A history of Greek mathematics - Wilbourhall.org

46 ARCHIMEDES
the curves touch at the point for which x = §a. If on the
other hand be 2 < -^fW\ it is proved that there are two real
solutions. In the particular case arising in Prop. 4 it is clear
that the condition for a real solution is satisfied, for the
m
expression corresponding to be 2 is 4r 3 , and it is only
m + n
m
4r
3
should be not greater than ^Wa 3 or
m + n
& z 7
which is obviously the case.
necessary that
4r 3 ,
(ii) Solution of the cubic by Dionysodorus.
It is convenient to add here that Eutocius gives, in addition
to the solution by Archimedes, two other solutions of our
problem. One, by Dionysodorus, solves the cubic equation in
the less general form in which it is required for Archimedes's
proposition. This form, obtained from (8) above, by putting
A'M = x, is
n
4r 2 :x 2 = (3r— x)
m + n r,
and the solution is obtained by drawing the parabola and
the
rectangular hyperbola which we should represent by the
equations
n r(3r~x) = y
2
and
m+n m+n 2 r2 = xy,
referred to A'A and the perpendicular to it through A as axes
of x, y respectively.
(We make FA equal to OA, and draw the perpendicular
AH of such a length that
FA:AH = CE:ED = (m + n):n.)
n