to the subsequent development of the concept of infinite series.
With the creation of the coordinate method and the dissemination of the concept of directed mechanical quantities (velocity, acceleration), the concept of negative number became clear and acquired intuitive content. On the other hand, complex numbers remained a by-product of the algebraic apparatus, and for the most part continued to be merely an object of unproductive disputes.
The discovery of the differential and integral calculus in the proper sense of the terms dates to the last third of the 17th century. Leibniz, who gave an extensive exposition of the fundamental ideas of the new calculus in articles published between 1682 and 1686, holds priority with respect to this discovery insofar as publication is concerned. But insofar as the time of obtaining the actual results is concerned, there is every reason to believe that priority rests with Newton, who arrived at the fundamental ideas of the differential and integral calculus in 1665-66. In 1669, the manuscript of Newton’s On Analysis by Means of Equations was submitted to the British mathematicians I. Barrow and J. Collins and was acclaimed by British mathematicians. The Method of Fluxions, in which Newton gave a definitive systematic exposition of his theory, was written in 1670-71 but was published only in 1736. Leibniz, however, began his investigations of infinitesimal analysis only in 1673.
Newton and Leibniz were the first to consider in general form the operations of the differentiation and integration of functions, which were basic to the new calculus. They established the relation between these operations (the Newton-Leibniz formula) and worked out a general uniform algorithm for them. However, Newton and Leibniz differed in their approach. The concepts of “fluent” (a variable quantity) and its “fluxion” (rate of change) are the basic concepts for Newton. To the direct problem of finding fluxions and the relations between fluxions for given fluents (differentiation and construction of differential equations), Newton contrasted the inverse task of finding fluents from given relations between fluxions, that is, the general problem of integrating differential equations. The problem of finding the antiderivative appears here as a particular case of integration of the differential equation dy/dx = f(x). This point of view was quite natural for Newton as the creator of mathematical physics: his calculus of fluxions was simply a reflection of the idea that the primary laws of nature are expressed by differential equations, and prediction of the course of the processes described by these equations requires their integration.
Leibniz focused on the problem of the transition from the algebra of the finite to the algebra of infinitesimals. He perceived the integral as the sum of an infinitely large number of infinitesimals, and regarded differentials—infinitesimal increments of variable quantities—as the fundamental concept of the differential calculus. In contrast, Newton, after introducing the corresponding concept of “moment,” strove to move away from it in his later works.
The publication of Leibniz’ works inaugurated a period of intensive collective development of the differential and integral calculus, the integration of differential equations, and the geometric applications of analysis by Jakob Bernoulli, Johann Bernoulli, G. L’Hôpital, and other mathematicians of Continental Europe. The modern approach to the study of mathematics whereby the results obtained are published immediately in journals and are used by other scientists in their own research soon after publication developed at this time.
In addition to analytic geometry, differential geometry developed in close connection with algebra and analysis. The foundations for the subsequent development of pure geometry, chiefly in the sense of the creation of the basic concepts of projective geometry, were laid in the 17th century. Also of particular importance were the investigations in number theory (B. Pascal, Fermat) and the development of 