  We generalise the celebrated Peierls' argument to study the stability of a
long-range interacting classical solid. Long-range interaction implies that all
the atomic oscillators are coupled to each other via a harmonic potential,
though the coupling strength decays as a power-law $1/x^{\alpha}$, where $x$ is
the distance between the oscillators. We show that for the range parameter
$\alpha <2$, the long-range interaction dominates and the one-dimensional
system retains a crystalline order even at a finite temperature whereas for
$\alpha \geq2$, the long-range crystalline order vanishes even at an
infinitesimally small temperature. We also study the effect of quantum
fluctuations on the melting behaviour of a one-dimensional solid at T=0,
extending Peierls' arguments to the case of quantum oscillators.
