  We describe the properties of the Ly$\alpha$ forest in the column density
range $\nhi \geq 2\times10^{12}$ cm$^{-2}$ based on 1056 lines in the
wavelength range 4300--5100\AA measured in extremely high S/N, $R=36,000$
spectra of four quasars. The column density distribution is well described by a
-1.5 power law to $2\times10^{12}$ cm$^{-2}$, below which limit confusion
becomes too severe to measure a spectrum of individual clouds. The distribution
of $b$ values shows a well-defined lower envelope with a cutoff at $b=20\kms$
corresponding to a cloud temperature of 24,000 K. There is only a very small
fraction (less than 1%) of narrow line clouds which cannot be identified with
metal-lines. From modeling the Ly$\alpha$ absorption lines as complexes of
clouds each with thermal broadening corresponding to $b_c$ we find the $b$
distribution can be understood if there is a mean of 3.25 clouds per absorption
line with a spread in velocity centroids characterized by a dispersion of
$10.75\kms$.
