Verse 25 · On the Eclipses of the Moon. · Surya Siddhanta · siva.sh

siva.sh/surya-siddhanta/4/25

राशित्रययुताद् ग्राह्यात् क्रान्त्यम्शैर् दिक्समैर् युताः। भेदे +अन्तराज् ज्या वलना सप्तत्यङ्गुलभाजिता॥

sanskrit

From the place of the (said) body increased by 3 signs find the declination, (which is called Áyana or solstitial valaná). Find the sum or difference of the degrees of this declination and those of the latitudinal valaná, when those valanas are of the same name or of contrary names: (the result is called sphuṭa or true valaná). The sine of the true valaná divided by 70 gives the valaná in digits.[2] [2. In the projection of eclipses, after drawing the disc of the body to be eclipsed, the north and south and the east and west lines, which lines will of course represent the circle of position passing through the body (supposed on the ecliptic) and the secondary to that circle at the given place, to find the direction of the line representing the ecliptic in the disc of the body on which the knowledge of the exact directions of the phases of the eclipses depends, it is necessary to know the angle formed by the said secondary and the ecliptic. This angle or that are of a great circle, 90° from the place of the body which is intercepted between the said secondary or the prime vertical and the ecliptic is called the valana or variation (of the ecliptic). And as it is very difficult to find this arc at once, it is divided into two parts of which the one is that portion of the great circle (90° from the place of the body) which is intercepted between the Prime vertical and the Equinoctial and the other is that portion of the same circle which is intercepted between the Equinoctial and the ecliptic; these two portions are called the Áksha valana and the Áyana-valana respectively. The Áksha valana is called the north or south according as the Equinoctial circle meets the great circle (90° from the place of the body) on the north or south of the prime vertical eastward of the body; and it is evident from this that on the northern latitudes when the body is in the eastern or western hemisphere the Áksha valana will be the north or south respectively. And the Áyana-valana is called the north or south according as the ecliptic meets the said great circle on the north or south of the Equinoctial to the east of the body, and hence it is evident that when the declination of the body's place increased by 8 signs is north or south the Áyana-valana will be the north or south respectively. From the sum of these valanas when they are of the same name or from the difference between them when of contrary names the arc which is intercepted between the prime vertical and the ecliptic is found and hence it will be north or south according as the ecliptic meets the said great circle on the north or south of the prime vertical eastward of the body and it is sometimes called the spashta or rectified valana. An image should appear at this position in the text. To use the entire page scan as a placeholder, edit this page and replace "{{missing image}}" with "{{raw image।English translation of the Surya Siddhanta and the Siddhanta Siromani by Sastri, 1861.djvu/56}}". Otherwise, if you are able to provide the image then please do so. For guidance, see Wikisource:Image guidelines and Help:Adding images. Let A {\displaystyle A} be the place of the body; B G C L {\displaystyle BGCL} the great circle 90° from it; B A G {\displaystyle BAG} the eoliptio; D E F {\displaystyle DEF} the Equinoctial; E {\displaystyle E} the Equinoctial point; G H L {\displaystyle GHL} the prime vertical; H {\displaystyle H} the intersecting point of the prime vertical and the Equinoctial, and hence the east or west point of the Horizon and therefore G H {\displaystyle GH} equivalent to the zenith distance in the prime vertical. Then the arc G D {\displaystyle GD} = the Áksha valana, D B {\displaystyle DB} = the Áyana-valana, and G B {\displaystyle GB} = the spashta or rectified valana. These arcs can be found as follows, Let L {\displaystyle L} = latitude of the place, n {\displaystyle n} = the zenith distance in the prime vertical, l {\displaystyle l} = the longitude of the body, e {\displaystyle e} = the obliquity of the eoliptio, d {\displaystyle d} = the declination of the body, x {\displaystyle x} = the Áksha valana, {\displaystyle y} = the Áyana-valana, and z {\displaystyle z} = the rectified valana. Then in the spherical triangle D H G {\displaystyle DHG} sin ⁡ G D H : sin ⁡ D H G = sin ⁡ G H : sin ⁡ G D {\displaystyle \sin GDH:\sin DHG=\sin GH:\sin GD}: here, sin ⁡ G D H = sin ⁡ B D E = cos ⁡ d {\displaystyle \sin GDH=\sin BDE=\cos d}, sin ⁡ D H G = sin ⁡ L {\displaystyle \sin DHG=\sin L}; and sin ⁡ G H = sin ⁡ n {\displaystyle \sin GH=\sin n}, ∴ cos ⁡ d : sin ⁡ L = sin ⁡ n : sin ⁡ x {\displaystyle \therefore \cos d:\sin L=\sin n:\sin x}. ∴ sin ⁡ x {\displaystyle \therefore \sin x} or the sine of the Áksha valana = sin ⁡ L sin ⁡ n cos ⁡ d {\displaystyle {\frac {\sin L\sin n}{\cos d}}}in which the Radius is used for cos ⁡ d {\displaystyle \cos d} in the text. This valana is called north or south according as the point D be north or south of the point G. And in the triangle D E B {\displaystyle DEB} sin ⁡ B D E : sin ⁡ B E D = sin ⁡ B E : sin ⁡ D B {\displaystyle \sin BDE:\sin BED=\sin BE:\sin DB}, or cos ⁡ d : sin ⁡ e = cos ⁡ l : sin ⁡ y {\displaystyle \cos d:\sin e=\cos l:\sin y} ∴ sin ⁡ y {\displaystyle \therefore \sin y} or the sine of the Áyana valana = sin ⁡ e × cos ⁡ l cos ⁡ d {\displaystyle {\frac {\sin e\times \cos l}{\cos d}}}in which the Radius is used for cos ⁡ d {\displaystyle \cos d} in the text. This valana is called north or south; according as the point B be north or south of the point D. And the rectified valana G B = G D + D B {\displaystyle GB=GD+DB}, when the point D lies between the points G and B, but if the point D be beyond them, the rectified valana will be equal to the difference between the Aksha and Ayana valanas. This also is called north or south as the point B be north or south of the point G. To mark the sine of the spashta valana in the projection of the eclipse it is reduced to the circle whose radius is 49 digits in the text. i. e. R : s i n x {\displaystyle R:sinx} = 49: reduced sine of the valana: ∴ reduced sine of the valana = 49 sin ⁡ x R = 49 sin ⁡ x 3438 = sin ⁡ x 70 {\displaystyle {\frac {49\sin x}{R}}={\frac {49\sin x}{3438}}={\frac {\sin x}{70}}} This reduced sine in digits is denominated the valaná in the text. B. D.]

english translation

rAzitrayayutAd grAhyAt krAntyamzair diksamair yutAH। bhede +antarAj jyA valanA saptatyaGgulabhAjitA॥

hk transliteration by Sanscript

siva.sh/surya-siddhanta/4/25

राशित्रययुताद् ग्राह्यात् क्रान्त्यम्शैर् दिक्समैर् युताः। भेदे +अन्तराज् ज्या वलना सप्तत्यङ्गुलभाजिता॥

sanskrit

From the place of the (said) body increased by 3 signs find the declination, (which is called Áyana or solstitial valaná). Find the sum or difference of the degrees of this declination and those of the latitudinal valaná, when those valanas are of the same name or of contrary names: (the result is called sphuṭa or true valaná). The sine of the true valaná divided by 70 gives the valaná in digits.[2] [2. In the projection of eclipses, after drawing the disc of the body to be eclipsed, the north and south and the east and west lines, which lines will of course represent the circle of position passing through the body (supposed on the ecliptic) and the secondary to that circle at the given place, to find the direction of the line representing the ecliptic in the disc of the body on which the knowledge of the exact directions of the phases of the eclipses depends, it is necessary to know the angle formed by the said secondary and the ecliptic. This angle or that are of a great circle, 90° from the place of the body which is intercepted between the said secondary or the prime vertical and the ecliptic is called the valana or variation (of the ecliptic). And as it is very difficult to find this arc at once, it is divided into two parts of which the one is that portion of the great circle (90° from the place of the body) which is intercepted between the Prime vertical and the Equinoctial and the other is that portion of the same circle which is intercepted between the Equinoctial and the ecliptic; these two portions are called the Áksha valana and the Áyana-valana respectively. The Áksha valana is called the north or south according as the Equinoctial circle meets the great circle (90° from the place of the body) on the north or south of the prime vertical eastward of the body; and it is evident from this that on the northern latitudes when the body is in the eastern or western hemisphere the Áksha valana will be the north or south respectively. And the Áyana-valana is called the north or south according as the ecliptic meets the said great circle on the north or south of the Equinoctial to the east of the body, and hence it is evident that when the declination of the body's place increased by 8 signs is north or south the Áyana-valana will be the north or south respectively. From the sum of these valanas when they are of the same name or from the difference between them when of contrary names the arc which is intercepted between the prime vertical and the ecliptic is found and hence it will be north or south according as the ecliptic meets the said great circle on the north or south of the prime vertical eastward of the body and it is sometimes called the spashta or rectified valana. An image should appear at this position in the text. To use the entire page scan as a placeholder, edit this page and replace "{{missing image}}" with "{{raw image।English translation of the Surya Siddhanta and the Siddhanta Siromani by Sastri, 1861.djvu/56}}". Otherwise, if you are able to provide the image then please do so. For guidance, see Wikisource:Image guidelines and Help:Adding images. Let A {\displaystyle A} be the place of the body; B G C L {\displaystyle BGCL} the great circle 90° from it; B A G {\displaystyle BAG} the eoliptio; D E F {\displaystyle DEF} the Equinoctial; E {\displaystyle E} the Equinoctial point; G H L {\displaystyle GHL} the prime vertical; H {\displaystyle H} the intersecting point of the prime vertical and the Equinoctial, and hence the east or west point of the Horizon and therefore G H {\displaystyle GH} equivalent to the zenith distance in the prime vertical. Then the arc G D {\displaystyle GD} = the Áksha valana, D B {\displaystyle DB} = the Áyana-valana, and G B {\displaystyle GB} = the spashta or rectified valana. These arcs can be found as follows, Let L {\displaystyle L} = latitude of the place, n {\displaystyle n} = the zenith distance in the prime vertical, l {\displaystyle l} = the longitude of the body, e {\displaystyle e} = the obliquity of the eoliptio, d {\displaystyle d} = the declination of the body, x {\displaystyle x} = the Áksha valana, {\displaystyle y} = the Áyana-valana, and z {\displaystyle z} = the rectified valana. Then in the spherical triangle D H G {\displaystyle DHG} sin ⁡ G D H : sin ⁡ D H G = sin ⁡ G H : sin ⁡ G D {\displaystyle \sin GDH:\sin DHG=\sin GH:\sin GD}: here, sin ⁡ G D H = sin ⁡ B D E = cos ⁡ d {\displaystyle \sin GDH=\sin BDE=\cos d}, sin ⁡ D H G = sin ⁡ L {\displaystyle \sin DHG=\sin L}; and sin ⁡ G H = sin ⁡ n {\displaystyle \sin GH=\sin n}, ∴ cos ⁡ d : sin ⁡ L = sin ⁡ n : sin ⁡ x {\displaystyle \therefore \cos d:\sin L=\sin n:\sin x}. ∴ sin ⁡ x {\displaystyle \therefore \sin x} or the sine of the Áksha valana = sin ⁡ L sin ⁡ n cos ⁡ d {\displaystyle {\frac {\sin L\sin n}{\cos d}}}in which the Radius is used for cos ⁡ d {\displaystyle \cos d} in the text. This valana is called north or south according as the point D be north or south of the point G. And in the triangle D E B {\displaystyle DEB} sin ⁡ B D E : sin ⁡ B E D = sin ⁡ B E : sin ⁡ D B {\displaystyle \sin BDE:\sin BED=\sin BE:\sin DB}, or cos ⁡ d : sin ⁡ e = cos ⁡ l : sin ⁡ y {\displaystyle \cos d:\sin e=\cos l:\sin y} ∴ sin ⁡ y {\displaystyle \therefore \sin y} or the sine of the Áyana valana = sin ⁡ e × cos ⁡ l cos ⁡ d {\displaystyle {\frac {\sin e\times \cos l}{\cos d}}}in which the Radius is used for cos ⁡ d {\displaystyle \cos d} in the text. This valana is called north or south; according as the point B be north or south of the point D. And the rectified valana G B = G D + D B {\displaystyle GB=GD+DB}, when the point D lies between the points G and B, but if the point D be beyond them, the rectified valana will be equal to the difference between the Aksha and Ayana valanas. This also is called north or south as the point B be north or south of the point G. To mark the sine of the spashta valana in the projection of the eclipse it is reduced to the circle whose radius is 49 digits in the text. i. e. R : s i n x {\displaystyle R:sinx} = 49: reduced sine of the valana: ∴ reduced sine of the valana = 49 sin ⁡ x R = 49 sin ⁡ x 3438 = sin ⁡ x 70 {\displaystyle {\frac {49\sin x}{R}}={\frac {49\sin x}{3438}}={\frac {\sin x}{70}}} This reduced sine in digits is denominated the valaná in the text. B. D.]

english translation

rAzitrayayutAd grAhyAt krAntyamzair diksamair yutAH। bhede +antarAj jyA valanA saptatyaGgulabhAjitA॥

hk transliteration by Sanscript