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

Theory of Transversals. 71
72. The anharmonic ratio of a pencil (a/3y8) of four great
circles (see fig., Def. ii.) is equal to the anharmonic ratio
(AB CD) of the four points in which it is intersected by a
transversal.
DEM.-We have, equation (55),
sin A4
(a3, y) = (AB, C). sin B
sin A
and (a3, 8) = (AB, C).. sin
Hence, by division, (apt3y) = (ABCD). (293)
73. E]ach diagonal of a complete spherical quadrilateral is
divided harmonically by the two remaining diagonals.
Fig. 25.
DEM.-Let the quadrilateral be BCB'C'; AA', BB', CC' its
three diagonals. Let BB', CC' intersect in M. Join AM, and
produce to cut BC in A".
Now we have (Art. 70),
(AB, C') (BC, A") (CA, B') = - 1,
and (Art. 71),
(AB, C') (BC, A') (C, B') =+ 1.
Hence (BC, A") = - (BC, A');... ', A", B, C
are harmonic points. Therefore (A - A'"B C) is a harmonic
pencil. Hence the proposition is proved.