352 reviel netz
are problems solved in the most appropriate way? Th e task, then, is to show
how the off ered solution to the problem comes out naturally, given the
terms themselves. Th is is what the analysis does: it reaches the solution to
the problem, as a demonstrative consequence of the terms that the problem
had set out. Th us analysis need not discover a solution, nor prove its truth
(though this is a by-product of a successful analysis). Its aim may simply be
to display how the solution emerges naturally out of the conditions set out
by the problem. Th e aim of the proof in an analysis is not in its conclusion,
but in the process itself: it lays down a rational bridge leading from the
terms of the problem, to the solution off ered.
If this is true, then Diophantus should have similar expectations from
his own analyses. But in fact this goal of the analyses emerges from his
choice of the form itself. He avoided schoolroom algebraical presentations
with their take-it-or-leave-it approach: probably, within the overall expec-
tations of elite literate Greek mathematics, this could not do. Such texts
were driven by a culture whose central mode was persuasion, and the text
therefore had to display a rational, persuasive structure. But neither did
Diophantus aim primarily to show the reason why. He could easily have
chosen to adopt a strictly theoretical approach to numerical problems,
as, one may perhaps say, certain Arabic mathematicians did much later;
his fl uency in extending numerical problems and solving quite complex
ones suggest that, in sheer terms of mathematical intelligence, he was
quite capable of such a theoretical approach. But he did not aim at such.
He understood his task in a more limited way – not so much to open up a
new fi eld of theoretical inquiry, but rather to arrange a fi eld inherited from
the past. Th e only constraint was that this fi eld should display a rational,
persuasive structure: Diophantus’ analyses served just that. Instead of
the take-it-or-leave-it of lay and school algebra, Diophantus would have
rational bridges leading from the terms of the problems to their solutions.
Th us he would show that the solutions are not arbitrary, but arise naturally
given the terms set out by the problems. 2626 It is interesting to notice in this context the cases where Diophantus departs from the strict
analytic presentation. Th is happens, in particular, where he has to make some arbitrary choices
of numerical values. Th en he sometimes takes us into his confi dence, explaining the rational
basis for his next move. For example in v .2: ‘but 16 monads are not some arbitrary number,
but are a square which, added to 20 monads, makes a square as well. So I am brought to
investigate: which square has a fourth bigger than 20 monads, and taken together with twenty
monads makes a square. So the square results to be bigger than 80. But 81 is a square bigger
than 80... ’ – this entire discussion is there to explain why, in an arbitrary move, Diophantus
picks the numerical value 9 and none else. Th e choice is arbitrary; but Diophantus shows that it
is not irrational, and is somehow suggested by the values at hand.