# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/३६१

एतत् पृष्ठम् परिष्कृतम् अस्ति
165
CHAPTER VI-MIXED PROBLEMS.

The rule for arriving at the square root of (an unknown) number as increased or diminished by a known number:-

279${\displaystyle {\tfrac {1}{2}}}$.[1] The known quantity which is given is first halved and (then) squared and then one is added (to it). The resulting quantity either when increased by the desired given quantity or when diminished by the (same) quantity yields the square root (exactly).

An example an illustration thereof.

280${\displaystyle {\tfrac {1}{2}}}$. Here is a number which, when increased by 10 or diminished by the same 10, yields an exact square root. Think out and tell me that number, O mathematician

The rule for arriving at the two required square quantities, with the aid of those required quantities as multiplied by a known number, and also with the aid of (the same known number as forming the value of) the square root of the difference (between these products):-

281${\displaystyle {\tfrac {1}{2}}}$.[2] The given number is increased by one; and the given number is also diminished by one. The resulting quantities when halved and then squared give rise to the two (required) quantities. Then if these be (separately) multiplied by the given quantity, the squrare root of the difference between these (products) becomes the given quantity.

An example in illustration thereof.

282${\displaystyle {\tfrac {1}{2}}}$-283. Two unknown squared quantities are multiplied by 71. The square root of the difference between these (two resulting products is also 71. O mathematician, if you know the process of calculation known as citra-kuṭṭīkāra, calculate and tell me what (those two unknown) quantities are.

279${\displaystyle {\tfrac {1}{2}}}$.^  This is merely a particular case of the rule given in stanza 275${\displaystyle {\tfrac {1}{2}}}$ wherein a is taken to be equal to b.

281${\displaystyle {\tfrac {1}{2}}}$.^  Algebraically, when the given number is d,${\displaystyle \left({\tfrac {d+1}{2}}\right)^{2}{\text{ and }}\left({\tfrac {d-1}{2}}\right)^{2}}$ are the required square quantities