  Eccentricity of wide binaries is difficult to measure due to their long
orbital periods. With Gaia's high-precision astrometric measurements,
eccentricity of a wide binary can be constrained by the angle between the
separation vector and the relative velocity vector (the $v$-$r$ angle). In this
paper, by using the $v$-$r$ angles of wide binaries in Gaia Early Data Release
3, we develop a Bayesian approach to measure the eccentricity distribution as a
function of binary separations. Furthermore, we infer the eccentricities of
individual wide binaries and make them publicly available. Our results show
that the eccentricity distribution of wide binaries at $10^2$ AU is close to
uniform and becomes superthermal at $>10^{3}$ AU, suggesting two formation
mechanisms dominating at different separation regimes. The close binary
formation, most likely disk fragmentation, results in a uniform eccentricity
distribution at $<10^{2}$ AU. The wide binary formation that leads to highly
eccentric wide binaries at $>10^{3}$ AU may be turbulent fragmentation and/or
the dynamical unfolding of compact triples. With Gaia, measuring eccentricities
is now possible for a large number of wide binaries, opening a new window to
understanding binary formation and evolution.
