The Curvature of Space

By the curvature of a curve at a point we mean the limit of the ratio of the angle Δα between the tangents at the endpoints of an arc to the length Δα of that arc as the latter contracts to the point; that is, the limit (8.1)$$k = \frac{{d\alpha }} {{ds}} = \mathop {\lim }\limits_{\Delta s \to 0} \frac{{\Delta \alpha }}{{\Delta s}}. $$k is also the reciprocal of the radius of curvature at the point in question, that is, the radius of the osculating circle at that point. (The osculating circle at a point P of a curve is dfined as the limit of circles determined by three points on the curve as they tend to P.) The concepts of the curvature of a curve and of the osculating (literally “kissing,” from the Latin osculans) circle were already known to Leibniz.1 Leibniz also suggested the possibility of characterizing the curvature of a surface by means of an osculating sphere.2