181 Newton pictured it all 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 Treatise of the System of the World, a less mathematical treatment of the Principia. Principia. See John Roche, "Newton's See John Roche, "Newton's Principia Principia," in Fauvel et al., eds., Let Newton Be! Let Newton Be!, p.58.

CHAPTER 30. HIDDEN IN PLAIN SIGHT.

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

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

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

CHAPTER 31. TWO ROCKS AND A ROPE.

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

188 "In performing the experiment" "In performing the experiment": Ibid., p.32.

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

CHAPTER 32. A FLY ON THE WALL.

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

192fn One prominent historian calls it One prominent historian calls it: The historian was Salomon Bochner, in The Role of Mathematics in the Rise of Science 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, (Princeton, NJ: Princeton University Press, 1966), p.40. For more on the invention of the musical staff, see Alfred Crosby, The Measure of Reality The Measure of Reality: Quantification and Western Society Quantification and Western Society, 12501600 12501600 (New York: Cambridge University Press, 1997), pp.14244. (New York: Cambridge University Press, 1997), pp.14244.

193 "the greatest single step ever" "the greatest single step ever": Livio, Is G.o.d a Mathematician Is G.o.d a Mathematician?, p.86.

CHAPTER 33. "EUCLID ALONE HAS LOOKED ON BEAUTY BARE".

194 known today as Cartesian coordinates 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" "I do not enjoy speaking in praise": E. T. Bell, The Development of Mathematics The Development of Mathematics, p.139.

195 "a notable advance in the history" "a notable advance in the history": Alfred North Whitehead, Science and the Modern World 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" "The point about zero": Newman, ed., The World of Mathematics The World of Mathematics, vol.1, p.442.

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

197 Nor did it matter if the rock 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 If there were vacuums: b.u.t.terfield, The Origins of Modern Science The Origins of Modern Science, p.3.

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

198 "Only by imagining an impossible" "Only by imagining an impossible": A. Rupert Hall, From Galileo to Newton From Galileo to Newton, p.63. Hall cites the two pa.s.sages from Galileo that I quote in his brilliant discussion of abstraction in science. See ibid., pp.6364. 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 Albert of Saxony, a logician a logician: My discussion follows the one on pp.5255 of John Barrow's admirably lucid The Infinite Book The Infinite Book.

CHAPTER 35. BARRICADED AGAINST THE BEAST.

210 For decades mathematicians had all tried For decades mathematicians had all tried: Struik, A Concise History of Mathematics A Concise History of Mathematics, pp.1019.

CHAPTER 37. ALL MEN ARE CREATED EQUAL.

219 Abraham Lincoln asked his listeners 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" "The planet Mars comes close": Kline, Mathematics in Western Culture Mathematics in Western Culture, p.230.

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

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

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

223 "the ultimate ratio" "the ultimate ratio": Ibid., p.135.

223 "In mathematics the minutest" "In mathematics the minutest": Ibid., p.134.

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

224 "For science it cannot be" "For science it cannot be": George Berkeley, The a.n.a.lyst The a.n.a.lyst: or A Discourse Addressed to an Infidel Mathematician or A Discourse Addressed to an Infidel Mathematician (London, 1754), p.34. (London, 1754), p.34.

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

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

CHAPTER 38. THE MIRACLE YEARS.

226 Calculus was in the air 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 Isis 84, no. 2 (June 1993), pp.31038. 84, no. 2 (June 1993), pp.31038.

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

226 "The way to chast.i.ty" "The way to chast.i.ty": Gale Christianson, In the Presence of the Creator In the Presence of the Creator: Isaac Newton and his Times Isaac Newton and his Times, p.258.

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

227 "Read only the t.i.tles" "Read only the t.i.tles": Westfall, Never at Rest Never at Rest, p.98.

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

229 "All this "All this," he wrote: Ibid.

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

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

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

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

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

231 From his earliest youth From his earliest youth: Gale Christianson, "Newton the Man-Again."

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

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

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

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

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

CHAPTER 39. ALL MYSTERY BANISHED.

234 In fact In fact, though though, Leibniz felt Leibniz felt: Since G.o.d 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 constantly finding new aspects of G.o.d's perfection to admire. finding new aspects of G.o.d's perfection to admire.

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

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

236 "perhaps the most resolute champion" "perhaps the most resolute champion": Ernst Ca.s.sirer, "Newton and Leibniz," p.379.

CHAPTER 40. TALKING DOGS AND UNSUSPECTED POWERS.

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

237 "an apt.i.tude that was hard to find" "an apt.i.tude 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 Abbe de St. Pierre, on a Talking Dog."

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

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

238 "If controversies were to arise" "If controversies were to arise": Umberto Eco, The Search for the Perfect Language The Search for the Perfect Language, p.281. (See Chapter 14, "From Leibniz to the Encyclopedie. Encyclopedie.") 238 Today a diligent team 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" "Leibniz was one of the supreme": Russell, History of Western Philosophy History of Western Philosophy, p.581.