 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 in the fourth century B.C. to reject completely the legitimacy of applying arithmetic to the study of geometric quantities. In fact, in the theory of proportions and in the method of exhaustion, mathematicians of the fourth and third centuries B.C. did succeed in indirectly applying arithmetic to geometry. The centuries that followed did not bring a definitive resolution of the problem through the creation of a new fundamental concept (irrational number) but rather gradual oblivion, which became possible with the gradual loss of the concept of mathematical rigor. At this stage in the history of mathematics, however, the temporary rejection of mathematical rigor proved to be useful by opening up the possibility for the unhindered development of algebra, whose use, within the framework of the rigorous concepts of Euclid’s Elements, was restricted to the extremely confining “geometric algebra” of segments, areas, and volumes. Significant advances in this direction can be found in the Metrics of Hero of Alexandria (Heron). Independent and extensive development of real algebraic calculation, however, is found only in Diophantus’ Arithmetica, which is primarily devoted to the solution of equations. Classifying his investigations as pure arithmetic, Diophantus naturally restricted himself, in contrast to the utilitarian Hero, to rational solutions, thus excluding the possibility of geometric or mechanical applications of his algebra. In antiquity, trigonometry was perceived largely as a branch of astronomy and not as a part of mathematics. The requirements of complete rigor in formulations and proofs were not imposed on trigonometry just as they were not on Hero’s computational geometry. Hipparchus was the first to compile tables of chords that were the equivalent of our sine tables. The principles of spherical trigonometry were developed by Menelaus and Ptolemy.
In pure mathematics, the scholars of the last centuries of the ancient period (except for Diophantus) concentrated increasingly on providing commentary on those