Epoch of 248 Vararuci Mnemonics of Moon and Other Hitherto Unknown Signatures of Udayagiri in Indian Astronomy ƒ
K. Chandra Hariƒ Abstract
Astronomical significance of the Udayagiri archaeoastronomical structure of the times of Candragupta-II Vikramāditya had been the subject of many many papers in recent times. Little is known of Vararuci apart from the sole surviving work Candravākyās (248 mnemonics) for the computation of Moon. Present work is an attempt to demonstrate that the epoch of the mnemonics of Vararuci is 20 March 403 CE, date of great observational significance at Udayagiri passageway. Astronomical features of this date which may be rightly described as the Vararuci epoch of Indian Astronomy, brings out that the epoch of the 248 Candravākyas of Vararuci is 20 March 403 CE when the apogee of the Moon was 900. Moon had a declination of 24019’ while setting precisely at the due west point. Also, Vararuci’s 248 day scheme of Moon’s anomalistic computation is shown to have disticnt differences with the Babilonian anomalistic methods. Decrement of 10 in the maximum velocity of the Vararuci scheme (as compared to Babylonian data) is shown to be a signature of the 20 March 403 CE epoch where in the first anomalistic cycle had apogee coinciding with the quadratures of Moon leading to reduced eccentricity and lesser speed in the cycle. Planetary positions of the epochs 402 and 403 CE have been presented to demonstrate that the observations at these epochs may have laid the foundation stone for the emergence of Siddhāntic astronomy in India. Special attention is drawn to the possible observation of Moon during the years 402 and 403 CE at Udayagiri which may have led the indegenous value of 40.5 for the maximum latitude of Moon. Epoch Kali 3503(elapsed) had given integral number of revolutions for Moon with the coincidence of vernal equinox and new moon while Kali 3504 (elapsed) had given integral number of revolutions for apogee to pave the way for Siddhāntic mean motions and astronomical use of the synodic super conjunction on 18 February -3101CE, 06:00 ZT. 16 April 403CE, 23:00 ZT is shown to had a nodal distance close to 900 and thus latitude sufficient to occult the η -Leonis -Leonis before conjuncting Regulus longitudinally. Declination of Moon on 20 March 403 CE and 16 April suggest that the setting Moon could be observed on these dates in alignment with the passageway at Udayagiri. Discussion render support to the legendary information on Vararuci as the court astronomer of Vikramāditya.Connection of the only surviving work of Vararuci, viz. the G ī rn rn a śreyādi V ākyas with the Udayagiri epoch of 402 CE and the period of Candragupta-II Vikramāditya renders a reliable and convincing identity for Vikramāditya and fixes the date of Vararuci to be the junction period of 4th-5th century CE.
ƒ
K. Chandra Hari, Institute of Reservoir Studies, ONGC, Ahmedabad-5. 1
I. Introduction Vararuci, the legendary astronomer reputed to have authored the g ī rn rn aśreyādi Candra-vākyas (alpha-numeric expression of the 248 daily velocities of the anomalistic cycle of moon) and the propounder of the Kat apay apayādi alpha-numeric notation is believed to have lived in the 4th century CE. Little else is known of the astronomer regarding his date, place and works except for the Candravākyas and the Yogyādi Sūryavākyas. Present work attempts to explore the indigenous anomalistic computational method of Vararuci with a view to understand the computational model and if possible to glean the epoch of its origin as well as the indigenous character of the method. Vararuci’s son Mel attol attol Agnihotri according to Keral olpatti olpatti had performed the first Agnihotram (Vedic sacrifice) for the Kalidina of “Ya-jna-sthā-nam-sam-ra-ks yam” yam” i.e. 1270701 days which corresponds to 14 February 378, 06:00 ZT, expired day 13th February was Māgha amāvāsya of the amānta reckoning. This date adds credence to the legends which place Vararuci in 4th century CE, nearly 100 years before
a. Vararuci’s method of computation as seen in Āryabhat a.
the only surviving work Candravākyas resembles Babylonian methods of Seleucid Era but this resemblance does not rule out the possibility of originality in adapting the Babylonian method to the later times when Vararuci lived.
Candravākyas of Vararuci Mnemonics of Vararuci1 or the alpha-numeric or kat apay apayādi expressions of the 248 day cycle of anomaly represent the true motion of moon in 9 anomalistic cycles of 27 days and 13h18m34s.45 i.e. 27.5545654 days. True moon is computed by successively eliminating khan d as as (epochs) where the moon had a precise apogee conjunction. Original Kalikhan d a used by Vararuci is lost with the passage of time and successive revision of methods with larger Kalikhan d as as where the apogee conjunction served the method better. Major steps involved are: 1. Kalidinam - Kalikhan d am am or Kalidays for the epoch at which true moon is known. Corresponding longitude of moon is called Khan d a-dhr a-dhr uvam uvam ( k k0) 2. From the balance days, cycles of 12372 is removed. (Let r represent the cycles of 12372 and k1 the respective longitudinal arc. 3. From the balance less than 12372, cycles of 3031 days are removed. (Let s be the cycles of 3031 days and k2 the respective longitudinal arc). 4. From the rest 248 days are removed i.e. t cycles of 248 days with arc = k3 5. True longitude of moon2 = k0 + r*k1+s*k 2 + Vākya
2
Original epoch of Vararuci, viz., Kalikhan d am am and k0 are lost now. Kunhan Raja edition of the Candravākyas make use of the Kalikhan d am am of 1741650 as indicated using the alpha-numeric expression ‘amitayavotsuka’ in the method outlined to find out the Vākya for a particular day from the Kalidina (Kali-day count). Epoch of 1741650 correspond to Saturday,16 July 1667 CE, 06:00 ZT for which tropical longitude of moon was 56 014’. Ayānāmśa of the Kerala tradition was 19005’ and so the sidereal longitude was 37009’. Text edited and published by Raja gives the position of moon as ‘kaulat abh abhūpālatanaya’ and yields the numeric value k0 as 36031’41.5”. Also, Raja gives other values as shown in Table-1 below: th
Table-1: Revised 17 century Elements of Vararuci’s Moon Computational Model Symbol
Days
Signs
Deg.
Min.
Sec.
Sec./60
Value in Decimals
k0
1741650
1
6
31
41
31
36.529491
k1
12372
9
27
48
9
44
297.804537
k 2
3031
11
7
31
10
16
337.520185
k 3
248
0
27
43
28
39
27.726250
Theory Underlying the Anomalistic Model
Theory underlying the methods Indian (Pancasiddhantika) and Babylonian meet with elaboration in the work of Jones 3 . Moon has its minimum velocity at the apogee and this minimum velocity undergoes variation with respect to a mean velocity ((µ µ) of 13010’35”in a cycle of 27.55 days because of the major inequality of around 6 degrees. Maximum and minimum velocities as found in Babylonian System B tablets were 15 016’05”and 11005’05”. Recurrence of the daily velocities demanded an integral number of days constituted of anomalistic cycles and the basic relations employed were – 9 Anomalistic cycles = 248 days 110 Anomalistic cycles = 3031 days 449 Anomalistic cycles = 12372 days The Vararuci technique that has survived after its revision in the 16 th century4 has incorporated the elements of Āryabhat ī ya as may be understood from the method prescribed for computing ī ya the samkrama based on the Śaka Years. But the mnemonics of moon’s 248 daily velocities have survived and those shall be the focus of the present study. 248 values of the 9 anomalistic cycles have been given in Appendix-I.
G ī rn rn aśreyādi Vākyas and Babylonian 248 Day Scheme Table-2 summarizes the important features of the Vararuci and Babylon schemes in contrast to the typical modern astronomical values. 3
Table-2: Characteristics Characteristics of the 248 Day Schmes Description
Modern
Vararuci
Babylon ACT 190
Deg
Min.
Sec.
Deg.
Min.
Sec
Deg.
Min
Sec
Minimum
11
49
20
11
57
0
11
6
35
Maximum
15
16
52
14
19
0
15
14
35
Average
13
10
39
13
10
33
13
10
35
Step
--
--
--
--
--
--
0
18
Period
--
248 Days
248 Days
A comparison of the Vararuci Vakyas with the Babylonian 5 Moon motions (δλ ) of the 248 day scheme is presented in Table-2 for the initial 10 days. Contrast of the complete spectrum of velocities is shown in plot.1. While the Babylonian scheme begins at a minimum velocity 11:07:10 and advances in 18’ steps, the scheme of Vararuci is anchored on the minimum value of 12:03 which has more observational content as may be noted from the numerical values and the plot. Minimum and maximum velocities of the Vararuci scheme differ by nearly 10 from the phenomena recorded in Babylonian 248 day scheme. Table-3: Example Data of Vararuci and Babylonian 248 Day Shemes
Day No:
Cumulative Arc Vararuci Babylon
Daily Motion Vararuci Babylon
R
D
M
R
D
M
S
R
D
M
R
D
M
1
0
12
3
0
11
7
10
12
3
0
11
7
10
2
0
24
9
0
22
32
20
12
6
0
11
25
10
3
1
6
22
1
4
15
30
12
13
0
11
43
10
4
1
18
44
1
16
16
40
12
22
0
12
1
10
5
2
1
19
1
28
35
50
12
35
0
12
19
10
6
2
14
9
2
11
13
0
12
50
0
12
37
10
7
2
27
3
2
24
8
10
12
54
0
12
55
10
8
3
10
33
3
7
21
20
13
30
0
13
13
10
9
3
24
9
3
20
52
30
13
36
0
13
31
10
10
4
7
58
4
4
41
40
13
49
0
13
49
10
This deviation of the Vararuci scheme rules out any borrowing of the scheme from Babylon sources albeit the possibility exists that Vararuci may owe the idea of a 248 day scheme from some older Siddhanta of his times indebted to Babylonian sources. Mean velocity in both the schemes are very close (viz., 13010’35” for Babylonian and 13010’33” for the Vararuci scheme) as may be understood from the symmetry of the plot.
4
Plot-1: Contrast of Vararuci and Babylon Schemes 16
15
y a d r14 e p n o 13 i t o M 12 n o o M
V B
11
10 0
50
100
150
200
250
300
Day Number Inference Possible on the Vararuci Scheme Velocity profile of the Vararuci scheme is distinctly different from the Babylonian as may be noted from the difference in the minimum and maximum values of the function. When contrasted with the modern data also Vararuci’s maximum velocity is less by nearly a degree while Babylonian observational data is close to modern maximum velocity of moon. This distinctly different signature of Vararuci scheme has to find an explanation either as am ancient approximation/error or as contributed by identifiable astronomical factors. 1. Discrepancy in the Vararuci’s maximum lunar velocities shows that the velocity profile did not undergo serious observational verifications during the one-and-a-half millennium existence of the technique in India as an expeditious technique to compute true moon. May be the emergence of the Siddhāntas giving better accuracy to moon lessened the importance of the 248 day scheme. 2. If we consider that the Indian method was an approximation based on theoretical and arithmetical methods, then Vararuci could have obtained maximum and minimum velocity symmetrical to the avaerage value. Equation of centre or the first anomaly of moon in Indian astronomy had been ≈ 50 and by halving it Vararuci could have fixed the amplitude as 2.5 degrees in deciding the maximum and minimum velocities of moon. 3. How could Vararuci miss nearly 1 degree of motion in the daily velocity of moon despite the much discussed Indian access to Babylonian and Greek source materials on astronomy? 5
4. What is the source of the strange originality that we see in Vararuci Vākyas? Little information has survived about the date and works of Vararuci with the passage of time and so we are left with no information to understand the mind of the astronomer or the phenomena that inspired him to author the strange 248 day scheme. We are only aware of the legends that Vararuci was the court astronomer of Candragupta-II, Vikramāditya and one among the famed Navaratnas along with Kāl idas. idas. This single streak of legendary information when examined against the astronomical epoch of Udayagiri viz., 20 March 402 CE when the vernal equinox coincided with new moon and the lunar phenomena around the subsequent yearly epoch of 20 March 403 CE, we meet with illuminating details on the scheme o f Vararuci.
G ī rn rn aśreyādi Vākyas and Udayagiri Epoch The significance of the epoch 20 March 402 CE to the Indian Siddhāntic astronomy and the observational aspects of the passageway at Udayagiri have been the subject of the two earlier works by the present author6,7. Present work is a sequel that identifies the epoch of 20 March 403 CE as the Indian astronomical epoch of Vararuci based on the epochal signatures that can be deciphered in the G ī rn rn aśreyādi Candravākyas. Appendix-1 presents the contrast of the 248 day velocities of moon in contrast to the modern astronomical velocities for the 248 days considered from 06:00 ZT on 20 March 403 CE. Plot-2 presents the difference between Vararuci and modern true daily motions of moon (Modern values – Vararuci values) against the Day Number from 20 March 402 CE for 248 days. Vararuci’s 248 lunar velocities can be explained scientifically, by considering the peculiarities of the Indian tradition of observation and verification of luni solar parameters through eclipse observations. Modern - Vararuci Daily Motion of Moon 1.5
1.0
0.5
e c n e r 0.0 e f f i D
0
30
60
90
120
1 50
180
2 10
240
-0.5
-1.0
-1.5
Day Number from 20 March 403 CE
6
Eclipse oriented parameters were precise only at syzygies and at quadratures and octants the 2nd and 3rd anomaly alters the equation of centre as: λ = L + 060.288*sin A + 10.274*sin (2DA)+0.658*sin 2D, where λ is the true longitude of moon, L mean longitude, A anomaly and D is the elongation . Plot-3 presents the impact of evection and variation along with the velocity differences (Modern – Vararuci). It is apparent from the plot that the difference of Vararuci’s lunar motions with the modern values is due to lack of the 2 nd and the 3rd inequalities. Evection had a negative maximum of 1.27 and the Variation had been almost nil on the 6 th day when moon was at syzygy and as moon moved away from syzygy to last quarter and the difference M-V became zero at the last quarter. At new moon again evection got cancelled out as Vararuci equation of centre was the sum of first and second anomalies at syzygies. Interplay of evection and variation as well as observational errors possible during ancient times explains the 248 lunar motions given by Vararuci.
Plot-3: 2nd and 3rd Inequalities of Moon 1.5
1.0
0.5
s e i t i l a0.0 u q e n I -0.5
0
50
10 0
15 0
200
25 0
Evection
-1.0
Variation M-V -1.5
Day Number from 20 March 403 CE
Signature of the 403 CE Epoch. What is unique in Vararuci’s data is the decrement nearing 10 in the maximum velocity of moon as compared to modern as well as Babylonian data as shown in Table-3 above. Moon attains maximum velocity at perigee and the data relevant to maximum velocities of moon are shown in Table-4: 7
Anomaly Cycle 1
Date and ZT
Sun λ
1868341.34251
403/03/29 23:45
9.216913
Moon at Velocity Perigee λ Modern Vararuci 254.5667 14.215 14.32
2
1868368.37102
403/04/26 00:26
35.32337
250.7341
14.626
14.32
18
3
1868396.42163
403/05/24 01:39
62.14091
260.6418
15.029
14.32
17
4
1868424.75502
403/06/21 09:39
89.12785
274.265
15.275
14.30
16
5
1868453.16267
403/07/19 19:26
116.2907
288.8807
15.291
14.32
15
6
1868481.50065
403/08/17 03:33
143.6796
302.6246
15.059
14.32
14
7
1868509.50192
403/09/14 03:34
171.1584
311.9266
14.638
14.32
12
8
1868535.95882
403/10/10 14:32
197.5384
300.159
14.224
14.32
9
9
1868561.11824
403/11/04 18:22
222.952
270.4726
14.426
--
4
Julian Date (ZT)
Tithi 21
In anomaly cycle 1, moon had been close to the first quarter at apogee and the perigee on 29 March 23:45 ZT. Maximum velocity of the moon in the cycle at perigee had been 14°13' only (30.03.403CE, 06:00 ZT) in complete agreement with Vararuci. Perigee had the longitude of 254.57 degrees with 21st tithi i.e. moon moving away from syzygies to last quarter. Apogee at the first quarter meant that the major axis of moon’s orbit was perpendicular to the direction of sun and less eccentricity for the moon’s orbit leading to a decrement in moon’s maximum velocity at perigee. In the first and last cycles Vararuci scheme has the maximum velocity close to the modern values in view of the above cited reason viz. the line of apsides perpendicular to the direction of sun. On the other hand, in the 5th cycle, the lunar phase was full moon in conjunction with perigee leading to a stretch out of the ellipse by solar graviation and thus increased eccentricity giving rise to a high value of velocity at the perigee. 3rd, 4th and 6th cycles show the same effect to a lesser degree. Plot-4 is illustrative of the different cycles and tithis of perigee that decided the velocity contribution from evection. Plot-4:
Moon Motion at Perigee vs. Tithi
16.0
y a 15.5 D r e p15.0 n o o M14.5 f o y 14.0 t i c o l e 13.5 V
5 6
4 3
7
2
9
1 8
13.0
0
2
4
6
8
10
12
14
16
18
20
22
Tithi
8
It becomes therefore apparent that the Vararuci mnemonics are not strictly based on continuous observations for the 9 anomalistic cycles or 248 days. Apogee conjunction on 17 March 403, 08:33 ZT had been at a longitude of 880.674 i.e. first quarter Moon conjunction with the apogee where the contribution from evection negatively contributed to make the first inequality 060.288*Sin A as 50.10*Sin A. This was true only for the 1st anomalistic cycle but Vararuci and Indian astronomy till the days of Vat eśvara and Lal l a had this deficiency in computing the moon. Evidence of Siddhā Siddhāntic Apogee and Kali 3504 (Elapsed)
Another striking aspect of the epoch is that the apogee of the moon was at 90 0, the position that the Siddhāntas ascribe to apogee. Therefore the epoch of 403 must have an integral number of revolutions upto 403 CE or Kali 3504 (elapsed). We can see that 4320000 :488219:: 3504:396 i.e. we get the orbital period of apogee as 8.8485 years. It is apparent from the above that the epoch of 20 March 403 CE had played a critical role in the fixing of the apogee of moon with reference to the Kaliyugādi. Observational aspects of the setting Moon on this date at Udayagiri presents to us some interesting features as may be noted in table-5 below: Table-5: Moonset at Udayagiri on 20 March 403 CE 20 March 403 CE Time: ZT
Azimuth
Altitude
Topocentric Declination
2:15
291.76
11.88
24.50
2:25
292.54
9.80
24.47
2:35
293.34
7.74
24.44
2:45
294.18
5.69
24.41
2:55
295.05
3.66
24.38
3:05
295.94
1.64
24.35
3:15
296.87
-0.36
24.33
On 20 March 403 CE, the setting moon had a declination less than 24 030’ and azimuth around 2940 to suggest that Moon did set at the same point as the set point of the sun on summer solstice. This in turn suggests the possibility of lunar alignments for the archaeoastronomy structure at Udayagiri. Moon’s latitude at the time was around 50 and given the inaccuracy of ancient measurements, it can be inferred that the siddhāntic value of β = 4.50 also had its genesis in the Udayagiri observations of 20 March 403 CE. As the moon had been approaching the 900 latitudinal limit from the node (460.5), observation during 20-21 March were best suited to fix the latitude of moon. It is apparent that the epochs 402 CE and 403 had great opportunities for observation of 9
planets with respect to vernal point. Salient features of the 402 CE epoch has undergone discussion in a previous work and the features discussed as above of Vararuci Vākyas and apogee of moon suggest that planetray observations have continued there subsequently. Table-6 presents the planetary positions at the two observational epochs at which the Siddh āntic astronomy probably took shape: Table-6: Planetary Positions 402 – 403 CE, 20 March 06:00 ZT, at 23N31, 75E45: Grahas
20 March 402 CE
20 March 403 CE
Longitude λ Latitude β Longitude λ Latitude β Sun
359.96
0.00
359.72
0.00
Moon
1.33
(-) 04°35'
123.33
04°58'
Mars
123.10
03°10'
358.79
(-) 00°41'
Mercury
333.93
(-) 02°21'
343.81
(-) 02°14'
Jupiter
115.42
01°00'
147.65
01°33'
Venus
41.83
01°49'
349.44
(-) 01°27'
Saturn
301.40
(-) 00°42'
312.35
(-) 01°08'
Moon-Node
66.10
--
45.50
--
Moon-Apogee
53.00
--
90.00
--
It may be noted: 1. At the time of vernal equinox we find that in 403 CE, the apogee of Moon was at 90 0. We find a period relation 4320000:488219:: 3504:396 here. Latitude of Moon was respectively (-) 04035’ and (+) 04058’ in the successive epochs and this has led to the sidddhāntic maximum latitude of 040.5, subject to the errors possible with the Indian observations. Moon had set due east on 20 March 403 CE at Udayagiri where the passageway aligned to observe such horizon phenomena. In the subsequent lunation, on 16 April 403 CE, Moon’s maximum latitude could be observed against the occultation of η-Leonis and longitudinal conjunction of setting Moon with the ecliptic star Regulus. Moon’s longitudinal conjunction at 23:11 ZT had a longitude of 127047’ and latitude 05012’. η-Leonis had an ecliptic longitude of 1250.64 and latitude of 04028’ and it can be inferred that the occultation of η-Leonis and conjunction with Regulus had enabled the astronomers at Udayagiri to fix the latitude of Moon as 40.5 instead of the Greek value of 50 that may have been available to them. It may be noted that the Ascending Node was at 47 026’ and hence for any astronomer interested in watching the latitude of Moon against an ecliptic reference star would have chosen the Regulus conjunction of Moon having a nodal distance of 900. Moon-Jupiter conjunction on 21 March March 403 CE also had the nodal distance required to watch the maximum latitude of Moon. 10
Declination of Moon on 20 March 403CE and 16 April 403 CE had been suitable for observing the setting moon aligned with the direction of the passageway at Udayagiri. 2. 20 March 402 CE epoch had the period relations 4320000:57753336::3503:46831 for the sidereal revolutions and 4320000:53433336::3503: 43328 for the synodic revolutions. 3. Mars was heliacally set and thus 3504 elapsed years had an integral number of synodic and sidereal revolutions of Mars. 4320000: 2296824::3504:1863. 4. Mercury and Venus were also set heliacally to yield 4320000:17937020::3504:14549 and 4320000:7022388:: 3504:5696.
the
relations:
5. In the case of Jupiter, approach to opposition could have enabled the astronomers to 4320000:364224:: 3504:295.5 or to fix the period with their arithmetical and algebraic techniques. 6. Saturn was heliacally set on 25 January 403 CE and heliacally rose on 20 March 403 CE at 03:41 ZT with an elongation of ≈ of ≈ 480 and with continued observations from 20 March 402 to 403 CE, Indian astronomers led by Vararuci could have easily worked out the Siddhāntic model of computations. 7. Latitude of Jupiter strikes our attention with the maximum of apparent latitude 1 0.5 on 20 March 403 CE. The following orbital elements in contrast to that of Āryabhat ī ya is ī ya striking by their closeness to modern values and those of later Siddh āntas like ya8. Āryabhat ī ī ya Table-7: Siddhā Siddhāntic and Modern Elements
Āryabhat a 500 CE
Modern values
Date 0
0
Node
Apogee
i
Mars
1.863
37.22
126.73
1.5
40
118
Mercury
6.971
29.43
232.68
2
20
210
Jupiter
1.393
84.31
169.30
1
80
180
Venus
3.378
62.40
289.08
2
60
90
Saturn
2.544
99.60
241.61
2
100
236
403/03/20 06
i
0
0
Node
0
0
Apogee
It may be noted that the Siddhāntic values of the Ascending Node were more accurate at the epoch 20 March 403 CE and the maximum latitude given for planets (Jupiter 10, Mercury 20 etc) were the results of observations observations during 402-03 CE.
Conclusions Examination of the G ī rn rn aśreyādi vākyas of Vararuci against the backdrop of the astronomical epoch of Udayagiri viz 20 March 402 CE suggests that the 248 day scheme of computation of true Moon had an indegenous origin and the epoch of its derivation may be identified as 20 March 403 CE. 9 Anomalistic cycles have been examined and the distinct features of the Vararuci scheme in contrast to the Babylonian scheme as well as the Moon velocities from modern algorithms have been explained as due to the use of only one inequality of Moon by the Indian astronomer. Perturbation of the longitudes because of evection has been shown to be the cause of the 10 decrement in the maximum velocity of Moon in the 248 24 8 day scheme of Vararuci. 11
Conjunction of the first quarter phase with apogee at 900 can be identified as an epochal signature of Udayagiri epoch of Kali 3504 in the Moon motions of Vararuci. 20 March 403 CE or the expiry of the Kali year 3504 had been remarkable also for the observation of Moon at Udayagiri because of its setting at the same place as that of the setting point of the summer solstice Sun. Solar alignment of the Udayagiri structure may also had lunar alignment and provisions to observe Moon accurately. Moon had been at the maximum latitude at the above epoch and in subsequent lunations offered observation against the ecliptic star Regulus and Jupiter so that latitude can be observed and measured with somekind of ancient astronomical artefacts. Planetary conjunctions significant to the context of the Siddh āntic computational models have been discussed with reference to the epochs of 20 March 402 and 403 CE – suggesting that observations have been carried through out the year to fix the integral number of revolutions or mean motions of planets indegenously rather than borrowing the same from Greek or Babylonian sources.
References 1
Vararuci-Candravakyas, C. Kunhjan Raja, Adyar, Numerical values of the 248 g ī rn rn aśreyādi vākyas in contrast to the corresponding modern values is given in Appe ndix-I.
2
Raja, Kunhan, C., Candravākyas of Vararuci, Vararuci, Pamphlet of Adyar Library, Madras. Madras. p.7: The verse quoted by Raja is –
+ʨÉiɪɴÉÉä äiºÉÖ iºÉÖ Eò½þ Eò½þÒxÉÆ tÖ MÉhÉÆ MÉhÉÆ ®úºÉMÉèÊ®ú Ê®úE ò&è EÖ ò±ÉÒxÉÉR ó MÉè&* &* nä ù´Éäxpè xpè ù®úÊ{É ½þi´ÉÉ iÉÎSUô¹]Æ õ ¦É´ÉÊiÉ ´ÉÉCªÉºÉÆJªÉä JªÉäxnù xnùÉ&** &ä ** “Kalidina reduced by the Kalikhan d am1741650 am1741650 represents the epoch for which Vākya computation is based. Days since the Kali epoch of 1741650, devoid of cycles of 12372 and 3031 represents the Vākya of moon for the day” 3
Jones, Alexander, The Development and Transmission of 248-day schemes for lunar motion in ancient astronomy. Archive for History of Exact Sciences 29 (1983) 1-36.
4
The Kalikhandam employed in finding the particular vakya for a Kalidinam makes use of the apogee conjunction on
5
Neugebauer, O., Astronomical Cuneiform Texts – III, Springer Verlag, New York, (1983), p.131 6 Hari, Chandra, K., Astronomical Alignment of Iron Pillar and Passageway at Udayagiri and Date of Sanakanika Inscription, Inscription, Under submission. 7
Hari, Chandra, K., 20 March 402 CE at Udayagiri: Candra Gupta-Vikram āditya Epoch of Indian Astronomy, Under submission
8
Shukla, KS., Sarma, KV., Āryabhat ī ya, Indian National Science Academy, N. Delhi (1976), ī ya, pp.18-20
Appendix –I 12
Vararuci Velocities & Modern Cumulative Speeds of Anomalistic Cycle Day Day Day Modern Vararuci Diff: Modern Vararuci Diff: Modern Vararuci Diff: No. No. No. 1
12.12
12.05
0.07
43
13.86
14.28
-0.42
85
12.10
12.13
-0.03
2
12.35
12.10
0.25
44
13.60
14.17
-0.56
86
12.30
12.27
0.04
3
12.64
12.22
0.42
45
13.35
14.03
-0.68
87
12.57
12.45
0.12
4
12.94
12.37
0.58
46
13.10
13.85
-0.75
88
12.91
12.65
0.26
5
13.25
12.58
0.67
47
12.86
13.62
-0.75
89
13.31
12.90
0.41
6
13.53
12.83
0.70
48
12.64
13.37
-0.73
90
13.75
13.17
0.59
7
13.78
12.90
0.88
49
12.43
13.10
-0.67
91
14.21
13.42
0.79
8
13.96
13.50
0.46
50
12.25
12.85
-0.60
92
14.64
13.67
0.97
9
14.10
13.60
0.50
51
12.09
12.60
-0.51
93
14.99
13.90
1.09
10
14.18
13.82
0.36
52
11.96
12.40
-0.44
94
15.21
14.07
1.15
11
14.21
14.00
0.21
53
11.87
12.23
-0.36
95
15.27
14.20
1.07
12
14.21
14.17
0.04
54
11.84
12.10
-0.26
96
15.14
14.30
0.84
13
14.17
14.28
-0.11
55
11.86
12.05
-0.19
97
14.86
14.32
0.54
14
14.11
14.32
-0.21
56
11.94
12.05
-0.11
98
14.46
14.28
0.18
15
14.01
14.30
-0.29
57
12.10
12.10
0.00
99
14.01
14.18
-0.18
16
13.88
14.22
-0.34
58
12.33
12.20
0.13
100
13.54
14.07
-0.53
17
13.71
14.12
-0.41
59
12.63
12.35
0.28
101
13.11
13.85
-0.74
18
13.50
13.92
-0.42
60
13.00
12.57
0.43
102
12.73
13.65
-0.92
19
13.26
13.73
-0.48
61
13.41
12.78
0.63
103
12.42
13.38
-0.96
20
12.99
13.47
-0.47
62
13.84
13.05
0.79
104
12.19
13.13
-0.95
21
12.72
13.22
-0.49
63
14.25
13.32
0.94
105
12.02
12.88
-0.86
22
12.46
12.97
-0.50
64
14.61
13.55
1.06
106
11.91
12.62
-0.70
23
12.23
12.72
-0.49
65
14.88
13.80
1.08
107
11.85
12.43
-0.58
24
12.04
12.48
-0.44
66
15.01
13.98
1.03
108
11.83
12.23
-0.40
25
11.91
12.30
-0.39
67
15.00
14.17
0.83
109
11.85
12.13
-0.29
26
11.85
12.15
-0.30
68
14.85
14.25
0.60
110
11.88
12.05
-0.17
27
11.86
12.08
-0.23
69
14.59
14.32
0.27
111
11.95
12.03
-0.08
28
11.95
12.03
-0.08 -0.08
70
14.25
14.30
-0.05 112
12.04
12.08
-0.04 -0.04
29
12.12
12.07
0.06
71
13.88
14.25
-0.37 113
12.17
12.20
-0.03
30
12.37
12.13
0.23
72
13.50
14.12
-0.61 114
12.35
12.33
0.02
31
12.68
12.28
0.39
73
13.15
13.95
-0.80 115
12.58
12.53
0.04
32
13.03
12.47
0.56
74
12.84
13.75
-0.91 116
12.86
12.77
0.09
33
13.40
12.68
0.72
75
12.57
13.50
-0.93 117
13.21
13.02
0.19
34
13.77
12.93
0.84
76
12.36
13.25
-0.89 118
13.61
13.28
0.32
35
14.10
13.20
0.90
77
12.18
12.98
-0.80 119
14.04
13.53
0.51
36
14.37
13.45
0.92
78
12.05
12.73
-0.69 120
14.48
13.77
0.71
37
14.54
13.70
0.84
79
11.95
12.52
-0.57 121
14.87
13.98
0.88
38
14.62
13.90
0.72
80
11.88
12.32
-0.44 122
15.15
14.13
1.02
39
14.60
14.08
0.52
81
11.84
12.17
-0.33 123
15.28
14.25
1.03
40
14.49
14.22
0.28
82
11.84
12.08
-0.24 124
15.23
14.30
0.93
41
14.32
14.30
0.02
83
11.88
12.03
-0.16 125
14.99
14.30
0.69
42
14.10
14.32
-0.21 -0.21
84
11.96
12.07
-0.11 -0.11 126
14.61
14.25
0.36
13
Contd.p2. Day Modern No. 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168
14.14 13.64 13.16 12.74 12.39 12.13 11.96 11.86 11.82 11.84 11.90 11.98 12.09 12.21 12.35 12.51 12.70 12.93 13.21 13.53 13.88 14.25 14.59 14.87 15.03 15.04 14.88 14.57 14.16 13.70 13.23 12.80 12.44 12.16 11.97 11.86 11.84 11.88 11.97 12.10 12.26 12.44
Vararuci
Diff:
Day Modern No.
14.13 13.98 13.77 13.53 13.28 13.02 12.77 12.53 12.33 12.20 12.18 11.93 12.05 12.13 12.23 12.43 12.62 12.88 13.13 13.38 13.65 13.68 14.23 14.18 14.28 14.32 14.30 14.20 14.07 13.90 13.67 13.42 13.17 12.90 12.65 12.45 12.25 12.15 12.07 12.03 12.08 12.17
0.01 -0.34 -0.60 -0.79 -0.89 -0.88 -0.81 -0.68 -0.51 -0.36 -0.29 0.05 0.04 0.08 0.12 0.08 0.09 0.05 0.07 0.14 0.23 0.56 0.36 0.69 0.75 0.72 0.58 0.37 0.09 -0.20 -0.44 -0.62 -0.73 -0.74 -0.68 -0.59 -0.41 -0.27 -0.10 0.07 0.18 0.27
169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210
12.62 12.80 12.98 13.18 13.38 13.61 13.85 14.09 14.32 14.51 14.62 14.62 14.52 14.29 13.98 13.60 13.20 12.82 12.48 12.21 12.01 11.89 11.86 11.91 12.02 12.19 12.41 12.65 12.90 13.15 13.38 13.58 13.76 13.91 14.03 14.12 14.19 14.22 14.21 14.15 14.02 13.84
Vararuci
Diff:
12.32 12.52 12.73 12.82 13.42 13.50 13.75 13.95 14.12 14.25 14.30 14.32 14.25 14.17 13.98 13.80 13.55 13.32 13.05 12.78 12.57 12.35 12.20 12.10 12.05 12.05 12.10 12.23 12.40 12.60 12.85 13.10 13.37 13.62 13.85 14.03 14.17 14.28 14.32 14.30 14.22 14.08
0.30 0.28 0.25 0.36 -0.03 0.11 0.10 0.14 0.20 0.26 0.32 0.31 0.27 0.13 -0.01 -0.20 -0.35 -0.50 -0.57 -0.58 -0.56 -0.46 -0.34 -0.19 -0.03 0.14 0.31 0.42 0.50 0.55 0.53 0.48 0.39 0.29 0.18 0.09 0.03 -0.06 -0.10 -0.15 -0.19 -0.24
Day Modern No. 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248
13.60 13.33 13.03 12.73 12.45 12.21 12.03 11.91 11.87 11.91 12.02 12.21 12.46 12.77 13.10 13.44 13.76 14.04 14.25 14.38 14.42 14.39 14.29 14.15 13.97 13.78 13.58 13.37 13.16 12.95 12.73 12.53 12.33 12.15 12.01 11.91 11.86 11.88
Vararuci
Diff:
13.90 13.70 13.45 13.20 12.93 12.68 12.47 12.28 12.13 12.07 12.03 12.08 12.15 12.30 12.48 12.72 12.97 13.22 13.47 13.73 13.92 14.12 14.22 14.30 14.32 14.28 14.17 14.00 13.82 13.60 13.33 13.07 12.83 12.58 12.37 12.22 12.10 12.05
-0.30 -0.37 -0.42 -0.47 -0.48 -0.47 -0.44 -0.37 -0.26 -0.16 -0.01 0.12 0.31 0.47 0.62 0.72 0.80 0.82 0.78 0.64 0.51 0.27 0.08 -0.15 -0.34 -0.50 -0.59 -0.63 -0.66 -0.65 -0.60 -0.54 -0.51 -0.43 -0.36 -0.31 -0.24 -0.17
14
