tists 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. New concepts, which did not fit into the old formal-logical categories of mathematics, were justified since they corresponded to the relationships of the real world. For example, the “reality” of the concept of the derivative stemmed from the “reality” of the concept of velocity in mechanics; therefore, the problem was not whether this concept could be justified logically but only how this was to be done.
The mathematical achievements of the 17th century began with the discovery of logarithms by J. Napier, who published his tables in 1614. In 1637, Descartes published Geometry, which contains the foundations of coordinate geometry and a classification of curves in which curves were subdivided into algebraic and transcendental. The real roots of an equation of any degree were investigated in algebra in close relation to the possibility of representing the roots of an equation P(x) = 0 by the points of intersection of the curve y = P(x) with the axis of the abscissas (Descartes, Newton, M. Rolle). P. Fermat’s investigations of maxima and minima and his search for tangents to curves in essence contained the methods of the differential calculus, but the methods themselves had not yet been isolated or developed. Another source of infinitesimal analysis was the development of the method of “indivisibles” by Kepler (1615) and B. Cavalieri (1635), both of whom applied this method to the determination of the volumes of solids of revolution and other problems. Thus, the foundations of the differential calculus were essentially created in geometric form.
A parallel development was the study of infinite series. The properties of very simple series, beginning with the geometric progression, were studied by J. Wallis (1685). N. Mercator (1668) expanded In (1 + x) in a power series. Newton obtained (1665-69) the binomial formula for any exponent and the power series expansions of the functions ex, sin x, and arc sin x. Nearly all 17th-century mathematicians, for example, Wallis, Huygens, Leibniz and Jakob Bernoulli, contributed 