  The problem of finding small sets that block every line passing through a
unit square was first considered by Mazurkiewicz in 1916. We call such a set
{\em opaque} or a {\em barrier} for the square. The shortest known barrier has
length $\sqrt{2}+ \frac{\sqrt{6}}{2}= 2.6389\ldots$. The current best lower
bound for the length of a (not necessarily connected) barrier is $2$, as
established by Jones about 50 years ago. No better lower bound is known even if
the barrier is restricted to lie in the square or in its close vicinity. Under
a suitable locality assumption, we replace this lower bound by $2+10^{-12}$,
which represents the first, albeit small, step in a long time toward finding
the length of the shortest barrier. A sharper bound is obtained for interior
barriers: the length of any interior barrier for the unit square is at least $2
+ 10^{-5}$. Two of the key elements in our proofs are: (i) formulas established
by Sylvester for the measure of all lines that meet two disjoint planar convex
bodies, and (ii) a procedure for detecting lines that are witness to the
invalidity of a short bogus barrier for the square.
