A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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
and tana) =
or
sin h = (cos 8' cos t — sin 8' tan 0) cos 0,
k
cos
cos 8' sin t
r + (cos 8' cos t — sin 8' tan 6) sin 6
cos 8 / sin £
sin 8' cos + 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
292 TRIGONOMETRY
seeing that Diodorus wrote his Analcmma in
the next century.
The other alternative source for Hipparchus's spherical
trigonometry is the Menelaus-theorem applied to the sphere,
on which alone Ptolemy, as we have seen, relies in his
Syntaxis. In any case the Table of Chords or Sines was in
full use in Hipparchus's works, for it is presupposed by either
method.
The Planisphaerium.
With the Analemma of Ptolemy is associated another
work of somewhat similar content, the Planisphaerium.
This again has only survived in a Latin translation from an
Arabic version made by one Maslama b. Ahmad al-Majriti,
Cordova (born probably at Madrid, died 1007/8) ;
of
the translation
is now found to be, not by Rudolph of Bruges, but by
'Hermannus Secundus', whose pupil Rudolph was; it was
first published at Basel in 1536, and again edited, with commentary,
by Commandinus (Venice, 1558). It has been
re-edited from the manuscripts by Heiberg in vol. ii. of his
text of Ptolemy. The book is an explanation of the system
of projection known as stereographic, by which points on the
heavenly sphere are represented on the plane of the equator
by projection from one point, a pole ;
Ptolemy naturally takes
the south pole as centre of projection, as it is th£ northern
hemisphere which he is concerned to represent on a plane.
Ptolemy is aware that the projections of all circles on the
sphere (great circles— other than those through the poles
which project into straight lines—and small circles either
parallel or not parallel to the equator) are likewise circles.
It is curious, however, that he does not give any general
proof of the fact, but is content to prove it of particular
circles, such as the ecliptic, the horizon, &c. This is remarkable,
because it is easy to show that, if a cone be described
with the pole as vertex and passing through any circle on the
sphere, i.e. a circular cone, in general oblique, with that circle
as base, the section of the cone by the plane of the equator
satisfies the criterion found for the subcontrary sections ' ' by
Apollonius at the beginning of his Conies, and is therefore a
circle. The fact that the method of stereographic projection is
so easily connected with the property of subcontrary sections