rean 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 later served as the starting point for Archimedes in his development of the method of infinitesimals. In the fourth century B.C., the age of a certain subordination of mathematics to restrictions imposed by idealist philosophy commenced in an atmosphere of political reaction and the declining power of Athens. Here, the science of numbers is strictly separated from the “science of calculation,” and geometry from the “science of measurement.” Assuming the existence of incommensurable segments, areas, and volumes, Aristotle imposed a general ban on the application of arithmetic to geometry. Geometric constructions were limited to those that could be carried out with compass and straightedge. The investigations of Eudoxus of Cnidus, which were connected with the trend toward logical analysis of the foundations of geometry, may be considered the single most significant achievement of the mathematicians of the fourth century B.C.
HELLENISTIC AND ROMAN ERAS. For seven centuries after the third century B.C., Alexandria was the center of scientific and particularly mathematical investigations. Here, Greek mathematics reached its zenith amid an atmosphere of the fusion of various world cultures, major state requirements, extensive building, and governmental patronage of science that was unprecedented in its scope. Despite the spread of Greek education and scientific interests throughout the entire Hellenistic and Roman world, Alexandria, with its Museum, the first research institution in the modern sense of the word, and with its libraries, attracted nearly all the major scholars. Of the mathematicians mentioned below, only Archimedes remained faithful to his homeland of Syracuse. The first century of the Alexandrian era (third century B.C.) was distinguished by the highest intensity of mathematical creativity. Euclid, Archimedes, Eratosthenes, and Apollonius of Perga belonged to this century.
In his Elements, Euclid collected and subjected to definitive logical treatment the achievements of the previous period in geometry.