A history of Greek mathematics - Wilbourhall.org

;'
'
462 DIOPHANTUS OF
ALEXANDRIA
The methods of Diophantus.
It should be premised that Diophantus will have in his
solutions no numbers whatever except rational ' ' numbers
he admits fractional solutions as well as integral, but he
excludes not only surds and imaginary quantities but also
negative quantities. Of a negative quantity per se, i.e. without
some greater positive quantity to subtract it from, he
had apparently no conception. Such equations then as lead
to imaginary or negative roots
he regards as useless for his
purpose ; the solution is in these cases dSvparos, impossible.
So we find him (V. 2) describing the equation 4 = 4a; + 20 as
droiros, absurd, because it would give x = — 4.
He does, it is
true, make occasional use of a quadratic which would give
a root which is
positive but a surd, but only for the purpose
of obtaining limits to the root which are integers or numerical
fractions ;
he never uses or tries to express the actual root of
such an equation. When therefore he arrives in the course
of solution at an equation which would give an ' irrational
result, he retraces his steps, finds out how his equation has
arisen, and how he may, by altering the previous work,
substitute for it another which shall give a rational result.
This gives rise in general to a subsidiary problem the solution
of which ensures a rational result for the problem itself.
It is difficult to give a complete account of Diophantus's
methods without setting out the whole book, so great is the
variety of devices and artifices employed in the different
problems. There are, however, a few general methods which
do admit of differentiation and description, and these we proceed
to set out under subjects.
I. Diophantus's treatment of equations.
(A)
Determinate equations.
Diophantus solved without difficulty determinate equations
of the first and second degrees ; of a cubic we find only one
example in the Arithmetica, and that is a very special case.
(1) Pure determinate equations.
Diophantus gives a general rule for this case without regard
to degree. We have to take like from like on both sides of an