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Infinitesimal calculus'''Infinitesimal calculus''' was independently invented by both Gottfried Leibniz|Leibniz and Isaac Newton|Newton in the 1660s, drawing on the work of such mathematicians as Isaac Barrow|Barrow and Rene Descartes|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 a number of authors, most notably Michel Rolle and
George Berkeley|Bishop Berkeley. Several mathematicians, including Colin Maclaurin|Maclaurin, attempted to prove the soundness of using infinitesimals, but it would be 150 years later, due to the work of Augustin Louis Cauchy|Cauchy and Karl Weierstrass|Weierstrass, where a means was finally found to avoid mere "notions" of infinitely small quantities, that the foundations of differential and integral calculus were made firm. In his work Weierstrass formalized the concept of Limit of a function|limit which eliminated the need for infinitesimals. Eventually due to the work of 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.
Varieties of infinitesimal calculus
- Differential calculus|Differential and Integral calculus|integral calculus: together, the original ''infinitesimal calculus'', due to Newton and Leibniz.
- Standard calculus (based on the approach of Weierstrass)
- Non-standard calculus (based on Robinson's approach to infinitesimals)
Category:Calculus
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