A history of Greek mathematics - Wilbourhall.org

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.