A history of Greek mathematics - Wilbourhall.org

r
r
;
ON THE SPHERE AND CYLINDER, I 41
convenience, a segment less than a hemisphere and, by the
same chain of argument (Props. 38, 40 Corr., 41 and 42), proves
(Prop. 44) that the volume of the sector of the sphere bounded
by the surface of the segment is equal to a cone with base
equal to the surface of the segment and height equal to the
radius, i. e. the cone with base w . AP 2 and height r (Fig. 2).
It is noteworthy that the proportions obtained in Props. 21,
22 (see p. 39 above) can be expressed in trigonometrical form.
If 4?i is the number of the sides of the polygon inscribed in
the circle, and 2n the number of the sides of the polygon
inscribed in the segment, and if the angle AOP is denoted
by a, the trigonometrical equivalents of the proportions are
respectively
w
2n 2n
v
IT « IT • 77" 7T
(1) sin f-sin- h ... +sm(2?i— 1) -— = cot —
' 2n 4n
( . oc . 2oc . . .oc)
(2) 2 -J sin - +sin h ... + sm in —
In
\)-\ + sina
n n)
= (1 — cos oc) cot
v
'
Thus the two proportions give in effect a summation of the
series
sin + sin 2 6 + .
. . +
sin (n — 1) 0,
both generally where nO is equal to any angle oc less than n
and in the particular case where n is even and 6 = ir/ n.
Props. 24 and 35 prove that the areas of the circles
2n
equal to
the surfaces of the solids of revolution described by the
polygons inscribed in the sphere and segment are the above
series multiplied by Inr 2 sin — and nr
2
4 n & n
and are therefore 4 77-
TT
. 2 sin — respectively
2 cos —
2
and n
. 2 cos — (1— cos a)
'
4 n 2n
respectively. Archimedes's results for the surfaces of the
sphere and segment, 47rr 2 and 27rr 2 (l — cos a), are the
limiting values of these expressions when n is indefinitely
increased and when therefore cos — and cos — become
4ti
2n
oc
unity. And the two series multiplied by 4 77-
2
sin— and
In