Daniel Bernoulli, d’Alembert and the inoculation of smallpox (1760)

  • Nicolas Bacaër


In 1760 Daniel Bernoulli wrote an article modeling smallpox. In his time there was much controversy around inoculation, a practice that could protect people but could also be deadly. He used Halley’s life table and some data concerning smallpox to show that inoculation was advantageous if the associated risk of dying was less than 11%. Inoculation could increase life expectancy at birth up to three years. D’Alembert criticized Bernoulli’s work, which was the first mathematical model in epidemiology.


Life Expectancy Increase Life Expectancy Publication Delay Susceptible People Paris Academy 
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Further reading

  1. 1.
    d’Alembert, J.: Onzième mémoire, Sur l’application du calcul des probabilités à l’inoculation de la petite vérole. In: Opuscules mathématiques, Tome second, pp. 26–95. David, Paris (1761). Google Scholar
  2. 2.
    Bernoulli, D.: Réflexions sur les avantages de l’inoculation. Mercure de France, 173–190 (1760). Also in: Die Werke von Daniel Bernoulli, Band 2, pp. 268–274. Birkhäuser, Basel (1982).
  3. 3.
    Bernoulli, D.: Essai d’une nouvelle analyse de la mortalité causée par la petite vérole et des avantages de l’inoculation pour la prévenir. Hist. Acad. R. Sci. Paris, 1–45 (1760/1766). English translation: Rev. Med. Virol. 14, 275–288 (2004).
  4. 4.
    Dietz, K., Heesterbeek, J.A.P.: Daniel Bernoulli’s epidemiological model revisited. Math. Biosci. 180, 1–21 (2002) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Duvillard, E.E.: Analyse et tableaux de l’influence de la petite vérole sur la mortalité à chaque âge. Imprimerie Impériale, Paris (1806). Google Scholar
  6. 6.
    Lambert, J.H.: Contributions mathématiques à l’étude de la mortalité et de la nuptialité (1765 et 1772). INED, Paris (2006).
  7. 7.
    Laplace, P.S.: Théorie analytique des probabilités. Courcier, Paris (1812). Google Scholar
  8. 8.
    Straub, H.: Bernoulli, Daniel. In: Gillespie, C.C. (ed.) Dictionary of Scientific Biography, vol. 2, pp. 36–46. Scribner, New York (1970) Google Scholar
  9. 9.
    Tent, M.B.W.: Leonhard Euler and the Bernoullis. A K Peters, Natick, Massachusetts (2009) MATHGoogle Scholar
  10. 10.
    Voltaire: Lettres philosophiques. Lucas, Amsterdam (1734).

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.IRD (Institut de Recherche pour le Développement)BondyFrance

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