228 THE GRAMMAR OF SCIENCE 



units in M,M, to the number in PP, in Fig. i6. But if 



4 5 450 



0,K be drawn parallel to MO, to meet P,0, in K, this 



'-'4 ^ 5^^5 5 "'S ' 



ratio is that of KQ^ to O^K, or is the slope of the chord 

 O^Q. to the vertical line ^■^^^.■ Thus the slope of any 

 chord of the curvative-chart to the vertical measures the 

 mean curvature of the corresponding portion of the curve 

 in Fig. 14. When we make the chord Q^Q,^ smaller and 

 smaller by causing Q^ to move towards Q^, the mean cur- 

 vature becomes more and more nearly the mean curvature 

 at and about P^ ; but as on p. 216 the chord becomes 

 more and more nearly the tangent at O^. As we have 

 defined actual curvature to be the limit to the mean 

 curvature in a vanishingly small length of curve beyond 

 P^ (see Fig. 14), we see that the actual curvature at P is 

 the slope to the vertical of the tangent O^S at the corre- 

 sponding point Q^ of the curvature-chart. This slope, 

 and accordingly the actual curvature, is therefore a 

 measurable quantity at each point of any curve.' 



^ 15. — The Relation between Curvature and Normal 



Acceleration 



Returning again to Figs. 14 and 15, we note that the 



mean curvature over the length P,P^ is the ratio of the 



^ 45 



number of angle units in L^NL. to the number of length 

 units in the element of curve P^P-. Now the speed in 



1 The mean curvature over any arc ab of a circle centre O is the ratio 

 of the angle between the tangents at its extremities, or — what is the same 



thing, since the tangents are perpendicular to the 

 radii Oa and Oh — of the angle aOb at the centre to 

 the arc ab. But we have seen in the footnote, p. 

 227, that the measure of this angle in radians is 

 the ratio of the arc fli^ to the radius. Hence it follows 

 that the mean curvature of a circle is equal to the 

 inverse of the radius (or unity divided by the radius). 

 As this mean curvature is therefore independent of 

 Yyq j_ the length of the arc, it follows that the actual cur- 



vature at each point must be the same and be equal 

 to the inverse of the radius. Since the radius of a circle can take every value 

 from zero to infinity, a circle can always be found which has the same amount 

 of bending as a curve at a given point, and thus fits it more closely at that 

 point than a circle of any other radius. The radius of this circle is termed 

 the raditts of citrvatitre of the curve at the given point. Hence the curvature 

 of a curve is the inverse of its radius of curvature. 



