A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples. By John Casey.

68 Various Applications.
We may remark that
(AB, X) = (AB, X') = (A'B', X)=- ('B, X);
since it follows that when an arc AB meets another XYX',
it is indifferent whether we take for the ratio (AB, X) or
(AB, X').
DEF. XIX.-If A, B, X, Y be four points on the same great
circle, the ratio of the two ratios of section (AB, X) (AB, Y) or
sinXA sin YA.
sn XA..: -_n is called the anharmonic ratio of the four points.
sin XB sin YB
If the ratio = - 1, the points X, Y divide AB harmonically.
For example, the two bisectors of an angle of a triangle divide
the opposite side harmonically.
With four points on an arc of a great circlc, the same as with four points
on a right line, we can, as in Sequel to Euclid, p. 127, form six anharmonic
ratios, any one of which may be called the anharmonic ratio of the points.
DEFINITION XX.- When three arcs of great circles a, /3, y pass
through the same point 1M, and are intersected in A, C, B by thegreat
M
Fig. 23.
circle described, with M as pole, the ratio of section (af/, y)
sin CA sin (ya)
sin (B szin (yP)'