Surya Siddhanta
Progress:73.0%
भक्ता फलाख्यम् तद्वर्गसम्युक्तकरणीपदम् । फलेन हीनसम्युक्तम् दक्षिणोत्तरगोलयोः॥
The square-root, (just found), diminished or increased by the Phala according as the Sun is south or north of the equinoctial, becomes the Koṇa-śanku[1] or the sine of altitude of the Sun when situated at an intermediate vertical (intersecting the Horizon at the N. E. and S. W. or N. W. and S. E. points). Then a R h : p {\displaystyle {\frac {aR}{h}}:p} (the sine of the Sun's altitude when he is at the prime vertical) = cos l : sin l {\displaystyle \cos l:\sin l} = e {\displaystyle e} (equinoctial shadow): 12; ∴ p = 12 a R h e {\displaystyle \therefore p={\frac {12aR}{he}}}; and ∴ p : R = 12 : x {\displaystyle \therefore p:R=12:x} (the hypothenuse of the Sun's shadow when he reaches the prime vertical): ∴ x = 12 R p = 12 R × h e 12 a R = h e a {\displaystyle \therefore x={\frac {12R}{p}}=12R\times {\frac {he}{12aR}}={\frac {he}{a}}}; supposing the Sun's declination to undergo no change during the day. [1. This is demonstrated thus. Let e {\displaystyle e} = the equinoctial shadow, a {\displaystyle a} = the sine of amplitude, k {\displaystyle k} = the Karaṇí f {\displaystyle f} = the Phala, and x {\displaystyle x} = the Koṇa-śanku.]
english translation
bhaktA phalAkhyam tadvargasamyuktakaraNIpadam । phalena hInasamyuktam dakSiNottaragolayoH॥
hk transliteration by Sanscriptयाम्ययोर् विदिशोः शङ्कुर् एवम् याम्योत्तरे रवौ । परिभ्रमति शङ्कोस् तु शङ्कुर् उत्तरयोस् तु सः॥
If the sun be south of the prime vertical, then the Koṇa-śanku will be south-east or south-west, but if he be north of it, then it will be north-east or north-west. The square-root of the difference between the square of the Koṇa-śanku and that of the radius, is called the Drịgjyá or the sine of the zenith distance.
english translation
yAmyayor vidizoH zaGkur evam yAmyottare ravau । paribhramati zaGkos tu zaGkur uttarayos tu saH॥
hk transliteration by Sanscriptतत्त्रिज्यावर्गविश्लेषान् 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 Sanscriptछायाकर्णौ तु कोणेषु यथास्वम् देशकालयोः । त्रिज्योदक्चरजायुक्ता याम्यायाम् तद्विवर्जिता ॥
Add or subtract the sine of the ascensional difference to or from the radius according as the Sun is in northern or southern hemisphere. The result is called the Antyá. From the Antyá subtract the versed sine of the time from noon (reduced to degrees); Multiply the remainder by the cosine of the declination.
english translation
chAyAkarNau tu koNeSu yathAsvam dezakAlayoH । trijyodakcarajAyuktA yAmyAyAm tadvivarjitA ॥
hk transliteration by Sanscriptअन्त्या नतोत्क्रमज्योना स्वहोरात्रार्धसङ्गुणा । त्रिज्याभक्ता भवेच् छेदो लम्बज्याघ्नो +अथ भाजितः॥
Divide the product, (thus found), by the radius; the quotient is called the chheda; the chheda multiplied by the cosine of latitude and divided by the radius becomes the Śanku[1] or the sine of the Sun's altitude (at the given time). [1. This will be manifest thus. Let l = {\displaystyle l=} latitude of the place north of the equator; d = {\displaystyle d=} the Sun's declination; a = {\displaystyle a=} the ascensional difference, t = {\displaystyle t=} the time from noon in degrees, and x = {\displaystyle x=} the Sun's altitude. Then we have the equation which is very common; sin x = cos t . cos l . cos d ± R . sin l . sin d . R 2 {\displaystyle \sin {x}={\frac {\cos t.\cos l.\cos d\pm R.\sin l.\sin d.}{R^{2}}}}; = ( cos t . ± tan l . tan d R cos l . cos d R {\displaystyle ={\frac {(\cos t.\pm {\frac {\tan l.\tan d}{R}}\cos l.\cos d}{R}}}; = ( cos t ± sin a ) cos l . cos d R 2 {\displaystyle ={\frac {(\cos t\pm \sin a)\cos l.\cos d}{R^{2}}}}; or = ( R + sin a − vers t ) cos d R . cos d R {\displaystyle ={\frac {(R+\sin a-\operatorname {vers} t)\cos d}{R}}.{\frac {\cos d}{R}}}. It is to be observed here, that when the latitude of the place is north, the sin a {\displaystyle \sin a} becomes plus or minus according as the declination is north or south. B. D.]
english translation
antyA natotkramajyonA svahorAtrArdhasaGguNA । trijyAbhaktA bhavec chedo lambajyAghno +atha bhAjitaH॥
hk transliteration by Sanscript