A history of Greek mathematics - Wilbourhall.org

266 TRIGONOMETRY
(a) ' Menelaus s theorem ' for the sphere.
The first proposition of Book III is the famous Menelaus's
'
theorem ' with reference to a spherical triangle and any transversal
(great circle) cutting the sides of a triangle, produced
if necessary. Menelaus does not, however, use a spherical
triangle in his enunciation, but enunciates the proposition in
terms of intersecting great circles.
'
Between two arcs ADB,
AEG of great circles are two other arcs of great circles DFG
and BFE which intersect them and also intersect each other
in F. All the arcs are less than a semicircle. It is required
to prove that
sin CE sin CF sin DB ,
sin EA " sin FD sin BA
It appears that Menelaus gave three or four cases, sufficient
to prove the theorem completely. The proof depends on two
simple propositions which Menelaus assumes without proof;
the proof of them is given by Ptolemy.
(1) In the figure on the last page, if OD be a radius cutting
a chord AB in C, then
AC:CB = sin AD: sin DB.
For draw A 31, BN perpendicular to OD.
AG:GB = AM:BN
Then
= |(crd. 2.4D):i(crd. 2DB)
= sin AD: sin DB.
(2) If AB meet the radius OC produced in T, then
AT:BT = sin AC: sin BC.