A history of Greek mathematics - Wilbourhall.org

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THE ANALEMMA OF PTOLEMY 291
or tan VG = tan SC cos SGV in the right-angled spherical
triangle SVG.
Thirdly,
tan QZ = tan Z Y = ^p = ^p
•
^r, = tan < -—
.
that is, 7 i^r-r = —— tj-^ , which is Menelaus, Sphaerica,
'
tan#if sin #¥
z
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 2oc described above the horizon by a star of
given declination #', by a procedure equivalent to the formula
cos a = tan 8' 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 8', Zeuthen has shown that they may be expressed by the
formulae
sin h = (cos 8' cos t — sin 8' tan 0) cos 0,
and tana) =
k
cos
cos 8' sin t
r + (cos 8' cos t — sin 8' tan 6) sin 6
or
sin 8' cos
cos 8 / sin £
+ cos 8* cos £ sin
The statement therefore of A. v. Braunmtihl 1 that the
Indians were the first to utilize the method of projection
contained in
the Analemma for actual trigonometrical calculations
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 application
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. 13, 14, 38-41.
U 2