l 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 series was undertaken by D’Alembert. Euler, Lagrange, and particularly Legendre laid the foundations for the study of elliptic integrals—the first type of nonelementary function to be subjected to an intensive study.
Considerable attention was devoted to differential equations. In particular, Euler devised (1739, published 1743) the first method of solving a linear differential equation of any order with constant coefficients, D’Alembert examined systems of differential equations, and Lagrange and Laplace developed the general theory of linear differential equations of any order. Euler, G. Monge, and Lagrange laid the foundations of the general theory of first-order partial differential equations; the same was done for second-order equations by Euler, Monge, and Laplace. Of particular interest is the introduction into analysis of the expansion of functions into trigonometric series since it led to a controversy between Euler, D. Bernoulli, D’Alembert, Monge and Lagrange which centered on the concept of function. This controversy prepared the groundwork for the fundamental results of the 19th century concerning the relationship between functions defined by means of analytic expressions and functions defined in an arbitrary manner. Finally, a new branch of analysis—the calculus of variations—was created by Euler and Lagrange in the 18th century. On the basis of various achievements of the 17th and 18th centuries, De Moivre, Jakob Bernoulli, and Laplace formulated the basic principles of probability theory.
In geometry Euler brought the system of elementary analytic geometry to its culmination. Euler, A. Clairaut, Monge, and J. Meusnier laid the groundwork for the differential geometry of space curves and surfaces. Lambert developed the theory of perspective, and Monge gave descriptive geometry its final form.
The survey presented above shows that 18th-century mathematics, while proceeding from the ideas of the 17th century, far surpassed previous centuries in the scale of work. This flowering of mathematics was prima