350 
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 con 
gruence-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, C'D are two other great 
circles, both of which are at right angles to the plane of AG'c, 
but CDc is perpendicular to AA', while C'D is inclined to it at 
an acute angle. Let any other great circle AB'BA' through 
A 
A A' cut CD in any point B between C and D, and C'D in B'. 
Let the ‘ parallel ’ circle EB'e be drawn through B', and let 
G'c' be the diameter of the £ parallel ’ circle touching the great 
circle C'D. Let L, K be the centres of the ‘ parallel ’ circles, 
and let R, p be the radii of the ‘ parallel ’ circles CDc, G'c' 
respectively. It is required to prove that 
2R:2p> (arc CB): (arc G'B'). 
Let C'O, Ee meet in N, and join NB'. 
Then B'N, being the intersection of two planes perpendicu 
lar to the plane of AC'CA', is perpendicular to that plane and 
therefore to both Ee and C'O.