A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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)
n .
7T . AA'
Hence, from above, the surface of the inscribed solid is
A'B . AA' or tt . A'B .AM, and is therefore less than
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
40 ARCHIMEDES
less than a hemisphere, and Prop. 43 proves that the same
formula applies also to a segment greater than a hemisphere.
As regards the volumes different considerations involving
' solid rhombi ' come in. For convenience Archimedes takes,
in the case of the whole sphere, an inscribed polygon of 4n
sides (Fig. 1). It is easily seen that the solid figure formed
by its revolution is made up of the following : first, the solid
rhombus formed by the revolution of the quadrilateral AB0B f
(the volume of this is shown to be equal to the cone with base
equal to the surface of the cone ABB' and height equal to p,
the perpendicular from on AB, Prop. 18); secondly, the
extinguisher-shaped figure formed by the revolution of the
triangle BOG about AA' (this figure is equal to the difference
between two solid rhombi formed by the revolution of TBOB'
and TCOC respectively about A A', where T is the point of
intersection of GB, G'B' produced with A'A produced, and
this difference is proved to be equal to a cone with base equal
to the surface of the frustum of a cone described by BG in its
revolution and height equal to p the perpendicular from on
BG, Prop. 20)
; and so on ; finally, the figure formed by the
revolution of the triangle GOD about A A' is the difference
between a cone and a solid rhombus, which is proved equal to
a cone with base equal to the surface of the frustum of a cone
described by CD in its revolution and height p
(Prop. 19).
Consequently, by addition, the volume of the whole solid of
revolution is equal to the cone with base equal to its whole
surface and height p (Prop. 26). But the whole of the surface
of the solid is less than 4 nr 2 , and p