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.