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

THEODOSIUS'S SPHAERIGA 247
(Books XII and XIII) Euclid included no general properties
of the sphere except the theorem proved in XII. 16-18, that
the volumes of two spheres are in the triplicate ratio of their
diameters ; apart from this, the sphere is only introduced in
the propositions about the regular solids, where it is proved
that they are severally inscribable in a sphere, and it was doubtless
with a view to his proofs of this property in each case that
he gave a new definition of a sphere as the figure described by
the revolution of a semicircle about its diameter, instead of
the more usual definition (after the manner of the definition
of a circle) as the locus of all points (in space instead of in
a plane) which are equidistant from a fixed point (the centre).
No doubt the exclusion of the geometry of the sphere from
the Elements was due to the fact that it was regarded as
belonging to astronomy rather than pure geometry.
Theodosius defines the sphere as a solid figure contained
'
by one surface such that all the straight lines falling upon it
from one point among those lying within the figure are equal
to one another which ',
is exactly Euclid's definition of a circle
'
with solid inserted before figure and surface substituted
' ' ' ' '
for ' line '. The early part of the work is then generally
developed on the lines of Euclid's Book III on the circle.
Any plane section of a sphere is a circle (Prop. 1). The
straight line from the centre of the sphere to the centre of
a circular section is perpendicular to the plane of that section
(1, Por. 2 ;
cf. 7, 23); thus a plane section serves for finding
the centre of the sphere just as a chord does for finding that
of a circle (Prop. 2). The propositions about tangent planes
(3-5) and the relation between the sizes of circular sections
and their distances from the centre (5, 6) correspond to
Euclid III. 16-19 and 15; as the small circle corresponds to
any chord, the great circle (' greatest circle ' in Greek) corresponds
to the diameter. The poles of a circular section
correspond to the extremities of the diameter bisecting
a chord of a circle at right angles (Props. 8-10). Great
circles bisecting one another (Props. 11-12) correspond to
chords which bisect one another (diameters), and great circles
bisecting small circles at right angles and passing through
their poles (Props. 13-15) correspond to diameters bisecting
chords at right angles. The distance of any point of a great
248 TRIGONOMETRY
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 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 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 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-