  We introduce a technique to measure gravitational lensing magnification using
the variability of type I quasars. Quasars' variability amplitudes and
luminosities are tightly correlated, on average. Magnification due to
gravitational lensing increases the quasars' apparent luminosity, while leaving
the variability amplitude unchanged. Therefore, the mean magnification of an
ensemble of quasars can be measured through the mean shift in the
variability-luminosity relation. As a proof of principle, we use this technique
to measure the magnification of quasars spectroscopically identified in the
Sloan Digital Sky Survey, due to gravitational lensing by galaxy clusters in
the SDSS MaxBCG catalog. The Palomar-QUEST Variability Survey, reduced using
the DeepSky pipeline, provides variability data for the sources. We measure the
average quasar magnification as a function of scaled distance (r/R200) from the
nearest cluster; our measurements are consistent with expectations assuming NFW
cluster profiles, particularly after accounting for the known uncertainty in
the clusters' centers. Variability-based lensing measurements are a valuable
complement to shape-based techniques because their systematic errors are very
different, and also because the variability measurements are amenable to
photometric errors of a few percent and to depths seen in current wide-field
surveys. Given the data volume expected from current and upcoming surveys, this
new technique has the potential to be competitive with weak lensing shear
measurements of large scale structure.
