The Music of Pythagoras (56 page)

Copyright © 2008 by Kitty Ferguson

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*
The stories of the three biographers themselves are in
Chapter 13
.

*
There was an Olympic victor in the 588
B.C
. games whose name was Pythagoras of Samos, as recorded in the highly reliable lists of Eratosthenes, the famous librarian at Alexandria who first measured the circumference of the earth. Eratosthenes conjectured that this man and Pythagoras the philosopher were one and the same. In order for that to be true, Pythagoras the philosopher would have had to have been born a few decades earlier than is usually supposed, in the late seventh century. For the Olympian Pythagoras to have been Eurymenes (who might have adopted the name of his teacher as the son of Eratocles did), Pythagoras the philosopher would have had to have been earlier still. What the appearance of the name in the Olympic victory lists probably does mean is that the name Pythagoras was current on Samos before Mnesarchus decided to name his son in honor of the Pythian oracle at Delphi.

*
Both tunnel and harbor still exist, though the harbor is now hidden below sea level, under later construction. You can visit the tunnel and walk a good distance into it. In an ingenious design, the walkway is separate and above the watercourse.

*
Iamblichus linked the date with the Olympic victory of Eryxidas of Chalcis, his own home city. Diogenes Laertius agreed that it had to have been between 532 and 528
B.C
.

*
Milo is also known as Milon. His name has come to symbolize extraordinary strength. He was the most famous wrestler in the ancient world.

*
In the twenty-first century, 2,600 years later, the people of former Magna Graecia still do not totally identify with the modern, centralized Italy. Old attitudes and identities die hard.

*
Porphyry said he got this information from Dicaearchus.

*
In some of the remoter villages of those mountains, the people in the twenty-first century still speak a form of Greek that linguists identify as neither modern Greek nor the Byzantine Greek that arrived with Byzantine Christian Greeks in late antiquity and the early Middle Ages, but as an ancient form of the language that is spoken almost nowhere else in the world.

*
Diogenes Laertius took the story from the writing of Diodorus, a scholar of the first century
B.C
. who in turn got the story from the writing of Plato’s pupil Heracleides of Ponticus.

*
Scholars regard this quotation as likely to be genuinely early, because it made light of Pythagorean belief, rather than extol it as would have happened later, in an overly adulatory period.

^*
Think of having the lowest string tuned to C on the piano, the fourth string tuned to F above the C, the next to G a whole step above that, and then the top string tuned at C an octave above the lower C.

^*
Musical instruments and human voices, because of intricate differences in the way their structures resonate and amplify sound, emphasize or “bring out” certain overtones more than others, and that is what causes the great variety of sounds they make. That is how a trumpet ends up sounding like a trumpet while a clarinet sounds like a clarinet.

^†
On the piano, equivalent notes might be, for example, middle C (ground note); c (octave above that ground note); g (fifth above that octave); c (fourth above that g). For a demonstration using the piano: Press down gently on the c above middle C without allowing it to sound (removing the damper from the strings). Strike middle C (the ground note) and you will clearly hear the octave. Press carefully on the g above that octave. Strike middle C and you will hear that fifth above the octave. A piano is not tuned to the Pythagorean system, but it is close enough for you to hear these overtones.

^*
A gnomon is an instrument for measuring right angles, like the device used by carpenters called a “carpenter’s square.”

^*
Not all pyramids have only four sides. The Great Pyramid that Pythagoras may have seen in Egypt is not a pyramid of this sort. It has five sides: a square base and four triangular sides.

^*
In some later ancient mathematics, whose roots can be traced to the “Pythagorean” tradition and which by some scholars’ interpretation existed separately and in parallel with the Euclidean tradition, the number 2 also had no status as a “number.” It was not considered even or odd or prime. Like “1,” it was not a number at all, but the “first principle of number.”

^*
Heracleides Ponticus is not to be confused with the earlier Heraclitus who so severely criticized Pythagoras. Heracleides Ponticus lived in the fourth century
B.C
. and was a pupil of Plato.

^*
Part of their “present condition” was an economy that was more primitive than Croton’s. They used no coinage, and would not until more than a century later. See W. K. C. Guthrie (2003), p. 178 n.

^*
“Theorem” has implications, in modern terminology, that do not apply to the earliest knowledge of this rule. With that in mind, this book will nevertheless continue to use “theorem” to avoid seeming to mean something different from what everyone calls the Pythagorean theorem.

^†
There were more than one Apollodorus, but this one was probably Apollodorus of Cyzicus, who lived in the fourth century
B.C
.

^*
The claim has never been that Pythagoras discovered the right angle or right triangle, but that he discovered the relationship between the three sides of a right triangle—what we call the Pythagorean theorem.

*
You can think of 3–4–5 as 3 inches, 4 inches, and 5 inches, though it could just as well be centimeters, miles, parsecs or any other unit of measurement.

^†
Unfortunately, most of Babylon of the early second millennium
B.C
. cannot now be excavated because it is well below the water table.

*
The tablet is in the Iraq Museum in Baghdad, listed in the register as 55357.

*
This mechanism, used probably in preparing calendars for planting, harvesting, and religious observances, was discovered in the wreck of a Roman ship that sank off the island of Antikythera in about 65
B.C
. It was more technically complex than any known instrument for at least a millennium afterward.

^†
Political and social upheaval may have created disruptions. Or the fault may lie with modern scholarship, for few sites have been dug from these periods. They do not attract many scholars, partly because the documents are terribly difficult to decipher. Furthermore, as the very complicated cuneiform script gave way to alphabetic Aramaic, documents tended to be written on perishable and recyclable materials. The old Sumerian, Akkadian, and the cuneiform script were used for fewer purposes, mathematics apparently not being one of them, and even where cuneiform was used, it was often on wax-covered ivory or wooden writing boards that were erased for reuse or have not survived.

*
A rational number is a whole number or a fraction that is made by dividing any whole number by another whole number: ½, 4/5, 2/7, etc. An irrational number is a number that cannot be expressed as a fraction, that is, as a ratio of two whole numbers. The square root of 2 was probably found by Pythagoreans, working from their theory of odd and even numbers, possibly as early as about 450
B.C
., and surely by 420, fifty to eighty years after Pythagoras’ death. Plato knew of the square roots of numbers up to 17.

*
Though Heraclitus seems forthright and outspoken in the fragments about Pythagoras, he was known to be no easy read. His contemporaries dubbed him Heraclitus the Obscure and Heraclitus the Riddler. A story circulated in the time of Diogenes Laertius that when Socrates received a copy of a book by Heraclitus, he commented: “What I understand is splendid; and so too, I’m sure, is what I don’t understand—but it would take a Delian diver to get to the bottom of it.”

*
See
Chapter 9
.

*
For historians, one use of the word “fragment” is for a quotation or reference in the writing of another author who had access to material that has since disappeared.

*
Ancient authors (and later translators) also called “unlimited” “limitless.” They called its opposite “limiting,” “limit,” or “limited.”

*
Note that 10 is
not
a perfect number as the term is defined in modern mathematics. We will get to those later.

*
Tarentum was the only colony established by Sparta, and Plato greatly admired the Spartan system of government. However, the people who had colonized Tarentum in 706
B.C
. had come there under unusual circumstances and might not have shared Plato’s enthusiasm for Sparta. They were sons of officially arranged marriages uniting Spartan women with men who were not previously citizens. The purpose was to increase the number of male citizens who could fight in the Messenian wars. When the husbands were no longer needed as warriors, the marriages were nullified and the offspring forced to leave Sparta.

*
For an example of the use of movement in geometry, take a straight line, fasten down one end of it, and swing the other end about. The result is an arc. Take a right triangle and stand it upright with one of the sides serving as its base; swivel it around the upright leg and the result is a cone. (The ancient scholar Eudemus used this explanation in his description of Archytas’ solution.)

^†
A lengthy text is needed to understand it and is available in S. Cuomo, Ancient Mathematics, Routledge, 2001, pp. 58 and 59, and on the Internet at
http://mathforum.org/dr.math/faq/davies/cu/bedbl.htm

*
More generally, ratios such as 5:4, or 9:8, in which the larger number is one unit larger than the smaller (mathematicians call these superparticular or epimeric ratios), cannot be divided into two equal parts.

*
“Diatonic” refers to the scales now known as major and minor scales.

*
Plato was not the first to think of the planets moving on rings. Anaximander’s cosmos involved huge wheels, whose hollow rims were filled with fire. The Sun, Moon, stars and planets were glimpses of this fire, showing through at openings in the wheel rims. Similar ideas had surfaced elsewhere as well. After Plato, the idea was taken up by his pupil Eudoxus, who responded to Plato’s challenge to produce an analysis that would account for the appearances in the heavens with an explanation along the lines introduced by the Pythagoreans, involving a combination of movements of the sphere of stars and the planets. Eudoxus did this with a system not of concentric rings but of concentric spheres, and that was adopted by Aristotle and would dominate astronomy until the time of Tycho Brahe and Johannes Kepler.

*
Kepler discovered other regular solids, the “hedgehog,” for example, but they did not have all the characteristics of the original five.

*
“The Academy” also refers to the men associated with this school after Plato’s lifetime, including his successors as
scholarch
elected for life by a majority vote of the members. Aristotle was also associated with the Academy, first as a pupil and later as a teacher. In several transformations, still claiming descent from the original, the Academy lasted until the sixth century
A.D
. as a center of Platonism and neo-Platonism.

*
The writer Richard E. Rubenstein put it succinctly: “Plato did not hate the world, it simply reminded him of a better place” (Richard E. Rubenstein,
Aristotle’s Children: How Christians, Muslims, and Jews Rediscovered Ancient Wisdom and Illuminated the Dark Ages
[New York: Harcourt, 2003]).

*
The classical scholar Walter Burkert thought that the way Aristotle “occasionally plays off the Pythagorean doctrines against the Academy” makes “the conclusion unavoidable that he was using written sources without Academic coloring. Therefore he must have had at least one original Pythagorean document” (Burkert, 47).

*
For the ancient Greeks, including the Pythagoreans, 1 was neither even nor odd, and it was not a number. Number implied plurality—more than 1.

*
What emerged as a Platonic idea, the “Indefinite Dyad,” was not a Pythagorean concept. Aristotle spoke of no very important role for “Twoness” in Pythagorean doctrine.

*
The table of opposites was probably not meant to imply good (the left column) and evil (the right), though other, later such tables did. For example, for Plato’s Academy, “good” led off the left-hand column, and still later, Platonists, neo-Pythagoreans, and pseudo-Pythagorean writers rearranged the columns. Plutarch’s table was thoroughly Platonized: “Good” was on top and “Dyad” replaced plurality

*
A modern major or minor scale.

*
Plato did not call them that, though he was using them in the most Pythagorean-inspired of his dialogues.

*
Many called him Empedocles the Pythagorean, but except for agreeing about reincarnation, his ideas ran far from Pythagorean thinking.