# geometric problems of antiquity

**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

*A*for which cos

*A*is constructible, is cos (

*A*/3) constructible? (3′) Is the area π of a unit circle constructible? The number

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

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

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