sin A/sin (a/γ) = sin B/sin (b/γ) = sin C/sin (c/γ),
(12)
and
cos (a/γ) = cos (b/γ) cos (c/γ) + sin (b/γ) sin (c/γ) cos A,
(13)
with two similar equations.
Two cases arise, namely (I.) according as one of the four triangles has as its sides the shortest segments between the angular points, or (II.) according as this is not the case. When case I. holds there is said to be a “principal triangle.”[3] If all the figures considered lie within a sphere of radius ¼πγ only case I. can hold, and the principal triangle is the triangle wholly within this sphere, also the peculiarities in respect to the separation of points by a plane cannot then arise. The sum of the three angles of a triangle ABC is always greater than two right angles, and the area of the triangle is γ²(A + B + C − π). Thus as in hyperbolic geometry the theory of similarity does not hold, and the elements of a triangle are determined when its three angles are given. The coordinates of a point can be written in the form
sin (ρ/γ) sin Φ cos φ, sin (ρ/γ) sin Φ sin φ, sin (ρ/γ) cos Φ, cos (ρ/γ),
where ρ, Φ and φ have the same meanings as in the corresponding formulae in hyperbolic geometry. Again, suppose a watch is laid on the plane OXY, face upwards with its centre at O, and the line 12 to 6 (as marked on dial) along the line YOY. Let the watch be continually pushed along the plane along the line OX, that is, in the direction 9 to 3. Then the line XOX being of finite length, the watch will return to O, but at its first return it will be found to be face downwards on the other side of the plane, with the line 12 to 6 reversed in direction along the line YOY. This peculiarity was first pointed out by Felix Klein. The theory of parallels as it exists in hyperbolic space has no application in elliptic geometry. But another property of Euclidean parallel lines holds in elliptic geometry, and by the use of it parallel lines are defined. For the equation of the surface (cylinder) of equal distance (δ) from the line XOX is
(x² + w²) tan² (δ/γ) − (y² + z²) = 0.