![]() Newton initially accused Leibniz of plagiarism. These ideas led to the calculus of real infinitesimals, organized by Gottfried Wilhelm Leibniz. But he did not publish all these discoveries, and at that time, the use of the infinitesimal method was still in bad history and did not have a suitable aspect. In his other works, he used series expansions for functions, including fractional and non-exponential powers, so that it was clear that he understood the principle of Taylor's series. He used to solve problems such as the movement of the planets, the shape of the surface of a rotating fluid, the expansion of the globe at the poles (swelling at the poles), the movement of weight by sliding on a wheel, and many other problems that are in his work (the book Principia Mathematica written in 1687 AD) discussed, used the arithmetic method. In his works, Newton restated his ideas in such a way as to correspond to the method of the time, as he replaced the calculations of infinitesimals with their geometrical equivalents. The rule of multiplication and chain rule, the concepts of derivatives of higher orders and Taylor's series, and analytical functions were used by Isaac Newton using a strange notation to solve problems in mathematics and physics. The combination of these concepts was achieved by John Willis, Isaac Barrow and James Gregory, the latter two of whom proved the Second Fundamental Theorem of Arithmetic around 1670. This concept represented equality within an infinitesimal error term. Pierre de Fermat claimed that the concept of "as equal as possible" (he coined the word "adequality" for this concept with the help of the Latin language) was inspired by Diophantus. The formal study of calculus brought together Cavalieri's method of infinitesimals and calculus of finite differences, which was developed in Europe at the same time. These ideas were similar to the work of Archimedes in his treatise called "Method", but they believe that the mentioned treatise of Archimedes was lost in the 13th century and was rediscovered in the 20th century, so Cavalieri was not aware of its existence. He was the one who claimed that volumes and areas should be written as sums of volumes and areas with infinitesimally small sections. In Europe, fundamental work took place in the form of Bonaventura Cavalieri's treatise. In the 14th century, Indian mathematicians presented an unstable method similar to differentiation that could be applied to some trigonometric functions. ![]() He used the results of what we now call the integration of this function, such formulas for the sum of the square of integers and the fourth power also provided him with the possibility of calculating the volume of the parabola. In the Middle East, Ibn Haytham (Latin: Alhazen) (965-1040 AD) derived a formula for the sum of fourth powers. 212-287 BC) developed this idea further to invent a heuristic method that resembles the methods of integral calculus. 408-355 BC) used the method of Afna (who did something similar before discovering the concept of limit) to calculate areas and volumes, while Archimedes (ca. Calculation of volume and area is one of the purposes of integral calculus, which can be found in the Moscow papyrus (13th Dynasty of Egypt, around 1820 BC) But the formulas were simple recipes without any indication of a specific method, so that some of these recipes lacked the main ingredients.įrom the age of Greek mathematics, Eudoxus (c. But it does not seem that these ideas have led to a systematic and stable approach. In the ancient period, some ideas led to integral calculus.
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