n 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 who preceded them. The works of the commentators of that period, such as Pappus (third century) and Proclus (fifth century), for all their universality could not lead, amid the decline of the ancient world, to the consolidation into a unified science of the algebra of Diophantus, the trigonometry incorporated in astronomy, and Hero’s frankly nonrigorous computational geometry, all of which developed separately.
CHINA. The Arithmetic in Nine Chapters, which was compiled from earlier sources by Chang Ts’ang and Ching Ch’ou-ch’ang in the second and first centuries B.C., reveals that Chinese mathematicians had highly developed computational techniques and an interest in general algebraic methods. This work describes, in particular, methods of computing square and cube roots of integers. A large number of problems are formulated in such a way that we must conclude that the authors knew the method of solving a system of linear equations by successive elimination of unknowns and thought of the problems as examples clarifying that method. Interest in problems of the following type arose in China in connection with calendar calculations: When a number is divided by 3 the remainder is 2, when divided by 5 the remainder is 3, and when divided by 7 the remainder is 2; what is the number? Sun-tzu (between the second and sixth centuries) and Ch’in Chiu-shao (13th century) by way of examples provided a description of a regular algorithm for solving such problems. The result obtained by Tsu Ch’ung-chih (second half of the fifth century) that the ratio of the circumference of a circle to its diameter lies in the range 3.1415926 < π < 3.1415927 can serve as an example of the high level of development of computational methods in geometry. The studies of the Chinese dealing with numerical solutions of equations are remarkable. Problems in geometry leading to cubic equations first appeared in the works of the astronomer and mathematician Wang Hsiao T’ung (first half of the seventh century). Methods of solving fourth- and higher degr