[and] also, if
is any other point on the sphere,
We shall add here another theorem, which has appeared nowhere else, as far as we know, and which can often be used with advantage.
Let
be four points on the sphere, and
the angle which
and
make at their point of intersection. [Then we have]
The proof is easily obtained in the following way. Let
we have then
Therefore
Since each of the two great circles goes out from
in two opposite directions, two supplementary angles are formed at this point. But it is seen from our analysis that those branches must be chosen, which go in the same sense from
toward
and from
toward
Instead of the angle
we can take also the distance of the pole of the great circle
from the pole of the great circle
However, since every great circle has two poles, we see that we must join those about which the great circles run in the same sense from
toward
and from
toward
respectively.
The development of the special case, where one or both of the arcs
and
are
we leave to the reader.
6) Another useful theorem is obtained from the following analysis. Let
be three points upon the sphere and put