The Clockwork Universe (35 page)

171
“He discourses often amid fifteen”
: Ibid., p. 112.

171
“If reasoning were like hauling”
: Galileo,
The Assayer.

175
“Shut yourself up with some friend”
: Galileo,
Dialogue Concerning the Two Chief World Systems.
This discussion takes place on day two.

175
“A company of chessmen”
: Locke,
Essay Concerning Human Understanding
, p. 98.

Chapter 29. Sputnik in Orbit, 1687

179
“It has been observed that missiles”
: The passage is from Galileo's
Two New Sciences
, quoted in David Goodstein and Judith Goodstein,
Feynman's Lost Lecture
, p. 38.

181
Newton pictured it all
: Newton drew the diagram in the 1680s, but it was first published after his death, in
A Treatise of the System of the World
, a less mathematical treatment of the
Principia.
See John Roche, “Newton's
Principia
,” in Fauvel et al., eds.,
Let Newton Be!
, p. 58.

Chapter 30. Hidden in Plain Sight

182
“My aim is to show”
: Shapin,
The Scientific Revolution
, p. 33.

183fn
“Music
,
” Leibniz wrote
: Kline,
Mathematics in Western Culture
, p. 287.

183
“Galileo spent twenty years”
: Gillispie,
The Edge of Objectivity
, p. 42.

Chapter 31. Two Rocks and a Rope

187
Unlike most legends
: Crease,
The Prism and the Pendulum
, p. 31.

188
“In performing the experiment”
: Ibid., p. 32.

188
When television shows a diver
: Barry Newman, “Now Diving: Sir Isaac Newton,”
Wall Street Journal
, August 13, 2008.

Chapter 32. A Fly on the Wall

190
“I sleep ten hours”
: Alfred Hooper,
Makers of Mathematics
(Vintage, 1948), p. 209.

192fn
One prominent historian calls it
: The historian was Salomon Bochner, in
The Role of Mathematics in the Rise of Science
(Princeton, NJ: Princeton University Press, 1966), p. 40. For more on the invention of the musical staff, see Alfred Crosby,
The Measure of Reality
:
Quantification and Western Society
,
1250–1600
(New York: Cambridge University Press, 1997), pp. 142–44.

193
“the greatest single step ever”
: Livio,
Is God a Mathematician
?, p. 86.

Chapter 33. “Euclid Alone Has Looked on Beauty Bare”

194
known today as Cartesian coordinates
: Descartes' original presentation differed from the treatment that would become standard, but all the future changes were implicit in his version.

195
“I do not enjoy speaking in praise”
: E. T. Bell,
The Development of Mathematics
, p. 139.

195
“a notable advance in the history”
: Alfred North Whitehead,
Science and the Modern World
, p. 20. Scientists have now found that human infants and various nonhuman animals can count (they can distinguish between two M&Ms and three, for instance), but Whitehead's point was that it took a breakthrough to see that such concepts as “twoness” were worth identifying.

195
“The point about zero”
: Newman, ed.,
The World of Mathematics
, vol. 1, p. 442.

196
Descartes wrestled to make sense
: Helena M. Pycior,
Symbols
,
Impossible Numbers
,
and Geometric Entanglements
(New York: Cambridge University Press, 2006), p. 82.

197
Nor did it matter if the rock
: Eugene Wigner makes this point in his pathbreaking essay “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.”

198
If there were vacuums
: Butterfield,
The Origins of Modern Science
, p. 3.

198fn
The question of whether vacuums
: Russell Shorto
, Descartes' Bones
(New York: Doubleday, 2008), p. 25.

198
“Only by imagining an impossible”
: A. Rupert Hall,
From Galileo to Newton
, p. 63. Hall cites the two passages from Galileo that I quote in his brilliant discussion of abstraction in science. See ibid., pp. 63–64. My comment about mathematics and abstraction in the final sentence of this chapter is also a paraphrase of Hall's argument on his p. 63.

Chapter 34. Here Be Monsters!

202
Albert of Saxony
,
a logician
: My discussion follows the one on pp. 52–55 of John Barrow's admirably lucid
The Infinite Book
.

Chapter 35. Barricaded Against the Beast

210
For decades mathematicians had all tried
: Struik,
A Concise History of Mathematics
, pp. 101–9.

Chapter 37. All Men Are Created Equal

219
Abraham Lincoln asked his listeners
: Lincoln made his remark on October 15, 1858 (and in at least one earlier speech) in his last debate with Stephen Douglas. The complete text is at http://www.bartleby.com/251/72.html.

222
“The planet Mars comes close”
: Kline,
Mathematics in Western Culture
, p. 230.

223
Perhaps infinitesimals were real but
: Carl Boyer,
The History of the Calculus and Its Conceptual Development
, p. 213.

223
Leibniz tried to explain
: William Dunham,
The Calculus Gallery
, p. 24.

223
“an enigma rather than”
: Leibniz's puzzled disciples were James and John Bernoulli, quoted in Kline,
Mathematics
:
The Loss of Certainty
, p. 137.

223
“the ultimate ratio”
: Ibid., p. 135.

223
“In mathematics the minutest”
: Ibid., p. 134.

223 calculus
is the Latin
: Donald Benson,
A Smoother Pebble
:
Mathematical Explorations
, p. 167.

224
“For science it cannot be”
: George Berkeley,
The Analyst
:
or A Discourse Addressed to an Infidel Mathematician
(London, 1754), p. 34.

224
Leibniz
,
boundlessly optimistic
: Dunham,
The Calculus Gallery
, p. 24
,
and Kline,
Mathematics
:
The Loss of Certainty
, p. 140.

224
“Persist
,
” d'Alembert advised
: Kline,
Mathematics
:
The Loss of Certainty
, p. 162.

Chapter 38. The Miracle Years

226
Calculus was in the air
: Most Newtonian scholars, including Newton's most careful biographer, Richard Westfall, and the preeminent expert on Newton's mathematical work, D. T. Whiteside, argue emphatically that Newton achieved his mathematical breakthroughs essentially on his own. For a contrary point of view, arguing that the influence of the Cambridge mathematician Isaac Barrow on Newton has been downplayed, see Mordechai Feingold's “Newton, Leibniz, and Barrow, Too: An Attempt at a Reinterpretation,”
Isis
84, no. 2 (June 1993), pp. 310–38.

226
market called Stourbridge Fair
: Stourbridge Fair served as Bunyan's inspiration for Vanity Fair in
A Pilgrim's Progress.
See Edmund Venables,
Life of John Bunyan
(London: Walter Scott, 1888), p. 173.

226
“The way to chastity”
: Gale Christianson,
In the Presence of the Creator
:
Isaac Newton and his Times
, p. 258.

227
Newton “read it 'til”
: D. T. Whiteside, “Isaac Newton: Birth of a Mathematician,” p. 58.

227
“Read only the titles”
: Westfall,
Never at Rest
, p. 98.

228
“The same year in May”
: Ibid., p. 143.

229
“All this
,
” he wrote
: Ibid.

229
“If you haven't done”
: Author interview, in Edward Dolnick, “New Ideas and Young Minds,”
Boston Globe
, April 23, 1984.

230
“Age is
,
of course
,
a fever chill”
: Quoted in Dean Simonton,
Creativity in Science
(New York: Cambridge University Press, 2004), p. 68.

230
“I know that when”
: Barrow,
Pi in the Sky
, p. 165.

230
“Look at a composer”
: Author interview, in Dolnick, “New Ideas.”

231
“no old Men (excepting Dr. Wallis)”
: Westfall,
Never at Rest
, p. 139.

231
From his earliest youth
: Gale Christianson, “Newton the Man—Again.”

231
“difficulty & ill success”
: Christianson,
In the Presence of the Creator
, p. 260.

232
He took the Latin form
: Ackroyd,
Newton
, p. 39.

232
“I will give thee the treasures”
: Christianson,
Isaac Newton
, p. 58. The verse is Isaiah 45:3.

232
“The fact that he was unknown”
: Westfall,
Never at Rest
, p. 137.

232
“In 1665
,
as he realized”
: Ibid., p. 138.

Chapter 39. All Mystery Banished

234
In fact
,
though
,
Leibniz felt
: Since God was infinite, His creation was infinite as well, which meant that the process of finding new things to understand was never-ending. But this was a virtue, not a defect, because human happiness consisted in
constantly
finding new aspects of God's perfection to admire.

234
“I don't know what I may seem”
: Westfall,
Never at Rest
, p. 863.

234
“As a blind man has no idea”
: I. Bernard Cohen's translation of
Principia
(Berkeley: University of California Press, 1999), p. 428.

236
“perhaps the most resolute champion”
: Ernst Cassirer, “Newton and Leibniz,” p. 379.

Chapter 40. Talking Dogs and Unsuspected Powers

237
“In the century of Kepler”
: C. H. Edwards, Jr.,
The Historical Development of the Calculus
, p. 231.

237
“an aptitude that was hard to find”
: Leibniz's letter can be found at www.leibniz-translations.com, a marvelous website run by the English philosopher Lloyd Strickland. See http://www.leibniz-translations.com/dog.htm, “Account of a Letter from Mr. Leibniz to the Abbé de St. Pierre, on a Talking Dog.”

237
“a museum of everything”
: Wiener, “Leibniz's Project.”

238
“I have so much that is new”
: Stewart,
The Courtier and the Heretic
, p. 256.

238
“If controversies were to arise”
: Umberto Eco,
The Search for the Perfect Language
, p. 281. (See Chapter 14, “From Leibniz to the
Encyclop
é
die.
”
)

238
Today a diligent team
: Author interview with Lawrencc Carlin, philosophy department at the University of Wisconsin at Oshkosh, July 15, 2008.

238
“Leibniz was one of the supreme”
: Russell,
History of Western Philosophy
, p. 581.

239fn
Unbeknownst to Leibniz
: Harriot's work on the telescope is discussed by Albert Van Helden in his “Introduction” to Galileo's
Sidereal Messenger
, p. 9, and his mathematical work is discussed in the online journal
Plus
. See Anna Faherty, “Thomas Harriot, A Lost Pioneer,” at http://plus.maths.org/issue50/features/faherty/.

239
“A container shall be provided”
: George Dyson,
Darwin Among the Machines
, p. 37.

240
Leibniz's knowledge of mathematics
: Joseph E. Hofmann,
Leibniz in Paris 1672–1676
:
His Growth to Mathematical Maturity
, p.2.

240
“I read [mathematics] almost”
: Dunham,
The Calculus Gallery
, p. 21.

241
his correspondence alone consisted
: Stewart,
The Courtier and the Heretic
, p. 138.

242
At an elegant dinner party
: A. Rupert Hall,
Philosophers at War
, p. 54.

242
Or perhaps he decided
: The suggestions in this sentence and the next are from email correspondence with Simon Schaffer, a distinguished historian of science at Cambridge University, on September 27, 2009.

243
“6accdae13eff7i319n4o4qrr4s8t12ux”
: Westfall,
Never at Rest
, p. 265.

243
Leibniz made no mention
: Hall,
Philosophers at War
, p. 77.

Chapter 41. The World in Close-Up