A Brief Guide to the Great Equations

by Robert Crease

  22 Westminster, 1728. The poem is discussed in I. Bernard Cohen, Science and the Founding Fathers: Science in the Political Thought of Thomas Jefferson, Benjamin Franklin, John Adams, and James Madison (1995), pp. 285–87.

  23 H. Saint-Simon, in Henri Saint-Simon: Selected Writings, ed. K. Taylor (London: Croom Helm, 1975), pp. 78–79.

  INTERLUDE: That Apple

  1 See D. McKie and G. R. de Beer, ‘Newton’s Apple’, Notes and Records of the Royal Society of London 9 (1951), pp. 46–54.

  2 Westfall, Never at Rest: A Biography of Isaac Newton (New York: Cambridge University Press, 1988), p. 155.

  3 William Stukeley, Memoirs of Sir Isaac Newton’s Life, ed. A. Hastings White (London: Taylor and Francis, 1936), pp. 19–20.

  4 E. N. da C. Andrade, Sir Isaac Newton, His Life and Work (New York: Doubleday Anchor, 1950), p. 35.

  5 I. Bernard Cohen, ‘Newton’s Discovery of Gravity’, Scientific American, March 1981, p. 167.

  4 ‘The Gold Standard for Mathematical Beauty’: Euler’s Equation

  1 Ed Leibowitz, ‘The Accidental Ecoterrorist’, Los Angeles magazine, May 2005, pp. 100–105, 198–201.

  2 Quoted in Carl A. Boyer, A History of Mathematics (Princeton: Princeton University Press, 1985), p. 482.

  3 Marquis de Condorcet, ‘Eloge to Mr. Euler’, trans. J. Glaus, www.groups.des.st-and.ac.uk/~history/Extras/Euler_elogium.html.

  4 Martin Gardner, The Unexpected Hanging and Other Mathematical Diversions (Chicago, University of Chicago Press, 1961) has an excellent chapter (3) on e.

  5 R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics, vol. 1 (New York: Addison-Wesley, 1963) has excellent sections (22-5 and 22-6) on imaginary numbers and imaginary exponents.

  6 Quoted in Boyer, History of Mathematics, p. 493.

  7 Condorcet, ‘Eloge to Mr. Euler.’

  8 David M. Burton, The History of Mathematics (New York: McGrawHill, 1985), p. 503.

  9 We can retrieve 2x for any arbitrary x by multiplying x by the natural logarithm ln(2), and then exponenting: 2x = exln(2).

  10 Leonhard Euler, Introduction to Analysis of the Infinite, book 1, trans. J. D. Blanton (New York: Springer, 1988), p. 112. Euler first published this in Miscellanea Berolinensia 7 (1743), p. 179.

  11 G. H. Hardy, P. V. Seshu Aiyar, and B. M. Wilson, eds., Collected Papers of Srinivasa Ramanujan (New York: Chelsea Publishing Company, 1962), p. xi.

  12 This wonderful way of representing Euler’s formula is presented in L.W.H. Hull’s note, ‘Convergence on the Argand Diagram’, Mathematical Gazette 43 (1959), pp. 205–7. Many thanks to George W. Hart for pointing this out, and for suggesting the different fonts.

  13 Herbert Turnbull, quoting Felix Klein, ‘The Great Mathematicians’, in The World of Mathematics, vol. 1, ed. James R. Neuman (New York: Simon and Schuster, 1956), p. 151.

  14 Yet this is not the most general expression. Mathematicians have sometimes argued, for instance, whether π is defined most economically. That is, given all the 2πs found in maths and science, and the vast simplification that results by making π radians the length around a unit circle, are there not beauties and economies to making the fundamental constant here the ratio of the circumference to the radius? To put it another way, are there any examples of places where the beauties and economies lie with π? The most obvious candidate is eiπ + 1 = 0. At first sight, it would seem to subtract from the elegance of this equation to become eiπ/2 + 1 = 0. Yet mathematicians have discovered a twist. Suppose we use the symbol ψ to designate 2π. Then we can write a more beautiful and economical formula, of which Euler’s formula is just a special case: This is more general, because one of the square roots of 1 is 11. Euler’s formula is a special case of this equation similar to the way that the Pythagorean theorem is a special case of the law of cosines.

  INTERLUDE: Equations as Icons

  1 Larry Wilmore, quoted in The New York Times, April 15, 2007, section 4, p. 4.

  2 Len Fisher, ‘Equations for Everyday Living’, New Scientist, July 30, 2005; Simon Singh, ‘Lies, Damn Lies and PR’, New Scientist, August 20, 2005.

  3 For a discussion of this point, see William Steinhoff, George Orwell and the Origins of 1984 (Ann Arbor: University of Michigan Press, 1975), chapter XII.

  4 Eugene Lyons, writing about the Soviet Union’s first Five Year Plan, quoted in Steinhoff, George Orwell and the Origins of 1984, p. 172.

  5 Quoted in Robert A. Orsi, ‘2 + 2 = 5’, American Scholar 76 (Spring 2007), pp. 34–43.

  5 The Scientific Equivalent of Shakespeare: The Second Law of Thermodynamics

  1 Maxwell to Lord Rayleigh, 1870. James Clerk Maxwell, The Scientific Letters and Papers of James Clerk Maxwell, Vol. II: 1862–1873 (Cambridge: Cambridge University Press, 1995), p. 583.

  2 Wilhelm Wien, ‘A New Relationship Between the Radiation from a Black Body and the Second Law of Thermodynamics’, in Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 1893 pp. 55–62 at p. 62.

  3 Max Planck, ‘On an Improvement of the Wien’s Law of Radiation’, Verhandl. Dtsch. Phys. Ges. 2 (1900), p. 202.

  4 Kelvin, ‘Nineteenth Century Clouds over the Dynamical Theory of Heat and Light’, in Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light (London: Cambridge University Press, 1904), pp. 486–527.

  6 ‘The Most Significant Event of the Nineteenth Century’: Maxwell’s Equations

  1 P. M. Harman, ed., The Scientific Letters and Papers of James Clerk Maxwell, vol. 1 (Cambridge: Cambridge University Press, 1990), p. 254.

  2 James Clerk Maxwell, A Treatise on Electricity and Magnetism (New York: Dover, 1954), p. ix.

  3 William Thomson, Kelvin’s Baltimore Lectures and Modern Theoretical Physics, ed. R. H. Kargon and P. Achinstein (Cambridge: MIT Press, 1987), p. 206.

  4 J. C. Maxwell, ‘Essay for the Apostles on ‘Analogies in Nature’, ‘ in The Scientific Letters and Papers of James Clerk Maxwell, vol. 1, ed. P. M. Harman (Cambridge: Cambridge University Press, 1990), pp. 376–83.

  5 ‘On Faraday’s Lines of Force’, in The Scientific Papers of James Clerk Maxwell, vol. 1, ed. W. D. Niven (New York: Dover, 1965), pp. 155–229.

  6 Maxwell, The Scientific Papers, p. 207.

  7 In a Letter from M. Faraday to J. Maxwell, March 25, 1857, cited in Maxwell, Scientific Letters and Papers, p. 548.

  8 ‘On Physical Lines of Force’, in The Scientific Papers, p. 500.

  9 Maxwell, The Scientific Papers, p. 533.

  10 In 1868, Maxwell wrote a short paper, ‘A Note on the Electromagnetic Theory of Light’ (Scientific Papers II, pp. 137–43), in which he admits that in his previous work on electromagnetic phenomena the connection to light was ‘not easily understood when taken by itself’, and he restates the connection in ‘the simplest form’, in the form of four theorems – but these are not yet ‘Maxwell’s equations.’

  11 Cited in Dorothy M. Livingston, The Master of Light (New York: Scribner’s 1973), p. 100.

  12 J. Clerk Maxwell, ‘On a Possible Mode of Detecting a Motion of the Solar System through the Luminiferous Ether’, Nature 21, January 29, 1880, pp. 314–15.

  13 A good account of this meeting is given in B. J. Hunt, The Maxwellians (Ithaca: Cornell University Press, 1991), chapter 7.

  14 Quoted in E. T. Bell, Men of Mathematics (New York: Simon and Schuster, 1937), p. 16.

  15 Albert A. Michelson, ‘The Relative Motion of the Earth and the Luminiferous Ether’, American Journal of Science 22 (1881), p. 120.

  16 Quoted in Livingston, Master of Light, p. 77.

  17 D.S.L. Cardwell, The Organization of Science in England (London: Heinemann), p. 124n.

  18 Of special importance was the flux theorem. ‘At a time when work on Maxwell’s theory could easily have wandered off into purely mathematical elaborations, the discovery of the energy flux theorem focused attention firmly on the physical state of the field.’ Hunt, The Maxwellians, p. 109.

  19 Oliver
Heaviside, Electromagnetic Theory, vol. 1 (New York: Chelsea, 1971), p. vii.

  20 Oliver Heaviside, Electrical Papers, vol. 2 (New York: Chelsea, 1970), p. 525.

  21 Hunt, The Maxwellians, p. 122.

  22 Those are taken from the Appendix in Hunt’s The Maxwellians, ‘From Maxwell’s Equations to ‘Maxwell’s Equations’, ‘ p. 247.

  23 Heaviside, ‘On the Metaphysical Nature of the Propagation of Potentials’, Electrical Papers, vol. 2, pp. 483–85.

  24 Hunt, The Maxwellians, p. 128.

  7 Celebrity Equation: E=mc2

  1 Dalai Lama, The Universe in a Single Atom: The Convergence of Science and Spirituality (New York: Morgan Road, 2005), p. 59.

  2 Luce Irigaray, Parler n’est jamais neutre (Paris: Editions de Minuit, 1987), p. 110.

  3 For a brief discussion of some Maxwell ‘modifiers’, see Alfred Bork, ‘Physics Just Before Einstein’, Science 152 (1966), pp. 597–603.

  4 G. FitzGerald, ‘The Ether and the Earth’s Atmosphere’, Science 13 (1889), p. 390.

  5 Lorentz to Rayleigh, August 18, 1892, cited in John S. Rigden, Einstein 1905: The Standard of Greatness (Cambridge, MA: Harvard University Press, 2005), p. 82.

  6 G. FitzGerald to H. Lorentz, November 14, 1894, quoted in Abraham Pais, ‘Subtle Is the Lord’: The Science and Life of Albert Einstein (New York: Oxford, 1982), p. 124.

  7 One physicist I know remembers that time dilates in a rest frame by thinking the following: ‘Cosmic rays reach earth.’ In the rest frame, that is, cosmic rays have a lifetime that ordinarily is too short for them to travel long distances. But because from earth’s point of view they are moving at speeds close to the speed of light, time is dilated for them long enough for them to reach the ground.

  8 Arthur Eddington, ‘Gravitation and the Principle of Relativity’, Nature, vol. 101, 1918, pp. 15–17 (quote appears on p. 16).

  9 Pais, ‘Subtle Is the Lord’, p. 128.

  10 Carl Seeling to Einstein, March 11, 1952. Quoted in Ronald W. Clark, Einstein: The Life and Times (New York: World Publishing, 1971), p. 84.

  11 P. A. Schilpp, ed., Albert Einstein: Philosopher-Scientist (London: Cambridge University Press, 1970), p. 53.

  12 Quoted in Clark, Einstein, p. 84.

  13 Quoted in Pais, ‘Subtle Is the Lord’, p. 139.

  14 Emilio Segre, From X-rays to Quarks (New York: Dover, 1980), p. 84.

  15 A. Einstein, ‘On the Electrodynamics of Moving Bodies’, Annalen der Physik 17 (1905), pp. 891–921, in Albert Einstein, The Collected Papers of Albert Einstein, Vol. 2. The Swiss Years: Writings, 1900– 1909, trans. A. Beck (Princeton: Princeton University Press, 1989), doc. 23, pp. 140–71.

  16 Einstein to Conrad Habicht, June 30, 1905. In Collected Papers, vol. 5, pp. 20–21.

  17 Rigden, Einstein 1905, p. 112.

  18 Einstein, Collected Papers, vol. 2, doc. 24, p. 174.

  19 Abraham Pais writes of Einstein’s achievement: ‘In physics the great novelties were, first, that the recording of measurements of space intervals and time durations demanded more detailed specifications than were held necessary theretofore and, second, that the lessons of classical physics are correct only in the limit v/c << 1. In chemistry the great novelty was that Lavoisier’s law of mass conservation and Dalton’s rule of simply proportionate weights were only approximate but nevertheless so good that no perceptible changes in conventional chemistry were called for. Thus relativity turned Newtonian mechanics and classical chemistry into approximate sciences, not diminished but better defined in the process.’ Pais, ‘Subtle Is the Lord’, p. 163.

  20 This is, again, a requirement of objectivity or covariance that drives a new wedge between ordinary notions of objectivity and scientific ones.

  21 A. Einstein, ‘The Principle of Conservation of Motion of the Centre of Gravity and the Inertia of Energy’, Annalen der Physik 20 (1906), pp. 627–33, in Collected Works, vol. 2, pp. 200–206.

  22 A. Einstein, ‘On the Inertia of Energy Required by the Relativity Principle’, Annalen der Physik 23 (1907), pp. 371–84, in Collected Works, vol. 2, p. 249.

  23 A. Einstein, ‘On the Relativity Principle and the Conclusions Drawn from it’, Collected Works, vol. 2, pp. 286–87.

  24 A. Einstein, Einstein’s 1912 Manuscript on the Special Theory of Relativity (New York: Braziller, 1996), pp. 102–3, 109.

  25 ‘A. Einstein, E = mc2: The Most Urgent Problem of Our Time’, Science Illustrated (April 1946), pp. 16–17.

  26 M. Planck, quoted in Einstein, Collected Works, vol. 2, p. 287.

  27 Clark, Einstein, p. 101.

  28 Niels Bohr, Nature, February 29, 1936.

  29 Abraham Pais, J. Robert Oppenheimer: A Life, with supplemental material by Robert P. Crease (New York: Oxford, 2006), p. 44.

  30 The New York Times, August 7, 1945, p. 1.

  31 Henry D. Smyth, Atomic Energy for Military Purposes: The Official Report on the Development of the Atomic Bomb under the Auspices of the United States Government, 1940–1945 (Princeton: Princeton University Press, 1945).

  32 ‘A. Einstein, E = mc2’, Science Illustrated.

  INTERLUDE: Crazy Ideas

  1 Jeremy Bernstein, Science Observed (New York: Basic Books, 1982), p. 310.

  2 Bernstein once noted several characteristics of crackpots. They insist that their work has solved everything, they are humorless, they are sure everyone is out to steal their ideas, they are sure the media will be interested, they use a lot of capital letters. ‘Scientific Cranks’, in Science Observed, ch. 14.

  8 The Golden Egg: Einstein’s Equation for General Relativity

  1 Times of London, November 8, 1919, p. 1.

  2 Quoted in Abraham Pais, ‘Subtle Is the Lord’: The Science and Life of Albert Einstein (New York: Oxford, 1982), p. 124.

  3 A. N. Whitehead, Science and the Modern World (New York: Macmillan, 1954), p. 13.

  4 Quoted in Pais, ‘Subtle Is the Lord’, p. 179.

  5 Ibid., p. 178.

  6 A. Einstein, The Collected Works of Albert Einstein, vol. 2, trans. A. Beck (Princeton: Princeton University Press, 1989), pp. 301–2.

  7 Ibid., p. 310.

  8 A. Einstein to C. Habicht, December 24, 1907, in Collected Works, vol. 5, p. 47.

  9 Quoted in Ronald W. Clark, Einstein: The Life and Times (New York: World Publishing, 1971), p. 120.

  10 Emilio Segre, From X-rays to Quarks (New York: Dover, 1980), p. 85.

  11 Quoted in Pais, ‘Subtle Is the Lord’, p. 152.

  12 A. Einstein, ‘On the Influence of Gravitation on the Propagation of Light’, Annalen der Physik 35 (1911), pp. 898–908, in Collected Works, vol. 3, p. 379.

  13 A. Einstein to Willem Julius, August 24, 1911, in Collected Works, vol. 5, p. 199.

  14 A. Einstein to E. Freundlich, September 1, 1911, in ibid., p. 202.

  15 J. Earman and C. Glymour, ‘Relativity and Eclipses: The British Eclipse Expeditions of 1919 and their Predecessors’, Historical Studies in the Physical Sciences 11 (1980), p. 61.

  16 A. Einstein to E. Mach, June 25, 1913, in Collected Works, vol. 5, p. 340.

  17 Quoted in Pais, ‘Subtle Is the Lord’, p. 311.

  18 Ibid., p. 212.

  19 A. Einstein to L. Hopf, August 16, 1912, in Collected Works, vol. 5, p. 321.

  20 A. Einstein to A. Sommerfeld, October 29, 1912, in Collected Works, vol. 5, p. 324.

  21 Zeitschrift für Mathematik und Physik 62 (1913), pp. 225–61. The ‘hair’s breadth’ remark is from John Norton, ‘How Einstein Found His Field Equations: 1912–1915’, Historical Studies in the Physical Sciences 14:2 (1984), pp. 253–316.

  22 A. Einstein to H. Lorentz, August 16, 1913, in Collected Works, vol. 5, p. 352.

  23 Quoted in Clark, Einstein, p. 173.

  24 Ibid., p. 199.

  25 A. Einstein to A. Sommerfeld, November 28, 1915, in Collected Works, vol. 8, p. 152.

  26 Pais, ‘Subtle Is the Lord’, p. 253.

  27 A. Einstein to H. Lorentz, January 16, 1915, in
Collected Works, vol. 8, p. 179.

  28 Quoted in Pais, ‘Subtle Is the Lord’, p. 253.

  29 A. Einstein, ‘Explanation of the Perihelion Motion of Mercury from the General Theory of Relativity’, November 18, 1915, in Collected Works, vol. 6, p. 113.

  30 Ibid., p. 117.

  31 Independently, mathematician David Hilbert produced a similar equation.

  32 Quoted in Clark, Einstein, p. 200.

  33 A. Einstein to H. Lorentz, January 17, 1916, in Collected Works, vol. 8, p. 179.

  34 And the following year, in a paper called ‘Cosmological Considerations in the General Theory of Relativity’, Einstein tinkered with his basic field equation. He had noted that it seemed to suggest that the universe is expanding, so he subtracted from the left-hand side of the equation (Gμν) another tensor gμν, multiplied by a constant λ, whose value, he admitted, was ‘at present unknown.’ The result kept the general covariance, as well as what he evidently assumed was a finite universe. This turned his field equation

  into

  Einstein introduced this factor – the now-famous cosmological constant – purely as a fudge factor, to save what he thought was a prediction of his theory that the universe was expanding. Within a few years he would begin to question the necessity of this concept, and in 1931 removed the constant λ from the theory for good, later calling this fudge the ‘biggest blunder’ of his life. Seventy years later, to explain data from measurements of supernovae, astronomers restored it.