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

- 283 Downloads
- 1 Citations

## 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.

## Keywords

Continuum Limit Orbital Motion Mathematical Proof Central Force Centripetal Force## Notes

### Acknowledgments

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

## 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.CrossRefzbMATHMathSciNetGoogle 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.Google Scholar
- 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.Google Scholar - Bernard Cohen, I. 1971.
*Introduction to Newton’s Principia*. Cambridge, MA: Harvard University Press.CrossRefzbMATHGoogle 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.Google Scholar - Guicciardini, Niccolo. 1999.
*Reading the Principia, The debate on Newton’s mathematical methods for natural philosophy from 1687 to 1726*. Cambridge: Cambridge University Press.CrossRefGoogle 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.Google Scholar
- Herivel, John. 1965.
*The background to Newton’s Principia*. Oxford: Clarendon Press.zbMATHGoogle Scholar - Huygens, Christiaan. 1929. De Vi Centrifuga, in Ouvres Complètes de Christiaan Huygens XVI, The Hague, pp. 253–301.Google Scholar
- Nauenberg, Michael. 1994a. Hooke, orbital motion and Newton’s Principia.
*American Journal of Physics*62: 331–350.Google Scholar - Nauenberg, Michael. 1994b. Newton’s early computational methods for dynamics.
*Archive for the History of Exact Sciences*46: 221–252.Google Scholar - 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.CrossRefzbMATHMathSciNetGoogle 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.CrossRefzbMATHMathSciNetGoogle Scholar - Nauenberg, Michael. 2011. Proposition 10, Book 2, in the Principia, revisited.
*Archive for History of Exact Sciences*65: 567–587.CrossRefzbMATHMathSciNetGoogle Scholar - Nauenberg, Michael. 2012. Comment on “Is Newton’s second law really Newton’s”.
*American Journal of Physics*80: 931–933.CrossRefGoogle Scholar - Newton, Isaac. 1960.
*The correspondence of Isaac Newton*, vol. II, 1676–1687, ed. H.W. Turnbull. Cambridge, MA: Cambridge University Press.Google Scholar - Newton, Isaac. 1969.
*The mathematical papers of Isaac Newton*, Vol. 3, ed. D.T. Whiteside. Cambridge, MA: Cambridge University Press.Google Scholar - Newton, Isaac. 1974.
*The mathematical papers of Isaac Newton*, Vol. 6, ed. D.T. Whiteside. Cambridge, MA: Cambridge University Press.Google Scholar - Newton, Isaac. 1981.
*The mathematical papers of Isaac Newton*, Vol. 8, ed. D.T. Whiteside. Cambridge, MA: Cambridge University Press.Google Scholar - 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.Google Scholar
- 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