Infinitesimal calculus

Infinitesimal calculus was invented by Newton and Leibniz in the 1660s, drawing on the work of such mathematicians as Barrow and Descartes. It consisted of differential calculus and integral calculus, used for the techniques of differentiation and integration respectively.

The use of infinitesimal quantities in early calculus was not proven to be rigorous, and was fiercely criticized by numerous philosophers, most notably Bishop Berkeley. Several mathematicians, including Maclaurin, attempted to prove the soundness of using infinitesimals, but it was not until the work of Cauchy and Weierstrass, which found a means to avoid notions of infinitely small quantities, that the foundations of differential and integral calculus were made firm. In their work they formalized the concept of limit which eliminated the need for infinitesimals. Eventually due to the work of Cauchy and Weierstrass, it became common to base calculus on limits instead of infinitesimal quantities. The name "infinitesimal calculus" was commonly applied to it.

The use of infinitesimals quantities was given a rigorous logical foundation by Abraham Robinson in the 1960s. During the period from Weierstrass to Robinson, infinitesimals were widely considered to have been a regrettable mistake, as can be learned from many histories of mathematics written in this era . Robinson's work largely rehabilitated them.

Colloquially, it can be used to refer to the approach formalized by Weierstrass, which has also come to be known as the standard calculus.

Varieties of infinitesimal calculus

• Standard calculus (based on the approach of Cauchy and Weierstrass)
• Non-standard calculus (based on Robinson's approach to infinitesimals)