203. Adam, A ., and Tannery, P., 44 n. Alexander, H. G ., 29 n., 234 n. 19. Anderson, A ., 237. Andrade, E. N . da C., xiv. Antiperistasis, theory of, 125 n. 3. Archimedes, 41. Aristotle, i, 2, 38, 43 n., 124, 125 n. 3, 126 nn. 8 and 10. Aston, F., 96, 104. Axis, of body, 310 Def. 6; of circulation, 210 § 5, 312 Law I. Ball, W . W . R., xiv, i i n., 12 n., 16, 17 n., 23 0 ., 6 50 ., 70 n., 98 n., 102, 108 n., 117 n., 131 n. 2, 257 n. 2. Barrow, I., 2, 3 n., 15. Beeckman, E, 340., 36, 52 n., 132 n. 6, 303 n. 13. Birch, T ., 5 n., 103 n., 132 n. 6. Bodies, absolutely solid or hard, 143 Ax. 9, 10, 151 n. 4, 213; colliding, spring of, 4, relative velocity of, 4, 83, 213; definition of, 226 Def. 2, 310 Def. 5; descent of, in relative and absolute space, 310 Def. 4; distance between, 139 Def. 13; elasticity of, 151 n. 4; ex­ tended, motion of, xiii n., 77-86, collision between, 14 n., 32, 82-86, 90, 171-9 , 2 1 1 -1 5 , 2 1 7 -1 8 ; falling, law of, see under Descartes, Galileo, and New ­ ton; motion towards one another, 77, 139 Def. 12; nature of, 227 § i ; pair or system of, centre of motion of, see under Motion, centre of. Borelli, J. A., 59, 233 n. 10. Boyer, C. B., 2 n. Boyle, R., his receiver, 127 n. 22. Bradwardine, T ., 2 n. Brewster, D., xiii n., xiv, i n., 14, 75 n., 96 n., 97 n., 102, 108 n., 117 n. Brownover, S., 108 n., 246. Bulk, proportionality of motion to, 133 § 3, 135 n. 3; and weight, 3 1 1 Def. i i . Cajori, F., 25 n., 66 n., 68. Calculator, 2. Centrifugal force, absence from treatment of orbital motion in Principia, 73 ; and Descartes, 7, 40, 47, 54; and Pluygens, 15) 56, 7 1 ; as species of force of a

body to persevere in its motion, 56, 3 11 Def, 13, 314 n. 14, 320 Def. 14; due to diurnal and annual motions of the Earth compared with gravity, see under Force, of gravity; its replacement by centripetal force, 13, 55, 59,73,289 n. i, 290 n. 9; law of, 196, derivation of, 1 1 13, discovery of, 89, result equivalent to, 185; of the M oon from the Earth, 58, 72; of the planets from the Sun, 58, 72; physical origin of, 7; quantitative treatment of, 7 -1 3 ; question o f its existence w'ithin framework o f Newton­ ian dynamics, 56, 62-63. Centripetal force, 277-89, 300 Lemma 2, 302, 318 Def. 4; and orbital motion, 13, 55i 61, 73, 277-89; definition of, 28, 277 Def. I, 299 Def. i, 3 1 1 Def. 16, 318 Def. 5; different kinds of, 318 Def. 5; examples of, 318 Def. 5; general formula for its calculation in orbital motion, 17 0 ., 19, 279 Theor. 3; its real nature as opposed to centrifugal force, 13; its replacement o f centrifugal force, 13, 55, 59, 73, 289 n. i, 290 n. 9. law of, for motion in circle under force to centre, 278 Theor. 2, for motion in circle under force to circumferential point, 280 Prob. i, for motion in ellipse under force to centre, 281 Prob. 2, for motion in ellipse under force to focus, 281 Prob. 3, for motion in equiangular spiral under force to eye, 280 Prob. i Scholium-, uniform, 287 Prob. 7. Circle, tyranny of, 8, 52; uniform motion in, 45-48, 51-52, 145 Ax. 19, and Galileo, 37, 40, 46, and Hooke, 240, 242, 242 n. 2, deviational treatment of, 12-13, 195-8, 278 Theor. 2, endeavour from centre in {see under Endeavour), force from centre in, 9, polygonal treat­ ment of, 7-10 , 18, 129-30, 139 n. f, 146-8. Clagett, M ., 2 n., 3 n., 126 n. 7, 235 n. 32. Clarke, S., 29, 234 n. 19. Clerselier, 234 n. 23. Cohen, I. B., xiv. Collins, J., 15, 237, 238 n. 2. Collisions, 142 A x. 7, 8, 9, 10; and D es­ cartes, 3, 4, 48, 49, 52; and Huygens, 4-5, 52; and third law' of motion, 31,

332

INDEX

312 Law 3; between extended body and immovable surface, 83, 90, 170 Prop. 33, 179 Prop. 39, 40; between extended bodies, 14 0 ., 32, 82-86, 90, 1 7 1-7 , 2 1 1 -1 5 , 2 1 7 -1 8 ; elastic, 4, 137 Def. 5, 142 Ax. 9; oblique, 31, 159 A x. 119, 121, 122; perfectly elastic, 4, 5, 143 Ax. 9, 1 5 1 n. 5; perfectly inelastic, 3, 1326, 143 A x. 9; problem of, 3, 4, 5, 6, 49; see also Reflection. Comets, 99, 102, 284 Prob. 4 Scholium, 291 n. 37, 302. Conatus, see under Endeavour. Conduitt, J., 15 n., 1 7 0 ., 75, 97. Conic sections, 249-51. Copernican system, i i , 301 B. Costabel, P., 5 n. Cotes, R., xiii. Cousin, V., 233 n. 8. Cranston, M . W., 108 n., 246. Crew, H., 25. Crew, H., and Salvio, A., 37 n., 132 n. 5. Crombie, A. C., 126 n. 7. Curve, approximation to by broken line, 18, 21, 78, 115, 130, 145 Ax. 18, 151 n. 10, 247, 256 n. 14. Degree of great circle on Earth’s surface, 65Descartes, R., and, action required to create and destroy motion, 48, centri­ fugal force, 7, 40, 47, 54, conservation of motion, 3, 32, 49, 135 n. 4, definition of motion, 3, determination of motion, 8, 140 n. 6, horizontal pendulum, 132 n. 6, impossibility of vacuum, 126 n. 15, law o f falling bodies, 36 n., motion as a state, 160 n. 4, particle free to slide in revolving tube, 47, 54, principle of inertia, 30, 40, 43, 44-45, 235 n. 26, problem of collisions, 3, 4, 48, 49, 52, stability of planetary orbits, 59 n., 227 § 2, 233 n. 10. his, concept of conatus ( = endea­ vour) 46, 47, 52, correspondence, dynamical discussion in, 43 n., double theory o f motion, 125 § 3, 127 n. 19, 233 nn. 7 and 8, 234 n. 16, 310 Def. 9, 314 n., 8 double theory o f motion and Newton, 219, 227-30, 233 n. 8, in­ fluence on Newton, i, 29, 40, 41, 42-53, 150 n. I, 219, 234 n. 15, Principia Philosophiae and Newton, 226-35, Vortex theory, 42, 60 n., 65, 66 n., 68, 127 n. 19, 227 § 2, 228 n. 5, 233 n. 9, 234 n. 18.

Density, 25, 319 Def. 6, 320 n. 10. Determination, of motion, 3, 45, 137 Def. 4; of centre of motion, 78. Deviation, as measure of centrifugal ten­ dency in motion in a circle, 195; as measure of centripetal force, 19-22, 249 Prop. 2, 252-3, 255 n. 12, 279 Theor. 3, 290 nn. 9 and i i , 291 nn. 22 and 34; between actual and inertial paths, 12. Division, forbidden, 2, 3, 136 n. 9, 149, 215 n. 2. Drake, S., 37 n, Dugas, R., 5 n., 36. Duhem, P., xiv, 36 n., 126 nn. 7, 10, and

13Edleston, J., xiii, 98 n., 99 n., 102 n. Endeavour, along tangent, 51, 147 A x. 20; from centre in motion in circle, 7, 1 2 -

13, 48, SI, 52 , 54-64, 79 , 129 § I, 147 A x. 21, 22, 148 Ax. 24, 25, 195-8, 231 Def. 10, and absolute motion, 228 § 3, 310 Def. 9, and centre of motion in rotating body, 209 § 4, and Descartes, 46, 47, 52, and gravity, 231 Def. 10, 233 n. 10, as impeded force, 13, 55; of Earth from Sun compared with force of gravity, 197; of Moon from Earth, 69 n., compared with endeavour of Moon from Sun, 15, 197, 237, com­ pared with force of gravity at Earth’s surface, 196; of planets from Sun, 72, 197. Equiangular spiral, motion in, 280 Prob. i

Scholium. Equinoxes, precession of, 86. Ether, resistance of, 301, 302. Extension, 136 Def. i, 208 § i ; of powers, 232 Def. 12. Fiersz, M ., 29, 126 n. 16. Flamsteed, J., 108, 238; his correspon­ dence with Newton in winter of 1684/5, 103, 104, 107. Fluent, 2 n. Fluxions, 2, 17 n., 34 n., 66, 67, 93. Force, absolute, 318 Def. 5 ; accelerative, 20, 21, 22, 318 Def. 5; and change of motion, 48, 141 Ax. 3, 4, 6, 142 Ax. 5, 150 Ax. 23, 150 n. 2, 157 Ax. 108, 158 Ax. 1 14, 115, 159 Ax. 1 18, 208 § I, 231 Def. 5; and rest, 27, 231 Def. 5; approximation to, by discrete impulses, 18, 21, 70, 247, 252, 278 Theor. I ; by which a body endeavours from

I NDE X centre in motion in a circle, in half a revolution 8, 9, 147 A x. 22, in a whole revolution, 148 A x. 24; centrifugal, see centrifugal force; centripetal, see centri­ petal force; concept of, 5-6, 7, and Galileo, 36, 37; definition of, 138 Def. 9; equality of, between colliding bodies, 6, 142 Ax. 7, 8; exercised, of a body, 311 Def. 12, 320 Def. 14; impeded, and conatus, 13, 55; impressed, 28, and change of state, 312 Law 1 ,3 1 2 Law 3, 318 Def. 4, and change of motion, 299 Law 2, 312 Law 2, and motion of pro­ jectile, 123; impulsive, 18, 20, 21, 39, 140 n. 10, 247, 252, 278 Theor. i, 303 n. 10, 3 1 1 Def. 14; innate or internal, of body or matter, 26-28, 277 Def. 2, H yp. 2, 278 Theor. i, 286 Prob. 6, 299 Def. 2, Law i, 311 Def. 12, 312 Law' i, 318 Def. 3; instantaneous, 140 n. 10; motive, 22, 318 Def. 5; nature of, 231 Def. 5; of a body, see force, innate or internal; of a body’s motion, 156 Ax. 106, 161 n. 10, 3 1 1 Def. 13. of gravity, as centripetal force, 302, 318 Def. 5, compared with centrifugal force, 131 §§ 8, 9, 132 n. 8, at Earth’s surface compared with centrifugal force there due to diurnal motion, 10, i i , 183-6, 188, 196, at Earth’s surface com­ pared with endeavour of Earth from the Sun, 197, at Earth’s surface compared with endeavour of the Moon from the Earth, 196, compared with centripetal force on the Moon, 302, compared with the resistance of the air, 289, its diminution due to diurnal motion, 58, 62. of planets from the Sun, 72; on a planet, 16, 18; physical, 6; qualitative definition of, 5, 6; resisting, 277 Def. 3, Hyp. I, 299 Def. 3, 299 Law 5, 313 Law 6; varying inversely as square of distance, 19, 20, 23, 34 n., 248 Prop. 2, 251 Prop. 3, 281 Prob. 3, 282 Theor. 4, 284 Prob. 4, 286 Prob. 5. Gabbey, W. .A., 43 n., 52 n., 1 5 1 n. 4. Galilei, G., i, 2, 7, 29, 32, 3 5 -4 1; and, area under velocity-time graph, 33, 34 n., 38, 300 Lemma 2, assumption that distance moved in given time pro­ portional to magnitude of force acting, 37, concept of force, 36, 37, inertial motion, 37, 42, 52 n., Merton Rule, io n ., 38,41, motion ( = momentum) 37,

333

motion in a circle, 37,40, 46, principle of inertia, 35, 36, problem of percussion, 37, 38, rate of fall under gravity, 125, 127 n. 24, 186, 189, second law of motion, 35, 36, 37. his Dialogue, 10, 34 n., 36, 37, 38, 41, 74, 127 n. 24, 132 nn. 4 and 5, 183, 189, 198 n. 4, Discorsi, 2 n., 4, io n ., 34 n., 35 m, 36, 37, 38, 39, 40 m, 41, 91, 132 n. 5, 207 n. I, Discourse on Bodies in Water, 37, discovery of para­ bolic path of projectile, 35, 36, 39, 41, 312 Law 2, Siderius Nuncius, 125 n. 3, supposed use o f first and second laws of motion, 31, 33, 35, P law of falling bodies, i i , 12, 35, 36, 38, 41, 74, 184, 185, 195, 197 n. I, 207 nn. 6 and 13, 256n. 13; New ton’s homage to, 360., 53. Gassendi, P., 42, 43. Geometry, analytical versus synthetical, 3. Gravitation, inverse square law of, 57, 58, 62, 67 n., 70, 71, 72, 104, 106 n., 237 n. I ; and Bullialdus, 16 n .; deriva­ tion of, 13; tests against M oon’s motion, 13, 14, 23, 58, 59, 60, 65-76, 89-90, 92, 132 n. 2, 302. Gravity, acceleration due to, 203; and endeavour, 231 Def. 10, 233 n. 10; and levity, 43 n .; as species of centripetal force, 286 Prob. 5 Scholium, 318 Def. 5; centre of, 85, 135 n. 6, 310 Def. 6, 3 1 1 Def. 10; definition of, 231 Def. 10; efficacy of, on inclined plane, 203; force of, see force of gravity; natural, i , 121, 123, 126 n. 13; of body and its solidity, 125; rate of fall due to, i i , 88, 125, 127 n. 24, 185, 186, 187, 188, 206; uniform, 312 Law 2. Gregory, D., 17 n., 72 n., 92, 94, 192, 199. Gregory, J., 15 n., 66, 238 n. 2. Hall, A. R., 560., 72 n., 183, 192. Hall, A. R., and Hall, M . B., 29 n., 102, 1 17 n., 198, 208, 219. Halley, E., 62, 70, 73, 105, 108, 257, 283; his correspondence with Newton in 1686, 12, 16 n., 17, 22 n., 23 n., 57, 58, 59 n., 61, 62, 65 n., 71 n., 72, 74, 92 n., 97 n., 99, 104, 105, 106, 192, 237 nn. i and 3, 240 n. 2, 242 n. 2, discussion with Hooke and Wren in 1684, 97, experiment of pendulum clock at St. Helena, 58, 64, visits to Newton at Cambridge in 1684, 17 n., 23, 24, 75, 97, lo i, 102, 103, 104, 107, 1 15.

j :?4

INDEX

Herivel, J. W., 103 n., 183, 208. Hooke, R., 24, 74, 75, 108, 303; and, con­ troversy with Newton over inverse square law of gravitation, 57, 58, 62, 670., 7on., 71, 104, io6n., 237 n. i, con­ troversy with Newton over optics, 15; his discussion with Halley and Wren in 1684, 97, intervention of 1679, xiv, 13, 14. 3 4 n., 7 5 . 1 14. 1 17. 238-45, 303 n. 19, supposed discovery of parallax due to Earth’s annual motion, 238, 240 n. 2, theory of circular motion, 240, 242, 242 n. 2. Horrox, J., 291 n. 30. Huygens, C., 14, 60, 72 n., 87, n o , 112, 132 nn. 6 and 7, 198, 236, 237 nn. 2 and 3; and, centrifugal force, 6, 15, 56, 71, horizontal pendulum, 132 nn. 6 and 7, problem of collisions, 4-5, 52. Hyperbola, motion in, 284. Impetus, 27, 28, 231 Def. 7, 311 Def. 13, 318 Def. 3, 318 Def. 4, 320 n. 7; School, I. Impulse, see under Force, impulsive. Indivisibles, method of, 244. Inertia, i, 26-28, 318 Def. 3; as internal principle of conservation of rest or motion, 231 Def. 5; definition of, 231 Def, 8; principle of, 5, 28, 29-30, 35, 36, 43, 44-45, 141 A x. I, 2, 150 n. I, 153 A x. 100, 234 n. 15, 246 Hyp. i, 274 n. di, 277 Hyp. 2, 299 Law 1 ,3 1 2 Law I, 314 n. 21, for circular motion, 86; see also under Descartes and Galileo. Inertial movement, and Galileo, 37, 42, 52 n.; along tangent, 12, 18, 19; be­ tween impulses, 18. Intension and remission of forms, 235 n. 32. Intensity of power, 231 Def. ir. Isochronism of cycloidal pendulum, 203-4. Kepler, J., his first law o f planetary motion, 16, 19, 22, 34 n., 102, 290 n. 18, proposition corresponding to, 248 Prop. 2, 251 Prop. 3, 281 Prob. 3; his laws of planetary motion, 16, 42, 125 n. I, 281 Prob. 3 Scholiujn, 289 n. i; his second law of planetary motion, 16, 19, 22, 34 n., qr, 78 n., 115, 255 n. 9, 290 n. 7, proposition corresponding to, 246 Prop. I, 255 n. 9, 278 Thcor. i ; his third law of planetary motion, 13,

66 n., 67, 69, 125 n. 2, 132 n. 3, 197, 198 n. 16, 279 Corol. 5 Scholium, 282 Theor. 4, Kepler-motion, problem of, 13 n., 16 22, 34 n., 60, 61, 7 1 ; solution to, 24656, 278 T'heor. i, 281 Prob. 3, analytical rather than synthetical, 13 n., 17 n. Koyre, A., x iv n ., 16, 36, 42, 45 n., 57 n., 59 n., 70 n., I l l n., 126 n. 16, 240 n. Leibniz, G ., 5, 17 n., 29, 234 n. 19. Lever, principle of, 170 Prop. 34. Levity, 289. Locke, J., 106, 108-17 pcissim, 120, 246, 254-s, 290 n. 8. Lohne, J., 34 n. Lucasian Chair of Mathematics, 72 n., 92, 98, 192. Lucretius, 303 n. 14. Macclesfield Collection, 257. Mach, E., 25. Magirus, J., i, 380., 121. Alaier, A., 235 n. 32. Marci, M ., 52 n. Mass, and weight, 25, 127 n. 23, 231 § 8, 235 n. 24; centre of, and centre of motion, 79, 85, 215 n. 8. Matter, quantity of, 317 Def. i, 319 Def. 7. Mersenne, M ., 230. Merton, College, 2 n. 3, 235 n. 32; Rule, 10, 38, 41, 88. Alore, H., 126 n. 16. More, L. T ., xiv, 15 n., n o n. Moon. See under Endeavour, Gravitation, Motion. Moscovici, S., 38 n. Motion ( = Momentum), and, Descartes, 3, 8, 32, 49, 135 n. 4, Galileo, 37, power to persevere in state of movement, 137 Def. 3; angular, principle of, 84; change of, and force, see under Force, and change of motion; composition of, law, i4 n ., 18, 31-32, 35, 39-40, 41, 182, 209 § 3, 246 Hyp. 3, 278 Hyp. 3, 290 n. 6, 290 n. 8, 294, 299 Lemma i, 313 Lemma i ; conservation of, 3, 5,133 §§ 5, 6, 135 n. 4, 142 A x. 7, 8, 159 A x. 122, 218 n. 25; definition of, 3, 136 Def. 2, 3; gained or lost by reflection, 214 § 4, 215 § 5; proportionality to, bulk, 133 § 3, 135 n. 3, distance moved in given time, 134 § 7, 135 n. 8; quantity of, total, 3, 26, 49, 317 Def. 2, definition of, 3 11 Def. II, 317 Def. 2, 319 Def. 12.

INDEX Motion ( = M ovem ent); absolute, and en­ deavour from centre, 228 § 3, 310 Def. 9, its distinguishing from relative motion, 57, 62, 310 Def. 9, 314 n. 7; and, Descartes, 125, 127 n. 19, i6 o n . 4, 219, 227-30, 233 nn. 7 and 8, 234 n. 16, 310 Def. 9, 314 n. 8, impressed force, 123, natural gravity, 121, 123, quantity, 123, rest, relative natures of, 318 Def. 3, spirit, 123; as state of body, 153 Ax. 100, 156 Ax. 106, 160 n. 4. centre of, in dynamical sense, 77, 78-79, 80, 85, 138 Def. 10, 140 n. 12, 209 § 4, 215 n. 8, in kinematical sense, 77-78 , 138 Def. 10, 140 n. 12, 144 A x. 13, 14a, IS, 145 Ax. 16, 17, 18, 209 § 4, of extended body, motion of, 176, Prob. 37, of pair of bodies, 79, 80, 81, 84, 139 Def. II, 140 n. 13, 148 A x. 25, 209 § 4, of pair of bodies, motion of, 80, 162-70, of system of bodies, motion of, 85, 209 § 4, 216 n. 13, 299 Law 4, 313 Law 5. circular ( — rotation), and distin­ guishing between true and relative motion, 57, 62, 310 Def. 9, and prin­ ciple o f inertia, 312 Law i, angular quantity of, 78, axes of circulation of, 210 § 5, 312 Law I, combination of, 82, 210 § 7, continuation of, under no forces, 211 § 8, equator of circulation of, 210 § 5, measure of, 145 A x. 19, 209 § 5, 210 § 6, measurement of, in terms of quantity of motion, 79, poles of circulation of, 210 § 5, principle of inertia for, 86, radius of circulation of, 81, 210 § 5, real quantity of, 81, 209 § 5, resolution of, 78. definition of, 136 Def. i, 226 Def. 4, 310 Def. 9; determination of, 137 Def. 4, 144 Ax. 15, and Descartes, 8, 140 n. 6; first law of, see under Inertia, principle o f; in circle, see Circle, motion in; in curve, measurement of velocity of, 149, 152 n. 20; in cycloid, 203-7; inertial, 12, 18, 19, and Galileo, 37, 42, 52 n.; in ellipse, 7, 13, 16-20, 1 16, 130 § 3, 248-9, 251-4, 281-6; in hyperbola, 284, parabola, 284, in plena, 124, polygon, 9, II, resisting medium, 286-9; instantaneous direction of, 7 ; in straight line, measure of velocity ot, 149, 152 n. 20; in vacuo, 123-4; Kepler, see under Kepler-motion; laws of, 312-13, and Galileo, 31, 33, 35, corol­ laries to, 31-33, 315 nn. 25, 26 and 28.

3 35

of extended body, 77-86, 143-5; of globe, 122, 126 n. 7; of moon, 55, and test of inverse square law of gravitation, 13, 14, 23, 58, 59, 60, 65-76, 89-90, 92, 132 n. 2, 302; of bodies toward one another, 77, 139 Def. 12; of planets, see under Kepler and Planets; of plane­ tary heavens, 301 B; o f projectiles, 12 1 3, 302, and Galileo, 35, 36, 39, 41, 312 Law 2, in resisting medium, 287 Prob. 7 Scholium, relative to rotating Earth, 238-45, 303 ; of solar system, 32; orbital, and centripetal force, 13, 17 n., 19, 55, 61, 73, 277-89; power to per­ severe in, measure of, 208 § i ; relative, its distinguishing from absolute motion, 57, 62, 310 Def. 9, 314 n. 7, and con­ taining space, 299 Law 3, 313 Law 4; resolution of, 182, 209 § 3; second law of, 20, 22, 26, 30-31, 161 n. 19, 246 H yp. 2, 299 Law 2, 300, 312 Law 2, 3 15 0 . 22, and Galileo, 35, 36, 37; third law of, 31, i6 i nn. 16 and 18, 312 Law 3, 315 n. 23; translational, 78; uniform, 312 Law I ; uniformly accelerated, in a straight line, 7; violent, r, 43 n., 121-5. Newton, I, and, apparent loss of interest in dynamics betw een 1679 and 1684, 22, argument from simple to complex, 78 n., assumption that distance moved in given time proportional to force acting, 10, 22, attempted derivation of proportionality between force and change of motion, 152-61, attraction of homogeneous sphere at external point, 23, calculations relating to comets in August 1685, 99, 102, confusion betw'een motive and accelerative forces, 22, derivation of principle of lever, 170 Prop. 34, doctrine of intension and re­ mission of forms, 235 n. 32, drawing of axes of ellipse in error for conjugate diameters, 17 n., exact formulation in principle of general solution to a prob­ lem 80, 84, exact law' of falling bodies, 286 Prob. 5, explanation of invariable aspect of Moon from Earth, 60, 236, extension of inverse square law down to surface of the Earth, 23 n., extension of Galileo’s law, i i , 190 n. 5, to case of non-uniform motion, 20, 22, 38 n., 109, 278 Hyp. 4, 279 Theor. 3, 300 Lemma 2, 314 Lemma 2, geo­ metrical rather analytical proofs, 42, method of determining figures, 16 n..

336

INDEX

17 n., nature of body, 219, necessity for unique physical motion, 227 § 3, Royal Society of London, 94, 96, 97, 102, 103, 104, 105, 106, 107, 108, solution to Kepler’s problem, 285, the falling apple, xiii, 65 n., 92, vectorial nature of law of conservation of momentum, 218 n. 25. his absence from Cambridge due to plague, 92, Arithmetica Universalis, 292 n. 38, attitude to hypotheses, i i i , 1 12, belief in possibility of vacuum, 123-4, belief in centrifugal force, 5859, copy of Descartes’s Principia Philosophiae, 51, 127 n. 20, departure from Cartesian philosophy, 65, 67, drive towards generality, 77, 80, drive to­ wards quantitative treatment o f dy­ namics, 5, 41, 132 n. 9, imagination of an experiment to prove the diurnal motion of the Earth, 16, 238-45, 310 Def. 4, mathematical researches, 15, mother, 15, optical lectures, 98, optical researches, 14, 15, 140 n. 9, Propositiones de M otii, 23, 24, 96, 102, 105, physical thought, the medieval back­ ground to, 121, 125 n. 3, preparation for early discoveries in dynamics, mathematics and optics, 121, researches in dynamics in 1684, 24, realization of importance of direction of motion, 136 n. 10, stepbrother, 15, stepfather, 128; on, body and time, 219, space, 219, 228-30. Norwood, R., 66. Ockham, W., 112, 126 n. 17. Oldenburg, H., 56, 57, 59, 60, 71, 74, 106, 236, 237 n. 3. Optics, 14, 15, 98, 121, 140 n. 9. Oughtred, W., 68 n. 3. Paget, E., 17 n., 24, 93 n., 101-8, passim, 1 1 4 ,1 1 7 ,2 5 7 . Parabola, motion in, 284. Patterson, L. D ., xiii n. Pelseneer, J., xiii n., x iv n ., 16, 245 n. Pemberton, H., xiii, 23, 65, 66 n., 67, 68, 69, 70, 73, 74, 75, 92, 198 n. 8. Pendulum, cycloidal, 72 n., 204-5; hori­ zontal, I I , 91, 131, 132 nn. 6 and 7, 186, 187; vertical, i i , 25, 58, 64, 125, 131, 186, 187, 231 § 8, 3 1 1 Def. II, 319 Def. 7. Picard, J., 75. Picot, 44 n.

Place, definition of, 226 Def. i, 310 Def. 7. Planets, com m on centre of gravity of, 301B; determination of their orbits, 283 Scholium, 301B; force on 16, 18; force of, from Sun, 72; longitude of, 285; mean distances of from Sun in relation to their periods of revolution, 1 2 1 ; relative motion of, 301B; stability of their orbits, 59, 60, and Descartes 59 n., 227 § 2, 233 n. 10; their en­ deavour from Sun, 72, 197. Plato, 125 n. 3. Polygon, motion in, 9, i i . Portsmouth Collection, xiii n., xiv, xv, 65 n., 96, 106, 208, 257. Portsmouth Draft Memorandum, xiii n., 17 n., 66-67, 69, 70, 74, 92, 93 n., 125 n. 2, 191 n. 13. Projectile, motion of, see under Motion. Ptolemy, i i , 64. Qualities, absolute and relative, 312; method of prime and ultimate, 3; moments of, 311 Def. 162; representa­ tives of, 299 Def. 4. Reflection, of bodies, 4, 9, i i , 12, 18, 130 § 2, 137 Def. 5, 139 n. f, 142 A x. 9, 214 § 4, 215 § 5; optical, 181 n. 32. Refraction of body, 138 Def. 7. Relativity, Special Theory of, 160 n. 8. Resistance, of air, compared with force of gravity, 289; of ether, 301, 302; of medium, 277 Def. 3, 299 Def. 3, 302, 3 11 Def. 17, dependence on density and speed, 277 Hyp. i, 299 Law 5, 302, 313 Law 6; to change in state, 318 Def. 3. Resisting medium, motion in, 286-9. Resolution, of motion, 182, 209 § 3; of velocity, 140., 31, 39, 208 § 2. Rest, 231 Def. 5; and force, 27, 231 Def. 5; and motion, relative nature of, 318 Def. 3; Definition of, 226 Def. 3, 310 Def. 8; state of, 30. Restitution, coefficient of, 4. Rigaud, S. P., xiii, 15 n., 102 n., 238 n. 2, 257 n. 2, 291 n. 37, 292 n. 38. Rooke, L., 4. Roubiliac, L . F., 303 n. 14. Royal Society of London, and problem of collisions, 4, 5 n .; see also under Newton. Salusbury, T ., 35, 37 n., 91, 183, 189, 238 n. 3.

337

INDEX Santlllana, G . de, 37 n., 132 n. 4, 189. Sine, versed, 22, 65. Singer, C., 36. Slusius, 66. Smith, Rev. B., 128. Solar system, centre of gravity of, 85; motion of, 32. Space, 29, 208 § I, 219, 228-30; absolute, 309 Def. 3; of planetary heavens, its motion, 301B; relative, 310 Def. 4. Speed, definition of, 136 n. 9. Spirit, 123. Spring of colliding bodies, 4. Stability of lunar and planetary orbits, 59, 60. State, motion as, 153 A x. 100, 156 Ax. 106, 160 n. 4; perseverance of body in its, 154, 155, 156; resistance to change of, 318 Def. 3. Street, 125, 238 n. 2. Stukeley, W., xiii n., 65 n., 92. Sullivan, J. W. N., xiv. Swiftness, estimation of, 136 Def. 2.

Torricelli, E., 2, 238 n. 3. Toulmin, S., 29 n. Turnbull, H. W ., 183, 189. Tum or, E., 15 n. Vacuum, 126 n. 15, 123-4. Velocity, comparison of one w'ith another, 2; composition of, 31, 32; definition of, 3, 232 Def. 14, 311 Def. 10, 314 n. 9; measurement of, 208 § i, for motion in, circle 145 Ax. 19, curved path, 149, 152 n. 20, straight line, 149, 152 n. 20; relative, of approach and separation of two colliding bodies, 4, 83, 143 A x. 9, 10, 151 n. 5, 213; resolution of, 14 n., 31, 39, 208 § 2. Vortex theory, see under Descartes. Waard de, C ., 34 n., 303 n. 13. Wallis, J., 4, 5, 52, 108, 238 n. 2. Weight, and bulk, 3 1 1 Def. 1 1 ; distinction betw'een mass and, 127 n. 23, 231 § 8, 235 n. 24; proportionality of, to mass,

25Tendency, centrifugal, as species of im­ peded force, 5 5; of M oon along tangent to her path, 65. Tim e, 219, absolute, definition of, 309 Def. I ; relative, definition of, 309 Def. 2.

Whewell, W., 34 n. Whiston, W., xiii, 60 n., 65, 66 n., 6 7passim, i i 7 n . , 1980. 8. Whiteside, D . T ., 24 n., 34 n., 117 n. Woolsthorpe, 14. Wren, C., 4, 5, 52, 97, 242 n. 2.

PRIN TED AT

THE

IN

G REAT

U N IV E R S IT Y BY

PRINTER

VIVIAN TO

THE

BRITAIN

PRESS,

OXFORD

R ID L E R U N IV E R S IT Y