History of Calculus Timeline

By 1664 Newton had made his first important contribution by advancing the binomial theorem, which he had extended to include fractional and negative exponents.

Image source: Binomial theorem

Infinitesimal calculus was developed in full in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz independently of each other, building on the work of their contemporaries.

Image source: History of calculus

Newton formulated his calculus between 1664 and 1666, later describing this period as 'the prime of my age for invention.' A manuscript dated May 20, 1665 showed that Newton had already developed the calculus to the point where he could compute the tangent and curvature at any point of a continuous curve. During plague-induced isolation, the first written conception of fluxionary calculus was recorded in De Analysi per Aequationes Numero Terminorum Infinitas.

Image source: Isaac Newton

In an effort to give calculus a more rigorous explication and framework, Newton compiled in 1671 the Methodus Fluxionum et Serierum Infinitarum. It was not published until 1736.

Image source: Method of Fluxions

In 1672, Leibniz met the mathematician Huygens who convinced Leibniz to dedicate significant time to the study of mathematics.

Image source: Gottfried Wilhelm Leibniz

By 1673 Leibniz had progressed to reading Pascal's Traité des sinus du quart de cercle, and it was during his largely autodidactic research that Leibniz said 'a light turned on'.

In the manuscripts of 25 October to 11 November 1675, Leibniz recorded his discoveries and experiments with various forms of notation, including the dy/dx notation still used today.

This revised calculus of ratios continued to be developed and was maturely stated in the 1676 text De Quadratura Curvarum where Newton came to define the present day derivative as the ultimate ratio of change, defined as the ratio between evanescent increments purely at the moment in question.

Image source: Taylor series

The calculus of variations began with Newton's minimal resistance problem, which Newton formulated and solved in 1685 and published in his Principia in 1687. It was the first problem in the field to be formulated and correctly solved.

Image source: Calculus of variations

It was followed by the brachistochrone curve problem of Johann Bernoulli (1696), which Bernoulli solved using the principle of least time but not the calculus of variations, whereas Newton did solve it in 1697, pioneering the field with his work on both problems.

Image source: Brachistochrone curve

An argument over priority led to the Leibniz–Newton calculus controversy, which continued until the death of Leibniz in 1716. Initial accusations were made by students and supporters at the turn of the century, but after 1711 both men became personally involved, accusing each other of plagiarism. The dispute led to a rift in the European mathematical community lasting over a century.

Image source: Leibniz–Newton calculus controversy

The first proof of Rolle's theorem was given by Michel Rolle in 1691 using methods developed by the Dutch mathematician Johann van Waveren Hudde.

Image source: Rolle's theorem

Euler's contributions to the calculus of variations began in 1733, and his Elementa Calculi Variationum gave to the science its name.

Image source: Leonhard Euler

One of the first and most complete works on both infinitesimal and integral calculus was written in 1748 by Maria Gaetana Agnesi.

Image source: Maria Gaetana Agnesi

To Lagrange (1773) we owe the introduction of the theory of the potential into dynamics, although the name 'potential function' and the fundamental memoir of the subject are due to Green (1827, printed in 1828).

Joseph Louis Lagrange contributed extensively to the theory of the calculus of variations, and Adrien-Marie Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima.

Image source: Adrien-Marie Legendre

Antoine Arbogast (1800) was the first to separate the symbol of operation from that of quantity in a differential equation.

To this discrimination Brunacci (1810), Carl Friedrich Gauss (1829), Siméon Denis Poisson (1831), Mikhail Vasilievich Ostrogradsky (1834), and Carl Gustav Jacob Jacobi (1837) have been among the contributors.

Image source: Carl Gustav Jacob Jacobi

Francois-Joseph Servois (1814) seems to have been the first to give correct rules on the subject of operational symbols in differential equations.

Image source: François-Joseph Servois

Legendre's great table appeared in 1816, contributing significantly to mathematical analysis.

Cauchy early undertook the general theory of determining definite integrals, and the subject has been prominent during the 19th century. He also stated the mean value theorem in its modern form, along with Bernard Bolzano.

Image source: Augustin-Louis Cauchy

The name 'potential function' and the fundamental memoir of the subject are due to Green (1827, printed in 1828).

Image source: George Green (mathematician)

To the subject Lejeune Dirichlet has contributed an important theorem (Liouville, 1839), which has been elaborated by Liouville, Catalan, Leslie Ellis, and others.

Image source: Peter Gustav Lejeune Dirichlet

An important general work is that of Sarrus (1842) which was condensed and improved by Augustin Louis Cauchy (1844).

Charles James Hargreave (1848) applied these methods in his memoir on differential equations, and George Boole freely employed them.

Other valuable treatises were written by Strauch (1849), Jellett (1850), Otto Hesse (1857), Alfred Clebsch (1858), and Carll (1885), but perhaps the most important work of the century is that of Karl Weierstrass.

Image source: Karl Weierstrass