248 TRIGONOMETRY 
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circle from its pole is equal to the side of a square inscribed 
in the great circle and conversely (Props. 16, 17). Next come 
certain problems : To find a straight line equal to the diameter 
of any circular section or of the sphere itself (Props. 18, 19) ; 
to draw the great circle through any two given points on 
the surface (Prop. 20) ; to find the pole of any given circu 
lar section (Prop. 21). Prop. 22 applies Eucl. III. 3 to the 
sphere. 
Book II begins with a definition of circles on a sphere 
which touch one another ; this happens ‘ when the common 
section of the planes (of the circles) touches both circles ’. 
Another series of propositions follows, corresponding again 
to propositions in Eucl., Book III, for the circle. Parallel 
circular sections have the same poles, and conversely (Props. 
1, 2). Props. 3-5 relate to circles on the sphere touching 
one another and therefore having their poles on a great 
circle which also passes through the point of contact (cf. 
Eucl. III. 11, [12] about circles touching one another). If 
a great circle touches a small circle, it also touches another 
«small circle equal and parallel to it (Props. 6, 7), and if a 
great circle be obliquely inclined to another circular section, 
it touches each of two equal circles parallel to that section 
(Prop. 8). If two circles on a sphere cut one another, the 
great circle drawn through their poles bisects »the intercepted 
segments of the circles (Prop. 9). If there are any number of 
parallel circles on a sphere, and any number of great circles 
drawn through their poles, the arcs of the parallel circles 
intercepted between any two of the great circles are similar, 
and the arcs of the great circles intercepted between any two 
of the parallel circles are equal (Prop. 10). 
The last proposition forms a sort of transition to the portion 
of the treatise (II. 11-23 and Book III) which contains pro 
positions of purely astronomical interest, though expressed as 
propositions in pure geometry without any specific reference 
to the various circles in the heavenly "sphere. The proposi 
tions are long and complicated, and it would neither be easy 
nor worth while to attempt an enumeration. They deal with 
circles or parts of circles (arcs intercepted on one circle by 
series of other circles and the like). We have no difficulty in 
recognizing particular circles which come into many proposi-