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 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.
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