A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

MENELAUS'S SPHAERICA 265
by the triangle ABC with great circles
drawn through B to
meet AC (between A and G) in D, E respectively, and the
case where D and E coincide, and they prove different results
arising from different relations between a and c (a > c), combined
with the equality of AD and EG (or DC), of the angles
ABD and EBG (or DBG), or of a + c and BD + BE (or 2BD)
respectively, according as a + c< = or >180°.
Book II has practically no interest for us.
The object of it
is to establish certain propositions, of astronomical interest
only, which are nothing more than generalizations or extensions
of propositions in Theodosius's Sphaerica, Book III.
Thus Theodosius III. 5, 6, 9 are included in Menelaus II. 10,
Theodosius III. 7-8 in Menelaus II. 12, while Menelaus II. 11
is an extension of Theodosius III. 13. The proofs are quite
different from those of Theodosius, which are generally very
long-winded.
Book III.
Trigonometry.
It will have been noticed that, while Book I of Menelaus
gives the geometry of the spherical triangle, neither Book I
nor Book II contains any trigonometry. This is reserved for
Book III. As I shall throughout express the various results
obtained in terms of the trigonometrical ratios, sine, cosine,
tangent, it is necessary to explain once for all that the Greeks
did not use this
terminology, but, instead of sines, they used
the chords subtended by arcs of a
circle. In the accompanying figure
let the arc iDof a circle subtend an
angle a at the centre 0. Draw AM
perpendicular to OD, and produce it
to meet the circle again in A' . Then
sin a = AM/AO, and AM is \AA'
or half the chord subtended by an
angle 2 a at the centre, which may
shortly be denoted by J(crd. 2 a).
Since Ptolemy expresses the chords as so many 120th parts of
the diameter of the circle, while AM / AO — AA'/2A0, it
follows that sin a and J(crd. 2 a) are equivalent. Cos a is
of course sin (90° — a) and is therefore equivalent to % crd.
(180°-2a).
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.