Routledge History of Philosophy Volume IV

170 RENAISSANCE AND SEVENTEENTH-CENTURY RATIONALISM
Moreover, the prism experiment shows that the effect does not depend on the
angle of incidence and that one refraction is sufficient for its production. Finally,
Descartes calculates from the refractive index of rainwater what an observer would
see when light strikes a drop of water at varying angles of incidence, and finds
that the optimum difference for visibility between incident and refracted rays is
for the former to be viewed at an angle of 41°–42° and the latter at an angle of
51°–52°, 29 which is exactly what the hypothesis predicts.
This procedure is similar to that followed in the Dioptrics, and in some
inspects to that followed in the Geometry. It is above all an exercise in problemsolving,
and the precedent for such an exercise seems to have been developed in
Descartes’s work in mathematics. Indeed, the later parts of the Rules turn
towards specifically mathematical considerations, and Rules 16–21 have such
close parallels with the Geometry that one can only conclude that they contain
the early parts of that work in an embryonic form. Rule 16 advises us to use ‘the
briefest possible symbols’ in dealing with problems, and one of the first things
the Geometry does is to provide us with the algebraic signs necessary for dealing
with geometrical problems. Rule 17 tells us that, in dealing with a new problem,
we must ignore the fact that some terms are known and some unknown; and
again one of the first directives in the Geometry is that we label all lines
necessary for the geometrical construction, whether these be known or unknown.
Finally, Rules 18–21 are formulated in almost identical terms in the Geometry. 30
There is something ironic in this, for one would normally associate a
mathematical model with a method which was axiomatic and deductive.
Certainly, if one looks at the great mathematical texts of Antiquity —Euclid’s
Elements or Archimedes’s On the Sphere and the Cylinder or Apollonius’s On
Conic Sections, for example—one finds lists of definitions and postulates and
deductive proofs of theorems relying solely on these. If one now turns to
Descartes’s Geometry, one finds something completely different. After a few
pages of introduction, mainly on the geometrical representation of the
arithmetical operations of multiplication, division and finding roots, we are
thrown into one of the great unsolved problems bequeathed by Antiquity—
Pappus’s locus problem for four or more lines, which Descartes then proceeds to
provide us with a method of solving.
Descartes’s solution to the Pappus problem is an ‘analytic’ one. In ancient
mathematics, a sharp distinction was made between analysis and synthesis.
Pappus, one of the greatest of the Alexandrian mathematicians, had distinguished
between two kinds of analysis: ‘theoretical’ analysis, in which one attempts to
discover the truth of a theorem, and ‘problematical analysis’, in which one
attempts to discover something unknown. If, in the case of theoretical analysis,
one finds that the theorem is false or if, in the case of problematical analysis, the
proposed procedure fails to yield what one is seeking, or one can show the
problem to be insoluble, then synthesis is not needed, and analysis is complete in
its own right. In the case of positive results, however, synthesis is needed, albeit
for different reasons. Synthesis is a difficult notion to specify, and it appears to