A history of Greek mathematics - Wilbourhall.org

DETERMINATE EQUATIONS 463
equation and neutralize negative terms by adding to both
sides, then take like from like again, until we have one term
left equal to one term. After these operations have been
performed, the equation (after dividing out, if both sides
contain a power of x, by the lesser power) reduces to Ax m = B,
and is considered solved. Diophantus regards this as giving
one root only, excluding any negative value as ' impossible '.
No equation of the kind is admitted which does not give
a ' rational ' value, integral or fractional. The value x = is
ignored in the case where the degree of the equation is reduced
by dividing out by any power of x.
(2) Mixed quadratic equations.
Diophantus never gives the explanation of the method of
solution which he promises in the preface. That he had
a definite method like that used in the Geometry of Heron
is proved by clear verbal explanations in different propositions.
As he requires the equation to be in the form of two positive
terms being equal to one positive term, the possible forms for
Diophantus are
(a) mx 2 +px = q, (b) mx 2 = px + q, (c) mx 2 + q=px.
It does not appear that Diophantus divided by m in order to
make the first term a square ; rather he multiplied by m for
this purpose.
cases in a form equivalent to
It is clear that he stated the roots in the above
/a -hV+ ^(ip 2 + mq) ,,. ip+