f 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 spherical trigonometry.
The period of elementary mathematics concluded (in Western Europe in the early 17th century) when mathematical interest shifted to the domain of the mathematics of variable quantities.
ANCIENT GREECE. The development of mathematics took an essentially different direction in ancient Greece than in the Orient. Although the level of Babylonian mathematics in the techniques of calculations, the art of solving algebraic problems, and the development of the mathematical tools of astronomy was reached and surpassed by the Greeks only in the Hellenistic era, mathematics in ancient Greece had entered a totally new stage of logical development much earlier. The need for clear mathematical proofs appeared, and the first attempts at a systematic construction of a mathematical theory were made. Mathematics, like all scientific and artistic endeavors, ceased to be impersonal as was the case in the ancient Orient. Now it was created by mathematicians who were known by name and who left mathematical works, which, unfortunately, have come down to us only in fragments preserved by later commentators.
In arithmetic, the Greeks considered themselves to be students of the Phoenicians, explaining the Phoenicians’ high development of arithmetic in terms of the requirements of their extensive commerce. Tradition links the beginnings of Greek geometry with the travels to Egypt in the seventh and sixth centuries B.C. of the first Greek geometers and philosophers Thales of Miletus and Pythagoras of Samos. In the Pythagorean school, arithmetic developed from the simple art of counting into number theory. Very simple arithmetic progressions (in particular, 1 + 3 + 5 + … + 2n — 1 =n2) were summed, the divisibility of numbers and different types of means (arithmetic, geometric, and harmonic) were studied, and the problems of number theory (such as the search for perfect numbers) were connected in the Pythagorean school to the mystical, magical nature ascribed to numerical relationships. In connection with the geometric Pythago