Pretty much all of basic geometry we learn in school is produced by constructions with straight edge and compass alone. ![]() They were designed to inform the less academic geometers of his time of the problems and the methods that had finally shown their insolubility. In 1895, the leading geometer of the late 19th and early 20th century, Felix Klein, found them important enough to make them the subject of popular lectures. The importance of the problems persisted. "Simultaneously with the gradual evolution of the Elements, the Greeks were occupying themselves with problems in higher geometry three problems in particular, the squaring of the circle, the doubling of the cube, and the trisection of any given angle, were rallying-points for mathematicians during three centuries at least, and the whole course of Greek geometry was profoundly influenced by the character of the specialized investigations which had their origin in the attempts to solve these problems." summarizes the role of these constructions as: Heath, in his expansive history of Greek mathematics. Thomas Heath, A History of Greek Mathematics. Much of the ingenuity of ancient geometers was devoted to finding these methods. Each construction is possible if geometers have access to curves and other devices not permitted by the use a straight edge and compass alone. To go beyond, required more sophisticated methods. To sophisticated geometers, these three problems represented the boundary of straight edge and compass methods. Whether the constructions were just very difficult or whether they were impossible in principle, was less of an issue. From the earliest times, however, it was recognized that the three constructions presented intractable problems for straight edge and compass constructions. ![]() It was only in the 19th century that proofs of their impossibility were provided. These apparently simple geometric constructions resisted all attempts at solution, by means of the standard procedures of ancient geometry, constructions with a straight edge and compass. For any given angle, construct lines that divide it into three equal angles. For a given cube, construct another cube that has twice its volume. For a given circle, construct a square of the same area.ĭuplicating the cube. Using a straight edge and compass only, they were: ![]() In the ancient Greek geometric tradition, three construction problems were prominent. Paradoxes of Impossibility: Classic Geometric Constructionsĭepartment of History and Philosophy of Science
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