226 



THE GRAMMAR OF SCIENCE 



point P^ to Pg on the path, and P^L^, PgL^ be the tangents 

 (p. 225) at P^, Pg respectively, then the direction of the 

 curve has continuously altered from P^L^ to PgL^. as we 

 traverse the length P^P^ of the curve. The angle between 

 these directions is L^NL^, and clearly the greater this angle 

 for a given length of curve P^P^, the greater will be the 

 amount of bending.^ The amount of angle through which 

 the tangent has been turned for a given length of curve 



Fig. 16. 



forms a fit measure of the total amount of bending in that 

 length. Accordingly we define the mean bending or mean 



curvature of the element of curve P.P. as the ratio of the 



4 5 



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

 of length in the element of curve P,P,. Thus the mean 

 curvature of any portion of a curve is the average turn of 

 its tangent per unit length of the curve. From the mean 

 curvature we can reach a conception of actual curvature as 

 a limit when the element of arc P^Pg is very small in just 



1 We are supposing here that the sense of the bending between P4 and Pg 

 does not change, that the curve is not like this : \J^. We can always ensure 

 that no such change takes place by taking a sufficiently small length of arc. 



