DUALITY (Lat. (Mantas, state of being two fold. from dualis, relating to two). A principle of geometry by which one proposition is trans formed into another through the interehange of a pair of elements; e.g. the proposition. If two triangles have two sides and the included mode of the one respectively equal to two sides and the included angle of the other, the triangles are congruent, may be transformed by the inter change of the words sides and angles into an other familiar proposition: If two triangles have two angles and the included side of the one respectively equal to two angles and the included side of the other, the triangles are congruent. This principle is often called the principle of reciprocity. and is extensively used in geometry. Although the reciprocal of a valid proposition is not of it-elf necessarily valid. it often new possible theorems for in vestigation. In plane geometry the dual ele ments most interchanged are: point—line; line—point; angles of a triangle (opposite),sides of the triangle: sides of a triangle— (opposite) sides of the triangle: pencil of lines—range of points: range of peints—peneil of lines.
Similarly there are dual propositions of solid geometry, as those formed by interchanging the words straight line and plane; e.g. two inter sooting straight lines determine a plane; two in t Pr-yeti planes determine a Atniiyht Such propositions also exist. one in plane geome try and the other in solid geometry. a- when the terms and tribedral angle are inter changed: e.g. one vide of a triangle is less that' the sum of the other two; one of a trilo drat angle is Iv— than the sum of the other two. The term duality appears in the projective geometry of ergonne (1813), and the develop ment of the 'principle of duality' in another work by the saino.ant'lor This prin ciple is a powerful agent of modern geometry, and it has been :Tidied to various properties of curves.