ce 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 for every ten minutes of angles with an accuracy of up to 1/604 and tables of tangents, and established the law of sines for spherical triangles. Nasir al-Din al-Tusi (Nasir-Eddin) further developed spherical trigonometry and al-Kashi gave a systematic account of the arithmetic of decimal fractions, which he rightly considered to be more convenient than sexagesimal fractions. In connection with problems of the extraction of roots, al-Kashi verbally formulated Newton’s binomial formula and found the rule for forming the coefficients Cmn = (Cn−1)m + (Cn−1)m−1 In the Treatise on the Circle (c. 1427), al-Kashi, found the value of π to 17 decimal places by computing the perimeters of inscribed and circumscribed polygons with 3 × 228 sides. In connection with the computation of extensive tables of sines, al-Kashi gave an extremely advanced iterative method for the numerical solution of equations.
WESTERN EUROPE BEFORE THE 16TH CENTURY. For Western European mathematics the 12th to 15th centuries were essentially a period of the mastery of the heritage of the ancient world and the Orient. In spite of the fact that no new significant results were established during this period, the general character of European mathematical culture at that time was distinguished by a number of substantial progressive features that made possible the rapid development of mathematics in subsequent centuries. The sophisticated requirements of the politically independent members of the Italian bourgeoisie, who were rapidly growing wealthy, led to the creation and extensive dissemination of textbooks that combined a practical general orientation with extreme detail and scientific orientation. In the 12th century, less than 100 years after the appearance of the first Latin translations of Greek and Arabic mathematical works, Leonardo of Pisa (Fibonacci) wrote Liber abaci (Book of the Abacus; 1202) and Practica geometriae (The Practice of Geometry; 1220), which expounded on arithmetic, mercantile arithmetic, algebra, and geometry. These books en