ough iY at right 
,nd the cone AEF 
e sections of the 
centres of gravity 
out A, the section 
>f the segment of 
n the same way, 
ere it is, balances, 
)G together, when 
ivity at H; and, 
3r, i.e. the middle 
^q-segmt. ADC). 
and 8 that of the 
F+8), 
8), 
[DC. 
G 2 . 
which is the result stated by Archimedes in Prop. 8. 
The result is the same for the segment of a sphere, 
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 
8=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 ABBi.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 is, 
not at H. 
Therefore formula (1) above will not serve. But we found 
that MN. NQ = (iYP 2 + NQ 2 ), 
whence MN 2 : (iYP 2 + NQ 2 ) = (NP 2 + NQ 2 ): NQ 2 ; 
therefore HA : AN = (.NP 2 + NQ 2 ): NQ 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 
AG at a distance |AG from A. Let X be the required centre 
of gravity of the segment. Then, taking moments about A, 
we have V. HA = 8. AX + V. %AG, 
V{AA'-%AG) = 8.AX 
3 A A ' 
= V( 2 An l) AX, from above.