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

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
250 TRIGONOMETRY
the required numerical ratios a new method had to be invented,
namely trigonometry.
No actual trigonometry in Theodosius.
It is perhaps hardly correct to say that spherical triangles
are nowhere referred to in Theodosius, for in III. 3 the congruence-theorem
for spherical triangles corresponding to Eucl.
I. 4 is practically proved ; but there is nothing in the book
that can be called trigonometrical. The nearest approach is
in III. 11, 12, where ratios between certain straight lines are
compared with ratios between arcs. ACc (Prop. 11) is a great
circle through the poles A, A' ; CDc, CD are two other great
circles, both of which are at right angles to the plane of ACc,
but CDc is perpendicular to AA\ while CD is inclined to it at
an acute angle. Let any other great circle AB'BA' through
A A' cut CD in any point B between C and D, and CD in B'.
Let the ' parallel ' circle EB'e be drawn through B\ .and let
Cc r be the diameter of the ' parallel ' circle touching the great
circle CD. Let L, K be the centres of the ' parallel ' circles,
and let R, p be the radii of the ' parallel ' circles CDc, Cc f
respectively.
It is required to prove that
2R:2p> (arc CB) :
(arc CB r ).
Let CO, Ee meet in N, and join NB'.
Then B'N, being the intersection of two planes perpendicular
to the plane of ACCA f , is perpendicular to that plane and
therefore to both Ee and CO.