Descartes considered that he needed Euclid's parallel postulate:
Descartes identified space and
the extension of matter, so geometry was, for him, about real physical space. But *geometric space, for Descartes, had to be Euclidean. This is because the theory of parallel
lines is crucial for Descartes' analytic geometry - not for Cartesian coordinates, which
Descartes did not have, but because he needed the theory of similar figures in order
to give meaning to expressions of arbitrary powers of x [23, p. 197]. Descartes was
the first person to justify using such powers. But an expression like x4 for Descartes
is not the volume of a 4-dimensional figure, but a line, which can be defined as the
fourth proportional to the unit line, x, and x3. That is, l/x = x3/x4. They are all
lines, and since all powers of x are lines, they can all be constructed geometrically -
but only if we have the theory of similar triangles, for which we need the theory of
Judith V. Grabiner, 'Why Did Lagrange "Prove" the Parallel Postulate?', The American Mathematical Monthly, Vol. 116, No. 1 (Jan., 2009), pp. 3-18: 7.
[23, p. 197] = L. Hodgkin, A History of Mathematics, Oxford: OUP, 205: 197.
Edit - Ram - added superscripts e.g. x3