A history of Greek mathematics - Wilbourhall.org

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
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
on
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