270 
TRIGONOMETRY 
It follows that this proposition was known before Mene 
laus’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 con 
clude that both the anharmonic property and Menelaus’s 
theorem with reference to the sphere were already included 
in some earlier text-book ; 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 
Menelaus, it 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. 
(<5) Propositions analogous to Fuel. VI. 8. 
Two theorems following, III. 6, 8, have their analogy in 
Eucl. VI. 3. In III. 6 the vertical angle A of a spherical 
triangle is bisected by an arc of a great circle meeting BG in 
D, and it is proved that sin ED/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 BG in D and 
E, and the proposition proves the harmonic property 
sin BE _ sin BD 
sin EG sin DC 
III. 7 is to the effect that, if arcs of great circles be drawn 
through B to meet the opposite side AG of a spherical triangle 
in D, E so that i ABD = / EBG, then 
sin EA. sin AD _ sin 2 AB 
sin DC. sin CE ~ sin 2 BG 
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 
Menelaus. 
1 Pappus, vii, pp. 870-2, 874.