Skip to main content

The Jesuits and the Method of Indivisibles


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).

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4


  1. 1.

    This observation occurs already in Grabiner (2014). In a review of Alexander Paulos rather surprisingly also conflates these terms (Paulos 2014).

  2. 2.

    An anonymous reviewer kindly improved my transcription of this hand-written Latin text.

  3. 3.

    Andersen represents the distinction anachronistically—by her own admission—by treating F as a sum, either ∑lΔh or ∑l.

  4. 4.

    In De Quadratura Arithmetica Leibniz writes, “… it serves to lay the foundations for the whole method of indivisibles in the soundest possible way” (quoted in Arthur 2008, 20–21). In a manuscript of 1673 he describes his characteristic triangle as constitutive of the analysis of indivisibles (Probst 2008, 104–105).

  5. 5.

    These reflections were not published until 1919, but were probably familiar to Barrow and Wallis (De Gandt 1995, 186).

  6. 6.

    Such a principle was clearly articulated and used explicitly by Leibniz, but it appears already in Nicholas of Cusa in the mid fifteenth century (Boyer 1949, 92, 93 and 217ff.).

  7. 7.

    Thus, they can be ordered with respect to one another. For example, the width of indivisible HI in Fig. 3 is less than the width of the indivisible HG.

  8. 8.

    Mysticism figures prominently in Pascal’s work. Hence his familiar remark, “The heart has its reasons, which reason does not know” (1909–14, Sect. 277, paragraph 55).

  9. 9.

    Barrow’s compatriot, Wallis, took the same position: “Now this is not to be so understood, as if those Lines (which have no breadth) could fill up a Surface; or those Plains or Surfaces, (which have no thickness) could compleat a Solid. But by such Lines are to be understood, small Surfaces, (of such a length, but very narrow)” (1685, 285–6, quoted in Malet 1997, 69).

  10. 10.

    III.16 is relatively isolated, although Euclid uses it in Book IV (e.g., IV.6).

  11. 11.

    For a fuller version of Grassi’s reply see Redondi (1987, 335–340). The quoted remark is on 336.

  12. 12.

    In the Catholic view, Aquinas had established that transubstantiation is the only doctrine consistent with both reason and scripture (Summa Theologica III, Qu.75–7). McCue describes a different scenario, in which Scotus’s Commentary (ca. 1300) is the decisive event (405ff.). Scotus argued that Aquinas’s philosophical arguments were unconvincing, but the authority of the Lateran Council (1215), was decisive; and according to Scotus, the Council declared for transubstantiation, and Ockham came to the same conclusion (410ff.). Both Scotus and Ockham found consubstantiation to be more philosophically satisfying. McCue argues that, ironically, Scotus misread the Lateran text.

  13. 13.

    In Two New Sciences Galileo conflates mathematical and physical atoms in attempting to explain cohesion by means of the resistance of a void (1974, 24ff.; cf. Opere VIII, 64ff.). These atoms are “unquantifiable parts … infinitely many indivisibles” (33; cf. Opere VIII, 72). In The Assayer, however, the minute particles responsible for taste, odor, and sound have shape and size (1957, 275–6). Particles of fire, which are responsible for heat, have shape and size as well, but they are susceptible to resolution into “truly indivisible atoms,” at which point “light is created” (277–8). It is unclear whether Galileo regarded these atoms as dimensionless, like Cavalieri’s indivisibles.

  14. 14.

    As a result Alexander’s attempt to portray Valerio as “an early victim of the Jesuit war against the infinitely small” is similarly unconvincing (cf. 126, 127).

  15. 15.

    See, e.g., Ferrone and Firpo (1986) and Festa (1991). I offer no opinion on this issue.

  16. 16.

    Here Galileo anticipates Locke’s primary/secondary quality distinction (Locke 1975, 2.8.9–10, 134–5), a notoriously difficult distinction to draw (cf. Macintosh 1976). For an insightful and defensible discussion of the distinction, see Smith (1990).

  17. 17.

    Here the phrase “in the sensible body” translates the Italian “corpo sensitivo,” in place of Drake’s “in consciousness.” Drake’s translation suggests, misleadingly, that secondary qualities are mental states, whereas Galileo intends that they are physical states (cf. Buyse 2015, 31).

  18. 18., Session XIII, Canon 2, accessed August 9, 2016.

  19. 19.

    While this is the Catholic tradition, it is less clear that it’s also part of the Aristotelian tradition, for Aristotle insists, “Clearly then it is in virtue of this category [substance] that each of the others [quality, affection, etc.] is. Therefore that which is primarily and is simply must be substance” (1984, Met. VII, 1028a29–30). This point was not lost on Luther: “Aristotle speaks of subject and accidents so very differently from St. Thomas that it seems to me this great man is to be pitied not only for attempting to draw his opinions in matters of faith from Aristotle, but also for attempting to base them upon a man whom he did not understand” (Luther 1959, 28; quoted in McCue 1968, 415).

  20. 20.

    Otherwise there could be no individuals persisting over time.

  21. 21.

    According to Weisheipl, this doctrine arose specifically to accommodate the Eucharist (1963, 157–8n). In Aquinas’s account of the Eucharist the accident of quantity is independent of the substance of Christ’s body, but the other accidents inhere in that quantity (1947, III, 77, 5).

  22. 22.

    Leibniz had other reasons to exclude indivisibles from his calculus, of course. Indivisibles were non-quantities according to Leibniz, while mathematics was the science of quantities.

  23. 23.

    I’m grateful to Chris Zedick and Mikhail Katz for helpful discussion of this topic.


  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.

  5. Aristotle (1984). The complete works of Aristotle. Barnes (Ed.). Princeton: Princeton University.

  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.

  8. Bascelli, T. (2015). Torricelli’s indivisibles. In Jullien ( 2015), 105–136.

  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).

  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.

  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.

  18. Ferrone, V., & Firpo, M. (1986). From inquisitors to microhistorians. The Journal of Modern History, 58, 485–524.

    Article  Google 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.

    Article  Google 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.

  24. Galilei, G. (1962). Dialogue concerning the two chief world systems (Drake Trans.). Berkeley: University of California.

  25. Galilei, G. (1974). Two new sciences. In Drake (Ed.), Madison: University of Wisconsin.

  26. Goldenbaum, U., & Jesseph, D. (Eds.). (2008). Infinitesimal differences: Controversies between Leibniz and his contemporaries. Berlin: Walter de Gruyter.

    Google Scholar 

  27. Grabiner, J. (2014). “Review of Alexander 2014” at

  28. Gregory, B. (2012). The unintended reformation. Cambridge: Harvard.

    Book  Google Scholar 

  29. Heath, T. (1956). The thirteen books of Euclid’s elements. New York: Dover.

    Google Scholar 

  30. Jullien, V. (Ed.). (2015). Seventeenth century indivisibles revisited. Boston: Birkhäuser.

    Google Scholar 

  31. Leibniz, G. (1920). Early mathematical manuscripts of Leibniz. In Child (Ed.). Chicago: Open Court.

  32. Leibniz, G. (1989). Philosophical essays (Ariew and Garber Trans.). Indianapolis: Hackett.

  33. Locke, J. (1975). An essay concerning human understanding. In Nidditch (Ed.). Oxford: Clarendon Press. (Fourth edition originally published in 1700).

  34. Luther, M. (1959). Luther’s works, American edition. In Bachmann (Ed.), (Vol. 36). Philadelphia: Fortress Press.

  35. Macintosh, J. (1976). Primary and secondary qualities. Studia Leibnitiana, 8, 88–104.

    Google Scholar 

  36. Malet, A. (1997). Barrow, Wallis, and the remaking of seventeenth century indivisibles. Centaurus, 39, 67–92.

  37. Matthews, M. (2009). Teaching the philosophical and worldview components of science. Science & Education, 18, 697–728.

    Article  Google Scholar 

  38. McCue, J. (1968). The doctrine of transubstantiation from Berengar through Trent: The point at issue. The Harvard Theological Review, 61, 385–430.

    Article  Google Scholar 

  39. Palmerino, C. (2003). Two jesuit responses to Galileo’s science of motion: Honore Fabri and Pierre Le Cazre. Feingold, 2003, 187–227.

    Google Scholar 

  40. Pascal, B. (1910). Of the geometric spirit. In Eliot (Ed.), Letters Thoughts & Minor Works (pp. 427–444). New York: P.F. Collier and Son.

  41. 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.

  42. Probst, S. (2008). Indivisibles and infinitesimals in early mathematical texts of Leibniz. In Goldenbaum and Jesseph ( 2008), 95–106.

    Google Scholar 

  43. Radelet-DeGrave, P. (2015). Kepler, Cavalieri, Guldin. Polemics with the departed. Jullien, 2015, 57–86.

    Google Scholar 

  44. Redondi, P. (1987). Galileo heretic. Rosenthal (Trans.). Princeton: Princeton University. (Originally published in Italian in 1983).

  45. Schubring, G. (2005). Conflicts between generalization, rigor, and intuition. New York: Springer.

    Google Scholar 

  46. Smith, A. (1990). Of primary and secondary qualities. The Philosophical Review, 99, 221–254.

    Article  Google Scholar 

  47. Sorabji, R. (1983). Time, creation, and the continuum. Cornell: Cornell University.

    Google Scholar 

  48. Tacquet, A. (1651) Cylindricorum et annularium, lib. IV (Antwerp).

  49. Torricelli, E. (1919–1944) Opere di evangelista torricelli. In Loria & Vassura (Ed.), (Vol. 4). Faenza: Montanari.

  50. Vanpaemel, G. (2003). Jesuit science in the Spanish Netherlands. In M. Feingold (Ed.), (pp. 389–432).

  51. 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 

Download references

Author information



Corresponding author

Correspondence to David Sherry.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sherry, D. The Jesuits and the Method of Indivisibles. Found Sci 23, 367–392 (2018).

Download citation


  • Indivisibles
  • Infinitesimals
  • Jesuit science
  • Eucharist
  • Euclid
  • Galileo
  • Cavalieri
  • Torricelli
  • Pascal
  • Barrow