Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

by Alexander, Amir

“They demonstrate everything in which they see a dispute”: See Clavius, “In disciplinas mathematicas prolegomena,” in Christopher Clavius, ed., Euclidis elementorum libri XV (Rome: Bartholemaeum, 1589), p. 5. Clavius’s Euclid, including the “Prolegomena,” was first published in 1574.

  “The theorems of Euclid”: See Clavius, “Prolegomena,” p. 5, translation from Lattis, Between Copernicus and Galileo, p. 35.

  “the first place among the other sciences should be conceded to mathematics”: Ibid.

  the angles in the same segment are all equal to one another: These theorems appear in Euclid’s Elements as propositions I.5, I.47, and III.21.

  Proposition 32: See the instance in Dana Densmore, ed., Euclid’s Elements, the Thomas L. Heath Translation (Santa Fe, NM: Green Lion Press, 2002), pp. 24–25.

  “Geometrical architect for all”: The quote is from Antonio Possevino,

  Biblioteca selecta (1591), quoted in Crombie, “Mathematics and Platonism,” pp. 71–72.

  “mathematical disciplines are not proper sciences”: Pereira continued: “to have science is to acquire knowledge of a thing through the cause on account of which the thing is; and science (scientia) is the effect of demonstration: but demonstration … must be established from those things that are ‘per se,’ and proper to that which is demonstrated, but the mathematician neither considers the essence of quantity, nor treats its affections as they flow from such essence, nor declares them by the proper causes on account of which they are in quantity, nor makes his demonstrations from proper and ‘per se’ but from common and accidental predicates.” Benito Pereira, De communibus omnium rerum naturalium principiis (Rome: Franciscus Zanettus, 1576), I.12, p. 24. Quoted in Crombie, “Mathematics and Platonism,” p. 67.

  If one seeks strong demonstrations, one must turn elsewhere: For more on Pereira and the “Quaestio de certitudine mathematicarum,” see Paolo Mancosu, “Aristotelian Logic and Euclidean Mathematics: Seventeenth-Century Developments in the Quaestio de Certitudine Mathematicarum,” Studies in History and Philosophy of Science 23, no. 2 (1992): 241–65; Paolo Mancosu, Philosophy of Mathematics and Mathematical ractice in the Seventeenth Century (New York and Oxford: Oxford University Press, 1996); Gatto, “Christoph Clavius’ ‘Ordo Servandus,’” esp. pp. 239–42; and Lattis, Between Copernicus and Galileo, pp. 34–36.

  The subject of mathematics is matter itself: The complete quote is “because the mathematical disciplines discuss things that are considered apart from any sensible matter—although they are themselves immersed in matter—it is evident that they hold a place intermediate between metaphysics and natural science. For the subject of metaphysics is separated from all matter, both in the thing and in reason; the subject of physics is in truth conjoined to sensible matter, both in the thing and in reason; whence, since the subject of the mathematical disciplines is considered free from all matter—although it [i.e., matter] is found in the thing itself—clearly it is established intermediate between the other two.” Clavius, “Prolegomena,” p. 5. Translation from Peter Dear, Discipline and Experience (Chicago: University of Chicago Press, 1995), p. 37.

  “It will contribute much”: The passage is from Clavius, “Modus quo disciplinas mathematicas in scholis Societatis possent promoveri,” quoted in Crombie, “Mathematics and Platonism,” p. 66.

  Clavius also made positive suggestions: The discussion and the quotes can be found in Clavius, “Modus quo disciplinas mathematicas,” p. 65.

  “Ordo servandus”: On Clavius’s “Ordo servandus,” see Gatto, “Christoph Clavius’ ‘Ordo Servandus,’” esp. pp. 243–46.

  Ratio studiorum: On mathematics in the “Ratio studiorum,” see Cosentino, “Math in the Ratio Studiorum,” pp. 65–66.

  writing new textbooks: For Clavius’s textbooks and publications, see Gatto, “Christoph Clavius’ ‘Ordo Servandus,’” esp. pp. 243–44, and Ugo Baldini, “The Academy of Mathematics of the Collegio Romano from 1553 to 1612,” in Feingold, ed., Jesuit Science and the Republic of Letters, esp. appendix C, pp. 74–75.

  a mathematics academy at the Collegio Romano: On Clavius’s academy at the Collegio Romano, see Baldini, “The Academy of Mathematics.” On Acquaviva’s decree, see Gatto, “Christoph Clavius’ ‘Ordo Servandus,’” p. 248.

  the Jesuits never deviated from their commitment to Euclidean geometry: On the Jesuits’ commitment to mathematics as both a key to understanding physical reality and as a model science to be emulated, see Rivka Feldhay, Galileo and the Church: Political Inquisition or Critical Dialogue? (Cambridge: Cambridge University Press, 1995), p. 222.

  the Dominicans could boast of no comparable accomplishment: On the Jesuit-Dominican struggle for intellectual and theological supremacy, see Feldhay, Galileo and the Church.

  “some would rather be blamed by Clavius”: Riccioli’s comment is quoted in Eberhard Knobloch, “Sur la vie et l’oeuvre de Christophore Clavius (1538–1612),” Revue d’histoire des sciences 41, no. 3–4 (1988): 335.

  His many admirers outside the Society: On Tycho Brahe’s and the Archbishop of Cologne’s opinions of Clavius, see ibid., pp. 335–36. On Commandino and Guidobaldo, see Mario Biagioli, “The Social Status of Italian Mathematicians, 1450–1600,” History of Science 25 (1989): 63–64.

  “a German beast with a big belly”: Clavius’s detractors are quoted in Knobloch, “Sur la vie et l’oeuvre de Christophore Clavius,” pp. 333–35.

  some new results in the theory of combinations: On Clavius’s innovations in this field, see ibid., pp. 343–51.

  a strict defender of the old orthodoxy: On Clavius as a defender of orthodoxy, see Lattis, Between Copernicus and Galileo, as well as Knobloch, “Sur la vie et l’oeuvre de Christophore Clavius.”

  he knew of Viète’s groundbreaking work: On the absence of Viète’s analysis from Clavius’s Algebra, see Baldini, “The Academy of Mathematics,” p. 63.

  “The theorems of Euclid and the rest of the mathematicians”: Clavius, “Prolegomena,” p. 5, translation from Lattis, Between Copernicus and Galileo, p. 35.

  even suspect the Jesuits of innovating: Acquaviva’s warning is quoted in Feingold, “Jesuits: Savants,” p. 18.

  3. Mathematical Disorder

  the Jesuits celebrated Galileo: The story of Galileo’s astronomical discoveries and his triumphant visit to Rome and the Collegio Romano is summarized in Stillman Drake, ed. and trans., Discoveries and Opinions of Galileo (New York: Anchor Books, 1957).

  Discourse on Floating Bodies: Discourse is discussed ibid., pp. 79–81.

  Letters on Sunspots: The Letters are discussed and largely translated ibid., pp. 59–144.

  “Letter to the Grand Duchess Christina”: Discussed and translated ibid., pp. 145–216.

  “I call the lines so drawn ‘all the lines’”: Cavalieri to Galileo, December 15, 1621. The letter can be found in Bonaventura Cavalieri, Lettere a Galileo Galilei, ed. Paolo Guidera (Verbania, Italy: Caribou, 2009), pp. 9–10.

  As early as 1604: Galileo’s early work on infinitesimals is cited in Festa, “La querrelle de l’atomisme,” p. 1042; and Festa, “Quelques aspects de la controverse sur les indivisibles,” p. 196.

  Galileo was still occupied with the paradoxes of the continuum: See Festa, “La querrelle de l’atomisme,” p. 1043.

  Dialogue on the Two Chief World Systems: The standard English edition is Galileo, Dialogue on the Two Chief World Systems, trans. Stillman Drake (Berkeley: University of California Press, 1962), first published in Florence in 1632.

  the mathematical continuum was modeled on physical reality: In addition to the discussion in the Discourses, Galileo presented a similar view of the continuum in his commentary on Antonio Rocco’s Filosofiche esercitazioni in 1633. See François de Gandt, “Naissance et métamorphose d’une théorie mathématique: La géométrie des indivisibles en Italie,” Sciences et techniques en perspective, vol. 9, 1984–85 (Nantes: Université de Nantes, 1985), p. 197.

  he had experimented extensively with indivisibles: On Valer
io’s work on indivisibles, see Carl B. Boyer, The History of the Calculus and Its Conceptual Development (New York: Dover Publications, 1949), pp. 104–106; and the entry “Luca Valerio” in The MacTutor History of Mathematics archive mathematical biographies, http://www-history.mcs.st-andrews.ac.uk/Biographies/Valerio.html.

  It follows, Salviati concludes: Salviati’s discussion of the law of falling bodies can be found in Galileo, Dialogues Concerning Two New Sciences, ed. Henry Crew and Alfonso de Salvio (Buffalo, NY: Prometheus Books, 1991), pp. 173–74. In the standard Italian Edizione Nazionale, it is on pp. 208–209.

  “I am proud, and will always be”: Quoted in Enrico Giusti, Bonaventura Cavalieri and the Theory of Indivisibles (Bologna: Edizioni Cremonese, 1980), p. 3n9.

  “his singular inclinations and ability”: Quoted ibid., p. 3n10.

  “I am now in my own country”: Cavalieri to Galileo, July 18, 1621, quoted ibid., p. 6n18.

  “to the great wonder of everyone”: The quote is from Cavalieri’s biographer Girolamo Ghilini, quoted ibid., p. 7n19.

  he approached the Jesuit fathers: Cavalieri to Galileo, August 7, 1626, quoted ibid., p. 9n26.

  “few scholars since Archimedes”: Galileo to Cesare Marsili, March 10, 1629, quoted ibid., p. 11n30.

  “like cloths woven of parallel threads”: Cavalieri’s comparison of indivisibles to threads in a cloth and pages in a book, and his discussion of the metaphors, are found in his Exercitationes geometricae sex (Bologna: Iacob Monti, 1647), pp. 3–4.

  proposition 19: Cavalieri, Geometria indivisibilibus libri VI, proposition 19, pp. 437–39. For discussions of this proof, see de Gandt, “Naissance et métamorphose d’une théorie mathématique,” pp. 216–17; and Margaret E. Baron, The Origins of the Infinitesimal Calculus (New York: Dover Publications, 1969), pp. 131–32.

  Archimedes had used his own ingenious approach: On Archimedes’s calculation of the area enclosed in a spiral, see Baron, The Origins, pp. 43–44.

  Cavalieri’s calculation of the area enclosed inside a spiral: The calculation is presented as proposition 19 in Cavalieri, Geometria indivisibilibus libri VI, p. 238.

  “It can be said that with the escort of good geometry”: Cavalieri to Galileo, June 21, 1639, in Galileo, Opere, vol. 18, p. 67, letter no. 3889. Quoted and translated in Amir Alexander, Geometrical Landscapes (Stanford, CA: Stanford University Press, 2002), p. 184.

  “all the lines”: On the importance of the concepts of “all the lines” and “all the planes” in Cavalieri’s work, see de Gandt, “Naissance et métamorphose d’une théorie mathématique,” and François de Gandt, “Cavalieri’s Indivisibles and Euclid’s Canons,” in Peter Barker and Roger Ariew, eds., Revolution and Continuity: Essays in the History and Philosophy of Early Modern Science (Washington, DC: Catholic University of America, 1991), pp. 157–82; and Giusti, Bonaventura Cavalieri.

  Cavalieri’s younger contemporary: The biographical information on Torricelli is from Egidio Festa, “Repères biographique et bibliographique,” in François de Gandt, ed., L’oeuvre de Torricelli: Science Galiléenne et nouvelle géométrie (Nice: CNRS and Université de Nice, 1987), p. 8.

  “I was the first in Rome”: Torricelli to Galileo, September 11, 1632, quoted ibid.

  “how the road you have opened”: Castelli to Galileo, March 2, 1641, quoted ibid., p. 9.

  a long and fruitful correspondence with French scientists: On Torricelli’s correspondence with his French colleagues, see Armand Beaulieu, “Torricelli et Mersenne,” in de Gandt, ed., L’oeuvre de Torricelli, pp. 39–51. On his connections with Italian Galileans, see Lanfranco Belloni, “Torricelli et son époque,” in de Gandt, ed., L’oeuvre de Torricelli, pp. 29–38; on the barometer, see Festa, “Repères,” pp. 15–18, and P. Souffrin, “Lettres sur la vie,” in de Gandt, ed., L’oeuvre de Torricelli, pp. 225–30.

  Opera geometrica: Torricelli’s Opera geometrica can be found in volume 1 of Gino Loria and Giuseppe Vassura, eds., Opere di Evangelista Torricelli (Faenza: G. Montanari, 1919–44). An Italian translation is in Lanfranco Belloni, ed., Opere scelte di Evangelista Torricelli (Turin: Unione Tipografico-Editrice Torinese, 1975), pp. 53–483.

  “De dimensione parabolae”: For discussions of “De dimensione parabolae,” see François de Gandt, “Les indivisibles de Torricelli,” in de Gandt, ed., L’oeuvre de Torricelli, pp. 152–53; and in de Gandt, “Naissance et métamorphose,” pp. 218–19.

  that the ancients possessed a secret method: Torricelli discusses this idea in the Opera geometrica, esp. in Loria and Vassura, eds., Opere, vol. 1, pp. 139–40. The Italian translation is in Belloni, ed., Opere scelte, p. 381.

  “the Royal Road through the mathematical thicket”: See Loria and Vassura, eds., Opere, vol. 1, p. 173. Quoted in de Gandt, “Les indivisibles de Torricelli,” p. 153.

  “We turn away from the immense ocean of Cavalieri’s Geometria”: Loria and Vassura, eds., Opere, vol. 1, p. 141.

  Torricelli’s directness made his method far more intuitive: See de Gandt, “Naissance et métamorphose,” p. 219.

  three separate lists of paradoxes: On Torricelli’s lists of paradoxes, see de Gandt, “Les indivisibles de Torricelli,” pp. 163–64.

  the simplest one captures the essential problem: Torricelli’s basic paradox is presented in a treatise entitled “De indivisibilium doctrina perperam usurpata,” in Loria and Vassura, eds., Opere, vol. 1, part 2, p. 417.

  “that indivisibles are all equal to each other”: Torricelli’s discussion of unequal indivisibles can be found in Loria and Vassura, eds., Opere, vol. 1, part 2, p. 320. It is quoted in de Gandt, “Les indivisibles de Torricelli,” p. 182.

  “semi-gnomons”: The diagrams here are derived from de Gandt, “Les indivisibles de Torricelli,” p. 187.

  to calculate the slope of the tangent: For a discussion of Torricelli’s method of tangents, see ibid., pp. 187–88, and idem, “Naissance et métamorphose,” pp. 226–29. De Gandt’s exposition is based on Torricelli’s Opere, ed. Loria and Vassura, vol. 1, part 2, pp. 322–33.

  4. “Destroy or Be Destroyed”: The War on the Infinitely Small

  “one of the best books ever written on mathematics”: Quoted in H. Bosmans, “André Tacquet (S. J.) et son traité d’arithmétique théorique et pratique,” Isis 9 (1927): 66–82.

  “either legitimate or geometrical”: André Tacquet, Cylindricorum et annularium libri IV (Antwerp: Iacobum Mersium, 1651), pp. 23–24.

  geometry formed the core of Jesuit mathematical practice: On the persistence of the Euclidean tradition among Jesuit mathematicians through the eighteenth century, see Bosmans, “André Tacquet,” p. 77. Not coincidentally, some of the most popular textbooks on Euclidean geometry in that era were composed by Jesuits, including Honoré Fabri, Synopsis geometrica (Lyon: Antoine Molin, 1669); and Ignace-Gaston Pardies, Elémens de géométrie (Paris: Sébastien Maire-Cramoisy, 1671). Both textbooks were published repeatedly in the seventeenth and eighteenth centuries.

  the struggle between geometry and indivisibles: Tacquet, Cylindricorum et annularium, pp. 23–24.

  the continuum is infinitely divisible: Benito Pereira, De communibus omnium rerum naturalium principiis (Rome: Franciscus Zanettus, 1576). On Pereira’s discussion of the composition of the continuum, see Paolo Rossi, “I punti di Zenone,” Nuncius 13, no. 2 (1998): 392–94.

  “great confusion and perturbation”: Quoted in Feingold, “Jesuits: Savants,” p. 30.

  The first decree by the Revisors General: The Revisors’ condemnations of 1606 and 1608 can be found in the Jesuit archive ARSI (Archivum Romanum Societatis Iesu), manuscript FG656 A I, pp. 318–19.

  Luca Valerio of the Sapienza University: The book was Luca Valerio, De centro gravitatis solidorum libri tres (Rome: B. Bonfadini, 1604). On Valerio’s use of infinitesimal methods, see Carl B. Boyer, History of the Calculus (New York: Dover Publications, 1947), pp. 104ff.

  experimenting with indivisibles: On Galileo’s 1604 experimentation with indivisibles see Festa, “Quelques
aspects de la controverse sur les indivisibles,” p. 196.

  “the Archimedes of our age”: Galileo, Dialogues Concerning Two New Sciences, p. 148.

  “the continuum is composed of indivisibles”: The first condemnation of 1615, dated April 4, is in ARSI manuscript FG 656A II, p. 456. The second, dated November 19, is in manuscript FG 656A II, p. 462.

  Valerio had misread the signs: On Valerio’s rise and fall, see David Freedberg, The Eye of the Lynx: Galileo, His Friends, and the Beginnings of Modern Natural History, (Chicago: University of Chicago Press, 2002), esp. pp. 132–34, as well as the online MacTutor biography of Valerio by J. J. O’Connor and E. F. Robertson at http://www.gap-system.org/~history/Biographies/Valerio.html. On his use of infinitesimals, see Boyer, History of the Calculus, pp. 104–106.

  By relying on their hierarchical order: On tensions between Jesuit intellectuals and the hierarchy’s efforts to control their scholarly work, see Feingold, “Jesuits: Savants”; and Marcus Hellyer, “‘Because the Authority of My Superiors Commands’: Censorship, Physics, and the German Jesuits,” Early Science and Medicine 1, no. 3 (1996): 319–54.

  he settled for a curt permission by the Jesuit provincial of Flanders: In contrast, when Tacquet published his Cylindricorum et annularium four years later, also in Flanders, his license stated that his work had been read and approved by three mathematicians of the Society. St. Vincent’s book carried no such endorsement.

  St. Vincent’s experience typifies the Jesuit attitude toward indivisibles: On Gregory St. Vincent’s troubles, see Feingold, “Jesuits: Savants,” pp. 20–21; Herman Van Looey, “A Chronology and Historical Analysis of the Mathematical Manuscripts of Gregorius a Sancto Vincentio (1584–1667),” Historia Mathematica 11 (1984): 58; Bosmans, “André Tacquet,” pp. 67–68; also Paul B. Bockstaele, “Four Letters from Gregorius a S. Vincentio to Christopher Grienberger,” Janus 56 (1969): 191–202.

  For the Jesuits, the choice could hardly have been worse: On the changing political and cultural climate in Rome surrounding the election of Urban VIII, see Pietro Redondi, Galileo Heretic (Princeton, NJ: Princeton University Press, 1987), esp. pp. 44–61 and 68–106.