A Cabinet Of Greek Curiosities

by J. C. McKeown

  Plato did not want mathematics to be part of philosophy, but a preliminary training for it, just like grammar and rhetoric. That is why he had inscribed over the door of his lecture room: “Let no one come in without an understanding of geometry” (Ps.-Galen On the Divisions of Philosophy 2).

  Alexander the Great asked Menaechmus the geometrician to explain geometry to him in an abbreviated way, but Menaechmus replied, “My lord, throughout the country there are Royal Roads and roads for ordinary people, but in geometry there is only one road for everyone” (Stobaeus Anthology 2.31.115). The Royal Road, though established long before, was developed by Darius I to maintain rapid contact with the far-flung parts of the Persian empire. The same story is told of Ptolemy I and Euclid.

  Someone who had begun to study geometry with Euclid asked him, when he had learned the first theorem, “What benefit do I get from learning this?” Euclid called his slave and said to him, “Give him half a drachma, for he has to make a profit from whatever he learns” (Stobaeus Anthology 2.31).

  Skill in language has a small, but essential, role to play in education, no matter what the subject may be, for the manner in which something is explained affects its comprehensibility. But fine speaking is not so very important and is really just a show put on to appeal to the listener. After all, no one uses grand language when teaching geometry (Aristotle Rhetoric 1404a).

  Archimedes deliberately wrote as briefly as seemed possible about the most complex geometrical and arithmetical concepts. He clearly had such love for those disciplines that he could not tolerate any extraneous material being mixed with them (Pappus The Mathematical Collection 8.1026).

  The surviving corpus of Greek mathematics is fairly large, but it is almost entirely lacking in literary appeal of any sort. Archimedes seems particularly austere, for he wrote in a form of Doric (the broad dialect of his native Syracuse), which struck speakers of Attic as being somewhat uncouth.

  When Archimedes was concentrating on his drawing board, his attendants had to drag him bodily away, so that they could undress him and anoint him; even then he would continue drawing diagrams in the oil on his skin (Plutarch Old Men in Public Affairs 786b).

  Dionysiodorus, the well-known geometrician from Melos, was not reliable, but I cannot bring myself to omit this instance of Greek vanity…. When he died, a letter was said to have been found in his tomb, written to those above by Dionysiodorus himself. It stated that he had gone down from his tomb to the deepest region of the earth, a distance of 42,000 stades (Pliny Natural History 2.248). Despite this autopsy, and allowing for variations in the length of a stade, Dionysiodorus’s figures for the diameter of the earth, and hence the circumference also, are about 25 percent too large.

  Courtiers are like counters on an abacus. Just as the person doing the calculation can decide that a counter is worth a mere copper coin at one moment, but a whole talent at the next, so it depends on a single nod from a king whether his courtiers are prosperous or lie groveling at his feet (Polybius Histories 5.26).

  Archimedes jumped out of his bath, yelling, “I have found it” [ερηκα, heureka] as if divinely inspired or possessed, and off he went shouting that over and over again. But we’ve never heard of any such passionate cries of “I’ve eaten it” or “I’ve kissed her” from a glutton or a sex-addict, even though there are and always have been countless thousands of such decadent people. We are disgusted by those who reminisce too passionately about meals, for deriving too much enjoyment from petty and inconsequential pleasures. But we share the enthusiasm of scientists like Archimedes and agree with Plato’s verdict on mathematics, that even though people may neglect it through ignorance and lack of experience, “it forces us to accept its importance because of the delight it brings” [Republic 528c] (Plutarch A Pleasant Life Is Impossible on Epicurean Principles 1094c).

  Whereas time causes grief and other emotions to alter and cease, when has the mere passage of time ever persuaded anyone that he has had enough of “twice two are four” or “all radii of a circle are equal” and made him change his mind about such beliefs and give them up? (Galen On the Doctrines of Hippocrates and Plato 4.7.43).

  Archimedes’s Sand-Reckoner calculated the number of grains of sand that would fill up the entire universe. He fixed on a total of 8 × 1063 grains. His calculation was based on the largest available unit of numbers, the myriad (= 10,000).

  Xenocrates declared that the number of syllables that can be produced by combining the letters of the alphabet is 1,002,000,000,000 (Xenocrates frg. 89).

  Odd numbers are more perfect than even ones, for they have a beginning, an end, and a middle, but even numbers lack a middle (Stobaeus Anthology 1.5).

  Always ensure that there is an odd number of sheep in your flocks, for this has some natural power to keep them safe and secure (Farm Work 18.2).

  Theodorus of Samothrace said that Zeus laughed for seven days without a pause after he was born, and that is why seven was considered a perfect number (Photius Lexicon 152b).

  • There are seven phases of the moon: twice crescent, twice half moon, twice gibbous, and once full.

  • The Bear and the Pleiades have seven stars.

  • The equinoxes are seven months apart [by inclusive reckoning], as are the solstices.

  • The soul has seven parts: the five senses, the voice, and the reproductive part.

  • The body has seven complete parts: the head, the neck, the chest, two feet, and two hands.

  • There are seven internal organs: the stomach, the heart, the lung[s], the liver, the spleen, and the two kidneys.

  • There are seven openings in the head: two eyes, two ears, two nostrils, and one mouth.

  • We see seven things: body, distance, shape, size, color, movement, and position.

  • There are seven variations in the voice: sharp, deep, circumflected, rough, smooth, long, and short.

  • There are seven types of movement: up, down, forward, back, right, left, circular.

  • There are seven vowels: α, ε, η, ι, ο, υ, ω.

  • The lyre has seven strings.

  • Plato composed the soul from seven numbers in the Timaeus.

  • There are seven ages of life: infant, child, adolescent, young man, adult, elder, and old man, and the change from each age to the next takes place every seven years.

  (Anatolius On the First Ten Numbers 11; he comments that “all things are seven-loving,” but gives similar, if rather shorter, lists for each of the other numbers from one to ten)

  A seven-string lyre.

  Why should numbers be equated with causes? There are seven vowels, a musical scale comprises seven chords, there are seven Pleiades, animals lose their teeth when they are seven years old (at least, some do, but others do not), there were seven heroes who fought against Thebes. So, is it the nature of the number seven that ensures that there were seven heroes, or that the Pleiades are made up of seven stars? Or could it be that there are seven heroes because Thebes had seven gates, or for some other reason? We count seven stars in the Pleiades, and twelve in the Bear, but people elsewhere count more stars in both (Aristotle Metaphysics 1093a, arguing against the Pythagoreans’ tendency to see numbers as the origin of things).

  The day was divided into twelve equal segments. The number twelve was chosen because it is the most useful. It can be divided into halves, thirds, quarters, sixths, and twelfths, a quality possessed by no lower number and by no higher number until twenty-four (Galen On the Diagnosis and Cure of the Sins of the Mind 5.83).

  NUMBER GAMES

  The twenty-four letters of the alphabet were used to denote numbers, with three additional characters, the archaic letters digamma, koppa, and sampi, used for 7, 90, and 900.

  α

  1

  β

  2

  γ

  3

  δ

  4

  ε

  5

  ζ

  6

  Ϝ

  7r />
  η

  8

  θ

  9

  ι

  10

  κ

  20

  λ

  30

  μ

  40

  ν

  50

  ξ

  60

  ο

  70

  π

  80

  90

  ρ

  100

  σ

  200

  τ

  300

  υ

  400

  φ

  500

  χ

  600

  ψ

  700

  ω

  800

  Ϡ

  900

  Hence, for example, the word ἄλφα (alpha) has a value of 532 (1 + 30 + 500 + 1), while the word βῆτα (beta) has a value of 311 (2 + 8 + 300 + 1). There was a certain fascination in matching words and phrases that had the same numerical equivalents. Oxyrhynchus Papyrus 3239 contains a list of such “isopsephic” pairs. For example:

  The principle was applied to prophecy:

  Seeing a weasel in a dream signifies an evil and tricky woman and a law-suit, for “law-suit” (δίκη, dike) and “weasel” (γαλῆ, gale) are isopsephic (42).

  (Artemidorus Interpretation of Dreams 3.28.20)

  Seeing an old woman in a dream foretells death to a sick person, since γρας (graus, “old woman”) adds up to 704 and ἡ ἐκφορά (he ekphora, “the funeral”) adds up to 704. An old woman symbolizes a funeral in any case, since she is going to die in the not very distant future.

  (Artemidorus Interpretation of Dreams 4.24.5)

  Suetonius preserves the most famous isopsephism, current in Rome after Nero killed his mother, Agrippina (Life of Nero 39):

  Nέρων/ἰδίαν μητέρα ἀπέκτεινε Neron/idian metera apekteine Nero/own mother killed

  50 + 5 + 100 + 800 + 50 = 1,005

  10 + 4 + 10 +1 + 50 + 40 +8 + 300 + 5+ 100 + 1 + 1+ 80 + 5 + 20+ 300 + 5 + 10 + 50 + 5 = 1,005

  There are a number of stylish isopsephic epigrams in the Greek Anthology by the mathematician and astronomer Leonides of Alexandria (6.321, 6.322, 6.324–29, and 9.344–56). In 6.329, addressed to Agrippina in celebration of her birthday, the letters in both couplets add up to 7,579: might Leonides have had something to do with the isopsephic charge of matricide?

  Leonides’s first sequence in Book Six of the Greek Anthology is interrupted by an epigram of a perhaps even more ingenious type, by the otherwise unknown Nicodemus of Heraclea. It is palindromic, making the same sense and preserving the same strict meter whether it is read forward or backward. Nicodemus’s other epigrams, 6.314–20 and 9.53, are also all palindromic. I have never heard of anyone nowadays attempting such a poem, however brief.

  Greek Anthology 12.6, one of many homosexual epigrams by Strato of Sardis, is a wittily forlorn and modestly isopsephic lament:

  πρωκτός (proctos, rectum) and χρυσός (chrysos, gold) have the same numerical value [1,570]. I discovered this once when I did a simple calculation.

  Homer Iliad 7.264 and 7.265 both add up to 3,508:

  ἀλλ’ ἀναχασσάμενος λίθον εἵλετο χειρὶ παχείῃ

  κείμενον ἐν πεδίῳ μέλανα τρηχύν τε μέγαν τε

  all’ anachassamenos lithon heileto cheiri pacheie

  keimenon en pedio melana trechun te megan te

  But giving ground he grasped a stone in his strong hand,

  a black one, both rough and big, lying on the plain.

  This monumentally useless discovery was made by an unknown scholar in the post-Renaissance period, and Aulus Gellius (Attic Nights 14.6) seems to imply that other such pairs of lines were known in antiquity. It has recently been determined, by computer search, that there are four more such pairs in the Iliad (1.490–91 = 3,833, 5.137–38 = 4,580, 6.162–63 = 3,221, 19.306–307 = 2,848) and three in the Odyssey (13.245–46 = 2,885, 18.388–89 = 3,511, 18.401–402 = 3,515) (J. L. Hilton Classical Journal 106 [2011] 386).

  Plutarch’s schoolteacher friend, Zopyrio, shows sound common sense when he says that it is a mere coincidence that there is an equal number of syllables (seventeen) in the first lines of the Iliad and Odyssey, and likewise in their final lines (sixteen) (Plutarch Table Talk 739a). As Quintilian says about compulsively detailed enquiries into the arcane aspects of mythological stories, “there are some things that it is to a teacher’s credit not to know” (Education of the Orator 1.8).

  XVIII

  SCIENCE AND TECHNOLOGY

  Atomic theory goes back a long way, for it was devised by Mochus of Sidon before the Trojan War

  (Posidonius frg. 285).

  NATURAL SCIENCE

  We do not consider any of our senses as actually being wisdom, even though they are our most authoritative sources of detailed information. They do not tell us the “why” about anything. For example, they do not tell us why fire is hot; they tell us simply that it is hot (Aristotle Metaphysics 981b).

  Things burn more quickly in winter than in summer, for the summer makes fire weaker, just as the sun also does, and just as fire itself makes light weaker, whereas winter and the coldness of the surrounding air concentrate fire. Anything that is concentrated is strong—that is why light in lamps carries farther (Theophrastus On Fire frg. 12).

  To whiten a genuine pearl, the Indians use this method. They give it to a rooster with its feed in the evening and search for it in its excrement in the morning. The pearl has been cleaned in the bird’s gizzard and is as bright as it ever had been (Stockholm Papyrus 60).

  Celandine root dyes fabric a gold color, by cold dyeing. But celandine is expensive, and the same result can be obtained with pomegranate root. If wolf’s milk is boiled and then reduced to a powder, it dyes fabric yellow (Stockholm Papyrus 139). It seems reasonable to wonder how this particular quality of wolf’s milk was discovered and how dyers obtained supplies.

  A tunnel dug by Eupalinus of Megara on the island of Samos in the 6th century B.C. carried a water supply about half a mile through a mountain for over a thousand years. It was rediscovered in the 1880s and is regarded as one of the greatest feats of ancient Greek engineering.

  The earth’s core consists almost entirely of frost and ice (Plutarch On the Principle of Cold 953e).

  [In the Red Sea] the sun does not spread its light before it rises, as it does with us. They say that, when it is still pitch-dark night, the sun shines out suddenly and unexpectedly over the middle of the sea, like a very fiery lump of coal, shooting out huge sparks…. Quite opposite phenomena are said to occur in the evening: the sun seems to illuminate the world for not less than two hours, or even three, after it has set, and this is regarded by the natives of that region as the most pleasant time of the day, since the temperature goes down when the sun sets (Diodorus Siculus The Library 3.48).

  The Epicureans make the foolish claim that the sun consists of atoms, and that it is born with the day and dies when the day dies (Servius on Virgil Aeneid 4.584).

  Heraclitus says, “The sun is as broad as a human foot, and does not grow bigger: for if it becomes broader, the Furies, allies of Justice, will find it out” (Column 4.5–9 of the 4th-century B.C. Derveni Papyrus, the oldest of all surviving Greek papyri, and one of the few substantial papyrus texts found outside Egypt).

  If someone reached the limit of the sky and stretched out his hand, to where would he stretch it out? Surely not into nothing, for nothing that exists is in nothingness. But he will not be prevented from stretching out his hand, for it is not possible to be prevented by nothingness (Archytas of Tarentum, as cited at Eudemus frg. 65).

  Now that all the richer, softer soil has been washed away, only the bare ground is left, like the bones of a diseased body. In former times … the plains were full of fertile soil and there was abundant timber in the mountains … in parts of which there
is now only food enough for bees…. The annual rainfall used to make the land fruitful, for the water did not flow off the bare earth into the sea…. Where once there were springs, now the shrines [to the deities of the stream] are all that is left (Plato Critias 110e, on the deforestation of Attica).

  We should not be too confident in dismissing as incredible the theory that India is connected to the region near the Pillars of Heracles, with the ocean forming a single unit. Those who support this opinion point out that the occurrence of elephants at both these extreme parts of the earth proves that they are connected (Aristotle On the Heavens 298a). At Meteorology 362b, Aristotle observes that it is possible to travel right around the world, the only potential obstacle being the sheer extent of the ocean.

  In earlier times, all islands wandered about and had no foundations (Scholion to Apollonius of Rhodes Argonautica 3.42).

  Why is it warmer when the sky is cloudy than when it is not? Is it because, as people said in antiquity, the stars are cold? (Ps.-Aristotle Problems 939b).