Starogrčka geometrijska algebra

U radu opisujemo probleme vezane uz geometriju kojima su se bavili starogrčki matematičari, a koji se moderno mogu interpretirati algebarski. U pokušajima rješavanja koristili su razne metode, no valjana geometrijska konstrukcija dopuštala je samo korištenje konstrukcija ravnalom i šestarom. U prvom poglavlju navodimo neke klasične konstrukcije ravnalom i šestarom koje su bile poznate među grčkim matematičarima. Drugo poglavlje opisuje neke pokušaje rješavanja tri klasična problema. U trećem poglavlju naglasak je na drugoj knjizi Euklidovih Elemenata. Četvrto poglavlje sadrži dokaz nesumjerljivosti stranice i dijagonale kvadrata te način na koji je Apolonije opisivao konike. Rad završava opisom Descartesovog povezivanja geometrije i algebre, dokazom ekvivalencije konstrukcija ravnalom i šestarom s rješavanjem algebarskih kvadratnih jednadžbi te dokazom da tri klasična problema nisu rješivi konstrukcijama ravnalom i šestarom.This thesis describes geometric problems that ancient Greek mathematicians were trying to solve and which can be interpreted geometrically. In their attempts they used different methods, but only solutions that used straightedge and compass were considered valid. In the first chapter we describe some of the classical straightedge and compass constructions that were known to Greek mathematicians. The second chapter describes some attempts to solve three classical problems. In the third chapter the emphasis is on the second book of Euclid’s Elements. The fourth chapter contains a proof of the incommensurability of the side and diagonal of a square and Appolonius’ approach to conics. The thesis concludes with Descartes’ connection of geometry and algebra, with proof that straightedge and compass constructions are equivalent to solving algebraic quadratic equations and with proofs of the impossibility of solving the three classical problems with ruler and compass constructions