Surya Siddhanta
तत्त्रिज्यावर्गविश्लेषान् मूलम् दृग्ज्याभिधीयते । स्वशङ्कुना विभज्याप्ते दृक्त्रिज्ये द्वादशाहते॥
Multiply the (said) sine of the zenith distance and the radius by 12 and divide the products by the Koṇa-śanku (above found); the quotients will be the shadow (of the gnomon) and its hypothenuse (respectively, when the Sun will come on an intermediate vertical) at the proper place and time. Then, 12 : e = x : e 12 x = {\displaystyle 12:e=x:{\frac {e}{12}}x=} Śaukutala (as shown in the note on 7th Śloka)); and since it is manifest from the same note that the Śaukutala applied with the sine of amplitude by addition or subtraction according aa the Sun is south or north of the equinoctial, becomes Bhuja (i. e. the sine of the Sun's distance from the prime vertical), ∴ e 12 x ± a = {\displaystyle \therefore {\frac {e}{12}}x\pm a=} Bhuja; but when the Sun is N. E., N. W., S. E., or S. W., it is equidistant from the prime vertical and the meridian. Therefore the hypothenuse of a right-angled triangle, of which one side is the Bhuja and the other equal to it, is the sine of the zenith distance. ∴ h y p . ) 2 = 2 ( e 12 x ± a ) 2 = e 2 72 x ± a e 3 x + 2 a 2 {\displaystyle \therefore \mathrm {hyp.} )^{2}=2({\frac {e}{12}}x\pm a)^{2}={\frac {e^{2}}{72}}x\pm {\frac {ae}{3}}x+2a^{2}}. Now, since the square of the sine of the zenith distance added to the square of the sine of the altitude is equal to the square of the radius, ∴ x 2 + e 2 72 x 2 ± a e 3 x + 2 a 2 = R 2 {\displaystyle \therefore x^{2}+{\frac {e^{2}}{72}}x^{2}\pm {\frac {ae}{3}}x+2a^{2}=R^{2}}; or ( e 2 + 72 ) x 2 ± 24 a e x = 72 R 2 − 144 a 2 {\displaystyle (e^{2}+72)x^{2}\pm 24aex=72R^{2}-144a^{2}}; ∴ x 2 ± 24 a e e 2 + 72 x = 72 R 2 − 144 a 2 e 2 + 72 = 144 ( 1 2 R 2 − a 2 ) e 2 + 72 {\displaystyle \therefore x^{2}\pm {\frac {24ae}{e^{2}+72}}x={\frac {72R^{2}-144a^{2}}{e^{2}+72}}={\frac {144({\frac {1}{2}}R^{2}-a^{2})}{e^{2}+72}}}. Now, in the foregoing equation it will be observed that the value of the side containing the known quantities is what has been already spoken of under the name of Karaṇí, and that the half of the co-efficient of x {\displaystyle x} is what has been already spoken of under the name of Phala, ∴ x 2 ± 2 f x = k {\displaystyle \therefore x^{2}\pm 2fx=k}, which gives x = f 2 + k ± f {\displaystyle \textstyle x={\sqrt {f^{2}+k}}\pm f}. B. D.
english translation
tattrijyAvargavizleSAn mUlam dRgjyAbhidhIyate । svazaGkunA vibhajyApte dRktrijye dvAdazAhate॥
hk transliteration by Sanscript