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 “favorite 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).
2. The discussion of Archytas’ work is based on Kahn, p. 40 ff. Most of the quotations from Archytas are drawn from Kahn’s book.
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. Archytas’ description of a bull-roarer, or rhomboi, is in W. K. C. Guthrie (2003), p. 227.
7. Aristotle, quoted in ibid., p. 335.
8. 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.”
9. 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.
10. 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 favor 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, On The Republic, Book 2, 28, 29. Reprinted in translation by Niall Rudd, in Cicero: The Republic and The Laws (Oxford, U.K.: Oxford University Press, 1998), pp. 43–44.
2. Quoted in Kahn, pp. 89–90.
3. From Pliny, Natural History 34.26. As retold in Kahn, p. 86.
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, On the Commonwealth, quoted from the edition translated by George Holland Sabine and Stanley Barney Smith (New York: The Liberal Arts Press, 1929), pp. 114–15.
13. Ibid., p. 206.
14. Cicero, On Divination, quoted in Barnes, p. 165–66.
15. Cicero, “Scipio’s Dream,” from On the Republic, translated by Cyrus R. Edmonds and Moses Hadas, in The Basic Works of Cicero (New York: Random House, 1951), p. 165.
16. Ibid., p. 166.
17. Ibid., p. 166.
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 conte
mporaneity 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. Dodds (1963), p. 259; quoted in Kahn, p. 118.
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. Pliny, Natural History, 2:20, translated 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 by the author from Stephenson, p. 31.
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. 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. From The Epistle on Music of the Ikhwan al-Safa’, translated by Amnon Shiloah; quoted from an excerpt reprinted in Godwin, p. 113.
6. Ibid., p. 113.
7. Ibid., p. 115.
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. The information about Aurelian comes from Godwin, p. 99.
11. Aurelian of Réôme, Musica Disciplina, translated by Joseph Ponte; quoted from an excerpt reprinted in Godwin, pp. 101–102.
12. Information about Eriugena comes from ibid., pp. 104–105.
13. John Scotus Eriugena, Commentary on Martianus Capella, translated by Joscelyn Godwin; quoted from an excerpt reprinted in ibid., p. 105.
14. Ibid., 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. Letter from Petrarch to Francesco Bruni, October 25, 1362, reprinted 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. 34
.
2. From the introduction to the excerpts from Petrarch in ibid., 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, reprinted in ibid., pp. 232–33.
8. G. Pico della Mirandola, Conclusiones sive Theses, edited and translated by Bohdan Kieszowski, reprinted in Godwin, p. 176. 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. See Cassirer et al., p. 245.
10. “Letter to Leo X,” quoted in Kahn, p. 158, from A. E. Chaignet, Pythagore et la philosophie pythagoricienne, vol. II (Paris, 1873), p. 330.
11. Leon Battista Alberti, The Ten Books of Architecture (Mineola, N.Y.: Dover replica Edition, 1987), Chapter 5 of Book 9.
12. Nicholas of Cusa, Of Learned Ignorance (1440). Quoted in Koenigsberger, p. 367.
13. 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.
14. Prefatory letter to De revolutionibus, Gesamtausgabe. Vol. II, De revolutionibus, p. 4, as quoted in Kuhn, Copernican Revolution, p. 142.
15. Mentioned in Brian L. Silver, The Ascent of Science (Oxford, U.K.: Oxford University Press, 1998), p. 177.
16. Book 1, Chapter 10 of De revolutionibus, Gesamtausgabe. Vol. II, De revolutionibus, p. 4, as quoted in Kuhn (1957), pp. 179–80.
17. 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).
18. See, for example, Rudolph Wittkower, Architectural Principles in the Age of Humanism (New York: Norton, 1971).
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