Skip to main content

On the deduction of Newton’s second law


We learned at high school and university that Newton’s second law is \(F=ma\). However, Newton never wrote this equation. Furthermore, there is no agreement among historians of science as to the equation that expresses Newton’s second law. On the other hand, Euler claimed to have discovered a principle of mechanics, which is \(F=ma\). The respective paper of Euler provides us with the means of clarifying the issue. We can compare Newton’s second law and Euler’s principle with each other and verify whether there are significant differences between both laws. This is the task of the present paper. The result is that Newton’s axiomatic system and technique of decomposing force differ from Euler’s.

This is a preview of subscription content, access via your institution.


  1. Katsikadelis, J.T.: Derivation of Newtons law of motion using Galileos experimental data. Acta Mech. 226, 3195–3204 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  2. Pourciau, B.: Newton’s interpretation of Newton’s second law. Arch. Hist. Exact Sci. 60, 157–207 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  3. Pourciau, B.: Is Newton’s second law really Newton’s? Am. J. Phys. 79, 1015–1022 (2011)

    Article  Google Scholar 

  4. Nauenberg, M.: Comment on Is Newton’s second law really Newton’s? by Bruce Pourciau. Am. J. Phys. 80, 931–933 (2012)

    Article  Google Scholar 

  5. Pourciau, B.: Reply to Comment on Is Newton’s second law really Newton’s? by Bruce Pourciau. Am. J. Phys. 80, 934–935 (2012)

    Article  Google Scholar 

  6. Truesdell, C.: A program toward rediscovering the rational mechanics of the age of reason. Arch. History. Exact. Sci 1, 3–36 (1960)

    MathSciNet  Article  MATH  Google Scholar 

  7. Cohen, I.B.: Newton’s second law and the concept of force in the Principia. In: Palter, R. (ed.) The Annus Mirabilis of Sir Isaac Newton, 1666–1966, pp. 143–191. M.I.T. Press, Cambridge (1970)

    Google Scholar 

  8. Dellian, E.: Die Newtonsche Konstante. Philos. Nat. 22, 400–405 (1985)

    Google Scholar 

  9. Maltese, G.: La Storia di \(F = ma\): la seconda legge del moto nel XVIII secolo, Firemze (1992)

  10. Roche, J.: What is momentum. Eur. J. Phys. 27, 1019–1036 (2006)

    MathSciNet  Article  Google Scholar 

  11. Maronne, S., Panza, M.: Euler reader of Newton: mechanics and algebraic analysis. Adv. Hist. Stud. 3, 68–81 (2014)

    Article  Google Scholar 

  12. Newton, I.: The Principia Translated by I Bernard Cohen and Anne Whitman. University of California Press, Berkeley (1999)

    MATH  Google Scholar 

  13. Newton, I.: Isaac Newton’s Philosophiae Naturalis Principia Mathematica, 3rd edn. Harvard University Press, Cambridge (1726/1972)

  14. Jammer, M.: Concepts of Force: A Study in the Foundations of Dynamics. Dover Publications, Inc., Mineola (1999)

    Google Scholar 

  15. Hecht, E.: Physics: Calculus. Brooks/Cole, Pacific Grove (2000)

    Google Scholar 

  16. Dransfeld, K., Kienle, P., Kalvius, G.M.: Physik, vol. I. Oldenbourg, München (2001)

    Google Scholar 

  17. Nolting, W.: Grundkurs: Theoretische Physik, vol. 1. Springer, Braunschweig (2005)

    MATH  Google Scholar 

  18. Demtröder, W.: Experimentalphysik, vol. 1. Springer, Berlin (2013)

    MATH  Google Scholar 

  19. Hahn, U.: Physik für Ingenieure 1. Oldenbourg, München (2015)

    Google Scholar 

  20. Katsikadelis, J. T.: Derivation of Newtons law of motion from Keplers laws of planetary motion. Arch. Appl. Mech. (2017)

  21. Euler, L.: Mechanica sive motus scientia analityce exposita, vol. I, II. In: Opera Omnia, serie II, Leipzig (1912)

  22. Euler, L.: Découverte d’un Nouveau Principe de Mécanique, Mémoires de l’académie des sciences de Berlin 6, 185-217. In: Opera Omnia, serie II, vol. 5, 81–108, Leipzig (1912)

  23. Suisky, D.: Euler as Physicist. Springer, Heidelberg (2009)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to R. Lopes Coelho.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Coelho, R.L. On the deduction of Newton’s second law. Acta Mech 229, 2287–2290 (2018).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: