Routledge History of Philosophy Volume IV

100 SCIENCE AND MATHEMATICS UP TO DESCARTES
The paradox suggested that the ancients had knowledge of a particularly fruitful
way of discovering new mathematical truths. As Descartes put it, ‘We perceive
sufficiently that the ancient Geometricians made use of a certain analysis which
they extended to the resolution of all problems, though they grudged the secret to
posterity.’ 12 Such suspicions were partially vindicated at the beginning of this
century (although without evidence of a grudge) by the discovery of a lost work
by Archimedes, known as the Method (Ephodos), in which the author showed
how to use theoretical mechanics and considerations of indivisibles in order to
discover new theorems about equating areas and volumes between different
plane and solid figures. These theorems were then open to proof by more rigorous
geometrical methods, particularly the reductiones ad absurdum involved in the
so-called method of exhaustion, in which inequalities of the relevant areas and
volumes were shown to lead to contradictions.
There were a few brief and vague ancient references to Archimedes’
‘method’, but rather more to the procedures of ‘analysis and synthesis’, whose
exact interpretation have caused much scholarly perplexity. In a famous passage
Pappus wrote:
Analysis is the path from what one is seeking, as if it were established, by
way of its consequences, to something that is established by synthesis.
That is to say, in analysis we assume what is sought as if it has been
achieved, and look for the thing from which it follows, and again what
comes before that, until by regressing in this way we come upon some one
of the things that are already known, or that occupy the rank of a first
principle. We call this kind of method ‘analysis’, as if to say anapalin lysis
(reduction backward). In synthesis, by reversal, we assume what was
obtained last in the analysis to have been achieved already, and, setting
now in natural order, as precedents, what before were following, and fitting
them to each other, we attain the end of the construction of what was
sought. This is what we call ‘synthesis’. 13
Pappus went on to distinguish theorematic (zetetikos) and problematic
(poristikos) analysis. The general thrust of the passages is clear. In theorematic
analysis we work backwards towards the first principles from which a theorem
follows, and in problematic analysis our goal is the solution of a problem, say the
finding of a figure whose area or other features will meet certain conditions,
while in synthesis we in some sense prove our results. But when we descend
towards the logical niceties a host of difficulties appear, 14 and these may
themselves have made the subject a particularly suitable candidate for
seventeenth-century transformation.
An especially important figure in this process was François Viète, who drew
on both the Diophantine ‘arithmetical’ tradition and more general algebraic
traditions, as well as those deriving from primarily geometrical works. Viète’s
most striking innovation in assuming a problem solved was to name the ‘unknown’