History of calculus

See also History of mathematics.

Though the origins of integral calculus are generally regarded as going no farther back than to the ancient Greeks, there is evidence that the ancient Egyptians may have harbored such knowledge amongst themselves as well (see Moscow and Rhind Mathematical Papyri). Eudoxus is generally credited with the method of exhaustion, which made it possible to compute the area and volume of regions and solids by breaking them up into recognizable shapes. Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. (See Archimedes on Spheres & Cylinders.) Thus, Archimedes and others after used integral methods throughout history.

Indian mathematician Bhaskara (1114-1185) was the first to conceive of differential calculus, using the "derivative" and inventing the basic idea of what is now known as Rolle's theorem. The 14th century Indian mathematician Madhava of Sangamagrama, along with other mathematicians of the Kerala school studied infinite series, convergence, differentiation, and iterative methods for solution of non-linear equations. Jyestadeva of the Kerala school wrote the first calculus text, the Yuktibhasa, which explores methods and ideas of calculus that were only repeated in seventeenth century Europe.

In 17th century Europe, Isaac Barrow, Pierre de Fermat, Blaise Pascal, John Wallis and others are said to have discussed the idea of a derivative. René Descartes introduced the foundation for the methods of analytic geometry in 1637, providing the foundation for calculus later introduced by Isaac Newton and Gottfried Leibniz, independently of each other. Leibniz and Newton are usually credited with the invention, in the late 1600s, of differential and integral calculus as we know it today, but mainly developed the fundamental theorem of calculus and worked on notation. Lesser credit for the development of calculus is given to Barrow, Descartes, de Fermat, Huygens and Wallis. A Japanese mathematician, Kowa Seki, lived at the same time as Leibniz and Newton and also elaborated some of the fundamental principles of integral calculus independently. [1]

Invention of Calculus

Isaac Newton

Gottfried Leibniz

Many of the results of Newton and Leibniz were known to mathematicians in Kerala, India almost 300 years previously. In 1835, Charles Whish published an article in the Transactions of the Royal Asiatic Society of Great Britain and Ireland, in which he claimed that the work of the Kerala school "laid the foundation for a complete system of fluxions." It was not until the 1940s however, that historians of mathematics verified Whish's claims, but their work is still underplayed in modern accounts of history of calculus, which holds Leibniz and Newton as it's inventors.

There has been considerable debate about whether Newton or Leibniz was first to come up with the important concepts of the calculus in Europe. The truth of the matter is that the ideas of calculus were a part of the mathematical knowledge of their day, and they independently put those pieces together in different but coherent ways. Common knowledge holds that Leibniz' greatest contribution to calculus was his notation, spending days trying to come up with the appropriate symbol to represent a mathematical idea. Newton's terminology and notation was less flexible than Leibniz's, yet it remained 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. It is now generally thought that Newton had discovered several ideas related to calculus earlier than Leibniz had. However, Leibniz was the first to publish in Europe. Today, modern history credits both Leibniz and Newton as having discovered calculus independently.

Newton provided a host of applications in physics, and his notation ${\dot {f}}(x)$ for the derivative of 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.

Leibniz based his work on the concept of infinitesimals, as opposed to the calculus of Isaac Newton, which is based upon the concept of the limit. Because infinitesimals were not put on a rigorous mathematical basis until the second half of the twentieth century, the delta-epsilon definition of limits and calculus became standard.

Controversy (Newton, Leibnitz... or Madhava?)

Madhava of Sangamagrama and the Kerala school were the first to come up with the important ideas of calculus in the 14th century and some [2] propose these ideas may have been transmitted to Europe by the 17th century. There is no evidence by way of relevant manuscripts but the evidence of methodological similarities, communication routes and a suitable chronology for transmission is hard to dismiss.

In the controversy between Newton and Leibniz, suggestions were made that the work of Leibniz was not independent, as he claimed, but influenced by reading copies of Newton's early manuscripts. That the Leibniz notation was original was common ground. There is evidence to show that Newton commenced work on the calculus about a decade before Leibniz did in 1676. Newton's work Method of Fluxions is presumed to be based on work carried out 1665-7, but it was not published until much later. Leibniz was in England in 1673 and again in 1676, and on the latter occasion did see some of Newton's manuscripts. In 1704 though, an anonymous pamphlet, later determined to have been written by Leibniz, accused Newton of having plagiarised Leibniz' work. A copy of one of Newton's very early manuscripts with annotations by Leibniz was found among Leibniz' papers after his death; the exact date when Leibniz first acquired this is unknown. A similar controversy arose in philosophy over whether or not Leibniz might have appropriated ideas of Baruch Spinoza.

It is often stated that the controversy isolated English-speaking mathematicians from those in continental Europe for many years; and that this set back British analysis (i.e. calculus-based mathematics) for a very long time. Newton's terminology and notation was less flexible than that of Leibniz, yet it was retained in British university teaching usage until the early 19th century. At that point the Analytical Society successfully lobbied for the introduction of Leibniz's notation in Great Britain.

Rigorous foundations

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:

$\int _{0}^{1}x^{n-1}(1-x)^{n-1}dx$

$\int _{0}^{\infty }e^{-x}x^{n-1}dx$

although these were not the exact forms of Euler's study. If n is an integer, it follows that $\int _{0}^{\infty }e^{-x}x^{n-1}dx=(n-1)!$ but if n is fractional it is a transcendent function. To it Legendre assigned the symbol $\Gamma$, 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 $\Gamma x$ and $\log \Gamma x$ 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 Hermann 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.