The East Mebon Sun and Venus diagrams were used for three centuries to structure the two largest Barays and their surroundings
9th and 10th centuries
We noted that the azimuth of the East Baray (-
A possible location could be the upper platform of this temple. The latter suggestion is probably correct because there is a relationship between this Sun diagram and the East-
It may seem strange the Khmer should have known and used the number “e” but we think it is possible for the following reasons:
1) According to http://hua.umf.maine.edu/China/astronomy/tianpage/0016dunhuang5405w.html , the neighhours of the Khmer, the Chinese, probably invented the "Mercator" projection during the 8th century. Now, the best way (probably the only one) to calculate this projection is to use the natural logarithms ( http://mathworld.wolfram.com/MercatorProjection.html )
2) The value of “e” is coded in the Pre Rup complex.
40. 57 m / 14. 93 m = 2. 717 (we used the median axes of the embankments).
308. 75 / 113. 5 = 2. 720
With 308. 75 m = distance from inner southern wall of second enclosure to northern embankment (median axis) of reservoir and 113. 5 m = north-
The diagrams for the Sun and Venus using the Midnight parameters were used together to create, seemingly, a remarkable geometrical construction. The location of the That Toc temple of Banteay Prei Nokor, located in the homeland of Jayavarman II, the founder of the Khmer dynasty, was determined to lie on a Sun (or Venus) apsides line running through the eastern embankment of the East Baray. Now, this apsides line crosses precisely the intersection of the Venus Sighra epicycle with the loxodrome running through the centre of both the Venus Manda epicycle and the centre of the epicycle of the Sun.
During the 10th century, the centre of the Prasat Sralao temple was located on the deferent of Venus.
This temple is seemingly a component of a huge complex which includes an arrangement of eight grid-
The deferent of the Sun runs through the centre of this arrangement.
The Pre Rup temple : the main site of the Venus diagram
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With respect to the true apsidal line, the error is small (smaller than 0.01°). It is worth noting the apsidal line from Banteay Prei Nokor is parallel to the radius of the Venus Manda epicycle and (as the "Midnight" system is involved) it is also parallel to the radius of the Sun Bhaskara epicycle. We show in the chapter on Mercury that a parallel apsidal line (ie parallel to the 22/04 apsidal lines of Sun and Venus) was used to determine, from the Bakong temple (Roluos area), the location of the ancient observer of Angkor Thom. Another parallel apsidal line also determines the location of the main cluster of monuments built in the Kulen Range from the Beng Mealea Sun. As the areas of Banteay Prei Nokor, Roluos and Kulen are related to Jayavarman II, we suggest the three apsidal lines are the homologs of the long trips of the king between his successive capitals during the early 9th century."
The loxodrome running through Prasat Sralao and the Venus location is tangent to the epicycle of the Beng Mealea Sun.
The longitude of the eastern dyke of the West Baray was determined after locating the midpoint of the lake (the site of the West Mebon to come) on the deferent of Venus (the West Baray was built during the first half of the century whilst the West Mebon was erected during the second half).
We noted above that the deferent of the Sun passes through the East
Mebon. Similarly the deferent of Venus passes through the West Mebon.
At the end of the 12th century, Jayavarman VII built the Ta Som temple on the meridian (blue line)of the centre of the Venus Manda epicycle.
The figure shows a loxodrome (green line) running through the centre of the Sighra epicycle of Venus and the centre of the Ta Som temple. We’ll see in the chapter about the Saturn’s diagram of Angkor Proper that this loxodrome crosses the centre of the Sighra epicycle of the latter planet. Morever, it’s azimuth (-
Jayavarman VII constructed the Ta Prohm temple on a loxodrome running from the summit of Phnom Bakheng to the Sun.
It should be remembered that the centre of this temple was also located on a tangent to the Sighra epicycle of Mercury running through the Krol Ko temple. Moreover, the loxodrome running from the centre of Ta Prohm to Mercury (-
The loxodrome from the centre of Ta Prohm to Bat Chum is nearly perpendicular (90.21°) to the loxodrome from the latter temple to East Mebon (orange lines on the following figure).
Ta Som temple
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Ta Prohm temple
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We discovered that the distance of 221.657 kam is probably related to the period we spoke about in the Pre Rup’s chapter ie the sidereal heliocentric period of revolution of Venus (0.6152 year).
221.657 / 360 = 0.6157
Sidereal period of Venus = 0.6152 year.
Error = 0.18 day
In other words, the difference of latitude (expressed in Khmer arcminutes) is equal to 360 sidereal revolutions of Venus (expressed in years). The use of the factor “360” is common in the tables of Aryabhata.
At first sight, this result could be due to chance but we think it is not the case because it is confirmed by the toponymy :
Phu = mountain
Moreover, it is worth remembering the distance of 221.657 kam was measured between two particular places:
The Bat Chum temple, built on the parallel of the Bakheng's crest, was consecrated on 953 ie 1 year after the East Mebon which is the main component of the Sun & Venus diagrams.
The Mountain of Venus
The northernmost Khmer sacred mountain, the Phu Phek mountain, stands in north-
The central tower of Banteay Kdei was not located exactly on the loxodrome from Bat Chum to Ta Prohm. As a consequence, the near right angle we described two paragraphs above is lowered from 90.21° to 89.82° if it is measured from Banteay Kdei instead of Ta Prohm. It is worth noting the mean angle is equal to 90.015°. So, the Banteay Kdei’s centre could have been located to correct the Ta Prohm's inaccuracy by creating an counteracting error of -
Jayavarman VII shortened the 10th century baray (Sras Srang) located to the east of Banteay Srei. He also built a nice landing stage on the western embankment of the reservoir. The overall proportions of the resulting Banteay Kdei-
Distance from south-
Distance from centre of landing stage to fourth enclosure = 154 m
Distance from fourth enclosure to third enclosure = 249 m
154 / 249 = 0.618
Venus sidereal period = 0.615 year
14 / 717 = 0.0195
Venus eccentricity (Manda ratio) = 0.0194
717 / 994 = 0.721
Sighra ratio of Venus = 0.722
The central tower of Banteay Kdei was located approximately halfway between the corresponding structures of Bat Chum and Ta Prohm. We have determined the precise position of this central tower : it was built 11 m to the north-
This created a difference of 22 m between the distances measured from the two other temples.
By dividing 22 m by the average of the latter distances, a nearly perfect Venus eccentricity can be calculated.
(1151 + 1129) / 2 = 1140 m
22 / 1140 = 0.0193
True Venus eccentricity = 0.0194
These distances were measured on « Google Earth ». We used the median axes of the enclosures (or gopuras)and the outer corners of the embankments.
This was measured on the 1/500 EFEO plan.
Jayavarman VII constructed also the Banteay Kdei temple between Ta Prohm and Bat Chum, to the west of the ancient reservoir called « Sras Srang. ». The south-
Last modification : 19 / 01 / 2009
Pre Rup : eastern second enclosure
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EFEO : 1/500 plan
EFEO : 1/500 plan
Courtesy of Moyra Farrington
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There is possibly a relationship between the number of towers and the “Midnight” eccentricity of Venus.
We noted that the latter can be calculated from the following formula:
Ev = 1 / (e x x 6)
With Ev = Venus eccentricity
e = 2.718 = Euler number (basis of natural logarithms)
This formula provides Ev = 0.0195 (Midnight value = 0.0194)
The Khmer arcminute was most probably used to determine the larger dimensions of the Angkor site :
Landing stage of Sras Srang (from EFEO plan)