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Gregory’s Sixth Operation


In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathematics under scrutiny. Following Ian Hacking, we point out that in the history of mathematics the amount of contingency is larger than is usually thought. As a case study, we analyze the historians’ approach to interpreting James Gregory’s expression ultimate terms in his paper attempting to prove the irrationality of \(\pi \). Here Gregory referred to the last or ultimate terms of a series. More broadly, we analyze the following questions: which modern framework is more appropriate for interpreting the procedures at work in texts from the early history of infinitesimal analysis? As well as the related question: what is a logical theory that is close to something early modern mathematicians could have used when studying infinite series and quadrature problems? We argue that what has been routinely viewed from the viewpoint of classical analysis as an example of an “unrigorous” practice, in fact finds close procedural proxies in modern infinitesimal theories. We analyze a mix of social and religious reasons that had led to the suppression of both the religious order of Gregory’s teacher degli Angeli, and Gregory’s books at Venice, in the late 1660s.

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  1. 1.

    Today scholars distinguish carefully between indivisibles (i.e., codimension one objects) and infinitesimals (i.e., of the same dimension as the entity they make up); see e.g., Koyré (1954). However, in the 17th century the situation was less clearcut. The term infinitesimal itself was not coined until the 1670s; see Katz and Sherry (2013).

  2. 2.

    This was an older order than the jesuits. Cavalieri had also belonged to the jesuat order.

  3. 3.

    Jean Bertet (1622–1692), jesuit, quit the Order in 1681. In 1689 Bertet conspired with Leibniz and Antonio Baldigiani in Rome to have the ban on Copernicanism lifted (Wallis 2012).

  4. 4.

    Translation: “But the fact that in Fermat’s equations those terms into which such things enter as squares or rectangles [i.e., multiplied by themselves or by each other] are eliminated but not those into which simple infinitesimal lines [i.e., segments] enter—the reason for that is not because the latter are something whereas the former are really nothing [as Nieuwentijt maintained], but because ordinary terms cancel each other out.”

  5. 5.

    The sources of such a proposal go back (at least) to A. Koyré who wrote: “Le problème du langage à adopter pour l’exposition des oeuvres du passé est extrêmement grave et ne comporte pas de solution parfaite. En effet, si nous gardons la langue (la terminologie) de l’auteur étudié, nous risquons de le laisser incompréhensible, et si nous lui substituons la nôtre, de le trahir.” (Koyré 1954, p. 335, note 3).


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M. Katz was partially supported by the Israel Science Foundation Grant No. 1517/12.

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Correspondence to Mikhail G. Katz.

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Bascelli, T., Błaszczyk, P., Kanovei, V. et al. Gregory’s Sixth Operation. Found Sci 23, 133–144 (2018).

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  • Convergence
  • Gregory’s sixth operation
  • Infinite number
  • Law of continuity
  • Transcendental law of homogeneity