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Infinitesimal calculus
 
 
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Infinitesimal calculus was independently invented by both Leibniz
Gottfried Leibniz

Gottfried Wilhelm Leibniz was a Germany polymath who wrote primarily in Latin and French language.He occupies an equally grand place in both the history of philosophy and the history of mathematics....
 and Newton
Isaac Newton

Sir Isaac Newton, Fellow of the Royal Society was an English people physicist, mathematician, Astronomy, Natural philosophy, Alchemy, and Theology and one of the the 100 in human history....
 in the 1660s, drawing on the work of such mathematicians as Barrow
Isaac Barrow

Isaac Barrow was an Kingdom of England scholar and mathematician who is generally given credit for his early role in the development of calculus; in particular, for the discovery of the fundamental theorem of calculus....
 and Descartes
René Descartes

Ren? Descartes , , also known as Renatus Cartesius , was a French philosophy, mathematician, scientist, and writer who spent most of his adult life in the Dutch Republic....
. It consisted of differential calculus
Differential calculus

Differential calculus, a field in mathematics, is the study of how function s change when their inputs change. The primary object of study in differential calculus is the derivative....
 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
George Berkeley

George Berkeley , also known as Bishop Berkeley, was an Irish people philosopher. His primary philosophical achievement was the advancement of a theory he called "immaterialism" ....
.






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Infinitesimal calculus was independently invented by both Leibniz
Gottfried Leibniz

Gottfried Wilhelm Leibniz was a Germany polymath who wrote primarily in Latin and French language.He occupies an equally grand place in both the history of philosophy and the history of mathematics....
 and Newton
Isaac Newton

Sir Isaac Newton, Fellow of the Royal Society was an English people physicist, mathematician, Astronomy, Natural philosophy, Alchemy, and Theology and one of the the 100 in human history....
 in the 1660s, drawing on the work of such mathematicians as Barrow
Isaac Barrow

Isaac Barrow was an Kingdom of England scholar and mathematician who is generally given credit for his early role in the development of calculus; in particular, for the discovery of the fundamental theorem of calculus....
 and Descartes
René Descartes

Ren? Descartes , , also known as Renatus Cartesius , was a French philosophy, mathematician, scientist, and writer who spent most of his adult life in the Dutch Republic....
. It consisted of differential calculus
Differential calculus

Differential calculus, a field in mathematics, is the study of how function s change when their inputs change. The primary object of study in differential calculus is the derivative....
 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
George Berkeley

George Berkeley , also known as Bishop Berkeley, was an Irish people philosopher. His primary philosophical achievement was the advancement of a theory he called "immaterialism" ....
. Several mathematicians, including Maclaurin
Colin Maclaurin

Colin Maclaurin was a Scotland mathematician. Due to changes in orthography since that time , his surname is alternatively written MacLaurin....
, attempted to prove the soundness of using infinitesimals, but it was not until the work of Cauchy and Weierstrass
Karl Weierstrass

Karl Theodor Wilhelm Weierstrass was a Germany mathematics who is often cited as the "father of modern mathematical analysis"....
, 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
Abraham Robinson

Abraham Robinson was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and transfinite number numbers were incorporated into mathematics....
 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
Calculus

Calculus is a branch of mathematics that includes the study of limit , derivatives, integrals, and infinite series, and constitutes a major part of modern university education....
.

Varieties of infinitesimal calculus

  • Differential
    Differential calculus

    Differential calculus, a field in mathematics, is the study of how function s change when their inputs change. The primary object of study in differential calculus is the derivative....
     and integral calculus: together, the original infinitesimal calculus, due to Newton and Liebniz.
  • Standard calculus
    Calculus

    Calculus is a branch of mathematics that includes the study of limit , derivatives, integrals, and infinite series, and constitutes a major part of modern university education....
     (based on the approach of Cauchy and Weierstrass)
  • Non-standard calculus
    Non-standard calculus

    In mathematics, non-standard calculus is the name for the modern application of infinitesimals, in the sense of non-standard analysis, to differential and integral calculus....
     (based on Robinson's approach to infinitesimals)