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Orbital motion and force in Newton’s \(\textit{Principia}\); the equivalence of the descriptions in Propositions 1 and 6

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Abstract

In Book 1 of the Principia, Newton presented two different descriptions of orbital motion under the action of a central force. In Prop. 1, he described this motion as a limit of the action of a sequence of periodic force impulses, while in Prop. 6, he described it by the deviation from inertial motion due to a continuous force. From the start, however, the equivalence of these two descriptions has been the subject of controversies. Perhaps the earliest one was the famous discussion from December 1704 to 1706 between Leibniz and the French mathematician Pierre Varignon. But confusion about this subject has remained up to the present time. Recently, Pourciau has rekindled these controversies in an article in this journal, by arguing that “Newton never tested the validity of the equivalency of his two descriptions because he does not see that his assumption could be questioned. And yet the validity of this unseen and untested equivalence assumption is crucial to Newton’s most basic conclusions concerning one-body motion” (Pourciau in Arch Hist Exact Sci 58:283–321, 2004, 295). But several revisions of Props. 1 and 6 that Newton made after the publication in 1687 of the first edition of the Principia reveal that he did become concerned to provide mathematical proof for the equivalence of his seemingly different descriptions of orbital motion in these two propositions. In this article, we present the evidence that in the second and third edition of the Principia, Newton gave valid demonstrations of this equivalence that are encapsulated in a novel diagram discussed in Sect. 4.

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Notes

  1. For this force, Newton coined the name centripetal, seeking the center, in contrast to Huygens’ name centrifugal, fleeing the center (Westfall 1980, 411).

  2. In spite of their importance, these lemmas are often ignored in discussions of the Principia.

  3. De motu corporum in gyrum (On the motion of bodies) (Newton 1974, 30–74) was a nine page tract which Newton wrote, in the Fall of 1684, in response to the visit of Edmond Halley, who asked Newton if he knew the orbit for an inverse square attractive force.

  4. For other presentations, see Francois de Gant, Force and Geometry in Newton’s Principia (de Gant 1995), Niccolo Guicciardini, Reading the Principia (Guicciardini 1999), and I, B. Cohen, A Guide to Newton’s Principia (Newton 1999).

  5. Hooke’s tract was also reviewed in the Philosophical Transaction of the Royal Society 101, (1674) 11–13.

  6. Huygens kept his result secret for 15 years, until finally publishing it in his Horologium Oscillatorum in 1674 (Huygens 1929, 253).

  7. Newton applied this relation for the acceleration to deduce from Kepler’s harmonic law for planetary motions (the square of the periods of the planets is proportional to the cube of their radii) that the gravitational force on the planets dependence inversely on the square of the distance from the Sun. He also verified that the ratio of the gravitational force at the surface of the earth to the gravitational force on the moon was “very nearly” proportional to the square of the ratio of the distance of the moon to the radius of the earth. However, Newton obtained the value of about 4,000, instead of 3,600, because he had used an incorrect estimate of the earth’ radius (Herivel 1965, 196).

  8. There was another more subtle reason for this choice that is generally not pointed out. Leibniz had introduce the concept of first-order and higher-order differentials that were defined as differences between lower order ones. The second-order differential associated with the displacement due to impulses in Prop. 1 could be expressed as the difference of adjacent first-order differential representing the sides of the polygonal trajectory. But this connection cannot be made with the second-order deflection from tangential motion described in Prop. 6.

  9. For example, in discussing Prop. 1, Eric Aiton wrote that “representing a curve by a sequence of first-order parabolic arcs is equivalent to representing the curve by a polygon with second-order infinitesimal sides,” and that the sides of the polygon in Prop. 1 could be interpreted as second-order infinitesimals (Aiton 1989, 211). The same error also was made by Derek Whiteside (Newton 1974, 34–39, n. 19), and subsequently, it has propagated in the literature; for example, in his recent Guide to Newton’s Principia, I.B. Cohen “warmly” recommends this erroneous analysis (Newton 1999, 115). But in Prop. 1, the correct interpretation is that these polygon sides are first-order infinitesimals, as is demonstrated in Sect. 4.

  10. In a recent article in this journal (Arthur 2013, 580), Richard Arthur described Leibniz’s treatment of circular motion by introducing first- and second-order differentials of the radial distance, but these differentials are equal to zero.

  11. An English translation of the second edition is not available at the present time, but fortunately Props. 1 and 6 in the third edition are identically the same as in the second edition, and therefore, it is possibly to use instead the recent English translation of the third edition (Newton 1999). For an English version of the first edition, I have relied on the excellent translation of Mary Ann Rossi (Brackenbridge 1995, 235–267).

  12. It seems strange that Newton regarded inertial motion as motion acted on by an innate or inherent force. I. B. Cohen regards it “as the most puzzling of all the definitions in the Principia” (Newton 1999, 96–102). In definition 2 of De Motu, Newton states that “I call the force of a body or the force innate in a body by reason of which it endeavors to persist in its motion along a straight line,” a description that he maintained also in the subsequent three editions of the Principia.

  13. Actually, this “mighty impulse (impulso unico sed magno)” is a first-order infinitesimal quantity giving rise to an instantaneous infinitesimal change \(\delta v\) in velocity along the direction of the impulse. At \(C\), for example, \(\delta v=Cc/\delta t\), where \(\delta t\) is the infinitesimal time interval between impulses, and the displacement \(Cc\) is a second-order infinitesimal quantity.

  14. In his book, Never at rest, Richard S. Westfall commented that “the derivation could not have survived critical examination, and Newton could not have built the Principia on a foundation so uncertain” (Westfall 1980, 413). But in fact, Newton did precisely that, not changing his formulation by one iota, but adding lemmas and corollaries that clarified the meaning of his proposition.

  15. The Royal Astronomer, John Flamsteed, complained to Newton “ I am obliged to your kind concession of ye perusall of your papers, tho I believe I shall not get a sight of them till our common friend Mr Hooke & the rest of the towne have been satisfied” (Newton 1960, 405).

  16. For a full discussion of Hooke’s diagram and text see (Nauenberg 1994a, 332).

  17. For a sequence of finite periodic impulses at equal time intervals \(\delta t\), the time \(t_n\) at the \(n\;th\) vertex of the polygonal orbit is \(t_n=n\delta t\). According to Prop. 1, \(\delta t\) is proportional to the equal area of the triangles subtended by this orbit. Then, in the limit that \(\delta t \) vanishes, and \( n\) approaches infinity, \(t_n \rightarrow t \), where \(t\) is the time at a given point on the orbit that is kept fixed during this limiting process. Hence, the time \(t\) at a point on the continuum orbit is proportional to the area subtended by the corresponding arc of this orbit.

  18. Corollary 1 states that

    In non resisting media if the areas are not proportional to the times, [then] the forces are not directed along the path of the radii (Brackenbridge 1995, 245).

    and Corollary 2 adds,

    In all mediums, if the description of areas is accelerated, the forces do not tend towards the point where the radii meet but deviate forward [ or in consequentia] from it. (Newton 1999, 447)

    Cohen and Koyre (1971, 91) have pointed out that with some further revisions, Corollary 1 of the first editions plus the addition in the MS Errata became in the second edition the Corollary 1 to Prop. 2 (see also footnotes aa, bb, cc in I. B. Cohen, Guide to Newton’s Principia (Newton 1999, p. 447)) which reads

    In nonresisting spaces or mediums, if the areas are not proportional to the times, the forces do not tend toward the point where the radii meet, but deviate forward [or in consequentia] from it, that is, in the direction toward which the motion takes place provided that the description of the areas is accelerated; but if it is retarded, they deviate backward [or in antecedentia, i.e., in a direction contrary to that in which the motion takes place] (Newton 1999, 447).

  19. Here, Newton has set the stage to obtain a measure of the force in the continuum limit. Since according to Newton’s construction in Fig. 1, \(Cc\) has been taken parallel to \(SB\), it follows that for finite chords \(AB,BC\) the diagonal \(BV\) is along the line \(SB\). But in this corollary, Newton states that \(BV\) will “pass through the center of forces” in the continuum limit, while in his diagram, this is also true for finite time intervals. Evidently, Newton must have been aware that when \(AB\) and \(BC\) are finite chords of two arcs described in “equal time” intervals, the location of \(C\) is not generally the same as the location obtained by his geometric construction with finite chords, where equal time refers to time interval time elapsed between successive inertial motion along the chord, instead of orbital motion along the arc, see Fig. 3. In Newton’s geometrical construction for Prop. 1, equal times correspond to equal areas of the triangles bounded by the chords and the radii. But for continuous orbital motion and forces, which is the limit of vanishingly small triangles, equal times for finite arcs of the orbit is associated with equal areas of the sectors bounded by these arcs and the radial lines. Thus, for finite arcs, \(\overline{AB}\) and \(\overline{BC}\), given \(A\) and \(B\) the position of \(C\) obtained by the requirement that the areas of sectors \(ASB\) and \(BSC\) be the same, will in general differ from the position \(C\) obtained from Newton’s geometrical construction described in Prop. 1, where equal areas refer to the areas of the finite triangles \(ASB\) and \(BSC\). In Cor. 2, Lemma 3, however, Newton indicated that in the limit that “the width of the triangle becomes vanishingly small and the number goes to infinity” the sum of the areas of the triangles bounded by chords, approach the area of the sum of the sectors bounded by the corresponding arcs.

  20. In Prop. 1, Newtow did not prove that the displacements \(Cc, Ff\), etc., are second-order quantities and, therefore, that the ratios \(Cc/dt^2,Ff/dt^2\) are finite in the limit that these differentials vanish. This proof was given in Prop. 6 by applying Lemma 10, in the first edition, and lemma 11 in the second one, assuming that the curvature is finite.

  21. Newton aptly chose the Latin word sagitta for arrow to describe the various displacements of the midpoint of an arc from its chord. Andrew Motte, however, who in 1728 translated the Principia into English, gave it the mathematical expression versed sine, which applies only to the special case that the sagitta is normal to the chord.

  22. A proof that the deviations \(\delta ^2 l_i\) from inertial motion due to impulses are second-order differentials, requires the assumption that the arcs of the underlying orbital curve (left out of Newton’s diagram for Prop. 1 but included in Fig. 3) that determines the magnitude of these deviations have finite curvature. Then, a proof can be obtained by a straightforward extension of Lemma 11, considering the deviation from inertial motion from an extension of the chord, instead of the tangent, of an arc (see Appendix 3). For uniform circular motion, which has constant curvature, such a proof was carried out by Leibniz (Bertoloni Meli 1993, 80–81).

  23. The first-order differentials \(\mathrm{d}x,\mathrm{d}x'\) are not equal, but defined to be transversed during equal time intervals \(\mathrm{d}t\).

  24. At the time, Hermann, Varignon and others, failed to recognize that Prop. 1 demonstrates the validity of Kepler’s area law. In Cartesian differential coordinates, it takes the form \(\mathrm{d}(x\mathrm{d}y-y\mathrm{d}x)=0\). A one line proof is obtained by also expressing the displacement \(ED\) in terms of the second-order differential \(\mathrm{dd}y\), which implies that \( \mathrm{dd}y=(y/x)\mathrm{dd}x\) (Nauenberg 2010, 277). But neither Hermann nor his contemporaries noticed this relation, and it took him another 6 years before he developed a proof, based on Prop. 6 instead of Prop. 1, of this fundamental theorem (see Appendix 2). He published it in his main work, Poronomia, and claimed that it was the first valid proof of Prop. 1 (Guicciardini 1999, 211–215).

  25. In polar coordinate \(r,\theta \), the area \(\mathrm{d}A\) of a differential triangle with a vertex at the center of coordinates, base of length \(r\), and height \(r \mathrm{d}\theta \) is

    $$\begin{aligned} \mathrm{d}A=(1/2) r^2\mathrm{d}\theta . \end{aligned}$$
    (2)

    In Cartesian coordinates, \(r=\sqrt{x^2+y^2},\) and \(\tan \theta =y/x.\) Hence \(\mathrm{d}(\tan \theta \))= \(\mathrm{d}\theta /{\cos }^2\theta =(x\mathrm{d}y-y\mathrm{d}x)/x^2\), and substituting \(r^2\cos ^2\theta \) for \(x^2\), obtain

    $$\begin{aligned} \mathrm{d}A=(1/2)(x\mathrm{d}y-y\mathrm{d}x) \end{aligned}$$
    (3)

  26. According to N. Guicciardini, (Guicciardini 1999, 207), Hermann obtain this expression by “focusing” on Prop. 6. But it is important to recognize that, like Leibniz, he first applied Prop. 1 to obtain a relation for the force in terms of the displacement from the extension of the chord of an arc of the orbital curve, rather than from its tangent as described in Prop. 6.

  27. Newton considered here the accelerative centripetal force or simply the acceleration, as opposed to the motive centripetal force that appears in the Principia, Definition 6, as the conventional definition \(-\) force = mass times acceleration. But this distinction is often ignored by commentators of Prop. 6, although it is evident that this proposition has nothing whatsoever to do with mass.

  28. The constant of proportionality is the inverse of the angular momentum \(l\) for unit mass, \(l=vSY\). where \(v\) is the velocity, and \(SY\) is the component of \(SP\) perpendicular to the direction of the velocity, i.e., \(\delta t=(1/l)QT\times SP\) (see Cor. 1 of Prop. 1). For Hermann’s proof that \(l\) is a constant which is based only on Prop. 6 see Appendix 2.

  29. For circular motion, it was first applied by Huygens in 1659 to derive the central acceleration \(a\) for a body moving with uniform velocity \(v\) in a circular orbit of radius \(\rho \). Huygens approximated the arc \(QP\) of the circle by the arc of the parabola \(QR= (1/2\rho )RP^2\) with vertex at \(P\). Then setting \(RP=v\delta t\), and \(QR=(1/2)a\delta t^2\), he found \(a=v^2/\rho \). In Prop. 6, the factor \(1/2\) is missing.

  30. In the first edition of the Principia Newton referred to Lemma 10 for this relation, but he did not invoke it in the second edition, referring instead to Lemma 11. In this lemma. Newton gave a proof that \(QR=QP^2/2\rho \), when the force is directed along the radius of curvature \(\rho \). For the case of a circle, his proof corresponds to Euclid’s Prop. 36 in Book 3. Since \(QP=v\delta t\), in this case \(QR=a\delta t^2\), which is the content of Lemma 10.

  31. We have added the sagitta \(PX\) to a corrected version of this diagram in Fig. 7.

  32. In his new statement of Prop. 6, Newton defined the centripetal force at a given point \(P\) of the orbit as the limit of the ratio \(sagitta/\delta t^2\), where the sagitta was first introduced in Cor. 4 of the version of Prop. 1 in the second edition of the Principia. In Prop. 6, Newton describes the saggita for a small arc of the orbit, with \(P\) located on the middle of the arc, as a line from \(P\) which bisects the cord of the arc. A subtle point here is what Newton meant by the middle of the arc where the centripetal force is defined. The answer is obtained by turning to Prop. 1, Cor. 4 where the arc describing the sagitta is defined by two adjacent arcs transversed in equal time intervals. Hence, the middle of the arc is not the geometrical middle, which is obtained by dividing an arc into two arcs with the same length.

  33. Around 1705, the expression for the central force in terms of the radius of curvature was obtained independently by the mathematician Abraham De Moivre, who then showed it to Newton. But Newton replied that he had already obtained a similar formula (Guicciardini 1999, 226).

  34. After Lemma 11, Newton devotes a Scholium to discuss curves for which the radius of curvature become infinite at certain points.

  35. We have superimposed this circle on Newton’s diagram as it appeared in the third edition of the Principia. The original tangent line \(ZPY\) at \(P\) had been drawn incorrectly, and an improved line is shown as a dashed line \(Z'PR'\) (The center \(C\) that is obtained by the intersection with the line perpendicular to \(ZPY\) at \(P\) gives a grossly exaggerated radius of curvature which does not contain the chord \(PV\)).

  36. For uniform motion with velocity \(v\) on a circle of radius \(\rho \), by 1669 Newton already had shown that the central acceleration \(a=v^2/\rho \). Since according to Prop. 1, \(v \propto 1/SY\), then for \(\alpha =0\), \(1/a \propto SY^2 \times PV \propto 2\rho /v^2 \).

  37. But already in Prop. 28, Book3, in the first edition of the Principia, Newton applied his curvature relation to obtain the shape of the lunar orbit due to the perturbation of the sun.

  38. The vertices \(A,B,C,c\) in Fig. 3, become \(B,P,Q,c\), respectively, in Fig. 7.

  39. Cor. 2 starts with,

    If chords \(AB\) and \(BC\) of two arcs successively described by the same body in equal times in nonresisting spaces ...

  40. To obtain the point c, first extend the line \(QR\) and then find its intersection with the extension of the chord \(BP\) of an arc \(\overline{BP}\) such that \(Pc=BP\).

  41. Such a proof is obtained by replacing the tangent line in Lemma 11 with the extension of a chord, as shown in Fig. 7 (see Appendix 3). In Cor. 4 of Prop. 1, Newton asserted that \(PX=(1/2)Qc\), which is evident from his diagram for Prop. 1.

  42. In the literature, it is customary to represent Newton’s second law of motion in two forms: (1) F \(\propto \delta v\) for an impulsive force and (2) F \(\propto \delta v/\delta t\) for a continuous force (Newton 1999, 116). But in Prop. 1, we have shown that for 1) the correct expression is \(\delta I \propto \delta v\), where \(\delta I\) is an infinitesimal first-order impulse, and that \(F=\delta I/ \delta t \propto \delta v/\delta t\). In the continuum limit, this is the same expression for force as in 2).

  43. According to Michel Blay, “it is nevertheless difficult to account for the exact relation between ‘the force acting at once with a great impulse’, and the centripetal force acting ‘uninterrupted”’ (Blay 2001, 233). This quote is repeated in Cohen’s guide to Newton’s Principia (Newton 1999, 71,Note 73), but the supposed difficulty is resolved by the realization that the impulse is not “great,” as Newton misleadingly wrote in Prop. 1, but it is a first-order infinitesimal impulse \(\delta I =F\delta t \), where \(F\) is the force in the continuum limit.

  44. In definition 7, Newton wrote that

    the accelerative quantity of centripetal force is the measure of this force that is proportional to the velocity that is generated in a given time (Newton 1999, 407).

  45. In this sense, Newton’s and Leibniz’s approach were not “equivalent in practice,” as has been emphasized by N. Guicciardini (Guicciardini 1999, 250–255).

  46. Varignon, Johann Bernoulli, de Moivre, Keill and Cotes, were able to translate Prop. 6 into differential calculus after discovering that the second-order displacement from the tangent could be expressed in terms of the radius of curvature of the orbit, before Newton included this connection in the second edition of the Principia, in Cor. 3 of this proposition (Guicciardini 1999, 223, 220).

  47. I am indebted to N. Guicciardini for calling my attention to this interesting Scholium.

  48. For similar derivations by Bernoulli and by Varignon see reference (Nauenberg 2010, 285–289).

  49. First published in the 1713 (second) edition of the Principia in Prop. 6 Cor. 3.

  50. Hermann claimed that he was the first to give a proof of the area law (Guicciardini 1999, 216), which is valid provided it is qualified that his proof was based on Prop. 6. instead of Prop. 1.

  51. Hermann’s original derivation is given in reference (Guicciardini 1999, 211–215).

References

  • Aiton, Eric. 1989. Polygons and parabolas: Some problems concerning the dynamics of planetary orbits. Centaurus 31: 207–221.

    Google Scholar 

  • Arthur, Richard T.W. 2013. Leibniz’s syncategorematic infinitesimals. Archive for History of Exact Sciences 67: 553–593.

    Article  MATH  MathSciNet  Google Scholar 

  • Blay, Michel. 2001. Force, continuity, and the mathematization of motion in the seventeenth century. In Isaac’s Newton’s natural philosophy, ed. Jed Z. Buchwald, and I. Bernard Cohen, 225–248. Cambridge, MA: The MIT Press.

    Google Scholar 

  • Brackenbridge, J. Bruce. 1995. The Key to Newton’s Dynamics with and English translation from the Latin of the first three sections of the 1687 edition of the Principia, by Mary Ann Rossi.

  • Brackenridge, J. Bruce, Nauenberg, Michael. 2002. Curvature in Newton’s dynamics. In Cambridge Companion to Newton, ed. I. Bernard Cohen, and George E. Smith, 85–137. Cambridge: Cambridge University Press.

  • Bernard Cohen, I. 1971. Introduction to Newton’s Principia. Cambridge, MA: Harvard University Press.

    Book  MATH  Google Scholar 

  • Bertoloni Meli, Domenico. 1993. Equivalence and priority: Newton vs. Leibniz. Oxford: Clarendon Press.

    Google Scholar 

  • de Gant, Francois. 1995. Force and Geometry in Newton’s Principia, translated by C. Wilson. Princeton: Princeton Univ. Press.

  • Guicciardini, Niccolo. 1999. Reading the Principia, The debate on Newton’s mathematical methods for natural philosophy from 1687 to 1726. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Gunther, R.T. 1930. Early Science in Oxford (Oxford University Press, Oxford 193) vol VI, p. 265; T. Birch The History of the Royal Society of London (Royal Society, London, 1756–1757), pp. 91–92.

  • Herivel, John. 1965. The background to Newton’s Principia. Oxford: Clarendon Press.

    MATH  Google Scholar 

  • Huygens, Christiaan. 1929. De Vi Centrifuga, in Ouvres Complètes de Christiaan Huygens XVI, The Hague, pp. 253–301.

  • Nauenberg, Michael. 1994a. Hooke, orbital motion and Newton’s Principia. American Journal of Physics 62: 331–350.

  • Nauenberg, Michael. 1994b. Newton’s early computational methods for dynamics. Archive for the History of Exact Sciences 46: 221–252.

  • Nauenberg, Michael. 2000. Newton’s Portsmouth perturbation method and its application to Lunar motion. In The foundation of newtonian scholarship, ed. R. Dalitz, and M. Nauenberg. Singapore: World Scientific.

    Google Scholar 

  • Nauenberg, Michael. 2003. Kepler’s area law in the Principia: Filling in some details in Newton’s proof of Proposition 1. Historia Mathematica 30: 441–456.

    Article  MATH  MathSciNet  Google Scholar 

  • Nauenberg, Michael. 2005a. Robert Hooke’s seminal contribution to orbital dynamics. Physics in Perspective 7: 4–34.

    Google Scholar 

  • Nauenberg, Michael. 2005b. Curvature in orbital dynamics. American Journal of Physics 73: 340–348.

    Google Scholar 

  • Nauenberg, Michael. 2010. The early application of the calculus to the inverse square force problem. Archive for History of Exact Sciences 64: 269–300.

    Article  MATH  MathSciNet  Google Scholar 

  • Nauenberg, Michael. 2011. Proposition 10, Book 2, in the Principia, revisited. Archive for History of Exact Sciences 65: 567–587.

    Article  MATH  MathSciNet  Google Scholar 

  • Nauenberg, Michael. 2012. Comment on “Is Newton’s second law really Newton’s”. American Journal of Physics 80: 931–933.

    Article  Google Scholar 

  • Newton, Isaac. 1960. The correspondence of Isaac Newton, vol. II, 1676–1687, ed. H.W. Turnbull. Cambridge, MA: Cambridge University Press.

  • Newton, Isaac. 1969. The mathematical papers of Isaac Newton, Vol. 3, ed. D.T. Whiteside. Cambridge, MA: Cambridge University Press.

  • Newton, Isaac. 1974. The mathematical papers of Isaac Newton, Vol. 6, ed. D.T. Whiteside. Cambridge, MA: Cambridge University Press.

  • Newton, Isaac. 1981. The mathematical papers of Isaac Newton, Vol. 8, ed. D.T. Whiteside. Cambridge, MA: Cambridge University Press.

  • Newton, Isaac. 1999. Principia, third edition. A new translation by I. Bernard Cohen and Anne Whitman with a Guide to Newton’s Principia by I. Bernard Cohen. University of California Press, Berkeley.

  • Pourciau, Bruce. 2004. The importance of being equivalent: Newton’s two models of one-body motion. Archive for History of Exact Sciences 58: 283–321.

    Google Scholar 

  • Pugliese, P. 1989. Robert Hooke and the dynamics of motion in a curved path. In Robert Hooke: New studies, ed. M. Hunter, and S. Schaffer. Woodbridge: Boydell Press.

    Google Scholar 

  • Westfall, Richard S. 1980. Never at rest: A biography of Isaac Newton. Cambridge, MA: Cambridge University Press.

    Google Scholar 

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Acknowledgments

I would like to thank Niccolò Guicciardini for many interesting comments on several of the topics covered here.

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Authors and Affiliations

  1. Department of Physics, University of California, Santa Cruz, CA, 95064, USA

    Michael Nauenberg

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Communicated by Niccolò Guicciardini.

Appendices

Appendix 1: Prop. 6, Cor. 3, on curvature and force in polar coordinates

In 1671, Newton obtained the radius of curvature \(\rho \) of a general curve in coordinates that are equivalent to polar coordinates \(r,\theta \). Written in the language of Leibniz, in terms of the first-order differentials \(\mathrm{d}r,\mathrm{d}\theta \) and the second-order differential \(\mathrm{dd}r\), it takes the form (Newton 1969, 169–173; Nauenberg 2005b, 345),

$$\begin{aligned} \frac{1}{\rho \cos ^3(\alpha )}= \frac{2\mathrm{d}r^2-r\mathrm{dd}r+r^2\mathrm{d}\theta ^2}{r^3\mathrm{d}\theta ^2}, \end{aligned}$$
(5)

where

$$\begin{aligned} \cos (\alpha )=\frac{r\mathrm{d}\theta }{\sqrt{r^2 \mathrm{d}\theta ^2+\mathrm{d}r^2}}, \end{aligned}$$
(6)

and \(\alpha \) is the angle between the radial line and the normal to the tangent of the curve at \(r,\theta \), shown in Fig. 6.

Since \(SY=r \cos (\alpha )\), it follows from Prop. 6, Cor. 3, thatFootnote 48

$$\begin{aligned} F\propto \frac{1}{SY^2 \times PV}= \frac{1}{2r^2 \rho \cos ^3(\alpha )}, \end{aligned}$$
(7)

where \(F\) is the central force (acceleration). Substituting Eq. 5 leads to the modern equation of classical mechanics,

$$\begin{aligned} r^2 F \propto \left( \frac{\mathrm{d}^2}{\mathrm{d}\theta ^2}+1\right) \frac{1}{r}, \end{aligned}$$
(8)

apart from the constant of proportionality. Setting \(\mathrm{d}t=(1/2)(SP \times QT)/l\), where \(l\) is the constant angular momentum for unit mass, one finds that this constant is \(l^2\).

By expressing the second-order displacement \(QR\) and the first-order line \(QT\) (see Fig. 6) in polar coordinates, this equation can also be obtained from Newton’s basic expression for \(F\) in Prop. 6,

$$\begin{aligned} r^2 F\propto \frac{QR}{ QT^2}. \end{aligned}$$
(9)

But such a derivation hides the important role of curvature in Newton’s dynamics.

Appendix 2: Hermann’s derivation of Kepler’s area law

It is of historical interest that in 1716, Jacob Hermann obtained a derivation of the generalized Kepler area law that Newton had given for an impulsive central force in Prop. 1 of the Principia, by starting, instead, with his expression for a continuous central force in Prop. 6. For this purpose, he applied the relation for this force in terms of the radius of curvature of the trajectory.Footnote 49 Referring to the diagram, Fig. 8, associated with this proposition, he set the differential time interval \(\delta t=PQ/v\), where \(v\) is the velocity at \(P\), and then demonstrated that the ratio \(l=SP\times QT/\delta t \) is a constant, corresponding to the angular momentum (for unit mass) \(l=v\times SY\). In essence, Hermann’s proof was a consistency check, filling a gap in Newton’s demonstration of the equivalence of Prop. 6 with Prop. 1 that was lacking in the first edition of the Principia.Footnote 50 For completeness, we present a brief derivation of Hermann’s proof in somewhat different form.Footnote 51

Referring to Fig. 8, which corresponds to Newton’s diagram for Prop. 6, Fig. 6, but with the addition of a line \(QY'\) tangent at \(Q\), and the normal \(SY'\) to this line, let \(p=SY\), \(p'=SY', \delta p=SY'-SY, q=PY\) and \(\delta s= PQ\). Then

$$\begin{aligned} \delta p=-q \delta \phi , \end{aligned}$$
(10)

where \(\delta \phi \) is the angle between the tangential lines \(QY\) and \(QY'\) and also between the corresponding perpendicular lines \(PC\) at \(P\), and \(QC\) at Q, which intersect a \(C\). Hence, \(PC=QC=\rho \), the curvature radius of the arc \(PQ\) and \(\delta s= \rho \delta s\). We have

$$\begin{aligned} \delta v=F_T \delta t \end{aligned}$$
(11)

where \(F_T\) is the tangential component of the force. Applying the expression for this component in terms of the radius of curvature \(\rho \),

$$\begin{aligned} F_T=\frac{v^2}{\rho }\frac{q}{p}, \end{aligned}$$
(12)

and substituting \(\delta t=\delta s/v=\rho \delta \phi /v \) in Eq. 11,

$$\begin{aligned} \mathrm{d}v=\frac{vq}{p}\delta \phi . \end{aligned}$$
(13)

Finally, substituting this relation for \(\delta \phi \) in terms of \(\delta v\) in Eq. 10,

$$\begin{aligned} \delta p=-\frac{p}{v}\delta v \end{aligned}$$
(14)

which implies that the angular momentum \(l=pv\) is a constant of the motion.

Fig. 8
figure 8

The diagram for Prop. 6 in the second and third edition of the Principia, with the addition of the line \(QY'\) tangent at \(Q\), the normal \(SY'\) to this line, and the radius of curvature \(\rho \) at \(P\) with center at \(C\)

Full size image

To understand why the radius of curvature \(\rho \) does not appear in Eq. 13, we express the second-order displacement \(QR\) from inertial motion in the form

$$\begin{aligned} QR=\frac{\delta s \delta \phi }{2 \cos \alpha } \end{aligned}$$
(15)

where \(\alpha \) is the angle between the lines \(SP\) and \(SY\). Then, by Prop. 6,

$$\begin{aligned} F_T=\frac{2QR \sin \alpha }{\delta ^2t}, \end{aligned}$$
(16)

and

$$\begin{aligned} \delta v = F_T \delta t =\frac{\delta s \delta \phi }{\delta t} \tan \alpha = v \delta \phi \frac{q}{p} \end{aligned}$$
(17)

which corresponds to Eq. 13.

Appendix 3: An extension of Lemma 11

In this appendix, we discuss Lemma 11 for Prop. 6 and an extension for its application to the continuum limit in Prop. 1. In Fig. 9, associated with this lemma, Newton has drawn the tangent \(AD\) at a point \(A\) of an arc \(\overline{AB}\) of a curve and the perpendicular \(AG\) to this tangent. The line \(DB\) is the perpendicular from \(B\) intersecting the tangent at \(D\), \(AB\) is the chord of the arc, and \(BG\) is a perpendicular to \(AB\) intersecting \(AG\) at \(G\). The vertices \(d,b\) and \(g\) and corresponding lines are similar to those associated with \(D,B\) and \(G\). If \(\overline{AB}\) is the arc of a circle with radius \(AJ/2\), then \(G=g=J\) and

$$\begin{aligned} DB=\frac{AB^2}{AG}, \quad db=\frac{Ab^2}{Ag} \end{aligned}$$
(18)

For a general arc \(\overline{AB}\), in the limit that \(B\) and \(b\) both approach \(A, BG\) and \(bg\) intersect at \(J\), where \(AJ/2\) is the radius of curvature of the arc at \(A\).

Fig. 9
figure 9

Newtons diagram for Lemma 11, with an extended arc \(\overline{B'b'A}\), and lines \(B'AD', b'Ad'\) superimposed in dotted lines

Full size image

In the absence of the tangent line \(AD\), as is the case in Prop. 1, Newton’s proof is readily generalized by assuming that \(A\) is a point approximately in the middle of an arc \(\overline{B'AB}\) and \(b'Ab\). Let \(AD'=AB\) be the extension of the chord \(AB'\) and \(Ad'\) the corresponding extension of the chord \(Ab'\). Then, if \(\overline{B'AB} \) is the arc of a circle,

$$\begin{aligned} D'B=2\frac{AB^2}{AG}, \quad d'b=2\frac{Ab^2}{Ag}, \end{aligned}$$
(19)

Likewise, for a general arc \(\overline{B'AB} \) the proof is the same as the previous one.

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Nauenberg, M. Orbital motion and force in Newton’s \(\textit{Principia}\); the equivalence of the descriptions in Propositions 1 and 6. Arch. Hist. Exact Sci. 68, 179–205 (2014). https://doi.org/10.1007/s00407-014-0136-6

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