A Brief Guide to the Great Equations

by Robert Crease

  10 To name one, Otto Neugebauer, who first deciphered the Pythagorean triplets of Plimpton 322, cited old Babylonian tablets as ‘sufficient proof that the ‘Pythagorean’ theorem was known more than a thousand years before Pythagoras.’ Otto Neugebauer, The Exact Sciences in Antiquity (Providence: Brown University Press, 1993), p. 36.

  11 Francis M. Cornford, Before and After Socrates (Cambridge: Cambridge University Press, 1972), pp. 72–73.

  12 American Mathematical Monthly 1, no. 1 (January 1894), p. 1.

  13 Elisha S. Loomis, The Pythagorean Proposition: Its Proofs Analysed and Classified. Publ. by The Masters and Wardens Association of the 22nd Masonic District of the Most Worshipful Grand Lodge of Free and Accepted Masons of Ohio, 1927; and The Pythagorean Proposition: Its Demonstrations Analysed and Classified (Ann Arbor, MI: Edwards Brothers, 1940). He ended the first book thus: ‘FINAL THOUGHT: Is it an all-embracing truth? The generalization of the Pythagorean Theorem so as to conform to and include the data of geometries other than that of Euclid, as was done by Riemann in 1854, and later, 1915, by Einstein in formulating and positing the general theory of relativity, seems to show that the truth implied in this theorem is destined to become the fundamental factor in harmonizing past, present and future theories relative to the underlying laws of our universe.’

  14 For instance, geometric proof 32 of the second edition ‘is credited to Miss E. A. Coolidge, a blind girl’; geometric proof 68 is ‘the first ever devised in which all auxiliary lines and all triangles used originate at the middle point of the hypotenuse of the given triangle. It was devised and proved by Miss Ann Condit, a girl, aged 16 years, of Central Junior-Senior High School, South Bend, Ind., Oct. 1938. This 16-year-old girl has done what no great mathematician, Indian, Greek, or modern, is ever reported to have done’; geometric proof 69 ‘is original…devised by Joseph Zelson, a junior in West Phila., Pa., High School, and sent to me by his uncle… It shows a high intellect and a fine mentality’, and Loomis adds that ‘this proof and the one before ‘are evidences that deductive reasoning is not beyond our youth’; geometric proofs 252–55 ‘show high intellectual ability, and prove what boys and girls can do when permitted to think independently and logically’; and regarding algebraic proof 93, Loomis remarks that it is ‘by Stanley Jashemski, age 19, of Youngstown, O., June 4, 1934, a young man of superior intellect.’

  15 Loomis, 1927 edition, p. 99.

  16 Loomis, 1940 edition, p. 269.

  17 Eli Maor, The Pythagorean Theorem: A 4,000-Year History (Princeton: Princeton University Press, 2007), p. xiv.

  18 Galileo, Galileo on the World Systems, trans. M. A. Finocchiaro (Berkeley: University of California Press, 1997), p. 97.

  19 G.W.F. Hegel, Hegel’s Philosophy of Nature, vol. 1, ed. and trans. M. J. Petry (New York: Humanities Press, 1970), p. 228.

  20 I thank my colleague David Dilworth for this observation.

  INTERLUDE: Rules, Proofs, and the Magic of Mathematics

  1 Oliver Byrne, ed. The First Six Books of the Elements of Euclid (London: William Pickering, 1847). It is available online at http://www.math.ubc.ca/people/faculty/cass/Euclid/byrne.html.

  2 David Socher, ‘A Cardboard Pythagorean Teaching Aid’, Teaching Philosophy 28, 2005, pp. 155–61.

  3 George MacDonald Fraser, Quartered Safe Out Here: A Recollection of the War in Burma (London: HarperCollins, 1992), p. 150.

  4 From pp. 9–11 in the opening autobiographical sketch of Albert Einstein: Philosopher-Scientist, ed. Paul Arthur Schilpp (London: Cambridge University Press, 1970).

  2 ‘The Soul of Classical Mechanics’: Newton’s Second Law of Motion

  1 Thus the subjective experience individuals have of themselves as a bodily centre of action was metaphorically projected into nonhumans as one of their properties. This illustrates what philosopher Maxine Sheets-Johnstone calls ‘the living body serv[ing] as a semantic template’, a process which, she points out, is key to the emergence of many early scientific concepts. It is a classic case of the use of analogical thinking, or the use of the familiar to understand the unfamiliar. What is remarkable in this instance is that the familiar has its basis in the tactile-kinesthetic experiences of bodily life; thus, in a corporeal template. For the connection between early ideas of force and religious ideas, see Max Jammer, Concepts of Force: A Study in the Foundations of Dynamics (Cambridge: Harvard University Press, 1957), chapter 2.

  2 Nearly all modern editions of Aristotle include the pagination of a standard edition published in 1831, and when making references it is standard practice – and highly efficient given the numerous editions and translations – to cite the name of Aristotle’s book followed by the chapter and, sometimes, line. This reference is to On the Heavens, Book I, ch. 3, lines 270 b 13–17.

  3 That unchanging things in the heavens, he concluded, move ‘with a ceaseless, circular movement’ is clear ‘not only to reason, but also in fact.’ Metaphysics, 1072a21.

  4 The full set of such rules of thumb can be found in Physics, Book VII, ch. 5, Physics, Book VIII, ch. 10, and On the Heavens, Book I, ch. 7. They include: ‘Half the force will move the same body half the distance in the same time’; ‘The same force will move a body half as heavy twice the distance’; ‘Twice the resistance halves the distance’; ‘The thicker the medium, the more slowly a body falls in it’; ‘The heavier the body, the faster it falls.’ It is tempting nowadays to express these rules mathematically. Later commentators, looking back at Aristotle from the vantage of thousands of years later, paraphrased Aristotle’s statements on motion as follows: Given the same time and force, the distance traversed by an object is inversely proportional to the resistance; and given the same distance and force, the time is directly proportional to the resistance. Often these were simplified still further in mathematical notation, combining distance and time as velocity (V), and representing F as force and R as resistance:

  V α F/R

  (velocity is proportional to force divided by the resistance)

  and

  V α W/R

  (velocity is proportional to weight divided by the resistance)

  But that would be misleading, and misrepresent what he saw. He knew there were exceptions and even areas where the rules did not apply. He knew, for instance, that the connection between force and speed was not smoothly varying – for while 50 people could push a ship half as far in the same time as 100 people, 1 person couldn’t push it at all. If the force equaled the resistance, the movement is clearly zero, but his rules suggested otherwise. And he believed that the speed of an object increases as it gets closer to its natural place, which is not reflected in his rules.

  5 Aristotle had no sense of uniform motion. He was less interested in the stages of motion – uniform, accelerated, uniformly accelerated – than in where the moved object came from and where it was heading. Thus he had no idea of speed as a particular instant versus average speed. Speed, to him, was overall speed, the time it takes something to complete a movement, and he noted that some movements take more time than others. ‘Velocity as a technical scientific term to which numerical values might be assigned’, Lindberg notes, ‘was a contribution of the Middle Ages.’ David C. Lindberg, The Beginnings of Western Science: The European Scientific Tradition in Philosophical, Religious, and Institutional Context, 600 B.C. to A.D. 1450 (Chicago: University of Chicago Press, 1992), p. 60.

  6 On the Heavens, Book I, ch. 8.

  7 Ibid., Book III, ch. 2.

  8 Aristotle’s two views here are from the fourth and eighth books of the Physics. For an analysis see Marshall Clagett, The Science of Mechanics in the Middle Ages (Madison: University of Wisconsin Press, 1959), pp. 505–9.

  9 See Clagett, The Science of Mechanics, pp. 258–61.

  10 ‘Impetus is a thing of permanent nature distinct from the local motion in which the projectile is moved…’ Quoted in Lindberg, The Beginnings of Western Science, p. 303.

  11 ‘[I]t is unnecessary to posit intelligences as the move
rs of celestial bodies… For it could be said that when God created the celestial spheres, He began to move each of them as He wished, and they are still moved by the impetus which He gave to them because, there being no resistance, the impetus is neither corrupted nor diminished.’ Quoted in Simon Oliver, Philosophy, God and Motion (New York: Routledge, 2005), p. 152.

  12 He proved, for instance, the mean speed theorem, that a uniformly accelerating body (a car accelerating from 0 to 60 mph in a minute, we might say) covers the same ground as a body moving at the mean uniform speed (30 mph for a minute). From Jammer, Concepts of Force, p. 66.

  13 ‘[I]t is necessary that points, lines, and surfaces, or their properties be imagined… Although indivisible points, or lines are non-existent, still it is necessary to feign them.’ Marshall Clagett, ed. and trans., Nicole Oresme and the Medieval Geometry of Qualities and Motions (Madison: University of Wisconsin Press, 1968), p. 165.

  14 See, for instance, I. B. Cohen, The Triumph of Numbers: How Counting Shaped Modern Life (New York: W. W. Norton, 2005).

  15 For Galileo’s use of force see Richard Westfall, Force in Newton’s Physics: The Science of Dynamics in the Seventeenth Century (New York: Elsevier, 1971), chapter 1 and Appendix A.

  16 Westfall, Force in Newton’s Physics, pp. 41–4`2.

  17 Jammer, Concepts of Force, p. 120.

  18 I. Bernard Cohen, ‘Newton’s Second Law and the Concept of Force in the Principia’, in The Annus Mirabilis of Sir Isaac Newton 1666–1966 ed. Robert Palter (Cambridge, MA: MIT Press, 1971), p. 171.

  19 Isaac Newton, The Principia: Mathematical Principles of Natural Philosophy, trans. I. Bernard Cohen and Anne Whitman (Berkeley: University of California Press, 1999), p. 407.

  20 Newton, Principia, p. 409.

  21 Herbert Butterfield, quoted by Oliver, Philosophy, God, and Motion, p. 168.

  22 Cohen, ‘Newton’s Second Law’, p. 143. Cohen goes on to describe the evolution of Newton’s ideas leading up to the Principia, saying that ‘[T]he Second Law may serve as a particularly fascinating index to Newton’s achievement in the Principia because it reveals to us how Newton was able to generalize his physics from the phenomenologically based dynamics of collisions and blows to the debatable realm of central forces, of gravitational attraction, and hence of continuous forces generally’ (p. 160).

  23 As Newton writes in the Preface, the purpose of the book is ‘to discover the forces of nature from the phenomena of motions and then to demonstrate the other phenomena from these forces.’ He will use the forces, for instance, to deduce ‘the motions of the planets, the comets, the moon, and the sea.’ Principia, p. 382.

  24 Voltaire to Pierre-Louis Moreau de Maupertuis, October 1932, in Voltaire’s Correspondence, vol. 2, ed. T. Besterman (Geneva: Voltaire Institute and Museum, 1953), p. 382.

  INTERLUDE: The Book of Nature

  1 Galileo, ‘The Assayer’, in Discoveries and Opinions of Galileo, trans. S. Drake (New York: Doubleday, 1957), pp. 237–38.

  3 ‘The High Point of the Scientific Revolution’: Newton’s Law of Universal Gravitation

  1 Another interesting idea came from the mathematician Pappus of Alexandria (third century ad), who proposed a way of treating gravity as if it were an Aristotelian ‘pulling’: find out how much pull it takes to move a weight on a plane, he said, and then tilt the plane to find out how much more pull it takes to move the weight upward. Jammer, Concepts of Force, p. 41.

  2 Lindberg, The Beginnings of Western Science, p. 275.

  3 See Lynn Thorndike, ‘The True Place of Astrology in the History of Science’, ISIS 46 (1955), p. 273.

  4 Nicoletto Vernias, De gravibus et levibus, Venice 1504, cited in Jammer, Concepts of Force, p. 67.

  5 ‘I think that gravity’, Copernicus wrote, ‘is nothing else than a certain natural appetition given to the parts of the earth by divine providence of the Architect of the universe in order that they may be restored to their unity and to their integrity by reuniting in the shape of a sphere. It is credible that the same affection is in the sun, the moon, and other errant bodies in order that, through the agency of this affection, they may persist in the rotundity with which they appear to us.’ De Revolutionibus, book 1, chapter 9.

  6 The idea that the planets travel elliptical paths was a fundamental break with long-held principles. Trust – in Tycho and in Copernicus’s heliocentric idea – enabled Kepler to appreciate the significance of the excess of Brahe’s data over the theory, and to question assumptions held since ancient times. Trust made the discrepancy meaningful, and directed his suspicion to the right place. This was not the first time that trust played a central role in a major scientific discovery, nor will it be the last.

  7 E. A. Burtt, The Metaphysical Foundations of Modern Science (Garden City, NY: Doubleday, 1954), p. 64.

  8 Johannes Kepler, Mysterium Cosmographicum, trans. E. J. Aiton (Norwalk, CT: Abaris Books, 1999), p. 203.

  9 In 1666, Italian physiologist Alfonso Borelli (1608–1679) proposed an explanation for the motions of the moons of Jupiter, involving an interaction of several forces, that invited application to planetary motions, suggesting that the same laws govern both the planets and their moons, and the sun and the solar system.

  10 Carl B. Boyer, ‘Boulliau, Ismael’, entry in the Dictionary of Scientific Biography, vol. 2 (New York: Scribner’s, 1970), pp. 348–49.

  11 Robert Hooke, ‘Lectiones Cutlerianae’, in R. T. Gunther, Early Science in Oxford, vol. 8, 1908, pp. 27–28.

  12 Hooke to Newton, November 24, 1679. Isaac Newton, The Correspondence of Isaac Newton, Vol. II, 1676–1687, ed. H. W. Turnbull (Cambridge: Cambridge University Press, 1960), p. 297.

  13 Writes Newton’s biographer Westfall, ‘Few periods have held greater consequences for the history of Western science than the three to six months in the autumn and winter of 1684–5.’ R. S. Westfall, Never at Rest: A Biography of Isaac Newton (New York: Cambridge University Press, 1988), p. 420.

  14 Feynman, Lectures on Physics, tape 13, no. 1, side 1.

  15 I. Bernard Cohen, Scientific American, March 1981.

  16 ‘Newton was the one who elevated Kepler’s law of areas to the status it enjoys today.’ Cohen, Scientific American, March 1981, p. 169.

  17 Quoted in I. Bernard Cohen, Birth of a New Physics (New York: W. W. Norton, 1985), p. 151.

  18 Ibid., p. 236.

  19 This was a remarkable development, but one whose pattern recurs in the history of science: Newton’s early work had been motivated by Kepler’s laws; he assumed that they were an accurate description of nature and they led him to a deep insight, yet the insight implied that Kepler’s laws were wrong, and allowed Newton to predict deviations from Kepler’s laws. This development illustrates how the human mind bootstraps itself in science, engaging in a back-and-forth interaction between two realms – our experience of nature and our models of it, how nature appears and the concepts through which we encounter it, and the way that this process changes both how nature appears and our concepts. Philosophers call such a process hermeneutics, a fancy term for interpretation, but it merely expresses a basic scientific procedure, a process often hidden because we tend to fix our eyes on nature rather than on the process. But without it, science would be trivial or impossible.

  20 As he writes to Bentley, ‘Gravity must be caused by an agent acting constantly according to certain laws; but whether this agent be material or immaterial, I have left to the consideration of my readers.’ I. Newton The Correspondence of Isaac Newton, Vol. III, 1688–1684, ed. H. W. Turnbull (Cambridge: Cambridge University Press, 1961), p. 254.

  21 This made historian Marjorie Nicolson wonder whether ‘Newton felt that his formulation of the law of gravitation was not so much the beginning of something new as the climax of something very old.’ She continued: ‘Here was the ultimate proof that the microcosm does reflect the macrocosm, that there is a repetition, interrelationship, interlocking between parts and whole, long surmised by classical, medieval,
Renaissance scientists, poets, mystics: the law that governs the planets and restrains the stars in their macrocosmic courses is the same law that controls the falling of a weight from the Tower of Pisa or the feather from the wing of a bird in the little world, of which man still remains the centre.’ Marjorie Hope Nicolson, The Breaking of the Circle: Studies in the Effect of the ‘New Science’ Upon Seventeenth-Century Poetry (New York: Columbia University Press, 1960), p. 155.