Page "Infinitesimal" Paragraph 3

The use of infinitesimals by Leibniz relied upon heuristic principles, such as the Law of Continuity: what succeeds for the finite numbers succeeds also for the infinite numbers and vice versa ; and the Transcendental Law of Homogeneity that specifies procedures for replacing expressions involving inassignable quantities, by expressions involving only assignable ones.

The 18th century saw routine use of infinitesimals by mathematicians such as Leonhard Euler and Joseph Lagrange.

Augustin-Louis Cauchy exploited infinitesimals in defining continuity and an early form of a Dirac delta function.

As Cantor and Dedekind were developing more abstract versions of Stevin's continuum, Paul du Bois-Reymond wrote a series of papers on infinitesimal-enriched continua based on growth rates of functions.

Du Bois-Reymond's work inspired both Émile Borel and Thoralf Skolem.

Borel explicitly linked du Bois-Reymond's work to Cauchy's work on rates of growth of infinitesimals.

Skolem developed the first non-standard models of arithmetic in 1934.

A mathematical implementation of both the law of continuity and infinitesimals was achieved by Abraham Robinson in 1961, who developed non-standard analysis based on earlier work by Edwin Hewitt in 1948 and Jerzy Łoś in 1955.

The hyperreals implement an infinitesimal-enriched continuum and the transfer principle implements Leibniz's law of continuity.

The standard part function implements Fermat's adequality.