joyed 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 achievements of 15th-century Central Asian mathematicians and that in actuality the major new ideas that determined the subsequent development of European mathematics were to appear only in the 17th century. However, in the 16th century it seemed that a new era was beginning in mathematics with the discovery of algebraic solutions of cubic equations (S. dal Ferro, c. 1515, and later and independently N. Tartaglia, c. 1530) and quartic equations (L. Ferrari, 1545), which for centuries had been considered impossible. G. Cardano investigated cubic equations and discovered the irreducible case in which the real roots of an equation are expressed in terms of complex numbers. This forced Cardano to acknowledge, albeit quite tentatively, the advantage of calculations involving complex numbers. Algebra was further developed by F. Vieta (Viete), the founder of true algebraic literal calculation (1591; before him letters were used only for unknowns). The concept of perspective, which was developing in geometry even before the 16th century, was set forth by the German artist A. Diirer (1525). In 1585, S. Stevin worked out the rules of arithmetic operations for decimals.
RUSSIA UNTIL THE 18TH CENTURY. Russian mathematical education in the ninth to 13th centuries was on the level of the most culturally advanced countries of the Orient and Western Europe. However, it was set back for a long time by the Mongolian invasion. In the 15th and 16th centuries, society’s requirements for mathematical knowledge increased significantly in connection with the strengthening and economic growth of the Russian state. Numerous manuscript handbooks of arithmetic and geometry containing rather extensive information necessary for practical activities (trade, taxation, construction, the use of artillery) appeared in the late 16th century and particularly in the 17th century.
A system of number symbols resembling the Greco-Byzantine system and based on the Slavic alphabet came into use in Ancient Rus’. Slavic numeration appears in Russian mathematical li