 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 technical applications of mathematics remained in the sphere of activity of mathematicians. For example, Euler worked on problems in shipbuilding and optics, Lagrange created the foundations of analytic mechanics, and Laplace, who considered himself primarily a mathematician, was also a major astronomer and physicist.
In the 18th century many outstanding results were obtained in mathematics. Through the works of Euler, Lagrange, and A. Legendre, number theory became a systematic science. Lagrange gave (1769, published 1771) a general solution of indeterminate quadratic equations. Euler established (1772, published 1783) the law of quadratic reciprocity. He also used the zeta function in the study of prime numbers, thus giving rise to analytic number theory.
By means of expansions into continued fractions, Euler proved (1737, published 1744) that e and e2 are irrational and J. Lambert demonstrated (1776, published 1768) that π is irrational. In algebra, G. Cramer introduced (1750) determinants for the solution of systems of linear equations. Euler considered it an empirically established fact that every algebraic equation has a root of the form . The conviction grew that imaginary expressions, not only in algebra but also in analysis, in general are always reducible to the form D’Alembert proved (1748) that the absolute value of a polynomial cannot have a nonzero minimum (the D’Alembert lemma), considering this as proof that every algebraic equation has a root. The formulas of A. De Moivre and of Euler relating the exponential and trigonometric functions of a complex variable led to further expansion of the applications of complex numbers in analysis. Newton, J. Stirling, Euler, and Laplace laid the foundations of the calculus of finite differences. B. Taylor discovered (1715) a formula for expanding an arbitrary function into a power series.
For 18th-century mathematicians, especially Euler, series became one of the most powerful and flexible tools of analysis. The serious study of the condition for the convergence of se