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

PLANE FIGURES 91
“The mathematician who wishes to tell the area or the altitudes
of a quadrilateral without knowing a diagonal, is either a fool or a
blunderer.”¹
Rule for finding an approximate value of the square root of
a non-square number :
118. Of the non-square number as multiplied by some
large square number, extract the square root, neglecting the
remainder ; and divide that by the square root of the multi-
plier.²
That is to say, multiply the non-square number by some large
square number (chosen at pleasure) ; then find out the square root of
that product, neglecting the remainder ; and then divide that square
root by the square root of (the square number, which was taken as)
the multiplier.
We illustrate this rule by the following two examples :
Ex. 1. Show that √10 = 3⅐ approximately.
Choose 49 for the square number. Then multiplying 10 by 49 , we
get 490. Extracting the square root of 490, we get 22 as the square root
and 6 as the remainder. Neglecting the remainder, and dividing 22 by
the square root of 49, i.e., by 7 , we get 3⅐ .
Ex. 2. Find the square root of 3 , correct to 2 decimal places.
Choose 1000000 for the square number. Then multipling 3 by
1000000 , we get 3000000 . Extracting its square root we get 1732 as
the square root and 175 as the remainder. Neglecting the remainder, and
dividing 1732 by the square root of 1000000, i.e., by 1000, we get 1·732.
Hence the square root of 3 correct to 2 decimal places is 1·73 .
¹ कर्णज्ञानेन विना चतुरश्रे लम्बकं फलं यद्वा ।
वक्तुं वाञ्छति गणको योऽसौ मूर्खः पिशाचो वा ॥ (MS, xv. 70)
Also see L (ASS), pp. 159–160.
² For similar rules, see MSi, xv. 55, or Siśe, xiii. 36 ; L (ASS),
p. 132, vs. 140 ; GK, II, p. 33, lines 4-7.