  We show that there exists a rational change of coordinates of Painlev\'e's P1
equation $y''=6y^2+x$ and of the elliptic equation $y''=6y^2$ after which these
two equations become analytically equivalent in a region in the complex phase
space where $y$ and $y'$ are unbounded. The region of equivalence comprises all
singularities of solutions of P1 (i.e. outside the region of equivalence,
solutions are analytic). The Painlev\'e property of P1 (that the only movable
singularities are poles) follows as a corollary. Conversely, we argue that the
Painlev\'e property is crucial in reducing P1, in a singular regime, to an
equation integrable by quadratures.

Classical Analysis and ODEs;Complex Variables