Euler’s Discovery and Resolution of D’Alembert’s Paradox

  • William W. HackbornEmail author
Conference paper
Part of the Proceedings of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques book series (PCSHPM)


This article makes a case for Euler as the first discoverer of what has come to be known as d’Alembert’s paradox. Suppose a body is immersed in an unbounded fluid and moves with constant velocity relative to the fluid, which is otherwise undisturbed: d’Alembert’s paradox asserts that, contrary to experimental evidence, the fluid exerts no drag force on the body (in the direction opposite to the body’s motion) if the fluid is inviscid and incompressible. Euler demonstrates this, for a two-dimensional body or an axisymmetric body whose axis aligns with its motion, in his extensive 1745 commentary on New Principles of Gunnery, a book published in 1742 by Benjamin Robins. After a rigorous analysis, Euler recognizes that the absence of a drag force conflicts with experience for fluids like air and water, and he uses Robins’ experiments with musket balls to explain this anomaly as a consequence of greater fluid pressure fore of the body than aft of it, due to a corresponding fore–aft asymmetry in the density of the fluid. Essentially, he resolves the apparent paradox by removing the assumption of the fluid’s incompressibility.



The author gratefully acknowledges that all the images used herein were scanned by and are used at the courtesy of, the University of Calgary, Military Museums Library and Archives.


  1. Batchelor GK (2000) An Introduction to Fluid Dynamics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  2. Birkhoff G (1950) Hydrodynamics: a Study in Logic, Fact and Similitude. Princeton University Press, PrincetonzbMATHGoogle Scholar
  3. Bernoulli D (1738) Hydrodynamica, sive de viribus et motibus fluidorum commentarii. StrasbourgGoogle Scholar
  4. Brown H (1777) The true principles of gunnery investigated and explained. Nourse, LondonGoogle Scholar
  5. D’Alembert JLR (1752) Essai d’une nouvelle théorie de la résistance des fluides. ParisGoogle Scholar
  6. Darrigol O, Frisch U (2008) From Newton’s mechanics to Euler’s equations. Phys D 237:1855–1869, doi: 10.1016/j.physd.2007.08.003MathSciNetCrossRefGoogle Scholar
  7. Euler L (1745) Neue Grundsätze der Artillerie, aus dem englischen des Herrn Benjamin Robins übersetzt und mit vielen Anmerkungen versehen. Berlin. Also E77 in Opera Omnia, Ser 2, 14:1–409. Birkhäuser, BaselGoogle Scholar
  8. Euler L (1753) Recherches sur la veritable courbe que décrivent les corps jettés dans l’air ou dans un autre fluide quelconque. Mem de l’acad des sci de Berlin 9:321–352Google Scholar
  9. Grimberg G, Pauls W, Frisch U (2008) Genesis of d’Alembert’s paradox and analytical elaboration of the drag problem. Phys D 237:1878–1886, doi: 10.1016/j.physd.2008.01.015MathSciNetCrossRefGoogle Scholar
  10. Hackborn WW (2006) The science of ballistics: mathematics serving the dark side. Proceedings of the CSHPM/SCHPM 31st Annual Meeting, 19:109–119Google Scholar
  11. Hackborn WW (2016) On motion in a resisting medium: a historical perspective. Am J Phys 84:127–134, doi: 10.1119/1.4935896CrossRefGoogle Scholar
  12. Hoffman J, Johnson C (2010) Resolution of d’Alembert’s Paradox, J Math Fluid Mech, 12:321–334, doi: 10.1007/s00021-008-0290-1MathSciNetCrossRefGoogle Scholar
  13. McMurran S, Rickey VF (2008) The impact of ballistics on mathematics. Proceedings of the 16th ARL/USMA Technical Symposium. West Point NYGoogle Scholar
  14. Newton I (1726) Philosophiae Naturalis Principia Mathematica, 3rd edn. In Cohen IB, Whitman A (ed, tr 1999) The Principia: Mathematical Principles of Natural Philosophy. University of California Press, BerkeleyGoogle Scholar
  15. Robins B (1742) New Principles of Gunnery. Nourse, LondonGoogle Scholar
  16. Smith GE (1999) Another way of considering Book 2: some achievements of Book 2. In Cohen IB (ed), A Guide to Newton’s Principia, a preface to Cohen IB, Whitman A (ed, tr, 1999) The Principia: Mathematical Principles of Natural Philosophy. University of California Press, BerkeleyGoogle Scholar
  17. Steele BD (1994) Muskets and pendulums: Benjamin Robins, Leonard Euler, and the ballistics revolution, Tech Cult 35:348–382Google Scholar
  18. Truesdell C (1954) Rational Fluid Mechanics, 1687–1765: Editor’s Introduction to Euler L, Opera Omnia Ser 2, 12:IX–CXXV. LausanneGoogle Scholar
  19. Vincenti WG, Bloor D (2003) Boundaries, Contingencies and Rigor: Thoughts on Mathematics Prompted by a Case Study in Transonic Aerodynamics, Soc Stud Sci, 33:469–507Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of AlbertaAugustana CampusCamroseCanada

Personalised recommendations