# Page:Popular Science Monthly Volume 21.djvu/522

We come now to the most difficult branch of the subject, that of curved surfaces and of curved space. The curvature of a plane curve at any point is the limit of the ratio of the length of the curve to the difference in direction of the initial and terminal tangents. Its differential expression is ${\displaystyle \textstyle {D_{t}s}}$ or ${\displaystyle \textstyle {\frac {D\times \!\,^{2}y}{[1+(D\times \!\ y)^{2}]{\frac {3}{2}}}}}$. To get the curvature of a curved surface at any point, we slice it up by planes normal to it at that point. On each of these planes it will describe a curve. These curves will have different curvatures at the original point. The reciprocal of the product of the greatest and least of these is called by Gauss the measure of curvature. This name he also applied to an analogous function of the co-ordinates of a point in space. The expression, for a plane curve, of the curvature is the reciprocal of the radius of the circle of closest possible contact at the point investigated. Hence, some have argued that transcendental geometry was inconsistent, in that it talked about the curvature of a space where there were not Euclidean straight lines, hence no radii, and nothing to refer the curvature to. This argument is open to other answers, but it is enough to say that the measure of curvature has no necessary connection with radii.