We find an invariant characterization of planar webs of maximum rank. For
4-webs, we prove that a planar 4-web is of maximum rank three if and only if it
is linearizable and its curvature vanishes. This result leads to the direct
web-theoretical proof of the Poincar\'{e}'s theorem: a planar 4-web of maximum
rank is linearizable. We also find an invariant intrinsic characterization of
planar 4-webs of rank two and one and prove that in general such webs are not
linearizable. This solves the Blaschke problem ``to find invariant conditions
for a planar 4-web to be of rank 1 or 2 or 3''. Finally, we find invariant
characterization of planar 5-webs of maximum rank and prove than in general
such webs are not linearizable.Comment: 43 page
