  We calculate the $S$-wave charmonium spectroscopy using the Hamiltonian with
the non-relativistic QCD (NRQCD) potential. The logarithmic factor $\ln \mu r$,
appearing in the next-to-leading order QCD loop corrections to the potential,
is expanded about $r=1/\mu$, where $\mu$ corresponds to the typical charmonium
scale. The resulting potential characterized by the Coulombic and linear
components is consistent with the form of the Cornell potential. We obtain
$\chi^2$ fitting results for the masses of the $S$-wave charmonium states,
$\eta_c(1^1S_0)$, $J/\psi(1^3S_1)$, $\eta_c(2^1S_0)$, and $\psi(2^3S_1)$ in
remarkable accordance with data. Our results successfully account for the
hyperfine splitting for the 1S state as well as for the 2S state. We further
use the three best fit parameters: the charm quark mass $m_c$, coupling
constant $\alpha_s$ and the corresponding scale $\mu$ to predict the $S$-wave
mass spectrum with $n\leq 6$. The hints for results are discussed.
