APLANATIC LENS

A D: C D; and when this property holds the line A D is said to be harmonically divided in B and C. The term harmonical applied in this sense, first occurs, we believe, in Lahire's Conic Sections. The characteristic property of the anharmonic ratio is the following: — If from any point out of the given straight line four straight lines a, b, c, d, be drawn through the four points respectively, and if sin (a b) denote the sine of the angle included between a and b (and so with the others), then the double or anharmonic ratio is equal to [sin (a 6): sin (6 c)] x ( A B: B C) x (C D: D A) x [sin (c d): sin (d a)]. These angles being entirely independent of the position of the straight line in which the four points A, B, C, D are taken, it follows that if any straight line whatever be drawn to intersect the four straight lines a, b, c, d respectively in the points A', B', C, D', then the anharmonic ratio (A' B': B' C) x(CD':DA') will be the same as (AB:BC)x (CD: DA). From this general property some very remarkable relations of geometrical figures have been deduced. (Steiner, Si/stem at is che Entwickelung der Abhangigkeit geometrischer Gestalten von einander; Berlin, 18.32; Chasles, Apergu Hist or i que des Mdlh>des en Geometric {Mimoires de Bruxelles), 1837.) [s. 1365]