APPENDIX
A
ANCIENT
HINDU LUNI-SOLAR VARIANTS
PRELIMINARY
CONSIDERATIONS
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:
| INTEGER MONTHS |
INDIAN |
MODERN |
| 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:
THE
INCREASE IN THE
DURATION OF THE YEAR
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]
THE INCREASE IN THE
DURATION OF THE MEAN SIDEREAL MONTH
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.
APPENDIX B
BABYLONIAN
LUNI-SOLAR VARIANTS
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.
ADDITIONAL INDICATORS
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.
Return to the Main Page
