The Music of Pythagoras (51 page)

We send our tiny beeps into the far distant reaches of space, certain that any intelligent beings out there, no matter how “other” they may be in some respects, could not have failed to discover what our world did . . . sure that our little signaled evidence of rationality will look familiar to them. In spite of the still unsolved mysteries—and the possibility that they may never be solved—our Pythagorean ideal of the unity and kinship of all being tells us this must be so.

Pythagoras . . . are you there?

The proof for the Pythagorean theorem that Jacob Bronowski thought may have been used by Pythagoras.
¹

Start with a right triangle.

Create a square using four triangles identical to that one, but rotated, so that the “leading points” of the triangles point to the four points of the compass (north, south, east, and west), and the long side of each triangle ends at the leading point of its neighbor:

What you now have is a square based on the long side of the original triangle—the “square on the hypotenuse.” It is this total area that must equal the sums of the squares of the other two sides, if the Pythagorean theorem is correct. As you proceed, remember that however you rearrange these five shapes, their total area stays the same. So, rearrange them into the following shape. Place a rod across your design and look at it carefully. You will see that you have two squares, and they are the squares on the other two sides of the triangle. Using no numbers, you have proved the Pythagorean theorem.

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.

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.

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.

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

1
. Ibid, p. 377.

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

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.

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.