direction from
to
is the same as that from
to
negative, if the contrary of one of these conditions hold; positive again, if the contrary of both conditions be true. In other words, the surface is considered positive if we go around the circumference of the figure
in the same sense as
negative, if we go in the contrary sense.
If we consider now a finite part of the line from
to
and denote by
the values of the angles at the two extremities, then we have
the sign of the area being taken as explained.
Now let us assume further that, from the origin upon the curved surface, infinitely many other shortest lines go out, and denote by
that indefinite angle which the first element, moving counter-clockwise, makes with the first element of the first line; and through the other extremities of the different curved lines let a curved line be drawn, concerning which, first of all, we leave it undecided whether it be a shortest line or not.
If we suppose also that those indefinite values, which for the first line were
be denoted by
for each of these lines, then
is capable of being represented in the same manner on the auxiliary sphere by the space
Since evidently
the space
If the bounding line is also a shortest line, and, when prolonged, makes with
the angles
if, further,
denote the same at the points
that
did at
in the line
then we have
but
therefore
The angles of the triangle
evidently are