252
TRIGONOMETRY
V
We may contrast with this proposition of Theodosius the
corresponding proposition in Menelaus’s Sphaerica (III. 15)
dealing with the more general case in which O', instead of
being the tropical point on the ecliptic, is, like B', any point
between the tropical point and D. If R, p have the same
meaning as above and r x , r 2 are the radii of the parallel circles
through B' and the new C', Menelaus proves that
sin a Rp
sin a' r x r 2 ’
which, of course, with the aid of Tables, gives the means
of finding the actual values of a or a' when the other elements
are given.
The proposition III. 12 of Theodosius proves a result similar
to that of III. 11 for the case where the great circles AB'B,
AG'C, instead of being great circles through the poles, are
great circles touching ‘ the circle of the always-visible stars ’,
i.e. different positions of the horizon, and the points O', B' are
any points on the arc of the oblique circle between the tropical
and the equinoctial points ; in this case, with the same notation,
4 is! : 2 p > (arc BC) : (arc B'G').
It is evident that Theodosius was simply a laborious com
piler, and that there was practically nothing original in his
work. It has been proved, by means of propositions quoted
verbatim or assumed as known by Autolycus in his Moving
Sphere and by Euclid in his Phaenomena, that the following
propositions in Theodosius are pre-Euclidean, I. 1, 6 a, 7, 8, 11,
12, 13, 15, 20 ; II. 1, 2, 3, 5, 8, 9, 10 a, 13, 15, 17, 18, 19, 20, 22 ;
III. lb, 2, 3, 7, 8, those shown in thick type being quoted
word for word.
The beginnings of trigonometry.
But this is not all. In Menelaus’s Sphaerica, III. 15, there
is a reference to the proposition (III. 11) of Theodosius proved
above, and in Gherard of Cremona’s translation from the
Arabic, as well as in Halley’s translation from the Hebrew
of Jacob b. Machir, there is an addition to the effect that this
proposition was used by Apollonius in a book the title of
which is given in the two translations in the alternative
