* 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.
* Scholars such as Kahn think these men were not fictional and that their words reflected a much older line of Pythagorean speculation.
* In search of the source of Iamblichus’ lists of Pythagoreans, Burkert believed he had narrowed down the possibilties, conclusively, to Aristoxenus (Burkert, p. 105, n. 406).
* When someone asked what the practical use of one theorem was, Euclid turned aside to his slave, sniffed, and muttered, “He wants to profit from learning, give him a penny.” The Pythagorean aphorism was “A diagram and a step (an advance in knowledge), not a diagram and penny.”
† The three surviving books in which he included material about the Pythagoreans are Metaphysics, Physics, and On the Heavens.
† Recall that the regular solids each fit neatly into a sphere, and the fifth is close to being a sphere.
† Copernicus would point to Heracleides Ponticus as an ancient precedent when he presented his hypotheses in the sixteenth century. The Earth also rotated in Plato’s Timaeus, and the idea was probably not original with either man, for Philolaus and possibly earlier Pythagoreans thought part of the apparent movement of the heavens was caused by the movements of the earth. Copernicus also referred to Hicetas and Ecphantus of Syracuse.
* The Elements was translated by B
oethius in about A.D. 480, but not until A.D. 1120, when Athelhard of Bath translated it again, this time from Arabic into Latin, did mathematicians begin to appreciate its worth.
* Cicero’s life, and his political life, began when Rome was a republic and ended after the assassination of Julius Caesar and the beginning of the reign of Octavian (Caesar Augustus). He was a strong supporter and defender of the republic and strove on its behalf during the civil wars.
* The Romans continued to use this formation effectively through the years of their republic and in the expansion of their Empire.
† Alcibiades’ reputation for lack of discipline and unscrupulousness was later used to support the charges brought against Socrates of corrupting the youth of Athens, which resulted in Socrates’ death sentence.
* Pliny lost his life when his insatiable curiosity about natural phenomena tempted him too close to the erupting Vesuvius.
* Cicero made several references to this celestial phenomenon that had appeared in the year 129 B.C. The scientific name is parhelion, in the vernacular a mock sun or sun dog. The appearance is of two extra suns, one on each side of the Sun. This happens when the Sun is shining through a thin mist of hexagonal ice crystals falling with their principal axes vertical. If the principal axes are arranged randomly in a plane perpendicular to the Sun’s rays, the appearance is of a halo around the Sun.
* Timaeus of Locri was the central character in Plato’s Timaeus, but there was no real person by that name. Writings attributed to him cannot be considered examples of Pythagorean doctrine. They are an interpretation of Plato’s Timaeus, from the first century B.C. or the first century A.D.
* Diogenes Laertius copied the excerpt not from the original but from an earlier author named Alexander Polyhistor who in turn–this was in the first half of the first century B.C.–copied it from a still older book.
† In view of all the other anachronisms in the Notebooks, scholars have ruled out the possibility that they were, after all, authentically early and primitively foreshadowed Aristotle’s cosmos.
* One clue has turned out to be a red herring: the suggestion that inclusion of superstition and “marvelous” events in a work represented more “primitive” thinking and dated the material earlier. Tales about a talking river or being in two places at the same time indicated that what you were reading was authentically early, so it was claimed. However, the late fourth century and the third, second, and first centuries B.C. and the early A.D. centuries were as accepting of magic, marvels, and portents as the fifth and sixth centuries B.C. had been—arguably more so. Such elements were expected in the biography of an important leader. Aristotle wrote during this period, when people may have been more ready to believe in a golden thigh than their fore-bears would have been at the time of Pythagoras. Clement of Alexandria, an eminent Christian scholar of the second and early third centuries A.D., described a “standard educational curriculum... astrology, mathematics, magic, and wizardry”—a quadrivium that would seem appropriate for Harry Potter’s Hogwarts School. “The whole of Greece,” Clement lamented, “prides itself on these as supreme sciences” (Clement of Alexandria, Stromateis 2.1.2. 3–4. Quoted in translation in J. Robert. Wright, ed., Ancient Christian Commentary on Scripture, Old Testament IX [Downers Grove, Ill.: Intervarsity Press, p. 18]). For Diogenes Laertius, Porphyry, and Iamblichus, the fact that material included the miraculous did not invalidate the information or call the source into question. There was probably a mystical or magical element to the earliest Pythagoreanism, but late Greek, Alexandrian, and Roman writers were eager to report and exaggerate it. It is difficult to see through the veil of a superstitious age and judge how skeptical an earlier era was, but it is clear that one cannot decide that information was more authentically ancient simply because it included more of the “marvelous.”
* There was a legend about a Christian Philo, even a Bishop Philo, and a story in which he met the Apostle Peter.
† Shakespeare found the stories of Antony and Cleopatra, Timon of Athens, and Coriolanus in the Lives, and sometimes used Plutarch’s words virtually verbatim or changed them (as he read them in translation) only as much as was necessary to transform them into verse.
* Nicomachus was also intrigued by a pseudo-science called gematria that was not Pythagorean but originated with the ancient Babylonians and survived in ancient Greece and the Hellenistic period. In gematria, each letter of the alphabet had a numerical value. A word could be spelled in numbers. Sargon II, in the century before Pythagoras’ birth, had the wall of Khorsabad built to a measurement that was the numerical equivalent of his name—16,283 cubits. The name for the Gnostic divinity Abraxas had the numerical value of 365, the number of days in a solar year. Nicomachus did not claim that gematria was a Pythagorean practice, and it was not.
* Plotinus used and developed Numenius’ thoughts so extensively that he was accused of plagiarism. A colleague came to his rescue by writing an entire book to point out the differences between the two.
* This scale adds up to more than an octave, a problem easily corrected by changing the interval between Saturn and the stars to a half tone, as music theorists in later antiquity corrected Pliny. A half tone (half step) is the interval between one key and the next—black or white—on a piano.
* Along with all other pagan schools, the Academy would close in 526, two years after Boethius died, by order of the Byzantine emperor Justinian.
* Nestorian Christians were a group that originated in Asia Minor and Syria in the fifth century A.D. and stressed the human nature of Christ. There are still many thousands of them; today called the Church of the East, the Persian Church, or the Assyrian or Nestorian Church. Most Nestorians live in Iraq, Syria, and Iran.
† It is indicative of the cosmopolitan mix of religions and ideas in the Middle Ages in Islamic regions of the world that Hunayn’s writing, reflecting ancient pagan ideas and coming from a Christian who lived and worked in Islamic Baghdad, survived mainly because of a twelfth/thirteenth-century Hebrew translation by Judah al-Harizi.
* By “perfect number” they did not mean what the Pythagoreans had meant when they identified 10 as the perfect number. A perfect number by more modern standards (found already in Nicomachus) is a number the sum of whose divisors equals the number. The number 6 is the smallest perfect number: 1 + 2 + 3 = 6.
* A planet’s period is the time it takes to complete one orbit.
* Aurelian was reading from a mistranslation of the Book of Job.
* Regino’s description sounds very much like Aristotle’s, which means he must indeed have gotten it through Boethius. Regino lived before the reintroduction of Aristotle to Latin Europe.
* In the mid-twentieth century, there was still one expert, Vincenzo Capparelli, who was convinced that Pythagoras invented Arabic numerals (Vincenzo Capparelli, La sapienza di Pitagora [Padua: CEDAM, 1941]).
* Most who used an abacus were still using Roman numerals, the English exchequer as late as the sixteenth century! (H. G. Koenigsberger, Medieval Europe, 400–1500 [Harlow, England: Longman Group, 1987], p. 202.)
* T. S. Eliot echoed those sentiments when he suggested that to those who say we shouldn’t read the old authors since we know so much more than they did, we should answer, “And they are what we know.”
* The “Chaldean Oracles,” written in verse in the second century A.D. by a man named Julianus the Theurgist and his son, combined Babylonian and Persian beliefs with Platonic and neo-Platonic philosophy and became an important religious book for neo-Platonists.
† “Chaldean” in this case meaning Babylonian.
* The great Andrea Palladio was to write four.
* Alberti’s most important buildings included, in Florence, the Palazzo Rucellai, the Rucellai Chapel, the Annunziata, and the façade of the Maria Novella church; in Rimini, the Tempio Malatestiano; and in Mantua, the churches of San Sebastiano and San Andrea.
* Ecphantus the Pythagorean lived in the fourth century B.C. There is s
ome suspicion that he may have been only a fictional character in one of Heracleides’ dialogues, but Copernicus thought he was a historical person, and most modern scholars tend to agree.
* A regular polygon is a flat shape in which all edges are the same length. For example: the triangle, square, pentagon, hexagon, etc. ad infinitum.
* A regular polyhedron is a solid shape in which all the edges have the same length and all the faces the same shape. The Pythagorean or Platonic solids are the regular polyhedra.
† When astronomers of Kepler’s time and earlier spoke of the “spheres,” they did not mean the planets. The Ptolemaic view of the cosmos had the planets traveling in transparent “crystalline spheres,” nested within one another like the layers of an onion and centered on the Earth. Though Kepler and Mästlin discussed spheres in their correspondence about Kepler’s new idea, Kepler (like his predecessor Tycho Brahe) did not believe there were actual glasslike spheres that one could crash through in a space vehicle. Thinking about them in a geometrical sense, not as physical reality, was nevertheless helpful in visualizing the movements of the planets.
The Music of Pythagoras Page 39