THEODOSIUS’S SPHAERICA
24 7
(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 doubt
less 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) corre
sponds 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