THE ANALEMMA OF PTOLEMY 
291 
or tan VC = tan SC cos SGV in the right-angled spherical 
triangle SVC. 
Thirdly, 
tan QZ = tan Z'Y' = jy?jr, 
S'F' O'F' _ , 1 
0'T'~ tan t • cos ^; 
OT' 
that is, 7i—,, which is Menelaus, Sphaerica, 
tan SM sin BM 1 
III. 3, applied to the right-angled spherical triangles ZBQ, 
MBS with the angle B common. 
Zeuthen points out that later in the same treatise Ptolemy 
finds the arc .2 a described above the horizon by a star of 
given declination S', by a procedure equivalent to the formula 
cos a — tan S' tan 0, 
and this is the same formula which, as we have seen, 
Hipparchus must in effect have used in his Commentary on 
the Phaenomena of Eudoxus and Aratus. 
Lastly, with regard to the calculations of the height h and 
the azimuth co in the general case where the sun’s declination 
is S', Zeuthen has shown that they may be expressed by the 
formulae 
sin h = (cos S' cos t — sin S' tan 0) cos 0, 
and 
or 
tan co = ——A 
cos S' sin t 
sin S' 
COS 0 
+ (cos S' cos t — sin S' tan 0) sin 0 
cos S' sin t 
sin S' COS 0 + COS S' COS t sin 0 
The statement therefore of A. v. Braunmuhl 1 that the 
Indians were the first to utilize the method of projection 
contained in the Analemma for actual trigonometrical calcu 
lations with the help of the Table of Chords or Sines requires 
modification in so far as the Greeks at all events showed the 
way to such use of the figure. Whether the practical applica 
tion of the method of the Analemma for what is equivalent 
to the solution of spherical triangles goes back as far as 
Hipparchus is not certain; but it is quite likely that it does. 
1 Braunmuhl, i, pp. 18, 14, 38-41. 
U 2