The history of calculus

The History of Calculus
Calculus, being a difficult subject, requires much more than the intuition and genius of one man. It took the work and ideas of many great men to establish the advanced concepts now known as calculus. Of the many mathematicians involved in the discovery of calculus, Gottfried Wilhelm von Liebniz and Sir Isaac Newton were the most important. Together, they established thebasic principles of calculus, and, with the help of other mathematicians, it was refined using the concept of the limit. The development of calculus can be thought of as being in three periods; Anticipation, Development, and Rigorization. During the Anticipation, various mathematicians provided the stepping stones to build the concepts of calculus. During the Development, Newton and Liebnizdeveloped the main concepts and principles used today. In the Rigorization, various mathematicians used the concept of the limit to give concrete meaning to the principles developed my Leibniz and Newton.
The history of calculus falls into several distinct time periods, most notably the ancient, medieval, and modern periods. The ancient period introduced some of the ideas of integral calculus, butdoes not seem to have developed these ideas in a rigorous or systematic way. Calculating volumes and areas, the basic function of integral calculus, can be traced back to the Egyptian Moscow papyrus (c. 1800 BC), in which an Egyptian successfully calculated the volume of a pyramidal frustum. From the school of Greek mathematics, Eudoxus used the method of exhaustion, which prefigures the concept ofthe limit, to calculate areas and volumes while Archimedes developed this idea further, inventing heuristics which resemble integral calculus. The method of exhaustion was later used in China by Liu Hui in the 3rd century AD in order to find the area of a circle. It was also used by Zu Chongzhi in the 5th century AD, who used it to find the volume of a sphere.
In AD 499 the Indian mathematicianAryabhata used the notion of infinitesimals and expressed an astronomical problem in the form of a basic differential equation. This equation eventually led in the 12th century to develop an early derivative representing infinitesimal change, and he described an early form of "Rolle's theorem". Around AD 1000, the Islamic mathematician Ibn al-Haytham (Alhazen) was the first to derive the formulafor the sum of the fourth powers, and using mathematical induction, he developed a method that is readily generalizable to finding the formula for the sum of any integral powers, which was fundamental to the development of integral calculus. In the 12th century, the Persian mathematician Sharaf al-Din al-Tusi discovered the derivative of cubic polynomials, an important result in differentialcalculus. In the 14th century, Madhava of Sangamagrama, along with other mathematician-astronomers of the Kerala school of astronomy and mathematics, described special cases of Taylor series, which are treated in the text Yuktibhasa.
In the modern period, independent discoveries in calculus were being made in early 17th century Japan, by mathematicians such as Seki Kowa, who expanded upon the method ofexhaustion. In Europe, the second half of the 17th century was a time of major innovation. Calculus provided a new opportunity in mathematical physics to solve long-standing problems. Several mathematicians contributed to these breakthroughs, notably John Wallis and Isaac Barrow. James Gregory proved a special case of the second fundamental theorem of calculus in 1668.
Gottfried Wilhelm Leibnizwas originally accused of plagiarism of Sir Isaac Newton's unpublished works, but is now regarded as an independent inventor and contributor towards calculus. Leibniz and Newton pulled these ideas together into a coherent whole and they are usually credited with the independent and nearly simultaneous invention of calculus. Newton was the first to apply calculus to general physics and Leibniz...