Chapter Five shifts focus from the history of formal science to the history of natural science, including medicine. In doing so, it builds on the argument from the previous chapter that science is a process of thematisation in which informal and indeterminate knowledge is thematised and articulated in a more formal and determinate way. This suggests, however, that scientists only discover what they already know. Indeed, both SSK and Heidegger attribute a circle to scientific reasoning. Yet, this circle is not vicious: it was already recognised by the second-century Greek physician, Galen, and became a topic of concentrated interest for physicians at the University of Padua during the Renaissance. In their view, the production of formal out of informal knowledge is an importantly empirical process. Heidegger argues that, in the early-modern period, this process was transformed into what he calls mathēsis. The rules guiding empirical inquiry were now ‘mathematicised,’ that is, consolidated as a coherent set of basic principles, a ‘basic blueprint.’ Mathēsis is a ‘reciprocal relation’ between empirical work with things, on the one hand, and the metaphysical projection of the thingness of things, on the other. Heidegger thus viewed early modern science as combining both mathematical and empirical elements. Kochan compares his account to the respective metaphysical and empiricist accounts of the historians of science Alexandre Koyré and Peter Dear. He argues that mathēsis was, above all, a transformation in the role played by Aristotelian “final causes” in early-modern natural philosophy. This challenges the historiographic commonplace, also accepted by SSK practitioners, that final causes were abolished from the new natural philosophy. Extrapolating from Heidegger’s work, Kochan argues that final causes, despite seventeenth-century rhetoric to the contrary, remained central to early-modern scientific practice.