NOTATION AND DEFINITIONS 
461 
in three terms is clearly assumed in several places of the 
Arithmetiea, but Diophantus never gives the necessary ex 
planation 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 inde 
terminate 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.