 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 enjoyed great success.
By the end of this era, with the invention of the printing press, textbooks came into still wider use. During this period the universities became the centers of theoretical scientific thought. The progress of algebra as a theoretical discipline and not merely as a collection of practical rules for the solution of problems is evidenced by a clear understanding of the nature of irrational numbers as the ratios of incommensurable quantities (the English mathematician T. Bradwardine [first half of the 14th century] and N. Oresme [mid-14th century]) and especially in the introduction of fractional exponents (Oresme) and negative and zero exponents (the French mathematician N. Chuquet [end of the 15th century]). Here too arose the first ideas, which anticipated the following era, of infinitely large and infinitesimal quantities. The broad scope of scientific research during this era was reflected not only in numerous translations and editions of Greek and Arabic works but also in such undertakings as the compilation of extensive seven-place trigonometric tables by Regiomontanus (J. Müller). Mathematical symbols were greatly improved. Scientific criticism and polemics developed. The search for solutions of difficult problems, which was encouraged by the custom of public competitions, led to the first proofs of unsolvability. Thus, Leonardo of Pisa proved in the work Flos (Flower; 1225), a collection of problems that he proposed and brilliantly solved, that neither rational nor simple quadratic irrationalities (of the type , etc.) could be solutions of the equation x3 + 2x2 + 10x = 20.
WESTERN EUROPE IN THE 16TH CENTURY. The 16th century was the first century of Western Europe’s superiority over ancient Greece and the Orient. This was true in astronomy (N. Copernicus’ discovery) and mechanics (the first investigations of Galileo appeared at the end of the century). On the whole, this was also the case in mathematics despite the fact that in some areas European science still lagged behind the achievem