A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

so that
ON CONOIDS AND SPHEROIDS 63
hS n
= ah(h + 2h+...+nh) + h{h 2 + (2h) 2 +...+(nh) 2 }.
The limit of this latter expression is what we should write
nb
Jo
(ax + x 2 ) dx = b 2 (%a + §6),
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,
(frustum in BF) :
(frustum on base QQ')
2
= BD 2 : QM
= AD.A'D:AM.A'M;
but, in order to reduce the summation to the same as that in
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 f — a, CD = \c.
Then AD.A'D = ±a 2 -±c 2 ,
and AM .
Thus in this case we have
(frustum BF) :
and
(frustum BF) :
A'M = \a 2 - (Jc + rhf {DM = rh)
= AD . A'D-{c . rh + (rh) 2 }
= cb + b 2 -{c.rh + (rh) 2 }.
(inscribed figure)
= n(cb + b 2 ) : [n(cb + b 2 )
- 2^{c .
rh + (rh) 2 }]
(circumscribed figure)
= n (cb + b 2 ) : [n (cb + b2 )
- S^" 1 {c.rh + (rh) 2 } ].
And, since b = nh, we have, by means of Prop. 2,
n(cb + b 2 ) : [n(cb + b 2 ) -^{c. rh + (rh) 2 }]
>(c + 6):{c + 6-(i C + |6)}
> n(cb + b 2 ) : [n(cb + b 2 )
^^ ln ~ 1 {c . rh + (rh) 2 }].
64 ARCHIMEDES
The conclusion, confirmed as usual by the method of exhaustion,
is that
(frustum BF) : (segment
of spheroid) = (c + b) : { c + b - (%v + J b)
= (c + 6):(|c + #6) 5
whence (volume of segment) : (volume of cone ABB')
=:(|c + 26):(c + 6)
= (3GA-AD):(2GA-AD), since CA = ±c + b.
As a particular case (Props. 27, 28), half the spheroid is
double of the corresponding cone.
Props. 31, 32, concluding the treatise, deduce the similar
formula for the volume of the greater segment, namely, in our
figure,
(greater segmt.) : (cone or segmt.of cone with same base and axis)
= (CA + AD):AD.
On Spirals.
The treatise On Spirals begins with a preface addressed to
Dositheus in which Archimedes mentions the death of Conon
as a grievous loss to mathematics, and then summarizes the
main results of the treatises On the Sphere and Cylinder and
On Conoids and Spheroids, observing that the last two propositions
of Book II of the former treatise took the place
of two which, as originally enunciated to Dositheus, were
wrong; lastly, he states the main results of the treatise
On Spirals, premising the definition of a spiral which is as
follows:
'
If a straight line one extremity of which remains fixed be
made to revolve at a uniform rate in a plane until it returns
to the position from which it started, and if, at the same time
as the straight line is revolving, a point move at a uniform
rate along the straight line, starting from the fixed extremity,
the point will describe a spiral in the plane.'
As usual, we have a series of propositions preliminary to
the main subject, first two propositions about uniform motion,