  Precisely measured electron-proton elastic scattering cross sections [Phys.
Rev. Lett. {\bf 105}, 242002 (2010)] are reanalyzed to evaluate their strength
for determining the rms charge radius ($R_{\rm E}$) of the proton. More than
half of the cross sections at lowest $Q^2$ are fit using two single-parameter
form-factor models, with the first based on a dipole parametrization, and the
second on a linear fit to a conformal-mapping variable. These low-$Q^2$ fits
extrapolate the slope of the form factor to $Q^2$=0 and determine $R_{\rm E}$
values of approximately 0.84 and 0.89~fm, respectively. Fits spanning all
$Q^2$, in which the single constants are replaced with cubic splines at larger
$Q^2$, lead to similar results for $R_{\rm E}$. We conclude that the scattering
data is consistent with $R_{\rm E}$ ranging from at least 0.84 to 0.89~fm, and
therefore cannot resolve the discrepancy between determinations of $R_{\rm E}$
made using muonic and electronic hydrogen-atom spectroscopy.
