DUALITY. A statement ca pable of two different meanings, both of them true, one obtained from the other, by simply inter changing two words, is an illus tration of the principle of duality.
An important application of the principle is found in projective geometry. In the plane this is accomplished by interchanging the words "point" and "line"; it is well illustrated by the theorem of Pappus, which may be stated as follows : Given any two straight lines u, u' in the plane ; choose any three points A, B, C on u, and any three points A', B', C', on u'. The three points of intersection AB', A'B; AC', A'C; BC', B'C lie on a straight line a". The dual theorem is: Given any two points U, U'; draw any three lines a, b, c through U and a', b', c' through U'. The three lines joining the points ab', a'b; ac', a'c; bc', b'c all pass through a point U". If the first proof has been established, the second follows by duality, since the determining elements of lines in terms of points are identical with those of points in terms of lines. The principle was first recognized by Poncelet in the Journal fur Mathematik (1829), and by Gergonne in the Annales de Mathernatiques pures et appliquees (1825-27), and first generally applied by Steiner in his Systematische Entwickelungen, (1832) .
In geometry of three dimensions there is a corresponding duality between points and planes. In this case the line is self dual, as it is determined by any two distinct points on it or by any two dis tinct planes through it.
Many other illustrations of the principle can be given. A geometry can be constructed in the plane by replacing the word "line" wherever it occurs in a proposition by the word "circle," if the circle associated with any given line is constructed as follows : Given a fixed circle C with cen tre 0. Let a given line meet it in A, B. Draw the circle through A, B, 0. After this has been done for every line, think of the point 0 being removed from the plane. The resulting system of incomplete circles furnishes a non euclidean interpretation of plane geometry. The line-sphere trans formation of Lie is an illustration of a complete duality between lines and spheres in space. Much of higher geometry is concerned with the principal of duality; every new application practically doubles the extent of existing knowledge.