Pythagorus
Page 39
Number 41 in The Catalogue of Lamprias, a list of Plutarch’s works that probably was compiled in the fourth century A.D.
This was how Porphyry reported Moderatus’ views: see Kahn, p. 105.
As quoted and/or paraphrased by Porphyry: see Kahn, p. 106.
Theon of Smyrna, Mathematics Useful for Understanding Plato, excerpted as Appendix 1 in K. S. Guthrie.
W. K. C. Guthrie (2003), p. 406.
Proclus, The Elements of Theology, revised and edited by E.R. Dodds (Oxford: Oxford University Press, 1963), p. 259.
Numenius fragment #2; quoted in Kahn, p. 122.
Numenius fragment, no number; quoted in Kahn, p. 122.
Numenius fragment #52; quoted in Kahn, p. 132.
Johannes Kepler, Johannes Kepler Gesammelte Werke, Max Caspar et al., eds. (Munich: Deutsche Forschungsgemeinschaft, and the Bavarian Academy of Sciences, 1937– ), vol. 6, p. 289.
Author’s rendition of Pliny, Natural History, 2:20, using translation by Bruce Stephenson, p. 24, in Stephenson The Music of the Heavens: Kepler’s Harmonic Astronomy (Princeton, N.J.: Princeton University Press, 1994).
Stephenson, p. 29. Stephenson cites Von Jan, ‘Die Harmonie der Sphären’, Philologus 52 (1893).
Simplified from Stephenson, p. 31. Piano notes are author’s addition.
Stephenson, p. 37.
Chapter 13: The Wrap-up of Antiquity
E. R. Dodds, The Greeks and the Irrational (Berkeley, Calif.: University of California Press, 1951), p. 296.
Ibid., p. 285.
Description of Rome in this era is based on Michael Grant, History of Rome (London: Faber & Faber, 1978), p. 284 ff.
Edward Gibbon, Decline and Fall of the Roman Empire, Chapter X. Quoted in Russell (1945), p. 287.
Description taken from Grant, p. 294.
Ibid.
Plotinus quoted in Dodds, pp. 285–86.
Ibid., pp. 286–87.
Kahn, p. 134.
Dodds, p. 287.
Ibid.
Fox, p. 190.
John 1:1–5, 9–12, 14. Paraphrased from the Oxford New International Version translation.
Augustine, City of God, translated by Gerald Walsh et al. (Garden City, N.Y.: Doubleday, 1958), pp. 241–42.
H. J. Blumenthal and R. A. Markus, eds., Neo-Platonism and Early Christian Thought: Essays in Honour of A. H. Armstrong (London: Variorum Publications, 1981), p. 90. This concept is the ‘soma-sema formula’.
Information about this period comes from H. G. Koenigsberger, Medieval Europe: 400–1500 (London: Longman, 1987).
Most of the information about Macrobius is from Stephenson, pp. 38–41.
Quoted in Richard E. Rubenstein, Aristotle’s Children: How Christians, Muslims, and Jews Rediscovered Ancient Wisdom and Illuminated the Dark Ages (New York: Harcourt, 2003), p. 62, from Josef Pieper, Scholasticism: Personalities and Problems of Medieval Philosophy, translated by Richard and Clara Winston (New York: McGraw-Hill, 1964), p. 30.
Chapter 14: ‘Dwarfs on the Shoulders of Giants’
For information about the Pythagorean tradition as it emerged in the Islamic world, I have relied on Joscelyn Godwin, The Harmony of the Spheres: A Sourcebook of the Pythagorean Tradition in Music (Rochester, Vt.: Inner Traditions International, 1993).
Quotations from Hunayn’s Nawadir al-Falasifa are from excerpts translated by Isaiah Sonne and reprinted in Godwin, pp. 92–98.
Hunayn, quoted in Godwin, p. 92.
Quoted in 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. 176.
Amnon Shiloah, The Epistle on Music of the Ikhwan Al-Safa (Baghdad, 10th century), No. 3 in Documentation & Studies, Hanoch Avenary, ed. (Tel Aviv: Tel Aviv University, School of Jewish Studies, 1978), p. 43. Shiloah’s translation.
Ibid., p. 43.
Ibid., p. 45.
Ibid.
From Al-Hasan Al-Katib, Kitah Kamal Adal Al-Gina’, translated by Amnon Shiloah; quoted from an excerpt reprinted in Godwin, p. 122.
See Aurelian of Réôme, The Discipline of Music (Musica Disciplina), translated by Joseph Ponte (Colorado Springs: Colorado College Music Press, 1968).
Ibid., pp. 22–23.
Information about Eriugena comes from Godwin, pp. 104–05; John J. O’Meara, Eriugena (Oxford: Clarendon Press, 1988); and Hans Liebeschütz, ‘The Place of the Martianus Glossae in the Development of Eriugena’s Thought’ in John J. O’Meara and Ludwig Bieler, eds., The Mind of Eriugena, Papers of a Colloquium, Dublin 14–18 July 1970 (Dublin: Irish University Press, 1970).
John Scotus Eriugena, Commentary on Martianus Capella, translated by Joscelyn Godwin; quoted from an excerpt reprinted in ibid., p. 105. Eriugena said he was led to ‘daring cosmological theories’ by Martianus Capella (aka Martin the Irishman), O’Meara, p. 30.
Godwin, p. 106.
John Scotus Eriugena, Periphyseon or De Divisione naturae, quoted in ibid., p. 104.
Regino of Prüm, ‘Epistola de harmonica institutione’, the introduction to his book about plainsong melodies, Tonarius. This excerpt translated by Sister Mary Protase LeRoux and reprinted in Godwin, p. 110.
Ibid., p. 111.
For the emergence of the universities, see Thomas Kuhn, The Structure of Scientific Revolutions (Chicago: University of Chicago Press, 1962), p. 102.
The archbishop’s translation project is described at length in Rubenstein.
Information about the Seven Liberal Arts is from Koenigsberger, p. 199 ff.
As discussed by Burkert, beginning on p. 386.
Information from Burkert, p. 406, including footnote 31.
For the Ars Geometrae supposedly composed by Boethius, see Burkert, p. 406.
Koenigsberger, p. 202.
Quoted in Koenigsberger, p. 201.
This information comes in part from a website of the University of Notre Dame Jacques Maritain Center: (http://maritain-nd.edu) Ralph McInerny, A History of Western Philosophy, vol. 2, part III, chapter IV.
John Hedley Brooke, Science and Religion: Some Historical Perspectives (Cambridge, U.K.: Cambridge University Press, 1991), p. 45.
Ibid., p. 25.
Chapter 15: ‘Wherein Nature shows herself most excellent and complete’
From a letter from Petrarch to Francesco Bruni, written in 1361, excerpted in Morris Bishop, Petrarch and His World (London: Kennikat Press: 1973), p. 351–52, and in Francis Petrarch, Letters of Old Age, vol. 1, Books I–IX (Sen 1,6), translated by Aldo S. Bernardo, Saul Levin, and Reta A. Bernardo (Baltimore: Johns Hopkins University Press, 1992), pp. 27–28.
From the introduction to the excerpts from Petrarch in Ernst Cassirer, 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), p. 25.
Petrarch, On His Own Ignorance, reprinted in ibid., p. 92.
Ibid., p. 94.
Ibid., p. 24.
Marsilio Ficino, Five Questions Concerning the Mind, reprinted in ibid., pp. 209–210.
Pico della Mirandola, On the Dignity of Man, translated by Charles Glenn Wallis (Indianapolis: The Library of the Liberal Arts, Bobbs Merrill Educational Publishing, 1965), p. 12.
G. Pico della Mirandola, Conclusiones sive Theses, translated by the author. For an attempt to make some sense out of this list, and connections with Plato, Nicomachus, Ptolemy, and, indeed, Oscar Wilde, see Godwin, p. 447.
‘Letter to Leo X’, quoted in Kahn, p. 158, from A. E. Chaignet, Pythagore et la philosophie pythagoricienne, vol. II (Par
is, 1873), p. 330.
Leon Battista Alberti, The Ten Books of Architecture (Mineola, N.Y.: Dover replica Edition, 1987), Chapter 5 of Book 9.
Nicholas of Cusa, Of Learned Ignorance (1440). Quoted in Koenigsberger, p. 367.
Prefatory letter to De revolutionibus, Gesamtausgabe. Vol. II: De revolutionibus. Kritischer Text, eds. H. M. Nobis and B. Sticker (Hildesheim, Germany: 1984), p. 4, as quoted in T. S. Kuhn, The Copernican Revolution: Planetary Astronomy in the Development of Western Thought (Cambridge, Mass.: Harvard University Press, 1957), p. 137.
Prefatory letter to De revolutionibus, Gesamtausgabe. Vol. II, De revolutionibus, p. 4, as quoted in Kuhn, Copernican Revolution, p. 142.
Mentioned in Brian L. Silver, The Ascent of Science (Oxford, U.K.: Oxford University Press, 1998), p. 177.
Book 1, Chapter 10 of De revolutionibus, Gesamtausgabe. Vol. II, De revolutionibus, p. 4, as quoted in Kuhn (1957), pp. 179–80.
Andrea Palladio, I quattro libri dell’ architettura. In a reproduction of the Isaac Ware 1738 English edition: The Four Books on Architecture (New York: Dover, 1978).
See, for example, Rudolph Wittkower, Architectural Principles in the Age of Humanism (New York: Norton, 1971).
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’
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.
Godwin, pp. 104–105.
Kepler, Harmonice mundi, Book V, in Gesammelte Werke, vol. 6, p. 289.
Stephenson is an extraordinarily thorough and invaluable guide through the labyrinth of Kepler’s Harmonice mundi.
Plato, Timaeus (London: Penguin, 1965), p. 15, p. 50n.
Kepler, Harmonice mundi, in Gesammelte Werke, vol. 6, p. 289.
This list has been simplified and summarised by the author, from Chapter 3, Book 5 of Kepler’s Harmonici mundi. See, for instance, The Harmony of the World by Johannes Kepler, translated by E.J. Alton, A.M. Duncan, J.V. Field (Philadelphia: American Philosophical Society, 1997), where the list in its long entirety is pp. 405–16.
For Kepler’s complete table, see p. 150 in Stephenson.
For a much more detailed explanation, see Stephenson, p. 171.
Kepler, Harmonice mundi, in Gesammelte Werke, vol. 6, p. 323.
Kepler, Harmonice mundi, in Gesammelte Werke, vol. 6, p. 356.
Stephenson points out that Kepler had read Proclus’s Platonic/neo-Pythagorean hymns.
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
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.
The episode having to do with Galileo’s father is retold in Silver, p. 176.
Barrow, p. 127.
Silver, p. 158.
Ibid., p. 177; Bronowski (p. 234) also mentions Newton’s attribution to Pythagoras.
Quoted in Barrow, p. 127.
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).
Joseph Addison, paraphrase of Psalm 19:1–6. Hymn 409 in The Hymnal 1982, according to the use of the Episcopal Church.
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.
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).
From Jevons, Principles of Science, quoted in Lindberg, pp. 371–72, n. 15.
Chapter 18: Janus Face
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.
Whitehead, Alfred North, and Bertrand Russell, Principia Mathematica, 3 vols. (Cambridge, U.K.: Cambridge University Press, 1910, 1912, 1913).
Aristotle, quoted in Russell (1945), p. 136.
Frege’s labours 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).
Bertrand Russell, ‘How to Read and Understand History’, in Understanding History and Other Essays (New York: Philosophic Library, 1957).
Barrow, p. 293.
Bertrand Russell, ‘The Value of Free Thought’, in Understanding History.
Chapter 19: The Labyrinths of Simplicity
Quoted in Kitty Ferguson, Prisons of Light: Black Holes (Cambridge, U.K.: Cambridge University Press, 1996), p. 114.
Bryan Appleyard, ‘Master of the Universe: Will Stephen Hawking Live to Find the Secret?’ Sunday Times (London)
Richard Feynman, QED: The Strange Theory of Light and Matter (Princeton, N.J.: Princeton University Press, 1985), p. 4.
John Archibald Wheeler, Journey into Gravity and Spacetime (New York: Scientific American Library, 1990), p. xi.
Barrow, p. 129.
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.
Richard A. Kerr, ‘Listening to the Music of the Spheres’, Science 1991, 253: 1207–1208.
P. Demarque and D. B. Guenther (1999) ‘Helioseismology: Probing the Interior of a Star’, Proceedings of the National Academy of Sciences 96: 5356–69.
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 Organisation for Astronomical Research in the Southern Hemisphere, or European Southern Observatory.
Marcia Bartusiak, Einstein’s Unfinished Symphony: Listening to the Sounds of Space-time (Washington, D.C.: National Academies Press, 2000).
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
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
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
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
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.
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
Diane Richards, ‘Listening to Northern Lights’, Astronomy, Dec. 2001, p. 63.
Appendix
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.
Alton, E.J., A.M. Duncan, and J.V. Field, translators. The Harmony of the World by Johannes Kepler. Philadelphia: American Philosophical Society, 1997.
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.
Baqir, Taha. ‘An Important 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.