The rectification of quadratures as a central foundational problem for the early Leibnizian calculus

Transcendental curves posed a foundational challenge for the early calculus, as they demanded an extension of traditional notions of geometrical rigour and method. One of the main early responses to this challenge was to strive for the reduction of quadratures to rectifications. I analyse the arguments given to justify this enterprise and propose a hypothesis as to their underlying rationale. I then go on to argue that these foundational concerns provided the true motivation for much ostensibly applied work in this period, using Leibniz’s envelope paper of 1694 as a case study.