The authors have calculated the number of self-avoiding polygons on the square lattice to 56 steps, and the caliper size to 54 steps. Analysis of the generating function permits estimates of the connective constant, mu =2.638 1585+or-10-6 and the critical exponents alpha =0.500 06+or-0.000 06 and v=0.753+or-0.007. The singularity structure of the polygon generating function is found to be consistent with a correction to the scaling exponent Delta =1.5, as predicted by Nienhuis (1982, 1984). The confluent part, however, maps into the additive analytic term due to the value of the exponent alpha .