A history of Greek mathematics - Wilbourhall.org

THEODOSIUS'S SPHAERIGA 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 Capricornus)
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 equidistant
from the equator rise and set in equal times.
In like
manner Theodosius (III. 8) in effect takes equal and contiguous
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 corresponding
arcs of
the equator intercepted between the latter great
circles are unequal and that, of the said arcs, that corresponding
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 fundamental
problems in Greek astronomy ; and in order to find