al 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 Pythagorean theorem, a method of obtaining infinitely many triples of “Pythagorean numbers,” that is, sets of three integers satisfying the equation a2 + b2 = c2, was found. In geometry, the problems that Greek geometers of the sixth and fifth centuries B.C. worked on after mastering the Egyptian heritage also arose naturally from the very simple requirements of construction, surveying, and navigation. Such, for example, were the problems of the relationship between the lengths of the legs and hypotenuse of a right triangle (which is expressed by the Pythagorean theorem), the relation between the areas of similar figures, squaring the circle, trisection of an angle, and duplication of the cube. However, the approach to these problems was necessarily new owing to the growing complexity of the object under study. Not confining themselves to approximate, empirically determined solutions, Greek geometers sought exact proofs and logically complete solutions of a problem. The proof of the incommensurability of the diagonal of a square with its side can serve as a clear example of this new trend. In the second half of the fifth century B.C. the philosophical and scholarly life of Greece was concentrated in Athens. Hippias of Elis and Hippocrates of Chios worked here. The first systematic textbook of geometry is ascribed to Hippocrates of Chios. A developed system of geometry that did not disregard such logical subtleties as proofs of the familiar congruence criteria for triangles undoubtedly had already been created by that time. The discovery of all five regular polyhedrons—perhaps the most noteworthy achievement of geometry in the fifth century B.C. and the result of the search for ideal simple bodies that could serve as the building blocks of the universe—was a reflection of the first, albeit purely speculative, attempts in mathematics to give a rational explanation of the structure of matter. At the turn of the fourth century B.C. Democritus created, on the basis of atomistic concepts, a method of determining volume that l