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

Stereographic Projection. 125
and the angle made by the circumcircle of abc with the chord
bc is equal to A (Euc.. III. XXI.). Hence the angle between
the arc bc and its chord is equal to E.
Cor. 1.-If the radius of the sphere be unity, the sides of the
rectilineal triangle can be expressed in terms of the spherical
triangle. Thus, evidently,
ab = tan A OB = tan - AB = tan c. (418)
ac = tan AOC= tan A C = tan b. (419)
Again, bc: ab: sin bac: sin acb: sin A: sin (C-E);:n n
bc: tan-c:: n:.;
sin b. sinc 2 sin a. sin - b. cos O c
bc= sin I a (420)
cos - b cos - c
Similarly, b'c cos i. (421)
cos b sin mc
The equation (420) may be got from equation (417) by putting k2 = 2. Thus2 chord B C sin a
bc =
OB. OC cosi b cos c'
and (421) from (420), by the substitution of ~ 78.
Cor. 2.-If the angles of the rectilineal triangle abc be
denoted by a, fi, y, we have
a=.A, /f =B-E, y= C-E. (422)
ExERCISES.-XXXII.
1. Prove the fundamental formula (13) by stereographic projection.
From the triangle abc we have
(bc)2 = (ca)2 + (ab)2 - 2 (ca) (ab) cosdA,
and substitute from equations (418)-(420).
2. Prove Napier's Analogies.
tWehave an ( - ) ac - ab
We have av t -eY)
tan ~ a =ac + ab'
And substitute frorn equations (418)-(422).