Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow

TL;DR

This paper explores the historical development of the concepts of limit and shadow, highlighting how Fermat, Leibniz, Euler, and Cauchy used approximate equality and negligible terms, connecting their ideas to modern infinitesimal theories.

Contribution

It clarifies the historical evolution of limits and shadows, linking classical methods to contemporary infinitesimal frameworks and demonstrating their application to real functions.

Findings

Historical methods align with modern infinitesimal concepts

Application to decreasing rearrangements of functions

Bridges historical and modern analytic ideas

Abstract

Fermat, Leibniz, Euler, and Cauchy all used one or another form of approximate equality, or the idea of discarding "negligible" terms, so as to obtain a correct analytic answer. Their inferential moves find suitable proxies in the context of modern theories of infinitesimals, and specifically the concept of shadow. We give an application to decreasing rearrangements of real functions.