THEODOSIUS’S SPHAERICA 
249 
tions. A particular small circle is the circle which is the 
limit of the stars which do not set, as seen by an observer at 
a particular place on the earth’s surface; the pole of this 
circle is the pole in the heaven. A great circle which touches 
this circle and is obliquely inclined to the ‘ parallel circles ’ is the 
circle of the horizon; the parallel circles of course represent 
the apparent motion of the fixed stars in the diurnal rotation, 
and have the pole of the heaven as pole. A second great 
circle obliquely inclined to the parallel circles is of course the 
circle of the zodiac or ecliptic. The greatest of the ‘ parallel 
circles’ is naturally the equator. All that need.be said of the 
various propositions (except two which will be mentioned 
separately) is that the sort of result proved is like that of 
Props. 12 and 13 of Euclid’s Phaenomena to the effect that in 
the half of the zodiac circle beginning with Cancer (or Capri 
cornus) equal arcs set (or rise) in unequal times; those which 
are nearer the tropic circle take a longer time, those further 
from it a shorter; those which take the shortest time are 
those adjacent to the equinoctial points; those which are equi 
distant from the equator rise and set in equal times. In like 
manner Theodosius (III. 8) in effect takes equal and con 
tiguous arcs of the ecliptic all on one side of the equator, 
draws through their extremities great circles touching the 
circumpolar ‘ parallel ’ circle, and proves that the correspond 
ing arcs of the equator intercepted between the latter great 
circles are unequal and that, of the said arcs, that correspond 
ing to the arc of the ecliptic which is nearer the tropic circle 
is the greater. The successive great circles touching the 
circumpolar circle are of course successive positions of the 
horizon as the earth revolves about its axis, that is to say, 
the same length of arc on the ecliptic takes a longer or shorter 
time to rise according as it is nearer to or farther from the 
tropic, in other words, farther from or nearer to the equinoctial 
points. 
It is,' however, obvious that investigations of this kind, 
which only prove that certain arcs are greater than others, 
and do not give the actual numerical ratios between them, are 
useless for any practical purpose such as that of telling the 
hour of the night by the stars, which was one of the funda 
mental problems in Greek astronomy; and in order to find