 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 degree equations were given by the 13th- and 14th-century mathematicians Ch’in Chiu-shao, Li Yeh, Yang Hui, and Chu Shih-chieh.
INDIA. The flowering of Hindu mathematics dates to the fifth to 12th centuries. The most important mathematicians of this period were Aryabhata, Bhaskara, and Brahmagupta. The Hindus made two important contributions. The first consisted in introducing into wide use the modern decimal system of notation including the systematic use of zero to designate the absence of units of a given order. The origin of the numbers used in India, now called Arabic, has not been fully elucidated. The second and more important contribution was the creation of an algebra that freely used not only fractions but irrational and negative numbers. However, negative solutions were usually considered impossible in the interpretation of solutions of problems. In general, it should be noted that while both fractions and irrational numbers were associated from the very time of their appearance with the measurement of continuous quantities, negative numbers arose chiefly out of the intrinsic requirements of algebra and only later (in the 17th century) acquired independent significance. In trigonometry the introduction of the sine, cosine, and inverse sine curves should be noted.
CENTRAL ASIA AND THE MIDDLE EAST. The Arab conquests and the brief consolidation of vast territories under the power of the Arab caliphs led to a situation in which the scholars of Central Asia, the Middle East, and the Iberian Peninsula used the Arabic language from the ninth to 15th centuries. Science developed in the commercial cities amid a cosmopolitan atmosphere and governmental support of major scientific undertakings. In the 15th century the work of Ulug Beg, who assembled more than 100 scholars at his court and observatory in Samarkand and who organized astronomical observations, the computation of mathematical tables, and other endeavors that long remained unsurpassed, were the brilliant culmination of this era.
In Western European scien