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(supremum) (Mukhtara and Patni 1975, vv. 1352, pp. 14-49). This twenty one-fold number system12 was used by the Jainas to demonstrate their Karma theory and cosmology. The other dravyamāna is upamāpramāna (simile-measure). It is of eight kinds. They are playa, sagara, sūcyangula, pratarangula, ghanangula, jagacchreņi (or jagatśreṇi), lokapratara (or jagatpratara), and loka (Jain, Mahendra 1999, sūtra 3.38, pp. 206-208; for details also see Mukhtara and Patni 1975, vv. 12 and 92-112, pp. 13 and 86-109).
(number-measure) is alaukika ganita if we confine ourselves to samkhyapramäna alone, and all of the four lokottara mānas (post-worldly measures) are alaukika ganita if our concern is both samkhyāpramāna and its area of application.
5. DISCUSSION
5.1. No directive, except the literal meaning of laukika ganita and lokottara ganita, has been issued by the Jainas, which shall help us to determine what mathematical thought will be placed in the category of laukika ganita and what one in the category of lokottara ganita. This intricacy will get illustrated if the following example is paid attention. The formula,
This twenty one-fold number system seems to have been said alaukika ganita by Todaramala (1720-1767 CE). Almost one century
earlier than him it was called lokottara ginative), for finding the volume of a sphere (post-worldly reckoning) by Hemaraja in his Ganitasära (Jain, Anupam 1988b, v. 4, p. 56). The same seems to have been discussed in the Alaukika-ganita ("Non-worldly mathematics") of an anonymous author, of which copy is said to have been preserved in Pañcayati Mandira, Delhi (Jain, Anupam 1988a, p. 25). "Gommaṇasāra grantha mem upayogi alaukika ganita ki kucha samjñām kā khuläsä (Eng. Revelation of some terms, applied in the book Gommaṭasära, of nonworldly mathematics)", a write-up inserted just after the foreword (prāgnivedana) into the Gommaṭasara (Karmakanda) of Nemicandra, edited by Khubachand Jain, reads that samkhyapramāna (number-measure) of twenty one-fold and upamäpramäna (simile-measure) of eight kinds along with kṣetramana (spacemeasure) that contains pradeśa (indivisible part of space (ākāśa)), kālamāna (time-measure) that contains samaya (indivisible part of time), and bhāvamāna (thought-measure) that contains avibhāgapraticcheda (indivisible correspondingsection of omniscience (kevalajñāna)) pertain to alaukika ganita (Jain, Khubchanda 1986/1913, pp. 6-11). On the basis of the above facts, the present author is of the opinion that samkhyāpramāna
whose diameter is d, referred to by Mahāvīra (c. 850 CE) in the Ganita-sara-sangraha (Padmavathamma, 2000, v. 8.28%, pp. 612-613) is also found in the Trilokasara of Nemicandra (c. 981), that too employed in the process of finding the first asamkhyāta (innumerate) (Mukhtara and Patni 1975, v. 19 first hemistich, p. 25). Theorization of the school into the two classes does not put any hurdle to accept that the formula is a content of the treatises of the both classes. On the other hand, it is very difficult to explain whether the formula is laukika or lokottara if the area of its application is taken into consideration.
12 For understanding this system it is suggested to read Singh, Navjyoti 1991, pp. 209-232.
Laukika ganita and lokottara ganita are the two divisions of mathematics in the school while the canonical class and the exclusive class are the two divisions of the school. The canonical class does not stand for lokottara ganita although most of the latter are the contents of the treatises belonging to the former. Similar is the case of the exclusive class and laukika ganita.
5.2. Though the list of the mathematicians of the exclusive class is smaller than that of the canonical
