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

ON THE SPHERE AND CYLINDER, II 45
46 ARCHIMEDES
E so that AD:DE= HH'-.H'M or (m + n):n. We have
the curves touch at the point for which x = §a. If on the
then OA = AD and OA+AM = MD, so that (8) reduces to
other hand be 2 < -^fW\ it is proved that there are two real
AD:DE = (A A'* : A'M 1 (AD
) . : MD),
solutions. In the particular case arising in Prop. 4 it is clear
or MD : DE = A A' 2 : A'M 2 .
Now, says Archimedes, D is given, since AD = OA. Also,
m + n
AD : DE being a given ratio, DE is given. Hence the problem
reduces itself to that of dividing A'D into two parts at
m + n
necessary that
& z 7
M such that
4r 3 , which is obviously the case.
MD : (a given length) = (a given area) : A'M 2 .
(ii) Solution of the cubic by Dionysodorus.
That is, the generalized equation is of the form
It is convenient to add here that Eutocius gives, in addition
x 2 (a — x) = be 2 , as above.
to the solution by Archimedes, two other solutions of our
problem. One, by Dionysodorus, solves the cubic equation in
(i) Archimedes's own solution of the cubic.
the less general form in which it is required for Archimedes's
Archimedes adds that, ' if the problem is propounded in this
A'M = x, is
general form, it requires a Siopicr/ios [i.e. it is necessary to
n
4r 2 :x 2 = (3r— x)
investigate the limits of possibility], but if the conditions are
m + n r,
added which exist in the present case [i.e. in the actual
and the solution is obtained by drawing the parabola and
problem of Prop. 4], it does not require a Siopta-fios' (in other
words, a solution is always possible). He then promises to
give at the end ' ' an analysis and synthesis of both problems
[i.e. the Sioptcrfjios and the problem itself]. The promised
solutions do not appear in the treatise as we have it, but
Eutocius gives solutions taken from an old book ' ' which he
managed to discover after laborious search, and which, since it
was partly written in Archimedes's favourite Doric, he with
fair reason assumed to contain the missing addendum by
Archimedes.
In the Archimedean fragment preserved by Eutocius the
above equation, x 2 (a — x) = be 2 , is solved by means of the intersection
of a parabola and a rectangular hyperbola, the equations
equations
the rectangular hyperbola which we should represent by the
of which may be written thus
n r(3r~x) = 2
n
y and
e m+n 2
x = 2 — y, (a—x) y — ab.
m+n 2 = r2 xy,
a
referred to A'A and the perpendicular to it through A as axes
The Siopio-fios takes the form of investigating the maximum
of x, y respectively.
possible value of x 2 (a — x), and it is proved that this maximum
(We make FA equal to OA, and draw the perpendicular
value for a real solution is that corresponding to the value
AH of such a length that
x = §a. This is established by showing that, if be = 2 -g^a?, FA:AH = CE:ED = (m + n):n.)
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
4r
3
should be not greater than ^Wa 3 or
proposition. This form, obtained from (8) above, by putting