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