Newton's Easy Quadratures ``Omitted for the Sake of Brevity'

In the 1687 Principia, Newton gave a solution to the direct problem (given the orbit and center of force, find the central force) for a conic-section with a focal center of force (answer: a reciprocal square force) and for a spiral orbit with a polar center of force (answer: a reciprocal cube force). He did not, however, give solutions for the two corresponding inverse problems (given the force and center of force, find the orbit). He gave a cryptic solution to the inverse problem of a reciprocal cube force, but offered no solution for the reciprocal square force. Some take this omission as an indication that Newton could not solve the reciprocal square, for, they ask, why else would he not select this important problem? Others claim that ``it is child's play' for him, as evidenced by his 1671 catalogue of quadratures (tables of integrals). The answer to that question is obscured for all who attempt to work through Newton's published solution of the reciprocal cube force because it is done in the synthetic geometric style of the 1687 Principia rather than in the analytic algebraic style that Newton employed until 1671. In response to a request from David Gregory in 1694, however, Newton produced an analytic version of the body of the proof, but one which still had a geometric conclusion. Newton's charge is to find both ``the orbit' and ``the time in orbit.' In the determination of the dependence of the time on orbital position, t(r), Newton evaluated an integral of the form ∫dx/x <sup>n</sup> to calculate a finite algebraic equation for the area swept out as a function of the radius, but he did not write out the analytic expression for time t = t(r), even though he knew that the time t is proportional to that area. In the determination of the orbit, θ (r), Newton obtained an integral of the form ∫dx/√(1−x<sup>2</sup>) for the area that is proportional to the angle θ, an integral he had shown in his 1669 On Analysis by Infinite Equations to be equal to the arcsin(x). Since the solution must therefore contain a transcendental function, he knew that a finite algebraic solution for θ=θ(r) did not exist for ``the orbit' as it had for ``the time in orbit.' In contrast to these two solutions for the inverse cube force, however, it is not possible in the inverse square solution to generate a finite algebraic expression for either ``the orbit' or ``the time in orbit.' In fact, in Lemma 28, Newton offers a demonstration that the area of an ellipse cannot be given by a finite equation. I claim that the limitation of Lemma 28 forces Newton to reject the inverse square force as an example and to choose instead the reciprocal cube force as his example in Proposition 41. (Received August 14, 2002) Published online March 26, 2003 Communicated by G. Smith     

Leonhard Euler's ‘anti-Newtonian’ theory of light

Leonhard Euler was the leading eighteenth-century critic of Isaac Newton's projectile theory of light. Euler's main criticisms of Newton's views are surveyed, and also his alternative account according to which light is a wave motion propagated through the aether. Important changes are identified as having occurred between 1744 and 1746 in Euler's thinking about the way in which a wave such as he supposed light to be is propagated through a medium. Paradoxically, in view of Euler's overtly anti-Newtonian stand, these amount to Euler abandoning his early, Malebranchian notions about the physical basis of wave propagation, in favour of the ideas set out by Newton in Book II of his Principia.     

<Emphasis Type="Italic">Newton's Argument for Proposition 1 of the</Emphasis> Principia

The first proposition of the Principia records two fundamental properties of an orbital motion: the Fixed Plane Property (that the orbit lies in a fixed plane) and the Area Property (that the radius sweeps out equal areas in equal times). Taking at the start the traditional view, that by an orbital motion Newton means a centripetal motion – this is a motion ``continually deflected from the tangent toward a fixed center' – we describe two serious flaws in the Principia's argument for Proposition 1, an argument based on a polygonal impulse approximation. First, the persuasiveness of the argument depends crucially on the validity of the Impulse Assumption: that every centripetal motion can be represented as a limit of polygonal impulse motions. Yet Newton tacitly takes the Impulse Assumption for granted. The resulting gap in the argument for Proposition 1 is serious, for only a nontrivial analysis, involving the careful estimation of accumulating local errors, verifies the Impulse Assumption. Second, Newton's polygonal approximation scheme has an inherent and ultimately fatal disability: it does not establish nor can it be adapted to establish the Fixed Plane Property. Taking then a different view of what Newton means by an orbital motion – namely that an orbital motion is by definition a limit of polygonal impulse motions – we show in this case that polygonal approximation can be used to establish both the fixed plane and area properties without too much trouble, but that Newton's own argument still has flaws. Moreover, a crucial question, haunted by error accumulation and planarity problems, now arises: How plentiful are these differently defined orbital motions? Returning to the traditional view, that Newton's orbital motions are by definition centripetal motions, we go on to give three proofs of the Area Property which Newton ``could have given' – two using polygonal approximation and a third using curvature – as well as a proof of the Fixed Plane Property which he ``almost could have given.' (Received August 14, 2002) Published online March 26, 2003 Communicated by G. Smith     

Borelli’s edition of books V–VII of Apollonius’s Conics,and Lemma 12 in Newton’s Principia

To solve the direct problem of central forces when the trajectory is an ellipse and the force is directed to its centre, Newton made use of the famous Lemma 12 (Principia, I, sect. II) that was later recognized equivalent to proposition 31 of book VII of Apollonius’s Conics. In this paper, in which we look for Newton’s possible sources for Lemma 12, we compare Apollonius’s original proof, as edited by Borelli, with those of other authors, including that given by Newton himself. Moreover, after having retraced its editorial history, we evaluate the dissemination of Borelli's edition of books V-VII of Apollonius’s Conics before the printing of the Principia.

     

The Newtonian Myth

I examine Popper’s claims about Newton’s use of induction in Principia with the actual contents of Principia and draw two conclusions. Firstly, in common with most other philosophers of his generation, it appears that Popper had very little acquaintance with the contents and methodological complexities of Principia beyond what was in the famous General Scholium. Secondly Popper’s ideas about induction were less sophisticated than those of Newton, who recognised that it did not provide logical proofs of the results obtained using it, because of the possibilities of later, contrary evidence. I also trace the historical background to commonplace misconceptions about Newton’s method.     

Visiting Newton's atelier before the Principia, 1679–1684

The worksheets that presumably contained Newton's early development of the fundamental concepts in his Principia have been lost. A plausible reconstruction of this development is presented based on Newton's exchange of letters with Robert Hooke in 1679, with Edmund Halley in 1686, and on some clues in the diagram associated with Proposition 1 in Book 1 of the Principia that have been ignored in the past. A graphical construction associated with this proposition leads to a rapidly convergent method to obtain orbits for central forces, which elucidates how Newton may have have been led to formulate some of his most fundamental propositions in the Principia.     

Proposition 10, Book 2, in the <Emphasis Type="Italic">Principia</Emphasis>, revisited

In Proposition 10, Book 2 of the Principia, Newton applied his geometrical calculus and power series expansion to calculate motion in a resistive medium under the action of gravity. In the first edition of the Principia, however, he made an error in his treatment which lead to a faulty solution that was noticed by Johann Bernoulli and communicated to him while the second edition was already at the printer. This episode has been discussed in the past, and the source of Newton’s initial error, which Bernoulli was unable to find, has been clarified by Lagrange and is reviewed here. But there are also problems in Newton’s corrected version in the second edition of the Principia that have been ignored in the past, which are discussed in detail here.     

Proposition II (Book I) of Newton’s <Emphasis Type="Italic">Principia</Emphasis>

After preparing the way with comments on evanescent quantities and then Newton’s interpretation of his second law, this study of Proposition II (Book I)— Proposition II Every body that moves in some curved line described in a plane and, by a radius drawn to a point, either unmoving or moving uniformly forward with a rectilinear motion, describes areas around that point proportional to the times, is urged by a centripetal force tending toward that same point. —asks and answers the following questions: When does a version of Proposition II first appear in Newton’s work? What revisions bring that initial version to the final form in the 1726 Principia? What, exactly, does this proposition assert? In particular, what does Newton mean by the motion of a body “urged by a centripetal force”? Does it assert a true mathematical claim? If not, what revision makes it true? Does the demonstration of Proposition II persuade? Is it as convincing, for example, as the most convincing arguments of the Principia? If not, what revisions would make the demonstration more persuasive? What is the importance of Proposition II, to the physics of Book III and the mathematics of Book I?     

From centripetal forces to conic orbits: a path through the early sections of Newton’s Principia

In this study, we test the security of a crucial plank in the Principia’s mathematical foundation, namely Newton’s path leading to his solution of the famous Inverse Kepler Problem: a body attracted toward an immovable center by a centripetal force inversely proportional to the square of the distance from the center must move on a conic having a focus in that center. This path begins with his definitions of centripetal and motive force, moves through the second law of motion, then traverses Propositions I, II, and VI, before coming to an end with Propositions XI, XII, XIII and this trio’s first corollary. To test the security of this path, we answer the following questions. How far is Newton’s path from being truly rigorous? What would it take to clarify his ambiguous definitions and laws, supply missing details, and close logical gaps? In short, what would it take to make Newton’s route to the Inverse Kepler Problem completely convincing? The answer is very surprising: it takes far less than one might have expected, given that Newton carved this path in 1687.     

Newton’s “satis est”: A new explanatory role for laws

In this paper I argue that Newton’s stance on explanation in physics was enabled by his overall methodology and that it neither committed him to embrace action at a distance nor to set aside philosophical and metaphysical questions. Rather his methodology allowed him to embrace a non-causal, yet non-inferior, kind of explanation. I suggest that Newton holds that the theory developed in the Principia provides a genuine explanation, namely a law-based one, but that we also lack something explanatory, namely a causal account of the explanandum. Finally, I argue that examining what it takes to have law-based explanation in the face of agnosticism about the causal process makes it possible to recast the debate over action at a distance between Leibniz and Newton as empirically and methodologically motivated on both sides.     

Emilie du Châtelet’s Institutions de physique as a document in the history of French Newtonianism

This paper discusses the contribution of Madame Du Châtelet to the reception of Newtonianism in France prior to her translation of Newton’s Principia. It focuses on her Institutions de physique, a work normally considered for its contribution to the reception of Leibniz in France. By comparing the different editions of the Institutions, I argue that her interest in Newton antedated her interest in Leibniz, and that she did not see Leibniz’s metaphysics as incompatible with Newtonian science. Her Newtonianism can be seen to be in the course of development between 1738 and 1742 and it was shaped by contemporary French debates (for example the vis viva controversy) and the achievement of French Newtonians like Maupertuis in confirming his theories. Her Institutions therefore is linked to the same drive to disseminate Newtonianism undertaken by popularisations such as Voltaire’s Elements de la philosophie de Newton and Algarotti’s Newtonianismo per le dame.     

Isaac Newton and the problem of the earth's shape

In this article I discuss the theory of the earth's shape presented by Isaac Newton in Book III of his Principia. I show that the theory struck even the most reputable continental mathematicians of the day as incomprehensible. I examine the many obstacles to understanding the theory which the reader faced — the gaps, the underived equations, the unproven assertions, the dependence upon corollaries to practically incomprehensible theorems in Book I of the Principia and the ambiguities of these corollaries, the conjectures without explanations of their bases, the inconsistencies, and so forth. I explain why these apparent drawbacks are, historically considered, strengths of Newton's theory of the earth's shape, not weaknesses.     

Newton's Contribution to the Science of Heat

The contents of Scala Graduum Caloris are described, supplemented by unpublished material. Both temperature measurements by his linseed oil thermometer and those based upon his law of cooling are shown to be reasonably accurate to 300°C, but above that value they are much too low. The apparent agreement and the deviation are explained by the differences between the assumptions that Newton made in deriving his law of cooling and the conditions in which he used it. Newton's attempts to link terrestrial and celestial science in applications in the Principia are shown to fail from his confounding the concepts of temperature, heat and radiant intensity and his ignorance of most factors affecting the temperature rise of irradiated materials. Other comments on varied aspects of heat, mainly published in the Queries, are set out and analysed. His originality is assessed.     

Émilie Du Châtelet's interpretation of the laws of motion in the light of 18th century mechanics

Émilie Du Châtelet is well known for her French translation of Newton's Philosophiae Naturalis Principia Mathematica. It is the first and only French translation of Newton's magnum opus. The complete work appeared in 1759 under the title Principes mathématiques de la philosophie naturelle, par feue Madame la Marquise Du Chastellet. Before translating Newton's Principia, Du Châtelet worked on her Institutions de physique. In this book she defended the Leibnizian concept of living forces – vis viva. This paper argues that both of these works were part of a critical transformation and consolidation of post-Newtonian mechanics in the early 18th century, beyond Newton and Leibniz. This will be shown by comparing Du Châtelet's translation of Newton's axioms with her own formulations of the laws of motion in light of Thomas Le Seur's and François Jacquier's Geneva edition which holds a special place among the several editions of the Principia that appeared in the early 18th century.     

The early application of the calculus to the inverse square force problem

The translation of Newton’s geometrical Propositions in the Principia into the language of the differential calculus in the form developed by Leibniz and his followers has been the subject of many scholarly articles and books. One of the most vexing problems in this translation concerns the transition from the discrete polygonal orbits and force impulses in Prop. 1 to the continuous orbits and forces in Prop. 6. Newton justified this transition by lemma 1 on prime and ultimate ratios which was a concrete formulation of a limit, but it took another century before this concept was established on a rigorous mathematical basis. This difficulty was mirrored in the newly developed calculus which dealt with differentials that vanish in this limit, and therefore were considered to be fictional quantities by some mathematicians. Despite these problems, early practitioners of the differential calculus like Jacob Hermann, Pierre Varignon, and Johann Bernoulli succeeded without apparent difficulties in applying the differential calculus to the solution of the fundamental problem of orbital motion under the action of inverse square central forces. By following their calculations and describing some essential details that have been ignored in the past, I clarify the reason why the lack of rigor in establishing the continuum limit was not a practical problem.     

Euler, vis viva, and equilibrium

Euler’s ‘On the force of percussion and its true measure’, published in 1746, shows that not only had the issue of vis viva not been settled, but that the concepts of inertia and even force were still very much up for grabs. This paper details Euler’s treatment of the vis viva problem. Within those details we find differences between his physics and that of Newton, in particular the rejection of empty space and reduction of all forces to the operation of inertia through contact. One can further see how Euler’s philosophy of science embraced explanation through mechanisms and equilibrium conditions.     

Newton's Propositions on Comets: Steps in Transition, 1681–84

Isaac Newton's closest approach to a system of the world in the critical period 1681–84 is provided in a set of untitled propositions concerning comets. They drastically revise his position maintained against Flamsteed in 1681 and may signal his adoption of a single comet solution for the appearances of 1680/1. Points of agreement and difference with the key pre-Principia texts of 1684–85 are analysed. He shows substantial control of the phenomena of tails which change very little in mechanical detail throughout his subsequent work. An emerging theory of gravitation brings planets, their satellites, and comets under the same laws of motion, yet retains a celestial vortex and includes a singular proposition in lieu of the usual formulation of Keplers area law. The analysis raises questions on a number of issues of recent Newtonian scholarship ranging from his achievement following his correspondence with Robert Hooke in 1679 to his veneration of the wisdom of the ancients. (Received September 7, 1999)