Chapter 4.
From Babylon to Copernicus
~~~~~~~~~~~~~~~~
1. Among the most famous
of past astrologers have been the Babylonians.
The religion and science of the ancient Babylonians, especially of
their soothsayers, worshippers of Bel (Marduk), were bound to the stars.
There was much concern with the foretelling of human destiny.
The notion of a connection
between astral bodies and human destinies appears to have been part of a
central concept that the cosmos contains nothing fundamentally dead or
inimical. The observations made
by Babylonian astronomer-priests reflect a longing to establish precisely the
interdependence between stars and earth
and man. S. Giedion says:
"In an often retold dream of that great figure of
the early period, Gudea of Lugash, the goddess Nisibis appeared to him
not only as the goddess of intelligence, wisdom, mathematics, and writing; she
also 'bore the tablet of the good star' -- in other words, she was
simultaneously goddess of astrology."
(S. Giedion, The Beginnings of Architecture, 1964, p. 9, 19,
138-139.)
2. Édouard Dhorme says of
the early Mesopotamians: "For
the Sumerians and Akkadians, the sky was, in effect, a great map on which
their destiny was inscribed. Men
called the constellations 'the writing of heaven' or 'the writing of the
firmament'." The experience of the night side of life, and the feeling of
being utterly at the mercy of destiny, permeated Mesopotamian existence.
Later, the Greeks took over the idea of destiny, without being led into
the deep pessimism already revealed in the depressing adventures of Gilgamesh,
around 2600 B.C. This interest in destiny was closely linked with a desire to
fathom in advance the will of the gods. The
stars were identical with the deities. They influenced all happenings and were thus guides to man's
fate. Everything depended on
whether the initiate was able to read the decisions of the gods from the
movements of the stars. It has
not been clearly proven just when this sort of belief in the stars arose.
But it must be closely linked with an anthropomorphization of the
universe, and thus it must have found its form shortly before or at the
beginnng of historical times....."
(Êdouard
Dhorme, Les Religions de Babylonie et d'Assyrie, 2nd edition, 1949, p.
282, p. 138-140.)
3. The Mesopotamians built
awe-inspiring structures called ziggurats, towers composed of series of
terraces joined by steps, with temples on top, probably containing places for
making sacrifices. "Both ziggurat and pyramid derive their existence,"
says Giedion, "from man's awakened urge toward the vertical as a symbol
of contact with the deity, contact with the sky... The notion of a ladder between heaven and earth was
marvelously portrayed." (l.c.,
p. 219, 225.) The tower of Babel
in the Bible is probably the great ziggurat at Babylon.
The word "Babel" means "gate of the God" in
Akkadian. There is a
similar-sounding word in Hebrew which means "confusion." Perhaps there is a pun in the Biblical story of the tower of
Babel concerning the confusion of tongues.
4. Relatively late in
their history, certain Babylonians were also pioneers in mathematical
astronomy. However, they made
accurate celestial observations for a long time before they developed their
mathematical astronomy. Simplicius,
for example, in his commentary on Aristotle's De caelo (6th century A.D.)
speaks of a sequence of observations sent by Callisthenes to Aristotle (4th
century B.C.) which had extended over 1903 years.
(Referred to by Marguerite Rutten in La Science des Chaldéens,
1970, p. 89-90). We may take with
a grain of salt, Rutten says, the assertion of Iamblichus (c. 250-330 A.D.)
that the Babylonians had observed the stars for 72,000 years.
5. Did the Babylonians'
astronomy grow out of their astrology, or vice versa -- or did they grow up
together? Otto Neugebauer says,
comparing astronomy and astrology: "It has often been said that astronomy
originated from astrology. I see
no evidence for this theory..... The best description of the true situation
might be the statement that we know equally little about the origin of
astrology or astronomy and that the relative influence of these two
disciplines on one another is largely a matter of conjecture."
(Otto Neugebauer, The Exact Sciences in Antiquity, 1957, p.
168.)
6. Rutten quotes Strabo,
the geographer (c. 60 B.C.-20 A.D.): "There
is in Babylonia a caste or colony of indigenous philosophers called "Chaldeans"
who concern themselves chiefly with astronomy.
Some also specialize in casting horoscopes, but they do not have the
approval of the others." (Rutten,
ibid., p. 89). According to
Rutten, this proves that alongside the astrologer-diviners there were true
astronomers, in the modern sense of the word.
Unfortunately, one can construe Strabo's statement to mean that some of
the philosophers frowned on personal astrology concerning individuals, as
contrasted with omen astrology, concerning nations or peoples, or natural
phenomena.
7. Neugebauer saw no
evidence that astronomy grew out of astrology, but Édouard
Dhorme did. He says:
"It was inevitable that a close relationship be established
between observation of the stars and the calendar, which gives measurements of
the celestial vault. The astrologers were in this way led to study the lives of
the gods not only in space, but also in time.
It was necessary for them to take note of the celestial phenomena which
gave to each day of the month and of the year its peculiar physiognomy.
The necessity of avoiding errors and giving a mathematical precision to
the results obtained quickly caused the synthesis of astrological observations
to be transformed into an exact science.
In this way, astronomy detached itself from astrology.
The religious apparatus which surrounded the calculations of the
diviners ended by passing into the background.
The divination tables were only empirical findings, but they continued
to answer to the need of the human soul to probe into the darkness of the
future. Astrology acquired a new
expansive force by separating itself from its indigenous culture.
It is in this way that it penetrated into Asia Minor, in particular
among the Hittites, and from there as far as Greece and Rome, where the
Chaldeans distinguished themselves as drawers of horoscopes and
fortune-tellers." (Dhorme, ibid., p. 288-289.)
8. Despite the fact that
the Babylonian astrologer/ astronomers are customarily said to have been
priests (Herodotus called them this), some Babylonians may have taken a
relatively secular attitude toward the stars.
A. Laurent says: "In Egypt, most of the books which treated science were
considered sacred books, composed and revealed by the gods themselves.
The Chaldeans, and later their disciples the Assyrians, attributed a
less elevated origin to their similar books.
For them, they were simply the fruit of the experience of educated men
and of generations of patient observers.
In particular, the treatises on divination (astrology, the science of
omens, haruspicy, etc.) appear to us, in fact, quite like the work of a number
of scholars who, through the centuries, have recorded from day to day the
relations which seemed to them to exist between the events of political or
private life and different sidereal or terrestrial phenomena.
Neither the Chaldeans nor the Assyrians did anything to obscure the
human origins of these treatises." (A.
Laurent, La Magie et la Divination chez les Chaldeo-Assyriens, 1894, p.
58.)
9. Observations of the
stars have long been connected with determination and maintenance of
calendars. Dhorme, speaking of
this relation, attributes to the Babylonians a calendar having a year of 12
months with 30 days each, plus a 5-day intercalary period. This
calendar, however, appears to have originated with the Egyptians.
Plutarch (c. 46-120 A.D.) says: "They
say that the Sun, when he became aware of Rhea's intercourse with Cronus,
invoked a curse upon her that she should not give birth to a child in any
month or any year; but Hermes, being enamoured of the goddess, consorted with
her. Later, playing at draughts
with the moon, he won from her the seventieth part of her illumination, and
from all the winnings he composed five days, and intercalated them as an
addition to the three hundred and sixty days.
The Egyptians even now call these five days intercalated and celebrate
them as the birthdays of the gods."
(Plutarch, "Isis and Osiris", in Plutarch's Moralia,
translated by Frank Cole Babbitt, 1936, v. 5, p. 31.)
10. Neugebauer says of the
Egyptian calendar of 12 30-day months plus 5 intercalated days that "this
calendar is, indeed, the only intelligent calendar which ever existed in human
history." (Otto Neugebauer, Exact Sciences in Antiquity, 1957,
p. 81.) He thus goes further than
Herodotus (c. 485-425 B.C.), who says that the priests of Egypt with whom he
talked "all agreed in saying that the Egyptians by their study of
astronomy discovered the solar year and were the first to divide it into
twelve parts --and in my opinion their method of calculation is better than
the Greek; for the Greeks, to make the seasons work out properly, intercalate
a whole month every other year, while the Egyptians make the year consist of
twelve months of thirty days each and every year intercalate five additional
days, and so complete the regular circle of the seasons." (Herodotus, The Histories, ii.4, translated by Aubrey
de Sélincourt,
1954, p. 130.) It may be that Dhorme confuses this Egyptian calendar with
the Babylonian lunar calendar in which some years have 12 months and others 13
months of 30 days each. This was
at first done irregularly, and later with 7 13-month years every 19 years (Neugebauer,
l.c., p. 102.) Such a 13th month
of 30 days can be considered to be an intercalation.
Dhorme, indeed, speaks of intercalating a month of 30 days into a 12
month calendar of 30 days each.
11. Did the Sumerians
already have astrology in early Mesopotamian culture?
O. R. Gurney says: "The
only clear evidence that the Sumerians already practised astrology comes from
the cylinder of Gudea (c. 2143-2124 BC).
In his first dream this ruler saw the goddess Nisaba studying 'a tablet
of the star (or stars) of heaven', which was interpreted to mean that she was
proclaiming 'the pure star for the building of the temple'.
In what way the star was thought to give such a sign is not explained.
From Mari, of the time of Hammurapi (c. 1780 BC), there is a letter
from the barû [professional omen inspector, a priest] Asqudum, which
is very revealing. The diviner
reports an eclipse of the moon; he knows that this is a bad omen, but no more,
proceeds to check the findings by haruspicy, and declares that after all the
outlook is favourable. Evidently
at this time haruspicy was the only reliable form of divination.....
It seems that it was not till much later that astrology rose to
prominence as a rival to haruspicy. That
it eventually did so is seen in some 600 reports on ominous events sent in to
the Assyrian king Esarhaddon (680-669 BC) from scholars posted in widely
distributed centres throughout the empire.
The great majority of these are astrological in character and are often
in response to an enquiry from the king as to the meaning of an ominous event.
Like the extispicy reports, they quote the relevant omens from the
handbook, here complete with the prediction, and a conclusion is drawn
regarding the general significance of the omen for the king, but never in
relation to a particular matter of policy.
Astrology could not be used, as extispicy was, to answer specific
questions. The officials who
write these reports are not barû priests but scholars with various
professional designations. One is
called 'scribe of "When Anu and Enlil" '. A special title which does not occur elsewhere is 'Chief of
the team of ten'."
12. "Horoscopic
astrology, the 12 signs of the zodiac, and the doctrine of the hypsomata were
a still later development. The earliest horoscope (now in Oxford) dates from 410 BC.
Two astrological manuals show drawings of the hypsomata, or positions
of greatest astrological influence: the
moon in Taurus, Jupiter in Cancer, Mercury in Virgo.
They date from the Seleucid period (after 300 BC).
The texts attached to these drawings have by now reached the refinement
of dividing each sign of the zodiac into twelve 'microzodiacs' of 2 1/2 days
each. This sophisticated
astrology, for which the 'Chaldeans' were renowned in the Roman world, was
only developed after the fall of Babylon to the Persians in 539 BC."
(O. R. Gurney, in Oracles and Divination, 1981, edited by
Michael Loewe and Carmen Blacker, p. 160-162.)
13. Samuel Angus makes the
claim that astrology made the Greek and Roman methods of inquiry into the
future antiquated. Augury and
haruspicy were practically abandoned.
Official oracles, like the one at Delphi, though revived under the
empire, had stiff competition, he says, from the Chaldaei and mathematici, as
well as from Christian and Gnostic apocalypses.
(Samuel Angus, The Mystery-Religions and Christianity, A Study in
the Religious Background of Early Christianity, 1925, p. 167.)
14. Prominent Greek
scientists such as the astronomer and mathematician Eudoxus (c. 390-340 B.C.)
and Theophrastus (c. 372-286 B.C.), student and successor of Aristotle,
studied the star-worship and astrological practices of the Babylonians.
According to Proclus (c. 412-485 B.C.) in his commentary on Plato's Timaeus,
Theophrastus, in his book On Signs, credited the Chaldeans of his time with a
theory with which they could predict "every event, and the life and death
of every person." (Pierre Duhem, Le Système
du Monde, 1913,
v. 2, p. 275.) Near the end of
the 3rd century B.C., professional astrologers from Babylonia set up business
among the Greeks. Michael Grant
tells us: "The first of
these practitioners was said to be the Babylonian priest Berossus, translator
of The Eye of Bel, who moved to Cos and founded an astrological school
on the island (c. 280 [B.C.]). But
it was not until after 200 that the movement reached the proportions of a
flood. This was the time when
Bolus of Mendes in Egypt (a country that had learnt its astrology from
Mesopotamia) compiled a treatise On Sympathies and Antipathies which
explained and justified the fictitious correspondence between heavenly bodies
and human beings. His book became
one of the most influential best-sellers of all time.
Another successful work was an astrological textbook, probably written
c. 150-120, which went under the probably fictitious Egyptian names of
Nechepso and Petosiris." (Michael Grant, From Alexander to Cleopatra,
TheHellenistic World, 1982, p. 214-222).
15. These beliefs fit
easily into Stoic doctrines, and the Stoics maintained astrological doctrines
from early on. It was, as we said
earlier, an understandable outgrowth of dismay at a world which seemed to be
rules by chance and fickle fortune. One
of the leaders of the Stoic school, Diogenes 'the Babylonian' from Seleucia on
the Tigris (d. 152 B.C.), maintained that the souls of men and women contain a
spark of the power that rules the heavens.
Grant says of this Diogenes: "Building
on his forerunner Cleanthes' veneration of the sun and the celestial bodies,
[he] became the traitor withing the gates who welcomed astrology for its
apparently convincing proof of this 'Sympathy of all Creation'."
Another Stoic, Panaetius of Rhodes (c. 185-109 B.C.) rejected the idea
that the sun, moon and stars causally affect the affairs of the world,
although he was willing to accept the validity of divination. But soon afterwards an influential Stoic, Posidonius of
Apamea in Syria (c. 135-50 B.C.), welcomed the basic astrological principles
as keys to the harmony of the universe.
16. Some believers in such
principles allowed a limited scope for free will, but nevertheless considered
themselves to be ruled by the unchanging and inescapable heavenly spheres,
which predestine all that happens. Others
revolted against a pitiless mechanical inevitability and sought means to
circumvent or reduce the oppressiveness of the astral powers.
This required finding out what the powers had in store, and how to
arrange one's activities to avoid their most hostile intentions.
For this, experts were needed: professional
astrologer/astronomers. These
became an influential group, who provided numberless believers with a
principal interest, consolation and excitement.
They cast horoscopes, in which the future destiny of a person was
worked out from the positions of heavenly bodies at the time of his or her
birth. The astrologer/
astronomers not only prophesied future destinies, but also counseled people on
how to outwit what had been destined. They mixed a kind of science with a kind of magic.
17. In science, as in
religion, a kind of submission seems to be required to some degree to what
there is and must be, while with magic a there is customarily intent
to dominate, to manipulate the gods, or the way nature works, or to
interfere with fate. With
technology, including applications of science, we often try to manipulate
nature. But with magic, we try to
change the will of the gods, or the laws of nature.
Magic rests on the assumption that we are not underlings in ways that
science or religion profess. Not
even the sky is the limit. Belief
in the power of magical manipulations was widespread in Hellenistic times.
There were some who investigated the laws by which the stars move,
without trying to alter either the laws or the stars, but a man might be at
the same time an astronomer and an astrologer, and maybe a magician, too.
18. The Babylonians were
known to the Greeks and Romans not only as astrologers, astronomers and
magicians, but as diviners by other methods.
Writing about 161 or 162 A.D., the satirist Lucian tells how Menippus
makes a descent into Hades to find out the right way to live.
He finds that the good life is not that of the rich and powerful, nor
that of a philosopher, but the ordinary life of one who lives in the present
and laughs a lot. To make his
descent into Hades, Menippus says: "...
I resolved to go to Babylon and address myself to one of the Magi, the
disciples and successors of Zoroaster, as I had heard that with certain charms
and ceremonials they could open the gates of Hades, taking down in safety
anyone they would and guiding him back again.....
Well, springing to my feet, I made straight for Babylon as fast as I
could go. On my arrival, I
conversed with one of the Chaldeans, a wise man of miraculous skill, with grey
hair and a very majestic beard; his name was Mithrobarzanes.
By dint of supplications and entreaties, I secured his reluctant
consent to be my guide on the journey at whatever price he would. So the man took me in charge, and first of all, for
twenty-nine days [approximately a lunar month], beginning with the new moon,
he took me down to the Euphrates in the early morning toward sunrise, and
bathed me; after which he would make a long address which I could not follow
very well, for like an incompetent announcer at the games, he spoke rapidly
and indistinctly. It is likely,
however, that he was invoking certain spirits."
19. "Anyhow, after
the incantation he would spit in my face thrice and then go back again without
looking at anyone whom he met. We
ate nuts, drank milk, mead, and the water of the Coaspes, and slept out of
doors on the grass. When he considered the preliminary course of dieting
satisfactory, taking me to the Tigris river at midnight he purged me, cleansed
me, and consecrated me with torches and squills and many other things,
murmuring his incantation as he did so. Then
after he had be charmed me from head to foot and walked all about me, that I
might not be harmed by phantoms, he took me home again, just as I was, walking
backward. After that, we made
ready for the journey. He himself
put on a magician's gown very like the Median dress, and speedily costumed me
in these things which you see -- the cap, the lion's skin, and the lyre
besides; and he urged me, if anyone should ask my name, not to say Menippus,
but Heracles or Odysseus or Orpheus."
(Lucian, Lucian, v. 4, "Menippus", translated by A. M.
Harmon, 1925, p. 83-87.)
20. The ancient Chinese,
on the whole, seem not to have become as secular-minded as the Babylonians
about the stars. Edward Schafer says that for most early Chinese, even for the
most advanced authorities, astronomy was indistinguishable from astrology.
As understanding of stellar motions was refined, and more and more
aspects of the starry firmament were removed from the realm of conjecture,
doubt and fear into the realm of the known and predictable, this
identification remained. Comets,
meteors and supernovae remained terrible signals from the powers in space, and
it would be wrong to suppose that the inclusion of quite reliable ephemerides
in a medieval Chinese almanac means that movements of celestial objects had
become accepted as merely physical transits of the sky.
Schafer says: "There were certainly skeptics, but it appears that most
men, even well-educated men, continued to believe that a predictable Jupiter
remained an awful Jupiter." Moreover,
the Chinese devoted little energy to making geometrical models of the physical
universe which would account for their observations and arithmetical
calculations. "Indeed,"
says Schafer, "cosmology languished close to the borderlands of
mythology, and for many, perhaps most people, the two were identical."
The obliquity of the ecliptic, the precession of the equinoxes, and the
true length of the tropical year were discovered quite early, but this didn't
put the diviners out of work. (Edward
Schafer, Pacing the Void, T'ang Approaches to the Stars, 1977, p.
9-10.)
21. According to Schafer,
a remarkable feature of T'ang astronomy/astrology was the extent of Indian
influences on it. A similar
condition prevailed centuries later, Schafer remarks, during the Mongol
domination of China, when Islamic science prevailed in the office of the
Astronomer Royal at Peking. Schafer
says: "The extent of western
influences on Chinese astronomical and cosmological thought in early antiquity
is uncertain. Speculation on the
matter has in the past tended to resemble the lush growth of the hot-house or
the tropical forest: jungly tangles of colorful lianes and rattans whose stems
are confused and whose roots are doubtful.
A sober hypothesis by a professional Assyriologist of our own century
[E. Bezold] seems as fair as any other: native Chinese astronomy/astrology was
probably modified by the Babylonian by at least the sixth century B.C."
(Schafer, ibid.)
22. When did astronomy proper begin to develop, as we understand the term? It depends on what you count as astronomy. People must have known a fair bit about the repeating movements and appearances of sun, moon, planets and stars long before they were able to leave written records. Very likely they made use of observations of the skies to predict -- or try to predict -- when the seasons would change, when was a good time to plant or harvest, when floods and other natural catastrophes were liable to occur, where they would land when they set out to sea, and so on.
23. On the antiquity of astronomy, Mircea Eliade says: "Alexander Marshak [sic] has recently been able to demonstrate the existence, in the Upper Paleolithic, of a symbolic system of temporal notations, based on observations of the moon's phases. These notations, which the author terms 'time-factored', that is, accumulated over a long period, permit the supposition that certain seasonal or periodic ceremonies were fixed long in advance, as is the case in our day among Siberians and North American Indians. This systems of notations remained in force for more than 25,000 years, from the early Aurignacians to the late Magdalenian. According to Marshak, writing, arithmetic, and the calendar properly speaking, which make their appearance in the first civilizations, are probably connected with the symbolism with which the system of notations used during the Paleolithic is impregnated. Whatever may be thought of Marshak's general theory concerning the development of civilization, the fact remains that the lunar cycle was analyzed, memorized, and used for practical purposes some 15,000 years before the discovery of agriculture. This makes more comprehensible the considerable role of the moon in archaic mythology, and especially the fact that lunar symbolism was integrated into a single system comprising such different realities as woman, the waters, vegetation, the serpent, fertility, death, "rebirth," etc." (Mircea Eliade, A History of Religious Ideas, 1978, French 1976, v. 1, p. 22-23; cf. Alexander Marshack, 1972, The Roots of Civilization: The Cognitive Beginnings of Man's First Art, Symbol, and Notation, p. 81 ff.)
24. No one knows when gods
first appeared among men. Nobody
knows when people began to try to find out their wills.
Who knows which ideas about gods were derived from ideas about
the sun, moon and stars? Sextus
Empiricus says: "And
Aristotle said that the conception of Gods arose amongst mankind from two
originating causes, namely from events which concern the soul and from
celestial phenomena. It arose from events which concern the soul because of the
inspired states of the soul which occur in sleep and because of prophecies.
For, says he, when the soul is by itself in sleep, then it takes on its
true nature and prophecies and predicts the future.
And it is in this state also when it is being separated from bodies at
death..... Moreover (they derived
this conception) from celestial phenomena also; for when they beheld the sun
circling around in the day-time, and by night the orderly motion of the other
stars, they supposed some God to be the cause of such motion and
orderliness." (Sextus
Empiricus, c. 200 A.D., Against
the Physicists, i.20-22, also known as Adversus Dogmaticos, iii,
and Adversus Mathematicos, ix.; translation
by R. G. Bury, 1936, p. 11, 13).
25. Cicero reports that
the Stoic Cleanthes (c. 300-220 B.C.) gave four reasons to account for the
formation in men's minds of their ideas of gods:
"He put first the argument ... arising from our foreknowledge of
future events; second, the one drawn from the magnitude of the benefits we
derive from our temperate climate, from the earth's fertility, and from a
vast abundance of other blessings; third, the awe inspired by lightning,
storms, rain, snow, hail, floods, pestilences, earthquakes, and occasionally
subterranean rumblings, showers of stones and raindrops the colour of blood,
also landslips and chasms suddenly opening in the ground, also unnatural
monstrosities human and animal, and also the appearance of meteoric lights and
what are called by the Greeks 'comets,' and in our language 'long-haired
stars,'..... all of which alarming portents have suggested to mankind the idea
of the existence of some celestial and divine power. And the fourth and most potent cause of the belief he said
was the uniform motion and revolution of the heavens, and the varied groupings
and ordered beauty of the sun, moon and stars, the very sight of which was in
itself enough to prove that these things are not the mere effect of chance.
When a man goes into a house, a wrestling-school or a public assembly
and observes in all that goes on arrangement, regularity and system, he cannot
possibly suppose that these things come about without a cause: he realizes
that there is someone who presides and controls.
Far more therefore with the vast movements and phases of the heavenly
bodies, and these ordered processes of a multitude of enormous masses of
matter, which throughout the countless ages of the infinite past have never in
the smallest degree played false, is he compelled to infer that these mighty
world-motions are regulated by some Mind."
(Cicero, De natura deorum, translated by H. Rackham, 1933, p.
137-139.)
26. It is, then, small
wonder that celestial objects came to be regarded as having power over our
affairs. In omen or portent
astrology, attempts are made to use such objects to predict events of
importance to a country and its rulers. Omen astrology seems to have been indigenous to Babylonia,
although the Chinese may have developed their own version independently.
Bartel van der Waerden assigns the beginning of omen astrology to
before the reign of Hammurabi in Babylonia (about 1800 B.C.), and perhaps much
earlier. (Bartel van der Waerden,
Science Awakening II, The Birth of Astronomy, 1974, p. 49.)
27. Here's a sample:
"When Scorpio approaches the front of the Moon and stands, the
reign of the king will be long; the enemy will come, but his defeat will be
accomplished." (R. Campbell
Thompson, The Reports of the Magicians and Astrologers of Nineveh and
Babylon in the British Museum, the original texts, printed in cuneiform
characters, edited with translations, notes, vocabulary, index and an
introduction, 1900, v. 2, p. lxxi.) Another
example: "The month of Elul, 15th day, eclipse [of the moon]:
the son of the king kills his father and seizes the throne, and the
enemy advances and destroys the country.
The 16th day, eclipse of the moon:
the king of a foreign country the same [i.e., is killed by his son],
the king of the country of Hâti advances and seizes the throne.
Rains in the sky, abundance of water in the canals."
(A. Laurent, La Magie et la Divination chez les Chaldéo-Assyriens,
1894, p. 60.) Another:
"If Mars is visible in the month of Tammuz (June-July), the beds
of the soldiers will be empty." That
is, there will be a military expedition.
(Marguerite Rutten, La Science des Chaldéens,
1970, p. 95.)
28. Although there may
have been secular attitudes among Chaldean diviners, we may suppose they were
to some degree influenced by the prevailing religion.
In ancient Babylonia, the sun deity Marduk, the greatest of the
Babylonian gods and successor to the moon deity of the Sumerians, set the
celestial beings to moving and determined their courses.
Marduk articulated time into units, and the regularity of celestial
motions became a model for the life of men in society, and a powerful force on
the development of their government, work and cities.
The highest duty of the highest officials of Babylon, the priests, was
to observe and interpret the movements of the sun, moon and other celestial
objects. (Babylon, in the time of
Nebuchadnezzar (died 562 B.C.), was probably the greatest and most well
organized city in the world, estimated to support between 250,000 and 300,000
inhabitants. It was
Nebuchadnezzar who is reputed to have built the "tower of Babel",
and to have destroyed the Jewish temple in Jerusalem.
In Greece, this was about the time of Anaximander, one of the
pre-Socratic philosophers, perhaps the first person to ever make a geometric
model of the universe, or at any rate this appears to be the earliest we know
about.)
29. At the head of the
Babylonian and Assyrian panoply of gods is Anu.
"Anu," we are told, "was the son of Anshar and Kishar.
His name signified 'sky' and he reigned over the heavens...
Aided by his companion, the goddess Antu, he presided from above over
the fates of the universe and hardly occupied himself with human affairs.
Thus, although he never ceased to be universally venerated, other gods
finally supplanted him and took over certain of his prerogatives.
But the great god's prestige remained such that the power of these
usurper gods was never firmly established until they, too, assumed the name
Anu..... The entire course of
human life was ... regulated by the sovereign will of the gods, whose chief
attribute was deciding the fates of men.
We have already seen how highly the gods valued this privilege which
fell successively to Anu, Enlil, Ea and Marduk.
Although it was the supreme god who made the final decision, all could
discuss it. At the beginning of
every year, while on earth the festival of Zagmuk was being celebrated, the
gods assembled in the Upshukina, the Sanctuary of Fates.
The king of the gods in the later Babylonian period, Bêl-Marduk,
took his place on the throne. The
other gods knelt with fear and respect before him.
Removing from his bosom the Tablet of Fates, Bêl-Marduk
confided it to his son Nabu, who wrote down on it what the gods had decided.
Thus the fate of the country was fixed for the coming year." (Larousse
Encyclopedia of Mythology, 1959, p. 52-53, 63.)
30. If Anu is the chief
god, what was the status of his parents Anshar and Kishar?
The Larousse has it that Apsu (sweet water) and Tiamat (salt
water) were the fount of all things. The
first offspring of these were Lakhmu and Lakhamu, "rather vague
gods" who "seem to be a pair of monstrous serpents.
They gave birth to Anshar, the male principle, and to Kishar, the
female principle, who represented respectively, so some think, the celestial
and terrestrial worlds. In the
same way the Greek gods were born of the union of Uranus, the sky, and Gaea,
the earth. But while in Greek mythology Gaea played an important role,
Kishar does not appear again in the story." (ibid, p. 49-50.)
31. Thorkild Jacobsen
tells the same story like this, based on Old Babylonian copies of Sumerian
texts from the third millenium B.C. "An
ranked highest among the gods. His
name, borrowed by the Akkadians as Anum, is the Sumerian word for
"sky" and inherently An is the numinous power in the sky, the source
of rain and the basis for the calendar since it heralds through its changing
constellations the times of the year with their different works and
celebrations..... An's spouse was
the earth, Ki, on whom he engendered trees, reeds, and all other vegetation
..... There also seems to have
been a tradition that saw the power in the sky as both male and female and
distinguished the god An (Akkadian Anum) from the goddess An (Akkadian Antum)
to whom he
was
married. According to that view
the rains flowed from the sky goddess' breasts, or (since she was usually
envisaged in cow shape) her udder -- that is from the clouds.....
An had not only engendered vegetation, he was the father and ancestor
of all of the gods, and he likewise fathered innumerable demons and evil
spirits. Frequently he was
envisaged as a huge bull..... The
view of An as a major source of fertility, the "father who makes the seed
sprout," engenderer of vegetation, demons, and all the gods, led
naturally to the attribution of paternal authority to him.....
With the developing of social differentiation and the attitudes of
growing respect and awe before the ruler, a new sensitivity to the potential
in the vast sky for inducing feelings of numinous awe seems to have come into
being. The sky can, at moments
when man is in a religiously receptive mood, act as vehicle for a profound
experience of numinous awe, as may be instanced in our own culture."
32. Jacobsen quotes a
passage from William James's The Varieties of Religious Experience:
"I remember the night, and almost the very spot on the hilltop,
where my soul opened out, as it were, into the Infinite, and there was a
rushing together of the two worlds, the inner and the outer.
It was deep calling unto deep,-- the deep that my own struggle had
opened up within being answered by the unfathomable deep without, reaching
beyond the stars. I stood alone
with Him who had made me, and all the beauty of the world, and love, and
sorrow, and even temptation."
33. Jacobsen continues:
"To the ancient Mesopotamians what the sky might reveal was An,
its own inner essence of absolute authority and majesty -- might
reveal, but would not necessarily reveal, for in everyday moods the sky would
be experienced apart from the numinous power in it and would recede into the
category of mere things..... Since
human society is not the only structure based on authority and command (the
natural world is as well), all things and forces in the polity that is the
universe conform to An's will. He
is the power that lifts existence out of chaos and anarchy and makes it an
organized whole. As a building is
supported by and reveals in its structure the lines of its foundation, so the
ancient Mesopotamian universe was upheld by and reflected An's ordering will. His command is "the foundation of heaven and
earth."..... As the ultimate
source of all authority An was closely associated with the highest authority
on earth, that of kingship. The
royal insignia lie before An in heaven for him to bestow, and with them he
conveys not only the general powers of kingship but duties linked to his own
cosmic functions: responsibility for the calendar and for carrying out his
calendric rites. For example, his
new moon festivals ... were celebrated in all temples, and the New Year
festival at which the year seems to have been named from one of the king's
accomplishments. Through this
mandate, accordingly, the king becomes An's instrument for seeing to it that
the times do not get out of joint."
(Thorkild Jacobsen, The Treasures of Darkness, A History of
Mesopotamian Religion, 1976, p. 95-97.)
Thus the source and model of authority and order was the heavens.
34. Since the seasons and
other important events are to some degree related to movements of the moon,
sun and stars, it's reasonable to try to correlate as many events as we can
with these movements. For
example, the approximate time for the flooding of the Nile in ancient Egypt
was correlated with movements of the sun and stars.
Certain kinds of weather are correlated with the appearances of
constellations, including not only their positions but also atmospheric
effects. Martin Nilsson says that
the most widely read of all Hellenistic poems was the Phainomena of
Aratus, which was a book containing rules for predicting the weather in this
way. (Martin Nilsson, Geschichte der griechischen Religion, 1950, v. 2,
p. 56.)
35. The process goes on
today. Here is an excerpt,
entitled "Weather Prognosticator, from the Hagers-town Town and
Country Almanack for the year of our Lord 1989, p. 9:
"This table and the accompanying remarks are the result of many
years' actual observation; the whole being constructed on a due consideration
of the Sun and Moon, in their several positions respecting the earth; and
will, by simple inspection, show the observer what kind of weather will most
probably follow the entrance of the Moon into any of her quarters, and that so
near the truth as to be seldom or never found to fail."
36. Beliefs that our
father is in heaven, and that it is on earth as it is in the heavens, are
widespread. Claude Lévi-Strauss
argues that among Indians of central Brazil, certain myths which on the
surface may seem to have no connection with astronomy, are in fact concerned
with the alternation of seasons, and therefore a kind of year.
In particular, he considers the story of Asare, told among the Sherente
people, concerning the rape of a mother by her own sons (the youngest of whom
is Asare), thrashing of the sons by their father, the sons setting fire to
their parents who escape by turning into falcons, a journey by the sons which
includes the digging of a well which gushes so much water that it forms the
sea, and three or so escapes from an alligator with the help of woodpeckers,
partridges, fruit rinds and a skunk. The
myth concludes: "When the
sea was formed, Asare's brothers had at once tried to bathe.
Even today, toward the close of the rainy season, one hears in the west
the sound of their splashing in the water.
Then they appear in the heavens, new and clean, as Sururu, the Seven
Stars (the Pleiades)." (Claude
Lévi-Strauss, The Raw and the Cooked, 1969, translation by P. and D.
Weightman of Le cru et le cuit, 1964, p. 199-200, v. 1 of Mythologiques
(Introduction to a Science of Mythology)).
Lévi-Strauss quotes J. F. Oliveira to the effect that among the
Sherente, the year begins with the appearance of the Pleiades, which coincides
roughly with the beginning of the dry season. (p. 217.).
37. According to Lévi-Strauss:
"Classical antiquity associated Orion with rain and storms.
Now we have seen that in central Brazil, Orion is also associated with
water -- but terrestrial, not celestial water.
In Greek and Roman mythology Orion caused rain to fall.
As Asare, the thirsty hero, Orion makes water rise up from the depths
of the earth. It is easy to
understand, since it is an obvious cosmographical fact, that the same
constellation that casues rain to fall in the northern hemisphere should be a
harbinger of drought in the southern hemisphere: in the inland areas between
the equator and the Tropic of Capricorn, the rainy season corresponds
approximately to our autumn and winter, the dry season to our spring and
summer. The Asare myth faithfully
presents the "southern" version of this factual truth, since the
Pleiades and Orion which follows closely in their wake, are said to herald the
beginning of the dry season." (p. 226-227).
38. There is a problem
here, since in "in one hemisphere Orion is associated with celestial
water in accordance with meteorological experience, while in the other
hemisphere, without there being any possibility of establishing a connection
with experience, symmetry is preserved by means of an apparently
incomprehensible link between Orion and water which is chthonic in origin --
that is, celestial water conceived of, as it were, upside down." (p. 227)
Lévi-Strauss traces this opposition by way of a transformation of a
key myth of the Bororo people. He
says: "It is therefore clear
that the two myths, the one belonging to the Ancient World [of European
classical antiquity] and the other to the New [Bororo of central Brazil], are,
as I postulated, reflections of each other.
The apparent inversions arise simply from the fact that while both are
concerned with the dry season, one myth refers to the beginning (after the
rains) and the other to the end (before the rains)." (ibid., p. 239).
39. The point is that
myths which superficially are about incest, rape, arduous and dangerous
journeys, people turning into birds or other creatures, and the like, may turn
out to be descriptions of astronomical and associated seasonal phenomena.
However, in the view of Lévi-Strauss:
"In granting that myths have an astronomical significance, I do
not propose to revert in any way to the mistaken ideas characteristic of the
solar mythography of the nineteenth century.
In my view, the astronomical context does not provide any absolute
point of reference; we cannot claim to have interpreted the myths simply by
relating them to this context. The
truth of the myth does not lie in any special content.
It consists in logical relations which are devoid of content or, more
precisely, whose invariant properties exhaust their operative value, since
comparable relations can be established among the elements of a large number
of different contents."
40. "For instance, I
have shown that one particular theme, such as the origin of man's mortality,
occurs in myths that appear quite different from each other in subject matter,
but that in the last analysis these differences can be reduced to a variety of
codes, evolved on the basis of the different sense categories -- taste,
hearing, smell, feel, and sight ..... In
the preceding pages, I have been solely concerned [in interpeting the myths
astronomically] to establish the existence of a different code, also a visual
one, but whose lexical material consists of contrasted pairs drawn from a
stable periodicity of the year and, on the other, of the synchronic
arrangement of the stars in the sky. This
cosmographic code is no truer than any other; and it is no better, except from
the methodological point of view, as far as its operations can be checked from
without. But it is not impossible that advances in biochemistry may
one day provide objective references of the same degree of accuracy as a check
on the precision and coherence of the codes formulated in the language of the
senses. Myths are constructed on
the basis of a certain logicality of tangible qualities which makes no
clear-cut distinction between subjective states and the properties of the
cosmos." (p. 240.) Thus
different "codes" are different realizations of structures of human
physiology, and Lévi-Strauss weights the different codes equally.
41. We can wonder,
however, whether or not an astronomical code has a kind of priority.
According to many cosmologies, the stars and their ways precede the
living and their ways.
To what extent have we developed in consonance with celestial objects
and movements? To what extent are
our physiology and thoughts tied to the stars?
As described by Lévi-Strauss, among Indians of Brazil, fire for
cooking food is related to the sun: "The
mediatory function of cooking fire therefore operates between the sun and
humanity in two ways. By its
presence, cooking fire averts total disjunction, since it unites the
sun and the earth and saves man from the world of rottenness in which
he would find himself if the sun really disappeared; but its presence is also interposed;
that is to say, it obviates the risk of a total conjunction, which would would
result in a burned world (p. 293.)
Incest and cannibalism in the myths are linked with eclipses, and the
origin of diseases. (p. 297.)
42. "Starting from
the problem of the mythic origin of cooking," says Lévi-Strauss, "I
have been led to verify my interpretation of domestic fire as a mediatory
agent between sky and earth by reference to the myth describing incest between
blood relatives as the origin of the eclipse......
A myth about the origin of storms and rain [the one Lévi-Strauss
started with] led me to myths about the origin of fire and the cooking of
foodstuffs... I was able to establish that all these myths belong to one
and the same set ....." (p. 298, 300.)
Which explains which? Do
analogous actions of sun, moon and other stars explain or describe the origin
of cooking fires? Or does the
analogy of the origin of cooking fires explain or describe actions of the sun,
moon and stars? Are these
interchangeable? If not, which
takes precedence? Recall Seneca
on the Etruscans: "Since
they attribute everything to divine agency, they are of the opinion that
things do not reveal the future because they have occurred, but that they
occur because they are meant to reveal the future."
43. Besides some roughly
correct season and even (at times) weather forecasting, there were no doubt
successes in predicting such events as attacks by enemies, since, for example,
rulers probably tended to attack after harvests, when their troops were
well-supplied with food, and harvests are correlated with the seasons.
However, prediction by consulting objects in the sky of such things as
who would be victorious in a war was likely to have been more chancy, unless,
of course, the objects were arrows and spears.
Isaiah, it seems, spoke sarcastically when he said
"Come down and sit in the dust,
O virgin daughter of Babylon.....
You are wearied with your many counsels;
let them stand forth and save you,
those who divide the heavens,
who gaze at the stars,
who at the new moons predict
what shall befall you.....
they cannot deliver themselves
from the power of the flame."
(Chapter 47, The Bible, Revised Standard Version)
44. The mathematical
astronomy of the Babylonians underwent a considerable development between
about 539 B.C. and 331 B.C., during the reign of the Persians in Babylonia.
It is during this period, perhaps about 450 B.C., that personal
astrology, the casting of horoscopes according to birth dates, developed. There is an old tradition that horoscopy was introduced to
the Greeks by Berossos, a Babylonian priest who founded the first Greek school
of astrology on the island of Kos about 300 B.C.
However, it appears that we have Greek horoscopes from about 150 years
earlier. On the task of personal
astrology, Auguste Bouché-Leclercq, says: "The calculation of the length
of life, with an indication of the kind of death pre-assigned by the stars, is
the great work of astrology, the operation judged the most difficult by its
adepts, the most dangerous and damnable by its enemies."
(Auguste Bouché-Leclerq, L'Astrologie grecque,
1899, p. 404.)
45. Van der Waerden
summarizes the development of astrology in this part of the world in the 6th
century B.C. as follows: "We
have seen that, after
the fall of the Assyrian
empire (- 611) the old polytheism was being pushed aside by a new religious
movement which flooded in two mighty waves from Iran to the West.
The first wave was that of Zervanism,
which reached Greece about - 550.
The second was the worship of Ahura Mazda, which was proclaimed around
- 500 B.C. as the official religion of the Persian empire.
Connected with this was the doctrine of the celestial origin and
immortality of the soul. We have
also seen that the old Omen astrology was replaced, about the same time or
somewhat later, by a new zodiacal astrology, within which we have to
distinguish two further stages: primitive
zodiacal astrology and horoscopy.
The first is connected in the sources with Orphism, which in its turn
is most closely tied up with Zervanism. On
the other hand, horoscopy is closely connected with the doctrine of the
celestial origin of the soul; its existence can be demonstrated in Babylon
about - 450 and in Greece about - 440." (ibid., p. 183.)
The name of the god Zervan Akarana means "boundless time."
The Zervanists, whose sect appears to have been formed about the 4th
century B.C., were astral fatalists who believed that "all fortune, good
and ill, that befalls man, comes from the twelve [zodiacal signs] and the
seven [planets]". This quotation is given by van der Waerden (p. 162) from a
Persian book called Mainog-i Khirad or Menok i Khrat, written
sometime between 220 and 650 A.D.
46. By about 300 B.C., the
Babylonians had constructed tables, based on centuries of observations, with
which they could successfully predict lunar eclipses, and with which they
could at times rule out solar eclipses. A
basic underlying problem they were trying to solve is a form of one which
haunts mathematical astronomy to this day.
From one point of view, this is the problem of predicting the day on
which a new moon will occur. The days are determined by the movement of Earth with respect
to the sun (or vice versa), while new moons are determined by the movement of
the moon with respect to Earth. Thus
the combined motions of sun, moon, and Earth are involved.
The problem of predicting the movements of the sun, moon and Earth with
respect to one another, starting from Newton's laws of mechanics and
gravitation, is known today as the 3-body problem.
In some important respects, the 3-body problem is still unsolved,
although a great deal is known about some basic special cases, and there are
elaborate techniques for approximating solutions.
The Babylonian methods were a kind of approximation technique, based on
interpolation, inserting calculated values between observed
values in systematic ways. As far
as seems to be known at present, the first attempts to use geometry to model
the movements of celestial objects and relations between them were made by the
ancient Greeks in the 6th century B.C.
The Babylonians seem not to have made geometrical models for this
purpose, or at least none have been found.
47. We
have fragments of a geometric cosmology put forward by the philosopher
Anaximander in the 6th century B.C. Anaximander
may have been the first to undertake a project of this kind.
He appears to have pictured the sky as a complete sphere rather than an
inverted bowl or hemisphere. Spheres
were to become the basis of geometric cosmology for many centuries.
However, for some unknown reason, if we can trust the fragment we have
from so long ago, Anaximander seems to have proposed that the earth is a right
circular cylinder with the greatest curvature in the north-south direction.
Aside: "It was Henry Ibsen
who said that the value of a truth lasted about fifteen years, then it rotted
into error." (James Huneker, Old Fogy, 1913, quoted in A New
Dictionary of
48. The arithmetical
predictions of the Babylonians and the geometric
construction of the heavens by the classical Greek philosophers contrast in a
startling way with other cosmologies of that era in the Near East, and with
other ancient Greek cosmologies, in which the heavens are peopled with gods
who often act unpredictably and capriciously.
Geometric cosmologies were developed extensively by astronomers and
philosophers of nature during the next several centuries after the time of
Anaximander. Plato and Aristotle, in the 4th century B.C., made use of the
work of these pre-Socratic thinkers in developing their own cosmologies.
We find in the works of Plato and Aristotle the first extended and
detailed reports, which we still have today, of cosmologies based on geometry,
as developed by Eudoxus of Cnidus and other mathematical astronomers of the
time. They had enormous influence
on the development of Western cosmologies from the time they were composed.
The special kind of certainty which geometric models seem to reveal
about the movements of the heavens, blended with an older personification and
deification of heavenly objects, were, it appears, instrumental in the
development of astrology.
49. Geometric models in
astronomy developed hand in hand with geometry itself.
Eudoxus of Cnidus (4th century B.C.) is said to have been a student of
Plato. He was one of the great
astronomers, and also one of the great geometers, of his time.
Besides being the source of the mathematical astronomy of Aristotle, he
was, as we mentioned earlier, a possible supporter of astrology.
In astronomy, he developed an elaborate cosmology based on spheres
moving on spheres. In geometry,
he developed a theoretical and logically satisfying theory of magnitudes
corresponding to our real numbers. This
theory, which has been preserved in Euclid's geometry book, the Elements
(c. 300 B.C.) is much like one developed by the German mathematician Richard
Dedekind about the middle of the 19th century (as Dedekind himself stated).
This system is in use today. Eudoxus
seems also to have invented the method of exhaustion for finding areas and
volumes, a method which is much like an application of the definite integrals
of calculus we use today for this purpose, although not formulated as
generally. With this method, he
found an equivalent of our formulas for the area of a circle, and the volumes
of a right circular cylinder, sphere and cone.
50. The Elements of
Euclid was (or were) the principal introduction to geometry for over 2000
years, and the geometry it contained has had, and continues to have, many
terrestrial as well as celestial applications.
More than that, the Elements has served as a model of a kind of
attainment of certainty -–given the initial assumptions, the axioms and
postulates -- which people have often tried to extend to other domains besides
geometry. Euclid's method,
commonly known today as the axiomatic method,
was described, in one form, by Aristotle in his works on logic, especially in
the Posterior Analytics. It
appears that Eudoxus originated the self-conscious and explicit use of this
method, and so was one of the founders of a philosophical tradition of
thinking about thinking, and reasoning about reasoning.
The science of deductive logic founded by
Plato, and even more Aristotle, was based in important respects on
extrapolation from this method of the mathematicians.
51. It is curious, and
rather sobering, to notice that versions of
52.
There are modern
versions of Euclid's Elements in which certain logical deficiencies of
Euclid's _Elements have been removed.
A central one has been the Grundlagen der Geometrie (Foundations of
Geometry) of David Hilbert (1st edition, 1899; last edition during
Hilbert's lifetime, 1930; there have been two translations into English).
However, the spirit of Euclid maintained by Hilbert has given way to a
large extent to the use of numerical coordinates, based on the analytic (or
algebraic) geometry associated with the name of Descartes.
We no longer make children associate how they see with how they reason
in the direct way Euclid did, but rather with how they count, and this is
usually presented in books in colorful language and with colorful pictures.
Stephen Leacock may have had an explanation for the way elementary
geometry books in schools look today, when he said: "To make education
attractive! There it is! To
call in the help of poetry, of music, of grand opera, if need be, to aid in
the teaching of the dry subjects of the college class room..... Here, for
example, you have Euclid writing in a perfectly prosaic way all in small type
such an item as the following: "A
perpendicular is let fall on a line BC so as to bisect it at the point C,
etc., etc.," just as if it were the most ordinary occurrence in the
world. Every newspaper man will
see at once
"AWFUL CATASTROPHE
PERPENDICULAR FALLS HEADLONG
ON A GIVEN POINT
The Line at C said to be completely bisected
President of the Line makes Statement
etc., etc., etc."
(Stephen Leacock, "Education Made Agreeable", from Moonbeams
from the Larger Lunacy_ 1915,
p. 155, 159.) The best translation into English of Euclid's Elements
is by Thomas Heath (1925, reprinted by Dover, 1956 and later).
Heath provides copious notes to guide one
53. To apply the axiomatic
method found in Euclid's geometry, one starts from basic statements usually
called axioms or postulates (although hypotheses or assumptions
would amount to about the same), taken as true for purposes of reasoning
(though in some applications, they may not be true, or true enough), and using
some rules of logic, derives chains of statements linking the axioms to other
statements, called theorems, which are then also taken to be true, and
then may be regarded, if one chooses, as axioms themselves.
These chains of statements make up proofs of the theorems.
Sometimes the term propositions is used instead of theorems,
but often propositions are taken to be statements to be proved, if
possible, rather than statements already proved. Thus a proposition may turn out to be true or false or
undecided or even undecidable in a certain sense, depending on whether or not
a proof or counterexample or neither has been found, and on whether or not a
proof or counterexample can be found within the given axiomatic system.
Since axioms are not proved, but taken as a basis for application of
the method, problems arise of deciding on the validity of the axioms and their
theorems when making applications. If
the axioms or theorems are meant to be applied to the movements of physical
objects, on Earth or in the heavens, one way to test their validity is by
using them to make predictions about the places and shapes of physical
objects, and seeing whether or not the predictions come true, at least to
within some margin of error taken to be allowable.
From this point of view, geometry is an empirical science, perhaps the
earliest such science. However,
some philosophers have held that the axioms of geometry are statements about
the way people, or their minds or brains, are constituted, and especially
about the way we are constrained to see the world with our eyes.
It is from this, one may maintain, that many of the axioms of geometry
get the peculiar certainty they have.
54. From another point of
view, the Elements of Euclid is a treatise on the five regular solids:
the tetrahedron, cube, octahedron, dodecahedron and icosahedron.
The last "book" or chapter of the Elements treats these
solids, and a good deal of what went before
55. Another famous astronomer and geometer of ancient Greece was
Apollonius, who worked in the early part of the 3rd century B.C.
Apollonius had a major influence on the development of astronomy by
virtue of his mathematical model of the solar system based on eccentric and
epicyclic motions. An eccentric
motion is one which takes
place with a constant speed on a circle, but is referred to a point inside the
circle other than the center of the circle.
An epicyclic motion is one which takes place on a circle
rotating at a constant speed about its center, with this center on another
circle also rotating at a constant speed.
Among other things, Apollonius seems to have shown that any eccentric
motion can be interpreted as an epicyclic motion, and conversely.
The major mathematical work of Apollonius concerned the mathematical
figures known as conic sections, which had been discovered by earlier
mathematicians. The conic
sections are cut out when a plane is passed through a complete right circular
cone. Aside from certain special
cases, known as degenerate conics, the conic sections comprise the ellipses
(including the circles), the parabolas, and the hyperbolas. One of the songs of Gilbert and Sullivan is about the
practicality of conic
"I am the very model of a modern Major-General;.....
I'm very well acquainted, too, with matters mathematical,
I understand equations, both the simple and quadratical,
About the binomial theorem, I'm teeming with a lot of news,
With many cheerful facts about the square of the hypotenuse.....
I quote, in Elegiacs, all the crimes of
Heliogabalus!
In conics I can floor peculiarities
parabolous."
(W. S. Gilbert and Arthur Sullivan, The Pirates of Penzance, 1880, Act 1.)
An
easy way to generate ellipses is to shine a flashlight on a flat surface like
a desk or table, and tilt the flashlight back and forth.
The cone in this case is the light generated by the flashlight, and the
plane being passed through the cone is the desk top.
The lighted spot is then in the form of an ellipse (to a good
approximation), though sometimes just the boundary of the lighted spot is
called an ellipse. You can also generate the beginnings of an hyperbola by
holding a flashlight lengthways on a wall.
To see one way conic sections could have been used by the ancient Greeks,
consider a person looking with one eye at the sun (but only very briefly).
A cone can be formed with its apex at the person's eye, using as generators
rays from the eye to points on the circumference of a circular
disk representing the sun. An
imaginary plane through this part of the cone, which meets all thexd
generators of the cone, but doesn't go through the eye, will have an ellipse
in common with the cone. Here one
should take ellipse to mean a curve, like the boundary of the
flashlight spot. If one takes the
generators to be rays from an eye to all the points on the
circumference and inside of a circular disk reprenting the sun, then
one would get an ellipse in
the sense of a flat region, like the entire flashlight spot.
If the plane is imagined to contain the center of the moon, we have the
beginning of a mathematical model for representing a lunar eclipse. It is likely that a primary motive and use for study of conic
sections by the ancient Greeks was to provide models for such astronomical
phenomena as eclipses.
56. Kepler, in developing
his cosmology of the solar system in the late 1500's and early 1600's, used
the mathematics of conic sections as developed by Apollonius in deriving his
three planetary laws, which became part of the basis for Newton's law of
gravity and its application to our solar system.
Newton, in the latter 1600's, showed that if two bodies in the universe
are sufficiently isolated from other bodies, then the paths they will follow
because of the gravitational attraction between them will be conic sections.
The simplest case is when one body is much smaller than the other, e.g.
a comet moving around the sun. The
theory of Newton predicts that if we ignore the influence of the moon, other
planets, etc., and regard the sun as fixed, then the orbit of the earth around
the sun is an ellipse with the sun at one focus.
This had already been projected and verified by Kepler for Mars and the
sun, as the simplest curve consistent with the observations of Tycho Brahe and
Kepler's own planetary laws. Thus
an essential part of our modern view of the solar system rests, by way of
Kepler, on the regular solids, discovered some 2400 or 2500 years ago by
members of a tradition of mathematics founded by the classical Greeks, and on
the geometry subsequently developed or formulated by such mathematicians and
astronomers as Eudoxus, Euclid, and Apollonius.
57. Kepler's contemporary Galileo, also very influential on Newton,
made much of the mathematics of Archimedes, one of the other great
mathematicians of antiquity, who worked somewhat later than Apollonius in the
3rd century B.C. Galileo often
referred to Archimedes using such phrases as "the divine Archimedes"
or the "superhuman Archimedes" (in Italisn).
Archimedes extended the work of Eudoxus on volumes of spheres, right
circular cylinders and right circular cones, and found a very accurate
approximation to the number which is the ratio of the circumference of a
circle to its diameter, the number we call π.
He also described a method of expressing larger and larger whole
numbers, formulated and proved laws of equilibrium for levers and floating
bodies, and a method of finding the area of a parabolic segment.
He seems to have been the first to introduce mathematical methods into
the study of forces in the universe, although there is a precedent in
Aristotle's works.
58. In astronomy,
Archimedes attempted to calculate the volume of the universe.
He used two proposals for the radius of the universe.
One which was conventionally accepted in the time of Archimedes was
that the radius of the universe is the radius of our solar system.
The other was based on the proposal by Aristarchus, an astronomer
roughly contemporaneous with Archimedes, some 1800 years before Copernicus,
that the sun is the center about which the earth revolves , and that the
radius of Earth's orbit is negligible compared with the radius of the
spherical surface on which the fixed stars lie.
In this spirit, Archimedes assumed that the ratio of the radius of our
solar system is to the radius of the universe as the ratio of the radius of
Earth's orbit is to the radius of our solar system. Archimedes calculated that with this assumption, the universe
would have room for no more than 1063 grains of sand, whereas with
the conventional radius it would have room for no more than 1053
grains of sand (Otto Neugebauer, A History of Ancient Astronomy, 1975,
Part Two, p. 646).
59. Another great Greek
astronomer was Hipparchus, who lived in the 2nd century B.C.
Building on the earlier work of Eudoxus and other astronomers, he
developed an elaborate cosmology using spheres moving on spheres, but the
system of Hipparchus was simpler and at the same time more comprehensive than
the one which had grown out of the work of Eudoxus.
Hipparchus also accumulated quite accurate observations of the relative
positions and motions of the main celestial objects visible without
magnification. By virtue of some
tables of ratios which he used in his work, he is often regarded as the
originator of trigonometry. He
also extended the work of Aristarchus on calculating the distances of our moon
and sun from Earth.
60. Even before the
decline of Greek political power in the 3rd century B.C., a school of Greek
astronomers had arisen in Alexandria, Egypt, in the midst of a culture much
older than that of the Greeks. It
was in Alexandria, about 140 A.D., that Ptolemy wrote his Megale
mathematike syntaxis or "Great mathematical treatise",
later known as the Almagest, from an Arabic form of a Greek word
meaning "the greatest". This
work was a synthesis and extension of the whole astronomical tradition which
had been initiated by the Greeks some 750 years before, and the Babylonians
even earlier than that. To feel
how long 750 years is, we have 2000 - 750 = 1250, so working backward about
750 years from our own time, we come to the year 1250, some 300 years before
the time of Copernicus, who may be regarded in part as a continuer of the
Greek astronomy, and in part as the introducer of a revolutionary new view in
astronomy. In the year 1250, European scholars were still trying to come
to terms with the Greek geometry and cosmologies with which they had lost
contact for a long period (perhaps itself about 750 years), during which
European scholarship in matters we now consider to be subjects of the sciences
had slowed down for various economic, religious and other reasons.
61. In Ptolemy's treatise
on the heavens, Earth is taken as the center of the physical universe.
Ptolemy offers a number of arguments based on the physics of his time
that this is so. For example,
Ptolemy says: "... the revolving motion of the earth must be the most
violent of all motions associated with it, seeing that it makes one revolution
in such a short time [a day]; the results would be that all objects not
actually standing on the earth would appear to have the same motion, opposite
to that of the earth; neither clouds nor or other flying or thrown objects
would ever be seen moving toward the east, since the earth's motion toward the
east would always outrun and overtake them, so that all other objects would
seem to move in the direction of the west and the rear.
But if they said that the air is carried around in the same direction
and with the same speed as the earth, the compound objects in the air would
none the less always seem to be left behind by the motion of both [earth and
air]; or if those objects too were carried around, fused, as it were, to the
air, then they would never appear to have any motion either in advance or
rearwards: they would always appear still, neither wandering or changing
position, whether they were flying or thrown objects.
Yet we quite plainly see that they do undergo all these kinds of
motion, in such a way that they are not even slowed down or speeded up at all
by any motion of the earth." (_Ptolemy's
Almagest, translated and annotated by G. J. Toomer, 1984, p.45.)
This argument by Ptolemy comes after a number of arguments in favor of
a spherical movement of the heavens, as the only motion consistent with the
phenomena, i.e. observations. He
argues also that the earth is spherical, and in the middle of the heavens.
These arguments are largely based on geometric relations which are
consistent with what is observed. Ptolemy's
theories were usually based on as accurate observations as were then
available.
62. According to Ptolemy, the moon, sun and then-known planets are observed to revolve about Earth in the order Moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn. The stars are taken to be at a great distance from Earth -- even at an "infinite distance" from Earth, whatever that might be taken to mean. Still, Ptolemy held that they revolved around Earth every day. They certainly appear to do this. Most of Ptolemy's treatise, about 600 of the approximately 650 pages in the translation by Toomer, is dedicated to elaborate mathematical procedures, calculations and tables which made it possible to predict, at any time, the future positions of the sun, moon and planets relative to Earth and the stars. On the whole, the accuracy was within the limits imposed by using measurements made with human eyes alone.