The History of Calculus

History of Calculus

• Calculus is a mathematical discipline that focuses on limits, continuity, derivatives, integrals, and infinite series.

• The origins of calculus can be traced back to ancient Greece, China, and the Middle East, where elements of calculus appeared.

• Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz, leading to the Leibniz-Newton calculus controversy.

• The word "calculus" comes from the Latin word for "small pebble" and originally referred to a method of computation.

• The Greeks used the method of exhaustion to calculate areas and volumes, while Archimedes developed heuristics that resemble the methods of integral calculus.

• Infinitesimal calculus was not put on a rigorous footing until the 17th century, when it was formalized by Cavalieri as the method of Indivisibles and eventually incorporated by Newton into a general framework of integral calculus.

• Indian mathematicians, such as Madhava of Sangamagrama, also made significant contributions to calculus, including the Taylor series and infinite series approximations.

• Johannes Kepler's work formed the basis of integral calculus, and Bonaventura Cavalieri developed the method of indivisibles, which he used to compute volumes and areas as the sums of infinitesimally thin cross-sections.

• European mathematicians, including Isaac Barrow, René Descartes, and Blaise Pascal, discussed the idea of a derivative, and Fermat introduced the concept of adequality, which was closely related to differentiation.

• The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed in Europe and Fermat's adequality.

• Newton and Leibniz independently developed the surrounding theory of infinitesimal calculus in the late 17th century, and Newton provided some of the most important applications to physics, especially of integral calculus.

• The development of calculus and its uses within the sciences have continued to the present day.The Development of Calculus: Newton and Leibniz

• By the 17th century, European mathematics shifted away from Hellenistic mathematics towards more modern works.

• Newton and Leibniz developed calculus independently, but with different approaches to the concept of change.

• Newton viewed calculus as the scientific description of motion and magnitudes, while Leibniz saw it as a metaphysical explanation of change.

• The core insight of calculus was the formalization of the inverse properties between the integral and the differential of a function.

• Newton began his mathematical training in Cambridge with Isaac Barrow, and made his first important contribution by advancing the binomial theorem.

• Newton's fluxional calculus was not published definitively, and he developed it through correspondence and smaller papers.

• Leibniz began his rigorous math studies later in life, and saw the tangent as a ratio between ordinates and abscissas.

• Leibniz embraced infinitesimals and wrote extensively about them, defining them as "less than any given quantity."

• The priority dispute between Newton and Leibniz led to a rift in the European mathematical community lasting over a century.

• Newton introduced the notation for the derivative of a function, while Leibniz introduced the symbol for the integral.

• Today, both Newton and Leibniz are credited with independently developing the basics of calculus.

• The work of both Newton and Leibniz is reflected in the notation used today.History of Calculus

• Calculus was independently developed by Sir Isaac Newton and Gottfried Leibniz in the 17th century.

• Calculus is a branch of mathematics that deals with the study of rates of change and their applications.

• The calculus of variations began with a problem of Johann Bernoulli in 1696, and Leonhard Euler first elaborated the subject.

• Niels Henrik Abel was the first to consider in a general way the question of what differential equations can be integrated in a finite form by the aid of ordinary functions.

• Antoine Arbogast was the first to separate the symbol of operation from that of quantity in a differential equation.

• The application of the infinitesimal calculus to problems in physics and astronomy was contemporary with the origin of the science.

• Among the many applications of analysis to physical problems are investigations of Euler on vibrating chords, Sophie Germain on elastic membranes, and Poisson, Lamé, Saint-Venant, and Clebsch on the elasticity of three-dimensional bodies.

• 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.

• Infinitesimal calculus was introduced into the social sciences, starting with Neoclassical economics.

• One of the first and most complete works on both infinitesimal and integral calculus was written in 1748 by Maria Gaetana Agnesi.

• The gamma function is an important function in calculus and is now called the gamma function.

• Karl Weierstrass's course on the theory may be asserted to be the first to place calculus on a firm and rigorous foundation.