Modern calculus was developed in 17th-century Europe by
Isaac Newton and
Gottfried Wilhelm Leibniz (independently of each other, first publishing around the same time). Elements of it first appeared in ancient Egypt and later Greece, then in China and the Middle East, and still later again in medieval Europe and India.
Ancient precursors Egypt and Babylon Calculations of
volume and
area, one goal of integral calculus, can be found in the
Egyptian Moscow papyrus (), but the formulae are simple instructions, with no indication as to how they were obtained.
Babylonians may have discovered the
trapezoidal rule while doing astronomical observations of
Jupiter.
Greece to calculate the area under a parabola in his work
Quadrature of the Parabola. Laying the foundations for integral calculus and foreshadowing the concept of the limit, ancient Greek mathematician
Eudoxus of Cnidus () developed the
method of exhaustion to prove the formulas for cone and pyramid volumes. During the
Hellenistic period, this method was further developed by
Archimedes (BC), who combined it with a concept of the
indivisibles—a precursor to
infinitesimals—allowing him to solve several problems now treated by integral calculus. In
The Method of Mechanical Theorems he describes, for example, calculating the
center of gravity of a solid
hemisphere, the center of gravity of a
frustum of a circular
paraboloid, and the area of a region bounded by a
parabola and one of its
secant lines.
China The method of exhaustion was later discovered independently in
China by
Liu Hui in the 3rd century AD to find the area of a circle. that would later be called
Cavalieri's principle to find the volume of a
sphere.
Medieval Middle East In the Middle East,
Hasan Ibn al-Haytham, Latinized as Alhazen (AD) derived a formula for the sum of
fourth powers. He used the results to carry out what would now be called an
integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a
paraboloid.
India Bhāskara II () was acquainted with some ideas of differential calculus and suggested that the "differential coefficient" vanishes at an extremum value of the function. In the 14th century, Indian mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric functions.
Madhava of Sangamagrama and the
Kerala School of Astronomy and Mathematics stated components of calculus. According to
Victor J. Katz, they, however, 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."
Modern Johannes Kepler's work
Stereometria Doliorum (1615) formed the basis 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.
Bonaventura Cavalieri, basing his work on Kepler's, The combination was achieved by
John Wallis,
Isaac Barrow, and
James Gregory, the latter two proving part of the
fundamental theorem of calculus around 1670. The
product rule and
chain rule, the notions of
higher derivatives and
Taylor series, and of
analytic functions were used by
Isaac Newton in an idiosyncratic notation which he applied to solve problems of
mathematical physics. In his works, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments that were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a
cycloid, and many other problems discussed in his
Principia Mathematica (1687). In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the
Taylor series. He did not publish all these discoveries, and at this time, infinitesimal methods were still considered disreputable. He is now regarded as an
independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for working with infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the
product rule and
chain rule, in their differential and integral forms. Unlike Newton, Leibniz put painstaking effort into his choices of notation. Today, Leibniz and Newton are usually both given credit for independently inventing and developing calculus. Newton was the first to apply calculus to general
physics. Leibniz developed much of the notation used in calculus today. A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus "
the science of fluxions", a term that endured in English schools into the 19th century. The first complete treatise on calculus to be written in English and use the Leibniz notation was not published until 1815. Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on both infinitesimal and
integral calculus was written in 1748 by
Maria Gaetana Agnesi.
Foundations In calculus,
foundations refers to the
rigorous development of the subject from
axioms and definitions. In early calculus, the use of
infinitesimal quantities was thought unrigorous and was fiercely criticized by several authors, most notably
Michel Rolle and
Bishop Berkeley. Berkeley famously described infinitesimals as the
ghosts of departed quantities in his book
The Analyst in 1734. Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz. Several mathematicians, including
Maclaurin, tried to prove the soundness of using infinitesimals, but it would not be until 150 years later when, due to the work of
Cauchy and
Weierstrass, a way was finally found to avoid mere "notions" of infinitely small quantities. The foundations of differential and integral calculus had been laid. In Cauchy's ''
Cours d'Analyse'', we find a broad range of foundational approaches, including a definition of
continuity in terms of infinitesimals, and a (somewhat imprecise) prototype of an
(ε, δ)-definition of limit in the definition of differentiation. In his work, Weierstrass formalized the concept of
limit and eliminated infinitesimals (although his definition can validate
nilsquare infinitesimals). Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called "infinitesimal calculus".
Bernhard Riemann used these ideas to give a precise definition of the integral. It was also during this period that the ideas of calculus were generalized to the
complex plane with the development of
complex analysis. In modern mathematics, the foundations of calculus are included in the field of
real analysis, which contains full definitions and
proofs of the theorems of calculus. The reach of calculus has also been greatly extended.
Henri Lebesgue invented
measure theory, based on earlier developments by
Émile Borel, and used it to define integrals of all but the most
pathological functions.
Laurent Schwartz introduced
distributions, which can be used to take the derivative of any function whatsoever. Limits are not the only rigorous approach to the foundation of calculus. Another way is to use
Abraham Robinson's
non-standard analysis. Robinson's approach, developed in the 1960s, uses technical machinery from
mathematical logic to augment the real number system with
infinitesimal and
infinite numbers, as in the original Newton-Leibniz conception. The resulting numbers are called
hyperreal numbers, and they can be used to give a Leibniz-like development of the usual rules of calculus. There is also
smooth infinitesimal analysis, which differs from non-standard analysis in that it mandates neglecting higher-power infinitesimals during derivations. The Hungarian polymath
John von Neumann wrote of this work, Applications of differential calculus include computations involving
velocity and
acceleration, the
slope of a curve, and
optimization. == Applications ==