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 provide logical justification for these ideas. Now they openly advocated the right to manipulate according to conventional rules mathematical expressions lacking direct meaning without considering the element of intuitive appeal and without providing any logical justification for the legitimacy of such manipulations. Among the older generation, Leibniz, who in reference to the integration of rational fractions by expanding them into imaginary expressions spoke in 1702 of the “marvelous intervention of the ideal world,” became increasingly inclined in this direction. The more realistically minded Euler did not speak of miracles but interpreted the legitimacy of operations involving imaginary numbers and divergent series as an empirical fact that was confirmed by the correctness of results obtained by similar transformations. Although work was initiated to provide rational clarification of the foundations of infinitesimal analysis, systematic logical substantiation of analysis was accomplished only in the 19th century.
The most prominent mathematicians of the 17th century often were also philosophers or experimental physicists; in the 18th century, however, mathematics became an independent profession. Eighteenth-century mathematicians came from diverse segments of society. They early revealed their mathematical abilities and moved rapidly along in their academic careers. For example, Euler, the son of a preacher in Basel, at the age of 20 was invited to the St. Petersburg Academy of Sciences as an adjunct, at 23 became a professor there, and at 39 became chairman of the physicomathematical division of the Berlin Academy of Sciences. Lagrange, the son of a French official, was a professor in Turin at 19, and at 30 chairman of the physicomathematical division of the Berlin Academy of Sciences. P. Laplace, the son of a French peasant, was a professor at a military school in Paris at 22 and a member of the Paris Academy of Sciences at 36. Mathematical natural science (mechanics, mathematical physics) and the technica