This essay proposes that mathematical biology can be used as a fruitful exemplar for the introduction of scientific principles to history. After reviewing the antecedents of the application of mathematics to biology, in particular evolutionary biology, I describe in detail a mathematical model of cultural diffusion based on an analogy with population genetics. Subsequently, as a case study, this model is used to investigate the dynamics of the early modern European witch-crazes in Bavaria, England, Hungary, and Finland. In the second part of the article, I sketch the methodological significance of this type of "scientific history" and, in particular, I identify three lessons that mathematical biology can contribute to historiography. The first lesson is on the fundamental distinction between an agent's purposes and structural social processes. I argue that mathematical modeling can be fruitfully applied to describe social processes, while agents' purposes ought to be addressed following a hermeneutic tradition. The second lesson is on the aim of mathematical modeling. Here I argue that the object of modeling, rather than being the prediction or retrodiction of events (a deductive-nomological approach), is the understanding of the factors involved in the dynamics of social processes (an analytic-descriptive approach). Finally, the third lesson is on the new understanding of science after the collapse of the standard view. In summary, while mathematical modeling can provide an extremely powerful approach to clarify the dynamics of certain macro-historical processes, scientific methods ought to be regarded as a complement to, not a substitute for, classical historiography.