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 maestri d'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 Niccolò Tartaglia several years later. The solution was published by Girolamo Cardano 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 Fridericus Gerhart 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 strongly influenced 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

Mathematics in the 17th and 18th centuries

The 17th century

I. Look through the passage quickly and say:

- what Kepler, Galileo, Descartes, and Newton worked at in the 17^{th} century

- what had replaced classical Greek geometry by the end of the 17th century

- whom Marin Mersenne in Paris informed of challenge problems and novel solutions

- what academies were founded in the 17^{th} century

The 17th century, the period of the scientific revolution, witnessed the consolidation of Copernican heliocentric astronomy and the establishment of inertial physics in the work of Kepler, Galileo, Descartes, and Newton. This period was also one of intense activity and innovation in mathematics. Advances in numerical calculation, the development of symbolic algebra and analytic geometry, and the invention of the differential and integral calculus resulted in a major expansion of the subject areas of mathematics. By the end of the 17th century a program of research based in analysis had replaced classical Greek geometry at the centre of advanced mathematics. In the next century this program would continue to develop in close association with physics, more particularly mechanics and theoretical astronomy. The extensive use of analytic methods, the incorporation of applied subjects, and the adoption of a pragmatic attitude to questions of logical rigour distinguished the new mathematics from traditional geometry.