“bring down the pride of the Jesuits”: This is from a letter by the Lincean Dr. Johannes Faber to Galileo from February 15, 1620. Quoted in Redondi, Galileo Heretic, p. 43.
the new Pope was amused and full of admiration: Cesarini to Galileo, October 28, 1623. Quoted in Redondi, Galileo Heretic, p. 49.
The plot misfired badly: On the Santarelli affair, see Bangert, A History of the Society of Jesus, pp. 200–201; and Redondi, Galileo Heretic, pp. 104–105.
Galileo even believed that he had been given implicit permission: See Redondi, Galileo Heretic, p. 50.
The young aristocrat was a rising star in Roman intellectual circles: On Pallavicino’s dissertation defense, see ibid., pp. 200–202. Father Grassi had been Galileo’s opponent in a dispute over the nature of comets and the chief target of The Assayer. His condemnation of the orthodoxy of Galileo’s atomism was contained in his 1626 book Ratio ponderum librae et simbellae, published under the pseudonym Lothario Sarsi.
Urban VIII had run out of options: On the political crisis in Rome in 1631, and on Cardinal Borgia’s attack on Urban VIII, see Redondi, Galileo Heretic, pp. 229–31.
Galileo’s friends were running for cover: Not all made it safely. In April of 1632, Giovanni Ciampoli, the most prominent Lincean in the Curia and personal secretary to the Pope himself, was given the impressive-sounding title of “Governor of Montalti di Castro” and exiled from Rome to the Apennines. He would never return.
Father Rodrigo de Arriaga of Prague: On Rodrigo Arriaga, his Cursus philosophicus, and his views on the infinitely small, see Rossi, “I punti di Zenone,” pp. 398–99; Hellyer, “‘Because the Authority of My Superiors Commands,’” p. 339; Feingold, “Jesuits: Savants,” p. 28; Redondi, Galileo Heretic, pp. 241–42; and John L. Heilbron, Electricity in the 17th and 18th Centuries (Berkeley: University of California Press, 1979), p. 107.
Quite possibly he was influenced by his friend Gregory St. Vincent: On Arriaga’s friendship with St. Vincent, see Van Looey, “A Chronology and Historical Analysis of the Mathematical Manuscripts of Gregorius a Sancto Vincentio,” p. 59.
“The permanent continuum can be constituted”: The Revisors’ decree is preserved as manuscript FG 657, p. 183, in ARSI (Archivum Romanum Societatis Iesu), the archive of the Society of Jesus in Rome. It is also reproduced in Egidio Festa, “La querelle de l’atomisme,” La Recherche 224 (September 1990): 1040; and quoted in French in Egidio Festa, “Quelques aspects de la controverse sur les indivisibles,” in M. Bucciantini and M. Torrini, eds., Geometria e atomismo nella scuola Galileana (Florence: Leo S. Olschki, 1992), p. 198. Special thanks to Professor Carla Rita Palmerino of Radboud University Nijmegen, in the Netherlands, for making available to me her notes from the Jesuit archives.
he found himself writing to Father Ignace Cappon: General Mutio Vitelleschi to Ignace Cappon, 1633, quoted in Michael John Gorman, “A Matter of Faith? Christoph Scheiner, Jesuit Censorship, and the Trial of Galileo,” Perspectives on Science 4, no. 3 (1996): pp. 297–98. Also quoted in Feingold, “Jesuits: Savants,” p. 29.
On February 3, 1640: ARSI manuscript FG 657, p. 481.
in January 1641: Ibid., p. 381. Cited and discussed in Festa, “Quelques aspects,” pp. 201–202.
On May 12, 1643: ARSI manuscript FG 657, p. 395.
In 1649: Ibid., p. 475.
Arriaga’s views on the continuum were unequivocally condemned: On Arriaga and the publishing history of his Cursus philosophicus, see Hellyer, “‘Because the Authority of My Superiors Commands,’” pp. 339–41.
Pallavicino was no ordinary novice: On Pietro Sforza Pallavicino and his career, see Redondi, Galileo Heretic, pp. 264–65; Hellyer, “‘Because the Authority of My Superiors Commands,’” p. 339; Festa, “La querelle de l’atomisme,” pp. 1045–46; Festa, “Quelques aspects,” pp. 202–203; and Feingold, “Jesuits: Savants,” p. 29.
the marchese still considered himself a progressive thinker: Redondi, Galileo Heretic, p. 265.
Pallavicino frequently came under the Revisors’ scrutiny: On Pallavicino’s conflicts with the Revisors and General Carafa, see Claudio Costantini, Baliani e i Gesuiti (Florence: Giunti, 1969), esp. pp. 98–101.
Pallavicino forged ahead, lecturing on his unorthodox views: Pallavicino hints at his troubles in Pietro Sforza Pallavicino, Vindicationes Societatis Iesu (Rome: Dominic Manephi, 1649), p. 225. Quoted and discussed in Festa, “Quelques aspects,” pp. 202–203.
“there are some in the Society who follow Zeno”: Superior General Vincenzo Carafa to Nithard Biberus, March 3, 1649. In G. M. Pachtler, SJ, ed., Ratio studiorum et institutiones scholasticae Societatis Jesu (Osnabrück: Biblio-Verlag, 1968), 3:76, doc. no. 41.
Ordinatio pro studiis superioribus: For the text of the Ordinatio, see G. M. Pachtler, SJ, ed., Ratio studiorum, vol. 3 (Berlin: Hofman and Comp., 1890), pp. 77–98. The sixty-five banned “philosophical” propositions are on pages 90–94, and an additional list of twenty-five banned “theological” propositions is on pages 94–96. For a discussion of the Ordinatio, its origins, and its effects, see Hellyer, “‘Because the Authority of My Superiors Commands,’” pp. 328–29. It is also mentioned in Feingold, “Jesuits: Savants,” p. 29; and Carla Rita Palmerino, “Two Jesuit Responses to Galileo’s Science of Motion: Honoré Fabri and Pierre le Cazre,” in M. Feingold, ed., The New Science and Jesuit Science: Seventeenth-Century Perspectives (Dordrecht: Kluwer Academic Publishers, 2003), p. 187.
“The succession continuum”: The propositions are listed in Pachtler, ed., Ratio studiorum, p. 92.
5. The Battle of the Mathematicians
“the three Jesuits, Guldin, Bettini, and Tacquet”: Stefano degli Angeli, De infinitis parabolis (Venice: Ioannem La Nou, 1659), under “Lectori Benevolo.”
It was also crucial to prove them mathematically wrong: On Guldin, Bettini, and Tacquet as the Society’s agents sent to combat the method of indivisibles, see Redondi, Galileo Heretic, p. 291.
Guldin was Clavius’s follower: For an excellent short biography of Guldin (and many other mathematicians), see the online MacTutor History of Mathematics archive, hosted by the University of St. Andrews in Scotland, at http://www-history.mcs.st-and.ac.uk/. See also Guldin’s biography authored by J. J. O’Connor and E. F. Robertson at http://www.gap-system.org/~history/Biographies/Guldin.html.
He first suggests that Cavalieri’s method is not in fact his own: On Guldin’s charge that Cavalieri derived his method from Kepler and Sover, see Giusti, Bonaventura Cavalieri, pp. 60–62; and Mancosu, Philosophy of Mathematics and Mathematical Practice, pp. 51–52.
“no geometer will grant him”: Paul Guldin, De centro gravitatis, book 4 (Vienna: Matthaeus Cosmerovius, 1641), p. 340.
This then leads Guldin to his final point: On Guldin’s mathematical criticisms of Cavalieri and his method, see Giusti, Bonaventura Cavalieri, pp. 62–64; and Mancosu, Philosophy of Mathematics, pp. 50–55.
“reasons that must be suppressed”: Guldin, De centro gravitatis, book 2 (Vienna: Matthaeus Cosmerovius, 1639), p. 3. Quoted in Bonaventura Cavalieri, Exercitationes geometricae sex (Bologna: Iacob Monti, 1647), p. 180, and quoted and discussed in Festa, “Quelques aspects,” p. 199.
Initially he intended to respond in the form of a dialogue: On Cavalieri’s plans for a dialogue and Rocca’s advice, see Giusti, Bonaventura Cavalieri, pp. 57–58.
None of this, he argues, has any bearing on the method of indivisibles: For Cavalieri’s claim to be agnostic on the subject of the composition of the continuum, see Mancosu, Philosophy of Mathematics, p. 54.
“relative infinity”: See Mancosu, Philosophy of Mathematics, p. 54; and Giusti, Bonaventura Cavalieri, p. 64.
“it is not necessary to describe actually”: Cavalieri, Exercitationes geometricae sex, part 3, “In Guldinum,” quoted in Giusti, Bonaventura Cavalieri, pp. 62–63.
“the hand, the eye, or the intellect?”: Guldin, De centro gravitatis, book 4, p. 344, quoted in Giusti, Bonaventura Cavalieri, p. 63.
Mario
Bettini, who inherited the mantle from Guldin: On Bettini, his place among the Jesuits and his relationship with Christoph Grienberger, see Michael John Gorman, “Mathematics and Modesty in the Society of Jesus: The Problems of Christoph Grienberger,” in Feingold ed., The New Science and Jesuit Science, pp. 4–7.
the author of two very long and eclectic books: Mario Bettini, Apiaria universae philosophiae mathematicae (Bologna: Io. Baptistae Ferronij, 1645); Mario Bettini, Aerarium philosophiae mathematicae (Bologna: Io. Baptistae Ferronij, 1648).
“were the Jesuit Fathers not here”: Cavalieri to Galileo, August 7, 1626, quoted in Giusti, Bonaventura Cavalieri, p. 9.
The move was ultimately blocked by the city’s senate: See Giusti, Bonaventura Cavalieri, pp. 9–10n26.
“infinity to infinity has no proportion”: Guldin, De centro gravitatis, book 4, p. 341, quoted in Mancosu, Philosophy of Mathematics, p. 54.
“‘what separates the false coin from the true’”: Bettini, Aerarium, vol. 3, book 5, p. 20. The quote is from Horace, Epistles, book 1.7, line 23: “Quid distent aera lupinis.”
“I respond to the counterfeit philosophizing”: Quoted in Stefano degli Angeli, “Appendix pro indivisibilibus,” in Problemata geometrica sexaginta (Venice: Ioannem la Nou, 1658), p. 295.
Cylindricorum et annularium: André Tacquet, Cylindricorum at annularium libri IV (Antwerp: Iacobus Meurisius, 1651).
the general’s response was surprisingly cool: On Tacquet and General Nickel, see Bosmans, “André Tacquet,” p. 72.
“a noble geometer”: Tacquet, Cylindricorum, pp. 23–24, quoted and discussed in Festa, “Quelques aspects,” pp. 204–205.
“nothing can be proven by anyone”: Tacquet, Cylindricorum, p. 23.
“I will always doubt its truth”: Ibid., p. 24, quoted and discussed in Bosmans, “André Tacquet,” p. 72.
Cavalieri’s student at Bologna: On Aviso and Mengoli, see Giusti, Bonaventura Cavalieri, pp. 49–50, as well as the Cavalieri and Mengoli entries in Charles Gillispie, ed., Dictionary of Scientific Biography (New York: Scribner, 1981–90).
Vincenzo Viviani (1622–1703): On Viviani, see Giusti, Bonaventura Cavalieri, p. 51, at J. J. O’Connor and E. F. Robertson, “Vincenzo Viviani,” MacTutor online biography at http://www-groups.dcs.st-and.ac.uk/history/Biographies/Viviani.html.
Antonio Nardi: On Nardi, see Giusti, Bonaventura Cavalieri, p. 51, as well as Belloni, “Torricelli et son époque,” pp. 29–38.
His first broadside: Angeli, “Appendix pro indivisibilibus.”
“can be called The Bee”: Angeli’s discussion of Bettini as a busy but unlucky bee can be found in his Problemata, pp. 293–95.
Everyone, he responds, except the Jesuits: Angeli’s polemic against Tacquet and his fellow Jesuit mathematicians is included in his preface to the reader in Stefano degli Angeli, “Lectori Benevolo,” in De infinitis parabolis.
“no advantage or utility to the Christian people”: On the papal brief of 1668 suppressing the three orders, see Sydney F. Smith, SJ, Joseph A. Munitiz, SJ, eds., The Suppression of the Society of Jesus (Eastbourne, UK: Antony Rowe Ltd., 2004), pp. 291–92. First published as a series of articles by Sydney Smith in The Month between February 1902 and August 1903.
the “Aquavitae Brothers”: See William Eamon, “The Aquavitae Brothers,” in http://williameamon.com/?p=552; and T. Kennedy, “Blessed John Colombini,” in The Catholic Encyclopedia (New York: Robert Appleton Company, 1910).
he had previously published no fewer than nine books: Angeli’s books were Problemata geometrica sexaginta (1658); De infinitis parabolis (1659); Miscellaneum hyperbolicum et parabolicum (1659); Miscellaneum geometricum (1660); De infinitorum spiralium spatiorum mensura (1660); De infinitorum cochlearum mensuris (1661); De superficie ungulae (1661); Accessionis ad stereometriam et mecanicam (1662); and De infinitis spiralibus inversis (1667). See Giusti, Bonaventura Cavalieri, p. 50n39.
Galileo was a brilliant public advocate for the freedom to philosophize: The quote is from Galileo Galilei, “Third Letter on Sunspots,” in Drake, ed., Discoveries and Opinions of Galileo, p. 134. A translation of the “Letter to the Grand Duchess Christina” can be found in the same volume, pp. 173–216.
Italy had been home to perhaps the liveliest mathematical community in Europe: On the early modern mathematical tradition in Italy, see Mario Biagioli, “The Social Status of Italian Mathematicians, 1450–1600,” History of Science 27, no. 1 (1989): 41–95.
6. The Coming of Leviathan
yet they went on digging: The story of the Diggers is told in Christopher Hill, The World Turned Upside Down (Harmondsworth: Penguin Books, 1975), chap. 7, “Levellers and True Levellers.” Quotes are from p. 110.
the Diggers soon followed up with a pamphlet: The pamphlet was called The True Levellers Standard Advanced, printed in 1649.
many other groups, and unnumbered individuals, emerged to take their place: For a detailed account of the radical sects of the English Revolution, see Hill, The World Turned Upside Down.
“a giddy hot-headed, bloody multitude”: The comment is by the Reverend Henry Newcombe, quoted in Christopher Hill, The Century of Revolution, 1603–1714 (New York: W. W. Norton and Company, 1982), p. 121.
“the gentry and citizens throughout England”: Pepys is quoted in Hill, Century of Revolution, p. 121.
“both Me, and Fear”: Quoted in Samuel I. Mintz, The Hunting of Leviathan: Seventeenth-Century Reactions to the Materialism and Moral Philosophy of Thomas Hobbes (Cambridge, UK: Cambridge University Press, 1970), p. 1.
“a little learning went a great way with him”: John Aubrey’s biography of Hobbes can be found as “Thomas Hobbes,” in Andrew Clark, ed., “Brief Lives,” Chiefly of Contemporaries, Set down by John Aubrey, between the Years 1669 and 1696 (Oxford: Clarendon Press, 1898), pp. 321–403. The quote is from p. 391.
a laudable record of sobriety: Aubrey’s report on Hobbes’s drinking can be found in his biography “Thomas Hobbes,” p. 350.
“prove things after my owne taste”: This is quoted in Mintz, Hunting of Leviathan, p. 2.
“the one who has opened to us the gate”: Ibid., pp. 8–9.
Thomas Harriot: On Harriot, see Alexander, Geometrical Landscapes.
tutor to the Prince of Wales: On Hobbes’s appointment as royal tutor and the opposition to it, see Mintz, Hunting of Leviathan, p. 12.
Leviathan: Thomas Hobbes, Leviathan, or the Matter, Forme, and Powers of a Commonwealth Ecclesiastical and Civil (London: Andrew Crooke, 1651).
“he cannot assure the power and means to live well”: See Hobbes, Leviathan, 11:2.
“solitary, poor, nasty, brutish, and short”: This famous quote appears ibid., 13:9.
“war of every man against every man”: Ibid., 13:13.
“the savage people in many places of America”: For Hobbes’s view of native Americans as living in the state of nature, see ibid., 13:11.
“more than consent, or concord”: Ibid., 17:13.
“This is the generation of that great LEVIATHAN”: Ibid.
“one person, of whose acts a great multitude”: Ibid.
“is but an artificial man”: Introduction ibid., p. 1.
blaming them directly for the onset of the civil war: Reflecting on the role of clergymen years later, Hobbes wrote that “the cause of my writing that book [i.e., Leviathan] was the consideration of what the ministers before, and in the beginning of, the civil war, by their preaching and writing did contribute thereunto.” See Thomas Hobbes, Six Lessons to the Professors of Mathematics, in Sir William Molesworth, ed., The English Works of Thomas Hobbes (London: Longman, Brown, Green, and Longmans, 1845), p. 335.
“a Civill Warr with the Pen”: Quoted in Steven Shapin and Simon Schaffer, Leviathan and the Air Pump (Princeton, NJ: Princeton University Press, 1985), p. 290.
Deciding which opinions and doctrines should be taught: Hobbes, Leviathan, 18:9.
7. Thomas Hobbes, Geometer
“made him in love with geomet
ry”: The account is from John Aubrey’s biography, “Thomas Hobbes,” p. 332.
Samuel Sorbière hailed him: Sorbière is quoted in Douglas M. Jesseph, Squaring the Circle: The War between Hobbes and Wallis (Chicago: University of Chicago Press, 1999), p. 6.
“the only science”: Hobbes, Leviathan, 4:12.
“there can be no certainty of the last conclusion”: Ibid., 5:4.
“Empusa”: All quotations in this passage are from Thomas Hobbes, “Elements of Philosophy, the First Section, Concerning Body,” in Molesworth, ed., The English Works of Thomas Hobbes, pp. vii–xii, “The Author’s Epistle Dedicatory.”
“no older than my own book”: Since Hobbes published De corpore in 1655, and De cive came out in 1642, true civil philosophy is no more than thirteen years old.
“revenge myself of envy by encreasing it”: Hobbes, “Elements of Philosophy,” pp. vii–xii.
“For who is so stupid”: Hobbes, Leviathan, 5:16.
“Physics, ethics, and politics”: Thomas Hobbes, dedicatory epistle to De principiis et rationcinatione geometrarum (London: Andrew Crooke, 1666), quoted in Jesseph, Squaring the Circle, p. 282.
“fright and drive away this metaphysical Empusa”: Hobbes, “Elements of Philosophy,” pp. vii–xii.
“because we make the commonwealth ourselves”: Thomas Hobbes, Six Lessons to the Professors of Mathematiques, One of Geometry, the Other of Astronomy, in the Chairs Set Up by the Noble and Learned Sir Henry Savile, in the University of Oxford (London: Andrew Crooke, 1656), reprinted in Molesworth, ed., The English Works of Thomas Hobbes, 7:181–356. The quote is from p. 184.
were as indisputably correct: On the geometrical power of the Leviathan’s decrees, see also Shapin and Schaffer, Leviathan and the Air Pump, p. 253.
“the skill of making and maintaining commonwealths”: Hobbes, Leviathan, 20:19.
it should have no unsolved, not to mention insoluble, problems: On Hobbes’s insistence that geometry should have no unsolved problems, see Hobbes, De homine, 2.10.5, quoted in Jesseph, Squaring the Circle, p. 221.
Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Page 37