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

ON THE SPHERE AND CYLINDER, II 43
This is expressed by Archimedes thus. If HM is the height
of the required cone,
HM:AM = (OA' + A'M):A'M, (1)
and similarly the cone equal to the segment A'BB' has the
height HM, where
HM :
A'M = (OA + AM) : AM.
(2)
His proof is, of course, not in the above form but purely
geometrical.
This proposition leads to the most important proposition in
the Book, Prop. 4, which solves the problem To cut a given
sphere by a plane in such a vjay that the volumes of the
segments are to one another in a given ratio.
Cubic equation arising out of II. 4.
If m : 7i be the given ratio of the cones which are equal to
the segments and the heights of which are h, h' , we have
Sr h\ , , /Zr s
,/dr — /i\ _ m ,, /Sr — h' h, \
\2r — h) ' ' "n \2r — h')
and, if we eliminate h' by means of the relation h + h! — 2r,
we easily obtain the following cubic equation in h,
tf-3h 2 r+ -A— =
m 3 0.
+ n
Archimedes in effect reduces the problem to this equation,
which, however, he treats as a particular case of the more
general problem corresponding to the equation
(r + h):b = c 2 :(2r-h) 2 ,
where b is a given length and c 2 any given area,
or x 2 (a — x) = be 2 , where x = 2r—h and 3r = a.
Archimedes obtains his cubic equation with one unknown
by means of a geometrical elimination of H, H' from the
equation HM = —
.
R'M, where HM, HM have the values
n
determined by the proportions (1) and (2) above , after which
the one variable point M remaining corresponds to the one
unknown of the cubic equation. His method is, first, to find
44 ARCHIMEDES
values for each of the ratios A'H' : H'M and H'H: A'H' which
are alike independent of H, H' and then, secondly, to equate
the ratio compounded of these two to the known value of
H'M.
ratio HH' :
(ex.) We have, from (2),
A'H : H'M = OA :
the
(OA + AM). (3)
(/?) From (1) and (2), separando,
AH:A3I= OA'iA'M, (4)
A'H' :
Equating the values of the ratio A'M :
A'M =0A: AM. (5)
AM
given by (4). (5),
we have 6A'
: AH = A'H' : OA -
= OH' : OH,
whence HH : OH = OH' : A'H', (since OA = OA')
or HH'.A'H' = OH 2 ,
so that HH' : A'H' = OH' 2 : ^i?' 2 .
(6)
But, by (5), OA' : A'H' = AM: A'M,
and, componendo, OH : A'H' — AA' : A'M.
By substitution in (6),
HH' : A'H = A A' 2 : A'M
Compounding with (3), we obtain
HH :
H'M = (A A' 2 : A 'M 2 ) (OA .
2 .
(7)
:
[The algebraical equivalent of this is
m + n 4 r 3
n "
(2r—h) 2 (r-{-h)
i
... , .
which reduces to
m + n 4r
= 3
—=-= =-z
on 3/rr — h 6
or h — z 3h 2 r-\ r = 3 0, as above.]
m + n
OA + AM). (8)
Archimedes expresses the result (8) more simply by producing
OA to D so that OA = AD, and then dividing AD at