ee 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 science the opinion long prevailed that the role of the “Arab culture” in mathematics consisted chiefly in the preservation and transmission to Western European mathematicians of the ancient Greek and Hindu mathematical discoveries. For example, the works of Greek mathematicians first became known in Western Europe through Arabic translations. In reality, however, the contributions to the development of science of the mathematicians who wrote in Arabic, and in particular of the mathematicians who were of the peoples of what is now Soviet Middle Asia and the Caucasus (Khorezm, Uzbek, Tadzhik, Azerbaijan), were much greater.
In the first half of the ninth century, Muhammad ibn Musa al-Khwarizmi gave the first exposition of algebra as an independent science. The term “algebra” was taken from the title of Khwarizmi’s work Al-jabr w’almugabala, from which European mathematicians of the early Middle Ages learned of the solution of quadratic equations. Omar Khayyam systematically studied and classified cubic equations and determined the conditions for their solvability (in the sense of the existence of positive roots). In his algebraic treatise Khayyam states that he searched long for an exact solution of cubic equations. The search of Central Asian mathematicians for such a solution was not successful. However, these mathematicians were well acquainted both with geometric methods (using conic sections) and approximate numerical methods of solution. While they adopted the Hindu decimal system and the use of zero, the mathematicians of Central Asia and the Middle East primarily used the sexagesimal system in major scientific calculations (apparently because of the sexagesimal division of angles in astronomy).
Trigonometry was developed extensively in connection with astronomical and geodetic projects. Al-Battani introduced the use of the trigonometric functions of sine, tangent, and cotangent. Abu-al-Wafa used all six trigonometric functions and verbally expressed the algebraic relations between them, computed tables of sines