Just as a debate about the fundamental nature of physical entities arose after Descartes, a similar issue arose after Newton. Like Descartes, but of course with very different epistemological and methodological considerations, Newton held that the most fundamental conserved quantity was “quantity of motion” or momentum. Leibniz opposed this, arguing instead for vis viva or “living force.” This controversy introduced two kinds of problems: 1) whether and how empirical proofs could be generated for metaphysical conceptions, and consequently 2) how to understand the relationship between metaphysics and experimental philosophy. These concerns were handled quite differently by two important philosophers: Gravesande and Du Châtelet. Their moves partly resolved older debates, but also partly reconfigured them into new questions we are still attempting to answer.

The Scientific Revolution completely transfigured the European intellectual landscape. Old divisions disappeared, while new fault lines emerged. Ancient philosophical sects had been replaced by new schools, featuring novel masters, disciples, and methodological commitments. However, the new schools still engaged in antagonistic discourse, attacking one another along new fronts—e.g., Cartesians against Gassendists, Newtonians against Leibnizians. This chapter presents the diverse philosophical camps that arose in the later stages of the Scientific Revolution by noting a shift in the use of the term ‘sect’. While it still signified something like an Ancient philosophical school for some, it could also take on a more negative polemical meaning, intended to disparage one’s opponents. Moreover, the individuals associated with the “sects” did not all faithfully subscribe to explicit, coherent, and systematic programs. On the contrary, declaring membership of a sect was as often a signal of opposition as of allegiance to a methodology or theory. Despite calls for conciliatory research programs, sectarian attitudes did not disappear by 1750, but delineated new battle lines between the Cartesians, the Leibnizians, and the Newtonians.

During the Scientific Revolution, philosophers wondered how best to understand space. Many debates revolved around the account advanced in Descartes’s Principles of Philosophy (1644), and this chapter treats it as a focal point. Descartes argued for a return to the Aristotelian view that there is no difference in reality between space and matter, entailing that empty space—space empty of matter—is impossible. Over the next century, all kinds of philosophers attacked this position, and this chapter takes their rejections of Cartesian space as a starting point for exploring alternative views. A varied selection of philosophers who reject Cartesian space are discussed, in chronological order: Henry More, Samuel Clarke, Isaac Newton, Catharine Cockburn, and Gottfried Wilhelm Leibniz. The sheer breadth of alternative theories of space they advance demonstrates the metaphysical richness of this era. Nonetheless, there is a deep agreement among their alternatives: all the accounts agree on the features of space. This base agreement set the scene for Kant’s theory of space, advanced after the Scientific Revolution ended.

During the seventeenth century, the advent of what were known as the “common” and “new” analyses fundamentally changed the landscape of European mathematics. The widely accepted narrative is that these analyses, analytic geometry and calculus (mostly due to Descartes and Leibniz, respectively), occasioned a transition from geometrical to symbolic methods. In dealing with the science of motion, mathematicians abandoned the language of proportion theory, as found in the works of Galileo, Huygens, and Newton, and began employing the Newtonian and Leibnizian calculi when differential and fluxional equations first appeared in the 1690s. This was the advent of a more abstract way of practicing mathematics, which culminated with the algebraic approach to calculus and mechanics promoted by Euler and Lagrange in the eighteenth century. In this chapter, it is shown that geometrical interpretations and mechanical constructions still played a crucial role in the methods of Descartes, Leibniz, and their immediate followers. This is revealed by the manner in which they handled equations and how they sought their solutions. The passage from proportions to equations did not occur in a single step; it was a process that took a century to reach completion.

In the last thirty years, both the belief that the mechanical philosophy is an adequate historical category and the conviction that it made a positive contribution to the sciences were deconstructed. Hence the question addressed in this chapter: What to do with the mechanical philosophy? The chapter begins with a terminological enquiry about ‘mechanical philosophy’ as an emic category, and compares the use of the term on the Continent and across the Channel. It is then suggested that we examine controversies in which mechanical philosophers, having defined themselves in opposition to other natural philosophers, made explicit their expectations with regard to physical explanations. Three such controversies are discussed: one about the motion of the heart (Descartes versus Plempius); one about the elasticity of the air (Boyle versus More); and one about the universal attraction of bodies (Huygens and Leibniz versus Newton). Finally, to counter negative evaluations of the mechanical philosophy, the chapter points out the cognitive advantages of structural explanations, to which the mechanical explanations belonged.

The differential equations as written by Leibniz and by his immediate followers look very similar to the ones in use nowadays. They are familiar to our students of mathematics and physics. Yet, in order to make them fully compatible with the conventions adopted in our textbooks, we have to change just a few symbols. Such “domesticating” renderings, however, generate a remarkable shift in meaning, making those very equations – when so reformulated – not acceptable for their early-modern authors. They would have considered our equations, as we write them, wrong and corrected them back, for they explicitly adopted tasks and criteria different from ours. In this chapter, focusing on a differential equation formulated by Johann Bernoulli in 1710,I evaluate the advantages and risks inherent in these anachronistic renderings.

Starting with a brief examination of Nicolas Fréret’s essay L’origine des Français et de leur établissement dans les Gaules, the third chapter looks at the onset of the eighteenth-century debate on the nation’s origins. The classic opposition between the Gallic and the Frankish theses is reassessed. In particular, the latter is considered in relation to the shift from the dominant legalist and royalist paradigm to the cultural and ethnic one. It will be argued that both Germanists and Romanists, by discussing the origins of France in ethnic terms, fuelled, independently of their immediate aims, a discourse that was subversive for it inevitably undermined the royalist national narrative and, therefore, monarchical authority as such. The chapter examines the writings of authors such as Fréret, the Père Daniel, René-Joseph de Tournemine, Joseph Vaissètte, the abbé de Trianon, Étienne Lauréault de Foncemagne, and Jean Baptiste Dubos.

By the turn of the twentieth century, many philosophers claimed that Kant’s philosophy of mathematics (and, indeed, his entire philosophy) was undermined by the real possibility of geometries in which Euclid’s axiom of parallels is false. It is surprisingly not well known that Kant – though never discussing parallel lines in his published work – did write a series of unpublished notes on the philosophical problems posed by parallel lines. This paper presents the argument and chief issues of these notes. Kant’s main point in these notes is that neither Euclid’s definition of parallel lines, nor the alternative definition proposed by Leibniz and Christian Wolff, fulfills the requirements on mathematical definitions that Kant explicates in the Critique of Pure Reason. Thus, these notes show that Kant did reflect on the problems posed by the theory of parallel lines and recognized that it was uniquely problematic. Moreover, Kant’s reflections yield new insights about his philosophy of mathematics. In particular, Kant’s attitude toward the theory of parallels demonstrates that he had a novel theory of mathematical definitions, axioms, and postulates – a theory that Kant’s contemporary readers have largely misunderstood.