Infinitesimal calculus

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Infinitesimal calculus was independently invented by both Leibniz and Newton 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 infinitesimal quantities was given a rigorous foundation by Abraham Robinson in the 1960s. Robinson's approach, called non-standard analysis, uses technical machinery from mathematical logic to create a theory of hyperreal numbers that interpret infinitesimals in a manner that allows a Leibniz-like development of the usual rules of calculus.

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

[edit] Varieties of infinitesimal calculus

Differential and integral calculus: together, the original infinitesimal calculus, due to Newton and Liebniz.
Standard calculus (based on the approach of Cauchy and Weierstrass)
Non-standard calculus (based on Robinson's approach to infinitesimals)