32 
ARCHIMEDES 
circles are sections made by the plane though N at right 
angles to A A' in the cylinder, the spheroid and the cone AEF 
respectively. 
Therefore, if HA A' be a lever, and the sections of the 
spheroid and cone be both placed with their centres of gravity 
at H, these sections placed at H balance, about A, the section 
MM' of the cylinder where it is. 
Treating all the corresponding sections of the segment of 
the spheroid, the cone and the cylinder in the same way, 
we find that the cylinder with axis AG, where it is, balances, 
about A, the cone AEF and the segment ADC together, when 
both are placed with their centres of gravity at H] and, 
if IE be the centre of gravity of the cylinder, i.e. the middle 
point of AG, 
HA : A W = (cylinder, axis AG): (cone AEF + segmt. ADC). 
If we call V the volume of the cone AEF, and S that of the 
segment of the spheroid, we have 
A A' 2 
(cylinder): (F+$) = 3 V. ~^- 2 : (F+ S), 
while 
HA-.AW = AA'-.^AG. 
AA' 
Therefore AA' :^AG = 3 V. : (F + 8), 
AA' 
and 
whence 
(F+$) = | F. 
AG 
S= f(|4£ - l) 
\2AG ) 
Again, let V' be the volume of the cone ADC. 
Then V:V'= EG*: EG 2 
BB' 2 
= T^*-AG*:DG*. 
But * DG 2 :AG. GA' = BB' 2 :AA' 2 . 
Therefore V:V' = AG 2 : AG. GA' 
= AG:GA'. 
It follows 
which is the 
The result 
proof, of com 
In the part 
or spheroid, t 
and it follows 
is 4 V', wheri 
volume of th< 
circumscribing 
In order m 
of a spheroid 
not at H. 
Therefore i 
that 
whence Ml 
therefore 
(This is sej 
in Prop. 9.) 
From this \ 
segment ADC 
equal to the c< 
Now the ce: 
AG at a distai 
of gravity of 
we have 
or y 
1523.2