The Mathematical Principles Underlying Newton's Principia Mathematica

The Mathematical Principles underlying Newton's Principia Mathematica Being the ninth Gibson Lecture in the History of Mathematics delivered within the University of Glasgow on 21st October 1969 by

D. T. WHITESIDE Assistant Director of Research Whipple Science Museum, Cambridge

PUBLISHED BY THE UNIVERSITY OF GLASGOW 1970

© COPYRIGHT 1970 SBN 85261 014 9

PRINTED BY THOMAS RAE LTD. GREENOCK SCOTLAND

THE MATHEMATICAL PRINCIPLES UNDERLYING NEWTON'S PRINCIPIA MATHEMATICA On the 18th of July 1733, half a dozen years after Isaac Newton's death, Dr William Derham (a close friend during his last years) observed that 'sr Is[aac] ... abhorred all Contests .... And for this reason, mainly to avoid being baited by little Smatterers in Mathematicks, he told me, he designedly made his Principia abstruse; but yet so as to be understood by able Mathematicians, who he imagined, by comprehending his Demonstrations, would concurr with him in his Theory'.(l) Forty years before, as Newton passed unseeingly by in the street at Cambridge, a nameless undergraduate had remarked sotto voce: 'There goes the man that writt a book that neither he nor anybody else understands.'(2) Manifestly, if it had been Newton's intention in the 1680's to make his treatise impossibly difficult for all but a tightly restricted elite to comprehend, in this one case at least he well succeeded. But when we go behind such hearsay and anecdote, we will find, somewhat surprisingly perhaps, that there is no trustworthy primary evidence that Newton did at any time deliberately contrive to render his Mathematical Principles of Natural Philosophy more esoteric and impenetrable than he need have done. I do not deny that this hallowed ikon of scientific history is far from easy to read. Quite bluntly, the logical structure of Newton's book is slipshod, its level of verbal fluency none too high, its arguments unnecessarily diffuse and .repetitive, and its content on occasion markedly irrelevant to its professed theme: the theory of bodies moving under impressed forces. But these faults are far from intentional and they can largely be excused by the very rapidity with which (in little more than two years from autumn of 1684) it was written and by its author's distinct lack of talent for writing in a popular way. Of such weaknesses no one was more conscious than Newton himself: to be sure, we now know that soon after his book appeared he reluctantly contemplated a grand revision of his work 5

which he never found time and energy to implement.(3) In default we must suffer the crudities of the text as Newton resigned it to us when we seek to master the Principia's complex mathematical content. To attain this understanding there is no royal road which can by-pass its conceptual difficulties, no novice's path which effectively softens its mathematical rigours. Newton himself, it is true, tried to lighten the heavy load of preparatory learning for such acquaintances as Trinity College's later guiding genius, Richard Bentley, who sought to achieve a limited understanding of but the Principia's basic propositions. Thus, having in 1691 given Bentley the eminently sensible advice that 'At y. first perusal of my Book it's enough if you understand y. Propositions W 1h some of y. Demonstrations weh are easier than the rest ... [then] pass on to y. 3d Book & when you see the design of that you may turn back to such Propositions as you shall have a desire to know', he was then led to compile for him a formidable mathematical reading list (4) which of necessity referred much more to recent works in geometrical and infinitesimal analysis than to ones in traditional algebra and classical geometry. Ineluctably, the hard essence of the Principia was thereby tacitly accepted as accessible to none but the sophisticated mathematician. In Newton's own lifetime only a handful of talented men working without distraction at the frontiers of current research-the Dutch scientist Christiaan Huygens, the German uomo universale Leibniz, the French priest Pierre Varignon, the Huguenot expatriate Abraham De Moivre and Newton's most able editor Roger Cotes-had, each in his own way, achieved a working knowledge of the Principia's technical content. Why so? If, as many still so often claim, the Principia was written in the familiar language of Greek geometry, why was it difficult for its 17th century reader to assimilate? Why, then as now, is a knowledge of Euclid, Archimedes and Apollonius so ineffectual in allowing the beginning student to penetrate its depths? Since at least the days of Whewell (that 19th century propagator of not a few Newtonian myths) the usual response to such queries has been to claim that Newton was blessed with an unparalleled insight into the structures of Greek geometry and with an unsurpassed facility and expertise in manipulating its elegant, archaic techniques. The view has been persuasively endorsed by G. L. Huxley in a recent Harvard 6

essayY) Nevertheless, such an interpretation has little foundation in historical reality. If you sift carefully through the full five hundred or so pages of the Principia's text (in any of its tens of editions) you will, I believe, trace a bare reference (in Book I, Proposition 79) to the main result of Archimedes' Sphere and Cylinder and a unique quotation of a proposition in Apollonius' Conics, while Pappus appears in the opening preface to the work merely as custodian (in the eighth book of his Mathematical Collection) of late Greek ideas on mechanical theory. Not to tantalise, the Apollonian citation (precisely, of Conics Ill, 17 and 18) occurs in Lemma 17 in Section 5 of Book I-but the reference is rather less straightforward than appears at first glance. The opening of this fifth section was, it is now clear, lifted bodily by Newton from an unpublished earlier treatise of his, the Solution of the Ancients' Problem of the Solid Locus, which deals with the semi-projective geometry of conics and was originally composed as a synthetic antidote to the rigorously analytical discussion of these second-degree curves in the second book of Descartes' Geometrie in 1637. Further, this Cartesian passage, with its liberal quotation of Commandino's 1588 Latin translation of Pappus' Collection, had inspired Newton to compose a critique of the Geometrie in which the geometrical construction of the conic through five given points set down by Pappus in the 13th proposition of his eighth book is preferred to Descartes' clumsy analytical variant.(6) In these circumstances it is tempting to conclude that Newton found his reference to Apollonius second-hand in Pappus' theorem rather th~n directly from his reading of the text of the Conics. The focus-directrix properties of a conic lavishly expounded in the Principia's preceding Section 4 are, in fact, a hasty last-minute addition to the body of the book made in late 1685 soon after Newton had seen the eighth book of Philippe de La Hire's newly published Conic Sections :(7) not only is this a modern borrowing, but the theory has no direct Greek antecedents. Lemma 16 of that same section (which resolves Apollonius' problem of constructing a circle tangent to three given circles) is accurately referred by Newton to Franc;ois Viete's French Apollonius of 1600,(8) itself but loosely based on the account given in Pappus' Collection of the lost Apollonian tract On Tangencies: Newton's method in the Principia, I need not say, is but 7

an elegant variant on Viete's scarcely classical solution of the threecircles problem. In general, I know no evidence which begins to hint that Newton ever seriously and painstakingly went through the translated Latin texts of Apollonius' Conics or any of the works of Archimedes; on the contrary, for his knowledge of conic properties he relied heavily in his younger days on the summary appended by Frans van Schooten to the first two Latin editions (issued in 1649 and 1659) of Descartes' Geometrie.(9) With regard to Lemma 12 of the Principa's first book (that 'All parallelograms circumscribed about any two conjugate diameters of a given ellipse or hyperbola are equal in area') William Whiston, his successor in the Lucasian Chair of Mathematics at Cambridge, wrote impressively of Newton's ability to perceive the subtlest mathematical truths 'almost by intuition'(1O) but omitted to mention, if ever he knew, that the theorem instanced was the 31st of the recently rediscovered seventh book of Apollonius' Conics. Newton's intuition may indeed have let him see that this Lemma is, in the case of the ellipse, all but self-evident on considering the general orthogonal projection at a fixed angle of a circle inscribed in a square; but clearly, if Whiston spoke true, his knowledge of the later books of Apollonius was far from perfect. On such frail foundations the claims made for Newton's masterly grasp of the totality of classical geometrical experience must rest. The very form of the Principia is, I do not need to say, heavily 'Euclidean', but Newton's 'Propositions', 'Theorems', 'Problems', 'Lemmas' and 'Scholia' are, let me equally insist, mere expository frameworks inherited from his enforced study, as a subdued Trinity College undergraduate, of Isaac Barrow's 1655 Cambridge edition of the Elements of Euclid, and they are manifestly retained in his own subsequent mathematical writings purely as a literary convenience. We should not allow this Grecian scaffolding to disguise the reality that the edifice beneath is, in its essentials, neither classically inspired nor classically built. A superficially more convincing alternative explanation was that strongly supported by Newton himself intermittently at the time (the decade from 1712) of his calculus priority dispute with the Leibnizians: this affirms that the arguments used in the central portions of the Principia are effectively fluxional analyses clothed in the heavy dress of traditional synthetic geometry. But be ever wary 8

of accepting uncritically at face value Newton's retrospective assessments of the nature and sequence of his scientific achievement. In the present instance, in any of the several simplistic senses in which Newton wished to suggest so personally convenient an interpretation to his early 18th century audience, this beguiling hypothesis eludes substantiation of its truth when we dare to test it. It is, may I say, futile to plough laboriously through the voluminous mass of Newton's extant papers (containing 10-15 million words at a conservative estimate) in search of manuscripts bearing dotted fluxional arguments which reappear, suitably recast in geometrical mould, in the pages of the first edition of the Principia. As Newton himself never forgot (though he tried cleverly to conceal it by various sly turns of phrase from his contemporaries) the standard dot-notation for fluxions was invented by him only in mid-December 1691, almost four and a half years after the Principia first appeared in London's bookshops. Fluxional re-workings of theorems in its first book are indeed still in existence; perhaps the most celebrated of these, a recasting by Newton of his basic measure of a central force, was first disclosed to the world in 1892 by Rouse Ball in a now forgotten essay.(ll) But without exception these are all second thoughts after the event and, even more importantly, are in every case something more than a denuded fluxional proof stripped of its obscuring geometrical cloak. This is not, of course, to maintain that Newton before 1691 did not employ other, less familiar types of fluxional notation. We find, for instance, that the literal fluxions p and q (for the instantaneous 'speeds' of variables x and y with respect to a uniformly 'flowing' independent variable of 'time') which he early came to elaborate in his October 1666 tract on limit-motions were invoked by him twenty years afterwards in still unpublished preliminary computations for determining the differential equation of the solid of revolution whose surface offers least resistance to linear flow;( 12) in the published theorem (Principia, Book 2, Preposition 35) this firstorder fluxional equation is, however, announced wholly without proof-almost as an act of God, indeed-and devoid of any attempt to transmute its analytical derivation into an 'acceptable' geometrical form. Elsewhere, the essence of a large treatise of Newton's on 'Curvilinear Geometry',(13) wherein the fluxion of an upper-case 9

fluent, A say, is represented by the corresponding lower-case equivalent, a, was some five years later inserted by Newton in his Principia as the celebrated 'fluxions' Lemma 2 of Book 2; but if you look closely through the sequel, this Lemma is appealed to only twice in the remaining two hundred pages of the work, both times to support not a fluxional argument but one developed by limit-increments. Again not to tantalise, I refer you to the immediately following Propositions 8 and 9 of the second book. Nowhere, let me repeat, are there to be found extant autograph manuscripts of Newton's, preceding the Principia in time, which could conceivably buttress the conjecture that he first worked the proofs in that book by fluxions before remoulding them in traditional geometrical form. Nor in all the many thousands of such sheets relating to the composition and revision of the Principia is there the faint trace of a suggestion that such papers ever existed. The conclusion seems inescapable. The published state of the Principia's text (one in which the geometrical limit-increment of a variable line-segment plays a fundamental role) is, with minor variants, precisely that in which it was first composed. During the two decades preceding the late summer of 1684 when Newton, in a foretaste of the mighty work to come, drafted the first version of his preliminary tract De Motu Corporum there is overwhelming evidence in the extant corpus of his papers of his developing expertise with (and, to be sure, firm preference for) arguments involving one or more orders of the infinitesimally small. To this ever-deepening mastery over the limit-increments of a mathematical variable, in his Principia itself Newton added the fundamental dynamical insight that a continuous force acting over an infinitesimal time may be computed in terms of the second-order deviation which it induces in a moving body from, or along, its instantaneous rectilinear inertial path (this is effectively his second Axiom of motion); or, equivalently, measured by the component of the instantaneous increment in the body's orbital speed which is directed to the centre of force. Both approaches make open use of vanishing limit-increments (of length and velocity respectively), and we will accordingly begin to appreciate the justice of Clifford Truesdell's claim that the Principia is 'a book dense with the theory and application of the infinitesimal calculus'.(14) This view of 10

the mathematical structure of Newton's work was, in fact, shared by a majority of those of his immediate successors who were able to penetrate its technical complexities: L'Hospital's observation in 1696 that the Principia is 'almost wholly of this [infinit~simal] calculus'( 15) is echoed down the years in remarks by many, but above all by the great Lagrange, who in 1797 affirmed of Book 2, Proposition 10 in particular that it is 'founded on the calculus of differences,.(16) This is no empty eulogy but a common recognition of a basic Newtonian technique: namely, with respect to a uniformly fluent 'related' variable the corresponding increments of all other 'conjoint' variables are to be evaluated by means of the expansion we now associate with Brook Taylor's name, but which was first stated in general form by Newton himself; the conditions of the problem to hand may then be reduced to a relationship between the various orders of the infinitesimally small which compound these increments, and the corresponding relationship between the fluent variables alone thenceaccurately or approximately as circumstance permits-accordingly determined. To fix ideas, if we 0 x suppose that the line-segments 0 X '----------~ and PY (departing from fixed origins o and P) have related lengths x and p....._ _ _ _ _ _ _ _~y. y = y" respectively, and that to the FIG. I. increment Xx =0, say, of OX = x there corresponds the increment Yy = y,,+ 0 - y" of PY, then dy 2 d 2y 3 d 3y YY=O·-dx +-!-o ·d-"2+!o ·d~+···' x x ft dfty where o . d~ is the n-th order infinitesimal of PY = Y with respect to base variable x. At a rough count some half of the problems posed in the Principia's three books are reduced to determining an appropriate equation involving two such fluent variables and their first-, second- and occasionally third-order infinitesimals and then computing the allied relationship which connects the fluent variables alone. In the simplest instance the infinitesimal equation will be first-order and, as we now say, in immediately integrable form, though the solution of more 11

general infinitesimal equations of higher orders was also known to Newton. Let me reaffirm my belief that the 'integration' of these prototype 'differential' equations is the central mathematical method employed by him in his Principia. Merely to make this assertion is not, however, enough. Let me attempt to justify it to the full by exploring in some detail the structure of a few of Newton's main theorems. The fundamental theme of the Principia is the examination of the motion in time of a body attracted by a general central force. The crucial insight on Newton's part which made such a mathematical discussion possible was to see that, by extending Kepler's areal law regulating the movements of the solar planets, the time of orbit of a body in an arbitrary central-force field could be measured by the area swept out by the radius vector joining force-centre to moving body. This fundamental theorem Newton rightly set as the opening proposition of his book. Let me retrace its familiar argument. Conceive that the arc AF traversed in given time by the body under the action of a force, varying uniquely with the distance from S, which 'pulls' it 'instantaneously' towards the centre S of force is split into a large number of arc-segments, AB, BC, CD, ... , EF each traversed in equal times; and correspondingly that the action of the continuous force is broken down into an equal number of discrete 'impulses' of

FIG. 2.

force, each equal in magnitude, striking the orbiting body at the points B, C, D, and so on continually towards the centre S. In other 12

words, if the body at B moves 'instantaneously' in the straight line Bc to c, the first force-impulse knocks it there along cC to C, yielding the line-segment BC as the direction and magnitude of its total motion; at C it moves 'instantaneously' in the direction BC to d in the second time interval, and is then struck along dD to D; and so on The argument then goes: in the limit as the number of divisions of the time of passage from A to F (and so of the divisions of the orbital arc AF) become infinitely great, any two adjoining radii vectores, say SB and SC, come to be indefinitely near, whence to sufficient accuracy-that dangerous phrase !-SB and Cc become parallel and the triangles BSC and BSc accordingly equal in area; but also the infinitesimal segments AB and BC traversed inertially in equal time-divisions are equal, and so the focal triangles BSc and ASB are equal in area; whence the triangles ASB and BSC are equal in area. The same holds for any adjacent pair of focal triangles centred on S. Hence each of the focal triangles ASB, BSC, CSD, DSE, ... whose end-segments AB, BC, CD, DE, ... are traversed by the orbiting body in equal intervals of time are equal in area, and consequently the total focal area SAB ... F measures the time of passage from A to F under the action of an arbitrary, continuously acting central force. I will not pretend that this plausible 'proof' of the generalised Keplerian law is anywhere near as simple or cogent as it appears at first glance: in particular, it is both mathematically and dynamically essential that the arc AF be itself traversed in infinitesimal time, and accordingly that the force-deviations cC, dD, eE, ... and so forth be each of second-order infinitesimal magnitude. But let me here glide over such subtleties since they are not discussed (though I would hesitate to say they were ignored) by Newton himself. This basic proposition is at once applied, in the foIlowing Proposition 6 of the Principia's first book, to derive a measure of the central force under which a given orbit is traversed round a given forcecentre. To summarise: if the force f(SP) acting over the infinitely small arc PQ in time dr, say, produces a total deviation RQ from the

13

S(o,o) FIG. 3 inertial path (the orbital tangent PR at P) and if QT be drawn perpendicular to SP, then the deviation RQ = t/(SP) . dt 2, where the time dt of orbit from P to Q is, by Newton's newly proved theorem, proportional to the focal sector SPQ = -!SP . QT since the points P and Q are taken to be infinitesimally distant. (Here again I slide over the subtlety that the deviation RQ is not in general a straight line, and may only accurately be assumed to be so when it is-as here, fortunately-taken to be of second-order infinitesimal magnitude.) At once RQ is proportional to f(SP) . Sp2 . QT2, whence the ratio RQ/ Sp2 . QT 2 is, in the limit as Q comes to coincide with P, a measure of the central force f(SP). Using this measure of what, in conscious mimicry of the physically non-existent vis centrifuga of his contemporary Huygens,< 17) Newton for the first time named 'centripetal' (centre-seeking) force, he was forthwith (in Propositions 7-13 following) easily able to compute the central attraction by which an orbiting body can be made respectively to traverse a circle round a point in its circumference, an equiangular (logarithmic) spiral round its pole and, most importantly, a general conic round its centre or a focus. The last, explored by Newton in Propositions 11-13 of Book 1 of the Principia, is his celebrated 'solution' to the problem posed by Kepler in his Astronomia Nova in 1609, when he propounded as his first hypothesis of planetary motion the observationally tested axiom that the solar planets move in ellipses round the sun set at a common focus. It is tempting to dwell on this historical aspect of Newton's eleventh proposition (one 14

which, as is well known, is the very raison d'etre for the Principia as a whole) but let me pass quickly on to consider the theoretical power of Newton's general measure of a central force. To do so, permit me (without published Newtonian precedent) to introduce the system of polar coordinates in which the force-centre S is the origin (0, 0), the general orbital point P(r, tjJ) and the infinitesimally distant point Q(r + dr, tjJ + dtjJ), with the polar angle asp = tjJ taken to be the base variable and its infinitesimal increment LPSQ (or dtjJ) =0. We readily determine that, to order 0 3 , QT2 = r2 • 0 2 and

with, of course, SP = r, so that in the limit as the time of passage from P to Q (and hence 0) becomes vanishingly small Newton's measure assumes the more familiar form

where 2

dtjJ

k = r . dt

is 'Kepler's' constant. Here nothing has been done except to cast Newton's geometrical measure into an equivalent analytical form: it has been given no new logical precision or structural redefinition. Such an analytical rewording does, however, allow us a decidedly clearer insight into one of the perennially thorny questions of Newtonian scholarship, one which asks whether it was in his power to solve the inverse problem of central forces: namely, to determine the totality of orbits traversible in a given force field. I believe we can affirm with confidence that Newton's measure would, in principle, allow him to do so in the inverse-square case at least. In this Keplerian instance, since f(SP) is proportional to I /SP 2 , in the analytical equivalent we may put f(r) = h/r 2 and, to solve the inverse problem, we need only show that the conic locus 15

11r = A+Bcos(q,+e) (A, Band e constant) referred to a focus as origin is the general

solution of the differential equation

1 (1) d2

h

-;. + #2 -;. = k2 (constant). Happily, we are spared from arguing that Newton was indeed capable of making such a deduction because he subsequently, in Proposition 41 of the Principia's first book, gave a complete solution of the problem of determining the resulting orbit in a given central-force field when the velocity and direction of initial motion at a point are known. Let me sketch this now classical approach, for it provides an excellent illustration of Newton's infinitesimal method in the Principia. Slightly to restate his argument (in the interest solely of brevity, I again add) assume that an orbiting body moving in a given centralforce field whose intensity at point I, varying with the distance CI from the centre C, is measured, say, by v I( CI) is fired from point V with speed S at an inclination ex to the radius vector CV. What is the resulting orbital arc VI? Suppose that the time of passage from V into I is t, and that the infinitesimal arc IK = d(arcVl) is c traversed in a further element of time dt, so that the speed lim dt-+zero

d(Vl) dt

at I is v and the acceleration in velocity over IK is accordingly dv dt

dv d(V/)

-=v. - -

16

As Newton establishes at some length (by an infinitesimal argument) in Propositions 39 and 40 preceding, the 'instantaneous' acceleration in the direction IK of orbit is due solely to the component of the central force f( CI) acting in that direction, that is, f( CI) diminished in the ratio IN d(CI) IK = d(VI)"

Hence there follows

f sv 2v. dv (or

V 2 _S 2)

=

fCl 2f(CI). d(CI) CY

and accordingly

is the measure of the instantaneous velocity v at the arbitrary point I of orbit. (In his Principia figure Newton incorporates this speed v in his diagram by defining a separate 'velocity' curve, but this is an elegant visual mnemonic which is inessential to the main argument, and I will henceforth ignore it.) Now, by the opening Keplerian proposition, the quantity C/2. d(L VC/)/dt is some constantNewton here takes it to be Q-while, in the figure, v2 or

f

S2+2

C1

f(CI). d(CI)

CY

/K is equal to ( dt)2, that is,

(d~l)r +(CI. d~~VCl)r. 17

On eliminating d(L VCI) and dt respectively between these equalities and integrating a second time, we deduce, retracing Newton's footsteps, that

and L VCI =

fcv C1

C1 2

•

J

Q

. d( C/).

CI

S2+2fcvf(CI). d(C/)-Q 2ICl 2

The first integral determines the time of orbit from V to I in terms of radius vector CI; the latter fixes the orbit (to within two arbitrary constants which are readily evaluated in terms of the initial speed S and angle Q( of projection at V) as the geometrical equivalent of a polar equation relating the angle VCI to the radius Cl. Unlike Johann Bernoulli, who in 1710 wished to claim the analytical form for himself as a major new discovery,(18) I can detect no essential difference between Newton's presentation and the now familiar algebraic equivalent in which one makes the trivial substitution of r, Rand fjJ for C/, CVand L VCI. In hindsight we may appreciate how much futile squabble Newton would have spared himself in after years if, in his Principia, he had now given in prime corollary to his Proposition 41 the particular instance of an inverse-square force, showing (by a couple of easy integrations well within his power) that the orbit is in all cases the focal conic llr = A +Bcos(fjJ +e). Already, however, in the scholium to Proposition 13 preceding he had outlined an ad hoc argument (one essentially involving two differentiations) which achieves that end. Now, evidently, he saw no need to repeat the result. But in his third corollary to the proposition, in a broad hint to the wise that his refusal to broach the inverse-square case stemmed in no way from his inability to resolve it mathematically, he briefly defined the orbits (general Cotesian spirals) comparably traversed in an inversecube force-field, though he tantalisingly gave his reader no proof of 18

his far from obvious constructions. To show the quality of the problem, if for fir) we substitute k/r 3 in the analytical version of the general polar equation of orbit and then carry through the two integrations, we find the resulting orbital curve to be the hyperbolicsecant spiral

!... = R

A.

- 1 A.)

J1

R

sech( -¢+cosh -

,

where A. and J1 are constants definable in terms of the instantaneous speed v, initial radius R, projection angle IX and central force g = k/ R3 at the point of projection. There are five distinct cases according as v2/gR lies between 0 and I; or is I; or lies between I and cosec 2 1X; or is cosec 2 1X; or is greater than cosec 2 1X. For simplicity of exposition, no doubt, Newton in his Principia corollary set the projection angle to be 90°, in which circumstance cases 2-4 degenerate into a circle orbit round the force-centre; in his Corollary 3 he needed therefore only to construct the first and fifth of these Cotesian spirals. (Earlier, in Proposition 9, he had separately treated the general second case: v2 = gR, that of the logarithmic spiral, from first principles.) Subsequently given all-but-trivial analytical redefinition by Pierre Varignon, Johann Bernoulli and Jakob Hermann, Newton's general theorem on central forces quickly (after 1700) became accepted as the standard text-book approach in all accounts of dynamics up to and including Laplace's Mecanique Celeste: with little modification it continues to hold that primary place today. From it all of Newton's remaining theorems on free and constrained motion in a nonresisting medium may be developed as mere offshoots. For example we may, on taking the arc VI to be s, obtain as condition for isochronous constrained motion in a force-field fer) the criterion that fer) . dr/ds = hs, h some constant. As an agreeable instance, Newton in Section 10 of his first book discusses the direct-distance case fer) = kr at some length, using an analogous limit-increment argument to identify the hypotrochoid R _ I(R2 + p2 - r2) . _ 1 pJ R2 + p2 - r2 ¢=--cos +sm - 1----:-'--R+p 2Rp R 2Rp 19

as the direct-distance tautochrone by deriving the SemI-IntnnsIC defining equation S2 = A(r2 -(R- p)2), A = 4Rp/(R+ p)2 with R, p the radii of the deferent circle and epicycle respectively. As a final illustration of the power of Newton's infinitesimal techniques in the Principia, and indeed of the breadth of his dynamical understanding, let me say something of the theorem (Book 2, Proposition 10), which deals generally with resisted motion in the simplest case where the force-centre is at infinity and the action of the force may accordingly be taken to be a constant gravitational 'pull', say g, acting vertically downwards. For ease of comprehending Newton's procedure allow me, unhistorically, to convert his Fermatian choice of co-ordinate variables into standard Cartesian equivalents x, y. We may then suppose that a missile, fired at F with known velocity and angle of p projection, traverses the orbital arc FGHJ under the action of two concurrently acF celerating and decelerating forces, one of constant gravity 9 and one of resistance, the latter varying with the density of the medium and the y body's instantaneous speed from point to point in the orbit. Take the resistance to be p at the point ________ ____ _____ x H(x, y) fixed by the c perpendicular coordinates AC = x, CH = Y with regard to some origin A, assumFIG. 5 ing the y-axis to be parallel to the downward 'pull' of gravity. Consider points G and Ion A~.

~

~~~~

Lx oro 20

~

the curve corresponding to equal increments and decrements BC = CD = 0 of the base (AC = x). At once, if T is the (indefinitely small) time of passage from G to H, and T' the corresponding time from H to I, then (to sufficient accuracy, we may prove) the speed at H is arc GH/T, that at I is arc HI/T', and the difference of these two velocities is due solely to the component of gravity acting along the curve (instantaneously in the direction of motion, that is) retarded by the force of resistance acting in the opposite direction. Whence

and so

!: = dy _ ~(iiI/T' g

ds g

GH/T).

T'

(In the limit as the increment 0 of the base vanishes, this may readily be put in the more familiar Leibnizian form p

dy

I dv

g= ds -g. dt

on taking v to be the instantaneous speed of the projectile at Hand t the time of passage from F to H.) In further argument, since the resistance acts along the arc GH in the direction of the tangent at G, the deviation at H vertically downwards, namely LH, is equal to tgT2; similarly, the succeeding deviation NI is equal to tgT,2. Hence we may set

J

Y-limJd 1 (~ ~ -NI)] HI-GH g - o-+zer ds 2NI LH .

P-

Now { DI :J:x+o} BG-Jx-o are the ordinates corresponding to the incremented abscissas

{~~:~~~} 21

and, on putting Q =dy dx'

R

=1- d 2; =1- dQ dx

dx

and

S=! d 3y =1 dR dx 3 dx'

we may accordingly evaluate DI = Y + oQ + 0 2 R + 0 3S + ... and BG=y-OQ+02R-0 3S+ ... , while the corresponding slopes at I and G are readily computed to be Q + 20R + 30 2S + ... and Q-20R+30 2S- ... . To 0(0 3), therefore,

and

J

NI 3S = 1+0 _+0(0 2 ). LH 2R

-

At once v2 , that is, lim (iii2) o-+-zero T'2

( X -JiI2).IS e ua I to = 0-- lim zero g 2NI q

g

1 +Q2 . x2R '

and finally

I!.. = 3SJi+Q""2 g 4R2 when t.he limit here too is taken. I will not go through the equivalent modern deduction which proceeds from 2

v =g

(dsjdx) 2 P (d 3yjdx 3)(dsjdx) to - =1d 2yjdx 2 g (d 2y/dx 2)2

22

,

but I would note that Johann Bernoulli once again claimed the simple analytical restatement of Newton's result as a major new finding of his own in 1719.(19) This further mention of Bernoulli reminds me that I have not yet admitted that I here expound the revised version of Newton's Proposition 10 which appeared for the first time in 1713 in the second edition of the Principia. If I may simplify a complicated sequence of events, in the version Newton published in 1687 he (in effect) carelessly substituted for the deviation LH at H from the tangent at G the deviation AG at G from the tangent at H: the substitution is not allowable since AG = 0 2R - 0 3S to 0(0 4 ) and so is not equal to LH to that necessary order. The consequence of Newton's oversight is to generate the result p

SJi+Q2

g

2R2

wrong by a factor of t, and so we find it set down in the Principia's first edition. Johann Bernoulli caught this error sometime before August 1710 when he reworked Newton's primary example of a semicircular path, but he could not, try as he might, fault the preceding general argument which produced it: in fact, it is easy to show in this case that Newton's uncorrected result leads to the absurd conclusion that, instantaneously along the curve, the resistance everywhere exactly balances the 'pull' of gravity, so that no accelerated motion along the curve is possible.(20) Bernoulli was then led heavily to support his nephew Niklaus' mischievous (if indeed not downright malicious) conjecture that Newton's proof was correct in its basic structure, the error entering subtly at some unspecified second stage when Newton came to re-interpret his primary measure of the ratio of resistance to gravity in terms of the coefficients Q, R and S: these, the Bernoullis claimed, Newton had blithely confused with the derivatives Q = dyJdx, 2R = d 2yJdx 2 and 6S = d 3yJdx 3 • It is, to borrow Lagrange's adjective,(21) a 'remarkable' comedy of errors that the substitution of Q--+Q, R--+2R and S--+6S converts into

23

the first edition's solution into that of the second: truly the jesting hand of coincidence at its most playful. However, the plausibility of the 'explanation' commended itself in Newton's day to many Continental scholars besides the two Bernoullis-even as late as 1798, despite Lagrange's definitive discussion of the question the previous year, to the capable (and honest) French mathematician Jean Trembley.(22) But to go back, Newton himself was told of the error in his first example (with its implied faulting of the argument of the general theorem) only in late September 1712 when he met Johann's nephew Niklaus during a short visit by the latter to London. At that moment the Principia's second edition, under Roger Cotes' able guiding hand, was all but ready for printing off, but Newton did not hesitate to hold the press until he was ready to substitute a corrected version of his Proposition 10. His still extant papers where he attempted to trace his earlier mistake (setting miscellaneous scraps aside, more than fifty sheets of them) reveal that for many days he could not, try as he might, fault his 1687 proof. Then, suddenly, he saw his mistake, broke off momentarily to draft a quick letter to Niklaus about it, and then returned to complete his demonstration of the corrected theorem. Subsequently, he formulated five distinct proofs before his perfectionist conscience was satisfied, even though in the ten pages available to him in the book waiting to be printed off, he found room for only one.(23) In the outcome Cotes' clever editorial knife neatly filled in all the cracks at the join and no obvious trace of the behind-scenes hurry and scurry which had gone on for a while was left to arouse public curiosity and loss of confidence when the revised text of the Principia finally appeared on sale in the spring of 1713. So much for a recent assertion(24) that 'one would be wrong in thinking that Newton knew how to formulate and resolve problems through the integration of differential equations'. I cannot here examine other examples which contradict it, but may briefly run through the most important. Several of the propositions in Book 2 appeal to logarithmic integrations whose justification is achieved by variations on a canonic limit-increment argument taken, without essential modification, from the little tract of Exercitationes Geometricce which James Gregory published at London in 1668. Double 24

integrals occur on more than one occasion: notably, as we have seen, in Proposition 41 of Book 1 and also (implicitly) in the following Proposition 71. If Newton had not suppressed his computations for Proposition 35 of Book 2 from the printed Principia, he would there have imparted a demonstration combining a partial differentiation (attained yet again in his preferred limit-increment form) followed by an integration to produce a first-order fluxional equation, stated geometrically in the printed theorem, which he then solved parametrically. Some of the arguments used in his discussion of lunar theory in Book 3 involve subtle approximations to first- and secondorders of the infinitesimal. Still more penetrating researches into the moon's path and its variations, both periodic and secular, were never finished and found their way into no published version of the Principia: one set of papers (partly printed in 1888) contains Newton's attempts to derive the annual advance of lunar perigee by an approximate geometrical solution of the restricted 3-body problem; sadly, he failed to realise that a viable approach to the problem requires the rounding-off to be made to third infinitesimal order (as to be sure is the case with the 1687 version of Proposition 10 of the second book) and in hindsight quietly 'cooked' one of the secondorder ways he had employed so as to make it 'yield' an impossibly accurate result.(25) But what proposition did Newton himself reckon to be the mathematical highlight of his Principia? An obvious choice, on the grounds of historical importance at least, could well have been the 'Propositio Kepleriana' (Book 1, Proposition II) where, as in the equivalent theorem in the De Motu Corporum of 1684 which it repeated, a secure foundation was for the first time given to Kepler's shrewd guess that the planets travel in ellipses round the sun. To be sure, if ever you visit the Lincolnshire country town, Grantham, where Newton went to Grammar School, you will see, dominating its main square, a fine statue of him (by Theed) which was inaugurated with due pomp and ceremony by Lord Brougham in 1858. Caught perhaps as he lectures, Newton there points dramatically with his right hand to a brandished scroll, on which the diagram of the 'Propositio Kepleriana' is deeply chiselled. Nearer to Newton's own day, if we can give credence to an account in The Postboy of 12 April 1731, his tomb in Westminster 25

Abbey in London was once inscribed (on the stone 'parchment' borne proudly by the cherubs who flank his effigy) with an unspecified 'Diagram' and a certain 'converging series'; but these, thanks to the polishers who cleaned the monument twenty years ago, are no longer visible to the naked eye. Contemporary portraits of Newton painted in his old age exist in abundance and on occasion he is depicted in these sitting before a half-open Principia; unhappily, when we take a closer look we will find that in all but one instance the painter has hidden the inner page from our gaze or filled it with meaningless squiggles. In the lone exception, an often-reproduced three-quarter view, perhaps by Enoch Seeman, completed just a few months before Newton's death early in 1727,(26) the sitter is portrayed with his hand resting on page 204 of the Principia's third edition: that, namely, on which is printed the central text and illustrating diagram for Proposition 81 of Book 1. Newton's choice-for surely it is his, and not just the artist's random selection of a pretty figure?-is at first sight wholly unexpected, but upon reflection we will begin to appreciate that here indeed is a theorem which can hold its own in beauty and power with any in the whole of Newton's magnum opus. I must here forbear to dwell on the detail of its structure (or on the sweep of the half dozen preceding pages which it climaxes) but may hint that, in effect, it employs a triple integral to evaluate the general 'pull' of a uniform sphere upon a point external to it. A peerless theorem indeed, and one very suitable to epitomise for posterity the wealth of mathematical ingenuity underlying the Principia's dynamical content. Let me, then, salute a man who, for all his several weaknesses, had not merely the ability and resource to create a system of force mechanics which endured for more than two centuries, but also the surpassing genius to fashion and refine the tool, the infinitesimal calculus, by which its potential riches could begin to be mined. The Principia remains untarnishably the queen of classics in the history of exact science.

26

NOTES (1) Derham to John Conduitt, 18 July 1733 (King's College, Cambridge. Keynes MS 133, 10.).

(0) An anecdote credited by Conduitt to Martin Foulkes (King's College, Cambridge. Keynes MS 130.6). (3) See David Gregory's Immutanda in Nova Elem: suorum editione ad mentem autoris Maio 1694 (University Library, Edinburgh. Gregory MS C42); compare Newton's Correspondence, 3, 1961, 384-6. (4) Now in Trinity College, Cambridge. R.4.47, first published by Joseph Edleston in his Correspondence of Sir Isaac Newton and Professor Cotes (London, 1850), Appendix, 274-5.

(6) William Whewell, History of the Inductive Sciences, Book II, Chapter VII (London,. 1837, 167); Philosophy of the Inductive Sciences, Part I, Book II (London,. 1840, 152). G. L. Huxley, 'Newtonian Studies II. Newton and Greek Geometry', Harvard Library Bulletin, 13, 1959, 354-61. (S) See ULC. Add. 3963.12: 'Solutio Problematis Veterum de Loco solido'; and Add. 4004, 89v-90v: 'Veterum Loca solida restituta'. (7)

Sectiones Conicre in Novem Libros distributre (Paris, 1685),

(0) Apollonius Gallus (Paris, 1600), read by Newton in late 1664 in Frans van Schoo ten's Francisci Vietre Opera Mathematica (Leyden, 1646), 325-38.

(0)

See Newton's Mathematical Papers, I, 1967,29-45.

(10) Memoirs of the Life and Writings of Mr. William Whiston . ... Written by himself (London, 1749), 39. (11) W. W. Rouse Ball, 'A Newtonian Fragment [now ULC. Add. 3965.2, 5r-6vJ relating to Centripetal Forces', Proceedings of the London Mathematical Society, 23, 1892,226-31.

(12)

ULC. Add. 3965.10, 107v/134v.

(13)

'Geometria Curvilinea' (ULC. Add. 3963.7, 46r-61v).

(14) An added note in the reprint of his 'A Program toward Rediscovering the Rational Mechanics of the Age of Reason' in Truesdell's Essays in thE' History of Mechanics (Berlin, 1968),99. (15) Guillaume de L'Hospital, Analyse des Injiniment Petits (Paris, 1696), Preface: ' ... lequel est presque tout de ce calcul'. (16) J.-L. Lagrange, Theorie des Fonctions Analytiques, contenant les Principes du Calcul d(lJerentiel (Paris, 1791), 19: 'une solution ... fondee sur la methode meme du Calcul differentiel'.

(11)

Horologillm Oscillatorium (Paris, 1673), 159-61.

27

(18) Histoire de I'Academie royale des Sciences. Annee M.DCC.X.Memoires (Paris, 1713), 523-6. (10)

Acta Eruditorum (May 1719),222-5.

(00) Johann Bernoulli to Leibniz, 12 August 1710 (Commercium Philosophicum, 2, Lausanne/Geneva, 1745,231-2).

(U)

Theorie des Fonctions Analytiques, 368.

(00)

Memoires de I'Ac. roy. des Sc. de Berlin. MDCCXCVIII (1801), 60-75.

(OS)

Newton's variant proofs are now scattered in ULC. Add. 3965.

(I.)

P. Costa bel in The Texas Quarterly, 10.3, 1967, 125.

(IS) ULC. Add. 3966.12, 102r-111 r, partly printed in A Catalogue 0/ the Portsmouth Collection . .. (Cambridge, 1888), Preface, Appx. 3.

(II) Now in the National Portrait Gallery, London; see David Piper's Catalogue 0/ Seventeenth Century Portraits . .. , 1625-17/4 (Cambridge, 1963), 249.

28