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________________ Dhruvarasi Takanika in Jaina Canons Dr. L. C. Jain* 1. Introduction At present the commentaries on the Suryaprajnapti are those due to Malayagiri', Amolaka Rsi2 and Ghasilala? Kohl had contributed a text which was revised by A. Olz. Similarly, the commentaries of the Candraprajnapti are available". These texts are from the fifth sub-composition ( upanga ) of the Svetambara Jaina School, in Ardhamagadhi Prakrt. 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 Prakrt and belongs to the Digambara Jaina School. Similarly in the same school the Dhavala commentary texts were composed by Virasena ( c. 9th century A. D. ) on the Satkhandagama texts which were composed by Puspadanta and Bhutabali (c. 2nd century A. D. or earlier ). The Gommatasara 10 texts of this school form the summary texts of the Satkhandagama texts, composed by Nemicandra Siddhantacakravarti (c. 10-11th century A. D.). The term Dhruvarasi appears in the above texts except the Gommatasara in which the term Dhruvahara 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 Dhruvarasi technique. 2. The Dhruvarasi Technique in the Tiloyapannatti In the Tiloypannatti, the technique of Dhruvarasi 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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________________ Dhruvarasi Takanika in Jaina Canons ekasatthi gunida pancasayajoyanani dasajutta / te acadala vimissa dhuvarasi nama caramahi // 122 11 ekatthisahassa atthavannutaram sadam taha ya/ igisatthie bhajide dhuvarasi pamanamuddittham // 123/1 31158 61 pannarasehim gunidam nimakarabimbappamanamavanijjam / dhuvarasido sesam viccalam sayalavihinam // 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 Dhruvarasi. Note: 510- is equal to 31158/61. This has been called the Dhruvarasi 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 Dhruvarasi. 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 dhruvarasi, 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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________________ Dr. L.C. Jain 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 Dhruvarasi in the Dhavala has been used for confirming the measure of the set of the illusive visioned bias (mithyadrsti jiva rasi ) through four analytical methods. However, this is altogether a different procedure for the use of the technique of a Dhruvarasi away from a convention of periodicity. Further, the Dhruvahara (pole divisor) concept in the Gommatasara, is given in details 13, where a geometric regression or gunahani is produced with the Dhruvahara as common-ratio. 3. The Dhruvarasi Treatment in the Suryaprajnapti and the Candraprajnapti Commentaries In various commentaries of the Suryaprajnapti and the Candraprajnapti, the Dhruvarasi 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 Dhruvarasi 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 Dhruvarasis 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 Vedanga 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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________________ Dhruvarasi Takanika in Jaina Canons 55 Transcription : Verse 1 "tettisam ca muhutta visatthibhaga ya do muhuttassa / cutti cunniyabhaga pavvikaya rikkha dhuvarasi // 1 // Translation Thirty three muhurtas (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 Dhruvarasi corresponding to the (sun)-constellation (yoga). Trancription : Verse 2 "iccha pavva gunao dhuvara sio ya sohanam kunasu / pusainam kamaso jaha ditthamanantananihir // 2/1 Translation The Dhruvarasi 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 Dhruvarasi is 33 + ( 2/62 ) + (34/(62 x 67 ) } muhurtas. Here the Dhruvarasi 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 muhurta or moves through 1830 half-mandalas or ahoratras 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) muhuratas. This is the Dhruvarasi required for the purpose. Now we pose the problem for the first parva, hence the Dhruvarasi 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 pusya constellation. Hence the sun remains in the aslesa for this much period. Just after
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________________ 56 Dr. L.C. Jain this, the first parva in form of the coming dark 15th of the sravana month comes to an end. Similarly the succeeding parvas are to be treated. For them the multiples of the Dhruvarasi are 2, 3, 4, 5,..........., 61, 62 respectively. The products form a geometric progression with the Dhruvarasi 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 Dhruvarasi (pole-set) appears to have been in use round about the fourth century B.C. when the Suryaprajnapti types of works in the Jaina School were possibly being compiled for the Karananuyoga group of study. The periodicity of natural phenomena and its calculations needed a group theoretic study and the Dhruvarasi 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 Yativrsabhacarya 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. Surapannatti ), Agamodaya Samiti, Bombay, 1919. This appears to have been compiled in c. 12th century A. D. 2. Suryaprajnapti Sutra, ed. Amolaka Rsi with Hindi translation, Sikandarabad (Dakshin ), c. Virabda 2446. 3. Suryaprajnapti Sutram, edited with Suryaprajnapti-prakasika, 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.)