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

OF
NOTATION AND DEFINITIONS 461
in three terms is clearly assumed in several places of the
Arithmetics, but Diophantus never gives the necessary explanation
of this case as promised in the preface.
Before leaving the notation of Diophantus, we may observe
that the form of it
limits him to the use of one unknown at
a time. The disadvantage is obvious. For example, where
we can begin with any number of unknown quantities and
gradually eliminate all but one, Diophantus has practically to
perform his eliminations beforehand so as to express every
quantity occurring in the problem in terms of only one
unknown. When he handles problems which are by nature
indeterminate and would lead in our notation to an indeterminate
equation containing two or three unknowns, he has
to assume for one or other of these some particular number
arbitrarily chosen, the effect being to make the problem
determinate. However, in doing so, Diophantus is careful
to say that we may for such and such a quantity put any
number whatever, say such and such a number; there is
therefore (as a rule) no real loss of generality. The particular
devices by which he contrives to express all his unknowns
in terms of one unknown are extraordinarily various and
clever. He can, of course, use the same variable y in the
same problem with different significations successively, as
when it is necessary in the course of the problem to solve
a subsidiary problem in order to enable him to make the
coefficients of the different terms of expressions in x such
as will answer his purpose and enable the original problem
to be solved. There are, however, two cases, II. 28, 29, where
for the proper working-out of the problem two unknowns are
imperatively necessary. We should of course use x and y;
Diophantus calls the first y as usual; the second, for want
of a term, he agrees to call in the first instance 'one unit',
i.e. 1. Then later, having completed the part of the solution
necessary to find x, he substitutes its value and uses y over
again for what he had originally called 1. That is, he has to
put his finger on the place to which the 1 has passed, so as
to substitute y for it.
This is a tour de force in the particular
cases, and would be difficult or impossible in more complicated
problems.
462 DIOPHANTUS
'
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