Just as the notion of _length_ was originally derived from the straight line, and extended to other curves by dividing them into infinitesimal straight lines, so the notion of _curvature_ was derived from the circle, and extended to other curves by dividing them into infinitesimal circular arcs. Curvature may be regarded, originally, as a measure of the amount by which a curve departs from a straight line; in a circle, which is similar throughout, this amount is evidently constant, and is measured by the reciprocal of the radius. But in all other curves, the amount of curvature varies from point to point, so that it cannot be measured without infinitesimals. The measure which at once suggests itself is, the curvature of the circle most nearly coinciding with the curve at the point considered. Since a circle is determined by three points, this circle will pass through three consecutive points of the curve. We have thus defined the curvature of any curve, plane or tortuous; for, since any three points lie in a plane, such a circle can always be described. If we now pass to a surface, what we want is, by analogy, a measure of its departure from a plane. The curvature, as above defined, has become indeterminate, for through any point of the surface we can draw an infinite number of arcs, which will not, in general, all have the same curvature. Let us, then, draw all the geodesics joining the point in question to neighbouring points of the surface in all directions. Since these arcs form a singly infinite manifold, there will be among them, if they have not all the same curvature, one arc of maximum, and one of minimum curvature[22]. The product of these maximum and minimum curvatures is called the _measure of curvature_ of the surface at the point under consideration. To illustrate by a few simple examples: on a sphere, the curvatures of all such lines are equal to the reciprocal of the radius of the sphere, hence the measure of curvature everywhere is the square of the reciprocal of the radius of the sphere. On any surface, such as a cone or a cylinder, on which straight lines can be drawn, these have no curvature, so that the measure of curvature is everywhere zero--this is the case, in particular, with the plane. In general, however, the measure of curvature of a surface varies from point to point.