  Descartes's geometry did indeed involve the breaking down of  complex problems into simpler ones. More significant, however, was  his use of algebra in developing ‘analytic’ geometry as  it came to be called, which allowed geometrical problems to be  transformed into arithmetical ones and more easily solved. In  representing the ‘unknown’ to be found by  ‘x’, we can see the central role played in  analysis by the idea of taking something as ‘given’ and  working back from that, which made it seem appropriate to regard  algebra as an ‘art of analysis’, alluding to the  regressive conception of the ancients. Illustrated in analytic  geometry in its developed form, then, we can see all three of the  conceptions of analysis outlined in   Section 1.1  above, despite Descartes's own emphasis on the decompositional  conception. For further discussion of this, see the supplementary  section on   Descartes and Analytic Geometry.  