  Using $e^+e^-$ annihilation data sets collected with the BESIII detector, we
measure the cross sections of the processes $e^+e^- \to e^+e^-$ and $e^+e^- \to
\mu^+\mu^-$ at fifteen center-of-mass energy points in the vicinity of the
$J/\psi$ resonance. By a simultaneous fit to the measured, center-of-mass
energy dependent cross sections of the two processes, the combined quantities
$\Gamma_{ee} \Gamma_{ee} / \Gamma_{\rm tot}$ and $\Gamma_{ee} \Gamma_{\mu\mu} /
\Gamma_{\rm tot}$ are determined to be ($0.346 \pm 0.009$) and ($0.335 \pm
0.006$) keV, respectively, where $\Gamma_{ee}$, $\Gamma_{\mu\mu}$, and
$\Gamma_{\rm tot}$ are the electronic, muonic, and total decay widths of the
$J/\psi$ resonance, respectively. Using the resultant $\Gamma_{ee}
\Gamma_{\mu\mu} / \Gamma_{\rm tot}$ and $\Gamma_{ee} \Gamma_{ee} / \Gamma_{\rm
tot}$, the ratio $\Gamma_{ee} / \Gamma_{\mu\mu}$ is calculated to be $1.031 \pm
0.015$, which is consistent with the expectation of lepton universality within
about two standard deviations. Assuming lepton universality and using the
branching fraction of the $J/\psi$ leptonic decay measured by BESIII in 2013,
$\Gamma_{\rm tot}$ and $\Gamma_{ll}$ are determined to be ($93.0 \pm 2.1$) and
($5.56 \pm 0.11$) keV, respectively, where $\Gamma_{ll}$ is the average
leptonic decay width of the $J/\psi$ resonance.
