Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (52 page)

1640–60:
The Interregnum. The civil war between King Charles I and Parliament leads to the execution of the king in 1649 and the establishment of a military dictatorship under Cromwell.

1640:
Hobbes, a Royalist, flees to Paris and joins Charles I’s court in exile, where he serves as mathematical tutor to the prince of Wales, the future Charles II.

1641:
The Jesuit mathematician Paul Guldin publishes
De centro gravitatis
, which contains an attack on Cavalieri and a systematic critique of his method.

1642:
Torricelli is appointed Galileo’s successor at the Medici court and professor of mathematics at the Florentine Academy.

1642:
Hobbes publishes his first philosophical work,
De cive
, in which he argues that only an absolute monarchy can save human society from chaos and civil war.

1644:
Torricelli publishes his most important work on infinitesimals, the
Opera geometrica
.

1644:
John Wallis is appointed secretary to the Westminster Assembly of Divines.

1645:
Wallis joins with other science enthusiasts to conduct and discuss scientific experiments. The group, known as the “Invisible College,” meets regularly for years.

1647:
Cavalieri responds to Guldin in his last work,
Exercitationes geometricae sex
. He dies shortly thereafter.

1647:
Death of Torricelli.

1648:
The Peace of Westphalia ends the Thirty Years’ War.

1648:
The Jesuit mathematician Mario Bettini denounces infinitesimals in his book
Aerarium philosophiae mathematicae
.

1648:
Pietro Sforza Pallavicino, Jesuit, nobleman, and future cardinal, is forced to publicly retract his advocacy of infinitesimals.

1649:
Charles I of England is executed.

1649:
Wallis appointed Savilian Professor of Mathematics at Oxford.

1649:
The Jesuit superior general, Vincentio Carafa, writes to the provinces to denounce infinitesimals.

1651:
The Jesuit mathematician André Tacquet, in
Cylindricorum et annularium libri IV
, declares that infinitesimals must be destroyed, or else mathematics will be destroyed.

1651:
Hobbes publishes
Leviathan
, in which he advocates a totalitarian state. He grounds his reasoning in geometry.

1651:
The Jesuits publish a list of permanently banned doctrines that includes infinitesimals.

1652:
Hobbes falls out with the court in Paris and returns to England.

1655:
Wallis publishes
De sectionibus conicis
.

1655:
Hobbes publishes
De corpore
, which includes “proofs” of ancient unresolved problems such as squaring the circle.

1655:
Wallis publishes
Elenchus geometriae Hobbianae
, in which he ridicules Hobbes and points out his mathematical mistakes.

1656:
Wallis publishes
Arithmetica infinitorum
.

1656:
Hobbes responds with
Six Lessons to the Professors of Mathematics
, wherein he retaliates by attacking Wallis’s use of infinitesimals, which he considers nonsensical and conducive to error, not truth.

1657–79:
Hobbes and Wallis criticize, ridicule, and hurl abuse at each other in dozens of books, pamphlets, and essays.

1658–68:
Stefano degli Angeli, professor of mathematics at the University of Padua, publishes eight works on infinitesimals, all of them openly ridiculing the Jesuit critics of infinitesimal mathematics.

1660:
Charles II is restored to the English throne.

1662:
The “Invisible College” receives a charter from Charles II and becomes the Royal Society of London.

1665:
The young Isaac Newton experiments with infinitesimals and develops a technique that will become known as the calculus.

1668:
The monastic order of the Jesuats, home to Cavalieri and Angeli, suppressed by papal decree.

1675:
Gottfried Wilhelm Leibniz develops his own version of the calculus.

1679:
Hobbes dies, mathematically discredited and politically isolated.

1684:
Leibniz publishes the first scholarly paper on the calculus, in the journal
Acta Eruditorum
.

1687:
Newton publishes the
Principia mathematica
, revolutionizing physics and establishing the first modern theory of the solar system. The work is based on the infinitesimal calculus and contains Newton’s first exposition of his method.

1703:
Wallis dies, lauded as a leading mathematician, a forerunner of the calculus, and a founder of the Royal Society.

NOTES

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Introduction

French courtier Samuel Sorbière
: On Samuel Sorbière’s visit to England, see his account in
A Voyage to England Containing Many Things Relating to the State of Learning, Religion, and Other Curiosities of That Kingdom
(London: J. Woodward, 1709), first published in Paris in 1664 as
Relation d’une voyage en Angleterre
. For the English reaction to Sorbière’s account, see Thomas Sprat,
Observations on M. de Sorbière’s Voyage into England
(London: John Martyn and James Allestry, 1665). A short biography of Sorbière is available in Alexander Chalmers,
General Biographical Dictionary
(London: J. Nichols and Son, 1812–17), 28:223. For a recent account of Sorbière’s career and especially his visit to England, see Lisa T. Sarasohn, “Who Was Then the Gentleman? Samuel Sorbière, Thomas Hobbes, and the Royal Society,”
History of Science
42 (2004): 211–32.

By his own testimony he was a “trumpeter”
: Sorbière’s description of himself as a “trumpeter” and not a “soldier” is found in his dedication to King Louis XIV in
A Voyage to England
.

“Hierarchy inspires People with Respect”
: Sorbière,
A Voyage to England
, pp. 23–24.

“the Royal Society be not some way or other blasted”
: Ibid., p. 47.

Sorbière insulted the Society’s patron
: Quoted in Sarasohn, “Who Was Then the Gentleman?,” p. 223.

“noxious in conversation”
: Sorbière,
A Voyage to England
, p. 41.

Hobbes, he wrote, was a courtly and “gallant” man
: For Sorbière’s praise of Hobbes, see see ibid., pp. 40–41. Sprat’s response is quoted in Sarasohn, “Who Was Then the Gentleman?,” p. 225.

“the indivisible Line of the Mathematicians”
: Sorbière,
A Voyage to England
, p. 93.

1. The Children of Ignatius

Machiavelli’s model of a cunning and brutal prince
: Machiavelli’s advice to the ideal ruler is contained in
The Prince
, first published in Italian in 1532.

“a soldier of God”
: For the founding of the Society of Jesus, see William V. Bangert,
A History of the Society of Jesus
(St. Louis, MO: Institute of Jesuit Sources, 1972). The quote is on page 21.

the Society’s growth
: Ibid., p. 98; R. Po-Chia Hsia,
The World of Catholic Renewal
(Cambridge: Cambridge University Press, 1998), p. 32.

“
a nursery of great men”
: The story of Montaigne’s visit to the Collegio Romano is told in Bangert,
History of the Society of Jesus
, p. 56.

their activities in France
: The Jesuits’ troubles in France are discussed in Bangert,
History of the Society of Jesus
, esp. pp. 120–21.

“to be obedient to the true Spouse of Christ”
: Ignatius of Loyola, “The First Rule,” in “Rules for Thinking, Judging, and Feeling with the Church,” in “The Spiritual Exercises,” in
Spiritual Exercises and Selected Works
(Mahwah, NJ: Paulist Press, 1991), p. 111.

“What I see as white”
: Loyola, “The Thirteenth Rule,” in “The Spiritual Exercises,” p. 213.

Big Brother
: George Orwell,
1984
, part 3, chap. 2.

“all authority is derived from God”
: An excellent discussion of the Jesuit ideal of obedience can be found in Steven Harris, “Jesuit Ideology and Jesuit Science,” Ph.D. dissertation, University of Wisconsin–Madison, 1988, pp. 54–57, and also in Bangert,
History of the Society of Jesus
, p. 42.

Neatness, cleanliness, and order
: The importance of neatness for Jesuits is discussed in Hermann Stoeckius,
Untersuchungen zur Geschichte der Noviziates in der Gesellschaft Jesu
(Bonn: P. Rost & Co., 1918). Quoted in Harris, “Jesuit Ideology,” p. 83.

“If the heretics should see”
: Favre’s letter is quoted in Bangert,
History of the Society of Jesus
, p. 75.

“Talis quus sis”
: The quote is from Francis Bacon,
The Advancement of Learning
, Book 1, III.3. Bacon is quoting from Plutarch’s life of the Spartan king Agesilaus.

the Jesuit intervention proved decisive
: On the Jesuits’ work in Germany, Belgium, and Poland, see Hsia,
The World of Catholic Renewal
, chap. 4, “The Church Militant.”

“Your holy order”
: Pope Gregory XIII’s address to the Jesuits can be found in Bangert,
History of the Society of Jesus
, p. 97.

Rubens was a devout Catholic
: On Rubens and the Jesuits, see Hsia,
The World of Catholic Renewal
, pp. 128, 154.

absolute authority
: On the Jesuits’ ideas of authority derived from abso-

lute truth and expressed in the Church hierarchy, see Rivka Feldhay, “Authority, Political Theology, and the Politics of Knowledge in the Transition from Medieval to Early Modern Catholicism,”
Social Research
73, no. 4 (2006): 1065–92.

2. Mathematical Order

“the parts of mathematics that a theologian should know”
: Ignatius’s advice on the teaching of mathematics is quoted in Giuseppe Cosentino, “Mathematics in the Jesuit Ratio Studiorum,” in Frederick J. Homann, SJ, ed. and trans.,
Church Culture and Curriculum
(Philadelphia: St. Joseph University Press, 1999), p. 55.

“to keep what remains, and restore what was lost”
: Polanco’s letter is quoted in Cosentino, “Mathematics in the Jesuit Ratio Studiorum,” p. 57.

“a sort of hook with which we fish for souls”
: Nadal’s pronouncement is quoted in M. Feingold, “Jesuits: Savants,” in M. Feingold, ed.,
Jesuit Science and the Republic of Letters
(Cambridge, MA: MIT Press, 2003), p. 6.

Ignatius regarded him as well-nigh infallible
: On Ignatius and the setting of the curriculum at Jesuit colleges, see Cosentino, “Mathematics in the Jesuit Ratio Studiorum,” p. 54.

“anyone suspect us of trying to create something new”
: Pereira’s and Acquaviva’s views on innovation are quoted in Feingold, “Jesuits: Savants,” p. 18.

Legem impone subactis: The impress of the Parthenic Academy is discussed in Ugo Baldini,
Legem impone subactis: Studi su filosofia e scienza dei Gesuiti in Italia 1540–1632
(Rome: Bulzoni, 1992), pp. 19–20.

On April 12 he was received as a novice into the Society of Jesus by Ignatius of Loyola himself
: On Clavius’s early years in the Society of Jesus, see James M. Lattis,
Between Copernicus and Galileo: Christoph Clavius and the Collapse of Ptolemaic Astronomy
(Chicago: University of Chicago Press, 1994), chap. 1.

Clavius was self-taught
: On Clavius’s mathematical education, see Lattis,
Between Copernicus and Galileo
, pp. 16–18.

Even years later he was still fighting
: On Clavius’s fight for recognition for mathematics professors in the Society, see A. C. Crombie, “Mathematics and Platonism in the Sixteenth-Century Italian Universities and in the Jesuit Educational Policy,” in Y. Maeyama and W. G. Saltzer, eds.,
Prismata
(Wiesbaden: Franz Steiner Verlag, 1977), pp. 63–94, esp. p. 65.

his status in the rigid hierarchy of the order
: On Clavius’s career at the Collegio Romano, see Cosentino, “Math in the Ratio Studiorum”; Crombie, “Mathematics and Platonism,” pp. 64–68; and Lattis,
Between Copernicus and Galileo
, chap. 1.

“intolerable to all the wise”
: Quoted in J. D. North, “The Western Calendar: ‘Intolerabilis, Horribilis, et Derisibilis,’” in G. V. Coyne, M. A. Hoskin, and O. Pedersen, eds.,
Gregorian Reform of the Calendar: Proceedings of the Vatican Conference to Commemorate its 400th Anniversary, 1582–1982
(The Vatican: Pontificia Academia Scientarum, 1983), p. 75.

it based its recommendations largely on Lilius’s suggestions
: On Lilius and the calendar commission, see Ugo Baldini, “Christoph Clavius and the Scientific Scene in Rome,” in Coyne, Hoskin, Pedersen, eds.,
Gregorian Reform of the Calendar
, p. 137.

Clavius would emerge as the public spokesman for the new system
: Clavius’s report on the calendar commission was published as Christopher Clavius,
Romani calendarii a Gregorio XIII restituti explication
, first published in 1603. Clavius also wrote various pamphlets refuting critics such as Joseph Justus Scaliger, Michael Maestlin, and François Viète.