Chapter 10 briefly comments the influence of Euclid’s theory of magnitude and the Eudoxian theory of proportions on Kant’s thought in light of advances in mathematics from the early seventeenth century to the late eighteenth century. Despite these advances, mathematics was still often viewed as a science of magnitudes and their measure. The chapter also indicates further work to be done to understand Kant’s philosophy of geometry, arithmetic, algebra, and analysis.

Kant’s reworking of the Euclidean theory of magnitudes and reformation of the Leibnizian-Wolffian metaphysics of quantity are in service of his project of explaining the foundations of mathematical cognition and the mathematical character of experience. The previous chapters revealed that Kant’s account is fundamentally mereological. The categories of quantity allow us to represent the part–whole relations among magnitudes, and Kant’s understanding of the role of the categories of quantity, the nature of composition, and the definitions of extensive and intensive magnitudes are all mereological. This introduces a gap between Kant’s mereological account of magnitudes and Euclid’s notion of magnitude for the latter is implicitly defined by its role in the theory of proportions – a richer, mathematical notion of magnitude. This prompts a closer look at what makes Euclid’s understanding of magnitudes mathematical. This chapter argues that Euclid’s geometry presupposes a tacit theory of measurement that is general, pure, and concrete, a theory that crucially depends on the relation of equality. It traces these presuppositions through the Euclidean tradition. It then argues that Kant also tacitly assumed the theory of measurement, but that he was aware of the crucial role that equality plays in bridging the gap between mereology and mathematics.