  We perform a new analysis of electron-proton scattering data to determine the
proton electric and magnetic radii, enforcing model-independent constraints
from form factor analyticity. A wide-ranging study of possible systematic
effects is performed. An improved analysis is developed that rebins data taken
at identical kinematic settings, and avoids a scaling assumption of systematic
errors with statistical errors. Employing standard models for radiative
corrections, our improved analysis of the 2010 Mainz A1 Collaboration data
yields a proton electric radius $r_E = 0.895(20)$ fm and magnetic radius $r_M =
0.776(38)$ fm. A similar analysis applied to world data (excluding Mainz data)
implies $r_E = 0.916(24)$ fm and $r_M = 0.914(35)$ fm. The Mainz and world
values of the charge radius are consistent, and a simple combination yields a
value $r_E = 0.904(15)$ fm that is $4\sigma$ larger than the CREMA
Collaboration muonic hydrogen determination. The Mainz and world values of the
magnetic radius differ by $2.7\sigma$, and a simple average yields $r_M=
0.851(26)$ fm. The circumstances under which published muonic hydrogen and
electron scattering data could be reconciled are discussed, including a
possible deficiency in the standard radiative correction model which requires
further analysis.
