The Analyst - Influence

Influence

Two years after this publication, Thomas Bayes published anonymously "An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of the Analyst" (1736), in which he defended the logical foundation of Isaac Newton's calculus against the criticism outlined in The Analyst. Colin Maclaurin's two-volume Treatise of Fluxions published in 1742 also began as a response to Berkeley attacks, intended to show that Newton's calculus was rigorous by reducing it to the methods of Greek geometry (Grabiner 1997).

Despite these attempts calculus continued to be developed using non-rigorous methods until around 1830 when Augustin Cauchy, and later Bernhard Riemann and Karl Weierstrass, redefined the derivative and integral using a rigorous definition of the concept of limit. The concept of using limits as a foundation for calculus had been suggested by d'Alembert, but d'Alembert's definition was not rigorous by modern standards (Burton 1997). The concept of limits had already appeared in the work of Newton (Pourciau 2001), but was not stated with sufficient clarity to hold up to the criticism of Berkeley.(Edwards 1994)

In 1966, Abraham Robinson introduced Non-standard Analysis, which provided a rigorous foundation for working with infinitely small quantities. This provided another way of putting calculus on a mathematically rigorous foundation that was in a similar spirit to the way calculus was done before the (ε, δ)-definition of limit had been fully developed.