We know that the curvature of a circle at every point of its circumference is a constant given by , where is the radius of the circle. But if in place of a circle we trace any arbitrary curve on our plane, the curvature will vary from point to point along the curve. The curvature of the curve at given point is then defined by the a certain circle which is tangent to the curve at the point . As a matter of fact there exist an indefinite number of circles of varying radii lying tangent to the curve at ; but among these circles one stands out prominently in that it is, so to speak, more perfectly tangent than all the others. Whereas the tangent circles intersect the curve in two points coinciding at , this privileged circle intersects it in three such points. It is called [Pg 93] the osculating circle (osculare meaning to kiss in Latin). The curvature of the curve at is defined by the curvature of its osculating circle at . Calling the radius of this osculating circle, the curvature of the curve at is thus given by (Fig. VI). Fig. VI We must now pass to the curvature of a surface. At a point on the surface where the curvature is to be computed we trace a normal to the surface. Then through this normal we trace a plane, which of course intersects the surface along a plane curve. We assume this normal plane to revolve round the normal as axis and we thus obtain a series of plane curves of intersection defined by the normal plane and surface. Each one of these curves passing through the point possesses a definite curvature at , and this curvature can be computed through the medium of the corresponding osculating circle. Here a geometrical fact is evidenced. It is found that in the general case there exist two remarkable positions of the intersecting normal plane, perpendicular to one another and therefore sectioning the surface along two curves (1) and (2) orthogonal to each other at . The curvature of one of these curves, say, the curve (1), is less than that of all other curves obtained by revolving the plane round the normal at , while the curvature of the second curve (2) is greater than all others. These two curvatures are called the two principal curvatures of the surface at the point and are designated by and respectively.