On the Invariance of the Tropical Zodiac -- Part I

by Robert Hand

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This is an inquiry into a fundamental problem that has to do with the tropical versus sidereal zodiac controversy and how it pertains to the practice of ancient astrology. It involves the very ancient practice of granting periods of years to the signs of the zodiac according to rising times. First of all let us define what is meant by the term "rising times."

As each sign rises in the east a certain number of degrees of right ascension pass over the meridian. This number of degrees constitutes the "rising times" of the sign. The period was assigned to each sign at the rate of one degree per year. For example at 40 degrees north the tropical sign Aries rises while 18d 06m pass over the meridian. Taurus rises while 21d 47m pass over the meridian, etc.

For the technically inclined another way of defining rising times is to compute the oblique ascension of the beginning and end of each sign and to subtract the oblique ascension or O.A. of 0 degrees of the sign from the O.A. of 30 degrees of that same sign. Oblique ascensions will be discussed below.

The computation of rising times has a great antiquity. According to Neugebauer in the History of Ancient Mathematical Astronomy (HAMA) the Babylonians in an early system of computations known as "System A" created such a scheme of rising times, although the times were computed not according to modern trigonometric methods as described above but by a numerical series such as that which follows: Starting with the first 30 degree segment which got 20 degrees of ascension, each 30 degree segment was given exactly 4 degrees more of rising than the preceding segment up to the sixth segment. The sixth and seventh segments have the same rising times, and subsequent segments decrease at the rate of 4 degrees until the twelfth segment has the same rising as the first. The longest rising times of the sixth and seventh 30 degree segment were each 40 degrees. We call these 30 degree segments rather than signs because according to Neugebauer these 30 degree segments began not at 0 Aries but at 10 degrees of Aries which was defined as being the location of the vernal point in the zodiacal signs. While it is tempting to assume that what we have here is measure of the vernal point in terms of a sidereal zodiac, the problem is that scholars do not agree that this vernal point was regarded as moving. The Babylonians may very well have regarded it as the permanent location of the vernal point in the zodiac. For the most part scholars do not believe that the Babylonians at this stage were aware that the vernal point precessed. According to these scholars Hipparchos the Greek discovered precession centuries after the development of System A. Other scholars such as Cyril Fagan and others who for the most part would not be regarded as "mainstream" [not necessarily to be interpreted as a criticism on my part], do believe that the Babylonians knew about precession.

Pursuant to this discussion let us acknowledge one thing at this point in the discussion. If we take Neugebauer's account of these rising times at face value, then we must accept that at some level that the Babylonians who developed this system were aware of the distinction between these 30 degree "signs" and 30 degree segments measured from the vernal point. It is not quite so clear that the Babylonians were conscious "Siderealists." However, whether or not the Babylonians who developed System A were conscious "Siderealists" is not central to our discussion.

Below is a table of the rising times of the 30 degree divisions from the vernal point as given in the Babylonian System A.

System A:
RSum RSun Long.Daylight
R120 20Ar 1012:00
R224 44Ta 1013:20
R328 72Ge 1014:08
R432 104Cn 1014:24 = Max.
R536 104Le 1014:08
R640 108Vi 1013:20
R740 220Li 1012:00
R836 256Sc1010:40
R932 288Sg 109:52
R1028 316Cp 109:36 = Min.
R1124 340Aq 109:52
R1220 360Pi 1010:40

The table is taken more or less directly from Neugebauer's HAMA with slight modifications in the notation. The first column simply refers to the 12 segments. The second column marked "R" contains the rising times as defined above of each of these 30 degree segments of the ecliptic. The third column gives a running total of the rising times computed from the rising of the vernal point. The fourth column gives the longitude in the zodiac in use (presumably a sidereal one) of the beginning of each 30 degree segment measured from the vernal point at 10 degrees of Aries. The fifth column gives the length of daylight that occurs when then Sun is in the particular degree. Note that when the Sun is at the vernal and autumnal points the daylight is precisely 12 hours; the daylight reaches a maximum of 14h 24m when the Sun is exactly 90 degrees later at the summer solstice point, and a minimum of 9h 36m when the Sun is at winter solstice point. Also note that the ratio of the longest daylight period to the shortest daylight period is precisely 3:2. Latitudes in the ancient world were measured according to the ratio of longest day to shortest day.

Another school of Babylonians developed a second system of rising times which are associated with a "System B." In this system the 30 degree divisions are made from a vernal point which is defined as 8 degrees of Aries. This is apparently a later measurement of the vernal point in the sidereal zodiac. Below is a table of rising times. The values are different from the previous table, but the logic of the table is the same.

RSum RSun Long.Daylight
R121 218 Ar12:00
R224 458 Ta13:12
R327 728 Ge14:00
R433 1058 Cn14:24 = Max.
R536 1418 Le14:00
R639 1808 Vi13:12
R739 2198 Li12:00
R836 2558 Sc10:48
R933 2888 Sg10:00
R1027 3158 Cp9:36 = Min.
R1124 3398 Aq10:00
R1221 3608 Pi10:48

We include this table of System B for the sake of completeness, but it will not figure in our discussion as much as System A except in one regard, the definition of the vernal point as being 8 degrees of Aries.

Note that in both tables opposite 30 degree segments have rising times which total 60 degrees. Also note the following: the first and twelfth segments have the same rising times; so do the second and eleventh segments, the third and tenth segments, and so forth. This symmetry is very important for reasons that we will disclose below. First however let us look at how well the rising times of System A reflect the actual rising times of the 30 degree segments as computed according to modern methods.

Rising Times of the Tropical Signs Using Modern Values Versus the Values of System A:
SegmentsR1-R12R2-R11R3-R10R4-R9R5-R8R6-R7
System A20 0024 0028 0032 0036 0040 00
501 B.C.E.20 1823 4229 4034 4936 0735 24

The last row is computed for 501 B.C.E. using modern trigonometric methods for Babylon. The agreement is quite good for R1-R12, R2-R11, and R5-R8. The fit is cruder for the other segments. It is also worth noting that the rising times computed for these 30 degree segments measured from the vernal point (which of course correspond to the tropical signs) are quite stable over time. Below is a table showing the rising times computed using modern methods for 501 B.C.E. and 2000 C.E. for the Babylon.

Rising Times of the Tropical Signs at Babylon
Ar-Pi Ta-AqGe-CpCn-Sg Le-Sc Vi-Li
501 B.C.E. 20 18 23 42 29 4034 49 36 0735 24
2000 C.E. 20 28 23 4829 4034 4236 0135 21

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