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Early Egyptian astronomy seems to have been of a practical origin. It was noticed that Sirius was the "herald of the flood" (p. 8). "The flooding of the Nile over its banks is the most important event in the Egyptian agricultural year. It gives new life to the parched land. This event is heralded some weeks beforehand by a striking event in the firmament, namely the first visibility of Sirius in the morning sky." (p. 9). Other practical advice based on stars include: "when strong Orion begins to set, then remember to plough"; and "fifty days after the solstice is the right time for men to go sailing" (p. 12). The stars were also used to tell time at night. "In the course of the centuries these Stars of Time became Gods of Time and Destiny." (p. 14). "'From their might derives everything that humanity encouters in the way of disasters,' says the revelation of Hermes Trismegistos." (p. 29). "According to Hermes Trismetgistos the decans can also be called 'horoskopoi'---hour indicators. The decan that rises in the hour of the birth of a child determines the nature of the child." (p. 32).
Babylonian astronomy, on the other hand, seems to be linked to, and largely dictated by, astrology as far back as the record goes. "The oldest cuneiform texts giving the positions of the planets in the zodiac date from the second half of the fifth century B.C. To just this period, and to Babylon too, belongs the oldest horoscope that has been preserved." (p. 2). Of course Babylonian astronomy is much older than this, but precise knowledge of planetary positions were not important as long as astrology was impersonal, perhaps for the reasons given below. Indeed, "Old-Babylonian astrology was not interested, or at least not in the first place, in the fate of the individual. Its principal interest was the well-being of the country. Its predictions concern the weather and the harvest, drought and famine, war or peace and of course also the fate of the Kings." (pp. 48-49).
The rationale for impersonal astrology may have included the following. "Just as the great Gods Sin (the moon) and Shamash (the sun) are obviously responsible for the regular procession of months, days and years, and thus influence our entire life, so it was thought that the Goddess Ishtar [Venus] communicates important things to us by her appearances and disappearances." (p. 57). Above we saw some examples of apparently important influences of the stars, in the spirit of which one will say things like "O Ursa major ... Put truth for me" (p. 58), as one prayer reads. A further consideration is the plausibility of the idea of a strictly periodic universe (of course the world would be periodic if it was determined by the heavens, which are paradigmatically periodic). As Eudemos was later to relate, "If we are to believe the Pythagoreans, I shall in the future, even as everything recurs according to the Number, again tell you tales here, holding this little stick in my hand, while you will sit before me as you do now; and likewise everything else will be the same." (p. 114). (Pythagoras' conception of the world owes much to Babylonian ideas, especially so his emphasis on number, which we shall see below is a very dear concept to the Babylonians. The periodicity at which the world repeats is presumably a common multiple of all planetary periods.)
The rationale for individual astrology seems to have included the following. The idea that the souls of the dead rise to the heavens is an old one. Not the first example is that "the inscription for the fallen at the battle of Potidea (-431) says: 'The aether will receive their souls, as the earth receives their bodies'" (p. 146). From here it is a rather short step to the idea that, as expressed for example "in Servius' commentary on Aeneid VI 714, the souls before birth go down through the planetary spheres, acquiring thereby from Saturn inertia, from Mars wrath, from Venus lust, from Mercury avarice, from Jupiter ambition" (p. 144). Another argument in support of this view is that the heavens are the paradigm of self-motion, which is not displayed by soulless objects. As Plato puts it: "the soul which has lost its wings is borne along until it gets hold of something solid, ... taking upon itself an earthly body, which seems to be selfmoving, because of the power of the soul within it" (p. 147, Phaedrus 246b-c).
As for mathematical astronomy, the Babylonian theory was decidedly arithmetical and instrumental as opposed to geometrical and realist. The foundation of the theory is periodicity data, which can be extremely accurate by averaging many years worth of observations. For example, the Babylonian value for the average lunar period is accurate to the second (p. 240). "Linear zigzag functions" fitted to such data formed the foundations of the theory, though ad hoc corrections were introduced whenever expedient, such as changes in slope or extra zigs and zags. An illustrative example of the commitment to this arithmetical-intrumeltal view is the fact that even eclipse magnitude was sometimes modelled by such a function (p. 239), even though the function is obviously nonsensical on most of its domain (viz. when there are no eclipses). Because of its character the theory was therefore remarkably accurate on observable, approximately zigzag-periodic phenomena (such as the positions of the sun and the moon) and phenomena derivable therefrom (such as lunar eclipses, though this was sometimes treated as a primitive observable in itself). The theory was however "not very good" (p. 278) on problems for which a geometric understanding would have been beneficial, such as planetary positions. For the same reason, the theory was "not very good for solar eclipses, because the Babylonians had no means of calculating the lunar parallax, which has a considerable influence on the magnitude of a solar eclipse" (p. 120). This is ironic since there is a famous story about Thales predicting a solar eclipse (which he must almost certainly have done on the basis of Babylonian theory). The ancient sources tell us that Thales predicted only that a solar eclipse would occur in a particular year, but this crude prediction was apparently sufficient to impress his countrymen (pp. 120-122).
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Van Der Waerden's Science Awakening II is comparably boring to Science Awakening I. Science Awakening I is the mathematics from Egyptian to Greek times. What I'm saying is the mathematics of these times is far more exciting than the astronomy which consists of motions of the moon, sun, and planets. Noticing there's planets at all is more exciting than knowing their motions. Knowing the twelve constellations is more exciting than knowing the motions themselves. It's pretty hard to make this exciting. You have to try pretty hard to make this interesting. But, if you try hard enough, you might find a few things here and there exciting!
In particular, the Babylonian astronomy game of determining the motions of the sun, moon, and planets from their periods. Period is not just one cycle of a circular motion from a fixed point. Period here means the planets after going around however many times, ends up at the same point. Perhaps an easier 20th/21st century example would be the motion of Mercury. One of the first confirmations of Einstein's General theory of Relativity was the odd pattern of the planet Mercury's orbit. Mercury of course has an elliptical orbit, and it just turns out to have a pattern, or a period where it comes back to the same point after so many elliptical orbits. So, Mercury's orbit looks a little bit of a smooth star shape(more complex than a five pointed star). Similarly, the motions of the moon, and planets from the perspective of viewers here on Earth, and in particular, the Babylonians, noticed there were patterns in the comings and goings in of the planets movement around the Earth(the geocentric view). They sought as many periods as they could. They were also interested in the "Great period" that stretches to 30,000 years. This idea was brought to the Greeks by a Babylonian - Berrosis. Carl Sagan mentions this guy in his Cosmos . . . about a great three volume history of the world, written by a Babylonian priest/astronomer Berrosis
Anyways, the Babylonians tried to be able to calculate where a planet would be by these periods. They had all kinds of mean values, and correction factors that Van Der Waerden clothes in modern simple one degree algebraic equations. Van Der Waerden also makes a pretty good case that the inspiration for the infinit series was Babylonian astronomy. They calculated some things like the risings of constellations from infinit series. The Greeks would go on to relate infinit series to proportions and come up with a vast theory of irrationals due to Thaetateous. This is found in book ten of Euclid's Elements.
Van Der Waerden notes that his account of Babylonian astronomy is only basics compared to Neugebauer's "Antiquity of Exact Sciences."
I've not mentioned much about Egyptian astronomy. Egypitan astronomy is mostly noticing Sirius, making a calendar by it, and how remarkably, the Nile flooded at the rising of Sirius. They have a mythology of deacons that is the degrees of nighttime. They don't have much. Mythology and Astronomy was almost more exciting in this book than the astronomy(until you kind of get the whole point about periods).
Van Der Waerden traces three periods of astrotheology. An early Omen period, a primitive zodiacal astrology, and a Horoscope astrology. He goes through some of the major historical points of various Egyptian and Mesopotamian cultures. Venus was discovered by a descendent king of Hammurabi; this was the high point astronomy of its day(about 1,700 B.C). He notes some interesting history of the Persians who were probably the high point of Babylonian astronomy before the classical Greeks took over. These are the kings of Xerxes, Darious, famous both in Greek history and the Bible. Van Der Waerden finds that Herodotus relates that Darious tried to wipe out the Chaldean astrologers. And then, later in Herodotuses Histories, he shows that Xerxes after Darious finds a place in the Persian empire for them to live comfortably. I find it interesting that Van Der Waerden notes all these things in Herodotus, and even much later in the book, the last chapter of the Book, he notes that the Greeks got their twelve constellations from the Egyptians. But, he doesn't note, after all the King Darious and Xerxes, and all the astrotheology he's poinint out, that right around the same paragraph of Herodotus saying the Greeks got the twelve constellations, Herodotus also says the Egyptians were the first to personify the twelve constellations - twelve constellations, twelve disciples of Jesus Christ. It seems clear to anybody who knows the twelve constellations and astrology, that the Gospels follow the twelve constellations pattern/story, and Jesus Christ is just a new Sungod, like all the sungods before him - the Egyptian Osirus, the Babylonians Marduk, Zoroastrianism, and Mithras, and the Greek Dionysius. But, Van Der Waerden misses this!