A history of Greek mathematics - Wilbourhall.org

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.
If m :
Cubic equation arising out of II. 4.
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 h'
s
,/dr — /i\ _ m ,, /Sr — 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 + n
3
= 0.
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 = —
.
n
R'M, where HM, HM have the values
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