Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (56 page)

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.

“The quadrature of the circle”
: The account is based on Jesseph,
Squaring the Circle
, pp. 22–26; and Archimedes, “Measurement of a Circle,” chap. 6 in E. J. Dijketerhuis,
Archimedes
(Princeton, NJ: Princeton University Press, 1987), pp. 222–23.

the three classical problems were simply insoluble
: On views on the solubility of the quadrature of the circle, see Jesseph,
Squaring the Circle
, pp. 25–26.

“that which has no parts”
: For Hobbes’s discussion of Euclid’s definitions, see
Six Lessons
, 7:201.

“If the magnitude of a body which is moved”
: Hobbes,
De corpore
, 2.8.11, reprinted in Latin in Thomas Hobbes,
Opera philosophica
(London: John Bohn, 1839), 1:98–99, more commonly known as Hobbes’s
Opera Latina
. The passage is translated in Jesseph,
Squaring the Circle
, pp. 76–77.

Points have a size, lines have a width
: On Hobbes’s view that points have size and lines have width in order to construct geometrical bodies, see Hobbes,
Six Lessons
, p. 318.

“conatus”
: For Hobbes’s discussion of his concepts “conatus” and “impetus,” see
De corpore
, 3.15.2, and
Opera Latina
, 1:177–78, both translated in Jesseph,
Squaring the Circle
, pp. 102–103.

“instead of saying that a line is long”
: Sorbière,
A Voyage to England
, p. 94.

“Every demonstration is flawed”
: Hobbes,
De principiis
, chap. 12, quoted in Jesseph,
Squaring the Circle
, p. 135.

“speake loudest in his praise”
: Ward published this anonymously in [Seth Ward],
Vindiciae academiarum
(Oxford: L. Litchfield, 1654), p. 57, quoted in Jesseph,
Squaring the Circle
, p. 126.

It was a trap, and Hobbes knew it
: On the beginnings of the war between Hobbes and Wallis, and Ward’s role in it, see Jesseph,
Squaring the Circle
, p. 126.

“problematically”
: The disclaimer is in Hobbes,
De corpore
(1655), p. 181. Translation from Latin in Jesseph,
Squaring the Circle
, p. 128. The titles of the chapters are in
De corpore
, p. 171.

were all gleefully exposed
: See John Wallis,
Elenchus geometriae Hobbianae
(Oxford: H. Hall for John Crooke, 1655).

“Solutions of Problems that have hitherto remained insoluble”
: Sorbière,
A Voyage to England
, p. 94.

The attempted proofs of the quadrature
: For a modern exposition of two of these proofs, see Jesseph, “Two of Hobbes’s Quadratures from
De corpore
, Part 3, Chapter 20,” in Jesseph,
Squaring the Circle
, pp. 368–76.

squaring of the circle is impossible
: For a fuller discussion of the impossibility of squaring the circle, see Jesseph,
Squaring the Circle
, pp. 22–28.

“solved some most difficult problems”
: Hobbes’s list of accomplishments is included in Aubrey, “Thomas Hobbes,” pp. 400–401. Translated in Jesseph,
Squaring the Circle
, pp. 3–4.

8. Who Was John Wallis?

“we
Tast
and
See
it to be so”
: John Wallis,
Truth Tried
(London: Richard Bishop for Samuel Gellibrand, 1643), pp. 60–61.

“Beside his constant preaching”
: The account of Wallis’s childhood and the quotes are from John Wallis, “Autobiography,” in Christoph J. Scriba, “The Autobiography of John Wallis, F.R.S.,”
Notes and Records of the Royal Society of London
25, no. 1 (June 1970): 21–23.

“
Mr Holbech
was very kind to me”
: Ibid., p. 25.

“scarce bred any man that was loyall to his Prince”
: The quote is from
The Autobiography of Sir John Bramston
, quoted in Vivienne Larminie, “Holbeach [Holbech], Martin,” in
Oxford Dictionary of National Biography
(Oxford: Oxford University Press, 2004–12).

“were scarce looked upon as
Accademical
studies”
: Wallis, “Autobiography,” p. 27.

“a pleasing Diversion at spare hours”
: Ibid.

“Knowledge is no Burthen”
: Ibid., p. 29.

the trial of Archbishop William Laud
: Agnes Mary Clerke, “Wallis, John (1616–1703),”
Dictionary of National Biography
, vol. 59 (1899).

A Serious and Faithful Representation: The pamphlet’s full title was
A Serious and Faithful Representation of the Judgements of Ministers of the Gospel within the Province of London, Contained in a Letter from Them to the General and His Counsel of War, Delivered to His Excellency by Some of the Subscribers, January 18, 1649
(printed in Edinburgh, 1703).

“now to be my serious study”
: Wallis, “Autobiography,” p. 40.

two mathematical treatises
: Wallis’s two treatises were
De sectionibus conicis
(Oxford: Leon Lichfield, 1655) and
Arithmetica infinitorum
(Oxford: Leon Lichfield, 1656), both included in Wallis,
Opera mathematicorum
(Oxford: Leonard Lichfield, 1656–57), vol. 2.

Presbyterian
referred to respectable clergymen
: Wallis wrote, “When they were called
Presbyterians
it was not in the sense of
Anti-Episcopal
, but
Anti-Independents
.” See Wallis, “Autobiography,” p. 35.

custos archivorum
: On Wallis’s election as keeper of the archives and Stubbes’s opposition, see Christoph J. Scriba, “John Wallis,” in Gillispie, ed.,
Dictionary of Scientific Biography
. On Stubbe and Hobbes, see Jesseph,
Squaring the Circle
, p. 12. On the demonization of Hobbes, see Mintz, “Thomas Hobbes,” in Gillispie, ed.,
Dictionary of Scientific Biography
.

he be burned at the stake
: John Aubrey, “Thomas Hobbes,” p. 339nc.

“Their first purpose”
: Thomas Sprat,
History of the Royal Society of London
(London: T.R., 1667), p. 53.

“such a candid and unpassionate company as that was”
: Ibid., pp. 55–56.

Now, with his new companions
: On Wallis’s involvement with the “Invisible College” during the Interregnum, and of the group’s diverse fields of interest, see Wallis, “Autobiography,” pp. 39–40.

“continued such meetings in
Oxford
”
: Wallis, “Autobiography,” p. 40.

Philosophical Transactions: The other candidate for “first scientific journal” is
Le journal des scavans
of the French Academy of Sciences, whose first issue appeared two months before
Philosophical Transactions
.

“gives us room to differ, without animosity”
: Sprat,
History of the Royal Society
, p. 56.

“they
work
and
think
in company”
: Ibid., p. 427.u
se their experience to reconstitute the entire body politic
: For more on the early Royal Society and its mission to recast English political life and prevent a return to the disastrous dogmatism of the Interregnum, see Shapin and Schaffer,
Leviathan and the Air-Pump
; Margaret C. Jacob,
The Newtonians and the English Revolution 1689–1720
(New York: Gordon and Breach, 1990), first published in 1976; James R. Jacob,
Robert Boyle and the English Revolution
(New York: Burt Franklin and Co., 1977); Barbara J. Shapiro,
Probability and Certainty in Seventeenth Century England
(Princeton, NJ: Princeton University Press, 1983); Steven Shapin,
A Social History of Truth: Civility and Science in Seventeenth-Century England
(Chicago: University of Chicago Press, 1995).

Cartesian philosophy
: The most concise summary of Descartes’s philosophy is contained in his eminently readable
Discourse on the Method
, first published anonymously in Leiden in 1637 as
Discours de la méthode
.

“more imperious, and impatient of contradiction”
: Sprat’s views on the dangers of dogmatism can be found in Sprat,
History of the Royal Society
, p. 33.

“slowness of the increase of knowledge amongst men”
: Ibid., p. 428.

“The
reason
of men’s contemning all
Jurisdiction
and
Power
”
: Ibid., p. 430.

“the most fruitful parent of
Sedition
is
Pride
”
: Ibid., p. 428–29.

a public statement of the goals and purpose of the Royal Society
: On Sprat and the grandees of the Royal Society, see the “Introduction” in Jackson I. Cope and Harold Whitmore Jones, eds.,
The History of the Royal Society by Thomas Sprat
(St. Louis, MO: Washington University Studies, 1958), esp. pp. xiii–xiv.

dogmatism leads to sedition
: In Sprat’s words, “it gives them fearless confidence in their own judgments, it leads them from contending in sport to opposition in earnest … in the
State
as well as in the
Schools
.” See Sprat,
History of the Royal Society
, p. 429.

“the influence of experiments is
Obedience to the Civil Government
”
: Sprat,
History of the Royal Society
, p. 427.

“that teaches men
humility
”
: On the beneficial effects of experimentalism, see ibid., p. 429.

“one great man”
: For the Royal Society’s idolization of Bacon, see ibid., p. 35.

Francis Bacon, Lord Chancellor to James I
: Bacon’s major works include
The Advancement of Learning
(1605),
Novum organum
(1620), and
New Atlantis
(1627).

True knowledge, worthy of
: True knowledge, in the Aristotelian scheme, was referred to as
scientia
, and required absolute certainty based on logical reasoning and ancient authority.

“the daintiness and pride of mathematicians”
: The quote is from Francis Bacon, “Of the Dignity and Advancement of Learning,” book 3, chap. 6, in James Spedding, ed.,
The Philosophical Works of Francis Bacon
, vol. 4 (London: Longman and Co., 1861), p. 370.

It was a conundrum that left the Society with an ambivalence toward mathematics
: On the early Royal Society’s ambivalence toward mathematics, and particularly Robert Boyle’s suspicion of the field, see Shapin,
A Social History of Truth
, chap. 7.

9. Mathematics for a New World

“It hath been my endeavour”
: John Wallis, “Autobiography,” in Scriba, “The Autobiography of John Wallis, F.R.S.,” p. 42.

“On Conic Sections”
: John Wallis,
De sectionibus conicis, nova methodo expositis, tractatus
(Oxford: Leon Lichfield, 1655). On the publication of this treatise, along with the
Arithmetica infinitorum
, see Jacqueline Stedall, trans.,
The Arithmetic of Infinitesimals, John Wallis, 1656
(New York: Springer-Verlag, 2004), p. xvii.

“any plane is made up, so to speak, of infinite parallel lines”
: Wallis,
De sectionibus conicis
, prop. 1, in Wallis,
Opera mathematica
(Oxford: Theatro Sheldoniana, 1695), p. 297.

“the area of the triangle is equal to the base times half the altitude”
: Wallis,
De sectionibus conicis
, prop. 3, in
Opera mathematica
, p. 299.

When criticized by Fermat
: Fermat’s criticisms are included in a wide-ranging correspondence of Wallis’s
Arithmetica infinitorum
, which Wallis published in 1658 under the title
Commercium epistolicum
. Fermat’s letters were published in French translation in volumes 2 and 3 of Paul Tannery and Charles Henri, eds.,
Oeuvres de Fermat
(Paris: Gauthiers-Villars et Fils, 1894–96).

“Mathematical entities exist”
:
The quote is from Wallis,
Mathesis universalis
(Oxford: Leon Lichfield, 1657), chapter 3; reprinted in Wallis,
Opera Mathematica
(Oxford: Sheldonian Theatre, 1695), p.21.

“it seems not in the power of the Will to reject”
: Ibid., pp. 60–61.

“a general proposition may become known by induction”
: Wallis,
Arithmetica infinitorum
(Oxford: Leon Lichfield, 1656), p. 1, prop. 1. Translation is from Stedall, trans.,
The Arithmetic of Infinitesimals
, p. 13.

study minute creatures under a microscope
: Hooke’s startling enlarged images of common insects and microbes invisible to the naked eye were published in Robert Hooke,
Micrographia or Some Physiological Descriptions of Minute Bodies Made by Magnifying Glasses
(London: John Allestry, 1667).