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History of calculus
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History of calculus

Although Archimedes and others have used integral methods throughout history, and a great many (Barrow, Fermat, Pascal, Wallis and others) had previously invented the idea of a derivative, Gottfried Wilhelm Leibniz and Isaac Newton are usually credited with the invention, in the late 1600s, of differential and integral calculus as we know it today. It should be noted that Kowa Seki, a contemporaneous Japanese mathematician, also elaborated these concepts.

Table of contents
1 Leibniz and Newton
2 Rigorous foundations
3 Integrals
4 Symbolic methods
5 Calculus of variations
6 Applications

Leibniz and Newton

Leibniz and Newton, apparently working independently, arrived at similar results. It is thought that Newton's discoveries were made earlier, but Leibniz' were the first to be published. Newton provided a host of applications in physics, and his notation for the derivative f with respect to x is still used in physics today, especially for derivatives with respect to time. Outside of physics it has mostly been displaced by the notation f'(x) for the derivative of f with respect to x. Also current is Leibniz's more flexible differential notation df/dx, again for the derivative of f with respect to x. Leibniz's notation is especially popular in the many situations when writing only f' would be ambiguous.

In 1704 an anonymous pamphlet, later determined to have been written by Leibniz, accused Newton of having plagiarised Leibniz' work. That claim is easily refuted as there is ample evidence to show that Newton commenced work on the calculus long before Leibniz can possibly have done. However, the resulting controversy led to suggestions that Leibniz may not have invented the calculus independently as he claimed, but may have been influenced by reading copies of Newton's early manuscripts. This claim is not so easily dismissed and there is in fact considerable circumstantial evidence to support it. Leibniz was not known at the time for his probity, and later admitted to falsifying the dates on certain of his manuscripts in an effort to bolster his claims. Furthermore, a copy of one of Newton's very early manuscripts with annotations by Leibniz was found among Leibniz' papers after his death, although the exact date when Leibniz first acquired this is unknown. It is also interesting to note that a similar controversy exists in philosophy over whether or not Leibniz may have appropriated the ideas of Spinoza in his writings on that subject.

The truth of the matter will never be known, and in any case is unimportant to anyone alive today. Leibniz' great contribution to calculus was his notation, and this is beyond doubt purely of Leibniz's invention. The controversy was unfortunate however in that it divided english-speaking mathematicians from those in Europe for many years. This set back British analysis (i.e. calculus-based mathematics) for a very long time. Newton's terminology and notation was clearly less flexible than that of Leibniz, yet it was retained in British usage until the early 19th century, when the work of the Analytical Society successfully saw the introduction of Leibniz's notation in Great Britain.

Rigorous foundations

The calculus was widely used, as it was a very powerful mathematical tool, but it was not until the mid-1800s that it was put on a rigorous foundation. For example, while the definition of the derivative itself has not changed since it was first introduced, it requires the notion of a limit. Newton, Leibniz, and their immediate successors interpreted limits intuitively instead of through precise definitions. This was standard practice at the time. Later, with the work of mathematicians like Augustin Louis Cauchy, Bernard Bolzano, and Karl Weierstrass, the foundations of calculus were clarified and made precise. The study of foundations eventually resulted in deep explorations of the concept of infinity by Georg Cantor and others.

Integrals

Niels Henrik Abel seems to have been the first to consider in a general way the question as to what differential expressions can be integrated in a finite form by the aid of ordinary functions, an investigation extended by Liouville. Cauchy early undertook the general theory of determining definite integrals, and the subject has been prominent during the 19th century. Frullani's theorem (1821), Bierens de Haan's work on the theory (1862) and his elaborate tables (1867), Dirichlet's lectures (1858) embodied in Meyer's treatise (1871), and numerous memoirs of Legendre, Poisson, Plana, Raabe, Sohncke, Schlömilch, Elliott, Leudesdorf, and Kronecker are among the noteworthy contributions.

Eulerian integrals were first studied by Euler and afterwards investigated by Legendre, by whom they were classed as Eulerian integrals of the first and second species, as follows:

although these were not the exact forms of Euler's study. If n is integral, it follows that

but if n is fractional it is a transcendent function. To it Legendre assigned the symbol , and it is now called the gamma function. To the subject Dirichlet has contributed an important theorem (Liouville, 1839), which has been elaborated by Liouville, Catalan, Leslie Ellis, and others. On the evaluation of

and  Raabe (1843-44), Bauer (1859), and
Gudermann (1845) have written. Legendre's great table appeared in 1816.

Symbolic methods

Symbolic methods may be traced back to Taylor, and the analogy between successive differentiation and ordinary exponentials had been observed by numerous writers before the nineteenth century. Arbogast (1800) was the first, however, to separate the symbol of operation from that of quantity in a differential equation. François (1812) and Servois (1814) seem to have been the first to give correct rules on the subject. Hargreave (1848) applied these methods in his memoir on differential equations, and Boole freely employed them. Grassmann and Hankel made great use of the theory, the former in studying equations, the latter in his theory of complex numbers.

Calculus of variations

The calculus of variations may be said to begin with a problem of Johann Bernoulli's (1696). It immediately occupied the attention of Jakob Bernoulli and the Marquis de l'Hôpital, but Euler first elaborated the subject. His contributions began in 1733, and his Elementa Calculi Variationum gave to the science its name. Lagrange contributed extensively to the theory, and Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. To this discrimination Brunacci (1810), Gauss (1829), Poisson (1831), Ostrogradsky (1834), and Jacobi (1837) have been among the contributors. An important general work is that of Sarrus (1842) which was condensed and improved by Cauchy (1844). Other valuable treatises and memoirs have been written by Strauch (1849), Jellett (1850), Hesse (1857), Clebsch (1858), and Carll (1885), but perhaps the most important work of the century is that of Weierstrass. His celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and unquestionable foundation.

Applications

The application of the infinitesimal calculus to problems in physics and astronomy was contemporary with the origin of the science. All through the eighteenth century these applications were multiplied, until at its close Laplace and Lagrange had brought the whole range of the study of forces into the realm of analysis. 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). The name "potential" is due to Gauss (1840), and the distinction between potential and potential function to Clausius. With its development are connected the names of Dirichlet, Riemann, Neumann, Heine, Kronecker, Lipschitz, Christoffel, Kirchhoff, Beltrami, and many of the leading physicists of the century.

It is impossible in this place to enter into the great variety of other applications of analysis to physical problems. Among them are the investigations of Euler on vibrating chords; Sophie Germain on elastic membranes; Poisson, Lamé, Saint-Venant, and Clebsch on the elasticity of three-dimensional bodies; Fourier on heat diffusion; Fresnel on light; Maxwell, Helmholtz, and Hertz on electricity; Hansen, Hill, and Gyldén on astronomy; Maxwell on spherical harmonics; Lord Rayleigh on acoustics; and the contributions of Dirichlet, Weber, Kirchhoff, F. Neumann, Lord Kelvin, Clausius, Bjerknes, MacCullagh, and Fuhrmann to physics in general. The labors of Helmholtz should be especially mentioned, since he contributed to the theories of dynamics, electricity, etc., and brought his great analytical powers to bear on the fundamental axioms of mechanics as well as on those of pure mathematics.