Metoda ekshaustije

Abstract

U prvom dijelu ovog rada prikazan je razvoj metode ekshaustije koji je započeo nešto prije helenizma u staroj Grčkoj, kad je Antifont, u pokušaju da riješi problem kvadrature kruga, predložio upisivanje pravilnih mnogokuta u krug. Ideju je formalno opisao Eudoks iz Knida propozicijom (Eudoksova lema) koju nalazimo u X. knjizi Euklidovih Elemenata, a njenu primjenu u geometriji nalazimo u XII. knjizi. Najveći doprinos razvoju ove metode dao je Arhimed iz Sirakuze, koji je pomoću ove metode odredio površine i volumene mnogih zaobljenih likove, odnosno tijela. Nakon Arhimeda sve do 17. stoljeća nema značajnijih rezultata u razvoju ove metode. U 17. stoljeću metoda ekshaustije postepeno se transformirala u moderni integralni račun čiji su utemeljitelji Isaac Newton i Gottfried Wilhelm Leibniz. Integralnim računom tako je zamijenjena antička, geometrijska metoda, praktičnijom i efikasnijom računskom metodom određivanja površina i volumena. U zadnjem dijelu ovog rada dan je kratki metodički osvrt ekshaustijskog pristupa u nastavi matematike na primjeru određivanja površine.In the first chapter of this thesis we present the development of method exhaustion, which started some time before the Hellenistic period in ancient Greece. Antiphon first attempted to solve the problem of the quadrature of the circle by inscribing regular polygons in circle. The idea was formally described by Eudoxus of Cnidos in a proposition (known as lemma of Eudoxus) which can be found the X. book of Euclid’s Elements. In the XII. book Euclid gives results obtained by application of this lemma to geometry. The most important contribution to this method was made by Archimedes of Syracuse, who used it to determine areas and volumes of curved plane and solid figures. After Archimedes there was no significant progress in development of this method up until the 17th century. In the 17th century, the method of exhaustion transformed into the modern integral calculus, whose founders are Isaac Newton and Gottfried Wilhelm Leibniz. In the last chapter we use on the example of area determination to give a short didactical comment on the application exhaustion approach in teaching of mathematics