INNOVATIONS IN MATHEMATICS: FROM THE "PHYSICAL MATHEMATICS" OF ARCHIMEDES TO THE "INTEGRAL CALCULUS"

The paper deals with the history and mechanisms of the novation in mathematics from Archimedes to Newton and Leibnitz that had ultimately led to invention of the integral calculus. Archimedes created the 'method' and managed to solve the basic problems, related with calculation of the areas and volumes of curvilinear figures. Although Kepler, Cavalieri, Fermat and other modern mathematicians followed the way and developed the method of Archimedes, the calculus in the proper sense of this word was finally invented by Newton and Leibnitz as a result of their 'reflexive thinking', when they realized that it was the calculus itself, not the class of problems it had been designed to solve, that should be considered the primal objective of their investigation. The article shows how the exact and natural sciences, such as astronomy, mechanics and optics, put questions to mathematics and how the latter respond to this questions with various form of innovative solutions.