Algebraic geometry 01 Algebraic curves, algebraic manifolds by I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov

By I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov

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We will give examples of these manifestations of the infinite in the earlier evolution of calculus (seventeenth and eighteenth centuries). 2 Seventeenth-Century Predecessors of Newton and Leibniz The Renaissance (ca. 1400–1600) saw a flowering and vigorous development of the visual arts, literature, music, the sciences, and—not least—mathematics. It witnessed the decisive triumph of positional decimal arithmetic, the introduction of algebraic symbolism, the solution by radicals of the cubic and quartic, the free use if not full understanding of irrational numbers, 38 5 Chapter 5 • Calculus: From Tangents and Areas to Derivatives and Integrals the introduction of complex numbers, the rebirth of trigonometry, the establishment of a relationship between mathematics and the arts through perspective drawing, and a revolution in astronomy, later to prove of great significance for mathematics.

The Renaissance also saw the full recovery and serious study of the mathematical works of the Greeks, especially Archimedes’ masterpieces. His calculations of areas, volumes, and centers of gravity were an inspiration to many mathematicians of that period. Some went beyond Archimedes in attempting systematic calculations of the centers of gravity of solids. But they used the classical “method of exhaustion” of the Greeks, which was conducive neither to the discovery of results nor to the development of algorithms.

And so we come to Pascal and the Chevalier de Méré. Pascal was intrigued by the Problem of Points proposed by de Méré and agreed to study it. Before long he had a solution. The problem was challenging and subtle, and so he wrote (in July 1654) to Fermat, the leading mathematician of France, asking if he would read his (Pascal’s) solution. Fermat obliged. Thus began the now famous Pascal–Fermat correspondence, lasting several months (July–November 1654), and resulting in the emergence of what turned out to be a most important mathematical discipline—probability.

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