Contribution of Ancient Jaina Mathematicians

Author(s): B S Jain
Publisher: Z_Deshbhushanji_Maharaj_Abhinandan_Granth_012045.pdf

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Thus the total population of the world is 264 x 282=296
=79, 228, 162, 514, 264, 337, 593, 543, 950, 336. This figure has twenty-nine digits and is divisible (by two) ninety-six times. All the above is conclusive to establish that the early Jainas knew the law of indices viz :
a" x a= a**
(a")" = amon where m, n may be integral or fractional.
11. Permutations and Combinations
It is a very important topic in mathematics. Its earlist use, as one of the several topics for discussion in mathematics, is traceable only from the time of the Jaina canon Sthananga-sutral (before 300 B.C.), The general formulae were given later by Mahavira (850 A.D.) in his GSS. In fact Mahavira? is the world's first mathemat!cian to give the general formulae
_n(n-1) (n--2) ..................(n-r+1)_ 1.2.3............................
r
r
n! ! (n-r)!
"Po=n(n-1) (n-2)...........................(n-r+1).
for the total number of combinations of n things taken r at a time and for the total number of permutations of the n things taken r at a time respectively.
In Anuyogadwara-sutra (about 100 B.C.), the number of permutations of six things is given by 1 X2 X3 X4 X 5 X 6. Silanka (9th cent. A.D.), the Jain commentator of the Anuyogadwara-sutra, has reproduced. from some mathematical texts, the rules for the permutations and combinations. The rule for determining the total number of transpositions that can be made with a specific number of things (bhede-samkhya-parijnanaya) is: "Beginning with unity upto the number of terms, multiply successively the (natural) numbers : That should be known as the result in the calculation of permutations and combinations (vikalpa-ganita)".
Thus the total number of permutations that can be made from r given different thing taken all at a time is
Pr = 1.2, 3. 4. 5. ..................... (r-1) = 1!. The other rules for finding the actual spread or representation (prastārānayanopaya) are: "The total number of permutations being divided by the last term, the quotient should be divided by the rest : They should be placed successively by the side of the initial term in the calculation of permutations and combinations."
Simple problems are stated in the Bhagwati-sūtras (300 B.C.). The corresponding Indian expressions used for the modern terms 'taken one at a time', 'taken two at a time', 'taken three at a time' etc. are respec
See Rule 747. 2. ,, GSS Chap. VI Rule 218, p. 94. 3. Rule 103, 115, 116 and others.
, Vide his Commentary on the Sutra.
Krtanga-sutra, Samaya dhyayana.
Anuyogadwara. Vers. 28. 5. See Sutra 314.
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