350 reviel netz
x and y , for sure, a confusing symbolism, as well (one would need to develop
procedures to diff erentiate ‘two numbers’ from ‘the second number’) – but
an eff ective symbolism nonetheless. Why did Diophantus not use it?
Because he had no use for it. Th e task Diophantus set himself did not call
for multiple symbols for multiple unknowns. He did not set out to produce
general proofs but rather to solve problems, where (with few exceptions) a
single unknown was to be found. Diophantus’ project aimed not to obtain
the generality of Euclidean theorems, but rather to solve problems, in a
manner expressing the rationality of the solution. Th is task defi ned, for
Diophantus, his choice of symbolism.
So let us then reframe accordingly our interpretation of Diophantus’
symbolism: not as a second-rate tool for the task of modern algebra, but,
instead, as the perfect tool for the task Diophantus set himself. I proceed to
discuss this task.Diophantus the analyst: choosing a mode of persuasion
Over and above the rigid structure of general enunciation followed by
particular problem, Diophantus follows a rigid form for each of the prob-
lems. We should now explain Diophantus’ motivations in choosing this
particular form (that he chose some rigid form – instead of allowing freely
varying forms for setting out problems – is of course natural given his deu-
teronomic project).
Th e basic structure of the Diophantine proposition, as is well known, is
that of analysis: that is, Diophantus assumes, for each proposition, that it
has already been solved. Typically, he then terms the hypothetically found
element ‘number’ (the ς with which we are familiar) and notes the conse-
quences of the assumption that the conditions of the problem are met (in
the case quoted above: 20, together with the number, is four times 100,
lacking the number). Th is is then manipulated by various ‘algebraic’ opera-
tions (roughly, indeed, those later used by al-Khwarizmi, in his algebra)
until the number comes to be defi ned as monads. Th is then is quickly
verifi ed in a fi nal statement where the terms are put ‘in the positions’. In
the Arabic Diophantus, besides the quick verifi cation one also has a formal
synthesis, repeating the argumentation of the analysis backwards so that
one sees that, given the solution, the terms of the problem cannot fail to
hold. Sesiano believes this may be due to Hypatia; alternatively, this could
be due to some Arabic commentator. In any case, the systematic addition
of the synthesis may serve as another example of how deuteronomic texts
seek the goal of completion.