ly 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 of astronomy. The Pythagorean theorem was known to the Babylonians.
Period of elementary mathematics. Mathematics arose as an independent science—with a clear understanding of the distinctness of its method and of the necessity of systematically developing its fundamental concepts and premises in a sufficiently general form—only after a large amount of specific material was accumulated in the form of separate methods of arithmetic calculations, methods of determining areas and volumes, and other procedures. With respect to arithmetic and algebra, this process may have started in Babylonia. However, this new trend, which consisted in the systematic and logically consistent construction of the foundations of the mathematical science, was fully defined in ancient Greece. The system of exposition of elementary geometry that was developed by the ancient Greeks served for two millennia as a model of the deductive construction of a mathematical theory. Number theory gradually grew out of arithmetic. The systematic study of quantities and measurement was created. The process of the formation (in connection with the problem of measuring quantities) of the concept of real numbers proved to be extremely protracted, owing to the fact that the concepts of irrational and negative numbers are among those more complex mathematical abstractions which, in contrast to the concepts of natural number, fraction, and geometric figure, are not an intrinsic part of prescientific human experience.
The creation of algebra as literal calculation concluded only at the end of the two-millennium period under consideration here. Special symbols for unknowns were developed by the Greek mathematician Diophantus (probably in the third century) and more systematically in India in the seventh century, but the representation of the terms of an equation by means of letters was introduced only in the 16th century by the French mathematician F. Vieta.
The development of geodesy and astronomy led early to the detailed elaboration of both plane and spheric