QUANTITATIVE studies of patterns of diversification and extinction throughout geological time have come to rely on databases assembled by Sepkoski(1). A new database has been presented by Benton(2), who argues that the increase in diversity is exponential, that "there is no need to assume equilibrium levels of global diversity, nor to apply logistic models to the investigation of past diversification patterns", and that mass extinctions played no special role in the diversification of life, However, we find that diversification can be predicted by a simple model in which long periods of growth obeying a logistic equation are sporadically interrupted by brief events leading to mass extinction, The initial value of diversity immediately after a mass extinction, initial growth rate and subsequent equilibrium level determine each segment of the growth curve, The larger mass extinctions that came after long periods without one seem to have been followed by larger equilibrium levels of diversity.