History of calculus
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See also History of mathematics.
Though the origins of 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. Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. (See Archimedes on Spheres & Cylinders (http://mathpages.com/home/kmath343.htm).) Archimedes and others after used integral methods throughout history, and a great many (Barrow, Fermat, Pascal, Wallis and others) are said to have discussed the idea of a derivative. Bhaskara (1114-1185), an Indian Mathematician, could be described as the "father" of differential calculus, giving an example of what is now called "differential coefficient" and the basic idea of what is now known as "Rolle's theorem".
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 Wilhelm Leibniz, independently of each other. 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, but mainly they are credited for developing the fundamental theorem of calculus and work on notation. Lesser credit for the development of calculus is given to Bhaskara, Barrow, Descartes, de Fermat, Huygens and Wallis, though de Fermat is sometimes described as the "father" of differential calculus. 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, though this was not known in the West at the time, and he had no contact with Western scholars. [1] (http://www2.gol.com/users/coynerhm/0598rothman.html)
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.
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Leibniz and Newton
There has been considerable debate about whether Newton or Leibniz was first to come up with the important concepts of the calculus. The truth of the matter will likely never be known. 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. Today, both Leibniz and Newton are generally considered to have discovered calculus independently.
In 1704 an anonymous pamphlet, later determined to have been written by Leibniz, accused Newton of having plagiarised Leibniz' work. 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 the controversy suggestions were 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. 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.
Newton provided a host of applications in physics, and his notation <math>\dot{f}(x)<math> 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.
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:
- <math>\int_0^1 x^{n-1}(1 - x)^{n-1}dx<math>
- <math>\int_0^\infty e^{-x} x^{n-1}dx<math>
although these were not the exact forms of Euler's study. If n is integral, it follows that <math>\int_0^\infty e^{-x}x^{n-1}dx = n!<math> but if n is fractional it is a transcendent function. To it Legendre assigned the symbol <math>\Gamma<math>, 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 <math>\Gamma x<math> and <math>\log \Gamma x<math> 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.fr:Histoire du calcul infinitésimal