CURVATURE. The curvature of a plane curve at any point is its tendency to depart from a tangent to the curve at that point. In the circle this deviation is constant, as the curve is per fectly symmetrical round its eentre. The curva ture of a circle varies, however, inversely as the radius—that is, it diminishes at the same rate as the radius increases. The reciprocal of the radius is therefore taken as the measure of the curvature of a circle. A straight line may be considered a circle of infinite radius and as hav ing no curvature, since = 0. The constancy of curvature in the circle suggests an absolute measure of curvature at any point in any other curve, for whatever be the curvature at that point a circle can be found of the same curva ture. The radius of this circle is called the radius of curvature for that point; and the circle itself the osculating circle. By means of this radius we may compare the curvatures at different points of the same curve or of different curves. In simple cases, as in the conic sections,
the measure or radius of curvature may be de termined geometrically, but it is usually neces sary to employ the calculus. The expression for the radius of curvature at any point (,v, y) of a curve is 3 [ 1 ± d.c 2 If the curve, instead of lying in a plane, twists in space, it is sometimes called a gauche curve or a curve of double curvature, and its curvature at any point may be measured by the radius of its osculating sphere at that point. The centre of the osculating circle or sphere is called the centre of curvature. The curvature of surfaces is de termined similarly to that of curves. Thus the measure of the curvature of the earth, commonly taken as the deviation of the line of apparent level from the line of true level—that is, from a line everywhere parallel to the surface of still water—is approximaIely eight inches per mile.