  Deligne's regularity criterion for an integrable connection $\nabla$ on a
smooth complex algebraic variety $X$ says that $\nabla$ is regular along the
irreducible divisors at infinity in some fixed normal compactification of $X$
if and only if the restriction of $\nabla$ to every smooth curve on $X$ is
fuchsian (i.e. has only regular singularities at infinity). The "only if" part
is the difficult implication. Deligne's proof is transcendental and uses
Hironaka's resolution of singularities.
  We present a purely algebraic proof of this implication which does not use
resolution beyond the case of plane curves. It relies upon a study of the
formal structure of integrable connections on surfaces with (possibly
irregular) singularities along a divisor with normal crossings.
