by Colin Pask
50. Simply Chaos
The last fifty years have seen a revision in our thinking about the output from deterministic systems and the way in which apparent randomness can be present in the physical world. The ideas are now widespread in various types of theories. The calculations used to explore the strange properties of the output generated by the logistic equation is a good example of the power of a relatively simple calculation. The further development and applications of the mathematics and the corresponding experiments have been extensive.
13.3 FINAL WORDS
I began this book by talking about the interplay between theory and observation and experiment. It is easy to have faith in a beautiful theory, as with Einstein and his theory of relativity; but we must always remember that experiment is the final arbiter, as Einstein also recognized when he explained that while “no amount of experimentation can ever prove me right; a single experiment can prove me wrong.”3 On the other hand, Sir Arthur Eddington (who might be called Einstein's disciple) turned things around when he suggested that “it is also a good rule not to put overmuch confidence in the observational results that are put forward until they are confirmed by theory.”4 Eddington might have had a lively debate with Fred Hoyle who said, “I don't see the logic of rejecting data just because they seem incredible.”5 Opinions may vary a little, but in the end, scientific methodology remains beautifully summarized in the words of Robert Millikan (taken from his address on receiving the 1923 Nobel Prize for his work on measuring the fundamental unit of electric charge):
Science walks forward on two feet, namely theory and experiment…. Sometimes it is one foot that is put forward first, sometimes the other, but continuous progress is only made by the use of both—by theorizing and then testing, or by finding new relations in the process of experimenting and then bringing the theoretical foot up and pushing it on beyond, and so on unending.6
I hope you can look back and see the various ways in which calculations and experiments have moved science along in cooperation.
Finally there are my ten choices for the great calculations. It may be a futile exercise, and I am sure every reader will come up with a different set. However, it does make one think about the role played by outstanding calculations in science, and I am pleased if you have been inspired to consider your own experiences. For example, some older readers might, like me, even remember using log tables at school or in college, although we may have forgotten what incredible labor-saving devices they were.
During my lifetime I have been privileged to see a return of Halley's Comet and observe a transit of Venus. On both occasions, I am sure that an appreciation of the superb background calculations enhanced the experience. Maybe we can all smile with pleasure when we read (in section 11.7.1) about Fritz Houtermans's boast to his girlfriend that he knew why the stars shine. Perhaps you too will recall events in your life that relate to some of the calculations I have described. If you have found my discussions of the calculations in some way stimulating, entertaining, and maybe even provocative, then I will feel satisfied.
CHAPTER 1: INTRODUCTION
1. Albert Einstein, “The Fundamentals of Theoretical Physics,” in Essays in Physics (New York: Philosophical Library, 1940).
2. Attributed to Kant, though it is not clear where he used these exact words. Certainly they are a good summary of the extensive discussions in his The Critique of Pure Reason (Chicago: Encyclopaedia Brittanica, 1990).
3. “Darwin, C. R. to Fawcett, Henry, 18 Sept [1861],” Darwin Correspondence Project, http://www.darwinproject.ac.uk/entry-3257 (accessed April 17, 2013).
4. Albert Einstein in a conversation reported in Werner Heisenberg, Physics and Beyond: Encounters and Conversations (London: George Allen & Unwin, 1971), p. 63.
5. Galileo in The Assayer (1623). See M. A. Finocchiaro, ed., The Essential Galileo (Indianapolis: Hackett Publishing, 2008).
6. For an introduction, see J. Baggott, Farewell to Reality: How Modern Physics Has Betrayed the Search for Scientific Truth (New York: Pegasus, 2013).
7. For a discussion and range of examples, see Monwhea Jeng, “A Selected History of Selection Bias in Physics,” American Journal of Physics 74 (2006): 578–83.
8. Thomas Malthus, Essay on the Principle of Population as It Affects the Future Improvement of Society (London: St Paul's, 1798).
9. Charles Babbage in a letter to the poet Tennyson in W. F. Bynum and R. Porter, eds., Oxford Dictionary of Scientific Quotations (Oxford: Oxford University Press, 2005), p. 34.
CHAPTER 2: ANCIENT MATHEMATICS
1. Euclid, The Thirteen Books of the Elements, trans. Sir Thomas Heath (New York: Dover, 1956).
2. Archimedes in Sir Thomas Heath, A History of Greek Mathematics, vol. 2, From Aristarchus to Diophantus (Oxford: Clarendon, 1921). See pages 50–56 for Archimedes's “Measurement of the Circle.”
3. Ibid.
4. Ibid.
5. Euclid, The Thirteen Books of the Elements.
CHAPTER 3: STEPS INTO MODERN MATHEMATICS
1. Keith Devlin, The Man of Numbers: Fibonacci's Arithmetic Revolution (London: Bloomsbury, 2011), p. 86.
2. John Napier, Mirifici Logarithmorum Canonis Descriptio (1614).
3. Ronald Calinger, A Contextual History of Mathematics (Upper Saddle River, NJ: Prentice Hall, 1999), p. 485.
4. Pierre-Simon de Laplace quoted in ibid.
5. Translation from the original Latin provided by Mark Hall.
6. See “Are These the Most Beautiful?” by David Wells in Mathematical Intelligencer 12 (1990): 37–41.
7. Pierre Simon de Laplace quoted in Robert A. Nolan, A Dictionary of Quotations in Mathematics (London: McFarland, 2002), p. 115.
8. Carl Friedrich Gauss quoted in Nolan, Dictionary of Quotations in Mathematics, p. 189.
9. Thomas Henry Huxley, “Scientific Education—Notes on an After-Dinner Speech,” Macmillan's Magazine (June 1868).
10. Howard Eves, Great Moments in Mathematics (before 1650) (Washington, DC: Mathematical Association of America, 1980), p. 116.
CHAPTER 4: OUR WORLD
1. Bishop James Ussher quoted in D. R. Oldroyd, Thinking about the Earth: A History of Ideas in Geology (Cambridge, MA: Harvard University Press, 1996), p. 49.
2. Sir William Thomson (Lord Kelvin), “On the Secular Cooling of the Earth,” Transactions of the Royal Society of Edinburgh 23 (1862).
3. Quoted in Ivan Ruddock, “Lord Kelvin, Science and Religion,” Physics World (February 2004): 56.
4. Thomas Henry Huxley, Quarterly Journal of the Geological Society, vol. 25 (1869), quoted in W. F. Bynum and R. Porter, eds., Oxford Dictionary of Scientific Quotations (Oxford: Oxford University Press, 2005), p. 530.
5. Ernest Rutherford (1904), quoted in Bynum and Porter, Oxford Dictionary of Scientific Quotations., p. 530.
6. R. D. Oldham, “On the Propagation of Earthquake Motion to Great Distances,” Philosophical Transactions of the Royal Society of London, Series A 194 (1900): 135–74.
7. R. D. Oldham, “The Constitution of the Interior of the Earth as Revealed by Earthquakes,” Quarterly Journal of the Geological Society 62 (1906).
8. Quoted in Martina Kölbl-Ebert, “Inge Lehmann's Paper: ‘P’ (1936),” Classic Papers in the History of Geology 24 (2001): 263.
9. Inge Lehmann, “Seismology in the Days of Old,” Eos Transactions American Geophysical Union 68 (1987): 33–35.
10. Quoted in Marijan Herak, “Mohorovičić Discontinuity,” Andrija Mohorovičić Memorial Rooms, http://www.gfz.hr/sobe-en/discontinuity.htm (accessed March 12, 2014).
11. Craig Jarchow and George Thompson, “The Nature of the Mohorovičić Discontinuity,” Annual Review of Earth and Planetary Science 17 (1989): 475.
12. Galileo Galilei, The Assayer (1623). See M. A. Finocchiaro, ed., The Essential Galileo (Indianapolis, IN: Hackett, 2008).
13. Galileo Galilei, Two New Sciences, trans. Stillman Drake (Madison: University of Wisconsin Press, 1974), p. 190.
14. I
bid., p. 197.
15. Ibid., p. 154.
16. Ibid., p. 77.
17. Ibid., p. 222.
18. Ibid., p. 268.
19. Ibid., p. 245.
20. Sir William Thomson (Lord Kelvin) and P. G. Tait, Principles of Mechanics and Dynamics; formerly titled Treatise on Natural Philosophy (New York: Dover, 1962). A republication of the last revised edition of 1912.
21. Bruce Parker, “The Tide Predictions for D-Day,” Physics Today 64 (September 2011): 38.
CHAPTER 5: THE SOLAR SYSTEM: THE FIRST MATHEMATICAL MODELS
1. G. J. Toomer, “Ptolemy,” in Dictionary of Scientific Biography, ed. C. C. Gillespie (New York: Charles Scribner's Sons, 1975), p. 187.
2. Ptolemy, Almagest, trans. G. J. Toomer (London: Duckworth, 1984), p. 57.
3. Isaac Newton quoted in D. T. Whiteside, “Newton's Lunar Theory: From High Hope to Disenchantment,” Vistas in Astronomy 19 (1976): 317–28.
4. From the introduction to M. Campbell-Kelly, M. Croaken, R. Flood, and E. Robson, eds., The History of Mathematical Tables (Oxford: Oxford University Press, 2003).
5. Quoted in C. M. Linton, From Eudoxus to Einstein: A History of Mathematical Astronomy (Cambridge: Cambridge University Press, 2004), p. 121.
6. Ibid., p. 126.
7. Quoted in O. Gingerich, “Kepler,” in Gillespie, Dictionary of Scientific Biography, p. 290.
8. Ibid., p. 295.
9. Ibid., p. 297.
CHAPTER 6: THE SOLAR SYSTEM: INTO THE MODERN ERA
1. Newton quoted in Colin Pask, Magnificent Principia: Exploring Isaac Newton's Masterpiece (Amherst, NY: Prometheus Books, 2013), p. 395.
2. Peter H. Cadogan, The Moon: Our Sister Planet (Cambridge: Cambridge University Press, 1981).
3. Quoted in I. B. Cohen, “Newton's Determination of the Masses and Densities of the Sun, Jupiter, Saturn, and the Earth,” Archive for the History of Exact Sciences 53 (1998): 83.
4. Daniel Defoe, A Visitation of the Plague (London: Penguin Books, 1986), originally published in London in 1722.
5. Isaac Newton, The Principia, bk. 3 (Amherst, NY: Prometheus Books, 1995), pp. 396–401.
6. Halley in a letter to Newton quoted in C. B. Waff, “Predicting the Mid-Eighteenth-Century Return of Halley's Comet,” in The General History of Astronomy, vol. 2, Planetary Astronomy from the Renaissance to the Rise of Astrophysics Part B: The Eighteenth and Nineteenth Centuries, ed. R. Taton and C. Wilson (Cambridge: Cambridge University Press, 1989), p. 70.
7. From an address by Joseph-Jérôme Lefrançois de Lalande quoted in Waff, “Predicting the Mid-Eighteenth-Century Return of Halley's Comet,” p. 69.
8. Quoted in N. R. Hanson, “Leverrier: The Zenith and Nadir of Newtonian Mechanics,” Isis 53 (1962): 361.
9. Quoted in M. Grosser, The Discovery of Neptune (New York: Dover, 1979), p. 93.
10. Quoted in A. Van Helden, “Measuring Solar Parallax: The Venus Transits of 1761 and 1769 and Their Nineteenth Century Sequels,” in The General History of Astronomy, vol. 2, Planetary Astronomy from the Renaissance to the Rise of Astrophysics Part B: The Eighteenth and Nineteenth Centuries, ed. R. Taton and C. Wilson (Cambridge: Cambridge University Press, 1989), p. 154.
11. Edmond Halley, “A New Method of Determining the Parallax of the Sun,” Philosophical Transactions 39 (1716): 454–56. Translation available at http://eclipse.gsfc.nasa.gov/transit/HalleyParallax.html (accessed April 12, 2014).
12. Data from C. M. Linton, From Eudoxus to Einstein: A History of Mathematical Astronomy (Cambridge: Cambridge University Press, 2004), p. 49 (see table 12.1).
13. A. Pais, Subtle Is the Lord: The Science and Life of Albert Einstein (Oxford: Oxford University Press, 1982), p. 253.
CHAPTER 7: THE UNIVERSE
1. Lucretius, De Rerum Natura, written around 50 BCE and translated by C. H. Sisson as The Poem on Nature (Manchester, UK: Carcanet, 2006). Reprinted with permission.
2. Digges quoted in E. R. Harrison, Cosmology: The Science of the Universe, 2nd ed. (Cambridge: Cambridge University Press, 2000), p. 492.
3. Kepler quoted in Harrison, Cosmology, p. 493.
4. Harrison, Cosmology, p. 505.
5. E. Hubble, “A Relation between the Distance and Radial Velocity among Extra-Galactic Nebulae,” Proceedings of the National Academy of Sciences 15 (1929): 168–73.
6. Isaac Newton, “General Scholium,” in The Principia, bk. 3 (Amherst, NY: Prometheus Books, 1995), p. 440.
7. J. A. Wheeler, Geons, Black Holes and Quantum Foam (New York: Norton, 2000).
8. R. J. A. Lambourne, Relativity, Gravitation and Cosmology (Cambridge: Cambridge University Press, 2010), p. 276.
9. Taken from a page of Fred Hoyle's notebook shown in Fred Hoyle: An Online Exhibition, http://www.joh.cam.ac.uk/library/special_collections/hoyle/ (accessed May 12, 2014).
10. Fred Hoyle, Astronomy and Cosmology: A Modern Course (San Francisco: W. H. Freeman, 1975), p. 402.
11. Fred Hoyle, Home Is Where the Wind Blows: Chapters from a Cosmologist's Life (Mil Valley, CA: University Science Books, 1994), p. 266.
12. Vera C. Rubin, “The Rotation of Spiral Galaxies,” Science 220, no. 4604 (1983).
13. J. P. Ostriker, P. J. E. Peebles, and A. Yahil, “The Size and Mass of Galaxies, and the Mass of the Universe,” Astrophysical Journal 193 (1974): L1–L4.
14. Vera C. Rubin, “Seeing Dark Matter in the Andromeda Galaxy,” Physics Today (December 2006).
15. John Michell, sec. 16 in “On the Means of Discovering Distance…In a Letter to Henry Cavendish,” Philosophical Transactions of the Royal Society of London 74 (1784): 35–57.
16. Freeman Dyson, “Chandaransekar's Role in 20th-Century Science,” Physics Today 63 (December 2010): 47.
CHAPTER 8: ABOUT US
1. Quoted in the biographical note preceding William Harvey, An Anatomical Disquisition on the Motion of the Heart and Blood in Animals, A Translation of the 1628 Original Exercitatio Anatomica de Motu Cordis et Sanguinis in Animalibus (Chicago: Great Books of the Western World, Encyclopedia Britannica, 1990).
2. Ibid.
3. Harvey, Anatomical Disquisition, p. 286.
4. Ibid.
5. Harvey in the dedication to his colleagues in Anatomical Disquisition.
6. Andrew Gregory, Harvey's Heart: The Discovery of Blood Circulation (Duxford, UK: Icon Books, 2001), p. 118.
7. James R. Newman in his commentary introducing the excerpt from Natural and Political Observations Mentioned in a following Index and Made upon the Bills of Mortality (published by John Graunt in London, 1662), in The World of Mathematics, vol. 3, ed. James R. Newman (London: George Allen and Unwin, 1990), p. 1416.
8. John Graunt in Natural and Political Observations Mentioned in a following Index and Made upon the Bills of Mortality (published by John Graunt in London, 1662), reproduced in The World of Mathematics, vol. 3, ed. James R. Newman (London: George Allen and Unwin, 1990), p. 1433.
9. Edmond Halley, “An Estimate of the Degrees of the Mortality of Mankind, Drawn from Curious Tables of the Births and Funerals at the City of Breslaw, with an Attempt to Ascertain the Price of Annuities upon Lives,” Transactions 17 (1693): 596–610.
10. Ibid., p. 602.
11. Quoted in Samir Okasha, “Population Genetics,” Stanford Encyclopedia of Philosophy, sec. 2, http://plato.stanford.edu/entries/population-genetics/ (accessed December 18, 2013).
12. G. H. Hardy, A Mathematician's Apology (Cambridge: Cambridge Canto Edition, 1992), p. 150.
13. George Bernard Shaw quoted in W. F. Bynum and R. Porter, eds., Oxford Dictionary of Scientific Quotations (Oxford: Oxford University Press, 2005), p. 547.
14. Geoffrey West and James Brown, “Life's Universal Scaling Laws,” Physics Today (September 2004): 38.
CHAPTER 9: LIGHT
1. Samuel Johnson quoted in James Boswell, Life of Johnson, vol. 3 (Oxford: Talboys, 1826).
2. Samuel Johnson, A Dictionary of the English Language (London: 1755).
3. Ole Roemer in “A Demonstration Concerning the Motion of Light” (1676) in Physical Thought from the Presocratics to the Quantum Physicists: An Anthology, ed. S. Sambursky (New York: Pica, 1975).
4. William Wordsworth, “My Heart Leaps Up,” in Norton Anthology of Poetry, 4th ed., ed. Margaret Ferguson, Mary Jo Salter, and John Stallworthy (New York: Norton, 1970), p. 728.
5. Isaac Newton, Opticks (New York: Dover, 1952), p. 175. A reprint of the original 4th ed. published in London in 1730.
6. John Keats, “Lamia,” in Poems of Science, ed. John Heath-Stubbs and Phillips Salman (Harmondsworth, UK: Penguin, 1984), p. 197.
7. John Donne, “An Anatomy of the World,” in Heath-Stubbs and Salman, Poems of Science, p. 79.
8. James Thomson, “To the Memory of Sir Isaac Newton,” in Heath-Stubbs and Salman, Poems of Science, p. 138.
9. Newton, Opticks, pp. 1–2.
10. Christiaan Huygens, Treatise on Light (1690). Available in many sources, such as vol. 34 of Great Books of the Western World (Chicago: Encyclopedia Britannica, 1952), chap. 1.
11. G. B. Airy, “On the Diffraction of an Object-Glass with Circular Aperture,” Transactions of the Cambridge Philosophical Society 5 (1835).
12. James Clerk Maxwell, A Treatise on Electricity and Magnetism, 3rd ed. (Oxford: Clarendon, 1904).
13. Heinrich Hertz, Electric Waves, as quoted in W. F. Bynum and R. Porter, eds., Oxford Dictionary of Scientific Quotations (Oxford: Oxford University Press, 2005), p. 279.
14. Albert Abraham Michelson, Light Waves and Their Uses, as quoted in Bynum and Porter, Oxford Dictionary of Scientific Quotations, p. 437.
15. Albert Einstein, “On a Heuristic Point of View about the Creation and Conversion of Light,” Annalen der Physik 17 (1905): 549–60. See Anna Beck, trans., The Collected Papers of Albert Einstein, vol. 2 (Princeton, NJ: Princeton University Press, 1989), p. 87.