Foundations of Science

, Volume 23, Issue 2, pp 367–392 | Cite as

The Jesuits and the Method of Indivisibles

  • David Sherry


Alexander’s Infinitesimal is right to argue that the Jesuits had a chilling effect on Italian mathematics, but I question his account of the Jesuit motivations for suppressing indivisibles. Alexander alleges that the Jesuits’ intransigent commitment to Aristotle and Euclid explains their opposition to the method of indivisibles. A different hypothesis, which Alexander doesn’t pursue, is a conflict between the method of indivisibles and the Catholic doctrine of the Eucharist. This is a pity, for the conflict with the Eucharist has advantages over the Jesuit commitment to Aristotle and Euclid. The method of indivisibles was a method that developed in the course of the seventeenth century, and those who developed ‘beyond the Alps’ relied upon Aristotelian and Euclidean ideals. Alexander’s failure to recognize the importance of Aristotle and Euclid for the development of the method of indivisibles arises from an unwarranted conflation of indivisibles and infinitesimals (Sect. 2). Once indivisibles and infinitesimals are distinguished, we observe that the development of the method of indivisibles exhibits an unmistakable sympathy for Aristotle and Euclid (Sect. 3). Thus, it makes sense to consider an alternative explanation for the Jesuit abhorrence of indivisibles. And indeed, indivisibles but not infinitesimals conflict with the doctrine of the Eucharist, the central dogma of the Church (Sect. 4).


Indivisibles Infinitesimals Jesuit science Eucharist Euclid Galileo Cavalieri Torricelli Pascal Barrow 


  1. Alexander, A. (2014). Infinitesimal: How a dangerous mathematical theory shaped the modern world. New York: Scientific American/Farrar, Straus and Giroux.Google Scholar
  2. Andersen, K. (1986). The method of indivisibles: Changing understandings. In A. Heinekamp (Ed.), 300 Jahre “Nova methodus” von G. W. Leibniz (1684–1984) (pp. 14–25). Stuttgart: Steiner.Google Scholar
  3. Andreae, J. (1979). Formula of concord: Epitome. Milwaukee: Northwestern Publishing. (Originally published in 1580).Google Scholar
  4. Aquinas, T. (1947). Summa theologica. The fathers of the English Dominican Province (Trans.). New York: Benziger Brothers.Google Scholar
  5. Aristotle (1984). The complete works of Aristotle. Barnes (Ed.). Princeton: Princeton University.Google Scholar
  6. Arthur, R. (2008). Leery bedfellows: Newton and Leibniz on the status of infinitesimals. Goldenbaum and Jesseph, 2008, 7–30.Google Scholar
  7. Barrow, I. (1734). The usefulness of mathematical learning explained and demonstrated (Kirkby, Trans.). London: Stephen Austen.Google Scholar
  8. Bascelli, T. (2015). Torricelli’s indivisibles. In Jullien ( 2015), 105–136.Google Scholar
  9. Boyer, C. (1949). The history of the calculus and its conceptual development. New York: Dover.Google Scholar
  10. Buyse, F. (2015). The distinction between primary properties and secondary qualities in Galileo Galilei’s natural philosophy. Working Papers of the Quebec Seminar in Early Modern Philosophy (Vol. 1, pp. 20–45).Google Scholar
  11. Calvin, J. (1953). Institutes of the christian religion (Vol. II). Grand Rapids: Wm. B. Eerdmans Publishing.Google Scholar
  12. Chareix, F. (2002). Le Mythe Galilée. Paris: Presses Univérsitaire de France.Google Scholar
  13. Dedekind, R. (1963). Essays on the theory of numbers. New York: Dover.Google Scholar
  14. DeGandt, F. (1995). Force and geometry in Newton’s principia, trans. Wilson. Princeton: Princeton University.Google Scholar
  15. Descotes, D. (2015). Two jesuits against the indivisibles. Jullien, 2015, 249–273.Google Scholar
  16. Dijksterhuis, E. (1961). The mechanization of the world picture. Oxford: Clarendon.Google Scholar
  17. Feingold, M. (Ed.). (2003). The New science and jesuit science: Seventeenth century perspectives. Dordrecht: Kluwer Academic.Google Scholar
  18. Ferrone, V., & Firpo, M. (1986). From inquisitors to microhistorians. The Journal of Modern History, 58, 485–524.CrossRefGoogle Scholar
  19. Festa, E. (1990). La querelle de l’atomisme: Galilée, Cavalieri et les jésuites. La Recherche, 21, 1038–1047.Google Scholar
  20. Festa, E. (1991). Galilée hérétique? Revue d’histoire des sciences, 44, 91–116.CrossRefGoogle Scholar
  21. Festa, E. (1992). Quelques aspects de la controverse sur les indivisibles. In M. Bucciantina et al. (Eds.), Geometria e atomismo nella scuola Galileiana (pp. 193–207). Florence: Leo S. Olschki.Google Scholar
  22. Fouke, D. (1992). Metaphysics and the eucharist in the early Leibniz. Studia Leibnitiana, 24, 145–159.Google Scholar
  23. Galilei, G. (1957). Discoveries and opinions of Galileo. In Drake (Ed.). New York: Random House.Google Scholar
  24. Galilei, G. (1962). Dialogue concerning the two chief world systems (Drake Trans.). Berkeley: University of California.Google Scholar
  25. Galilei, G. (1974). Two new sciences. In Drake (Ed.), Madison: University of Wisconsin.Google Scholar
  26. Goldenbaum, U., & Jesseph, D. (Eds.). (2008). Infinitesimal differences: Controversies between Leibniz and his contemporaries. Berlin: Walter de Gruyter.Google Scholar
  27. Gregory, B. (2012). The unintended reformation. Cambridge: Harvard.CrossRefGoogle Scholar
  28. Heath, T. (1956). The thirteen books of Euclid’s elements. New York: Dover.Google Scholar
  29. Jullien, V. (Ed.). (2015). Seventeenth century indivisibles revisited. Boston: Birkhäuser.Google Scholar
  30. Leibniz, G. (1920). Early mathematical manuscripts of Leibniz. In Child (Ed.). Chicago: Open Court.Google Scholar
  31. Leibniz, G. (1989). Philosophical essays (Ariew and Garber Trans.). Indianapolis: Hackett.Google Scholar
  32. Locke, J. (1975). An essay concerning human understanding. In Nidditch (Ed.). Oxford: Clarendon Press. (Fourth edition originally published in 1700).Google Scholar
  33. Luther, M. (1959). Luther’s works, American edition. In Bachmann (Ed.), (Vol. 36). Philadelphia: Fortress Press.Google Scholar
  34. Macintosh, J. (1976). Primary and secondary qualities. Studia Leibnitiana, 8, 88–104.Google Scholar
  35. Malet, A. (1997). Barrow, Wallis, and the remaking of seventeenth century indivisibles. Centaurus, 39, 67–92.Google Scholar
  36. Matthews, M. (2009). Teaching the philosophical and worldview components of science. Science & Education, 18, 697–728.CrossRefGoogle Scholar
  37. McCue, J. (1968). The doctrine of transubstantiation from Berengar through Trent: The point at issue. The Harvard Theological Review, 61, 385–430.CrossRefGoogle Scholar
  38. Palmerino, C. (2003). Two jesuit responses to Galileo’s science of motion: Honore Fabri and Pierre Le Cazre. Feingold, 2003, 187–227.Google Scholar
  39. Pascal, B. (1910). Of the geometric spirit. In Eliot (Ed.), Letters Thoughts & Minor Works (pp. 427–444). New York: P.F. Collier and Son.Google Scholar
  40. Paulos, J. (2014). The sixteenth century line of fire: Infinitesimal, a look at a sixteenth-century math battle. New York Times, April 7, 2014, D5.Google Scholar
  41. Probst, S. (2008). Indivisibles and infinitesimals in early mathematical texts of Leibniz. In Goldenbaum and Jesseph ( 2008), 95–106.Google Scholar
  42. Radelet-DeGrave, P. (2015). Kepler, Cavalieri, Guldin. Polemics with the departed. Jullien, 2015, 57–86.Google Scholar
  43. Redondi, P. (1987). Galileo heretic. Rosenthal (Trans.). Princeton: Princeton University. (Originally published in Italian in 1983).Google Scholar
  44. Schubring, G. (2005). Conflicts between generalization, rigor, and intuition. New York: Springer.Google Scholar
  45. Smith, A. (1990). Of primary and secondary qualities. The Philosophical Review, 99, 221–254.CrossRefGoogle Scholar
  46. Sorabji, R. (1983). Time, creation, and the continuum. Cornell: Cornell University.Google Scholar
  47. Tacquet, A. (1651) Cylindricorum et annularium, lib. IV (Antwerp).Google Scholar
  48. Torricelli, E. (1919–1944) Opere di evangelista torricelli. In Loria & Vassura (Ed.), (Vol. 4). Faenza: Montanari.Google Scholar
  49. Vanpaemel, G. (2003). Jesuit science in the Spanish Netherlands. In M. Feingold (Ed.), (pp. 389–432).Google Scholar
  50. Weisheipl, J. (1963). The concept of matter in fourteenth century science. In E. McMullin (Ed.), The concept of matter in Greek and medieval philosophy (pp. 147–169). Notre Dame: University of Notre Dame.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Northern Arizona UniversityFlagstaffUSA

Personalised recommendations