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moving westward). First visibility is also known as the heliacal rising, first appearance before the rising sun.

(2) Stationary point in the east (®). This marks the end of retrograde motion and the beginning of direct (eastward motion, toward the Sun). This is the instant of greatest western elongation (because the object is west of the Sun).

(3) Last visibility in the east (£). This marks the end of the morning star phase and just precedes what we call "superior conjunction."

(4) First visibility in the west (x), just after sunset. The first appearance as an evening star, east of the Sun, still in direct motion, but now moving away from the Sun.

(5) Stationary point in the west (Y). The end of direct and onset of retrograde motion and moving back toward the Sun. This is greatest eastern elongation.

(6) Last visibility in the west (W). The end of the evening star phase and heliacal setting. This phase precedes what we call "inferior conjunction."

For Mars, Jupiter, and Saturn, the progression is as follows:

(1) First visibility in the east (r), just before sunrise. The heliacal rise is followed by direct but slowing motion.

(2) First stationary point (®). This marks the onset of retrograde motion.

(3) Opposition (Q). Here, the planet rises acronychally (and sets cosmically) because it is opposite the Sun.

(4) Second stationary point (Y). This marks the end of the retrograde motion and the resumption of direct motion.

(5) Last visibility in the west (W), just after sunset. The heliacal setting precedes conjunction.

The planetary tablets bear pairs of columns of the date and the longitude for each of the configurations, with each successive cycle repeated in successive rows. The period of each column is the same, and it is based on observation of essentially horizon phenomena, except for the stationary points, for which the daily motion with respect to nearby stars, would be measurable. The period of repetition of the rows, i.e., the number of cycles until the table begins to repeat, indicates the time for the phenomena to arrive back at the same celestial longitudes as the first row. This period is an integer multiple of the time interval between successive repetitions of the same phenomenon; between each repetition, the phenomenon travels an arc length Dl along the ecliptic. The relationship can be expressed as

where

Prow is the interval of time between the two rows of data. P«sid» is the period to return to the same celestial longitude, that is, the same position on the zodiac (a kind of sidereal period conceived from a geocentric standpoint—to see how this can work, when the planetary configuration also repeats, refer to Figure 2.24b). Psyn is the synodic period, the time interval between successive occurrences of the same planetary configuration or phenomenon. Recall that a synodic period is the relative period of the planet with respect to the Earth. In the context of ancient Mesopotamia, it is the time to appear at the same configuration in the sky with respect to the position of the Sun.

Mathematically, to paraphrase Neugebauer,21 m is the smallest integer such that m repetitions of a certain phenomenon, say, r, corresponds to n "sidereal rotations" of r. In other words, after m occurrences of phenomenon r, that phenomenon appears again at the same place on the zodiac;

21 In Neugebauer's (1955/1983, p. 182) notation, II = m, and Z = n, all integers, and he refers to the number of complete mean synodic arcs, each of arc length Dl, contained in a complete circle, as P = 360/Al = II/Z. From this and (7.2), it follows that P = P-sid»/Psyn.

during that time interval, the phenomenon appeared around the sky in various zodiacal signs and required n complete circuits (360°, or in Babylonian notation, 6,0 degrees of rotation) to return to the same point among the stars.

Additional work is required to relate this motion of the phenomenon among the stars to the apparent motions of the planet and the Sun. The Sun and the planet must wind up at the same locations in m synodic years (during which interval, the configuration has repeated n times). Neugebauer's method of determining the motions is first to write the separate motions of the Sun and planet:

V mJ

Multiplying both sides by m,

The upper of (7.4) gives the time for the Sun to go through m repetitions of the phenomenon and therefore to get back to the same location with respect to the stars. The period for it to get back to the same place with respect to the stars (the right-hand side of this equation) is therefore given in sidereal years. The same motion for the planet in this interval allows us to equate the upper and lower parts of (7.4), so that in m occurrences

(im + n) years = (km + n) sidereal rotations. (7.5)

Now the idea is to find the smallest values of n and m that satisfy (7.5), which Neugebauer refers to as the fundamental period relation for the planet under consideration. Neugebauer uses this equation to establish mean motions. He establishes empirically that for the inferior planets, i = k and for superior planets, i = k + 1. Moreover, for Mercury, k = 0; so i = 0 and for Venus, k = 1, i = 1; for Mars, k = 1, i = 2; for Jupiter and Saturn, k = 0, i = 1. The argument for Mercury goes as follows: Mercury proceeds from, say, one r to the next G in ~1/3 year. The Sun does not travel a full circle in this time, hence, i = 0; but because Mercury is always very close to the Sun, k = 0 also. Similar reasoning can produce the values for the rest of the planets. Neugebauer uses these relations to determine the mean motions. The application of (7.5) to the inner planets gives the result that after m occurrences of a phenomenon, the number of rotations to come back to the same point on the zodiac is the same as the number of elapsed years, and there are m occurences in (n + km) years. For the outer planets, the number of years is the sum of the number of rotations and the number of occurrences of the phenomenon:

m occurrences over (km + n) rotations in (km + n + m) years.

With i and k known, the ratio m/n can be determined. This ratio is just the number of times the phenomenon can occur over a complete circle; Neugebauer calls this a period: P = m/n = 360/Al. Neugebauer (1955/1983, p. 283) gives an example of the role that these numbers played in the Babylonian planetary theory. In Tablet 801, Saturn is said to make 256 appearances in 265 years, in which time, it completes nine "rotations" = 3240°. Therefore, m = 256, and n = 9. We have from (7.5), (im + n) years = (km + n), so that with k = 0, i = 1, m + n years = n rotations, or 256 + 9 years for the nine rotations (the same result is achieved for the expression for the number of years vs. rotations explicitly for outer planets, given above). The number of repetitions (appearances at a particular configuration) in a full 360° is P = m/n = 28.4444... or in Babylonian notation, 28;26,40.

The mean "rate" follows: = -360 = 12.656, or 12;39,22.

At P

However, the speed of a planet in moving from one type of configuration (see Figures 2.23, 3.18, and 7.10) to another is not constant, due to: the eccentricity of the planet (see Table 2.9; §2.3.5); the varying distance from Earth, hence, apparent angular speed; the varying relationship between the orbital path of the planet and the ecliptic; and the motion of the Earth. Consequently, systems of interpolation involving step functions (System A) and zigzag functions (System B) were used, as per the lunar tables, to get "true synodic arcs" per time interval. Further details can be found in Neugebauer (1955/1983, Vol. II, pp. 284ff.).

Of the 300 astronomical texts, none appear to be earlier than the year 4 (~308 b.c.) of the Seleucid Era and the last is from 353 S.E. (~41 a.d.). All in all, the tables reveal the remarkable extent to which astronomical progress had been made in the Seleucid Era and furnish abundant evidence of the critical importance of astronomical data to Mesopotamia. As far as Babylonian interest in planetary motions is concerned, Neugebauer (1955/1983, p. 279) remarks that the main thrust seemed to be toward determining the sequences of the phenomena, and that the daily motions are secondary. Indeed, he notes further (1955/1983, fn. 5) that the Babylonian ephemerides would not have been much use to Hellenistic astrology, because the positions of planets in a zodiacal sign at the moment of birth are not immediately solvable from the ordinary planetary ephemerides; moreover, he states that "the characteristic phenomena listed in these ephemerides play no role whatsoever in astrological practice." This was certain to change, as we will see.

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