  We use a sample of 53 galaxy clusters at 0.03 < z < 0.1 with available masses
derived from the caustic technique and with velocity dispersions computed using
208 galaxies on average per cluster, in order to investigate the scaling
between richness, mass and velocity dispersion. A tight scaling between
richness and mass is found, with an intrinsic scatter of only 0.19 dex in mass
and with a slope one, i.e. clusters which have twice as many galaxies are twice
as massive. When richness is measured without any knowledge of the cluster mass
or linked parameters (such as r200), it can predict mass with an uncertainty of
0.29+/-0.01 dex. As a mass proxy, richness competes favourably with both direct
measurements of mass given by the caustic method, which has typically 0.14 dex
errors (vs 0.29) and X-ray luminosity, which offers a similar 0.30 dex
uncertainty. The similar performances of X-ray luminosity and richness in
predicting cluster masses has been confirmed using cluster masses derived from
velocity dispersion fixed by numerical simulations. These results suggest that
cluster masses can be reliably estimated from simple galaxy counts, at least at
the redshift and masses explored in this work. This has important applications
in the estimation of cosmological parameters from optical cluster surveys,
because in current surveys clusters detected in the optical range outnumber, by
at least one order of magnitude, those detected in X-ray. Our analysis is
robust from astrophysical and statistical perspectives. The data and code used
for the stochastic computation is distributed with the paper. [Abridged]
