A history of Greek mathematics - Wilbourhall.org

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THE METHOD 33
, AG/3AA'
It follows that S=V. ~^f (jjq ~ l)
= V
/ ^
%AA'-AG
A'G
= V. , iAA' + A'G
A'G
which is the result stated by Archimedes in Prop. 8.
The result is the same for the segment of a sphere. The
proof, of course slightly simpler, is given in Prop. 7.
In the particular case where the segment is half the sphere %
or spheroid, the relation becomes
S = 2 V\ (Props. 2, 3)
and it follows that the volume of the whole sphere or spheroid
is 4 V\ where V is the volume of the cone ABB' \ i.e. the
volume of the sphere or spheroid is
two-thirds of that of the
circumscribing cylinder.
In order now to find the centre of gravity of the segment
of a spheroid, we must have the segment acting where it
not at H.
Therefore formula (1) above will not serve. But we found
that MN. NQ = (i^P 2 + JVQ 2 ),
whence MJSf 2 : (iVT 2 + NQ 2 )
= (FP 2 + FQ 2 ) NQ :
therefore HA AN : = (NP 2 + NQ 2 NQ
) :
2 .
2 ;
(This is separately proved by Archimedes for the sphere
in Prop. 9.)
From this we derive, as usual, that the cone AEF and the
segment ADC both acting where they are balance a volume
equal to the cone AEF placed with its centre of gravity at H.
Now the centre of gravity of the cone AEF is on the line
A G at a distance fAG from A. Let X be the required centre
of gravity of the segment. Then, taking moments about A,
we have
or
V .HA = S.AX+V.iAG,
V(AA'-iAG) = S.AX
is,
= y(^~rn l)AX
y
from
above.
1523.2 D