GEOMETRIC PROBLEMS OF ANTIQUITY

three famous problems involving elementary geometric constructions with straight edge and compass, conjectured by the ancient Greeks to be impossible but not proved to be so until modern times. The three problems are: (1) the duplication of the cube, also known as the Delian problem because it is said to have originated with the task of constructing a cubical altar at Delos having twice the volume of the original cubical altar; (2) the trisection of an arbitrary angle; (3) the squaring, or quadrature, of the circle, i.e., the construction of a square whose area is equal to that of a given circle. These problems were solved in the 19th cent. by first transforming them into algebraic problems involving "constructible numbers." A constructible number is one that can be obtained from a whole number by means of addition, subtraction, multiplication, division, or extraction of square roots. The problems of antiquity correspond to the following algebraic problems: (1′) Is the cube root of 2 constructible? (2′) Given an angle A for which cos A is constructible, is cos (A/3) constructible? (3′) Is the area π of a unit circle constructible? The cube root of 2 is not constructible, since it involves a cube root. (Note, however, that roots that are powers of 2, e.g., 4th, 8th, 16th roots, are constructible because they can be expressed as combinations of square roots.) In problem (2′), certain special angles can be trisected, e.g., 90°, since both cos 90° and cos 30° are constructible, but for most angles this is easily shown to be impossible. Finally, the solution of problem (3′) did not come until 1882, when the German Ferdinand Lindemann showed that π is a transcendental number and thus cannot be expressed in terms of any roots of any rational numbers (see number). Although these problems cannot be solved using only straight edge and compass, the Greeks developed methods of solving them using higher curves.

See F. Klein, Famous Problems of Elementary Geometry (1956).

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