A Brief Guide to the Great Equations (41 page)

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.

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.

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.

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 2
^x
for any arbitrary
x
by multiplying
x
by the natural logarithm ln(2), and then exponenting: 2
^x
=
e
^x
ln(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
e
^i
π
+ 1 = 0. At first sight, it would seem to subtract from the elegance of this equation to become
e
^i
π/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.

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.

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.

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.

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