Greek mathematicians not only failed to develop any general rules for computing limits, but never even formulated the concept of the limit itself, on which their methods were based (even the general term "method of exhaustion" for these methods is a modern term). Would lead to a contradiction (his ideas were often motivated by "mechanical considerations" ). Is infinitely small, the inequality $ S \neq K/3 $ Exhaustion, method of), in which infinitesimal quantities are used merely to prove that two given magnitudes (or two ratios between given magnitudes) are equal.Ģ) More sophisticated problems involving the method of exhaustion, in which the required finite magnitude is obtained as the limit of a sum Three kinds of such problems were particularly important in the history of mathematics.ġ) The simplest problems, solved by the mathematicians of Ancient Greece by the method of exhaustion (cf. In order to grasp the importance of this method, it must be pointed out that it was not the infinitesimal calculus itself which was of practical importance, but only the cases in which its use resulted in finite quantities. Even though the method of "infinitely smalls" had been successfully employed in various forms by the scientists of Ancient Greece and of Europe in the Middle Ages to solve problems in geometry and in natural science, exact definitions of the fundamental concepts of the theory of infinitely-small functions were laid only in the 19th century. A term which formerly included various branches of mathematical analysis connected with the concept of an infinitely-small function.
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