ater 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. At the same time he first laid the foundations of a systematic theory of numbers, proving the infinite nature of the sequence of primes and constructing a complete.theory of divisibility. Of the geometric studies of Euclid not included in the Elements and of the works of Apollonius of Perga, the creation of a complete theory of conic sections was of the greatest importance for the subsequent development of mathematics. The chief contribution of Archimedes in geometry lies in the determination of various areas and volumes (including the areas of a parabolic segment and the surface of a sphere and the volumes of a sphere, spherical segment, and segment of a paraboloid) and centers of gravity (of, for example, a spherical segment and a segment of a paraboloid). The spiral of Archimedes is just one example of the transcendental curves that were studied in the third century B.C. After Archimedes, Alexandrian science no longer reached its former completeness and depth, although the growth of the volume of scientific knowledge continued; the rudiments of infinitesimal analysis contained in the heuristic methods of Archimedes were not developed further.
It should be stated that the interest in the approximate measurement of quantities and in approximate calculations that arose from applied needs did not lead mathematicians of the third century B.C. to reject mathematical rigor. All the numerous approximate extractions of roots and even all astronomical computations were accomplished with a precise indication of the limits of error, as was the case with the celebrated Archimedean determination of the circumference of a circle in the form of the faultlessly proved inequalities
where p is the circumference of a circle with diameter d. This distinct understanding that approximate mathematics is not “unrigorous” mathematics was later forgotten.
The lack of a conclusively formulated concept of the irrational number was a significant shortcoming of all mathematics in antiquity. As has been indicated, this fact led philosophy i