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Simply stated, our planet rotates on its axis from west to east once in twenty four hours with respect to the Sun during which time it simultaneously moves approximately one degree (also from west to east) on its annual orbital path. The time taken to complete one such revolution with respect to a fixed stellar reference point (the sidereal year) is 365.25636 days and hence the mean daily orbital motion is 360/365.25636 = 0;59,8,11,34,12,39,..degrees per day. However, the orbit of Earth is elliptical rather than circular, and Earth moves at its fastest when closest to the Sun (perihelion) and slowest when it is most distant (aphelion), thus it is either fast or slow with respect to the mean motion; this provides the basis for the equation of time. Moreover, Earth's axis is not at right angles to the plane of motion, but tilted some 23.5 degrees with the axis subject to a slow, continual variation caused by precession. In general terms the latter may be defined as the conical motion of the axis of a rotating, oblate spheroid; in our present context the rotating oblate spheroid is Earth itself. The annual amount of precession is very small, i.e., approximately 50 seconds of arc or 0;0'50" degrees = 1/72 of a degree, or 1 Babylonian "barley-corn." The time required to complete one complete wobble of Earth's axis at this theoretically convenient rate would consequently be: 360 degrees/0;0'50" = 25,920 years, and yes, there are that many "barleycorns" in one revolution. The sidereal year differs from the seasonal or tropical year (equinox to equinox) of 365.2421897 days by the annual amount of precession, but this last parameter is also subject to further modifications imposed by the combined motions of the Moon and the planets. Lastly, the return with respect to velocity is represented by the anomalistic year of 365.25964 days. Next, because of the one degree or so of Earth's daily orbital motion, rotation with respect to the fixed sidereal reference point is reached before Earth completes a full rotation on its axis; technically, the sidereal day is defined by The Oxford Dictionary of Astronomy (Oxford University Press, New York, 1997:426) as:
The interval of time between successive transits of the mean equinox, equal to 23 h 56 m 04s. Because of the precession of the equinoxes, the mean equinox is not a completely fixed sidereal point. As a result the sidereal day is 0.0084 s shorter than Earth's rotation period relative to the stars.
whereas the solar day as defined by the same source (p.432) is:
The interval between successive transits of the Sun across the observer's meridian, that is, the rotation of Earth with respect to the Sun. Strictly, this is the apparent solar day, which varies slightly during the year because of the equation of time. Its average length, the mean solar day, is 24 hours or 86400s. Because of Earth's orbital motion around the Sun, the solar day is about 4 minutes longer than the sidereal day, a discrepancy that adds up to one whole day in the course of a year.
Modern conversion factors for the various lengths of the day as given in The American Ephemeris Explanatory Supplement, p.695 are:

1 day of mean solar time = 1.00273790935 days of mean sidereal time = 24h 03m 56.555368s = 86636.555368 mean sidereal seconds.
1 day of mean sidereal time = 0.99726956633 days of mean solar time = 23h 56m 04.09054s = seconds = 866164.09054 mean solar seconds.

The norm for purposes of comparison may first be established from the following Hindu luni-solar relationship given by H.P. Blavatsky (The Secret Doctrine, Book 1-3-17), i.e., the latter writes:

"The Hindus state that 20,400 years before the age of Kali Yug, the first point of their Zodiac coincided with the vernal equinox, and that the sun and moon were in conjunction there. This epoch is obviously fictitious;* but we may inquire from what point, from what epoch, the Hindus set out in establishing it. Taking the Hindu values for the revolution of the sun and moon, viz., 365d. 6h. 12m.30s., and 27d. 7h. 43m. 13s., we have --
20,400 revolutions of the sun = 7,451,277d. 2h.
272,724 revolutions of the moon = 7,451,277d. 7h. "
The actual point being investigated by Mme Blavatsky is not our immediate concern, but it is necessary to be aware that although the given length for the mean sidereal month is very accurate (differing by a single second from modern) superior estimates for the length of the year were known (see Dwight W. Johnson's Exegesis of Hindu Cosmological Time Cycles and Glen R. Smith's 6000 Year Barrier for more information on this topic). It is also necessary to realize that both the year and the mean sidereal month are likely rounded values in any case, as indicated by the last place in their sexagesimal equivalents, i.e., the year is 365;15,31,15 days and 27;18,18,2,30 days respectively. Although unstated, the 20,400 year interval incorporates additional luni-solar constants as would be expected in a long-term period relation of this nature. Modern estimates are given to five decimal places in the Explanatory Supplement to the Astronomical Almanac (p.695); a comparison with the latter is shown below:
273821 Draconic Months 27.21222 Days 27.21222 Days
272724 Sidereal Months 27.32166 Days 27.32166 Days
252324 Synodic Months 27.55455 Days 27.55455 Days
270419 Anomalistic months 29.53059 Days 29.53059 Days

thus the period relation in question supplies the norm against which unusual data may be checked with confidence in the present cultural context.

Returning to the polar hypothesis, even it there were no other changes whatsoever, there would nevertheless be predictable changes to the length of the day and the year after a full 180-degree reversal. In other words, the mean solar day would become shorter than the mean sidereal day instead of vice-versa. The various conversion factors for the various lengths of the day are also provided by the American Ephemeris Explanatory Supplement, p.695, i.e.,

1 day of mean solar time = 1.00273790935 days of mean sidereal time = 24th 03m 56.555368s = 86636.555368 mean sidereal seconds.
1 day of mean sidereal time = 0.99726956633 days of mean solar time = 23h 56m 04.09054s = seconds = 866164.09054 mean solar seconds.

while the sidereal day is defined by The Oxford Dictionary of Astronomy (Oxford University Press, New York, 1997:426) as:

The interval of time between successive transits of the mean equinox, equal to 23 h 56 m 04s. Because of the precession of the equinoxes, the mean equinox is not a completely fixed sidereal point. As a result the sidereal day is 0.0084 s shorter than Earth's rotation period relative to the stars.
thus the rotation of Earth is 24h 00m 00.0084 seconds in mean sidereal time whereas the solar day (p.432) is defined as:
The interval between successive transits of the Sun across the observer's meridian, that is the rotation of Earth with respect to the Sun. Strictly, this is the apparent solar day, which varies slightly during the year because of the equation of time. Its average length, the mean solar day, 24 hours or 86400 s. Because of the Earth's orbital motion around the Sun, the solar day is about 4 min longer than the sidereal day, a discrepancy that adds up to one whole day in the course of a year.
However, after a reversal of the Poles, Earth's rotational motion will oppose the orbital motion as explained earlier. This means that the "day" would appear to have become shorter and hence the lengths of both the "year" and the "month" would appear correspondingly longer. With this in mind we now turn to an unusual Indian luni-solar period relationship released to the Internet in February (1998) by James.Q.Jacobs, (The Aryabhatya of Arayabhata) who recognized both its unusual nature and its significance.
The period in question is 4,320,000 years. The latter period and its decimal-reduced variants occur in many ancient contexts, including those of the Babylonians, the Egyptians and the Greeks. The complete 4,320,000-year period relationship in the Indian context is as follows:
"In a yuga the revolutions of the Sun are 4,320,000, of the Moon 57,753,336, of the Earth eastward 1,582,237,500, of Saturn 146,564, of Jupiter 364,224, of Mars 2,296,824, of Mercury and Venus the same as those of the Sun"
"2) of the apsis of the Moon 488,219. of (the conjunction of) Mercury 17,937,020, of (the conjunction of) Venus 7,022,388,... of the node of the moon westward 232,226 starting at the beginning of Mesa, at sunrise on Wednesday at Lanka.".
One would expect that the simple division of the number of days by the number of years would produce a value similar or at least close to that given by Mme Blavatsky, (365;15,31,15 or 365.258680555556 days) but instead we find a value that is exactly one day longer in company with a correspondingly longer mean sidereal month:

365;15,31,15 days = 7,451,277d.2h. divided by 20,400 years [Normal Year]
366;15,31,15 days = 1,582,237,500 days divided by 4,320,000 years. [Greater than Normal]

27.3216779295796 days = 7,451,277d.7h. divided by 272,724 revolutions [Normal Mean Sidereal Month]
27.3964693571987 days = 1,582,237,500 days divided by 57,753,336 months [Greater than Normal]

In other words, we find exactly what we are looking for, deviations from the norm that indicate the motions of the Sun and moon have both changed, and that their mean periods have also lengthened as a consequence of a shorter day. Moreover, we also find that the change is almost identical to that expected, i.e., both periods increase by a factor of 1.002737785 (or: 1;0,9,51,21,42,..from the division of 366;15,31,15 by 365;15,31,15.) The multiplication of this factor by 24 leads to 24;3,56,32,40,52,.., i.e., 24 hours; 3 minutes 56.54468627 seconds, or an excess over 24 hours of 3 minutes 56.54468627 seconds compared to the modern conversion excess of 3 minutes 56.555368 seconds Explanatory Supplement to the American Ephemeris and Nautical Almanac, p 695].
It can be seen from the Aryabhata relationship that the periods of the planets may be expressed in terms of revolutions with respect to those of the Sun, thus they are independent of the number of days, i.e., 4,320,000 years can be divided by the number of sidereal revolutions in all five cases. The results are accurate enough when compared with modern estimates expressed in years, but not as accurate when obtained from the number of days in the relationship. For example, the mean sidereal period of Mercury is 4,320,00 divided by 17,937,020 = 0.2408427 years - an excellent estimate. Applying Mme Blavatsky's value for the year (365;15,31,15) the latter expressed in days is in turn 87.9698801 days. However, the division of 1,582,237,500 days by 17,937,020 results in the poorer value of 88.210722 days. On the other hand, if the last value is divided by 366;15,31,15 days then the correct sidereal period is again obtained. This applies to all the planetary sidereal periods and thus we have the latter also expressed in terms of the changes to the length of the year. In passing it may be noted that the Aryabhata data can be readily normalized by the reduction of the number of days in the relationship by the stated number of years, i.e., 1,582,237,500 days - 4,320,000 days = 1,577,917,500 days which produces the "normal" year of 365;15,31,15 days.

One could consider the Aryabhata relationships to be an intellectual exercise rather than precise data as discussed above if it we not for the Babylonian textual data and the explicit methodology used in the latter. On the other hand, the value of the sidereal month (27.3964693571987 days) does not occur among the luni-solar mean periods listed by the Explanatory Supplement to the American Ephemeris and Nautical Almanac, nor is a particularly useful or common value in practice. Moreover, for the increased year of 366;15,31,15 days the corresponding mean synodic month is also high, i.e., 29.6114302 days as opposed to the modern and ancient norms of 29.530395 days.

James Q. Jacobs noted that the accuracy inherent in the Aryabhata period relationship could be understood in terms of axial rotations of Earth as opposed sidereal and solar rotations per se. He therefore found a point in time from this viewpoint that produced a close correspondence with the Aryabhata mean period for the sidereal month, arriving at 1604 BC for the date in question. Once again, if it were not for the polar hypothesis, this too would furnish an excellent explanation for the unusual nature of the relationships in the Aryabhata. However, it is just possible that the material was also intended to draw attention to the date in question in this manner. Could complex information be combined in this dual fashion? It depends, perhaps, not so much on capabilities per se, but rather on the need, the time, and the priorities assigned to the matter. We should also realize that the best minds of those times could conceivably have labored over perhaps centuries, if not longer to refine and perfect the messages that they wished to pass on. This may have been just one further example.



Now that we have some idea of the phenomenon under consideration we can prepare a short list of parameters and situations that should hopefully be present in whatever texts we might find in our search for documented understanding. Firstly, in so much as modern scholarly interpretations do not appear to embrace the inversion hypothesis we might expect to find that translators would have grave difficulties in understanding any material which concerns the phenomenon in question. We would therefore expect to find that they would be unable to interpret textual material that speaks of changes in the velocities of the sun and moon, that mention north and south, nor would they be able to make anything of corrections and differences mentioned in such contexts. Nor also, would they expect to find additional parameters beyond the norm that might accompany the delineations we are concerned with. We would also hope to find references to inexplicable reversals, i.e., unexplained references to Sunset instead of Sunrise, and similar substitutions between the two solstices and the two equinoxes. We might also find that certain parameters would be referred to as being "small" as indeed they are from a daily rotational viewpoint. And if we are especially fortunate, we might even find instructions that not only deal with such variations, but also link the methodology and values with those found in Megalithic Britain. In this last context we would therefore hope to find a concentration on the movements of the Sun and Moon, especially Sunrise, Sunset, Moonrise, Moonset and unexplained parameters for the two major luminaries preferable in the same puzzling context. Lastly, we might hope to find the factor 2.5 that is so strongly integrated into the stone circles and variants in Megalithic Britain, and also an understanding of the role played by such sites as Brodgar, Newgrange, Avebury, Stonehenge and others. Even without the above we should certainly bear in mind the fact that the following constants derived by Alexander Thom from the stone "circles" in Megalithic Britain are all finite, convenient parameters in base-60 as stated in Part I, i.e.,
One point of possible relevance here is the obvious convenience of the latter approximation over the more accurate ratio 22/7 in base-60, i.e., in sexagesimal notation 22/7 = 3+1/7 = 3;8,34,17,8,34,17,8,34,17,.. which is an inconvenient repeating sexagesimal number, as is its reciprocal. The more accurate 22/7 might still have been required, but as Thom suggests, it may have been reserved for the largest of the stones circles with the more convenient approximation 25/8 = 3+1/8 = 3;7,30 (reciprocal = 0;19,20) employed in less critical applications. In any event, for the latter estimate, four "pi" (25/2) is simply 12;30, i.e., the factor 12.5 mentioned above in the same context with the approximation 25/8. In addition, the other two working estimates determined by Thom - 3.1400 and 3.139 - are also convenient numbers in sexagesimal notation, i.e., 3;8,24 and 3;8,20,24 respectively. Lastly, the same convenience is again apparent in the conversion of the Megalithic Yard to base-60, i.e., 1 MY = 2.720 = 2;43,12.
and it is this notation that is employed with great competence throughout the Babylonian astronomical cuneiform texts. But before discussing the Babylonian astronomical texts in detail it is perhaps useful to recall the quotation provided by Graham Hancock mentioned in Part I [Fingerprints of the Gods, 1996:509]:
Likewise, in Mesopotamia, the Noah figure Utnapishtim was instructed by the god Ea, ' to take the beginning, the middle, and the end of whatever was consigned to writing and then bury it in the City of the Sun at Sippara ' [emphases supplied]
.... After the waters of the flood had gone, survivors were instructed to make their way to the site of the City of the Sun ' to search for the writings ', which would be found to contain knowledge of benefit to future generations of mankind
The above is no doubt best considered to be legend and indeed the astronomical material at hand dates from the Seleucid Era [310 BC - 75 AD] in any case. But it nevertheless indicates the kind of forethought involved, and although it may be coincidental, a number of the clay tablets we are interested in were indeed buried. Moreover, the use of clay tablets also meets some of the more important criteria outlined in Part I (except for size and portability, which may have been a necessity in this particular application), namely that the medium should be durable on one hand, yet neither perishable nor valuable on the other. Indeed, the only thing of value is the information itself, while the fact that the material has come down to us at all after being buried for so long attests to its durablility. Unfortunately, the state of preservation varies markedly, and it is in fact surprising just how much information has been recovered, even from quite badly damaged texts. But what is even more surprising is that in spite not crediting Babylonian astronomers with possessing a fictive planetary model that any real understanding has been gained at all, especially in the luni-solar context, given the greater complexity that attends this component. On the other hand, with such a prevailing viewpoint dominating the field, it is hardly surprising that the indicators listed above remained undetected. We find, for example, numerous unexplained luni-solar parameters in ACT No.200, a major astronomical cuneiform text described as "the most important text in this group." This Babylonian "procedure" text provides precisely the kind of instructions and data we would hope to encounter, i.e., unusual parameters associated with the motions of the sun and moon with complex concentrations on both sunset and sunrise. This group of texts, moreover, deals with one of the two main luni-solar systems known as System A - which of the two system is the one that incorporates inexplicably high lunar velocities linked directly to an unexpected set of additional lunar data. F.X. Kugler initially chose to interpret the latter in terms of the variable diameter of the moon, but one of the main reasons this suggestion was rejected by the noted authority Otto Neugebauer [1899-1996] was that the correlation was too good! (Astronomical Cuneiform Texts, Lund Humphreys, London, 1955; hereafter referred to as ACT). The reader should note that this was not simply one or two values, which could certainly be coincidental, but rather a complex variable function that maps the abnormally high variable velocity of the moon itself. However, this was not a consideration that bothered Neugebauer, who wrote (ACT, p.44):
Kugler proposed to see in (Column) Phi the apparent diameter of the moon, expressed in units corresponding to 15 minutes. This assumption leads indeed to a very close agreement with the actual values, but I am convinced that this agreement is purely accidental. My arguments against Kugler's hypothesis are as follows: (a) the extremely high accuracy of the agreement with the correct values of the lunar diameter goes far beyond the accuracy obtainable with ancient and mediaeval instruments. Because values can only be derived from actual observations, agreement with accurate modern values must be accidental and thus loses its force as an argument for interpretation; (b) Kugler has to assume units of 1/4 degree each which are nowhere else attested; (c) the little we know about the lunar diameter from the procedure texts does not support the assumption of a very accurate mean value of the lunar diameter and shows nowhere knowledge about its variability; (d) the use made of column phi for the computation of G does not support Kugler's hypothesis.
The above is not included here as an attack on Neugebauer per se, but to emphasize it is simply not known how the details for this function were obtained, what observational techniques were employed, and when or where the work was carried out. And this uncertainty remains irrespective of where the text was actually found and where it might apply in terms of latitude. Nor do we really know what degree of accuracy can be derived from the use of long sightlines and patient skill exercised at the critical northern latitudes already discussed. On the other hand, we now at least have some suspicion where the work might have been accomplished, even though it may well take a great deal more analysis before we can be at all sure. Nevertheless, we have the monuments, and it is just possible that we have part of the instructions; but even more importantly, we might also have some of the results to work with.

Continuing with the text in question, we also find the following line concerned with the mean motion of the Sun and an unexplained correction (ACT No.200, Section 9, p.196):

' [from South] to North or from North to South' 59,8 the mean velocity of the sun [ . . . . . . . . . . . ] . . . nappaltu (of) day(light)

i.e., the unusual term nappaltu is understood to mean a "difference" of some kind. The dual references concerning South to North and vice versa are immediately recognizable and appropriate from an inversion viewpoint, as we have noted a number of times already. Neugebauer states here that the 0;59,8 degrees is a better value than the 0;59,9 degrees that occurs in the second Babylonian system (System B), but from the Aryabhata material we can see that the latter is likely obtained from a rotational excess of 3;56,36, minutes while the former in turn represents a less convenient but more accurate excess of 3;56,32 minutes.
In other words, based on what has been gained from the analysis of the Aryabhata data and the use of one of the main parameters of Babylonian astronomy - the "year" (denoted here by Y) expressed as 12;22,8 mean sidereal months of 29;31,50,8,20 days we may proceed as follows. The ratio (Y+1)/Y determines the excess over 24 hours, which is found to be 3 minutes, 36 seconds, 32 sixtieths of a second, plus 36 sixtieths of the latter, etc., i.e., we obtain an excess of: 3;56,32,36,18,27,.. minutes expressed in sexagesimal notation. The Babylonians were more than adept at rounding or truncation their parameters, thus we can simply truncate to 3;56,32 which is still very accurate for an excess over 24 hours. The next step is also simple enough, but it has far reaching consequences nonetheless. If we convert the excess time in minutes to degrees of rotation (i.e., the excess over 360 degrees) the result is obviously going to be slightly less than one degree since the latter equals 4 minutes of time, then as now. Thus we now obtain from the truncated two sexagesimal place excess:

3;56,32 minutes excess time = 0;59,8 degrees excess axial rotation
Those who are familiar with the parameter in the given context will be aware that it is generally understood to be the mean value of a variable velocity function which tracks the daily solar velocity in monthly increments and decrements according to whether the function deals with the increasing or decreasing parts of the orbit.. The minimum velocity is 0;55,32 degrees and the maximum is 1;2,44 degrees known from other parts of the text, and if it were not for the unusual terminology, unknown parameters, and unattested operations that accompany it one might even tend to agree with Neugebauer that the function represents variable motion in terms of daily arcs. But with so much unexplained data in attendance this would not appear to be a very sensible interpretation, especially since this value mean value occurs with such loaded and unexplained phrases as that given above. We will also see that Neugebauer's single assignment sheds very light on the critical parts of the text that deal with unexplained changes to the solar and lunar velocities. Actually, the parameter is best considered to be both orbital and rotational, with the variable velocity function that attends it serving both, i.e., the first tracks the second. In other words, we now see how the daily rotational excess varies throughout the year, and also how it changes from month to month and even from day to day since the amount of the decrease or increase is also provided in one form or another. One could go at some length about this application, but before moving on to another interesting section in this text we might dwell on the term nappaltu, which Neugebauer discusses in his commentary to Section 8 of ACT No.200 leading up to the reference to South and North, i.e., he states (ACT, p.195-196):

"What follows (lines 7 to 10) contains the term nappaltu. This term has already occurred in Section 7, which also deals with the solar velocity. The phrase in line 7:
40 of the nappaltu (of) day(light) and night
has an exact parallel in the first half of a sentence in a text of the series "MUL-APIN" (VAT 9412) rev.III,5: 40 nap-pal-ti u4 u ge6 aná il-ma 2,40 nap-pal-ti gi-du8-a igi '0;0,40 of the nappaltu of day(light) and night multiply by 4, and you will see 0;2,40 of the nappaltu of the visibility.' From the context of this text it follows that 0;0,40 is the daily increase or decrease of the length of the watches of daylight or night measured in mana by means of a waterclock. To this is associated a daily increase or decrease of 0;2,40 degrees of the lunar visibility. I do not see how the change of the length of daylight is brought into relation with the solar velocity which is mentioned in line 7 and line 8. Line 8 begins with the words: 'which you predict' and it seems plausible to consider this as the end of a sentence. What follows makes no sense at all:
' days of (?) the sun velocity days' and then again '40 of the nappaltu (of) day(light)...'
In line 9 we have after the break at the end of line 8:
' [from South] to North or from North to South'
followed by:
59,8 the mean velocity of the sun [ . . . . . . . . . . . ] . . . nappaltu (of) day(light)
[ACT 200, Section 9, Lines 7-9, emphasis supplied]
In the above Neugebauer split the last line for his commentary. As a single line it is as follows:
' [from South] to North or from North to South' 59,8 the mean velocity of the sun [ . . . . . . . . . . . ] . . . nappaltu (of) day(light)
which is much more informative. Unfortunately, the state of the text and other uncertainties make it difficult to establish the full meaning as yet in this particular context; but suffice it to say here that there are other areas in ACT No.200 that can also be explored from the inversion viewpoint. In particular, there is Section 16, which contains further puzzling statements and perceived "errors" just as we might anticipate if the material pertains to the inversion hypothesis. Moreover, it also incorporates the multiplication factor 2.5 applied in a procedure at the equinox. The latter notwithstanding, this provides further understanding about such sites as Brodgar and Maeshowe in the Orkneys.
Briefly and simply, one can formulate the situation in terms of the inversion correction necessary after 30 days, i.e., approximately one month or the interval from one new moon to the next, but for a special theoretical case when the conjunction between the sun and the moon takes place at the equinox on one hand and at the precise time of sunset on the other. This is a highly theoretical situation, but one that nevertheless best illustrates the method to be used. In simplest terms, if the rotational excess is exactly 4 minutes (as opposed to 3 minutes, 56.556s etc.,) then the excess rotation would be simply one degree per day and thus after 30 days the excess rotation would add up to 30 degrees, which is one Babylonian "Danna". Now at the equinox there is 12 hours of daylight = 180 degrees, or 6 danna, and since there has been a total gain of 30 degrees, the conjunction in the inverted mode would now take place one danna before sunset since we are now dealing with the reversal of sunrise and sunset caused by the inversion. This the text calls "5 danna time." Actually the one degree per day is the simplest of all scenarios introduced here to explain the concept; the text employs a solar velocity of 0;57,56 degrees per day that pertains to the equinox and a constant of 5 degrees is also determined from the multiplication of the convenient value of 12 degrees daily lunar velocity by the factor 2.5. Thus the text states:

For 5 danna time
you shall take 5 degrees for the moon before sunset.

which causes Neugebauer to comment that: " the meaning of the remark 'before sunset' .. is not clear to me. Perhaps it is simply and error for 'after sunrise'..." Perhaps it is, but it certainly makes sense from the inversion viewpoint, and moreover, there is one other indicator that pertains to the daily solar velocity that also needs to be addressed. Neugebauer describes the operation in question as follows:

This means: if the daily solar velocity in (Aries) is 0;57,56, (degrees per day), then the velocity per large hour is 0;10 x 0;57,56, = 0;9,39,20. This amount is multiplied by 5 danna (previously found for the part of the day time until conjunction) and this would lead to 0;24,8,20 (degrees). The last digit is omitted and apparently referred to as "small" though I do not know of any similar case in mathematical or astronomical texts. [ACT No.200, Section 16, p.209]
The rareness of this occurrence is an alerting factor, for we can see precisely why it could be ignored if it was indeed concerned with rotational excess, i.e., as a rotational difference it represents a change of less than 1 second in 3,476; specifically the rounding from 0;24,8,20 to 0;24,8 (with the factor 2.5 also utilized here) corresponds to rotational velocities of 0;57,56 and 0;57,55,12 respectively. This results in a 3,475.2 seconds excess compared to the original value of 3,476; the change is indeed "small." Finally, there is one further indicator in Section 16 that is of some consequence, i.e., one of the reversed corrections that we might hope to find, in this instance concerning the equinoxes. The latter part of the section is almost completely destroyed, but Neugebauer nevertheless states towards the end of his commentary:
It seems, ... that line 27 contains the statement ' for 10 Taurus the correction (is) 0 ' whereas the value 0 is assigned to Aries 10 according to the list given in Section 15 rev. II,12. [ACT No. 200, Section 16, p.210]
Here we might suspect that there could be both a double application and/or a mixture of procedures to contend with. In fact it is possible that column phi could have a triple application in so much as it is also known to be used with the Saros Eclipse Cycle. On the other hand, this too would be effected by the inversion, thus the rotational aspect would also come into play here as well. But either way, we begin to see that Kugler may have been closer with his interpretation of the mysterious column Phi than Neugebauer, although perhaps neither was completely correct. The "5 danna time" we can understand to be a form of shorthand that refers to the monthly rotational excess, and moreover, we can further see that the 30 degrees or so could be expressed in terms of scaled lunar diameters, e.g., 30;00 degrees as a opposed to 0;30 degrees although of course the actual diameter is quite variable. Moreover, we can now see that one further puzzling term concerning diameters - "the 2 of the disc" - might in the same sense relate to lunar radii, and it is here that we are able to return to Maeshowe and Brodgar, with the former's emphasis on sunset on one hand and six hours of daylight at the winter solstice on the other. In other words, a place that might correspond, not to "5 danna time," but instead, to "5 hour time," to lunar radii, and the 14 or so days that elapse between Full and New moons. This is necessarily hypothetical, but it does provide a possible reason for this particular location and it can also be checked quantatively there and elsewhere for variations on the same theme.
The above treatment and analysis represents just one of the luni-solar texts and an initial focus. Moreover, it is not intended to be the same analysis as that given by Neugebauer, but rather it is the deduction of the method to be applied in the context of the inversion hypothesis.

Returning to the Babylonian material, it is necessary to note that there are numerous additional operations and parameters that may be applied to the proposal outlined above. First there is ACT No.211 (p.274-275) where we again meet unusual parameters and operations in unusual luni-solar contexts, i.e., Neugebauer finds non-standard values and multiplications in the first Section (omitted here) and then comments on the second section as follows: [ACT 211, Section 2, lines 5-8]
In line 5 a "circumference" and some "coefficient" are mentioned. The same words, again in unintelligible context, occur in rev. 2 and 1.
and after finding some common ground with later parameters, he is then disturbed to find a minimum value for one of the luni-solar velocities assigned to an unexpected location in the zodiac. A little further on he makes the following comments concerning the last line of the section:
No meaning whatever can I associate with line 8. The phrase ana-tar-sa ' opposite ' also appears in obv.13, and in rev. 6 and rev.10, but always, as far as can be seen, without the usual numerical data ( ' opposite x put y ' ).
Actually, Neugebauer translated line 8 as:
ana tar-sa 31,22,30 in 1[(?) . . [ . . . . . .
where the dotted lines and square brackets indicate places where the textual material becomes unreadable or is destroyed. Even with almost no knowledge of the methodology it is more than likely from this that some value is to be equated with "31,22,30" or, in our present context, a value is to be put opposite 0;31,22,30, in keeping with both the "circumference" and the "coefficient" mentioned earlier in the section. It is worth noting here that the mysterious and extraneous column Phi in System A is equated with specific lunar "velocities" in exactly this manner. For example, the maxima and minima of the two functions are known from ACT 200 Section 5. And this text, the reader will recall, is the one discussed initially; here Column Phi parameters are given first [ACT 200, Section 6, line 16 and 17, p.190]:
Opposite 2,17,4,48,53,20 you shall put 15,56,54,22,30 as velocity
Opposite 1,57,47,57,46,40 you shall put 11,4,4,41,15 as velocity.
with the first set pertaining to the maxima and the second to the minima.

But there are more uncertainties and unusual parameters in Section 4 of No.211. For example, we find [Neugebauer's translation, p.275]

" In line 12 the value ' zero ' is mentioned, in line 13 a monthly difference of 5. Line 15 says ' if 2 finger, predict 2,40 '. The meaning of this is obscure."

The next section (the fifth) contains further interesting information, for here we find operations and parameters that are closer to Kugler's initial assessment than any of Neugebauer's arguments: [p.275]

Line 1 mentions a ' coefficient ' and the ' duration (of a month) ' which usually means column G. Line 2 could be translated ' If . . . . circumference bypasses 35,20 35,20 . . . ' The only parameter 35,20 known to me is the value 0;35,20 (degrees) for the extremal apparent diameter of the moon according to Ptolemy. It is tempting to consider this is not a coincidence, but all details remain unexplained. In line 3 appears the number 2,13,20 which is otherwise known as the characteristic value of (Phi) but hardly has this meaning here. Cf., for this problem No. 200aa p.212 - No parameter 3,14,24 (line 4) is known to me from the lunar theory.
Neugebauer introduces the sixth and last section [p.275] with:
A beginning with tab is hard to explain. The term "eclipse magnitude" indicates at least the subject under discussion. What follows seems to be parallel to line 3 in Section 5.
which is hardly informative, since Neugebauer failed to provide an explanation for line 3 in the first place. As for the mention of "eclipse magnitude" this would appear to be one of the major areas that is in need of revision. It appears more likely that it concerns the apparent diameter of the moon and the corrections discussed above. One might suppose that is possible that "eclipse magnitudes" were under discussion, but it hardly seems a compatible fit with Neugebauer's fundamental premise that the Babylonians did not even possess a physical model of the Solar System. Or putting it another way, it is even harder to imagine how, let alone why they would be concerned with the magnitude of eclipses if they possessed no fictive model or any real understanding of the complexities of orbital motion, especially in luni-solar contexts.
Frankly, it is difficult to understand what Neugebauer was attempting to achieve by insisting on this unlikely and improbable thesis. He was a philologist and reputably an astronomer, as well as a thoroughly competent mathematician, and there can be no doubt that he was also one of the foremost authorities in this highly specialized field. But as for his being one its "most refulgent lights" in Thomas Taylor's true sense of the term, perhaps our present situation is akin to that encountered by the medieval Arab scholar Al-Bitruji, who in attempting to come to terms with the work of Claudius Ptolemy felt obliged to comment:

Ptolemy said in this place that retrograde motion according to the ancients is best explained if one considers the sun in the middle (of the spheres), which is most natural. But he did not give a reason for this being ' most natural, ' proving that he was not a natural philosopher though he was a mathematician.58
Lastly, it likely that much of the Babylonian material refers to the norm as opposed to the inverted state, but there still exist unexplained operations, constants and corrections as seen above, and until they are all satisfactorily resolved the matter cannot be considered settled.

Hopefully renewed interest, fresh outlooks, and a definite focus will add to our understanding of this material in general, the megalithic monuments, and the phenomenon in particular.

Copyright © 1998. John N. Harris, M.A.(CMNS). Last Updated on March 1, 2004.

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