ents of 15th-century Central Asian mathematicians and that in actuality the major new ideas that determined the subsequent development of European mathematics were to appear only in the 17th century. However, in the 16th century it seemed that a new era was beginning in mathematics with the discovery of algebraic solutions of cubic equations (S. dal Ferro, c. 1515, and later and independently N. Tartaglia, c. 1530) and quartic equations (L. Ferrari, 1545), which for centuries had been considered impossible. G. Cardano investigated cubic equations and discovered the irreducible case in which the real roots of an equation are expressed in terms of complex numbers. This forced Cardano to acknowledge, albeit quite tentatively, the advantage of calculations involving complex numbers. Algebra was further developed by F. Vieta (Viete), the founder of true algebraic literal calculation (1591; before him letters were used only for unknowns). The concept of perspective, which was developing in geometry even before the 16th century, was set forth by the German artist A. Diirer (1525). In 1585, S. Stevin worked out the rules of arithmetic operations for decimals.
RUSSIA UNTIL THE 18TH CENTURY. Russian mathematical education in the ninth to 13th centuries was on the level of the most culturally advanced countries of the Orient and Western Europe. However, it was set back for a long time by the Mongolian invasion. In the 15th and 16th centuries, society’s requirements for mathematical knowledge increased significantly in connection with the strengthening and economic growth of the Russian state. Numerous manuscript handbooks of arithmetic and geometry containing rather extensive information necessary for practical activities (trade, taxation, construction, the use of artillery) appeared in the late 16th century and particularly in the 17th century.
A system of number symbols resembling the Greco-Byzantine system and based on the Slavic alphabet came into use in Ancient Rus’. Slavic numeration appears in Russian mathematical literature until the early 18th century, but beginning in the late 16th century this numeration was gradually replaced by the decimal system now in use.
The oldest known mathematical work dates from 1136 and was written by the Novgorod monk Kirik. It is devoted to arithmetic-chronological calculations, which show that at this time it was already known in Rus’ how to solve the complex problem of determining the date of Easter each year, a problem that reduces to solving indeterminate linear equations over the integers. The arithmetic manuscripts of the late 16th century and of the 17th century contain, in addition to a description of Slavic and Arabic numeration, arithmetic operations involving positive integers and a detailed exposition of the rules of operation on fractions, the rule of three, and a solution of linear equations in one unknown using the rule of false position. For easier application of the general rules, many practical examples were discussed in the manuscripts and a counting board—the prototype of the Russian abacus—was described. The first, arithmetic, part of the celebrated Arithmetic of L. F. Magnitskii (1703) was organized in the same manner. Manuscripts on geometry, which for the most part pursued practical goals, contained descriptions of rules, often approximate, for determining areas and volumes and employed the properties of similar triangles and the Pythagorean theorem.
Period of the development of the mathematics of variable quantities. An essentially new period of mathematical development begins with the 17th century. “The Cartesian variable quantity was a turning point in mathematics. Because of this, motion and thus dialectics entered mathematics and for the same reason the differential and integral calculus became immediately necessary” (F. Engels in K. Marx and F. Engels, Soch., 2nd ed., vol. 20, p. 573). The range of quantitative relations and spatial forms now studied by mathematics is no longer exhausted by numbers, quantities, and geometric figures. This was due chiefly to the 