A history of Greek mathematics - Wilbourhall.org

ON THE SPHERE AND CYLINDER, I 39
whence, adding antecedents and consequents, we have
(Fig. 1) (BB' + 00'+... + EE') :
A A' = A'B : BA,
(Prop. 21)
(Fig. 2) {BR + CC+...+ \PP') :AM=A'B:BA. (Prop. 2 2)
When we make the polygon revolve about A A', the surface
of the inscribed figure
so obtained is made up of the surfaces
of cones and frusta of cones; Prop. 14 has proved that the
BF,
.
surface of the cone ABB' is what we should write tt . AB
and Prop. 16 has proved that the surface of the frustum
BOC'B' is tt.BC(BF+GG). It follows that, since AB =
BG = . . , .
the surface of the inscribed solid is
tt .AB {%BW + ^(BB' + GC')+ ...},
that is, tt . AB
(BB' + 00'+... + EE') (Fig. 1), (Prop. 24)
or tt.AB (BB' + CC+...+ ±PP') (Fig. 2). (Prop. 35)
Hence, from above, the surface of the inscribed solid is
n . A'B . AA' or tt . A'B .AM, and is therefore less than
7T . AA' 2 (Prop. 25) or tt . A'A . AM,
that is, tt . AP 2 (Prop. 37).
Similar propositions with regard to surfaces formed by the
revolution about AA' of regular circumscribed solids prove
that their surfaces are greater than it. A A' 2, and tt .AP 2
respectively (Props. 28-30 and Props. 39-40). The case of the
segment is more complicated because the circumscribed polygon
with its sides parallel to AB, BG ... DP circumscribes
the sector POP'. Consequently, if the segment is less than a
semicircle, as GAG', the base of the circumscribed polygon
(cc')
is on the side of GC' towards A, and therefore the circumscribed
polygon leaves over a small strip of the inscribed.
This
complication is dealt with in Props. 39-40. Having then
arrived at circumscribed and inscribed figures with surfaces
greater and less than tt. AA' 2 and tt. AP 2 respectively, and
having proved (Props. 32, 41) that the surfaces of the circumscribed
and inscribed figures are to one another in the duplicate
ratio of their sides, Archimedes proceeds to prove formally, by
the method of exhaustion, that the surfaces of the sphere and
segment are equal to these circles respectively (Props. 33 and
42); tt .AA' 2 is of course equal to four times the great circle
of the sphere. The segment is, for convenience, taken to be