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Dhruvarāśi Takanika in Jaina Canons
Dr. L. C. Jain* 1. Introduction
At present the commentaries on the Süryaprajñapti are those due to Malayagiri', Amolaka Rşi2 and Ghāsilāla? Kohl had contributed a text which was revised by A. Olz. Similarly, the commentaries of the Candraprajñapti are available". These texts are from the fifth sub-composition ( upānga ) of the Svetāmbara Jaina School, in Ardhamāgadhi Prākrt. According to Jacobi and Schubring they might have taken shape during the third and fourth century B. C. According to Needham and Lingo, they may go back to the latter part of the 1st millennium B.C. Certain descriptions are available in the contents rendered by Weber, Ind. Stud., 10, 254ff. ( 1868 ) and Thibaut?, JASB 40, comp. and also in his 'Astronomie' in the Grundias, p. 20ff., 29.
The 'Tiloyapannatti' was compiled by Yati Vrsabha ( c.5th century A. D. or earlier 18. This is in Sauraseni Prākrt and belongs to the Digambara Jaina School. Similarly in the same school the Dhavală commentary texts were composed by Virasena ( c. 9th century A. D. ) on the Satkhandāgama texts which were composed by Puspadanta and Bhūtabali (c. 2nd century A. D. or earlier ). The Gommatasāra 10 texts of this school form the summary texts of the Satkhandāgama texts, composed by Nemicandra Siddhāntacakravarti (c. 10-11th century A. D.).
The term Dhruvarāsi appears in the above texts except the Gommatasāra in which the term Dhruvahāra has been used. This appears to be an invariant or constant of the equation of motion and has been sometimes used as a parameter for generating a group of events which were periodic in character. The noted periodicity in the five-year yuga system of India was thus tackled through the Dhruvarāśi technique. 2. The Dhruvarāśi Technique in the Tiloyapannatti
In the Tiloypannatti, the technique of Dhruvarāśi has been applied for finding out the distance between the orbits of the moon and those of the sun from the Meru".
Description of the three relevant verses is as follows:
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Dhruvarāśi Takanika in Jaina Canons
ekasatthi gunidā pañcasayājoyanāni dasajuttā / te açadāla vimissā dhuvarāsi nāma cāramahi // 122 11 ekatthisahassā atthavannutaraṁ sadam taha ya/ igisatthie bhajide dhuvarāsi pamānamuddittham // 123/1
31158
61
pannarasehim gunidam nimakarabimbappamānamavanijjam / dhuvarāsido sesaṁ viccālam sayalavihiņam // 124 //
30318
61
Translation of the above verses is as follows:
Verse 122 : On multiplication of five hundred and ten yojanas by sixtyone, and adding forty-eight to the product, the result (as divided by the denominator sixty-one) becomes the extension of the orbital ground called the Dhruvarāsi.
Note: 510– is equal to 31158/61. This has been called the Dhruvarāśi
61 or the orbital field of the sun or the moon.
Verse 123: The quotient obtained on dividing thirty-one thousand and one hundred fifty-eight by sixty-one has been shown as the pole-set or Dhruvarāsi.
Note : The above verse has been elaborated in this verse.
Verse 124 : On multiplying the diameter of the moon by fifteen, the product is subtracted from the dhruvarāśi, the result is the measure of the interval of all the remaining orbits.
Note : Diameter of the moon is 56/61. Hence ( 56/61 ) x 15 = 840/61. Now one can find the interval between the remaining orbits as equal to (31158/ 61) - ( 840/61 ) or equal to ( 30318/61). Further Procedure
In the verse ahead, the following has been worked out. When (30318/61) is divided by 14, one gets the interval between every one of the orbits as
35 214
yojanas.
Now to this amount is added the moon's diameter 56/61 yojanas, getting the common difference
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36 179
yojanas.
Thus, the first orbit is at a distance of 44820 yojanas from the Meru. The second orbit is at a distance of
427
44820 + 36 yojanas from the Meru. The third orbit is at a distance of 44820 +2 ( 360 ) yojanas from the Meru.
179
427
This is carried on till the last orbit.
Similarly the method of the Dhruvarāśi in the Dhavalā has been used for confirming the measure of the set of the illusive visioned bias (mithyādrsti jiva rāśi ) through four analytical methods. However, this is altogether a different procedure for the use of the technique of a Dhruvarāśi away from a convention of periodicity. Further, the Dhruvahāra (pole divisor) concept in the Gommațasāra, is given in details 13, where a geometric regression or gunahāni is produced with the Dhruvahāra as common-ratio. 3. The Dhruvarāśi Treatment in the Süryaprajñapti and the Candraprajñapti Commentaries
In various commentaries of the Sūryaprajñapti and the Candraprajñapti, the Dhruvarāśi technique has been applied to calculate requisite sets which usually form an arithmetical or a geometrical sequence.
First of all the method is given to find out the measure of a Dhruvarāśi for solving a particular problem of astronomy. Then the process of obtaining the subsequent progression of desired results is given.
There are as many as twelve examples in which the use of different types of the Dhruvarāsis is given in the commentaries of the above texts. The muhurta is divided into 62 parts and further subdivided into 67 parts. Similarly other units are divided and subdivided. The Vedānga system of time, however, is different for divisions and sub-divisions.
One of the examples may be illustrated as follows:
Suppose it is required to find out the position of the sun when it is in yoga with a constellation at the instant of the end of a requisite parva (half-lunation). For this, the following verses appear in the commentaries 14
Det er
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Dhruvarăsi Takanika in Jaina Canons
55
Transcription : Verse 1
"tettisaṁ ca muhuttā visatthibhāgā ya do muhuttassa /
cutti cunniyabhägā pavvikaya rikkhā dhuvarāsi // 1 // Translation
Thirty three muhūrtas (plus) two parts out of sixty-two parts (plus) thirtyfour parts out of sixty-two into sixty-seven parts, is to be known as the half-lunation (parva ) from of the Dhruvarāśi corresponding to the (sun)-constellation (yoga). Trancription : Verse 2
"icchā pavva guņão dhuvarā sio ya sohanaṁ kuņasu /
pūsāiņaṁ kamaso jahā ditthamanantaņāņihir // 2/1 Translation
The Dhruvarāśi is multiplied by whatever is the requisite ( sequential number of the parva (half-lunation ); and then from the product are subtracted (the measure of ) the constellations as pusya, etc., in sequence, according to the omniscient vision. Explanation
Let the problem be as to in which sun-constellation does the first parva ends. For this the Dhruvarāśi is
33 + ( 2/62 ) + (34/(62 x 67 ) } muhurtas. Here the Dhruvarāśi is calculated as follows:
In all there are 124 parvas, out of which 62 are bright halves and 62 dark halves in course of five-year yuga or five sun-constellation yogas. Hence, for one parva (half-lunation ) we get 5/124 sun-constellation part of a yoga. Now this is multiplied by 1830 to convert it into the type of 67 parts getting (5x 1830)/124 or 4575/62. Noting that the sun moves 1830 celestial parts in a muhūrta or moves through 1830 half-mandalas or ahorātras in a five-year yuga, we convert 4575/62 into muhurtas by multiplying it by thirty. Thus, we get ( 4575 x 30 )/62 muhurtas
3+1 2/62) + 34/162 x 67) muhüratas. This is the Dhruvarāśi required for the purpose.
Now we pose the problem for the first parva, hence the Dhruvarāśi is multiplied by one. This product is to be subtracted by the period covered by the pusya constellation. Hence we get 33+ (2/62 ) + {34/( 62 x 67 ) } - 19 + ( 45/62) + {33/ ( 62 x 67 ) } = 13+ (19/62 ) + (1/( 62 x 67 ) ).
This period remains to be covered after the sun has passed over the puşya constellation. Hence the sun remains in the āśleşā for this much period. Just after
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Dr. L.C. Jain
this, the first parva in form of the coming dark 15th of the śravana month comes to an end.
Similarly the succeeding parvas are to be treated. For them the multiples of the Dhruvarāśi are 2, 3, 4, 5,..........., 61, 62 respectively. The products form a geometric progression with the Dhruvarāśi as the common ratio.
In the above system, it may be noted that a muhurta or 48 minutes set was sub-divided into 62 parts and each such part was further sub-divided into 67 parts. This system was slightly finer than the sexagesimal system of dividing an hour into, 60 minutes and each minute into sixty seconds. Concluding Remarks
The above probe into the technique of the Dhruvarāśi (pole-set) appears to have been in use round about the fourth century B.C. when the Süryaprajñapti types of works in the Jaina School were possibly being compiled for the Karanānuyoga group of study. The periodicity of natural phenomena and its calculations needed a group theoretic study and the Dhruvarāśi technique was an attempt towards it. From the several remaining examples it appears that progressions and regressions were the powerful tools for dealing with such periodic phenomena. It also appears that this technique might have played a decisive role in developing the later larger yuga system for the planetary motions whose account has been mentioned to have become extinct by Yativrsabhācārya in his Tiloyapannatti 15. This group theoretic yuga system seems to have been converted into the theory of epicycles in Greek later on. Mention may be made also of the work of Roger Billard on the yuga system of India through the computer16. Acknowledgement
The author is grateful to Dr. A. K. Bag for encouraging this research. Thanks are also due to Prof. S. C. Datt for providing research facilities at the Department of Physics, R. D. University, Jabalpur. References 1. Tika by Malayagiri, (Pr. Sūrapannatti ), Agamodaya Samiti, Bombay, 1919.
This appears to have been compiled in c. 12th century A. D. 2. Sūryaprajñapti Sūtra, ed. Amolaka Rşi with Hindi translation, Sikandarabad
(Dakshin ), c. Virābda 2446. 3. Sūryaprajñapti Sūtram, edited with Suryaprajñapti-prakāśikā, Sanskrit commen
tary of Ghasilal and Hindi, Gujarati translation by Kanhaiyalal, Vol. 1 (1981 ), Vol. 2 (1982), Ahmedabad.
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________________ Dhruvarasi Takanika in Jaina Canons 57 4. Kohl, J. F., Die Suryaprajnapti, Versuch einer Text geschichtee, Stuttgart, 1937 Revised by Author Olz, 1938. 5. Candraprajnapti Sutram, edited with Candraprajnapti-prakasika, Sanskrit commentary of Ghasilal by Kanhaiyalal, Rajkot, 1973. 6. Needham, J. and Ling, W., Science and Civilization in China, Cambridge, 1959, Vol. 3. 7. Thibaut, G., Astronomie, Astrologie and Mathematik ( der Inder ) in Grundiss der Indo-Arischen Philoloie and Altertumskunda, ed. G. Buhler and F. Kielhorn, Bd. 3, Heft 9. 8. Jadivasaha's 'Tiloyapannatti', ed. H. L. Jain and A. N. Upadhye, Part 2, Sholapur, 1951. 9. The Satkhandagama of Puspadanta and Bhutabali with Dhavala commentary of Virasena, Vol. 3 (1941 ) and Vol. 4 (1942), ed. H. L. Jain et. al., Amaraoti. 10. Gommatasara of Nemicandra Siddhantacakravarti, Vol. 2 (Jivakanda ) with Karnatakavrtti, Jivatattvapradipika and Hindi Trans., ed. A. N. Upadhye and K. C. Shastri, Bharatiya Jnanapith, New Delhi, 1979. 11. Tiloyapannatti, Vol. 2, Ch. 7, V. 122. 12. Jain, L. C., On Certain Mathematical Topics of the Dhavala Texts, IJHS, Vol. 2, 1976, No. 2, 1976, pp. 85-111. 13. Gommatasara, op. cit., pp. 628-648. 14. Cf.(5), pp. 394-398, Cf. also (3), pp. 129.133. 15. Tiloyapannatti, op. cit., Ch. 7, V. 458 et seq. 16. Cf. Billard, Roser, L'Astronomic Indienne, E' cole Francaise D'ex treme-Orient, Paris, 1971. The records are also available in computerized cassette. *Ex. Professor of Mathematics 677, Sarafa Jabalpur - 482 002 (M.P.)