... since the very idea of infinitesimal was foreshadowed by Cavalieri ( "limit") in 1635, then put forward in an indirect way by John Wallis ($1/\infty$) in 1655, and then formalized by Newton ( "$o$" ) in 1666, and by Leibniz still a few years later?

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    $\begingroup$ What "ban" ? Are you referring to Amir Alexander's novel ? If so, see the post Are infinitesimals Dangerous ? $\endgroup$ – Mauro ALLEGRANZA Jun 17 '19 at 11:36
  • $\begingroup$ On this complex issue, you can see : Vincent Jullien (editor), Seventeenth-Century Indivisibles Revisited (2015, Birkhauser). $\endgroup$ – Mauro ALLEGRANZA Jun 17 '19 at 11:47
  • $\begingroup$ @MauroALLEGRANZA, that post caused my question, the answers there are not satisfactory. $\endgroup$ – user157860 Jun 17 '19 at 11:54
  • $\begingroup$ Please edit the question to provide more context and describe your source of information. Otherwise every one who reads this question has to click around on links and read multiple answers to try to figure out what you mean by a ban, and whether there really was one. $\endgroup$ – Ben Crowell Jun 24 '19 at 13:51

See this question at math.se

The "infinitesimals" we are talking about (actually more correctly called "indivisibles") is the assertion that a plane area is made up of parallel line segments; or that a solid region is made up of parallel plane sections.

The Amir book is not a novel, it is historical research. (And it is very interesting!) The short answer (see my answer there): Jesuits prohibited indivisibles for use in their education system because indivisibles were viewed as contrary to Aristotle.

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  • $\begingroup$ I read the answers there, you all are implying that Alexander made a glaring mistake. The ban was on indivisibles , not on infinitesimals.The latter are not indivisible and compatible with Aristotle's and the Church's idea of continuum. $\endgroup$ – user157860 Jun 19 '19 at 11:56
  • $\begingroup$ @user157860 ... good point. I will rearrange my answer. Your question here, and all the discussion, would be more correct if it used the therm "indivisibles". Any mathematical theory of infinitesimals occurred after 1632, so of course infinitesimals were not banned in 1632. Then why did Alexander use "Infinitesimals" as the title of his book? You will have to see the book to find out. $\endgroup$ – Gerald Edgar Jun 19 '19 at 12:27
  • $\begingroup$ I read an excerpt and there is no ambiguity, Alexander is really convinced the ban was on infinitesimals: You might email him and check, my English and expertise on the issue is limited. Yet, Cavalieri's principle was not known in 1632, he discussed it with Galileo but it is highly unlikely that he spilled the beans to the jesuits $\endgroup$ – user157860 Jun 19 '19 at 13:57
  • $\begingroup$ I rearranged my answer here. The other answer was already about "indivisibles" . $\endgroup$ – Gerald Edgar Jun 21 '19 at 10:47
  • $\begingroup$ Many of the math.SE answers talk about the link to atomism and perceived contradiction to transubstantiation. Is that wrong? $\endgroup$ – Ben Crowell Jun 24 '19 at 13:53

The issue regards more indivisibles than infinitesimals and must be located in the context of the Early Modern European debate about the "revamping" of atomism.

See : Vincent Jullien (editor), Seventeenth-Century Indivisibles Revisited (2015, Birkhauser) for details about the works of Kepler (1609), Cavalieri (1635) and Guldin (1640).

Cavalieri developed his theory of geometry during the years 1620–1622.

According to Vincent Jullien's chapter dedicated to Indivisibles in the Work of Galileo :

on 7 May 1610, Galileo wrote, in a letter to the secretary of the Grand Duke of Tuscany, that he was planning a piece of work on the De Compositione continui. In February and March 1626, Cavalieri reminded him of the project: “do you remember the work on indivisibles that you had decided to write?”

Indivisibles are implicitly mentioned in part of the second day of the Dialogo (1632), at the beginning of the demonstration of the law of falling bodies.

And see : Galileo's Saggiatore (1623) and the reply by the jesuit Orazio Grassi (Libra, published under the name : Lotario Grassi)

asserting that Galileo's book advanced an atomic theory of matter, and that this conflicted with the Catholic doctrine of the Eucharist, because atomism would make transubstantiation impossible.

Grassi's second response to Il Saggiatore, the Ratio ponderum librae et simbellae (1626), focused mainly on doctrinal issues.

unlike The Assayer, which had recourse to the lethal polemical weapons of satire and the new philosophy, the Ratio used those no-less-lethal weapons of doctrinal and dialectical retort based on religious and philosophical orthodoxy.[Pietro Redondi, Galileo Heretic, Princeton University Press, 1987. p.191]

The jesuit mathematician Paul Guldin was an harsh critic of Cavalieri's method of indivisibles into his De centro gravitatis (or Centrobaryca, three volumes, 1635-41), on mathematical grounds.

See also Mordechai Feingold (editor), Jesuit Science and the Republic of Letters (MIT Press, 2002), page 28-29, for details about jesuit Rodrogo de Arriaga's Cursus philosophicus (Anversa, 1632) condemnation of 1632, concerning "mathematical atomism" and "the opinion on quantity made up of indivisibles".

I think that the modern source is Egidio Festa, La querelle de l'atomisme: Galilee, Cavalieri et les jesuites (1990).

The easily available sources (Wiki,etc.) about the condemnation of indivisibles of Galileo and Cavalieri (dated August 10, 1632), led the Revisors General of the Jesuits Jacob Bidermann all ref to Amir Alexander's book.

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  • $\begingroup$ As I stated in my question the method of indivisibles was not circulated before 1640- $\endgroup$ – user157860 Jun 17 '19 at 14:07
  • $\begingroup$ As to Galileo, his "Saggiatore" got the imprimatur of the Church, moreover he never shared Cavalieri's ideas. $\endgroup$ – user157860 Jun 17 '19 at 17:41

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