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Noindent the TEACHING of MATHEMATICS THE TEACHING OF MATHEMATICS nnoindent2010 commaTHE Vol TEACHING period XIII OF comma MATHEMATICS 1 comma pp period 5 1 endash 6 1 ON THE ANCIENT PROBLEM OF DUPLICATION nnoindentOF A CUBE2010 IN HIGH , Vol SCHOOL . XIII TEACHING , 1 , pp . 5 1 −− 6 1 Andrea Grozdani acute-c and Gradimir Vojvodi c-acute n centerlineAbstract periodfON The THE paper ANCIENT is devoted PROBLEM to exposition OF DUPLICATION of constructionsg with straightedge and compass comma constructible numbers and their position with respect to all algebra hyphen n centerline fOFTHE A CUBETEACHING IN HIGH OF MATHEMATICS SCHOOL TEACHING g ic numbers period2010 .. , Vol Although . 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Although the large number of constructions may be accomplished this way , we know of three problems dating from Greek era nnoindent that cannot be solved in that way : nquad duplication ( doubling ) of a cube −− to find a side of a cube whose volume is twice that of a given cube ; nquad tris ection of an angle −− to find one third of a given angle ; squaring a circle −− to construct the square that has the same area as a given circle . Unsolved problems of that kind initiated a completely new way of thinking −− how would it be possible to prove that certain pro b lems could not be s o lved $ ? $ nquad The nnoindent answer is in modern algebra and group theory . nquad The problem of solving algebraic equations dates far back in the past and for a long time it was the central content of algebra . Descriptions of solving certain simple algebraic equations had appeared nnoindent as early as 2000 years B . C , for example in Egypt , nquad during the Middle Dynasty , in the London papyrus known as Ahmess calculation , nquad and on Babylonian tiles , nnoindent approximately at the same time . nquad The Babylonians were able to solve quadratic equations , while in the XVI century Girolamo Cardano , Nicolo Tartaglia , Lodoviko 52 .. A period Grozdani acute-c sub comma G period Vojvodi c-acute nnoindentFerari comma52 n ..quad ScipioneA .del Grozdani Ferro and many $ nacute othersf werecg dealingf , g with$ G solving . 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