पाटीगणित (श्रीधराचार्य - प्राचीन संस्कृत टीका एवं प्रो. कृपाशंकर शुक्ल सविस्तार व्याख्या)

64 DETERMINATIONS
Rule for solving problems in which the visible quantity is the
remainder due to the subtraction of a fraction of the unknown
and also a multiple of the square root of the unknown
(bhâgamûlâgroddeśa) :¹
76. After having divided the pada (i.e., the co-efficient
of the square root of the unknown) and the visible quantity
(or ultimate remainder, agra) by one minus the fraction, add
the square of half the first quotient to the second quotient,
then take the square root of that, then increase that by half
the first quotient, and then multiply that by itself.²
This rule is meant to solve problems reducing to a quadratic
equation of the type
x - (a/b)x - p√x = d ,
where a/b is the 'fraction', p the 'pada', d the visible quantity, and √x
the positive square root of x.
The solution is correctly stated as
x = [ p / {2(1 - a/b)} + √{ (p / {2(1 - a/b)})² + d / (1 - a/b) } ]²
Ex. 100. One-third of a troop of monkeys together
with one-third of itself has gone to the tank ; the square root
(of the whole troop) is afflicted with thirst ; and the remain-
ing two monkeys are sitting under the mango tree. (What is
the number of monkeys in the troop ?)³
Rule for solving problems involving two visible quantities,
the square root of the original number, and remainders due
to subtraction of fractional multiples of remainders (ubhayâgra-
mûlaśeṣoddeśa):⁴
77. Take the continued product of units severally
diminished, as before, by the fractional multiples of remain-
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¹ Mahâvîra calls bhâgamûlâgra-jâti by the name mûla-jâti, Śrîpati by
the name mûlâgra-bhâga-jâti, and Nârâyaṇa by the name ṛṇâṁśavimûla-jâti.
² Cf. GSS, iv. 33; GT, p. 50 lines 27-28 (contd. on p. 51, lines 1-2).
Also see GK, I, p. 21, lines 2-6.
³ For other examples see GSS, iv. 34, 35, 36, 37, 38, 39; GT, p. 51,
lines, 11-14; p. 52, lines 16-19; p. 53, lines 17-20; GK, I, p. 23, lines 15-16,
18-19, and p. 24, lines 1-2.
⁴ Mahâvîra calls ubhayâgramûlaśeṣa-jâti by the name dviragraśeṣamûla-
jâti, and Śrîpati by the name ubhayâgradṛśya-jâti.