When does a hyperbola meet its asymptote? Bounded infinities, fictions, and contradictions in Leibniz

Abstract

In his 1676 text De Quadratura Arithmetica, Leibniz distinguished infinita terminata from infinita interminata. The text also deals with the notion, originating with Desargues, of the perspective point of intersection at infinite distance for parallel lines. We examine contrasting interpretations of these notions in the context of Leibniz's analysis of asymptotes for logarithmic curves and hyperbolas. We point out difficulties that arise due to conflating these notions of infinity. As noted by Rodriguez Hurtado et al., a significant difference exists between the Cartesian model of magnitudes and Leibniz's search for a qualitative model for studying perspective, including ideal points at infinity. We show how respecting the distinction between these notions enables a consistent interpretation thereof.

19 pages

References in corpus (6)

• Two-track depictions of Leibniz's fictions
• Leibniz on bodies and infinities: rerum natura and mathematical fictions
• Is pluralism in the history of mathematics possible?
• Historical infinitesimalists and modern historiography of infinitesimals
• Three case studies in current Leibniz scholarship
• Evolution of Leibniz's thought in the matter of fictions and infinitesimals

Cited by in corpus (2)

• Of pashas, popes, and indivisibles
• Episodes from the history of infinitesimals