590 VIII. ON THE STRUCTURE OF MATHEMATICS 

3. CURVATURE In modern scientific literature we hear often the fundamental term 'curvature' mentioned, and a few words about it will not be amiss. If we take two perpendicular lines X'OX and OY and select on OK a number of points A,
B, C, D., further and further away from 0 and describe arcs of circles with these points as centres with radii AO, BO, CO, DO., respectively Fig. 9, we find each successive arc flatter and closer to the line X'X than its predecessor. In other words, the larger the radius of our circle, the flatter its arc is. In the limit as the radius of the circle becomes indefinitely large, the arc approaches a straight line by intuition and by definition. We notice also that the curvature of each circle is uniform, that is, onevalued at every point; but that when we pass from one circle to another of different radius, the curvature changes.
If we consider a curve and two points on it, Mi and Mi, (Fig. 10) and draw two tangents at these points; then the angle between these two tangents will depend on two factors, the sharpness of the curve and the distance between
the points M\ and Mi. If we take the
points near enough and designate the
length of the arc between them by As,
the angle between the two tangents
by Ad, then the limiting value of the
ratio , as Mi approaches M\,
becomes, and is a measure of the
rate of change of the direction of the
tangent at M, as M moves along the
curve. Let us designate the rate at
which the tangent turns where the
point describes the curve with unit velocity as the curvature, or
but as k is essentially a positive number or zero we accept only the absolute
value of this ratio. To find we notice that tan








The reciprocal of the curvature is called the radius of curvature. The radius of curvature of a circle is its radius. The curvature of a curve is measured by the radius of the osculating circle, that circle which fits the curve the most closely in the neighbourhood of our point. 
