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

PTOLEMY'S ANALEMMA 135
meridian, NAMH the horizon, Z, the zenith and S the Sun.
Here the celestial longitudes of C, S and A are taken to be
known; hence ZC and CH are also known.
Now take ZCS for the triangle and HMA to be the trans-
versal ; we then have by Menelaus's theorem.
(sin ZH / sin HC) × (sin CA / sin AS) × (sin SM / sin MZ) = 1
or sin SM = (cos CZ × sin AS) / sin CA
It is thus clear that Ptolemy had no direct method for
connecting the Sun's altitude and the hour-angle. This method
is workable for the problem “given time, find the altitude” but
is not workable in the converse problem ; besides, the calcula-
tion of the longitudes of A and C is very cumbrous.
Again, when EA has been found out, taking ZHM for the
triangle and CSA for the transversal, we get,
(sin HA / sin Am) × (sin MS / sin SZ) × (sin ZC / sin CH) = 1, whence and thence HM,
the azimuth can be found. The method is here also cumbrous,
there being no direct connection between altitude and azimuth ;
besides the time-element is not avoided.
The Analemma of Ptolemy
and the Indian Method.
When the Sun's declination is zero and his hour-angle, is
H, Zeuthen¹ following the method of the ‘Analemma’ of
Ptolemy, as explained by Braunmühl² has deduced the following
equations :
(1) cos Z = cos H. cos ϕ
(2) tan α = tan H / sin ϕ
To these two, Heath following Braunmühl, adds
(3) ³tanZQ = tan H / cos ϕ
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1. Heath, Greek Mathematics, Vol. II, pp. 290-91.
   Zeauthen, Bibliotheca Mathematica, 13, 1900, pp. 23-27.
2. Braunmuhl , ibid, pp. 12-13.
3. The Indian form of this equatiom is R Sin ZQ = (R Sin H × R) / [√(R² - R²cos²H × R² Sin²ϕ) / R]
   Bhāskara's, Golādhyāya, Com. on VIII, 67.