Part 17 (1/2)
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 /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.