Notes

Chapter 1: The Long-haired Samian

  1. Iamblichus’ Pythagorean Life or Life of Pythagoras is available in translation by Thomas Taylor: Iamblichus Life of Pythagoras (Rochester, Vt.: Inner Traditions International, 1986). It and the biographical treatments by Porphyry and Diogenes Laertius are available in translation by 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). The Guthrie anthology also contains some of the pseudo-Pythagorean works.
  2. Diogenes Laertius’ and Porphyry’s ‘lives’ of Pythagoras are reprinted in K. S. Guthrie.
  3. Jacob Bronowski, The Ascent of Man (Boston: Little, Brown, 1973), p. 156.
  4. Quoted in Richard Buxton, ed., From Myth to Reason: Studies in the Development of Greek Thought (Oxford, U.K.: Oxford University Press, 1999), p. 74.
  5. Kurt A. Raaflaub, ‘Poets, Lawgivers, and the Beginnings of Political Reflection in Archaic Greece’, in Christopher Rowe and Malcolm Schofield, eds., The Cambridge History of Greek and Roman Political Thought (Cambridge, U.K.: Cambridge University Press, 2000), p. 51.
  6. Plato, Thaetetus, 174 A., quoted by Thomas L. Heath, Greek Astronomy (London: J. M. Dent and Sons, 1932), p. 1, and Arthur Koestler, The Sleepwalkers: A History of Man’s Changing Vision of the Universe (London: Hutchinson, 1959), p. 22.
  7. The story of Thales and the river Halys was one of those collected by Herodotus and included in his Histories I 75.3–5. Reprinted in Barnes, p. 10.
  8. Ian Shaw, Ancient Egypt: A Very Short Introduction (Oxford, U.K.: Oxford University Press, 2004). p. 12.
  9. Porphyry’s biography is reprinted in K. S. Guthrie, p. 124.

Chapter 2: ‘Entirely different from the institutions of the Greeks’

  1. For the information about what Pythagoras might have learned in Egypt, I have relied on David P. Silverman, ed., Ancient Egypt (New York: Oxford University Press, 1997).
  2. For information about Babylon in this era, I have relied on H. W. F. Saggs, Everyday Life in Babylonia and Assyria (Assyrian International News Agency, 1965); and Joan Oates, Babylon (London: Thames & Hudson, 1979). Speculation about the historical timing of Pythagoras’ abduction from Egypt is based on Saggs, p. 25. Modern scholarly knowledge about the city of Babylon during this period comes from a variety of sources: the biblical and Greek tradition, Nebuchadnezzar’s building inscriptions, business, legal and administrative records, and the excavation of the city, which together give a fairly clear picture of life in the Babylonian capital under Nebuchadnezzar II, though there are many details which we do not yet know and may never know.

Chapter 3: ‘Among them was a man of immense knowledge’

  1. Exhibits in the Museo Archeologico Nazionale di Croton suggest the appearance of the ancient city.
  2. Information about Achaea comes from N. G. L. Hammond, A History of Greece to 322 B.C. (Oxford, U.K.: Oxford University Press, 1986), pp. 13 and 118.
  3. Porphyry’s The Life of Pythagoras, reprinted in K. S. Guthrie, 1987, p. 135.
  4. Acts 17:21.
  5. Kurt A. Raaflaub, ‘Poets, Lawgivers, and the Beginnings of Political Reflection in Archaic Greece’, in Christopher Rowe and Malcolm Schofield, eds., The Cambridge History of Greek and Roman Political Thought (Cambridge, U.K.: Cambridge University Press. 2000), p. 57.
  6. Guthrie, William Keith Chambers, The Earlier Presocratics and the Pythagoreans, Vol.1 of A History of Greek Philosophy. (Cambridge, U.K.: Cambridge University Press, 2003), pp. 176–177. Guthrie refers to the historian C. T. Seltman.
  7. Ibid., pp. 176–77.

Chapter 4: ‘My true race is of Heaven’

  1. This overview of Greek beliefs about immortality and the manner in which Pythagorean doctrine fits into this context is based on the discussion in Guthrie (2003), beginning on p. 196, and on W. K. C. Guthrie, The Greeks and Their Gods (Cambridge, U.K.: Cambridge University Press, 1951).
  2. See Betrand Russell, The History of Western Philosophy (London: George Allen & Unwin, 1945).
  3. This is the way the verse in Homer was translated by Alexander Pope.
  4. The story was told by Diogenes Laertius and also by Diodorus in his Universal History X, quoted in Barnes, p. 34.
  5. See W. K. C. Guthrie (2003), pp. 201–202.
  6. Quoted in ibid, p. 199.
  7. Ibid, p. 199.
  8. Ibid.
  9. Eudemus, Physics, fragment quoted by Simplicius in Commentary on the ‘Physics’ 732.23–33. Quoted in Barnes, p. 35.
  10. See Barnes, pp. 167–68.
  11. Material is from Aulus Gellius, Attic Nights, Book IV, xi 1–13. Attic Nights was twenty volumes long, a compendium of miscellaneous knowledge. It and parts of it are available in a number of editions.
  12. For the discussion around the suggestion that the miraculous stories were meant to discredit Pythagoras, see Walter Burkert, Lore and Science in Ancient Pythagoreanism (Cambridge, Mass.: Harvard University Press, 1972), p. 146

Chapter 5: ‘All things known have number’

  1. Ibid, p. 377.
  2. W. K. C. Guthrie (2003), p. 178.

Chapter 6: ‘The famous figure of Pythagoras’

  1. Bronowski, p. 156.
  2. From ‘Surveying’ article in the Encyclopaedia Britannica, 2007, Online, 3 Mar. 2007 http://www.britannica.com/eb/article-51763, p. 2.
  3. This information comes from a conversation with John Barrow and from his book Pi in the Sky: Counting, Thinking, and Being (Oxford, U.K.: Clarendon Press, 1992), pp. 73–75. The information about Indian women and their doorstep paintings comes from personal experience in Kothapallimitta, South India, in 2000, and trying to do it myself.
  4. W. K. C. Guthrie (2003), p. 187.
  5. Except where otherwise footnoted, and except for some information about Tell Harmal, the information in these paragraphs about Babylonian mathematics comes primarily from Eleanor Robson, ‘Three Old Babylonian Methods for Dealing with Pythagorean Triangles’, Journal of Cuneiform Studies (1997) 49, pp. 51–72.
  6. Robson, ‘Mesopotamian Mathematics: Some Historical Background’, in Victor Katz, ed., Using History to Teach Mathematics: An International Perspective (Cambridge, U.K.: Cambridge University Press, 2000), p. 154
  7. Plimpton 322 is now in the collection of Columbia University, in New York City.
  8. See Taha Baqir, ‘An Important Mathematical Problem Text from Tell Harmal’, Sumer 6 (1950), pp. 39–55. Taha Baqir was curator of the Iraq Museum.
  9. Diagram and text reconstruction are from Robson, ‘Three Old Babylonian Methods’, p. 57.
  10. See, for example, Ross King, Brunelleschi’s Dome: How a Renaissance Genius Reinvented Architecture (London: Penguin, 2000).
  11. John Noble Wilford, ‘Early Astronomical Computer Found to Be Technically Complex’, New York Times, November 30, 2006.
  12. For discussion, see Robson, ‘Mesopotamian Mathematics: Some Historical Background’, pp. 154–55. The quotation is from Robson, ‘Influence, Ignorance, or Indifference? Rethinking the Relationship Between Babylonian and Greek Mathematics’, The British Society for the History of Mathematics, Bulletin 4 (Spring 2005), pp. 2, 3.
  13. Ibid., p. 14.
  14. Ibid., p. 10.
  15. See discussion in ibid, pp. 2, 3.
  16. Charles H. Kahn, Pythagoras and the Pythagoreans: A Brief History (Indianapolis: Hackett, 2001), p. 134.
  17. Marcus Vitruvius Pollio, De Architectura, Book IX. Vitruvius’ work is reprinted as Vitruvius: Ten Books of Architecture (Cambridge, U.K.: Cambridge University Press, 2001).
  18. Bronowski, p. 160.

Chapter 7: A Book by Philolaus the Pythagorean

  1. For the discussion of the acusmatici and the mathematici and the question about which were truer to the original teachings of Pythagoras, I have relied on Burkert, particularly the section entitled ‘Acusmatici and Mathematici’.
  2. The names of some mathematici have survived. One was Archytas of Tarentum, and he mentioned Eurytus of Tarentum as one of his predecessors. This was the same Eurytus linked with Philolaus in Plato’s Phaedo. Eurytus and Philolaus had students whose names Aristoxenus listed. They were from Chalcidice in Thrace and from Phlius.
  3. I have mostly followed Burkert regarding the authenticity of fragments of Philolaus; I have also relied on W. K. C. Guthrie (2003) in the discussion of Philolaus’ book and on Guthrie and Jonathan Barnes, Early Greek Philosophy (London: Penguin Books, 1987), for translations of quotations.
  4. Plutarch, On the Face in the Moon 929AB, quoted in Barnes, p. 89.
  5. From Plutarch, Pericles; passage quoted in full in Barnes, p. 92.
  6. Quoted from Aristotle’s Metaphysics, in W. K. C. Guthrie (2003), pp. 287–88.
  7. See W. K. C. Guthrie (2003) p. 248, for the seed of this idea.
  8. W. K. C. Guthrie (2003) wrote that Philolaus was Aristotle’s ‘favourite author’ (p. 260). The quotation is in W. K. C. Guthrie (2003), pp. 307–308.
  9. Quoted in ibid., p. 309.
  10. See ibid., p. 233, for the arguments each way concerning Alcmaeon’s dates. The quotation appears in W. K. C. Guthrie (2003), p. 313.
  11. Plato, Phaedo, quoted in ibid., p. 310.
  12. Quoted in ibid., p. 311.
  13. Quoted in ibid., p. 312.

Chapter 8: Plato’s Search for Pythagoras

  1. Information about Plato’s visits to Megale Hellas (as he would have called it) and Syracuse can be found in many sources. I have used W. K. C. Guthrie (2003), and Malcolm Schofield, ‘Plato and Practical Politics’, in Christopher Rowe and Malcolm Schofield, eds., The Cambridge History of Greek and Roman Political Thought (Cambridge, U.K.: Cambridge University Press, 2000). See also Plato’s own letter (the authenticity of which is debated) addressed to ‘Friends and Followers of Dion’, Letter #7 in his Letters.
  2. For information about Archytas and his work I have relied on Kahn, p. 40ff, and Carl Huffman, Archytas of Tarentum: Pythagorean, Philosopher and Mathematician King (Cambridge: Cambridge University Press, 2005). Unless otherwise noted, the quotations from Archytas are drawn from Kahn.
  3. Burkert, p. 68.
  4. Modern scholars such as Kahn have made a distinction that sets Archytas a little more apart from the earlier Pythagoreans but still keeps him in the tradition. Kahn described Archytas’ harmonic theory as ‘work of original genius . . . working in the Pythagorean musical tradition that is represented for us by the earlier theory of Philolaus’ (pp. 32–43).
  5. From Eudemus (also mentioned by Aristotle), quoted in Kahn, p. 43.
  6. Huffman, pp. 105–6.
  7. Archytas’ description of a bull-roarer, or rhomboi, is in W. K. C. Guthrie (2003), p. 227.
  8. Aristotle, quoted in ibid., p. 335.
  9. For this discussion and the question about which were truer to the original teachings of Pythagoras, I have relied on Burkert, particularly the section entitled ‘Acusmatici and Mathematici’.
  10. The original is Aristophon, fragment 12; see Burkert, p. 199, for the quotation. Burkert was not sure the Greek word used actually referred to the Pythagoreans.
  11. Quotations are from the musician Stratonicus and from Sosicrates, in Burkert, p. 202.

Chapter 9: ‘The ancients, our superiors . . .’

  1. Kahn, p. 50 ff, is especially helpful in interpreting Plato’s thought as it related to the Pythagoreans, and he pinpoints these two themes.
  2. The quotations from Plato’s Timaeus, unless otherwise noted, are taken from the translation by Desmond Lee (London: Penguin, 1977).
  3. From Plato’s Philebus, in Kahn, p. 14.
  4. Ibid., p. 58.
  5. From Plato’s Gorgias, Quoted in W. K. C. Guthrie (2003), p. 305.
  6. Book 7 of Plato’s Republic, quoted in ibid., p. 162.
  7. Plato, Timaeus, in the Stephanus edition (1578), p. 52.

Chapter 10: From Aristotle to Euclid

  1. W. K. C. Guthrie (2003), p. 331. I have closely followed Guthrie in his discussion of Aristotle’s reactions to the Pythagoreans. Where not otherwise noted, the quotations from Aristotle are drawn from Guthrie’s book.
  2. Burkert lists the writers in whose work fragments from Aristotle appear (pp. 28, 29).
  3. ‘What the sky encloses’ is a quotation from Burkert (p. 31), but he was paraphrasing Aristotle.
  4. It is no longer generally accepted that, as Burkert states, ‘like all pre-Socratics they take everything that exists in the same way, as something material’ (Burkert, p. 32). That does not apply correctly either to the Pythagoreans or to the other pre-Socratics.
  5. Burkert, pp. 45–46.
  6. W. K. C. Guthrie (2003), p. 259.
  7. The quotation is from Burkert (p. 431), in his discussion of these different possibilities, but he did not favour this choice.
  8. This discussion draws on W. K. C. Guthrie (2003), p. 266 ff, and Burkert, p. 68 ff.
  9. Ibid., p. 226n.
  10. Burkert’s paraphrase of Aëtius, in which he seems to have given only the final four words in direct quotation (Burkert, p. 70).
  11. Quoted in W. K. C. Guthrie (2003), pp. 263–64.
  12. Historical information about this era comes in part from Greg Woolf, ed., Cambridge Illustrated History of the Roman World (Cambridge, U.K.: Cambridge University Press, 2003); and from Paul Cartledge, ed., Cambridge Illustrated History of Ancient Greece (Cambridge, U.K.: Cambridge University Press, 1998).
  13. Specifically in the Elements, Book VII.
  14. One scholar, the German B. L. van der Waerden, insisted that there were writings before Archytas and long before Euclid that dealt with this same material. Kahn calls some of his claims ‘excessive’ (Kahn, p. 41n.). Propositions in Book II of the Elements have very early Babylonian precursors, as the Pythagorean theorem did, that Euclid probably was not aware of (Robson [2005], p. 4).
  15. For information and discussion, see Burkert, p. 432.
  16. This quotation is from Iamblichus, 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). Quoted in S. Cuomo, Ancient Mathematics (London: Routledge, 2001), pp. 236–37.

Chapter 11: The Roman Pythagoras

  1. Marcus Tullius Cicero, The Republic, Book II: xv, translated by G.G. Hardingham (London: Barnard Quaritch, 1884), pp. 137–9.
  2. Quoted in Kahn, pp. 89–90.
  3. From Pliny, Natural History 34.26.
  4. From the Pythagorean Notebooks, excerpt quoted in Diogenes Laertius’ The Life of Pythagoras, reprinted in K. S. Guthrie, pp. 148–49.
  5. Quoted in Simon Hornblower and Antony Spawforth, eds., The Oxford Companion to Classical Civilization (Oxford, U.K.: Oxford University Press, 1998), p. 245.
  6. Thomas Wiedemann, ‘Reflections of Roman Political Thought in Latin Historical Writing’, in Christopher Rowe and Malcolm Schofield, eds., The Cambridge History of Greek and Roman Political Thought (Cambridge, U.K.: Cambridge University Press, 2000), p. 526–27.
  7. Sextus Empiricus, quoted in Kahn, p. 84.
  8. Elizabeth Rawson, Intellectual Life in the Late Roman Republic (Baltimore: Gerald Duckworth & Co., 1985), p. 310.
  9. Cicero, Timaeus, ‘Introduction’. Quoted in Kahn, p. 73.
  10. Kahn, p. 88.
  11. Cicero, Vatinium 6. Quoted in Kahn, p. 91.
  12. Cicero, The Republic, Book I:x, p. 19.
  13. Ibid., Book III:xi, p. 245.
  14. Cicero, On Divination, quoted in Barnes, p. 165–66.
  15. Cicero, The Republic, Book VI:xviii, p. 363 (passages known as ‘Scipio’s Dream’).
  16. Ibid.
  17. Ibid., p. 365.
  18. Vitruvius (Marcus Vitruvius Pollio). De architectura, Book VI. Vitruvius’ work has been reprinted as Vitruvius: Ten Books of Architecture (Cambridge, U.K.: Cambridge University Press, 2001).
  19. Cesariano’s drawing for Vitruvius: ccat.sas.upenn.edu/george/vitruvius.html. The quotation is from Book IX of Vitruvius.
  20. Vitruvius, Book I.
  21. Ibid.
  22. The information about King Juba is from a footnote in Kahn (p. 90) to E. Zeller, Die Philosophie der Griechen in ihrer geschichtlichen Entwicklung I–III (Leipzig: Reisland 1880–92), p. 97.
  23. Lysis’s Letter to Hipparchus, quoted in Diogenes Laertius, The Life of Pythagoras, in K. S. Guthrie, pp. 141–55.
  24. Burkert dates the letter to the third century B.C. A. Staedele dates it to the first. Kahn, p. 75, mentions contemporaneity as Burkert’s suggestion, citing Walter Burkert, ‘Hellenistische Pseudopythagorica’, Philologus 105 (1961).
  25. Introduction to the Occelus piece in K. S. Guthrie, p. 203.
  26. Mentioned in Kahn (p. 79), where the footnote refers to a quotation in J. Dillon, The Middle Platonists (Ithaca, NY: Cornell University Press, 1977), p. 156n.
  27. See Bruno Centrone, ‘Platonism and Pythagoreanism in the Early Empire’, in Cambridge History of Greek and Roman Political Thought (Cambridge, U.K.: Cambridge University Press, 2000). In Centrone’s words (p. 568): ‘Here it is an artificial language, which only reproduces the commonest features of Doric.’
  28. Kahn, p. 76.

Chapter 12: Through Neo-Pythagorean and Ptolemaic Eyes

  1. For the discussion of the neo-Pythagorean philosophers and cults, I have relied on Kahn and Centrone.
  2. From the Pythagorean Golden Verses, reprinted in K. S. Guthrie, p. 164.
  3. Seneca, in a ‘Letter to Lucilius’ (108.17–21). Quoted in Kahn, p. 151.
  4. Philostratus, Life of Apollonius of Tyana, quoted in Robin Lane Fox, Pagans and Christians in the Mediterranean World from the Second Century A.D. to the Conversion of Constantine (London: Penguin Books, 1986), p. 245.
  5. Philostratus, in Fox, p. 248.
  6. Eudorus, quoted or paraphrased in Arius Didymus. Quoted in Kahn, p. 96.
  7. Information about the Alexandrian Jewish community is from Greg Woolf, ed., Cambridge Illustrated History of the Roman World (Cambridge, U.K.: Cambridge University Press, 2003), p. 277.
  8. Quoted in Kahn, p. 101.
  9. Centrone, p. 561n.
  10. Philo of Alexandria, De opific, quoted in Kahn, p. 100n.
  11. The two descriptions come from Harry Austryn Wolfson and Valentin Nikiprowetsky, as reported by Centrone, p. 561.
  12. Centrone, p. 561.
  13. Number 41 in The Catalogue of Lamprias, a list of Plutarch’s works that probably was compiled in the fourth century A.D.
  14. This was how Porphyry reported Moderatus’ views: see Kahn, p. 105.
  15. As quoted and/or paraphrased by Porphyry: see Kahn, p. 106.
  16. Theon of Smyrna, Mathematics Useful for Understanding Plato, excerpted as Appendix 1 in K. S. Guthrie.
  17. W. K. C. Guthrie (2003), p. 406.
  18. Proclus, The Elements of Theology, revised and edited by E.R. Dodds (Oxford: Oxford University Press, 1963), p. 259.
  19. Numenius fragment #2; quoted in Kahn, p. 122.
  20. Numenius fragment, no number; quoted in Kahn, p. 122.
  21. Numenius fragment #52; quoted in Kahn, p. 132.
  22. 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.
  23. 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).
  24. Stephenson, p. 29. Stephenson cites Von Jan, ‘Die Harmonie der Sphären’, Philologus 52 (1893).
  25. Simplified from Stephenson, p. 31. Piano notes are author’s addition.
  26. Stephenson, p. 37.

Chapter 13: The Wrap-up of Antiquity

  1. E. R. Dodds, The Greeks and the Irrational (Berkeley, Calif.: University of California Press, 1951), p. 296.
  2. Ibid., p. 285.
  3. Description of Rome in this era is based on Michael Grant, History of Rome (London: Faber & Faber, 1978), p. 284 ff.
  4. Edward Gibbon, Decline and Fall of the Roman Empire, Chapter X. Quoted in Russell (1945), p. 287.
  5. Description taken from Grant, p. 294.
  6. Ibid.
  7. Plotinus quoted in Dodds, pp. 285–86.
  8. Ibid., pp. 286–87.
  9. Kahn, p. 134.
  10. Dodds, p. 287.
  11. Ibid.
  12. Fox, p. 190.
  13. John 1:1–5, 9–12, 14. Paraphrased from the Oxford New International Version translation.
  14. Augustine, City of God, translated by Gerald Walsh et al. (Garden City, N.Y.: Doubleday, 1958), pp. 241–42.
  15. 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’.
  16. Information about this period comes from H. G. Koenigsberger, Medieval Europe: 400–1500 (London: Longman, 1987).
  17. Most of the information about Macrobius is from Stephenson, pp. 38–41.
  18. 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’

  1. 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).
  2. Quotations from Hunayn’s Nawadir al-Falasifa are from excerpts translated by Isaiah Sonne and reprinted in Godwin, pp. 92–98.
  3. Hunayn, quoted in Godwin, p. 92.
  4. 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.
  5. 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.
  6. Ibid., p. 43.
  7. Ibid., p. 45.
  8. Ibid.
  9. From Al-Hasan Al-Katib, Kitah Kamal Adal Al-Gina’, translated by Amnon Shiloah; quoted from an excerpt reprinted in Godwin, p. 122.
  10. See Aurelian of Réôme, The Discipline of Music (Musica Disciplina), translated by Joseph Ponte (Colorado Springs: Colorado College Music Press, 1968).
  11. Ibid., pp. 22–23.
  12. 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).
  13. 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.
  14. Godwin, p. 106.
  15. John Scotus Eriugena, Periphyseon or De Divisione naturae, quoted in ibid., p. 104.
  16. 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.
  17. Ibid., p. 111.
  18. For the emergence of the universities, see Thomas Kuhn, The Structure of Scientific Revolutions (Chicago: University of Chicago Press, 1962), p. 102.
  19. The archbishop’s translation project is described at length in Rubenstein.
  20. Information about the Seven Liberal Arts is from Koenigsberger, p. 199 ff.
  21. As discussed by Burkert, beginning on p. 386.
  22. Information from Burkert, p. 406, including footnote 31.
  23. For the Ars Geometrae supposedly composed by Boethius, see Burkert, p. 406.
  24. Koenigsberger, p. 202.
  25. Quoted in Koenigsberger, p. 201.
  26. 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.
  27. John Hedley Brooke, Science and Religion: Some Historical Perspectives (Cambridge, U.K.: Cambridge University Press, 1991), p. 45.
  28. Ibid., p. 25.

Chapter 15: ‘Wherein Nature shows herself most excellent and complete’

  1. 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.
  2. 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.
  3. Petrarch, On His Own Ignorance, reprinted in ibid., p. 92.
  4. Ibid., p. 94.
  5. Ibid., p. 24.
  6. Marsilio Ficino, Five Questions Concerning the Mind, reprinted in ibid., pp. 209–210.
  7. 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.
  8. 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.
  9. ‘Letter to Leo X’, quoted in Kahn, p. 158, from A. E. Chaignet, Pythagore et la philosophie pythagoricienne, vol. II (Paris, 1873), p. 330.
  10. Leon Battista Alberti, The Ten Books of Architecture (Mineola, N.Y.: Dover replica Edition, 1987), Chapter 5 of Book 9.
  11. Nicholas of Cusa, Of Learned Ignorance (1440). Quoted in Koenigsberger, p. 367.
  12. 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.
  13. Prefatory letter to De revolutionibus, Gesamtausgabe. Vol. II, De revolutionibus, p. 4, as quoted in Kuhn, Copernican Revolution, p. 142.
  14. Mentioned in Brian L. Silver, The Ascent of Science (Oxford, U.K.: Oxford University Press, 1998), p. 177.
  15. Book 1, Chapter 10 of De revolutionibus, Gesamtausgabe. Vol. II, De revolutionibus, p. 4, as quoted in Kuhn (1957), pp. 179–80.
  16. 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).
  17. See, for example, Rudolph Wittkower, Architectural Principles in the Age of Humanism (New York: Norton, 1971).
  18. 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. 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.
  8. For Kepler’s complete table, see p. 150 in Stephenson.
  9. For a much more detailed explanation, see Stephenson, p. 171.
  10. Kepler, Harmonice mundi, in Gesammelte Werke, vol. 6, p. 323.
  11. Kepler, Harmonice mundi, in Gesammelte Werke, vol. 6, p. 356.
  12. Stephenson points out that Kepler had read Proclus’s Platonic/neo-Pythagorean hymns.
  13. 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 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).
  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 Organisation 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.