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

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',
270 TRIGONOMETRY
tan AM \
sin BM sin MC / . , , , ,
smM'C'K
*
tan A'M'}
there are two spherical triangles ABC,
P
P'
in some earlier text-book ;
Menelaus, it
sin (a + b) _ sin (a' + &')
sin (a — b) sin (a/ — b')
the formula
sin (a + b) 1 + cos 6 T
~"
sin (a — b) 1 — cos C
sin BE sin BD
sin EC sin DC
the fact that
in D, E so that lABD = L EBC, then
sin EA . sin A D _ sin AB
2
sin DC .
viz. that, if ABCD, A'B'G'D' be two
sin AD sin BC sinA'D' sin B'C
'
sin DG sin AB " sin B'C' ' sin A'B'
Menelaus.
1
— ^i^r, — ~— TF77T* I
since both are equal to jttF' )'
sin B'M'
III. 5 proves that, if
A'B'C right-angled at A, A' and such that C—C, while 6
and 6' are less than 90°,
from which we may deduce 1
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,
transversals,
It follows that this proposition was known before Menelaus's
time. It is most easily proved by means of ' Menelaus's
Theorem', III. 1, or alternatively it may be deduced for the
sphere from the corresponding proposition in plane geometry,
just as Menelaus's theorem is transferred by him from the
plane to the sphere in III. 1. We may therefore fairly conclude
that both the anharmonic property and Menelaus's
theorem with reference to the sphere were already included
and, as Ptolemy, who built so much
upon Hipparchus, deduces many of the trigonometrical
formulae which he uses from the one theorem (III. 1) of
seems probable enough that both theorems were
known to Hipparchus. The corresponding plane theorems
appear in Pappus among his lemmas to Euclid's Porisms, 1
and
there is therefore every probability that they were assumed
by Euclid as known.
(8) Projoositions analogous to Eucl. VI. S.
Two theorems following, III. 6, 8, have their analogy in
Eucl. VI. 3. In III. 6 the vertical angle i of a spherical
triangle is bisected by an arc of a great circle meeting BG in
D, and it is proved that sin BD/ sin DC = sin BA/ sin AC;
in III. 8 we have the vertical angle bisected both internally
and externally by arcs of great circles meeting BC in D and
E, and the proposition proves the harmonic property
III. 7 is to the effect that, if arcs of great circles be drawn
through B to meet the opposite side AC of a spherical triangle
sin CE ~ sin 2 J30'
As this is analogous to plane propositions given by Pappus as
lemmas to different works included in the Treasury of
Analysis, it is clear that these works were familiar to
Pappus, vii, pp. 870-2, 874.