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MOST IMPORTANT SCIENTIFIC DISCOVERIES
Differential and integral calculus. History and essence of scientific discovery
Directory / The most important scientific discoveries Long before Newton и Leibniz many philosophers and mathematicians dealt with the question of infinitesimals, but limited themselves to only the most elementary conclusions. Even the ancient Greeks used the method of limits in geometric studies, by means of which they calculated, for example, the area of a circle. A special development was given to this method by the greatest mathematician of antiquity Archimedes, who discovered with its help many remarkable theorems. Kepler and in this respect came closest to Newton's discovery. On the occasion of a purely mundane dispute between a buyer and a seller over several mugs of wine, Kepler took up the geometric determination of the capacity of barrel-shaped bodies. In these studies one can already see a very clear idea of infinitesimals. Thus, Kepler considered the area of a circle as the sum of countless very small triangles, or, more precisely, as the limit of such a sum. Later, the Italian mathematician Cavalieri took up the same question. In particular, the XNUMXth-century French mathematicians Roberval did a lot in this area, Farm и Pascal. But only Newton and somewhat later Leibniz created a real method that gave a huge impetus to all branches of the mathematical sciences. According to Auguste Comte, differential calculus, or the analysis of infinitesimal quantities, is a bridge thrown between the finite and the infinite, between man and nature: a deep knowledge of the laws of nature is impossible with the help of one rough analysis of finite quantities, because in nature at every step - infinite, continuous, changing. Newton created his method based on previous discoveries made by him in the field of analysis, but in the most important issue he turned to the help of geometry and mechanics. When exactly Newton discovered his new method is not exactly known. From the close connection of this method with the theory of gravitation, one should think that it was developed by Newton between 1666 and 1669, and in any case before the first discoveries made in this area by Leibniz. “Newton considered mathematics to be the main tool for physical research,” notes V.A. Nikiforovsky, “and developed it for numerous further applications. After lengthy reflections, he came to the calculus of infinitesimals based on the concept of motion; mathematics for him did not act as an abstract product of the human mind He believed that geometric images - lines, surfaces, bodies - are obtained as a result of movement: a line - when a point moves, a surface - when a line moves, a body - when a surface moves. These movements are carried out in time, and for an arbitrarily small time a point , for example, an arbitrarily small path will pass.To find the instantaneous speed, the speed at a given moment, it is necessary to find the ratio of the increment of the path (in modern terminology) to the increment of time, and then the limit of this ratio, i.e., take the "last ratio", when the increment of time tends to zero.So Newton introduced the search for "last ratios", derivatives, which he called fluxions... ... The use of the theorem on the mutual inverseness of the operations of differentiation and integration, known even to Barrow, and the knowledge of the derivatives of many functions gave Newton the opportunity to obtain integrals (in his terminology, fluents). If the integrals were not directly calculated, Newton expanded the integrand into a power series and integrated it term by term. To expand functions into series, he most often used the binomial expansion discovered by him, and also applied elementary methods ... " The new mathematical apparatus was tested by the scientist already by the time of the creation of the main work of his life - "Mathematical Principles of Natural Philosophy". At that time, Newton was fluent in differentiation, integration, series expansion, integration of differential equations, and interpolation. “Newton,” continues V.A. Nikiforovsky, “made his discoveries before Leibniz, but did not publish them in a timely manner; all his mathematical works were published after he became famous. arbitrary exponent. In 1664, he prepared a manuscript entitled "The following sentences are sufficient to solve problems by motion", containing the main discoveries in mathematics. The manuscript remained in draft form and was not published until three hundred years later. In "Analysis by means of equations with an infinite number of terms", written in 1665, Newton expounded his results in the doctrine of infinitesimal series, in the application of series to the solution of equations... ...In 1670-1671, Newton began to prepare for publication a more complete work - "The Method of Fluxions and Infinite Series". It was not possible to find a publisher: at that time, books on mathematics brought a loss ... In the "Method of Fluxions" Newton's teaching acts as a system: the calculus of fluxions is considered, their application to determining tangents, finding extrema, curvature, calculating quadratures, solving equations with fluxions, which corresponds to modern differential equations". Only in 1704 did the first of all Newton's works on analysis come out - written by him in 1665-1666. Seven years later they published "Analysis using Equations with an Infinite Number of Terms". The "Method of Fluxions" saw the light only after the author's death in 1736. For a long time, Newton did not even suspect that the German Leibniz was successfully dealing with a similar problem on the continent. For the time being, highly appreciating the merits of each other, in the end, scientists became involved in a debate about the priority of the discovery of infinitesimal calculus. Gottfried Wilhelm Leibniz (1646–1716) was born in Leipzig. Leibniz's mother, taking care of her son's education, sent him to Nicolai's school, which at that time was considered the best in Leipzig. Gottfried spent whole days sitting in his father's library. He read Plato, Aristotle, Cicero, Descartes indiscriminately. Gottfried was not yet fourteen years old when he amazed his school teachers by showing a talent that no one suspected in him. He turned out to be a poet - according to the then concepts, a true poet could only write in Latin or Greek. At the age of fifteen, Gottfried became a student at the University of Leipzig. Officially, Leibniz was considered at the Faculty of Law, but the special circle of legal sciences far from satisfied him. In addition to lectures on jurisprudence, he diligently attended many others, especially in philosophy and mathematics. Wanting to complete his mathematical education, Gottfried went to Jena, where the mathematician Weigel was famous. Returning to Leipzig, Leibniz brilliantly passed the exam for a master's degree in "liberal arts and world wisdom", that is, literature and philosophy. Gottfried at that time was not even 18 years old. The next year, turning to mathematics for a while, he wrote "Discourse on Combinatorial Art". In the autumn of 1666, Leibniz left for Altorf, the university town of the small Nuremberg Republic. Here, on November 5, 1666, Leibniz brilliantly defended his doctoral dissertation "On Entangled Matters". In 1667, Gottfried went to Mainz to the elector, to whom he was immediately introduced. For five years, Leibniz held a prominent position at the Mainz court. This period in his life was a time of lively literary activity. Leibniz wrote a number of works of philosophical and political content. On March 18, 1672, Leibniz left for France on an important diplomatic mission. Acquaintance with the Parisian mathematicians in the shortest possible time delivered to Leibniz the information without which, for all his genius, he could never have achieved anything truly great in the field of mathematics. The school of Fermat, Pascal and Descartes was necessary for the future inventor of differential calculus. For Leibniz, real mathematics began only after visiting London in 1675. On his return to Paris, Leibniz divided his time between studies in mathematics and works of a philosophical nature. The mathematical direction more and more prevailed in him over the legal one, the exact sciences now attracted him more than the dialectic of Roman lawyers. In the last year of his stay in Paris in 1676, Leibniz worked out the first foundations of the great mathematical method known as "calculus". The facts convincingly prove that although Leibniz did not know about the method of fluxions, he was led to the discovery by Newton's letters. On the other hand, there is no doubt that Leibniz's discovery, in terms of generality, convenience of designation, and detailed development of the method, has become a tool of analysis much more powerful and popular than Newton's method of fluxions. Even Newton's compatriots, who for a long time preferred the method of fluxions out of national vanity, gradually adopted Leibniz's more convenient designations; as for the Germans and the French, they even paid too little attention to Newton's method, which in other cases has retained its significance to the present day. The mathematical method of Leibniz is in close connection with his later theory of monads - infinitesimal elements from which he tried to build the universe. Mathematical analogy, the application of the theory of largest and smallest quantities to the moral field, gave Leibniz what he considered a guiding thread in moral philosophy. Leibniz's political activities largely distracted him from mathematics. Nevertheless, he devoted all his free time to the processing of the differential calculus he invented, and between 1677 and 1684 managed to create a whole new branch of mathematics. In 1684, Leibniz published in the journal Proceedings of Scientists a systematic exposition of the principles of differential calculus. All the treatises he published, especially the last one, which appeared almost three years before the publication of the first edition of Newton's Principia, gave science such a huge impetus that at present it is difficult even to assess the full significance of the reform carried out by Leibniz in the field of mathematics. What was vaguely imagined to the minds of the best French and English mathematicians, with the exception of Newton, who had his own method of fluxions, suddenly became clear, distinct and generally accessible, which cannot be said about Newton's brilliant method. “Leibniz, in contrast to the concrete, empirical, prudent Newton,” writes V.P. Kartsev, “was a major systematist in the field of calculus, a daring innovator. phenomena. This ambitious and unrealistic project was, of course, unrealizable, but, having changed, it turned into a universal system of notation for small calculus, which we still use. He freely operates with signs ... which he rightly considers signs of inverse operations, and turns with them as freely and freely as with algebraic symbols.He easily operates with derivatives of higher orders, while Newton introduces fluxes of higher order strictly limitedly, if necessary for solving a specific problem. Leibniz saw in his differentials and integrals a general method, consciously sought to create a rigid algorithm for a simplified solution of previously unsolved problems. Newton, on the other hand, did not care at all about making his method public. His symbolism was introduced by him only for "internal", personal consumption, he did not strictly adhere to it. Here is the opinion of the Soviet mathematician A. Shibanov: "Bowing before the indisputable authority of their great compatriot, British scientists subsequently canonized every stroke, every smallest detail of his scientific activity, even the mathematical signs he introduced for personal use." "The tradition of reverence for Newton weighed heavily on English science, and his designations, clumsy compared to Leibniz's, hindered progress," agrees the Dutch scientist D.Ya. Stroyk. In a letter written in June 1677, Leibniz directly revealed to Newton his method of differential calculus. He did not respond to Leibniz's letter. Newton believed that the discovery belongs to him forever. It is enough that it was hidden only in his head. The scientist sincerely believed: timely publication does not bring any rights. Before God, the discoverer will always be the one who discovered first. Author: Samin D.K.
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