A history of Greek mathematics - Wilbourhall.org

268 TRIGONOMETRY
But, by the propositions proved above,
GK sin GE GL sin GF DT _ sin DB
KA ~ sin EA' LD ~~ sin FD* YA ~ sJn~BA '
therefore, by substitution, we have
sin CE __ sin (LP sin DB
sin EA " sin FD ' sin BA "
Menelaus apparently also gave the proof for the cases in
which J.Z),
(ri? meet towards A, G, and in which AD, GB are
parallel respectively, and also proved that in like manner, in
the above figure,
sin GA sin CD sin FB
sin AE sin DF sin BE
(the triangle cut by the transversal being here CFE instead of
ADG). Ptolemy 1 gives the proof of the above case only, and
dismisses the last-mentioned result with a similarly '
'.
(/3) Deductions from Menelaus s Theorem.
III. 2 proves, by means of I. 14, 10 and III. 1, that, if ABC,
A'B'G' be two spherical
triangles in which A — A', and G, G f
are either equal or supplementary, sin c/sin a = sin c'/sin a'
and conversely. The particular case in which G, (7 are right
angles gives what was afterwards known as the regula
'
quattuor quantitatum ' and was fundamental in Arabian
trigonometry. 2 A similar association attaches to the result of
III. 3, which is the so-called tangent ' ' or shadow-rule of the
' '
Arabs/ If ABC, A f B'G' be triangles right-angled at A, A', and
G, G f are equal and both either > or < 90°, and if P, P f be
the poles of AG, A'C, then
sin AB _ sinA'B' sin BP
sin AG ~ sin A'G' '
sin B'P' '
Apply the triangles so that G' falls on C, C'B' on GB as GE,
and C A' on GA as GD
;
then the result follows directly from
III. 1.
result becomes
Since sin BP — cos AB, and sin B'P' = cos A'B\ the
sin GA
tan AB
sin C'A' " ta^rZ 7^ 5
which is the ' tangent-rule ' of the Arabs. 3
1
Ptolemy, Syntax-is, i. 13, vol. i, p. 76.
2
See Braunmuhl, Gesch. der Trig, i, pp. 17, 47, 58-60, 127-9.
3
Cf. Braunmuhl, op. cit. i, pp. 17-18, 58, 67-9, &c.