A history of Greek mathematics - Wilbourhall.org

,
;
290 TRIGONOMETRY
both in the plane LSHG at right angles to the meridian
therefore arc SQ — arc UZ\
Hence all four arcs SV, VC, QC, QS are represented in the
auxiliary figure in one plane.
So far the procedure amounts to a method of grajMcally
constructing the arcs
required as parts of an auxiliary circle
in one plane. But Ptolemy makes it clear that practical
calculation followed on the basis of the figure. 1 The lines
used in the construction are SF= sint (where the radius =1),
FT=0Fsin(f), FG = OF sin (90° -0), and this was fully
realized by Ptolemy. Thus he shows how to calculate the
arc SZ, the zenith distance (= d, say) or its complement 8V,
the height of the sun (= h, say), in the following way. He
says in effect: Since G is known, and Z F / 0'G / = 90° — 0, the
ratios O'F' : F'T and O'F' : O'T' are known.
0'F f T)
[In fact 7^7777-, =
—, .
n
L
O'T' crd. (180° -20)'
of the sphere.]
- E— —rr, where D is the diameter
Next, since the arc MS or M'S' is known [
= £],
and therefore
the arc P'S' [= 90°-t], the ratio of
[in fact 0'F'/D= {crd.
(lS0-2t)}/2D.
It follows from these two results that
*
O'F' to D is known
/= crd.(l 8 0°-2Q .
erd
x
2D Yn
QO _
'
Lastly, the arc SV (= h) being equal to B'I\ the angle h is
equal to the angle 0'1'T in the triangle FQ'T'.
And in this
triangle O'F, the radius, is known, while O'T' has been found ;
and we have therefore
O'T' crd.(2/ (arc BV — arc B'X'), the figure gives
tan a) _ G
,
xg' s'F' _ s'F' err 1
Q
,
-jTy,- ,F rr
- tan t.
ffi ^
1
See Zeuthen in Bibh'otheca mathematics, h, 1900, pp. 23-7.