athematical method often recedes into the background.
Mathematics and technology. The principles of arithmetic and elementary geometry arose, as will be seen from the history section below, in response to the needs of everyday life; the subsequent creation of new mathematical methods and ideas took place under the influence of the mathematical natural sciences, such as astronomy, mechanics, and physics, whose development responds to practical needs. But the direct ties between mathematics and technology have frequently taken the form of application of existing mathematical theories to technological problems. Let us point out, however, examples of the rise of new general mathematical theories in response to direct technological requirements: the development of the method of least squares was connected with geodetic work; the study of many new types of partial differential equations first began with the solution of technical problems; and operational methods of solving differential equations were developed in connection with electrical engineering. The requirements of communications led to the formation of a new branch of probability theory—information theory. The problems of synthesizing control systems led to the development of new branches of mathematical logic. Along with the needs of astronomy, technical problems also played a decisive role in the development of the methods of approximate solution of differential equations. Many methods of approximate solution of partial differential equations and of integral equations were developed entirely to meet technological needs. The problem of rapidly obtaining numerical solutions became quite acute with the growing complexity of technical problems. Numerical methods are acquiring ever greater importance in connection with the possibilities opened up by computers for the solution of practical problems. The high level of theoretical mathematics has made it possible to develop rapidly the methods of computer science, which has played a major role in the solution of a number 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 p