The Music of Pythagoras

by Kitty Ferguson

  19. It was Victor Thoren who called attention to these specifics about the way Tycho carried out Pythagorean/Palladian ideals in the design of Uraniborg; see his The Lord of Uraniborg: A Biography of Tycho Brahe (Cambridge, Mass.: Cambridge University Press, 1990).

  Chapter 16: “While the morning stars sang together”

  1. Johannes Kepler, letter to Michael Mästlin, June 11, 1598, Johannes Kepler Gesammelte Werke, Max Caspar, Salther von Dyck, Franz Hammer and Volker Bialas, eds. (Munich: Deutsche Forschungsgemeinschaft and the Bavarian Academy of Sciences, 1937–), vol. XIII, p. 219.

  2. Godwin, pp. 104–105.

  3. Kepler, Harmonice mundi, Book V, in Gesammelte Werke, vol. 6, p. 289.

  4. Stephenson is an extraordinarily thorough and invaluable guide through the labyrinth of Kepler’s Harmonice mundi.

  5. Plato, Timaeus (London: Penguin, 1965), p. 15, p. 50n.

  6. Kepler, Harmonice mundi, in Gesammelte Werke, vol. 6, p. 289.

  7. For Kepler’s complete table, see p. 150 in Stephenson.

  8. For a much more detailed explanation, see Stephenson, p. 171.

  9. Kepler, Harmonice mundi, in Gesammelte Werke, vol. 6, p. 323.

  10. Kepler, Harmonice mundi, in Gesammelte Werke, vol. 6, p. 356.

  11. Stephenson points out that Kepler had read Proclus’s Platonic/neo-Pythagorean hymns.

  12. Kepler, in a letter to Vincenzo Bianchi, February 17. Letter number 827 in Gesammelte Werke 17.326.213–19. Quoted in Stephenson, p. 241.

  Chapter 17: Enlightened and Illuminated

  1. Galileo Galilei, Il Saggiatore, 1623. Quoted and translated in Daniel T. Max, The Family That Couldn’t Sleep (New York: Random House, 2006), p. 5.

  2. The episode having to do with Galileo’s father is retold in Silver, p. 176.

  3. Barrow, p. 127.

  4. Silver, p. 158.

  5. Ibid., p. 177; Bronowski (p. 234) also mentions Newton’s attribution to Pythagoras.

  6. Quoted in Barrow, p. 127.

  7. Quoted in ibid., p. 128, from G. Leibniz, The Philosophical Works of Leibniz, translated by G. Duncan (New Haven, Conn.: Tuttle, Morehouse and Taylor, 1916).

  8. Joseph Addison, paraphrase of Psalm 19:1–6. Hymn 409 in The Hymnal 1982, according to the use of the Episcopal Church.

  9. The paragraphs about how the image of Pythagoras was used by revolutionaries are based on James H. Billington, Fire in the Minds of Men: Origins of the Revolutionary Faith (New York: Basic Books, 1980). All quotations, unless otherwise noted, also come from quotations in his book.

  10. Information about Buonarroti comes from Elizabeth L. Eisenstein, The First Professional Revolutionist: Filippo Michele Buonarroti (1761–1837), A Biographical Essay (Cambridge, Mass.: Harvard University Press, 1959).

  11. From Jevons, Principles of Science, quoted in Lindberg, pp. 371–72, n. 15.

  Chapter 18: Janus Face

  1. The two books discussed in this chapter are Bertrand Russell, The History of Western Philosophy (London: George Allen & Unwin, 1945); and Arthur Koestler, The Sleepwalkers: A History of Man’s Changing Vision of the Universe (London: Hutchinson, 1959). All quotations are from these works except where otherwise footnoted.

  2. Whitehead, Alfred North, and Bertrand Russell, Principia Mathematica, 3 vols. (Cambridge, U.K.: Cambridge University Press, 1910, 1912, 1913).

  3. Aristotle, quoted in Russell (1945), p. 136.

  4. Frege’s labors were not wasted; his book is considered a classic. It is The Foundations of Arithmetic: A Logico-mathematical Enquiry into the Concept of Number, available in an edition translated by J. R. Austin (Oxford, U.K.: Blackwell, 1980).

  5. Bertrand Russell, “How to Read and Understand History,” in Understanding History and Other Essays (New York: Philosophic Library, 1957).

  6. Barrow, p. 293.

  7. Bertrand Russell, “The Value of Free Thought,” in Understanding History.

  Chapter 19: The Labyrinths of Simplicity

  1. Quoted in Kitty Ferguson, Prisons of Light: Black Holes (Cambridge, U.K.: Cambridge University Press, 1996), p. 114.

  2. Bryan Appleyard, “Master of the Universe: Will Stephen Hawking Live to Find the Secret?” Sunday Times (London)

  3. Richard Feynman, QED: The Strange Theory of Light and Matter (Princeton, N.J.: Princeton University Press, 1985), p. 4.

  4. John Archibald Wheeler, Journey into Gravity and Spacetime (New York: Scientific American Library, 1990), p. xi.

  5. Barrow, p. 129.

  6. These paragraphs about the new music of the spheres rely on information from Kristine Larsen, “From Pythagoras to WMAP: The ‘Music of the Spheres’ Revisited,” paper presented to the Society of Literature, Science, and the Arts (November 13, 2005), and published on the Internet (www.physics.ccsu.edu/larsen/wmap.html). The articles and papers cited below are all cited in Larsen’s paper.

  7. Richard A. Kerr, “Listening to the Music of the Spheres,” Science 1991, 253: 1207–1208.

  8. P. Demarque and D. B. Guenther (1999) “Helioseismology: Probing the Interior of a Star,” Proceedings of the National Academy of Sciences 96: 5356–69.

  9. ESO (May 15, 2002), “Ultrabass Sounds of the Giant Star Xi Hya.” http://www.eso.org/outreach/press-rel/pr-2002/pr-10-02.html. The ESO is the European Organization for Astronomical Research in the Southern Hemisphere, or European Southern Observatory.

  10. Marcia Bartusiak, Einstein’s Unfinished Symphony: Listening to the Sounds of Space-time (Washington, D.C.: National Academies Press, 2000).

  11. Steve Roy and Megan Watzke, “Giant Galaxy’s Violent Past Comes into Focus,” Harvard University press release, May 10, 2004. http://chandra.harvard.edu/press/04_releases/press_051004.html

  12. Don Savage, Steve Roy, and Megan Watzke, “Chandra ‘Hears’ a Black Hole for the First Time,” Harvard University press release, September 9, 2003. http://chandra.harvard.edu/press/03_releases/press_090903.html

  13. Mark Whittle, “Sounds from the Infant Universe.” Abstract for American Astronomical Society talk, June 3, 2004. http://www.astro.virginia.edu/-dmw8f/sounds/aas/aas_abs.pdf

  14. Mark Whittle, “Primordial Sounds: Big Bang Acoustics,” press release: American Astronomical Society Meeting, June 1, 2004. http://www.astro.virginia.edu/-dmw8f/sounds/aas/press_release.pdf

  15. Shaun Cole et al. (August 5, 2005), “The 2dF Galaxy Redshift Survey: Power-Spectrum Analysis of the Final Dataset and Cosmological Implications,” arXiv: astro-ph/0501174; Daniel J. Eisenstein et al. (January 10, 2005), “Detection of the Baryon Acoustic Peak in the Large-scale Correlation Function of SDSS Luminous Red Galaxies.” arXiv: astro-ph/0501171.

  16. Ron Cohen, “Ultimate Retro: Modern Echoes of the Early Universe.” Science News Online 167(3), Jan. 15, 2005. http://www.sciencenews.org/articles/20050115/fob1.asp

  17. Diane Richards, “Listening to Northern Lights,” Astronomy, Dec. 2001, p. 63.

  Appendix

  1. Bronowski, pp. 158–160.

  Bibliography

  Books I have used and that appear in the endnotes.

  Al-Hasan Al-Katib. Kitah Kamal Adal Al-Gina’. Translated by Joscelyn Godwin, from a previous French translation by Amnon Shiloah and excerpted in Godwin, Joscelyn, ed. The Harmony of the Spheres: A Sourcebook of the Pythagorean Tradition in Music. Rochester, Vermont: Inner Traditions International, 1993. P. 122.

  Alberti, Leon Battista. The Ten Books of Architecture. Dover Replica Facsimile Edition. Mineola, N.Y.: Dover, 1987.

  Appleyard, Bryan. “Master of the Universe: Will Stephen Hawking Live to Find the Secret?” Sunday Times (London), 1988.

  Augustine. City of God. Translated by Gerald Walsh et al. Garden City, N.Y.: Doubleday, 1958.

  Aulus Gellius. Attic Nights. Books I–V. Translator J. C. Rolfe. Cambridge, Mass.: Harvard University Press, 1927.

  Aurelian of Réôme. Musica Disciplina. Translated by Joseph Ponte. Colorado Springs: Colorado College Music Press, 1968. Pp. 20–23.

  Baqir, Taha. “An Im
portant Mathematical Problem Text from Tell Harmal.” Sumer 6 (1950).

  Barnes, Jonathan. Early Greek Philosophy. London: Penguin, 1987.

  Barrow, John. Pi in the Sky: Counting, Thinking, and Being. Oxford, U.K.: Clarendon Press, 1992.

  Bartusiak, Marcia. Einstein’s Unfinished Symphony: Listening to the Sounds of Space-time. Washington, D.C.: National Academies Press, 2000.

  Billington, James H. Fire in the Minds of Men: Origins of the Revolutionary Faith. New York: Basic Books, 1980.

  Blumenthal, H. J., and R. A. Markus, eds. Neo-Platonism and Early Christian Thought: Essays in Honour of A. H. Armstrong. London: Variorum Publications, 1981.

  Bronowski, Jacob. The Ascent of Man. Boston: Little, Brown, & Co., 1973.

  Brooke, John Hedley. Science and Religion: Some Historical Perspectives. Cambridge, U.K.: Cambridge University Press, 1991.

  Burkert, Walter. Lore and Science in Ancient Pythagoreanism. Cambridge, Mass.: Harvard University Press, 1972.

  Buxton, Richard, ed. From Myth to Reason: Studies in the Development of Greek Thought. Oxford, U.K.: Oxford University Press, 1999.

  Capparelli, Vincenzo. La sapienza di Pitagora. Padua: CEDAM, 1941.

  Cartledge, Paul, ed. Cambridge Illustrated History of Ancient Greece. Cambridge, U.K.: Cambridge University Press, 1998.

  Caspar, Max. Kepler. Translated and edited by C. Doris Hellman. Reissue, with references by Owen Gingerich and bibliographical citations by Gingerich and Alain Segonds. New York: Dover, 1993.

  Cassirer, Ernst, Paul Oskar Kristeller, and John Herman Randall, Jr., eds. The Renaissance Philosophy of Man: Selections in Translation. Chicago and London, University of Chicago Press, 1948, 1969.

  Centrone, Bruno. “Platonism and Pythagoreanism in the Early Empire.” In Christopher Rowe and Malcolm Schofield, eds. Cambridge History of Greek and Roman Political Thought. Cambridge, U.K.: Cambridge University Press, 2000.

  Cesariano. Drawing for Vitruvius: ccat.sas.upenn.edu/george/vitruvius.html.

  Cicero, Marcus Tullius. On the Commonwealth. Translated by George Holland Sabine and Stanley Barney Smith. New York: The Liberal Arts Press, 1929.

  ———. Cicero: The Republic and The Laws. Niall Rudd, translator. Oxford, U.K.: Oxford University Press, 1998.

  ———. “Scipio’s Dream.” In On the Republic. Translated by Cyrus R. Edmonds and Moses Hadas. In The Basic Works of Cicero. New York: Random House, Modern Library, 1951.

  Cohen, Ron. “Ultimate Retro: Modern Echoes of the Early Universe.” Science News Online 167 (3), Jan. 15, 2005. http://www.sciencenews.org/articles/20050115/fob1.asp

  Cole, Shaun, et al. August 5, 2005. “The 2dF Galaxy Redshift Survey: Power-spectrum Analysis of the Final Dataset and Cosmological Implications.” arXiv: astro-ph/0501174.

  Copernicus, Nicolaus. De Revolutionibus. Gesamtausgabe. Vol. II, De Revolutionibus. Kritischer Text. Eds. H. M. Nobis and B. Sticker. Hildesheim, Germany, 1984.

  Cuomo, S. Ancient Mathematics. London: Routledge, 2001.

  Demarque, P., and D. B. Guenther (1999) “Helioseismology: Probing the Interior of a Star.” Proceedings of the National Academy of Sciences 96: 5356–69.

  Diogenes Laertius. The Life of Pythagoras. Translated and reprinted in Kenneth Sylvan Guthrie. The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings Which Relate to Pythagoras and Pythagorean Philosophy. Grand Rapids: Phanes Press, 1987.

  Dodds, E. R. The Greeks and the Irrational. Berkeley, Calif.: University of California Press, 1951.

  Eisenstein, Daniel J. et al. January 10, 2005. “Detection of the Baryon Acoustic Peak in the Large-scale Correlation Function of SDSS Luminous Red Galaxies.” arXiv: astro-ph/0501171

  Eisenstein, Elizabeth L. The First Professional Revolutionist: Filippo Michele Buonarroti (1761–1837), a Biographical Essay. Cambridge, Mass.: Harvard University Press, 1959.

  The Epistle on Music of the Ikhwan al-Safa’. Translated by Amnon Shiloah, in Joscelyn Godwin, ed., The Harmony of the Spheres: A Sourcebook of the Pythagorean Tradition in Music. Rochester, Vt.: Inner Traditions International, 1993, p. 113.

  ESO (European Organization for Astronomical Research in the Southern Hemisphere, or European Southern Observatory) (May 15, 2002) “Ultrabass Sounds of the Giant Star Xi Hya.” http://www.eso.org/outreach/press-rel/pr-2002/pr-10-02.html

  Ferguson, Kitty. Prisons of Light: Black Holes. Cambridge, U.K.: Cambridge University Press, 1996.

  ———. Tycho and Kepler: The Unlikely Partnership That Forever Changed Our Understanding of the Heavens. New York: Walker, 2002.

  Feynman, Richard. QED: The Strange Theory of Light and Matter. Princeton, N.J.: Princeton University Press, 1985.

  Fox, Robin Lane. Pagans and Christians in the Mediterranean World from the Second Century A.D. to the Conversion of Constantine. London: Penguin Books, 1986.

  Frege, Gottlob. The Foundations of Arithmetic: A Logico-mathematical Enquiry into the Concept of Number. Translated by J. R. Austin. Oxford, U.K.: Blackwell, 1980.

  Godwin, Joscelyn, ed. The Harmony of the Spheres: A Sourcebook of the Pythagorean Tradition in Music. Rochester, Vt.: Inner Traditions International, 1993.

  Grant, Michael. History of Rome. London: Faber & Faber, 1978.

  Guthrie, Kenneth Sylvan. The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings Which Relate to Pythagoras and Pythagorean Philosophy. York Town, ME: Phanes Press, 1987. Contains Iamblichus’, Porphyry’s and Diogenes Laertius’ biographies of Pythagoras as well as some of the pseudo-Pythagorean works.

  Guthrie, William Keith Chambers. The Earlier Presocratics and the Pythagoreans. Vol 1 of A History of Greek Philosophy. Cambridge, U.K.: Cambridge University Press, 1962, 2003.

  ———. The Greeks and Their Gods. Cambridge, U.K.: Cambridge University Press, 1951.

  Hammond, N. G. L. A History of Greece to 322 B.C. Oxford, U.K.: Oxford University Press, 1986.

  Heath, Thomas L. The Thirteen Books of Euclid’s Elements I. Cambridge, U.K.: Cambridge University Press, 1926.

  Heath, Thomas L. Greek Astronomy. London: J. M. Dent & Sons, 1932.

  Hornblower, Simon, and Antony Spawforth, eds. The Oxford Companion to Classical Civilization. Oxford, U.K.: Oxford University Press, 1998.

  Hunayn. Nawadir al-Falasifa. Excerpts translated by Isaiah Sonne, in Joscelyn Godwin, The Harmony of the Spheres: A Sourcebook of the Pythagorean Tradition in Music. Rochester, Vt.: Inner Traditions International, 1993. Pp. 92–98.

  Iamblichus of Chalcis. Life of Pythagoras. Translated by Thomas Taylor. In Thomas Taylor. Iamblichus’ Life of Pythagoras. Rochester, Vt.: Inner Traditions International, 1986.

  ———. On Common Mathematical Knowledge 91.3–11. Translation in I. Mueller, “Mathematics and Philosophy in Proclus’s Commentary on Book I of Euclid’s Elements.” In J. Pépin and H. D. Saffrey, eds. Proclus, Lecteur et Interprète des Anciens. Paris: CNRS, 1987.

  John Scotus Eriugena. Commentary on Martianus Capella. Translated by Joscelyn Godwin from a ninth-century Bodleian Library manuscript and excerpted in Godwin, Joscelyn, ed. The Harmony of the Spheres: A Sourcebook of the Pythagorean Tradition in Music. Rochester, Vt.: Inner Traditions International, 1993. P. 105.

  Kahn, Charles H. Pythagoras and the Pythagoreans: A Brief History. Indianapolis: Hackett, 2001.

  Kepler, Johannes. Harmonice mundi. Translated by Eric J. Aiton, A. M. Duncan, and J. V. Field as Five Books of the Harmony of the World. Philadelphia: 1993.

  ———. Johannes Kepler Gesammelte Werke. Max Caspar, Salther von Dyck, Franz Hammer, and Volker Bialas, eds. Munich: Deutsche Forschungsgemeinschaft and the Bavarian Academy of Sciences, 1937–.

  Kerr, Richard A. “Listening to the Music of the Spheres.” Science 1991, 253: 1207–1208.

  King, Ross. Brunelleschi’s Dome: How a Renaissance Genius Reinvented Architecture. London: Penguin, 2000.

  Koenigsberger, H. G. Medieval Europe: 400–1500. London: Longman, 1987.

  Koestler, Arthur. The S
leepwalkers: A History of Man’s Changing Vision of the Universe. London: Hutchinson, 1959.

  Kuhn, Thomas S. The Copernican Revolution: Planetary Astronomy in the Development of Western Thought. Cambridge, Mass.: Harvard University Press, 1957.

  ———. The Structure of Scientific Revolutions. Chicago: University of Chicago Press, 1962.

  Larsen, Kristine. “From Pythagoras to WMAP: The ‘Music of the Spheres’ Revisited.” Paper presented to the Society of Literature, Science, and the Arts, November 13, 2005, and published on the Internet. www.physics.ccsu.edu/larsen/wmap.html

  Lee, Desmond. Introduction to Plato’s Timaeus. London: Penguin, 1965.

  Leibniz, G. The Philosophical Works of Leibniz. Translated by G. Duncan. New Haven, Conn.: Tuttle, Morehouse and Taylor, 1916.

  Lindberg, David C. 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.

  Max, Daniel T. The Family That Couldn’t Sleep. New York: Random House, 2006.

  McInery, Ralph. A History of Western Philosophy. Vol. 2. From the website of the University of Notre Dame Jacques Maritain Center. http://maritain.nd.edu

  Oates, Joan. Babylon. London: Thames & Hudson, 1979.

  Palladio, Andrea. The Four Books on Architecture. New York: Dover, 1978. Replica of the Isaac Ware English edition (1738) of I quattro libri dell’ architettura.

  Philostratus. Life of Apollonius of Tyana. Translated by F. C. Conybeare. Cambridge, Mass.: Harvard University Press, 1912.

  Pieper, Josef. Scholasticism: Personalities and Problems of Medieval Philosophy. Translated by Richard and Clara Winston. New York: McGraw-Hill, 1964.

  Plato. Gorgias. Translated by Chris Emlyn-Jones and Walter Hamilton. London: Penguin, 2004.

  ———. Protagoras and Meno. Translated by Robert C. Bartlett. London: Penguin, 1956.

  ———. The Republic. Translated by Desmond Lee. London: Penguin, 2003.

  ———. Timaeus. Stephanus edition. 1578.