f 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 the fundamental concepts of combinatorial analysis (Fermat, B. Pascal, Leibniz). The 17th century also saw works on probability theory (Fermat, B. Pascal), which culminated at the end of the century in a result of fundamental importance—the discovery of the simplest form of the law of large numbers (Jakob Bernoulli, published 1713). We should also note the construction by B. Pascal (1641) and Leibniz (1673-74) of the first calculating machines, which, incidentally, long had no practical consequences.
18TH CENTURY. In the early 18th century the general mode of mathematical research gradually changed. The successes of the 17th century, brought about primarily by methodological innovation, stemmed from the boldness and depth of the general ideas. This brought mathematics closer to philosophy. By the beginning of the 18th century, the new fields of mathematics created in the 17th century reached a level at which further advancement required skill in the use of the mathematical apparatus and inventiveness in the finding of surprising sophisticated ways of solving difficult problems. Of the two greatest mathematicians of the 18th century, L. Euler and J. Lagrange, the former is the more brilliant representative of this virtuosic trend. However, Lagrange, while perhaps yielding to Euler in the number and diversity of problems solved, combined masterful techniques with the broad generalizing concepts that were typical of the French mathematical school of the second half of the 18th century, which was closely connected with the extensive philosophical movement of French enlighteners and materialists. Fascination with the extraordinary power of the apparatus of mathematical analysis naturally led to the belief in the possibility of its purely automatic development and in the infallibility of mathematical calculations even when the symbols involved were meaningless. During the period of creation of infinitesimal analysis, mathematicians relied on ideas that carried strong intuitive conviction even though they were unable to