ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

GREEK METHODS IN SPHERICAL ASTRONOMY 127 triangle ABC, the relations :-- (i) sin a=sin b sin A (ii) sin c=tan a cot A (iii) cos b=cos a cos c (iv) tan c=tan b cos A The above are some of the Napier's rules for a right-angled spherical triangle, deducible from Menelaus's theorem¹. They are generally sufficient in the case of such triangles. In any spherical triangle, however, this theorem of Menelaus does not in any single step lead to any of the equivalents of the time- altitude or altazimuth equations in spherical astronomy. The ancient Indian methods, though none of them are so highly finished as Menelaus's theorem, yet are not less powerful in tackling the problems that arise in astronomy in connection with the apparent diurnal motion of the heavens. The Greek or Ptolemaic method presents no further points of interest except in its application. We now proceed to illustrate the ancient Indian methods and shall refer to the Ptolemaic method as occasion arises. Ancient Indian Methods in Spherical Astronomy In the Indian methods there is no general rule to follow. It is by properties of similar right-angled triangles that a fairly complete set of accurate formulae are obtained. These right- angled plane triangles are classified under the names,—'Krānti- kṣetras' (triangles of declination) and 'Akṣa-kṣetras' (triangles of latitude). We consider the following problems :— Problem :— To find the time of rising on the equator of a length l, of arc of the ecliptic measured from the first point of Aries. Let ω be the obliquity of the ecliptic and R. A. the right ascension corresponding to the longitude l, and δ the correspond- ing declination. The Indian form of the equation is :

1. Three more can be deduced similarly, namely, (v) sin c=sin b sin C (vi) sin a=tan c cot C (vii) tan a=cos C tan b.