A history of Greek mathematics - Wilbourhall.org

—
,
'
MENELAUS'S SPHAERICA 269
It follows at once (Prop. 4) that, if AM, A'M' are great
circles drawn perpendicular to the bases BG, B'C of two
spherical triangles ABC, A'B'C in which B = B',C — G',
sin BM sin MC / . ,
,
—
sin B'M'
, ,
tan
AM \
^i^r, — ~— TF77T* I
since both are equal to jttF' )'
smM'C'K
III. 5 proves that, if
*
tan A'M'}
there are two spherical triangles ABC,
P
P'
A'B'C right-angled at A, A' and such that C—C, while 6
and 6' are less than 90°,
sin (a + b) _ sin (a' + &')
sin (a — b)
from which we may deduce 1
sin (a/ — b')
the formula
sin (a + b) 1 + cos 6 T
sin (a — b)
~" 1 — cos C
which is equivalent to tan b = tan a cos C.
(y) Anharmonic property of four great circles through
one point.
But more important than the above result is
the proof assumes as known the anharmonic
property of four great circles
drawn from a point on a sphere in relation
to any great circle intersecting them
all,
viz. that, if ABCD, A'B'G'D' be two
transversals,
sin AD sin BC sinA'D' sin B'C
sin DG
'
sin AB " sin B'C' '
sin A'B'
the fact that
* Braunmiihl, op. cit. i, p. 18; Bjornbo, p. 96.