terature 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 explicit introduction into mathematics of the ideas of motion and change. The idea of the relation between quantities (for example, the value of a sum depends on the values of the addends) was already implicit in algebra. However, in order to encompass quantitative relations in the process of their change, the very relations between quantities had to be made an independent subject of study. Therefore, the concept of function, which subsequently played the same role of a primary and independent subject of study as the concepts of quantity or number had previously, moved into the foreground. The study of variables and of functional relations led to the fundamental concepts of mathematical analysis, which in explicit form introduce the idea of the infinite into mathematics, and to the concepts of limit, derivative, differential, and integral. Infinitesimal analysis was created, primarily in the form of the differential calculus and the integral calculus, which makes it possible to relate finite changes of variables to their behavior in the immediate vicinity of the individual values that they assume. The fundamental laws of mechanics and physics were written in the form of differential equations, and the problem of integrating these equations was advanced as one of the most important tasks of mathematics. The search for unknown functions defined by different types of conditions is the subject of the calculus of variations. Thus, in addition to equations in which the unknowns are numbers, there appeared equations in which the unknowns to be determined are functions.
The subject studied by geometry likewise expanded considerably with the penetration into geometry of the ideas of motion and transformation of figures. Geometry began to study motion and transformations as such. For example, in projective geometry projective transformations of a plane or space are themselves one of the main objects of study. The conscious development of these ideas dates only to the late 18th century and the early 19th. Much earlier, with