mber of very important practical problems, including those in the use of atomic energy and in space exploration.
A clear understanding of the position of mathematics as an independent science with its own subject and method became possible only after considerable factual material was accumulated. Such an understanding came about in ancient Greece in the sixth and fifth centuries B.C. It is customary to place the development of mathematics before that time in the period of the rise of mathematics and to date the beginning of the period of elementary mathematics to the sixth and fifth centuries B.C. During these two early periods, mathematical investigations dealt almost exclusively with an extremely limited stock of basic concepts that had arisen in the early stages of historical development in connection with the simple requirements of domestic life, which reduced to the counting of objects, calculation of the number of goods and areas of plots of land, determination of the size of the individual parts of architectural structures, measurement of time, mercantile calculations, and navigational measurements. The early problems of mechanics and physics (with the exception of certain studies conducted by the Greek scholar Archimedes [third century B.C.] already requiring the rudiment of infinitesimal calculus) could still be satisfied with this stock of basic mathematical concepts. Astronomy, which was entirely responsible, for example, for the early development of trigonometry, was the only science that systematically placed its own particular and very great demands on mathematics long before the extensive development of the mathematical study of natural phenomena in the 17th and 18th centuries.
In the 17th century the new requirements of the natural sciences and technology forced mathematicians to focus their attention on creating methods that would make it possible to mathematically study motion, the processes of the change of quantities, and transformations of geometric figures (for example, in projection). The period 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 large