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

MIXTURES. SPECIAL PROBLEMS 65
ders as the divisors of the pada as well as the (last) visible
quantity ; to the second quotient also add the first visible
quantity. After that apply the previous rule (starting with
‘add the square of half the first quotient’).¹
This rule applies to problems which reduce to the solution of the
quadratic equation of the type
(1 - a/b) (1 - c/d) (1 - e/f) (x - d₁) - p√x = d₂ ,
where a/b, c/d, e/f are the fractional multiples of remainders, p is the pada,
d₁, d₂ the first and last visible quantities, and √x denotes the positive
square root of x, as before.
According to the above rule, this equation should be first trans-
formed into the form
x - {p / [(1 - a/b)(1 - c/d)(1 - e/f)]} √x = {d₂ / [(1 - a/b)(1 - c/d)(1 - e/f)]} + d₁ ,
and then solved by the previous rule.
Ex. 101. After giving away one (out of a certain
number), then one-sixth of what remains, then one-fourth of
what remains after that, then one-third of what remains after
that, and then the square root of the original number, the
residue left is 5. (What is that number ?)²
Rule of inversion :
78. (Proceeding from the visible quantity backwards,
make) addition subtraction, subtraction addition, multi-
plication division, division multiplication, square square-
root, and square-root square : this is stated to be the method
of inversion.³
Ex. 102. Say what (is that number which) being multi-
plied by 5/2, then divided by 3, then squared, then increased
by 9, then reduced to its square root, and then diminished by
1, becomes 4.⁴
¹ Cf. GSS, iv. 47 ;GT, p. 54, lines 21-24.
² For other examples see GSS, iv. 48, 49, 50; GT, p. 55, lines 7-10;
p. 56, lines 8-11 ; p. 5 , lines 19-22.
³ Cf. Ā, ii. 28; BrSpSi, xviii. 14(17); GSS, vi. 286; MSi, xv. 23;
GT, p. 65, vs. 83; SiŚe, xiii. 13; L (ASS), p. 42, vs. 48; GK, I, p. 46,
lines 13-16.
⁴ For similar examples see GSS, vi 287; GT, p. 66, vs. 84; p. 67,
vs. 85; L (ASS), p. 43, vs. 50.