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

MIXTURES. SPECIAL PROBLEMS 61
two as multiplied by 2 and divided by 5 went away
towards the north, and three (cows) are left. (What is the
numerical strength of the herd ?)¹
Here the fractions (of the herd of cows) that went away in the
various directions are
½, ¼, and ⅖(½ - ¼)
i.e., ½, ¼, and ⅒ ;
and the visible quantity (i e., the number of cows left) is 3. Hence the
numerical strength of the herd
= 3 / [1 - (½ + ¼ + ⅒)] , i.e., 20 cows.
Rule for solving problems involving remainders due to sub-
traction of square root, etc. (mūlādiśeṣoddeśaka) : ²
75. When the visible quantity stands near a square
root, multiply the visible quantity by 4, then increase that by
the square of the pada (i.e., the co-efficient of the square root
of the unknown), then extract the square-root of that and
increase that by the pada, and then take the square of half of
that ; (and when the visible quantity stands near a fraction)
divide the visible quantity by one minus the fraction.³
The first part of this rule relates to the solution of the quadratic
equation of the type
x - p√x = d,
where p is the pada, d the visible quantity (dṛśya), and √x the positive
square root of x. The solution is correctly stated as
x = [ {√(4d + p²) + p} / 2 ]²

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¹ For similar examples see GSS, iv. 23-27; GT, p. 46, lines 20-25;
L (ASS), p. 53, vs. 55.
² Mahāvīra and Śrīpati have called mūlādiśeṣa-jāti by the name
śeṣamūla-jāti.
³ Cf. GSS, iv. 40; GT, p. 48, lines 24-25.