Similarity considerations indicate that the cross-sectional areas of the pyramids are equal at equal heights the volume of the pyramids are simply considered as the 'sums' of these areas hence, the equalities of the corresponding terms of the two sums prove that the sums themselves are equal as well.Sometime in the 5th century B.C. Euclid overcame this difficulty in his Elements by using the method of exhaustion.Īs reported by Archimedes, the 'atomistic' method for proving the above theorem used by Democritus (Fig. In deriving the volume of a pyramid, the main difficulty encountered by Euclid and by Eudoxus was to prove that two pyramids with equal heights and equal base areas have equal volumes. It is pointed out by Archimedes, in particular, that Democritus determined the volume of the pyramid prior to Eudoxus (even though he failed to give a rigorous proof of his results). infinitesimals in the modern sense of the word) - there also existed a more primitive, but more illustrative method, attributable to Democritus (4th century B.C.). The works of Archimedes (in particular, his Message to Eratosthenes) indicate that - prior to Archimedes' logically precise method for estimating areas and volumes with the aid of sums of a very large number of terms that are decreasing without limit (i.e. A fortiori, ancient science never produced anything resembling the modern algorithm of integral calculus, from which, as a result, in calculating a new integral by modern methods, one does not define it as a limit of sums, but uses much simpler and handier rules for the integration of functions belonging to different classes. 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 eq 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.

Working with infinitesimals