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tively Eka samyoga, dvika samyoga, trika samyoga etc. Although some methods of finding out the permutations and combinations of certain things were known by the time the Bhagwati-sutra was written, yet the definite formulation of any mathematical rule is traceable only from the time of the Anuyogadwara-sutra (about 100 B.C.).
In Sushrutasl medicinal work (about 600 B.C.), it is stated that out of six different rasas (viz. sweet, bitter, sour, saltish, hot, astrigent) 63 combinations can be obtained by taking the rasas one at a time, two at a time, three at a time etc. This gives the respective number of combinations 6, 15, 20, 15, 6 and 1, which obviously sum upto 63.
There are similar calculations of the groups that can be found out of the different instrument of senses (karanas), or of the selections that can be made out of a number of males, females and eunchs, of the permutations and combinations in various other things. In all the cases the results are given as could be obtained with the help of the above general formulae given by Mahavira (850 A. D.).
Thus the word vikalpa for combinations is traceable before the advent of Jainism. Although the notion of permutations and combinations is traceable in India even prior to Jainism, yet the credit goes to the early Jainas for the simple two reasons-firstly for treating the subject as a separate topic in mathematics and secondly for working out the general formulae by the time of Mahavira (850 A. D.).
12. CONCLUDING REMARKS
The original mathematical works of the Jains have not come to light and a considerable amount of search and research about the Jaina manuscripts is, therefore necessary. In fact, there are three main difficulties in the study of ancient Indian mathematics viz.
1. the difficulty of getting original works, some of which are not available in India, 2. the difficulty of the language--ancient mathematical works are in Sanskrit, and most of them
are in poetry and not in prose, which makes it all the more difficult to understand them
and lastly 3. the writers of original scientific treatises are generally very brief.
Their aim was to just indicate the general outline of procedure and to leave the details to be worked out by the interested worker in the field. Some writers have given bare rules without demonstrations or examples, and the whole thing is so condensed that it is often difficult to interpret their meaning by one who is not a mathematician and a Sanskritist at the same time.
1. See Sushruta Samihita, Chap. LXIII.
Rasabheda Vikalpadhaya. 2. See Jambudvipa Prajnapti. XX, Sutras 4, 5.
Anuyogadwara-Sutra. Sutras 76, 96, 126. AKNOWLEDGEMENTS : The author has the blessing of H. H. Rashtra Sant Muni (Dr.) Nagraj Ji D. Litt., and also of reverend Muni Shri Mahandra Kumar Ji. He is grateful to Professor D. S. Kothari, Chancellar Jawahar Lal Nehru University, New Delhi, nor his altruistic concern in giving kind and valuable suggestions thereby adding to the quality of the present venture. The author is indebted to Dr. Raghu Nath Sharma, Department of Sanskrit, University of Delhi, for some discussions concerning the above paper.
आचार्यरत्न श्री देशभूषण जी महाराज अभिनन्दन ग्रन्थ
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