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The Babylonian Theory of the Planets
By N. M. Swerdlow
PRINCETON UNIVERSITY PRESSCopyright © 1998 Princeton University Press
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Periodicity and Variability of Synodic Phenomena
Units of Distance and Location and of Time and Date
The Babylonian zodiac (lu-ma-me) is divided into twelve zodiacal signs (sing, lu-ma). Each sign is in turn divided into 30 us, a term with the general meaning 'length' interpreted, according to modern usage, as degree (°), each of which may then be divided sexagesimally to as many places as desired. The zodiac, its twelve equal signs, and the divisions of signs into u and its fractions are purely conventional, an abstraction intended to facilitate computation, as in the ephemerides, while retaining the names of constellations of stars of irregular lengths unsuitable for computation. It is by no means obvious, and in fact is very unlikely, that the observational texts, the Diaries, use signs of equal length—for how would one know where their limits lie?—rather than simply regions around the constellations, and it is the presence of the planets and phenomena in the constellations and near stars that are of ominous significance, as we have seen. In Greek astrology, on the other hand, which depends principally upon computed horoscopes, it is equal signs rather than irregular constellations that are significant. But even this has Babylonian prototypes in the, apparently computed, positions of planets in horoscopes published by Sachs (1952) and Rochberg (1998), which, however, also contain positions with respect to stars and constellations, presumably taken from observational records.
While it is conventional to call locations and distances measured in signs and us 'longitude', one should not take this to be a coordinate, in the sense of Ptolemy and modern astronomy, measured along the ecliptic or even a circle. Rather, it is more likely that the zodiacal signs are regions, for the purpose of computation of equal length, of a band or path around the heaven through which the sun, moon, and planets move, within which their phenomena take place, and any further divisions into u and smaller units are arithmetic and likewise for computation. The motion of a planet between the phenomena computed in the ephemerides is anything but an arc of a circle, and the us separating phenomena, which occur intermittently, are best thought of purely arithmetically rather than geometrically as parts of a circle divided into 360°, a Greek geometrical adaptation of a Babylonian arithmetical convention. There is but a single 'atypical' Text F (AT 1, 208-09) describing a second coordinate, a 'going up and down* that appears to be a zodiacally fixed latitude for Saturn, less likely Jupiter, and this is not an inclined circle but a truncated zigzag—which one must admit is a good approximation to an inclined circle—with 'up and down' in units of cubits and fingers, used more for observation than computation. Likewise, a more extended description of lunar latitude in atypical Text E (AT 1, 200-05) considers the motion of the node and treats latitude itself, not as the inclination of a circle, but as the width of the moon's path or road (dagal ma-la-ka), of 10 u or 6 cubits (12 u)—the latter also found in the procedure text ACT 200.6—in which the moon goes 'up and down' in cubits and fingers while passing by stars.
In any case, the principal object of the planetary theory is to determine, not the motion of a planet, but the date arid location of a phenomenon, for which the very notion of a defined path is without meaning. Although there were methods of interpolation, some very sophisticated, for finding the motion of the planet between phenomena, the phenomenon itself may be regarded simply as taking place in one location after another each time the planet and the sun reach a characteristic elongation. It is not quite appropriate to compare the phenomenon to a pseudo-planet with a continuous motion of its own, like the lunar nodes, the head and tail of the dragon, in Indian and Arabic astrology, because the phenomenon does not exist most of the time while the lunar nodes move continuously along the ecliptic between the eclipses that make them apparent. Nevertheless, we shall, for lack of any alternative term, refer to locations and distances in us through the zodiac as longitude, and measure them in degrees, but without thereby implying that they are measured along a circular coordinate. We shall also use the term arc, as in synodic arc, to mean simply an interval of distance, without implying an arc of a circle projected on to the celestial sphere.
Further, the us is a metrical unit for computation, not observation, for which a different unit, the kù (cubit), was used, divided into 24 or 30 u-si (finger). The conversion of the kù used for observation, principally of the distance of the moon and planets from fixed stars and from each other, and the us is uncertain; possible values for the kù are 2 or 2½ u, corresponding to 24 or 30 u-si, although as ued for observation the units may not be precisely convertible. Gerd Graßhoff (1998) has found an average value of about 2.4 u for the kù, suggesting a value of the kù of 2½ u, although the conversion still may have been neither precisely defined nor used. We have noted that in the atypical text for planetary latitude the distances 'up and down' are in cubits and fingers, while the lunar latitude text uses both u and cubits and fingers, with the moon's diameter taken as 12 fingers, which is here equal to 1 u, implying 1 kù = 2 u - 24 u-si, the same relation implicit in columns E and Ψ of lunar ephemerides for the computation of eclipses. But this defined kù may not be the same as the kù used for observation.
The u is also a unit of time: one day contains 6,0 u, now called 'degrees of time' or the like, which may then be divided sexagesimally to many places. The origin is, perhaps, a purely arithmetic, not practical, division of each of the three night watches into 1,0 u and, by analogy, of three divisions of daylight to make 6,0 u in all—this would be Epping's division of the day into 6 parts and Neugebauer's 6 'large hours'—or perhaps a more practical division of watches into 2 beru (danna), each of which is divided into 30 u, so that one day contains 12 beru or 6,0 u. The term beru is also used for 30 u of longitude, but not specifically in the sense of a zodiacal sign. Like the division of the zodiac, the u of time is principally a unit of computation, although indirectly it can also be measured. In lunar ephemerides, Column C gives the computed seasonal length of daylight in u, where the longest day M = 3,36 u and the shortest m = 2,24 u; thus M/m = 3/2, a reasonable approximation in small numbers for Babylon, however, and in whatever units, it may originally have been measured. The time in u of the visibility of the moon between the rising and setting of the sun and moon may likewise be computed, as in the lunar ephemerides or from the rules given in the procedure texts ACT 200 and 201, or observed by some measure of time, as by weight or volume of water in a water clock—the unit of weight, the mina, is for water essentially a linear function of volume, and so may be measured by volume—but not by direct measurement of an arc. Us are also used for the time of visibility of planets at their first and last appearances, which we shall take up later. A recent study by Stephenson and Fatoohi (1994) of 53 reports from ADT and other sources extending from -561 to -66 of durations of lunar eclipses in u has confirmed that, however it was measured, the u is very accurately equal to 4 minutes of time where one day equals 24 hours. When adopted by the Greeks and interpreted geometrically, these u of rising and setting times and length of daylight became degrees of oblique ascension of a circle, the celestial equator, divided into 360° so that 15° equal one hour. But as used by the Scribes, the u must be considered purely arithmetically as a unit of time, and not as a unit of a coordinate of right or oblique ascension, just as the division of zodiacal signs into 30 u does not indicate a coordinate of longitude.
The Babylonian lunar calendar contains months of 29 or 30 days that are, in principle, determined astronomically, either by observation or by computation from lunar ephemerides of the evening of the first visibility of the moon. However, for practical and commercial purposes, just as banks often use uniform months for computing interest, a conventional month of 30 days was used, which was conveniently carried over into the planetary ephemerides since it eliminates the problem of determining whether months between computed phenomena are 29 or 30 days long. This unit of time, which the Scribes did not distinguish from the day (umu, me), is now called by the Sanskrit term 'tithi' (τ), and like the u may be divided sexagesimally to as many places as desired, a division that is also for purposes of computation since, for example, the dates of first appearances and synodic times between first appearances may be computed to several places even though first appearances only take place in the morning or evening separated by integral numbers of days. We thus have the parallel metrological divisions for distance and time, 1 sign = 30 u and 1 month = 30 tithis, 12 signs = 6,0 u and 12 months = 6,0 tithis. Likewise for division of the day, 1 beru = 30 u and 1 day = 12 beru = 6,0 u. It happens that the sun moves about G per day, and thus about 1 u per tithi and 30 u per month, but these are in no way essential to the definitions, which are purely formal. Since the calendar month is 29 or 30 days long, the error in stating the 'day' of the month in tithis rather than days cannot exceed 1 day and is usually less. Like the u, the tithi is a unit for computation, not observation, for when dates are noted in observational records, they are days of the calendar month, not tithis of a conventional month. For the ominous significance of planetary phenomena and the effect of rituals and magic performed under the planets or stars depend upon the day of the lunar, that is, the calendar month, the days of which are themselves ominous, as we have noted. We have also seen that in many planetary omens only the month is taken to be significant, in which case all the more are the divisions of the month purely for the purpose of calculation, and a slight discrepancy of at most one day of no consequence.
The length of the solar year in the planetary theory is likewise taken without regard to the length of each month in days, for in terms of the month m, the year y is denned as y = 12;22,8m, a parameter also found in System A lunar theory. If m = 30τ,
y = 12:22,8 · 30τ = 6,11:4τ = 6,0τ + 11;4τ = 12m + 11;4τ.
where the 'epact' e = 11;4τ is the excess of the solar year over the 'lunar year' of 12m = 6,0τ. Within the complexities of a true lunar calendar with variable months, the ue of tithis and a denned epact over 12 months is equivalent to the uniform day count of the Egyptian calendar or the modern Julian day number, and greatly simplifies the reckoning of time since all months and years are now equal and related in a simple way. The only necessary adaptation to the calendar is the insertion of additional months in accordance with the calendrical cycle 19y = 19 · 12m + 7m = 235m, by which the same month of 30τ is transferred from the longer year of the planetary theory, y = 12:22,8m = 6,11;4τ, to the shorter calendar year, y = (235/19)m ≈ 12;22,6,20m = 6,11;3,10τ.
The Scribes' units of distance and time, of location and date, are economical and efficient, show a high degree of abstraction, and make no more assumptions depending upon observation and measurement than absolutely necessary. The irregularities of the lengths of zodiacal constellations and months are avoided by defining 1 sign = 30 u and 1 month = 30 tithis; the year is 12:22,8 months in the planetary theory and 12:22,6,20 months in the calendar. All relations are purely formal except for a single astronomical parameter, the relation of the year to the month, and even this is partly formal and defined differently for different purposes. In order to express time in the ordinary unit of the solar day, a second parameter derived from observation is required, the length of the mean synodic month in days. Like the relation of the year to the month, the Scribes employed different values of the relation of the month to the day, but it will suffice for our purpose to ue the approximate values
m = 30τ = 29;31,50d, 1τ = 0;59,3,40d ≈ 0;59d, 1d = 1:0,57, ... τ ≈ 1;1τ.
Using days instead of tithis, all relations for the year and month become more cumbersome and less precisely defined, and with true months of 29 or 30 days all but unmanageable. Fortunately, we shall work entirely in tithis, months of 30τ, and years of 12:22,8m = 6,11;4τ. In this way, j u s t as intervals of longitude are expressed in u, which are divided as required, signs s = 30 u, and zodiacal rotations r = 12s = 6,0 u, intervals of time are expressed in tithis, also divided as required, months m = 30τ, years y = 12m + e = 6,11;4τ, and, for calendrical purposes, lunar years of 12m = 6,0τ.
Dates and Locations of Phenomena in the Diaries: Observed and True Dates
The most important source for observations of planets is of course the Diaries, but in addition there are collections for individual planets for periods of years, of which all too few survive, that may be closer to what was actually used for the development of the planetary theory. The dates and locations of the planetary phenomena interpreted in the omen texts and computed in the ephemerides are reported in the Diaries, but not necessarily directly from observation for either date or location, and must be used with care to distinguish what is, in descending order of directness, observed, even measured, estimated, inferred, or computed. By far the most numerous observations of planets in the Diaries are of their distances 'above' or 'below' and 'in front of or 'behind' normal stars and each other, measured in cubits and fingers, which are found in the earliest Diary, ADT -651. Here are early examples:
-651 XII. The 19th, Venus stood in the region of Aries, 10 fingers behind Mars; the moon was surrounded by a halo, and α Scorpii stood in it. The 20th, Mars was 1 finger to the left of the front? of Aries; it came close.
-567 II. The 18th, Venus was balanced 1 cubit 4 fingers above a Leonis.
-567 III. At that time (the 1st), Mars and Mercury were 4 cubits in front of α [Leonis].
-567 XI. Night of the 23rd, [.... Mars?] was balanced above (sic) the small star which stands 3 1/2 cubits behind Capricorn.
Excerpted from The Babylonian Theory of the Planets by N. M. Swerdlow. Copyright © 1998 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents
Introduction, Planetary Omens, Observations, and Calculations
Ominous Phenomena in the Heavens 1
Observation of Phenomena 16
Calculation of Phenomena 23
Pt. 1 Periodicity and Variability of Synodic Phenomena
Units of Distance and Location and of Time and Date 34
Dates and Locations of Phenomena in the Diaries: Observed and True Dates 39
Periods and Their Errors 57
Synodic Arc, Synodic Time, and Their Relation 64
Pt. 2 Derivation of the Parameters for Synodic Arc and Time from the Dates of Phenomena
Theoretical and Empirical Considerations 73
System B 78
System A 88
Mercury: Systems A[subscript 1] and A[subscript 2] 104
Pt. 3 Alignment to the Zodiac, Initial Position, Elongation, Subdivision of the Synodic Arc and Time
Alignment to the Zodiac 135
Initial Position and Elongation 141
Elongation and Subdivision of the Synodic Arc and Time 147
Summary and Conclusion
App Alternative Methods of Deriving Parameters
Derivation of System A from the Number of Phenomena 183
Finding the Synodic Arc from the Direct Measurement of Longitude 186
Notation and Abbreviations
Index of Names
Index of Subjects
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