eriod of the mathematics of variable quantities begins with the use of variables in the analytic geometry of the French scientist R. Descartes and the creation of the differential and integral calculus.
The further expansion of the range of quantitative relations and spatial forms studied by mathematics led, at the start of the 19th century, to the necessity of consciously referring to the process of the expansion of the subject of mathematical investigations after setting the task of studying systematically, from a sufficiently general viewpoint, the possible types of quantitative relations and spatial forms. The creation of “imaginary geometry” by the Russian mathematician N. I. Lobachevskii, which subsequently found real applications, was the first significant step in this direction. The development of this type of investigation introduced such important new features in the structure of mathematics that it is natural to place the mathematics of the 19th and 20th centuries in the special period of modern mathematics.
Origin of mathematics. The counting of objects in the earliest stages of cultural development led to the creation of very simple concepts of the arithmetic of natural numbers. It was only on the basis of a developed system of oral reckoning that written numeration systems arose and the methods of performing the four arithmetic operations on natural numbers were gradually developed (only division was to present difficulties for a long time). The requirements of measurement (the amount of grain, the length of a road) led to the appearance of the names and designations for the simplest fractions and to the development of methods of performing arithmetic operations on fractions. In this way there accumulated material that gradually evolved into the oldest mathematical science—arithmetic. The measurement of areas and volumes and the requirements of construction and, somewhat later, of astronomy gave rise to the development of the rudiments of geometry. These processes progressed among many peoples largely independently and in parallel. The accumulation of arithmetic and geometric knowledge in Egypt and Babylonia was of particular importance to the subsequent development of science. In Babylonia, advanced techniques of arithmetic calculations were the basis of rudiments of algebra and, the requirements of astronomy, of rudiments of trigonometry.
Extant mathematical texts of ancient Egypt (the first half of the second millennium B.C.) consist primarily of examples dealing with the solution of individual problems and, at best, of prescriptions for their solution, which sometimes can be understood only by analyzing the numerical examples given in the texts. The term “prescription” for the solution of certain types of problems is precisely right, since there obviously was no mathematical theory in the sense of proofs of general theorems. The lack of any distinction between exact and approximate solutions attests to this. Nonetheless, the very stock of established mathematical knowledge was quite large, as can be inferred from well-developed construction techniques, the complexity of land relations, and the need for an accurate calendar.
There are more mathematical texts enabling us to assess mathematics in Babylonia than in Egypt. Babylonian cuneiform mathematical texts encompass the period from the second millennium B.C. to the rise and development of Greek mathematics. Babylonia of this period had acquired from the earlier Sumerian period a well-developed mixed decimal-sexagesimal numeration system that already incorporated positional notation (given symbols designated the same number of units of different sexagesimal orders). Tables of reciprocals reduced division to multiplication. In addition to tables of reciprocals, there were tables of products, squares, and square and cube roots. Of the achievements of Babylonian geometry, which exceeded the knowledge of the Egyptians, we note the well-developed measurement of angles and certain rudiments of trigonometry that apparently were connected with the development o