A history of Greek mathematics - Wilbourhall.org

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.