Estimation of the number of species at spatial scales too large to census directly is a longstanding ecological challenge. A recent comprehensive census of tropical arthropods and trees in Panama provides a unique opportunity to apply an inference procedure for up-scaling species richness and thereby make progress toward that goal. Confidence in the underlying theory is first established by showing that the method accurately predicts the species abundance distribution for trees and arthropods, and in particular accurately captures the rare tail of the observed distributions. The rare tail is emphasized because the shape of the species-area relationship is especially influenced by the numbers of rare species. The inference procedure is then applied to estimate the total number of arthropod and tree species at spatial scales ranging from a 6000 ha forest reserve to all of Panama, with input data only from censuses in 0.04 ha plots. The analysis suggests that at the scale of the reserve there are roughly twice as many arthropod species as previously estimated. For the entirety of Panama, inferred tree species richness agrees with an accepted empirical estimate, while inferred arthropod species richness is significantly below a previous published estimate that has been criticized as too high. An extension of the procedure to estimate species richness at continental scale is proposed.