A problem facing the excavator of a kiln site is that of reconsturcting the pottery found in order to describe the production of the kilns. An alternative approach to the physical reconstructon of vessels is presented. By using statistical models of the relationships between rim and base sherds, and geometric models of vessel forms, reconstructions can be made mathematically on the basis of a simple empirical classification of rim and base forms. Vessel categories are an outcome of the reconstruction process, in that no further classification is needed. The waster output of the site can be described in terms of a scheme of hypothetical vessels, which taken as a whole give a complete picture of the bulk of the output. Two advantages are claimed for this approach: (i) it enables the reconstruction of vessel forms to be carried out on very fragmentary material, an otherwise impossible, or at least enormous, task, (ii) it enables the waster output to be described efficiently and effectively.