A history of Greek mathematics - Wilbourhall.org

286 TRIGONOMETRY
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Thus AC is found, and therefore FC-AC or FA.
The lengths of BG, FG are found in I. 14, 16 by the same
method, the four intersecting great circles used in the figure
being in that case the equator EE', the ecliptic ZZ\ the great
circle PBCP' through the poles, and the great circle PKLP'
passing through the poles of both the ecliptic and the equator.
f
In this case the two arcs PL, AE are cut by the intersecting
great circles PC, FK, and Menelaus's theorem gives (1)
sin PL _ sin OP sin BF
sin KL ~ sirTBG '
sin FK'
But sinPZ=l, sin KL = sin BFG, sinCP=l, sinPZ = l,
and it follows that
sin BG= sin BF sin BFC,
corresponding to the formula for a triangle right-angled at C,
sin a = sin c sin A.
(2) We have
sin PK sin PB sin GF
'
sin KL sin BG sin FL
and sin PK = cos KL = cos BFC, sin PB = cos BG, sin FL = 1
so that
corresponding to the formula
tan BG = sin GF tan BFG,
tan a = sin b tan A.
While, therefore, Ptolemy's method implicitly gives the
formulae for the solution of right-angled triangles above
quoted, he does not speak of right-angled triangles at all, but
only of arcs of intersecting great circles. The advantage
from his point of view is that he works in sines and cosines
only, avoiding tangents as such, and therefore he requires
tables of only one trigonometrical ratio, namely the sine (or,
as he has it,
the chord of the double arc).
The Analcmma.
Two other works of Ptolemy should be mentioned here.
The first is the Analemma. The object of this is to explain
a method of representing on one plane the different points