Reasoning and symbolism in Diophantus 347
format - not a single monolith to start with. In fi tting his text into the
established elite Greek mathematical format, Diophantus had a certain
freedom.
Th e fi rst decision made by Diophantus was to keep the basic dichotomy
of presentation from standard Greek mathematics, with an arrangement
of a general statement followed by a particular proof. Th is indeed would
appear as one of the most striking features of the Greek mathematical
style. But most important, this arrangement is essential to the large-scale
transformation introduced by Diophantus. To produce a structure based
on rational completion, Diophantus needed to have something to complete
rationally: a set of general statements referring to each problem in terms
transcending the particular parameters of the problem at hand.
I therefore argue that Diophantus’ general statements can be under-
stood, at two levels, as a function of his deuteronomic project. He needs
the general statements so as to conform to the elite form of presentation
he sets out to emulate. Even more important, he needs them to provide
building blocks for his main project of systematization. Th e upshot of this
is that Diophantus does not need the general statements for the logical fl ow
of the individual problem. Th is is indeed obvious from an inspection of the
problems, where the general statements play no role at all.
Th is observation may shed some light on the major mathematical ques-
tion regarding Diophantus, that is, did he see his project in terms of provid-
ing general solutions? In some ways he clearly did. Th e clearest evidence is
in the course of the propositions (extant in Arabic only) vii .13–14. We are
given a square number N which is to be divided into any three numbers (i.e.
N= a + b + c ) so that either N+ a , N+ b , N+ c are all squares ( vii .13), or N− a ,
N− b , N− c are all squares ( vii .14). It is not surprising that, in both cases, we
reach a point in the argument where we are asked to take a given square
number and divide it into two square numbers 19 – the famous Fermatian
problem ii .8. Now, Diophantus (or his Arabic text) explicitly says that this
is possible for ‘It has been seen earlier in this treatise of ours how to divide
any square number into square parts.’ 20 Th ere, of course, the divided square
is a particular number, 16. (Th e particular number chosen as example in
vii .13–14 is 25.) Th is reference is hardly a late gloss, as the very approach
taken to the problem is predicated upon the reduction into ii .8. Indeed,
the natural assumption on the part of any reader familiar with elite Greek
19 By iteration, this allows us to divide a square number into any number of square numbers;
Diophantus, in fact, requires a division into three parts. Note however that even the basic
operation of iteration itself calls for a generalization of the operation of ii .8.
20 Sesiano 1982 : 166.