Mathematical Innovation and Tradition: The Cartesian Common and the Leibnizian New Analyses

During the seventeenth century, the advent of what were known as the “common” and “new” analyses fundamentally changed the mathematical landscape of European mathematics. The widely accepted narrative is that these seventeenth-century analyses, algebra and calculus, mostly due respectively to  Descartes and Leibniz, occasioned a transition from geometrical to symbolic methods; i.e., from Euclidean proportion theory to equations. In this chapter, I will show that geometrical interpretations and mechanical constructions still played a crucial role in the mathematical methods of Descartes, Leibniz, and their immediate followers.