  We provide a new proof of the near-critical scaling relation $\beta=\xi_1\nu$
for Bernoulli percolation on the square lattice already proved by Kesten in
1987. We rely on a novel approach that does not invoke Russo's formula, but
rather relates differences in crossing probabilities at different scales. The
argument is shorter and more robust than previous ones and is more likely to be
adapted to other models. The same approach may be used to prove the other
scaling relations appearing in Kesten's work.
