Calculus, originally called infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series. Many elements of calculus appeared in ancient Greece, then in China and the Middle East, and still later again in medieval Europe and in India. Infinitesimal calculus was developed in full in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz independently of each other. An argument over priority led to the Leibniz–Newton calculus controversy, which continued until the death of Leibniz in 1716. The development of calculus and its uses within the sciences have continued to the present.
Etymology
In mathematics education, "calculus" denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. The word is Latin for (the diminutive of , meaning ), a meaning which still persists in medicine. Because such pebbles were used for counting out distances,See, for example: tallying votes, and doing abacus arithmetic, the word came to mean a method of computation. In this sense, it was used in English at least as early as 1672, several years prior to the publications of Leibniz and Newton.
In addition to the differential calculus and integral calculus, the term is also used widely for naming specific methods of calculation. Examples of this include propositional calculus in logic, the calculus of variations in mathematics, process calculus in computing, and the felicific calculus in philosophy.
Early precursors of calculus
Ancient
Egypt and Babylonia
The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of volumes and areas, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (), but the formulas are only given for concrete numbers, some are only approximately true, and they are not derived by deductive reasoning. Babylonians may have discovered the trapezoidal rule while doing astronomical observations of Jupiter.
Greece
From the age of Greek mathematics, Eudoxus ( 408–355 BC) used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes ( 287–212 BC) developed this idea further, inventing heuristics which resemble the methods of integral calculus.Archimedes, Method, in The Works of Archimedes Greek mathematicians are also credited with a significant use of infinitesimals. Democritus is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone's smooth slope prevented him from accepting the idea. At approximately the same time, Zeno of Elea discredited infinitesimals further by his articulation of the paradoxes which they seemingly create.
Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat in his The Quadrature of the Parabola, The Method, and On the Sphere and Cylinder.MathPages — Archimedes on Spheres & Cylinders It should not be thought that infinitesimals were put on a rigorous footing during this time, however. Only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true. It was not until the 17th century that the method was formalized by Cavalieri as the method of Indivisibles and eventually incorporated by Newton into a general framework of integral calculus. Archimedes was the first to find the tangent to a curve other than a circle, in a method akin to differential calculus. While studying the spiral, he separated a point's motion into two components, one radial motion component and one circular motion component, and then continued to add the two component motions together, thereby finding the tangent to the curve.
China
The method of exhaustion was independently invented in China by Liu Hui in the 4th century AD in order to find the area of a circle., Chapter, p. 279 In the 5th century, Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere. Extract of page 27
Medieval Middle East
In the Middle East, Ibn al-Haytham, Latinized as Alhazen ( AD), extended Archimedes' method of exhaustion, finding the volume of the solid of revolution formed by rotating a parabola around a line perpendicular to its axis. To achieve this, Ibn al-Haytham invented a way of computing the sums of kth powers, which he applied for the sums of squares and the sums of fourth powers. Roshdi Rashed has argued that the 12th century mathematician Sharaf al-Dīn al-Tūsī must have used the derivative of cubic polynomials in his Treatise on Equations. Rashed's conclusion has been contested by other scholars, who argue that he could have obtained his results by other methods which do not require the derivative of the function to be known.
India
Bhāskara II ( 1114–1185) devised a way of working with infinitesimals applied to trigonometry. There is evidence of an early form of Rolle's theorem in his work, though it was stated without a modern formal proof. In his astronomical work, Bhāskara gives a result that looks like a precursor to infinitesimal methods: if then . This can be interpreted as the discovery that cosine is the derivative of sine, although he did not develop the notion of a derivative.
Some ideas on calculus later appeared in Indian mathematics, at the Kerala school of astronomy and mathematics. Madhava of Sangamagrama in the 14th century, and later mathematicians of the Kerala school, stated components of calculus such as the Taylor series and infinite series approximations. They considered series equivalent to the Maclaurin expansions of , , and more than two hundred years before they were studied in Europe. According to Victor J. Katz they were not able to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today."
A special case of Mean value theorem for inverse interpolation of the sine was described by Parameshvara, in his commentaries on Govindasvāmi and Bhāskara II.Radha Charan Gupta "A mean-value-type formula for inverse interpolation of the sine", Ganita 30 (1–2): 78—82.
Medieval Catholic Europe The mathematical study of continuity was revived in the 14th century by the Oxford Calculators and French collaborators such as Nicole Oresme. They proved the "Merton mean speed theorem". Oresme also developed the first proof of the divergence of the harmonic series. His work, along with the contemporaneous work of Richard Swineshead on a different series, marked the first appearance of infinite series other than the geometric series in mathematics. Japan Seki Takakazu developed *enri* (円理, “circle principles”), a mathematical system that addressed problems involving areas, volumes, and infinite series, broadly comparable in some aims to the problems later addressed by calculus in Europe.However Seki's investigations did not proceed from the same foundations as those used by Newton and Leibniz in Europe.Smith,
Takebe Katahiro extended the Enri principle to obtained power series expansion of in 1722, 15 years earlier than Euler.
Modern precursors Integrals Johannes Kepler's work on calculating volumes of shapes, and on finding the optimal shape of a wine barrel, were significant steps toward the development of integral calculus. Kepler developed a method to calculate the area of an ellipse by adding up the lengths of many radii drawn from a focus of the ellipse.
A significant work was a treatise inspired by Kepler's methods published in 1635 by Bonaventura Cavalieri on his method of indivisibles. He argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. He discovered Cavalieri's quadrature formula which gave the area under the curves of higher degree. This had previously been computed in a similar way for the parabola by Archimedes in The Method, but this treatise is believed to have been lost in the 13th century, and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. Cavalieri's work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first.
Torricelli extended Cavalieri's work to other curves such as the cycloid, and then the formula was generalized to fractional and negative powers by Wallis in 1656. In a 1659 treatise, Fermat is credited with an ingenious trick for evaluating the integral of any power function directly. Fermat also obtained a technique for finding the centers of gravity of various plane and solid figures, which influenced further work in quadrature.
Derivatives In the 17th century, European mathematicians Isaac Barrow, René Descartes, Pierre de Fermat, Blaise Pascal, John Wallis and others discussed the idea of a derivative. In particular, in and in distributed in 1636, Fermat introduced the concept of adequality, which represented equality up to an infinitesimal error term. This method could be used to determine the maxima, minima, and tangents to various curves and was closely related to differentiation.
Isaac Newton would later write that his own early ideas about calculus came directly from "Fermat's way of drawing tangents".
Fundamental theorem of calculus The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time, and Fermat's adequality. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving predecessors to the second fundamental theorem of calculus around 1670.
James Gregory, influenced by Fermat's contributions both to tangency and to quadrature, was then able to prove a restricted version of the second fundamental theorem of calculus, that integrals can be computed using any of a function's antiderivatives.See, e.g., Marlow Anderson, Victor J. Katz, Robin J. Wilson, Sherlock Holmes in Babylon and Other Tales of Mathematical History, Mathematical Association of America, 2004, p. 114.
The first full proof of the fundamental theorem of calculus was given by Isaac Barrow. Translator: J. M. Child (1916)Review of J.M. Child's translation (1916) The geometrical lectures of Isaac Barrow reviewer: Arnold Dresden (Jun 1918) p.454 Barrow has the fundamental theorem of calculus
Other developments
One prerequisite to the establishment of a calculus of functions of a real variable involved finding an antiderivative for the rational function . This problem can be phrased as quadrature of the rectangular hyperbola . In 1647 Gregoire de Saint-Vincent noted that the required function satisfied , so that a geometric sequence became, under , an arithmetic sequence. A. A. de Sarasa associated this feature with contemporary algorithms called logarithms that economized arithmetic by rendering multiplications into additions. So was first known as the hyperbolic logarithm. After Euler exploited , and was identified as the inverse function of the exponential function, it became the natural logarithm, satisfying .
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. The mean value theorem in its modern form was stated by Bernard Bolzano and Augustin-Louis Cauchy (1789–1857) also after the founding of modern calculus. Important contributions were made by Barrow, Huygens, and many others.