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" A model of trepidation is mentioned in rather cryptic terms in the Surya Siddhanta,
a model first brought to the notice of European historians by Davis and Colebrooke.
The [trepidation]amplitude oscillates between the limits + & - 27°, passing through
zero in the year 3600 complete of the Kali yuga*. The rate is 54” per year (the year
being the sidereal year as used in the Surya Siddhanta), and therefore the period
of a complete cycle is 7200 years. "
(Raymond Mercier : “Studies in the medieval conception of precession”)
Note: the trepidation is called « libration » by some authors
Compare the "Explicit" trepidation model with a more correct (and probably earlier)
"Implicit" model of precession described below.
* passing through the zero amplitude point after (each) 3600 years from the beginning
of the Kali Yuga.
2) The Metonic cycle allows observation, every 19 years, of a lunar event which
ruled over the beginning of Kaliyuga.
-The lunar event is the conjunction of the new Moon (first crescent) with the star
zeta Piscium ie the point zero of the Indian sidereal zodiac. It is mentioned in
the Surya Siddhanta and it could becalculatedby Aryabhata and his successors.
In the following reasoning, we’ll use the Gregorian calendar to be able to compare
distant events with a good accuracy. (the shift of this calendar with respect to
the seasons is indeed very small).
The mean angular distance between the new Moon and zeta Piscium (5°) is linked to
the constraint of observation (a short duration before Moon’s setting) and to the
precession(some 4° from 800 to 1100 CE)
During the Khmer epoch, as zeta Piscium was near the vernal equinox, the conjunction
of the new Moon with zeta Piscium was observed during March. At the beginning of
Kaliyuga (3102 BCE), it occured in January but, according to the explicit model of
precession (trepidation or libration) which was usually applied in this mythological
context, it happened, by convention, in March (the trepidation model implies indeed
that after 3600 years, the vernal equinox would have returned to the same day of
the calendar as it was in 3102 BCE).
During a period of some three centuries (ie with a precessional shift from 0 to
4°), a practical way to predict the dates of the conjunctions of Zeta Piscium with
the new Moon was to use a Metonic cycle ie a period of 19 tropical years including
235 lunations (synodic periods of 29.53 days) after which the Moon's phases occur
on the same date. In this case, the set of Metonic cycles could have begun on 19
/ 03 / 834. Now, according to the EFEO archeologist Claude Jacques, the king Jayavarman
II (blue insert here below), died in 834.
We discovered that, during the Angkorian epoch, two important ceremonies were probably
related to this Metonic cycle which began during the year of the death of Jayavarman
1) According to Claude Jacques (« Angkor, résidence des dieux ») the first pious
work of Rajendravarman, after the Koh Ker interval, was the restoration of the Baksei
Chamkrong temple. The new divinity of the sanctuary was consecrated, in the Gregorian
calendar, on 28 / 02 / 948 CE around 09:40 ( 834 + 6x19 = 948 ). The divinity was
a golden statue of Paramecvara. Now, "Paramecvara" is the posthumous name of Jayavarman
II. We’ll see below why the inscription’s date is 20 days before the conjunction
of the new Moon with zeta Piscium in the evening of 19 / 03 / 948. The age of the
Moon (9.97 days) on 28 / 02 / 948 AD at 09:40 is relevant in the context of our
study (see below).
2) Suryavarman II, the builder of Angkor Wat, was coronated on CE 1119 (834 + 15x19)
, six years after the beginning of his reign. The coronation took place, as usual,
at new year (ie the spring equinox), around the conjunction of the new Moon with
zeta Piscium on 20 / 03 (13 / 03 Julian).
True or alleged death of Jayavarman II
Founder of the Khmer, or Cambodian, Empire and outstanding member of the series of
rulers of the Angkor period (802–1431). Among Jayavarman II's accomplishments were
the deification of the Cambodian monarchy, the establishment of the devarāja cult
as the official state religion, and the reunification of the old kingdom of Chenla,
which he expanded and formed into the Khmer Empire.
Jayavarman II." Encyclopædia Britannica.
Encyclopædia Britannica 2009 Ultimate Reference Suite. Chicago: Encyclopædia Britannica,
Seven Metonic cycles after the new Moon of March 834 CE, on 19 / 03 / 967 CE (14
/ 03 Julian), the new Moon set near zeta Piscium. That means that the planetary gathering
of April 967 occured during a lunar month which began (7x19 years + 1 lunation) after
the event of March 834 CE (depicted here above).
It is worth remembering that any combination of integers allowing a calculation (from
a remote conventional reference time) of the date of an exceptional planetary configuration
was viewed by the astrologers as a sign of the gods.
We noted previously, about the Baksei Chamkrong’s inscription, that the Moon’s age
corresponding to 28 / 02 / 948 CE (23 / 02 Julian) at 09:40 was relevant in the
context of our study. The age was indeed 9. 97 days at this moment. It was not
very far from the age on 27 / 04 / 967 CE ( 22 / 04 Julian) at 23:57 ie 9. 81 days.
We suggest it is not fortuitous. The goal was probably to indicate the predicted
Moon’s age at the moment which, since the previous century, was the basis of the
planetary diagrams drawn virtually across Cambodia.
The oldest Saturn and Jupiter’s diagrams (22/04/967 CE) include two remote sacred
heights - the promontory of Preah Vihear, in the Dangrek Range, and the Vat Phou
Mountain, in present Laos - and two pre-Angkorian sites (Sambor Prei Kuk and Angkor
Borei). This was possible by using somefortuitous configurationsand, maybe, by
creating voluntarily some small astronomical errors. We’ll show that, in the planetary
diagrams, the Khmers located Jupiter at 0.5° from the true position they were probably
able to calculate (see chapter 6-B).
The later diagrams, using another date of reference (ie 15/04/967 CE) are the result
of another method. The Khmers were indeed able to create some relationships between
astronomical and geographical features by chosing a particular date. As the Sun was
always located due East from the observer (religious tradition), it was possible
to rotate the apsides lines by altering the date. This method explains The Angkorian
sacred hill of Chau Srei Vibol, as seen from Phnom Bok, indicates indeed the apogee
of the 15/04 Sun .
5) The crossing of zeta Piscium through winter solstice :
Jupiter and Saturn come into conjunction every twenty years. Such a conjunction took
place on 18/12/1303 CE in the Ophiucus constellation which corresponded, during that
epoch, with the tropical sign of the Archer. Kepler and other astrologers predicted
« fire from heaven » because this event began a series of conjunctions within the
triplex of tropical signs designated by the astrologers as the « Fiery Trigon »
The Ram, the Lion and the Archer are indeed, in the tropical zodiac, the « signs
of Fire ». It was the first time the conjunctions of Jupiter and Saturn occured inside
the triplex of «Fire » signs since the Khmer epoch. The triple conjunction of 967
CE occured indeed inside the tropical sign of the Ram (on the border of Aries and
On 16/01/988 , the conjunction was observed at the end of the Archer sign and, on
07/03/1008, it appeared inside the Maiden sign (an « Earth » sign: the rule is not
precise). The latter conjunction was triple. Finally, on 20/04/1027 CE (60 years
after 967), the conjunction was watched again in the Ram sign. It is worth noting
it occured at some degrees from the conjunction of 967 CE. Every 60 years, indeed,
the longitude of the conjunction is shifted eastward (prograde motion). It comes
back to the same sign after 794 or 795 years ( = one « J-S » period). It is worth
noting the conjunction’s location does not cross the whole zodiac. It is shifted
in the prograde direction (blue arrow) during some 13x60 years through more or less
120° (yellow zone). Then, after adding some 20 years, it comes back to the initial
sector in the retrograde direction (red arrow). More precisely, the conjunction
which was observed on 08/05/173 CE inside the Ram sign came back to the same sign
during 967 CE, after 13x59.62 years + 19 years.
Eight J-S periods befores 967 CE, during 5388 BCE (tables), the star zeta Piscium,
the zero point of the Indian sidereal zodiac, was located at 7.96° from the winter
solstice (longitude = 277.96°). We noted that, by using the precession’s rate of
the Surya Siddhanta, the Khmers could have imagined that zeta Piscium was located
precisely on the winter solstice during 5388 BCE (see calculation at left).
In this case, the implicit (correct) model of precession of the Surya Sddhanta would
have been used (insert).
During 967 CE, there were 95.46° between zeta Piscium and the winter solstice.
There were 2,320,545 Julian days between 22 / 04 / 967 CE (24:00) and 31 / 12 / 5388
BCE (24:00). This corresponds to 6353.441634 tropical years.
95.46 / 6353.441634 = 0.0150249 degree per year
0.0150249 * 3600 = 54.09 arcseconds per year (Surya Siddhanta: 54” per year)
It is worth noting again that, according to Burgess, the trepidation (libration)
model was probably a shrewdness imagined by later Indian scholars to keep true the
assumption of the equality of the tropical and sidereal years.
Khmer scholars probably knew precession caused a continuous retrograde motion of
the equinoxes. Our hypothesis (that zeta Piscium can be calculated, using the Indian
estimate for the rate of precession, to be located at the winter Solstice in 5388
BCE) could be an explanation as to why the Indian name for “precession” (ayanamça)
IMPLICIT MODEL OF PRECESSION OF SURYA SIDDHANTA
« It is evident , then, that the initial point of the sphere is regarded as the
movable point , and the equinox as the fixed one. . Now this is no less strange than
inconsistent with the usage of the rest of the treatise. Elsewhere zeta Piscium is
treated as the one established limit, from which all motion commenced at the creation,
and by reference to which all motion is reckoned, while here it is made secondary
to a point of which the position among the stars is constantly shifting, and which
hardly has higher value than a node , as which the Hindu astronomy in general treats
it (see p. 98). The word used to express the motion (lambate) is the same with that
employed in a former passage (i. 25) to describe the eastward motion of the planets,
and derivatives of which (as lamba, lambana, etc.) are not infrequent in the astronomical
language ; it means literally to « lag, hang back, fall behind: » here we have it
farther combined with the prefix pari, « about, round about, » which seems plainly
to add the idea of a complete revolution in the retrograde direction indicated by
it, and we have translated the line accordingly. This verse, then, contains no hint
of a libratory movement, but rather the distinct statement of a continuous eastward
The Surya Siddhanta: A Textbook of Hindu Astronomy
By Ebenezer Burgess, Phanindralal Gangooly
Translated par Ebenezer Burgess
Published by Motilal Banarsidass Publ., 1989
A) The vernal equinox crossed the ecliptic meridian of Alpheratz (alpha Andromedae)
during the year 967. Now, in the context of Surya Siddhanta, Alpheratz is the junction
star of the 25th Nakshatra (lunar mansion)
It is worth noting that between 499 CE, which Aryabhata determined as the end of
the first period of 3600 years of the Kali Yuga from its inception at the "primordial
super-conjunction" of 3102 BCE, and 967 CE, all the planets gathered in the trepidation
zone three times, on 08/04/670, on 08/04/730 and on 05/02/789 CE. However, the event
of 967 CE was the only gathering which happened integer multiples of 12 years after
3102 BCE, as well as 79 CE (Saka era) and 499 CE."
Moreover, it was the first gathering in the trepidation zone which occured after
the death of Jayavarman II in 834 CE (see number 4)
6) Other observations related to precession
4) The crossing of Pleiades through spring equinox at the beginning of the Indus
valley civilization :
The role played by the Pleiades (eta Tauri) or Krittika as indicator of the beginning
point of the yearly cycle in the primitive calendars from Kamchatka to the Cape
of Good Hope, from Australia to North and South America, is a widely confirmed fact.
Actually we cannot make an exception to this seemingly general rule in cultural
anthropology for any good reason in the case of the Indus Civilization.
From “Astronomy in the Indus valley civilization”
Syed Mohammad Ashfaque
As seen from Moenjodaro and Harappa, this heliacal rising was watched, respectively,
32 and 34 days after the equinox (table). According to Syed Mohammad Ashfaque, the
seal 2430 from Moenjodaro depicts the heliacal rising of the Pleiades cluster.
Although it was rather approximate and probably fortuitous, this correlation between
the heliacal rising of the Pleiades star cluster, as seen from Indus Valley around
2400 BCE, and the event of 22/04/967 CE would have called the Khmer astronomers'
attention.The Nakshatras system (lunar zodiac with 27 or 28 mansions) has indeed,
according to Syed Mohammad Ashfaque, a pre-Rig Vedic origin. It is a component of
the astronomical knowledge the Aryans learned from the Dravidians (insert). The Pleiades,
clearly identified by Professor Ashfaque on the seal 2430, became the "junction star"
of the first Nakshatra (Krittika) from 1181 BCE to circa 570 CE. At the latter epoch,
the Nakshatra "Asvini" (whose junction star was 13 Arietis) became the first point
of the lunar zodiac. It was a late attempt to take into account the precession.
So, as far as the Pleiades are concerned, there was probably a cultural continuity
between the pre-Rig Vedic and Classical epochs.
So far (as) astronomy is concerned, the most likely place where the Aryans learnt
the heavenly science from the Dravidians, appears to be the area of the Malwa plateau.
This seems to have happened more specifically in the ancient city of Ujjain, whose
reputation as a centre of astronomical studies in India from times immemorial is
attested from various literary accounts.
Ujjain remained the centre of astronomical studies even during the Classical and
Medieval periods of ancient Indian astronomy. By virtue of its supposed location
just on the meridian linking the Island of Ceylon with the North Pole, it came to
be treated as the "Cupola of the Earth" (al- Qubba) by some of the Islamic astronomers
and medieval European geographers."
B) We discovered that, on 22/04/967 AD at 23:57, Jupiter ran through the ecliptic
meridian of the calculated Indian sidereal zero point (calculations in insert). As
the same conjunction occured near the equinox of the reference year of 564 CE (insert
and link below), we suggest 22/04/967 was a stage of a Jupiter calendar used to measure
In the case of the CE 967 event, 23:56:40 is the moment when the hour angle of the
Sun was equal to 12 hours on the meridian of Phnom Bok.
In the case of the BCE 3102 event, 0:00:00 is the approximate local mean time
of Lanka (latitude = 0° ; longitude = 76° E )
The astronomical event of April 22 CE 967 may seem rather ordinary at first sight
it was most probably seen as remarkable by the Khmers for reasons related to:
1) The beginning of Kali Yuga:
The planets were gathered round the star zeta Piscium which is the initial point
of the Indian sidereal zodiac. This was also the case on February 17-18 3102 BCE
i.e. on the date which was used as the epoch of reference by the Indian astronomers
who established the tables of mean planetary longitudes. More precisely, the five
planets were gathered, during the events of 967 CE and 3102 BCE, inside the trepidation
zone ie inside the zone (horizontal yellow lines on star maps below) where, according
to the explicit model of precession of the Surya Siddhanta, the vernal equinox
was oscillating (see insert). The amplitude of the trepidation was nil on 21
/ 03 / 499 CE at Noon and on 18 / 02 / 3102 BCE at 00:00 (meridian of Lanka ie
the 76th eastern meridian). The sidereal reference point is 10 arcminutes East (ie
towards Aries) of zeta Piscium. Despite the fact that it was erroneous, this model
provided almost correct stellar longitudes for Vedic and post-Vedic periods (after
1200 BCE). As far as the Kali Yuga’s beginning was concerned, the successive gatherings
of the initial points of both zodiacs in the same area was probably a way to keep
true the assumption of the equality of the tropical and sidereal years. We’ll see
later the more correct “Implicit” model of precession was probably used by the Khmer
astronomers to determine the location of equinoxes and solstices during remote epochs.
So, the planetary gathering occurred during the second half of April 967, probably
on the date of the temple’s consecration. This “opening of the eye” occurred on April
22 (Julian calendar). The date, confirmed by Briggs, is remarkable. It is indeed
the date of the Sun’s zenithal culminationin northern Cambodia, i.e. in the central
area of the Khmer empire. In the Gregorian calendar, it is 27 / 04 (any year). It
is worth remembering that the wet season, the most important moment of the year for
the peasantry, is heralded by this zenithal culmination.
In Angkor Proper, the zenithal
culmination occurs one day earlier
on the 21/04, (see graphic on right)
It is interesting to note that the Khmer empire was large enough to provide a large
"bracket" of dates. On the 10th parallel, the Sun culminates at zenith on April 16
(Gregorian). It is May 8 (Gregorian) on the 17th parallel. In CE 967, the corresponding
Julian dates were April 11 and May 3. That means the date of consecration was the
“mean” date of zenithal culminations for people living between the 10th and 17th
parallel. Moreover, it is the date when the Sun crosses the zenith of Koh Ker, an
ancient Khmer city which was the “pivot” of the outer planets diagrams.
It is important to note the astronomical configurations of 15/04 and 22/04/967 CE
(Julian) occured, respectively, 30 days and 37 days after the spring equinox. These
are the dates of the Sun’s zenithal culminations on, respectively, the parallel of
Phnom Chissor (lat. = 11.18°) and the parallel of Koh Ker (lat.= 13.78°). Now, these
sites are related to planetary diagrams based on the corresponding dates of zenithal
- Koh ker (Sun at zenith on 22/04) was built on a location which played the role
of a “hub” in the oldest diagrams (22/04) of the outer planets.
- The temple of Phnom Chissor (Sun at zenith on 15/04) was built by Jayavarman I
as a component of a later Sun’s diagram (15/04) we mentioned here above about Chau
Srei Vibol. We noted that the Khmer name of Phnom Chissor, “Suryaparvata”, means
“Mountain of the Sun God”.
The event of 15/04/967 and 22/04/967 CE occured respectively 30 and 37 days after
the spring equinox. At the dawn of Indian civilization, around 2400 BCE, the astronomers
of Harappa and Moenjodaro observed the heliacal rising of the Pleiades star cluster
approximately one month after the spring equinox.
According to Syed Mohammad Ashfaque ("Astronomy in the Indus valley civilization"),
this observation was usual because the the Pleiades were located, during that epoch,
near the vernal equinox (insert).
“It is now known, with the discovery in 1936 in the eastern Gopura of the fourth
enclosure of the temple’s foundation stele, that Banteay Srei formed a whole, whose
style proves quite homogenous. Inscribed in 968, the first year of the reign of Jayavarman
V, the inscription gives, with the position of the sun, the moon and the planets,
the date of April-May 967 - the last year of Rajendravarman’s reign, under whom at
least some of the construction was probably started.”
This planetary gathering was studied by Michael Freeman and Asger Mollerup:
"The opening of the eye" of a Khmer temple was the date when the temple was consecrated.
Michael Freeman's books gives the dates and time for four such temple consecrations:
1) Preah Kor - 25 Jan 880 A.D. 2) Baksei Chamrong - 23 Feb 948 A.D. at 09:40, 3)
Banteay Srei - 22 April 967 A.D.and 4) Mangalartha - 28 April 1295 A.D. at 08:45.
(Amazingly going down to minutes!) “
According to the Surya Siddhanta, the sidereal zero point was located 10’ to the
east (prograde) of zeta Piscium. The ecliptic longitude of this zero point was nil
during the year 564 CE. The conjunction of Jupiter with the equinoctial Sun on 20/03/564
CE at 03:57 (Lanka) was the origin of the Sassanian astronomical reform. One week
earlier (13/03/564 CE at 12:52, Lanka meridian), Jupiter crossed the sidereal zero
point. As the event of 967 CE was predicted, the Khmers used most probably the precession
rate of the Surya Siddhanta (54 arcseconds per year) to determine the longitude of
the sidereal zero point during the latter year.
Elapsed Julian days between 13/03/564 CE at 12:52 (Lanka) and 22/04/967 CE at 23:57
(Angkor) = 147235. 3824
Elapsed years (of 365.25 days) = 403.1085
Longitude of sidereal zero point (according to Surya Siddhanta)
= 0 + 403.1085 x 54/3600 = 6. 0466°
Now, the longitude of Jupiter on 22/04/967 CE at 23:56 was 6. 0414°
It seems the Khmers calculated the ecliptic longitude of Jupiter with a great accuracy.
3) The topography and the pre-Angkorian sites:
According to one of the historians, Anianus, the interval between the creation and
the night of the Friday when the deluge commenced, is 2226 years 23 days and 4 hours.
This statement by Anianus is reported by Ibn-Albazyar in his Kitab-alkiranat (Book
of the Conjunctions)
From « The Great Year in Greek, Persian and Hindu Astronomy » by B.L. van der Waerden
It is worth noting that, according to the Hellenistic author Anianus, the Greek « Deluge »
(upper insert at right) occured 2226 years 23 days and 4 hours after the « Creation »
(lower insert at right) ie on 01/02/5328 BCE.
It is 60 years after the event described here above.
The duration between "Flood" (18/02/3102 BCE) and "Creation" is 2226.063427 years.
As Anianus (Anynus) was a Greek scholar, we have supposed he used the tropical year
of Hipparchus ie (365.25 - 1/300) days
The elapsed days between "Flood " and "Creation" was 813062.2465 days
JD of 18/02/3102 BCE (Julian) at 00:00 (Lanka) = 588465.29167
JD of "Creation" = -224596.9548
It is 01/02/5328 BCE at 18:05 on Lanka meridian. It is 15 minutes after sunset.
Sun's longitude = 273.6° (3.6° from winter solstice)
We think the Greeks tried to locate the Sun on the solstice of an epoch which was
important for some reason.
The Sun was at 5.17° from zeta Piscium but it is worth noting the error would have
been smaller in the Indian context :
Longitude of zeta Piscium in 564 CE = 0
Precession’s rate provided by Surya Siddhanta = 54 arcseconds per year
Elapsed years from 564 CE to 5328 BCE = 5891
5891 * 54/3600 = 88.365° = longitude of zeta Piscium in 5328 BCE (with implicit model
of Surya Siddhanta)
It is 1.635° from winter solstice (between the latter and the Sun)
That means the Sun was at 1.97° from the « Indian » zeta Piscium.
We suggest the Khmer astronomers believed there was a conjunction of the Sun with
zeta Piscium near the winter solstice at the moment of the Greek "Creation". This
should have occurred 8 periods of Jupiter-Saturn less 60 years before 22/04/967 CE.
Let me summarise my conclusions. After the discovery of trigonometry, tables were
calculated by trigonometrical methods. By the aid of these tables, several Hellenistic
authors tried to date the deluge by calculating conjunctions of Saturn and Jupiter
in the fourth millennium B.C. One of those attempts led to the discovery of an approximate
mean conjunction of all planets in 3102 B.C. Next, a new theory and new tables were
fabricated, based on the assumption of a mean conjunction of all planets in February
of this year, and of an exact repetition of all planetary positions at the end of
a certain World-Year. These tables were used, with corrections, in Sassanid Persia.
Aryabhata corrected the theory, replacing the Persian world-year of 360000 years
by a period twelve times as large.
From « The Great Year in Greek, Persian and Hindu Astronomy » by B.L. van der Waerden
The year when Rahu and Ketu swallowed the Sun over the craddle of Khmer civilization.
Most observations described here above could have been noticed anywhere in the Hindu
world. On the contrary, a particulary spectacular astronomical event was observed
only in the Khmer part of this world during the year 967 CE. A total eclipse of
Sun occured indeed on 10/07/967 CE. The umbra crossed Asia from the Mediterranean
Sea to the Pacific Ocean. The line of centrality did not cross Mesopotamia, Persia
and India but it ran through the northern part of the Khmer Empire. It crossed, in
particular the Vat Phou site, the craddle of the Khmer civilization. According to
the Indian mythology, the demons Rahu and Ketu were supposed to swallow the Sun.
The chart (to the right) is provided by the "Five Millennium Canon of Solar Eclipses:
-1999 to +3000"
It should be noted the Indians and Khmer astronomers were probably able to predict
program which calculates the solar eclipses by using the Surya Siddhanta algorithm.:
(prepared by T C.Hartley, Director of the Ernest B. Wright Observatory at the Department
of Physics at Hartwick College )
The Sun was fully eclipsed at 15:27. It was short (from 15:26:18 to 15:27:26) but
the sight before the Khmers' eyes was wonderful : Venus, Mercury, Sirius, Castor,
Pollux and Regulus could be watched. On the eastern horizon, Arcturus, Antares, Spica
and alpha Centauri were visible. Such a show over the cradle of Khmer civilization
was most probably interpreted as a divine sign. We are not sure the Khmer were able
to predict that Vat Phou was precisely on the line of centrality but, anyway, the
Vat Phou's show has probably convinced the Khmers they were right to use the year
967 AD as a reference for the planetary diagram.
The following sky maps (« Starry Night Pro ») show this eclipse was spectacular
for observers standing in Vat Phou