A history of Greek mathematics - Wilbourhall.org

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248 TRIGONOMETRY
circle from its pole is equal to the side of a square inscribed
Next come
in the great circle and conversely (Props. 16, 17).
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 circular
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
segments of the circles (Prop. 9).
intercepted
If there are any number of
great circles
parallel circles on a sphere, and any number of
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 propositions
of purely astronomical interest, though expressed as
propositions in pure geometry without any specific
reference
to the various circles in the heavenly sphere. The propositions
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-