Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (55 page)

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 geometry”
: 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.