292 
TRIGONOMETRY 
seeing that Diodorus wrote his Analemma in the next cen- of o 
tury. The other alternative source for Hipparchus’s spherical nius 
trigonometry is the Menelaus-theorem applied to the sphere, men 
on which alone Ptolemy, as we have seen, relies in his of C 
Syntaxis. In any case the Table of Chords or Sines was in attr 
full use in Hipparchus’s works, for it is presupposed by either Hip 
method. trea 
perl 
The Planisphaerimn. c ird 
With the Analemma of Ptolemy is associated another no ^ 
work of somewhat similar content, the Planisphaerium. J ec ^ ] 
This again has only survived in a Latin translation from an ( 
Arabic version made by one Maslama b. Ahmad al-Majriti, of ^ 
Cordova (born probably at Madrid, died 1007/8); the transla- °^ 1€ 
tion is now found to be, not by Rudolph of Bruges, but by mm 
‘Hermannus Secundus’, whose pupil Rudolph was; it was _G 
first published at Basel in 1536, and again edited, witli com- vlve 
mentary, by Commandinus (Venice, 1558). It lias been v 
re-edited from the manuscripts by Heiberg in vol. ii. of his lnsc 
text of Ptolemy. Tlie book is an explanation of the system ^79 
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 the northern P 
hemisphere which he is concerned to represent on a plane. late< 
Ptolemy is aware that the projections of all circles on the cent 
sphere (great circles—other than those through the poles how 
which project into straight lines—and small circles either wen 
parallel or not parallel to the equator) are likewise circles, Ptol 
It is curious, however, that he does not give any general cone 
proof of the fact, but is content to prove it of particular The 
circles, such as the ecliptic, the horizon, &c. This is remark- not 
able, because it is easy to show that, if a cone be described inco 
with the pole as vertex and passing through any circle on the clea: 
sphere, i. e. a circular cone, in general oblique, with that circle at t] 
as base, the section of the cone by the plane of the equator iron 
satisfies the criterion found for the ‘ subcontrary sections ’ by min 
Apollonius at the beginning of his Conics, and is therefore a x < 
circle. The fact that the method of stereographic projection is Skih 
so easily connected with the property of subcontrary sections nM