A history of Greek mathematics - Wilbourhall.org

ON THE SPHERE AND CYLINDER, II 45
E so that AD:DE= HH'-.H'M or (m + n):n. We have
then OA = AD and OA+AM = MD, so that (8) reduces to
or MD :
AD:DE = (A A'* : A'M
DE = A A' 2 : A'M
1
) . (AD
:
2
.
MD),
Now, says Archimedes, D is given, since AD = OA. Also,
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 such that
MD :
That is,
(a given length) = (a given area) : A'M
the generalized equation is of the form
x 2 (a — x) = be 2 , as above.
2 .
(i) Archimedes's own solution of the cubic.
Archimedes adds that, ' if the problem is propounded in this
general form, it requires a Siopicr/ios [i.e. it is necessary to
investigate the limits of possibility], but if the conditions are
added which exist in the present case [i.e. in the actual
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
of which may be written thus
e 2
x 2 = — y, (a—x) y — ab.
a
The Siopio-fios takes the form of investigating the maximum
possible value of x 2 (a — x), and it is proved that this maximum
value for a real solution is that corresponding to the value
x = §a. This is established by showing that, if be 2 = -g^a?,