ON CONOIDS AND SPHEROIDS 
63 
l let the axis 
let AA'—a. 
i, cylinder on 
i axis to the 
frustum BF 
so that 
hS n = ah (h+ 2h+ ... +nh) + h {h 2 + {2h) 2 + ... +{nh) 2 }. 
The limit of this latter expression is what we should write 
r>b 
{ax + x 2 ) dx = b 2 {%a +1 h), 
-0 
') 
and Archimedes’s procedure is the equivalent of this integration. 
III. In the case of the spheroid (Props. 29, 30) we take 
a segment less than half the spheroid. 
As in the case of the hyperboloid, 
b 
(frustum in BF): (frustum on base QQ') 
= BD 2 : QM 2 
rh + (rh) 2 }, 
= AD . A'D : AM. A'M; 
but, in order to reduce the summation to the same as that in 
%. rh + {rh) 2 }. 
Prop. 2, Archimedes expresses AM. A'M in a different form 
equivalent to the following. 
Let AD (=b) be divided into n equal parts of length h, 
and suppose that AA'= a, CD = \c. 
: {t a + ^nh) 
a. rh + {rh) 2 }. 
Then AD . A'D = ^a 2 — ^c 2 , 
and AM. A'M = i a 2 - {%c + rh) 2 {DM = rh) 
= AD . A'D— {c . rh + {rh) 2 } 
:{%a + ^nh), 
B') 
{AD+ 2CA)] 
= cb + b 2 — {c . rh + {rh) 2 }. 
Thus in this case we have 
(frustum BF): (inscribed figure) 
m. 
nt to proving 
is indefinitely 
i 
= n {cb + b 2 ) :[n {cb + b 2 ) — {c .rh-f- {rh) 2 } ] 
and 
(frustum BF): (circumscribed figure) 
= n{cb + b 2 ): \n{cb + h 2 ) — '2A l ~ x {c . rh + {rh) 2 ]'\. 
And, since h = nh, we have, by means of Prop. 2, 
n{cb + b 2 ) : \n{cb + b 2 ) — . rh + {rh) 2 }] 
> {c+b): {c + b — (1 c + § b)} 
2 +... + {nh) 2 ] 
> n{ch + b 2 ): \n{cb + b 2 ) —. rh + {rh) 2 }\