The Renaissance

Date: 2015-10-07; view: 505.

Italian artists and merchants influenced the mathematics of the late Middle Ages and the Renaissance in several ways. In the 15th century a group of Tuscan artists, including Filippo Brunelleschi, Leon Battista Alberti, and Leonardo da Vinci, incorporated linear perspective into their practice and teaching, about a century before the subject was formally treated by mathematicians. Italian maestrid'abbaco tried, albeit unsuccessfully, to solve nontrivial cubic equations. In fact, the first general solution was found by Scipione Del Ferro at the beginning of the 16th century and rediscovered by NiccolòTartaglia several years later. The solution was published by GirolamoCardano in his Ars magna in 1545, together with Lodovico Ferrari's solution of the quartic equation.

By 1380 an algebraic symbolism had been developed in Italy in which letters were used for the unknown, for its square, and for constants. The symbols used today for the unknown (for example, x), the square root sign, and the signs + and - came into general use in southern Germany beginning in about 1450. They were used by Regiomontanus and by FridericusGerhart and received an impetus in about 1486 at the University of Leipzig from Johann Widman. The idea of distinguishing between known and unknown quantities in algebra was first consistently applied by François Viète, with vowels for unknown and consonants for known quantities. Viète found some relations between the coefficients of an equation and its roots. This was suggestive of the idea, explicitly stated by Albert Girard in 1629 and proved by Gauss in 1799, that an equation of degree n has n roots. Complex numbers, which are implicit in such ideas, were gradually accepted about the time of Rafael Bombelli (d. 1572), who used them in connection with the cubic.

Apollonius' Conics and the investigations of areas (quadratures) and of volumes (cubatures) of Archimedes formed part of the humanistic learning of the 16th century. These studies strongly influenced the later developments of analytic geometry, the infinitesimal calculus, and the theory of functions, subjects that were developed in the 17th century

I. Read the passage and find the key sentence in each paragraph. Say how the paragraphs are built.

II. How is the passage built? Where is the key sentence to the whole passage?

III. Speak in short about what studies stronglyinfluenced the later developments of analytic geometry, the infinitesimal calculus, and the theory of functions, subjects that were developed in the 17th century

IV. Suggest the Russian counterparts for the following words and word combinations:

linear perspective, to treat the subject formally, to solve nontrivial cubic equations, a solution of the quartic equation, to find a general solution, to develop an algebraic symbolism, the symbols for the unknown, the square root sign, to come into general use, to receive an impetus, to distinguish between known and unknown quantities, relations between the coefficients of an equation and its roots, it is suggestive of the idea, a complex number, equation of degree n has n roots, the cubic, analytic geometry, the infinitesimal calculus, the theory of functions