 the creation of analytic geometry in the 17th century, the relationship of geometry to all other mathematics changed fundamentally: a universal method of translating the problems of geometry into the language of algebra and of analyzing and solving them by purely algebraic and analytic methods was found, and the possibility emerged of representing (illustrating) algebraic and analytic facts geometrically, for example, by graphic representation of functional relations.
In the 17th and 18th centuries algebra was largely devoted to consequences stemming from the possibility of studying the left-hand side of the equation P(x) = 0 as a function of the variable x. This approach made it possible to study the problem of the number of real roots and to develop methods of separating and approximately calculating these roots. In the complex domain this approach led the French mathematician J. D’Alembert to a proof of the fundamental theorem of algebra, which states that every algebraic equation has at least one root. Although not strictly rigorous, this proof was nevertheless sufficiently convincing for 18th-century mathematicians. Also significant in the 17th and 18th centuries were the achievements of “pure” algebra, which does not require the concepts of continuous change of quantities that were borrowed from analysis. (Suffice it to mention here the solution of arbitrary systems of linear equations by means of determinants, and the elaboration of the theory of the divisibility of polynomials and the elimination of unknowns.) However, the conscious separation of purely algebraic facts and methods from the facts and methods of mathematical analysis was typical only of the second half of the 19th century and of the 20th century. In the 17th and 18th centuries, algebra was construed largely as the first chapter of analysis, in which one is limited to algebraic relations and equations, in distinction to analysis, where one investigates arbitrary relations between quantities and the solution of arbitrary equations.
The scientists of the advanced Western European countries, above all I. Newton and G. von Leibniz, endeavored in the 17th century to create the new mathematics of variable quantities. In the 18th century the St. Petersburg Academy of Sciences, where many prominent foreign mathematicians (L. Euler, D. Bernoulli) worked, became one of the main centers of mathematical investigations, and the Russian mathematical school, which produced brilliant investigations in the early 19th century, gradually took shape.
17TH CENTURY. The new stage of mathematical development described above was closely linked with the creation in the 17th century of mathematical natural science, whose goal is the elucidation of the course of individual natural phenomena in terms of general, mathematically formulated natural laws. Throughout the 17th century truly profound and extensive mathematical investigations were conducted only in two fields of the natural sciences, namely mechanics and optics. In mechanics, Galileo discovered the laws of falling bodies (1632-38); J. Kepler, the laws of planetary motion (1609-19); and Newton, the law of universal gravitation (1687). In optics, Galileo (1609) and Kepler (1611) constructed telescopes; Newton developed optics on the basis of the corpuscular theory, and C. Huygens and R. Hooke on the basis of wave theory. Nevertheless, 17th-century rationalist philosophy advanced the idea of the universality of the mathematical method (Descartes, B. Spinoza, Leibniz), which imparts a certain brilliance to this essentially philosophical era in the development of mathematics.
Serious new mathematical problems were posed in the 17th century by navigation (the necessity of improving clock-making and of designing accurate chronometers) and by cartography, ballistics, and hydraulics. Seventeenth-century authors appreciated and insisted on emphasizing the great practical importance of mathematics. By relying on its close connection with natural science, 17th-century mathematics was able to rise to a new stage of development. N