  We describe an algorithm which directly determines the quintessence potential
from observational data, without using an equation of state parametrisation.
The strategy is to numerically determine observational quantities as a function
of the expansion coefficients of the quintessence potential, which are then
constrained using a likelihood approach. We further impose a model selection
criterion, the Bayesian Information Criterion, to determine the appropriate
level of the potential expansion. In addition to the potential parameters, the
present-day quintessence field velocity is kept as a free parameter. Our
investigation contains unusual model types, including a scalar field moving on
a flat potential, or in an uphill direction, and is general enough to permit
oscillating quintessence field models. We apply our method to the `gold' Type
Ia supernovae sample of Riess et al. (2004), confirming the pure cosmological
constant model as the best description of current supernovae
luminosity-redshift data. Our method is optimal for extracting quintessence
parameters from future data.
